Internal Friction of Materials Anton Puskar

i

INTERNAL FRICTION OF MATERIALS

iii

Anton Puškár

Transport and Communications Technical UniversityZilina, Slovak Republic

CAMBRIDGE INTERNATIONAL SCIENCE PUBLISHING

^

iv

Published byCambridge International Science Publishing7 Meadow Walk, Great Abington, Cambridge CB1 6AZ, UK

http://www.demon.co.uk/cambsci/homepage.htm

First published May 2001

© Anton Puškár© Cambridge International Science Publishing

Conditions of saleAll rights reserved. No part of this publication may be reproduced or trans-mitted in any form or by any means, electronic or mechanical, including pho-tocopy, recording, or any information storage and retrieval system, withoutpermission in writing from the publisher.

British Library Cataloguing in Publication DataA catalogue record for this book is available from the British Library

ISBN 1 898326 509

Production Irina StupakPrinted by PWP Acrolith Printing Ltd, Hertford, England

v

PREFACEThe complete absence of books characterising the internal frictionof materials, its external and internal aspects and the application ofmeasurements in various scientific and technical areas, especially inphysical metallurgy and threshold states of materials, has been theimpetus for the author to write this book. Without understandingthe principle and mechanisms of anelasticity and the effect ofvarious factors on internal friction, together with the application ofmethods of reproducible internal friction measurements, it is notpossible to solve the problems of the application of thesemeasurements as direct or indirect methods for the evaluation of thestructural stability of alloys, problems of cyclic microplasticity anddeeper understanding of processes associated with the response ofmaterials to single or repeated loading.

In addition to the original systematisation of the possibilitiesof using internal friction measurements in various sciences, thebook presents the latest theories and results together with practicalapproaches to the measurement and evaluation of the resultantrelationships

Anton Puškár

vii

CONTENTS

1 AIMS OF INTERNAL FRICTION MEASUREMENTS....... 1

2 NATURE AND MECHANISMS OF ANELASTICITY ......... 5

2.1 ELASTICITY CHARACTERISTICS........................................... 52.1.1 Effect on elasticity characteristics .................................................. 132.1.2 Elasticity characteristics of structural materials ............................. 272.1.3 Elasticity characteristics of composite materials ............................ 372.2 MANIFESTATION OF ANELASTICITY ................................. 432.2.1 Delay of deformation in relation to stress ....................................... 442.2.2 Internal friction ............................................................................... 502.2.3 Mechanisms of energy scattering in the material ............................ 532.3 DEFECT OF THE YOUNG MODULUS ................................... 62

3 FACTORS AFFECTING ANELASTICITY OF

MATERIALS ............................................................................... 793.1. INTERNAL FRICTION BACKGROUND................................. 803.1.1 The substructural and structural state of material .......................... 813.1.2 Vacancy mechanism........................................................................ 823.1.3 Diffusion-viscous mechanism ......................................................... 843.1.4 Dislocation mechanisms ................................................................. 853.1.5 The relaxation mechanism .............................................................. 873.2. EFFECT OF TEMPERATURE ON INTERNAL FRICTION .. 873.2.1 Mechanisms associated with point defects ..................................... 943.2.2 Dislocation relaxation mechanisms ................................................ 973.3 EFFECT OF STRAIN AMPLITUDE ....................................... 1043.3.1 The Granato–Lücke spring model ................................................ 1053.3.2 Thermal activation ........................................................................ 1073.3.3 Internal friction with slight dependence on strain amplitude......... 1093.3.4 Plastic internal friction ................................................................. 1143.4 EFFECT OF LOADING FREQUENCY .................................. 1143.5 EFFECT OF LOADING TIME ................................................. 1213.6 EFFECT OF MAGNETIC AND ELECTRIC FIELDS ........... 128

4 MEASUREMENTS OF INTERNAL FRICTION AND THE

DEFECT OF THE YOUNG MODULUS ............................. 1334.1 APPARATUS AND EQUIPMENT ........................................... 133

viii

4.2 EXPERIMENTAL MEASUREMENTS AND EVALUATION........................................................................................................... 143

4.2.1 Infrasound methods ...................................................................... 1444.2.2 Sonic and ultrasound methods ...................................................... 1514.2.3 Hypersonic methods ..................................................................... 1674.3 PROCESSING THE RESULTS OF MEASUREMENTS AND

INACCURACY ........................................................................... 1694.3.1 Inaccuracy caused by the design of equipment ............................. 1694.3.2 Inaccuracies of the measurement method ..................................... 1724.3.3 Errors in processing the measurement results ............................... 174

5 STRUCTURAL INSTABILITY OF ALLOYS .................... 181

5.1 DIFFUSION MOBILITY OF ATOMS ..................................... 1815.1.1 Interstitial solid solutions .............................................................. 1835.1.2 Substitutional and solid solutions ................................................. 1955.2 RELAXATION OF DISLOCATIONS ...................................... 1975.2.1 Low-temperature peaks ................................................................ 1975.2.2 Snoek and Köster relaxation ......................................................... 1985.2.3 Phenomena associated with martensitic transformation in steel ... 2045.2.4 Migration of solute atoms in the region with dislocations ............ 2055.3 RELAXATION AT GRAIN BOUNDARIES ............................ 2135.3.1 Pure metals ................................................................................... 2155.3.2 Solid solutions .............................................................................. 2165.3.3 Relaxation models ........................................................................ 2175.4 ANALYTICAL PROCESSING OF THE RESULTS OF

MEASUREMENTS .................................................................... 2195.4.1 Solubility boundaries .................................................................... 2195.4.2 Activation energy and diffusion coefficient .................................. 2215.4.3 Breakdown of the solid solution ................................................... 2235.4.4 Intercrystalline adsorption ............................................................ 2245.4.5 Transition of the material from ductile to brittle state .................. 2275.4.6 Relaxation movement of microcracks ........................................... 230

6 CYCLIC MICROPLASTICITY ............................................ 234

6.1 CRITICAL STRAIN AMPLITUDES AND INTENSITY OFCHANGES OF CHARACTERISTICS..................................... 237

6.1.1 Physical nature of the critical strain amplitude ............................. 2396.1.2 Methods of evaluating critical amplitudes .................................... 2456.2 CYCLIC MICROPLASTIC RESPONSE OF MATERIALS .. 2486.2.1 Dislocation density and the activation volume of microplasticity . 248

ix

6.2.2 Condensation temperature of the atmospheres of solute elements 2576.2.3 Deformation history ............................................................................... 2626.2.4 Cyclic strain curve ........................................................................ 2676.2.5 Temperature and cyclic microplasticity ........................................ 2766.2.6 Magnetic field and microplasticity parameters ............................. 2826.2.7 Saturation of cyclic microplasticity .............................................. 2906.3 FATIGUE DAMAGE CUMULATION ..................................... 2976.3.1 Hypothesis on relationship of Q–1 – ε and σ

a – N

f dependences ... 297

6.3.2 Deformation and energy criterion of fatigue life ........................... 3006.3.3 Effect of loading frequency on fatigue limit ................................. 311References ................................................................................................... 315Index ...................................................................................................... 325

xi

A - coefficient of anisotropyA

p- approximate coefficient of anisotropy

a - lattice spacingB - proportionality coefficientBHR - Blair, Hutchins and Rogers modelb - Burgers vectorb - fatigue life coefficientc - fatigue life exponentc

ikmn- elasticity constant

cm

- average concentrationc

0- initial concentration

cp

- heat capacity at constant pressureD - diffusion coefficientD⊥, D

||- diffusion coefficient normal and parallel to the dis-

locationd

z- grain size

E - modulus of elasticity (Young modulus) in tension orcompression

ED

- dynamic modulus of elasticity (Young modulus)E

N- non-relaxed modulus of elasticity

ER

- relaxed modulus of elasticityE

S- modulus of elasticity

Eef

- effective modulus of elasticity∆E/E - defect of modulus of elasticity (Young modulus)e - temperature coefficient of the change of the modulus

of elasticityF - coefficient of the shape of the hysteresis loopF⊥⊥⊥⊥⊥, F

||- force acting normal and parallel in relation to the

dislocationFR - Finkel’stein-Rozin relaxationG - shear modulus of elasticity∆G - difference of moduli of elasticityH - activation enthalpy - Planck’s constantI - magnetization currenti - interstitial atomK - bulk modulus of elasticityK

S- mean capacity of absorption of energy in the

microvolumek - Boltzmann’s constantk

I- coefficient of magnetomechanical bond

Lef

- effective length of the dislocation segmentL

n- length of the pinned dislocation segment

SYMBOLS

xii

Lp

- spacing of pinning pointsM - ratio of the extent of internal friction and the defect

of the modulus of elasticityN

f- number of cycles to fracture

n - coefficient of cyclic strain hardeningn

o- density of geometrical inflections

P - pressureQ

c- activation energy for creep

Q–1 - internal frictionQ–1

S- height of Snoek’s peak

Q–1SK

- height of Snoek and Köster peakQ–1

m- internal friction slightly depending on ε

Q–1o

- internal friction independent of ε, the so-calledbackground

Q–1p

- internal friction strongly dependent on ε, so-calledplastic friction

Q–1t

- internal friction strongly dependent on loading timeR - degree of dynamic relaxationR

e- yield stress

Rm

- ultimate tensile strengthSK - Snoek and Köster relaxation∆S - entropy differences - substitutional atoms

mnik- elasticity constant

ss - designation of a pair of substitutional atomsT - oscillation periodTF - thermal fluctuation relaxationT

h- homologous temperature

Tp

- ductile to brittle transition temperatureT

t- melting point

t - timeV - volumeV+ - activation volumev - vacancyv

d- dislocation velocity

vl

- velocity of the longitudinal wavev

t- velocity of the transverse wave

W - total energy supplied to the systemW

t- energy consumed by material up to fracture

Wk

- half energy of the formation of a double kink∆W - energy scattered in the material during a cycleZ

1, Z

2- total power of the exciter and power required to

overcome resistance in the exciterα - coefficient of intensity of dampingβ - widening of the peak of internal frictionγ - thermal conductivity coefficient

xiii

γkr1

- first critical strain amplitudeγ

kr2- second critical strain amplitude

δ - logarithmic decrement of dampingε

ac- total strain amplitude

εap

- plastic strain amplitudeε

c- total strain

εd

- additional strainε

e- elastic strain

εi

- fatigue ductility coefficientε

mn- strain tensor

εkr1

- first critical strain amplitudeε

kr2- second critical strain amplitude

εkr3

- third critical strain amplitudeε

p- plastic strain

εt

- strain at crack formationε

d- rate of change of additional strain

ΘD

- Debye temperatureΘ

E- Einstein temperature

χ - coefficient of proportionalityλ

1, λ

2- parameters of the ellipsoid of deformation

µ - Poisson numberυ - Debye frequencyρ - specific densityρ

a- density of active dislocation sources

ρd

- dislocation densityρ

n- density of stationary dislocations

ρp

- density of mobile dislocationsσ - normal stressσ

a- stress amplitude

σC

- fatigue limitσ

f- fatigue strength coefficient

σik

- stress tensorσ

K- physical yield limit

τ - shear stressτ

r- relaxation time

σε - relaxation time at constant strainϕ - phase shiftϕ

o- angle of deflection of the pendulum

Ψ - relative amount of scattered energyω - circular frequency∆ω - half width of the resonance peak at half its

height

1

Nature and Mechanisms of Anelasticity

1

AIMS OF INTERNAL FRICTIONMEASUREMENTS

The elasticity characteristics belong in the group of the importantparameters of solids because they are often used in the analyticalsolution of the problems of deformation and failure. The elasticityvalues are used in all engineering calculations and the design ofcomponents, sections and whole structures. In the development andapplication of specific materials and high-strength components pro-duced from them, it is necessary to consider the required rigidityand also the probability that a certain amount of energy will buildup in the system during service. Actual solids are characterised bythe scattering of mechanical energy in them, i.e. internal friction.This representation of the anelasticity of materials and their tran-sition from elastic to anelastic, microplastic or even plastic responseto external loading are the consequences of the effects of externalloading and the activity of various mechanisms and sources of scat-tering of mechanical energy in the material which may be charac-terised by relaxation, dislocational, mechanical and magneto-mechanical hysteresis. These mechanisms result in changes of thestructure-sensitive properties of materials and also the structure-sensitive component of the Young modulus which is still regardederroneously in a number of publications as a material constant. Theexternal factors, such as mechanical loading, temperature, the effectof the magnetic field, different frequency of the changes of loading,etc., lead to changes of the nature and mechanisms of the processesof scattering of mechanical energy in materials. The relationshipsbetween the changes of internal friction and the defect of the Youngmodulus with changes taking place in the material on the atomiclevel, on the level of a group of grains, in the volume of the loadedsolid or in a group of solids, have already been confirmed and veri-fied.

2

Internal Friction of Materials

The author of the book has developed an original system, Fig.1.1, which, using the currently available data, shows the possibili-ties of the application of internal friction measurements in differ-ent sciences, from the atomic size up to entire structural complexes.The effect of the external factors on the occurrence of threshold

FFFFFigigigigig.1.1.1.1.1.1.1.1.1.1

Internalfriction

Solid statephysics

Damage

Thresholdstate

of materials

Vibroacousticsof system

solid solutions

diffusion

thermal activ. param.

phase transformations

point defects

dislocation structure

grain boundaries

strain

thermal

cyclic loading

radiation

hydrogen

damping capacity

Young's modulus

micromechanicalcharacteristics

relaxation

additional loading

creep

cracking resistance

quality of system

noise in system

amplitude-frequencyspectrum of system

structural damping

aerodynamic damping

flaw inspection

vibrothermography

vibrotechnologies

3


states of materials and components [1] is associated with continu-ous changes of the response of the material to their effect. This isreflected in changes of internal friction [2]. After compiling andverifying a physical model explaining the nature of changes of in-ternal friction with the changes of the external factor it is possibleto conclude that the internal friction measurements can be used asan indirect method of the monitoring of processes taking place insolids. The evaluation and quantification of the dynamics of changesin solid solutions, during diffusion and phase transformations, areoften associated with the determination of the parameters of ther-mal activation, with the data on the point, line and area defects ofthe crystal lattice taken into account.

The changes of temperature, static and cyclic deformation, andalso radiation or the presence of hydrogen in the material are re-flected in the degradation of the characteristics of the material asa result of damage cumulation. Accurate measurements of internalfriction enable indirect quantification of this phenomenon.

When evaluating the threshold state of materials and structures,it is also essential to quantify the damping capacity, the defect ofthe Young modulus, microplastic characteristics, relaxation andadditional elasticity phenomena, etc. these processes are also accom-panied by changes of internal friction so that the internal frictionmeasurements can be used for examining the process and criticalcharacteristics of specific materials.

Vibroacoustic examination of a structure or machine under dif-ferent service conditions, by evaluation of the acoustic quality, noiseand amplitude–frequency spectrum makes it possible to proposemeasures for ensuring high reliability and safety of operation of thesystem, especially under resonance conditions. Design or aerody-namic damping of components, sections of the entire structure maybe utilised here.

The efficient selection of materials with the required damping ca-pacity improves the functional behaviour and reliability of opera-tion of the machine and decreases the ecological damage from vi-brations and noise of machines. In technical practice, flaw inspec-tion methods are used on a wide scale, but the possibilities of thesemethods have not as yet been exhausted. Vibrothermography is notyet used widely as a method for identification of the areas of pref-erential absorption and scattering of energy in a real solid. However,it represents a significant tool in the solution of problems of stress-strain heterogenities and concentrators in the solids, with one of theinternal friction mechanisms playing the dominant role. Vibro-tech-

Aims of Internal Friction Measurements

4


nologies are gradually introduced into various production and trans-port applications where the level of internal friction in certain com-ponents may be very low whereas in others it may be high, depend-ing on the system utilising vibrotechnology.

Taking the actual scale of this subject, in this book, special at-tention is given only to some selected problems, associated mainlywith physical metallurgy and threshold states.

5


2

NATURE AND MECHANISMS OFANELASTICITY

The solution of a set of problems associated with the nature, meas-urements, evaluation and application of information on the internalfriction and the defect of the Young modulus of materials is basedon a brief and functional characterisation of the elasticity param-eters of structural monoliths and composite materials, and also onthe evaluation of the effect of different internal and external factorson their magnitude. A special position is occupied by the effect offactors causing nonlinearity between stress and strain, i.e. anelasticbehaviour of the materials. This includes the explanation of the phe-nomenon of deformation lagging behind stress, irreversible scatteringof energy in materials and mechanisms by which the energy of vi-brations is irreversibly scattered in the materials. These processesare also reflected in the level of the Young modulus and the occur-rence of defects of the Young modulus, and this may influence theaccuracy of calculations of permitted stresses in components orwhole structures.

2.1 ELASTICITY CHARACTERISTICSUnder the effect of a generally oriented force the solid can changeits dimensions and shape. If the relative strain in a specific direc-tion is denoted by the strain sensor mn and the force per unit cross-section, causing this strain, is denoted as the stress sensor ik, then

ik ikmn mnc , (2.1)

mn mnik iks , (2.2)

where cikmn

and snmik

are the elasticity constants. Strain tensor εmn

6


and stress tensor σik

are the tensors of the second order. They canbe described by nine pairs of strain components or nine pairs ofstress components, and the unit volume is selected sufficiently smallto ensure that the strain of stress unit is the same everywhere.

Three pairs of components act in the direction of the x axis, i.e.σ

xx in the direction normal to the wall of the cube; these pairs are

oriented normal to the x axis. They include the normal stress whichis positive for tension and negative for compression. The secondpair of the components in the direction of the x axis acts on thewalls oriented normal to the y axis. This pair is denoted σ

xy and σ

yx.

The third pair of the components in the direction of the x axis actson the walls normal to the z axis. This pair is denoted σ

xz and σ

zx.

The second and third pair act in the given planes and tend to shiftthem mutually. They are denoted as tangential or shear stress. Sincethe elementary volume of the material does not rotate, the compo-nents must be in equilibrium, i.e. σ

xz = σ

zx, and they are denoted τ

xz,

and also σxy

= σyz

, denoted τxy

. Consequently, this gives a symmet-ric tensor of the second order with six components

ik

xx xy xz

xy yy yz

xz yz zz

. (2.3)

Every symmetric tensor of the second order has three main axes.Since the axes of the coordinates are regarded as synonymous withthe axes of the tensor, only the main stress σ

1 is obtained.

For an anisotropic medium, equation (2.1) can be expanded intoa system of linear equations, i.e. for stresses σ

xx, σ

yy, σ

zz, σ

yz, σ

xy,

strains εxx

, εyy

, εzz

, γyx

, γzx

and γxy

, using the elasticity constants cikmn

,as given for the first of six rows in the form

xx xx yy zz yz zx xyc c c c c c 11 12 13 14 15 16 , (2.4)

where γ is shear.Similarly, equation (2.2) can be expanded utilising the elasticity

constants smnik

, as given for the first of the six rows

xx xx yy zz yz zx xyc s s s s s 11 12 13 14 15 16 . (2.5)

7


Generally, the elasticity constant cikmn

(and also smnik

) has theform of a tensor with 39 components, where the first of the 6 rowsis c

11, c

12, c

13, c

14, c

15, c

16 (or s

11, s

12, s

13, s

14, s

15, s

16).

Proportionality is found between the elasticity constants c12

= c21

or s11

= s21

and, generally, cαβ = cβα, or sαβ = sβα. As a result of thissymmetry, the crystallographic system with the lowest symmetry(triclinic) has only 21 independent components, and the number ofindependent components in the orthorhombic system decreases to 9,in the tetragonal and diagonal systems it decreases to 6, and in thehexagonal system to 5.

For cubic crystals there are three elasticity constants, c11

, c12

andc

44. The elasticity constants c

ikmn and s

nmik are linked by the defined

relationships [3].The physically determined elasticity constants for technical ap-

plications are complicated. For crystallographic systems with thesymmetry higher than orthorhombic, the normal tensile stress in thedirection of the x axis (σ

xx) results in relative elongation ε

xx and two

reduction in areas εyy

and εzz

:

xxxx

yy xx zz xxE

112 13, , . (2.6)

If we consider the normal stress in the direction of the y and z axes,we obtain three moduli in tension (Young modulus) and six Poissonnumbers, of which only three are mutually independent, because

E E E E E E1

12

2

12

2

23

3

32

3

31

1

13 , , . (2.7)

Shear stress τyz

in the yz plane causes shearing γyz

. Consequently,τ

yz = G

1γ

yz. Similarly, in the xz plane, where τ

xz = G

2γ

xz and in the

xy plane, where τxy

= G3γ

xy.

The orthorhombic crystal in the system of the technical elastic-ity moduli has only nine independent elasticity characteristics, i.e.three tensile (Young) moduli E, three shear (Coulomb) moduli g, andthree Poisson numbers µ. For the cubic crystal, these are three char-acteristics (E, G, µ), and for an isotropic solid it is E, G, because

GE

2 1

. (2.8)

8


Equation (2.8) is also important because polycrystalline metals andalloys without a sharp texture behave as isotropic materials.

The bulk Young modulus K is defined as isotropic pressure Pdivided by the relative volume change caused by this pressure

KPV

V

E

3 1 2 . (2.9)

The values of the elasticity moduli (E, G), the bulk Young modu-lus (K) and Poisson number (µ) for a number of metals arepresented in Table 2.1. For rock and glass µ = 0.25, G = 0.4E ,K = 2/3E, for metals µ = 0.33, G = 3/8E, K = E, and for elastomersµ = 0.5, G = 1/3E, and the K/E ratio is high.

The Young modulus is closely linked with the velocity of propa-gation of sound in a metallic material. In the case of a longitudi-nal elastic wave, the velocity of propagation is

lE

(2.10)

and in propagation of a transverse elastic wave, the velocity ofpropagation is

t

G , (2.11)

where ρ is the specific density of the metallic material [4].This phenomenon is utilised in accurate measurements of E and

G, because measurements are taken without exchange of heat withthe environment which enables also the determination of adiabaticelasticity moduli which differ from the elasticity moduli obtainedunder isothermal conditions (for example, in the tensile test). Table2.1 gives values of v

1 and v

t for several metals.

The values of the elasticity constant make it possible to deter-mine accurately the anisotropic factor of the elastic properties ofmetallic materials from the equation

9


TTTTTaaaaabbbbble 2.1 le 2.1 le 2.1 le 2.1 le 2.1 Elasticity moduli and other characteristics of several metals at 20°C

foepyTlatem

E)aPG(

G)aPG( µ K

)aPG(V

l

sm( 1– )V

t

sm( 1– )

*lAaCiNuCdPgA

rItPuAbP

8.076.912.1323.5417.241

1.180.8259.961

1.883.73

3.624.71.988.458.156.920.412

0.161.136.31

43.013.013.053.093.083.062.044.024.044.0

5.772.710.0916.9311.2916.3010.0737.2724.571

8.84

5536–

4985627449546863

––

16338512

6213–

9123892278917761

––

9321068

**iLaN

KV

rCeFbRbNoM

aT

5.012.76.4

2.1317.9722.322

3.10.7010.0330.0917.393

0.47.27.13.840.201

9.6874.02.937.911

1.170.351

63.023.053.063.0

82.0

93.003.053.092.0

8.113.80.4

3.4515.6911.371

0.9515.2823.4911.803

90758703

–0006

–4606

–4015946674449135

12824341

–0872

–5233

–9802215393023482

***gM

iToCdC

8.444.4119.022

5.56

6.713.345.486.42

82.063.023.003.0

5.432.7013.091

2.26

5985362672850313

6723229294033661

****nInS

9.311.06

8.46.32

64.033.0

6.146.06

95420033

9079461

Comment: * - cubic face centred; ** - cubic body centred; *** - hexagonal closed-packed; **** tetragonal

Ac

c c

2 44

11 12

, (2.12)

where for an isotropic case A = 1. For some metals, the dependenceof the mechanical properties on the loading direction is shown inFig. 2.1.

10


The approximate coefficient of anisotropy of the elastic proper-ties can be determined as the ratio of the maximum and minimumvalues of the elasticity moduli. Table 2.2 gives the accurate (A) andalso approximate (A

p) anisotropy coefficients of the elastic proper-

ties of some metals.If E

max (= E

111) and E

min (= E

100) are available, it is possible to

determine E in the direction characterised by the angles α, β, γ tothe axes of the cube using the Weert’s equation in the form

13

1 1

100 111

2 2 2 2 2 2

E E E

cos cos cos cos cos cos . (2.13)

The equation can also be used for polycrystalline materials with atexture, if the latter is expressed by two or more ideal orientations.

FFFFFigigigigig.2.1..2.1..2.1..2.1..2.1. Directions of true stress at fracture of aluminium single crystal (a), elongationof aluminium single crystal (b), Young modulus of aluminium single crystal (c),Young modulus of iron single crystal (d), shear modulus of elasticity of iron singlecrystal (e) and Young modulus of magnesium single crystal in tension (f).

TTTTTaaaaabbbbble 2.2le 2.2le 2.2le 2.2le 2.2 Approximate (Ap) and accurate (A) anisotropy coefficients of elastic properties

foepyTlatem

Exam

ninoitcerid

>111<)aPG(

Enim

ninoitcerid

>001<)aPG(

Ap

Gxam

ninoitcerid

>001<)aPG(

Gnim

ninoitcerid

>111<)aPG(

Ap

Ap

lAuCeF

W

7.74.910.920.04

4.68.65.310.04

571.1078.2051.2000.1

9.27.78.115.51

5.21.31.65.51

31.105.239.100.1

2.13.34.20.1

11


The elasticity moduli are associated with the characteristics de-termined by the force influence of interaction of the atoms in thecrystal lattice linked with the thermal expansion coefficient, Debyetemperature, sublimation temperature, melting point, etc. These con-siderations show that the elasticity moduli can be determined ap-proximately, with an acceptable correlation factor, using the meas-ured values of these characteristics.

The relationship between the Young modulus E (or G) and themelting point of metal T

m has the form

T k A Et 1 , (2.14)

where k1 = 5K in determination of E, and k

1 = 8.5K in determina-

tion of G, where K is the bulk Young modulus, and A is the pro-portionality coefficient.

The relationship between the Young modulus, the volume coeffi-cient of thermal expansion β and the relative molar heat capacityat constant pressure c

p is determined by the equation

Kc

Vp

a

0 , (2.15)

where γ0 is a constant and V

a is the molar volume. At room tempera-

ture and elevated temperatures T, the approximate validity of thefollowing equation has been confirmed

=β

(2.16)

The Poisson number and constant γ0 are linked by the equation

2

30

0

, (2.17)

where η = 1.5 for fcc metals, and η = 0.945 for bcc metals.Debye temperature Θ

D is linked with the Young modulus by the

equation

12


DE

A

1682

12

1 6

/

, (2.18)

where ρ is specific density, and A1 is atomic density.

The elasticity moduli can also be determined from accurately re-corded results of tensile or torsion tests. However, the most accu-rate data are obtained by measuring the velocity of propagation ofelastic waves v

1 or v

t (equations 2.10 and 2.11).

The resonance methods are effective and accurate (error 0.5-0.8%) in the determination of the elasticity moduli. However, itshould be noted that the natural frequency of the longitudinal vibra-tions is an order of magnitude higher than the natural frequency ofthe bending vibrations. Increase of the loading frequency increasesthe intensity of relaxation processes. This is reflected in an increaseof temperature and the associated decrease of the Young modulus.This results in a systematic error in the measurement of the elas-ticity moduli by the resonance method. This shortcoming can beeliminated using the pulsed methods of Young modulus measure-ments. These methods are based on the measurement of the veloc-ity of propagation of a pulsed elastic wave in metal, and the wave-length is small in comparison with the dimensions of the solid.

The Poisson number can be determined from X-ray diffractionmeasurements of the lattice parameters of the metallic material.

The accuracy of the pulsed methods of measurement of the Youngmodulus is high (error is approximately 0.1%). However, thesemethods also have certain shortcomings. The most important prob-lem is the fact that when measuring the velocity of propagation ofa pulsed wave it is necessary to measure the Poisson number at aspecific moment of time. Procedural problems do not enable meas-urements of the Young modulus to be taken at higher temperatures.

The tabulated data on the elasticity moduli of metals and alloysare limited because they represent the characteristics at room tem-perature and do not describe the initial state of the material or itsthermal-deformation history. This shortcoming is partially eliminatedby a set of data [5] which contains the elasticity moduli for a largenumber of metals and alloys at elevated temperatures.

The elasticity constants and also technical elasticity moduli areinfluenced by a large number of external and internal factors.

13


2.1.1 Effect on elasticity characteristicsThe effect of temperature on the elasticity characteristics is asso-ciated with the thermal expansion of the material, i.e. with the tem-perature dependence of the atomic spacing. Analysis of this prob-lem has shown that the change of the elasticity moduli is not asso-ciated with absolute temperature, but is linked with homologoustemperature T

h = T/T

m, where T is the temperature at which the

Young modulus is determined.For the same homologous temperatures, the relative change of the

elasticity characteristics for many metals is the same (Fig. 2.2).This relationship is linked with the identical homologous tempera-ture dependence of the change of atomic spacings.

Increase of temperature results in a decrease of E, G and K. Thevalue of the Poisson number initially slowly decreases and then in-creases with a further increase of temperature; because of the dif-ferent thermal strain history of the material, the dependence is morecomplicated.

Decrease of temperature, like increase of pressure, results in thesame change of the atomic spacing in the crystal lattice. This showsthat the change of the Young modulus will be similar. The changeof bulk Young modulus K at absolute temperature from 0 to T is de-scribed by the equation

K

K

g T

0 3

, (2.19)

Th

E/E

0

FFFFFigigigigig.2.2..2.2..2.2..2.2..2.2. Dependence of relative values of the Young modulus of various metals onhomologous temperature, where E

0 is the Young modulus at 0 K.

14


where

gK

K

1

0

(2.20)

is the change of the Young modulus during deformation of the lat-tice by the value ε, K

0 is the bulk Young modulus at 0 K. Modu-

lus K is proportional to the curvature of the relief of the potentialenergy of the crystal lattice in the area with the atom.

Depending on the distance of the atom from the equilibrium po-sition, the curvature of the potential relief decreases as a result ofincrease of ε. This shows that g < 0, and the value of the modu-lus decreases with increasing temperature. Consequently

0

0 0

.3

p

a

gc TK

K V K

γ∆ = (2.21)

Equation (2.21) shows that the resultant value is strongly influ-enced by the value of c

p. As in the case of equation (2.21), it is

possible to write similar equations for the change of E or G. Likethe temperature dependence of the heat capacity at constant pres-sure, the temperature dependence of the elasticity moduli can bedivided into three ranges: low-temperature range, where T << Θ

D,

transition range, where ΘD

≤ T ≤ 0.5Tt, and high-temperature range,

where T > 0.5Tt.

In the low-temperature range, the coefficient of the effect of tem-perature on the change of the modulus e is proportional to t

0 g c

p,

and also proportional to (T/ΘD)3. In the entire temperature range the

dependence of the Young modulus on temperature has the shape K/K

0 ~ T4.Two cases can occur in the transition temperature range. If the

Debye temperature for a specific metal or alloy is significantlylower than 0.5T

t, then c

p ≈ 3R, where R is the gas constant, and

e = const. This shows that the modulus of elasticity increases pro-portionately with increasing temperature. When Θ

D is close to 0 or

higher than 0.5Tt, the value of c

p increases with increasing tempera-

ture and the dependence is ‘domed’ in the upward direction. The in-crease of temperature by one degree results in a decrease of theYoung modulus by 0.02–0.04%, with the approximation sufficientfor a wide range of the materials.

15


The selection of experimental dependences is very important; itis necessary to ensure that they describe with sufficient accuracy thechanges of the elasticity moduli of the materials. For example, inRef. 6, the temperature dependences of the Young modulus for VSt3steel in the form E = (21.68 – 67×10–4T)×104 MPa, where T is in°C, were verified for the temperature range from –70 to +70°C.

For metals with high melting points, for temperatures of up to2000°C, the authors of Ref. 7 published the empirical dependences:for vanadium in the form E = (12.8 – 9.61×10–4T)×104 or G =(4.88 – 8.48 × 10–4T)×104, for niobium E = (10 + 9.18×10–4T -4.11×10–7 T2)×104 or G = (3.12 + 9.9×10–5T )×104, and for tantalumE = (16.9–8.22×10–4T–1.66×10–7T2 )×104 or G = (7.74–1.73×10–4 T)×104, always in MPa.

For tungsten, we can use the equations in the form E = E0[(T

t -

T)1/Tt]0.4, or G = G

0[(T

t – T)1/T

t]0.263, E = E

0[(T

t – T)1/T

t]0.463, G =

G0[(T

t – T)1/T

t]0.465, where E

0 and G

0 are the moduli at 0 K.

On the basis of analysis of the elastic characteristics of 40 al-loys based on Fe, Ni, Cu and Al in the temperature range below500 K, it was shown [8] that, with the exception of Invar alloys,the temperature dependence of the Young modulus is described quiteaccurately by the following equation

0 ,

1E

T

E E

e

η= − Θ−

(2.22)

here ΘE is the Einstein temperature, η/Θ

E is the limiting value of

the tangent dE/dT to the E(T) dependence.At elevated temperatures, above 0.5T

m, the rate of decrease of the

Young modulus rapidly increases, and the temperature at which therapid decrease of the modulus starts is close to or identical with thetemperature of the start of increase of the high-temperature back-ground of internal friction.

There are several hypotheses explaining the rapid decrease of theYoung modulus in the high-temperature range. This may be causedby the nonlinear dependence of atomic forces on additional thermalstrain. Some hypotheses are based on the assumption according towhich this behaviour is the consequence of deformation due to dis-location movement. The hypotheses are supported by the assumptionaccording to which the mobility of dislocations increases with in-creasing temperature. This is reflected in an increase of the contri-

16


bution of dislocational anelasticity.The elasticity moduli also decrease with increasing internal fric-

tion. Measurements of the temperature dependence of the elasticitymoduli in the temperature range (0.5–0.7) T

m show that the activa-

tion energy of the change of the elasticity moduli is close to the ac-tivation energy for self-diffusion [9] which is close to the activa-tion energy of the relaxation process at the grain boundaries in poly-crystalline materials. In the temperature range close to the meltingpoint (0.95–0.97) T

m the Young modulus changes as a result of the

temperature maximum on the temperature dependence of internalfriction. Measurements of the Young modulus of Sn, Bi, Cd and Pbup to the melting point showed that in the vicinity of the meltingpoint the Young modulus rapidly decreases, and a slight increase ofthe Young modulus is recorded only in the case of Sn at tempera-tures higher than 0.98T

m [10]. This phenomenon was also reflected

in the arrest of the decrease of the Young modulus of specimens ofsintered iron with a tin filler in the vicinity of the melting point ofSn.

Acceleration of decrease of the Young modulus with increasingtemperature is also caused by relaxation processes taking place inthe process of gradual increase of external stress.

In the forties, Frenkel showed that a metal starts to melt whenthe Young modulus is 0. Theoretical calculation showed that theYoung modulus at the melting point is 0.7–0.75 of the modulus at0 K. The experimental dependences and also Fig. 2.2 show that withincrease of temperature up to the melting point the Young modulusdecreases by 40–60%. The difference between the theoretical calcu-lations and the results of measurements confirms the effect of high-temperature relaxation.

The form of the temperature dependence of the Young modulusat high temperatures may have a significant effect on the activationenergy of creep [11]. In steady-state creep, the creep rate is deter-mined by the equation

2

eQn RTA e

−ε = σ (2.23)

where A2

is the proportionality constant, σ is the acting stress, n isthe strain hardening coefficient, Q

c is the activation energy for

creep, and R is the gas constant. If we take into account the tem-perature dependence of the Young modulus, the equation has the

17


following form

( )

*

3 ,e

n Q

RTA eE T

− σε =

(2.24)

where A3 is the proportionality constant and Q∗

c is the modified ac-

tivation energy of creep which can be determined from the equation

( )* lnln1 1c

d E TQ R nR

dT T

δ ε= − − δ

(2.25)

Consequently

2* .c c c

nRT dEQ Q Q

E dT∆ = − =

The change of the activation energy of creep may be quite consid-erable. For example, in the case of Inconel alloy at a temperatureof 704°C, Q*

c = 551.7 kJ mol , ∆Q = 80.9 kJ mol, and at 1037°C,

Q = 251. 4 kJ⋅mol–1 and ∆Qc = 59.5 kJ⋅mol−1. This example shows

that in calculations it is important to take into account the infor-mation on the change of the Young modulus with increasing tem-perature.

External pressure and internal stress also influence the level ofthe Young modulus.

The increase of external pressure P results in increase of theYoung modulus. Up to a pressure of 5 GPa, we can use the follow-ing dependence

( )011 ,E E P= + χ (2.26)

where E0 is the Young modulus at the atmospheric pressure, and thevalue χ

1 varies from 10–1 to 10–2 GPa–1. Specific values of χ

1 for

various materials are presented in Ref. 5.The increase of the Young modulus with increasing hydrostatic

pressure is caused by a decrease of the atomic spacing in the crystal

18


lattice. This hypothesis has been verified theoretically and also bymeasurements [12].

The dependence of the Young modulus on stress is general notonly for the hydrostatic pressure conditions. The dependence of theYoung modulus on the state of the structure of the hardened mate-rial, observed in a large number of experiments, is usually inter-preted from the viewpoint of the magnitude, nature and distributionof internal stresses in the material [13]. Theoretical calculations ofthe dependence of the Young modulus on the microstrain of the crys-tal lattice ∆a/a were carried out by Levin [14]. The calculated andexperimental results are presented in Fig. 2.3.

Increasing microstrain decreases the Young modulus but the scat-ter of the measured Young modulus values increases. This is theresult of the scatter of distribution of the internal stresses in the ma-terial. The results show that the Young moduli of the materialswhich contain internal stresses are in fact random quantities forwhich it is possible to obtain the corresponding dependence of dis-persion S2 on microstrain ∆a/a. In the case of the experimentallydetermined change of dispersion moduli S2 it is necessary to takeinto account the dispersion caused by the measuring procedure andthis value characterises the inaccuracy of determination of the elas-ticity moduli of the metals and alloys.

FFFFFigigigigig.2.3..2.3..2.3..2.3..2.3. Dependence of Young modulus on microstrain ∆a/a for heat treated St3steel.

19


The change of the Young modulus with increasing pressure andtemperature has the same basis, i.e. the change of the atomic spac-ing, thus yielding the equation

3 ,C V T

E E EK

T T P

∂ ∂ ∂ = − α ∂ ∂ ∂ (2.27)

where C, V and T indicate that the values are determined at constantconcentration, volume, and temperature, and α is the coefficient oflinear thermal expansion [15].

The quantity (∂E/∂T)V is included in the equation due to the de-

pendence of the elasticity moduli of ferromagnetic materials on thedegree of arrangement of magnetic domains. At temperatures lowerthan T

c (Curie temperature) and for non-ferromagnetic materials,

(∂E/∂T)V = 0. Consequently

( )0

0

3 .T

T TT T

EE E K T dT

P

∂ − = − α ∂ ∫ (2.28)

For pure iron and binary alloys of iron with cobalt, nickel, chro-mium and molybdenum at T << T

c equation (2.28) describes satis-

factorily the experimentally measured values.The dependence of the Young modulus on the type and severity

of the stress state and, in particular, strain state is of principal im-portance not only for predicting the changes of the elastic materi-als in service but also for the development of suitable theoreticalmodels of the strength and plasticity behaviour of metals and alloys.

The level of the Young modulus is also influenced significantlyby the strain.

The addition of alloying elements to the main metal changes itselastic characteristics as a result of changes of the interatomicbonds in the alloys. This takes place in relation to the chemicalinteractions between foreign atoms as a result of the change of theconcentration of ‘free’ electrons (electron factor) or as a result ofthe effect of lattice defects and, consequently, lattice parameters,which is the result of different atomic radii of the added and mainmetal (dimensional factor). In reality, these factors can affect theYoung modulus in the same or opposite direction. Experiments showthat the dependence of the change of the Young modulus on the con-

20


centration of the alloying element (c) is the result of these factorsacting in the same direction. This result can be expressed by theequation in the form

ln3 .l

T T

E aE E K c

P c

∂ ∂ ∆ = ∆ − ∂ ∂ (2.29)

where ∆Ee

is the component of the change of the Young modulus asa result of the change of the number of electrons, a is the latticespacing.

The characteristic ∆Ee

has complicated form and depends onmany influences, such as the change of the energy of positive ionsin the electron ‘gas’, the change of the degree of overlapping ofelectron paths of the adjacent atoms, the zonal structure of themetal, etc.

The dependence of the Young modulus on the concentration of thealloying elements is determined by the characteristics of the atomsof the alloying element and the main metal. Examples for binarysystems with complete solubility of the metals in the solid state arepresented in Fig. 2.4.

For alloys of the metals which do not form intermetallic com-pounds together, the elasticity moduli are an additive property to afirst approximation. For the majority of non-transitional and sometransitional elements in the range of the mass concentration from 10to 20% and for Cu–Pt, Mo–W, Cu–Ni and other alloys, the resultsobtained by Köster show that the concentration dependence of theYoung modulus in the entire concentration range is almost linear.The contribution of the electron factor to the value of the Youngmodulus increases with increasing difference of the valency of thecomponents of the alloying and the main metal (Fig. 2.4a).

In most cases, the elasticity moduli of intermediate phases arehigher than those of pure components. The experimental results ob-tained for the Cu–Sn and Cu–Zn systems show, Fig. 2.5, that higherelasticity moduli of the intermetallic phases influence the changesof the elasticity moduli in a wide concentration range.

Compounds of metals and an intermetallic phase are character-ised by high elasticity moduli. In the case of carbides, it should notbe assumed that the Young modulus of carbon is E = 0.09 × 105

MPa. It is more accurate to consider the value for diamond, i.e.E = 1 × 105 MPa. For example, for titanium E = 1.1 × 105 MPa,and for TiC E = 0.46 × 105 MPa. For tungsten E = 3.96 × 105

21


FFFFFigigigigig.2.4..2.4..2.4..2.4..2.4. Effect of composition of alloys on their Young modulus.

FFFFFigigigigig.2.5..2.5..2.5..2.5..2.5. Change of the Young modulus of Cu–Zn and Cu–Sn in relation to compositionindicating the formation of intermetallic phases.

MPa, and for WC E = 6.7 × 105 MPa. These approaches are of in-formation nature.

In the systems with superlattices, the magnitude of the Youngmodulus depends on the state of the structure, for example, the val-

wt.%

22


ues for the disordered state of an alloy are presented in Fig. 2.6.After the formation of a superlattice, there are significant changesin relation to the initial condition.

The dependence of the elasticity moduli on the composition of thealloying for some transitional elements is complicated becausd-electrons take part in the formation of bonds in these metals. Cs,Ti, Mg and Fe are characterised by unstable electron configurations,reflected in allotropic transformations. Extrema on the E(c)dependences may also form when the transitional metal is the maincomponent (Fe–Al, Ti–Al) or it is also an addition (Cu–Ti), or both(Ti–Mo).

The complicated nature of the changes of binding forces in alloy-ing of transition metals has so far prevented the construction of aunified mechanism of the effect of foreign atoms on the elasticcharacteristics of the transition metals.

In ferrous alloys, it is possible to evaluate the effect of the elec-tron factor on the change of the Young modulus in alloying with dif-ferent elements [14]. The electron factor of the transition metals ofthe same period as iron (chromium, vanadium, cobalt, nickel, man-ganese) is almost 0 and, consequently, all changes of the elasticitymoduli in alloying are associated with the dimensional factor. Theatoms of the transitional elements from higher periods are charac-

FFFFFigigigigig.2.6..2.6..2.6..2.6..2.6. Change of the elasticity moduli in relation to the concentration of the solidsolution.

23


terised by non-zero electron factor values. The values of the elec-tron factor of the elements from the same period (for example, Ruand Rh or Re, Ir, Pt) are the same and increase with increase of thenumber of the period.

The dependence of the Young modulus on composition for iron-based alloys is complicated (Fig. 2.7 and 2.8). The most significantchanges of the E(c) dependence are recorded at small amounts of theadditions. This phenomenon is associated with the fact that the for-mation of the solid solutions of the transitional metals is accompa-nied by an increase of the number of electrons when interactiontakes place between s and d electrons. This increases the Youngmodulus. The increase of the concentration of the alloying elementis accompanied by an increase of the strength of the effect of thedimensional factor which increases the distance between the atoms.The determination of the changes of the Young modulus and theaccompanying change of the parameters of the crystal lattice withthe change of the composition of the alloy has made it possible toconclude that, after dissolving in iron, the alloying elements formadditional bonds and the effect becomes stronger with the distanceof the added element from iron in the periodic system of elements.

The temperature dependence of the Young modulus for binary

FFFFFigigigigig.2.7. .2.7. .2.7. .2.7. .2.7. Dependence of the Young modulus of iron-based alloys on the content ofdifferent additions (molar %).

24


FFFFFigigigigig.2.8. .2.8. .2.8. .2.8. .2.8. Effect of temperature on the dependence of the Young modulus on the temperatureof an Fe–Cr alloy.

alloys is characterised by the temperature coefficient which dependsonly slightly on the concentration of the added element. Conse-quently, the E(T) dependences with increasing content of the alloyingelement are almost parallel.

The increase of the carbon content of iron decreases the elasticitycharacteristics. In annealed steels, the largest increase of the Youngmodulus is recorded at low carbon concentrations (to 0.2 wt.%). Inquenched steels, the E(c) dependence decreases, even though theincrease of the carbon content accelerates the increase of the Youngmodulus in comparison with the annealed condition.

The dependence of the elasticity characteristics on compositionin the case of the Fe–Ni system is also influenced by prior tensiledeformation and annealing (Fig. 2.9).

In the steels, carbon is present in most cases in cementite Fe3C

and, consequently, the evaluated dependences E(c) are examinedmainly from the viewpoint of the effect of different factors on ce-mentite. Cementite is a ferromagnetic phase with T

C ≈ 210°C. Since

Armco iron has a conventional, almost linear dependence E(T), thedependence for carbon steels at T ≈ 200°C is characterised by asmall delay which changes to a local maximum with the increase ofthe carbon content of the alloy, for example in white cast iron. The

25


form of the previously plotted curves also enables extrapolation ofthe changes of the E(T) dependence for cementite.

Examination of the titanium alloys with different iron and alu-minium content [16] showed higher elasticity moduli and the in-crease of the Debye temperature with increasing content of the al-loying elements. If the different electron structures of the Al and Featoms are considered, it may be concluded that aluminium has asignificant effect on the nature of atomic interactions in titaniumalloys. Increase of the aluminium content increases the force char-acteristics of the atomic bonds; this may be one of the reasons forthe increase of the ultimate tensile strength when alloying titaniumwith aluminium from 480 MPa (for example, titanium alloy VT1-0) up to 1000 MPa (for example, titanium alloy VT5).

Investigations of the elasticity characteristics of alloys of tita-nium with molybdenum, vanadium and niobium shows that the for-mation of extrema on the E(c) curves may be attributed to the startand finish of transformation of the α-phase to β-phase, and in thecase of the titanium alloys with these elements, the dependences arequantitatively similar indicating the same nature of the mutual ef-fect of titanium alloys with these elements. The additions decreasethe elasticity moduli of the alloys with the α-phase and also with

FFFFFigigigigig.2.9.2.9.2.9.2.9.2.9. Dependence of the Young modulus and Poisson number on the compositionof ferrous and nickel alloys after cold tensile deformation (symbol 1) and after annealing(symbol 2).

wt.%

26


FFFFFigigigigig.2.10..2.10..2.10..2.10..2.10. Dependence of the Young modulus of alloys of Ti with Mo (a) and withV (b) after rapid cooling (1) and tempering (2).

α + β phases. Transition to the single-phase region (β) is associ-ated with the minimum on the E(c) dependence. Alloying of thesingle-phase (β) solid solutions results in a monotonic increase ofthe elasticity moduli up to the values typical of the pure compo-nents.

Figure 2.10 shows that α-Ti is characterised by higher values ofthe Young modulus in comparison with β-Ti. Differences are alsoevident in alloying. Whilst in the case of α-Ti the Young modulusdecreases, in the case of β-Ti the Young modulus increases with in-creasing concentration of the additions.

Rapid cooling of the alloy greatly changes the dependence of theYoung modulus on concentration, Fig. 2.10. The rapid decrease ofthe Young modulus in the range of low concentration is associatedwith the formation of supersaturated α′- and α″-solid solutions. Inthe case of the Ti–Mo alloy this takes place up to 6 wt.% Mo. Thesubsequent increase of the E(c) dependence is caused by the forma-tion of the martensitic phase ω which formed as a result of thetransformation of β to ω . The maximum of the E(c) dependencecorresponds to the highest volume content of the β-phase. The sec-ond minimum of the E(c) dependence can be explained by the in-crease of the content of the β-phase. This results in an increase of

wt.% Mo wt.% V

27


the volume fraction of the metastable cubic volume-centred struc-ture with less dense packing and, consequently, in a decrease of theYoung modulus. In the region where there is only the β-phase, theincrease of the Young modulus takes place as a result of the in-crease of the strength of the effect of alloying elements on thechange of the interatomic forces. In the region of the single-phaseβ-alloys there are no significant differences in the values of theelasticity moduli for the annealed and quenched condition.

The differences in the elasticity moduli are characterised by dif-ferent sensitivity to the composition of the alloy. In the Ti–Zn al-loy, the E(c) dependence is represented by a smooth curve with theminimum at c = 50 wt.%. In this range, K and µ change non-monotonically in relation to composition.

In alloys where the components form a mechanical mixture, spe-cifically Al–Be, Al–Si, Bi–Sn, Fe–Fe

3C, etc., the dependence of the

Young modulus on composition is linear. The Young modulus alsodepends on the distribution of the inclusions which is reflected inthe upper and lower boundary of the elasticity moduli of the me-chanical mixtures. Calculations show that at concentrationsc < 40 wt.%, the elasticity moduli are the additive characteristicsof the phases present. At a large amount of the second phase in thealloy, the calculated and measured values of the elasticity modulidiffer.

Ordering in the alloys also influences the elasticity moduli be-cause the effect of the electron factor is very strong. Ordering in-creases the interatomic forces, the electron factor has the positiveorientation and increases the level of the Young modulus. The de-pendence of the Young modulus on temperature in orderedalloys at temperatures below Kurnakov temperature T

k is character-

ised by a constant value of the temperature factor of the change ofthe Young modulus e. With the increase of temperature to the rangeT ≈ T

k, the dependence E(T) shows an anomaly in the form of a

jump or a sudden decrease of the Young modulus to the value cor-responding to the difference of the elasticity moduli of the orderedand disordered state of the alloy (Fig. 2.11).

2.1.2 Elasticity characteristics of structural materialsIn most cases, multiphase and multicomponent alloys with differentchemical composition are used in technical practice. The compli-cated phase composition and the differences in the degree ofmetastability of structural components result in large differences inextent of the utilisation of the processes of structural and phase

28


changes, ageing, recovery, etc. in the application of various mate-rials. Analysis of the effect of these factors on the elasticity char-acteristics is complicated owing to the fact that factors, determin-ing the actual value of the Young modulus, are also important.These processes determine the effective value of the interatomicforces by the change of the concentration of the alloying elementsin the solid solutions, the change of internal stresses at coherentphase boundaries, actual changes of the elasticity moduli of the in-dependent phases, the increase of the number of structural defects,etc. (the first group of factors).

The second group of factors includes the processes ofanelasticity; this is reflected in the structural sensitivity of the elas-ticity moduli. The change of the structure and phases of the alloysin quenching, tempering, ageing, etc. results in a change of the levelof the elasticity moduli; this change is large in comparison with thevalues given in the tables. Consequently, it is necessary to have notonly the values from various tables or material specifications, butalso data on the specific structural state and the relationships oftheir changes in the expected working conditions of the material.

The sensitivity of the elasticity moduli of the material to thechange of the structure and phases is the reason for the qualitativelynew thermal hysteresis of the moduli. In this case, the curves E(T)in heating and cooling differ. Hysteresis is found only in a specific

FFFFFigigigigig.2.11..2.11..2.11..2.11..2.11. Effect of temperature on the Young modulus of ordered CuZn and Cu3Au

alloys.

29


temperature range (Fig. 2.12, curves 2). Figure 2.12 shows thechanges of the Young modulus in relation to temperature for a steelwith 0.4 wt.% C (curves 1) and for white cast iron with 2.8 wt.%C and 1.4 wt.% Si (curves 2). Both materials were normalised inthe initial condition. In the temperature region below the eutectoidtransformation temperature (curves 1) the dependence E(T) isslightly nonlinear. In the eutectoid transformation temperature rangethere is a large decrease of the elasticity moduli associated with thestart of dissolution of cementite particles. Completion of the α→γtransformation results in a drop in the rate of decrease of the E(T)dependence. The large increase of E at temperatures of 1000–1050°C is interpreted by the equalisation of the composition of thephases as a result of the dissolution of the solutes built up at thegrain boundaries in the disappearance of the old grain boundariesduring the growth of austenite grains. With the reversed change oftemperature the extent of relaxation at the grain boundaries de-creases and the dependence E(T) is shifted higher in comparisonwith gradual heating. The curves E(T) in heating and cooling be-come identical below the eutectoid transformation temperature.

FFFFFigigigigig.2.12..2.12..2.12..2.12..2.12. Effect of temperature on the Young modulus of steel 40 (1) and whitecast iron (2).

30


The special feature of the E(T) dependence of cast iron is thepresence of a wide temperature range in which thermal hysteresisforms. The form of the E(T) dependence depends on the process ofbreakdown of cementite and ferrite formation. The most significantchanges are recorded in the temperature range 680–850°C. In heat-ing, the E(T) curve shows two distinctive steps associated with thestart (710–720°C) and finish (810–840°C) of eutectoid transforma-tion in white cast iron. The first stage corresponds to the transfor-mation of pearlite to austenite which is part of ledeburite, and alsoto the compressive stress in the transformation of ferrite to austeniteas a result of the differences in the coefficient of thermal expansionof eutectic cementite and pearlite. The second stage corresponds tothe completion of the breakdown of pearlite at the original austenitegrain boundaries where extensive relaxation of a large part of theresidual stresses could have taken place during the tests. The meas-urements of internal friction with the change of temperature showedthat every step on the E(T) curve corresponds to the internal fric-tion maximum. The relaxation mechanism is associated with thechange of the susceptibility of the grain boundaries to stress relaxa-tion with the formation and growth of new phase particles on them.

The elasticity characteristics of the materials change in relationto the method and conditions of heat treatment. The final results arethe consequence of the mutual effect of the factors which increaseor decrease the Young modulus. In quenching of carbon steels, thevolume fraction of cementite in the structure of steel decreases. TheYoung modulus of cementite is lower. Therefore, this process in-creases the Young modulus. In addition, quenching is accompaniedby martensitic transformation resulting in supersaturation of thedislocation density. This process decreases the Young modulus. Afteradding up the processes, the quenching of steels and cast irons de-creases the Young modulus. The magnitude of the decrease dependson the carbon content, Fig. 2.13. The increase of the carbon con-tent to 0.45 wt.% increases the relative change

1

z z

z

E E E

E E

∆ −=

of the steel after quenching from 900°C. A further increase of thecarbon content has no longer any effect on the value of ∆E/E

1. This

can be explained by the formation of the effective values of theYoung modulus with another influence. Increasing carbon content of

31


the steel decreases the temperature at which the martensitic trans-formation is completed. For steels with a carbon content higher than0.5 wt.%, the temperature of completion of the martensitic transfor-mation is lower than 0°C so that the structure of the quenched steelalso contains retained austenite. The Young modulus of austenite ishigher than that of ferrite, and this compensates the decrease of theactual values of the Young modulus as a result of the effect of thepreviously mentioned factors.

The increase of quenching temperature to 1050°C changes the na-ture of the changes of the Young modulus in relation to the carboncontent of the steel (Fig. 2.13, curve 2). Therefore, the increase ofE/E

1 may be caused by increase of the grain size, resulting in a de-

crease of E and, consequently, increase of ∆E/E1. In the case of

steels with a high carbon content, the decrease of E may also bereflected in the formation of various defects of the structure duringquenching.

Quenching temperature of the steel affects the Young modulus.The largest decrease of the Young modulus of the carbon steels isrecorded at the quenching temperatures higher than 740–750°C. Thisis accompanied by a large and rapid increase of the value of ∆E/

FFFFFigigigigig.2.13.2.13.2.13.2.13.2.13. Dependence of the Young modulus of steel on carbon content after quenchingfrom 900°C (1) or 1050°°°°°C.

wt.%C

32


E1 in the temperature range to 900°C, and the increase of ∆E/E

1 is

smaller in steels with a higher carbon content (U10A) than in thesteels with a lower carbon content (U8A). At temperatures higherthan 900°C the change of ∆E/E

1 in the case of U8A steel stabilises,

but in U10A steel the change continues to increase. This may beassociated with the formation of microcracks and also a decrease ofE

z when quenching from high temperatures.Quenching affects the temperature dependence of the Young

modulus [17]. Figure 2.14 shows that the Young modulus ofquenched U8A steel changes non-monotonically during heating. Inthe temperature ranges 50–150°C, 300–400°C, and 500–700°C, thetemperature coefficient of the change of the Young modulus de-creases. This indicates the increase of the rate of decrease of theYoung modulus, and at temperatures of 50–300°C and 400–500°Cthere is a large increase of e.

The maxima on the e(T) curves are distributed in the temperatureranges corresponding to the development of various transformationsduring tempering of steel. In the temperature range 50–150°C, wherethe value of e decreases, the ε-carbide, coherent with the motherphase, appears. The regions of internal stresses are characterised by

FFFFFigigigigig.2.14..2.14..2.14..2.14..2.14. Temperature dependence of E/E and of e for USA steel after quenchingfrom 800 °C (a) and tempering at 150° (b), 300 (c) and 450°C (d).

33


a decrease of the values of the Young modulus and by theincrease of the rate of decrease of the Young modulus in this tem-perature range. The increase of the degree of precipitation of theparticles of the ε-carbide is accompanied by a decrease of theextent of distortion of the crystal lattice. This is also reflected inthe increase of the Young modulus to the values corresponding tothe given temperature. The rate of decrease of the Young modulusslows down with increasing temperature and this decreases the valueof e.

Prior tempering at 150°C results in the precipitation of the ε-carbides and in the removal of distortions of the crystal latticeassociated with the formation of this carbide. Consequently, in sub-sequent heating at 50–150°C the height of the maximum on the e(T)dependence decreases. The large increase of the value of parametere in the temperature range 300–400°C is associated with the forma-tion of cementite. Prior tempering at 450°C results in the precipi-tation of cementite and the removal of distortions of the crystallattice associated with the formation of cementite. Subsequent heat-ing in this temperature range is accompanied by a continuous changeof the Young modulus. The values of the Young modulus of thequenched steel in the temperature range 200–300°C and the decreaseof the values of e in this range are associated with the breakdownof retained austenite.

The experiments with the changes of the retained austenite ofquenched steels showed that the increase of the retained austenitecontent results in a decrease of e in the temperature range 200-300°C. This phenomenon can be explained by the fact that the re-tained austenite is in the stressed state. It’s breakdown decreases thelevel of internal stresses thus decreasing the rate of decrease of theYoung modulus with increasing temperature. The precipitation ofcarbon in the solid solution of alpha iron also contributes to theprocess. When the retained austenite content of the quenched steelis increased, the change of the Young modulus with increasing tem-perature is more marked. The decrease of the temperature coefficientof the change of the Young modulus e in the temperature range 50–150°C with increasing quenching temperature is explained by theincrease of the number of coherent particles of the ε-carbide as aresult of the increase of the number of nucleation areas of the car-bide.

The alloying of steels increases the structural sensitivity of theirYoung modulus, especially after heat treatment. This is caused bythe increase of the metastability of the structural components. There

34


FFFFFigigigigig.2.15..2.15..2.15..2.15..2.15. Temperature dependence of the Young modulus of 70S2ChA steel afterquenching from 900°C in oil (a) and tempering for 1 h at 300 ° (b) and 400 °C (c).

are also changes in the form of the E(T) dependence, reflected in ad-dition changes of the curves of the temperature hysteresis of theelasticity moduli associated with the development of phase transfor-mations in the material.

The dependence of the Young modulus on temperature tempera-ture in alloyed chromium steels is shown in Fig. 2.15. At tempera-tures of 150–200°C and 350–500°C, the elasticity moduli of thequenched steels increase. The decrease of the rate of increase of theYoung modulus in tempering in the temperature range 150–200°C isassociated with the breakdown of martensite and the formation ofε-carbide. The second ‘plateau’ in the temperature range 350–500°Cis attributed to the transformation of carbides.

The Young modulus increases as a result of tempering thequenched steel at temperatures higher than 500°C, especially in al-loyed steels. This is associated with the fact that the addition ofalloying elements, such as Mo, Cr, V reduces the rate of the break-down of the supersaturated solid solution and increases the tempera-ture range in which this process takes place. The extension of thebreakdown time of martensite and the formation of pearlite increasethe Young modulus after tempering at high temperatures. The in-crease of the chromium content in the steel increases the intensity

35


of this phenomenon. The increase of the Young modulus at tempera-tures higher than 500°C is also influenced by the process of coa-lescence of the carbide particles. This is reflected in a higher de-gree of equilibrium of the alloy.

Whilst the quenched U8A steel is characterised by a monotonicdecrease of the value of E with increasing temperature, the 70S2Kh2and 60S3KhMVA steels, quenched from a temperature of 900°C,show an increase of the Young modulus by 1.2–1.5% in a specifictemperature range. If the temperature in subsequent tempering isincreased, the intensity of this defect decreases, Fig. 2.15c. In thealloy quenched steels in the temperature range 200–500°C the val-ues of the dependence E(T) are lower than for the quenched carbonsteels, although outside this temperature range there are no differ-ences in the values of the E(T) dependence. This is explained by theshift of the region of transition of the metastable to stable carbideof the cementite type. The process of breakdown of the super-saturated solid solution of carbon in α-iron is accompanied by theformation and growth of finelly dispersed particles of metastablecarbides, coherent with the matrix. This process is associated withthe formation of high internal stresses which contribute to a furtherdecrease of the Young modulus. The formation of the stable carbideswith the disruption of coherence of the carbides and the matrix de-creases the efficiency and number of defects of the crystal latticeand E increases. Silicon may also exert an effect; this elementchanges the nature of interatomic forces as a result of a change ofthe fraction of covalent bonding in the alloys.

The increase of the elasticity moduli of the materials with struc-tural inhomogeneity in the transition from the metastable to stablephase is probably a general phenomenon which takes place not onlyin steels because similar behaviour was found in Cu–Be, Cu–Ti–Crand Cu–Ni–Sn systems containing metastable intermetallic phases.

With the change of temperature, the Young modulus is greatlyaffected by combined heat treatment and plastic deformation. If un-der normal conditions the change of the Young modulus after plasticdeformation is not high, for example, in U8 steel it does not exceed2.5%, in the case of alloyed steels these changes are considerablylarger. For alloy steel 1Cr18Ni9Ti after quenching from 900°C and40% deformation, the change of the Young modulus ∆E/E is 10–11%. In nickel martensitic steels after deformation and ageing, thedecrease of E in comparison with the condition of the steel aftertempering at high temperature is as high as 14%. Evaluation of theelasticity characteristics of 36NKhTYu alloy steel after quenching

36


and ageing showed that the decrease of the Young modulus of thematerial is associated with the change of the texture during heatingto the quenching temperature as a result of recrystallisation. Afterageing the Young modulus increases. This phenomenon is control-led by the process of short-range ordering and by the formation ofnuclei of the γ′-phase with a size of 1–1.5 nm. Continuation of age-ing results in a small increase of the Young modulus. This is ex-plained by the anisotropy of the elasticity characteristics of theparticles of the γ′-phase during their growth.

The elasticity moduli are influenced not only by the type andnumber of the particles of the strengthening phase but also by theuniformity of distribution of this phase in the volume of the mate-rial. For example, the Young modulus of a high-speed tool steeldecrease in the case of the nonuniform distribution of the carbidephase, or the moduli of elasticity of steels for stamping dies de-crease as a result of the formation of small areas with free ferrite.This increases the structural homogeneity of the material.

The effect of the main metallic mass and the shape of graphiteon the change of the Young modulus with increasing temperature forgrey and high-strength cast iron was investigated in Ref. 18. Theanomalies of the E(T) dependence are caused mainly by the number,dimensions and shape of graphite and also, to a lesser degree,whether the main metallic mass is ferritic or pearlitic.

The method of measuring the E(T) dependence can also be usedin examination of the kinetics of arrangement of the silicon atomsin the main the metallic mass of ferritic grey cast iron. The decreaseof the Young modulus after quenching when the solid solution is dis-ordered, and also the anomalous increase of the temperature depend-ence of the Young modulus in the temperature range 350–550°C areexplained by the breakdown of the superlattice. Evaluation of the re-covery of the elasticity moduli after quenching grey cast iron fromdifferent temperatures made it possible to determine the criticalvalue (approximately 775°C) separating the characteristic changesof the elasticity moduli. In quenching from temperatures in the rangefrom Curie temperature to 775°C, the solid solutions of iron andsilicon in the matrix are characterised by the disordered distributionof the solute atoms resulting in a decrease of the elasticity moduli.Quenching from 775°C and higher temperatures results in the rapiddecrease of the Young modulus as a result of more extensive defectsin the structure formed in the process of quenching from higher tem-peratures.

37


2.1.3 Elasticity characteristics of composite materialsSignificant changes of the elasticity moduli are caused by the re-laxation of stresses in materials with high structural inhomogeneity,especially in metals with pores. The effect of porosity on the levelof the elasticity moduli is also observed in the case of many met-als in the cast condition (alloys with a high melting point, sinteredsystems, composites, etc.).

The Young modulus of the material with porosity p, denoted E(p)

,can be determined from the empirical equation in the form

( ) ( ) ( )0 1 ,m

pE E p= − (2.30)

where m is the experimental factor. Increase of porosity is accom-panied by a decrease of the Poisson number; this is often ignoredwhen comparing the mechanical properties of dense and porous ma-terials. In fact, the µ–p dependence is almost the same as that forthe dense material.

Frantsevich et al. [5] recommend the following equation

2/ 3

01

,1

p b

pE E

ap

−=+ (2.31)

where a, b are the experimental coefficients taking into account theeffect of stress concentration in the vicinity of non-spherical pores.

The elastic characteristics of the material, representing a me-chanical mixture of certain types of particles with different defor-mation, can be evaluated from the deterministic and statistical view-point.

In the first case, the mechanical model of a composite materialis represented by a solid in which the interaction of the componentsdepends on their mutual displacement. The equation of motion is de-rived for the displacement of the matrix and the filler. The equationcontains factors and variables which depend on the heterogeneity pa-rameters of the material. Some models solve the propagation of pla-nar waves when: 1. The stress tensor of one component depends onthe strain tensors of other components and on the tensors of thepartial initial volume strains; 2. A force pulse is transferred fromone component to another; this pulse depends on the uniaxial dis-placement and the deformation of the volumes of components.

38


In the second case, in the statistical theory of random fields inthe mechanics of inhomogeneous deformation, it is necessary tosolve a large number of problems of the elastic equilibrium ofinhomogeneous media. This is carried out on the basis of the sta-tistical characteristics of the stress and strain states of the solids.

In inhomogeneous media, the mean stress at a specific point ofthe solid depends not only on the deformation at this point but alsoon the deformation at all other points of the solid. A special situ-ation arises in the intermediate layer which in the homogeneous me-dia is represented by, for example, the surface of the solid, but forinhomogeneous media this region is found more often.

The composite materials based on metals include systems withdispersion strengthening and materials reinforced with particles orfibres. In all cases, the material contains a metallic or alloy matrixwith other phases, particles of fibres with different distribution [19].Usually, the second phase is regarded as stronger and more rigidthan the matrix. The elasticity characteristics of a two-componentcomposite can be described, for the upper limit, using the equation

1 1 2 2hE E c E c= + (2.32)

and for the lower limit

1 2

2 1 1 2

,s

E EE

E c E c=

+ (2.33)

where E1, E

2 are the Young moduli of the filler and the matrix, c

1,

c2 are the volume fractions of the components.The expression for E

h corresponds to the case with the same de-

formation in the two components, and the expression for Es for the

same stresses. The difference between Eh and E

s decreases if calcu-

lations are carried out using the variance principles, the method ofthe theory of random processes, etc.

The calculated and experimentally determined elasticity modulifor powder composites based on iron, molybdenum and tungstenwith the content of an copper–iron binder to 35% are compared inTable 2.3.

The deviation of the experimental values of the Young modulusto higher values is explained by the fact that the expressions for E

h

and Es do not take into account factors such as the interaction of

39


TTTTTaaaaabbbbble 2.3le 2.3le 2.3le 2.3le 2.3 Calculated and experimental values of the Young modulus of compositematerials with different matrix

%relliffo

foeulavdetaluclaC E 01( 5 )aPMfoseulavfostnemirepxE E

01( 5 )Cº(foserutarepmetta)aPM

Eh

Es

Em

Eν 02 002 004 006 008

*55203

578.1578.1578.1

867.1867.1867.1

039.1808.1977.1

–––

059.1048.1318.1

08.177.137.1

07.116.165.1

06.133.143.1

84.1––

**02522363

786.2786.2786.2786.2

312.2312.2312.2312.2

945.2644.2642.2821.2

694.2642.2428.1683.1

183.2031.2129.1047.1

083.2031.2029.1527.1

33.260.209.196.1

33.269.168.156.1

61.278.187.195.1

***5283

731.3–

014.2–

––

––

558.2071.2

37.250.2

37.250.2

56.200.2

55.259.1

Comment: * - iron; ** molybdenum; *** - tungsten

the elastic fields of the inclusions and the matrix, the localnonuniformity of the stress state, the elastic properties of the com-ponents, etc. More accurate calculations of the elastic characteris-tics of the composites can be carried out taking into account thestatistical nature of the distribution of heterogenities, the mutualeffect of the components and correlations between their elasticfields.

The theory of isotropic deformation of elastic solids with randomheterogenities, using the elasticity factors ν and λ, gives

( )3 2.mE

ν λ + ν=

λ + ν (2.34)

The method of calculating Em is suitable for composites with a small

volume fraction of the filler. For the volume fraction of the fillerlarger than 20% we can use the following equation

( ) ( )( )( ) ( )

0

1,

1 1

kc

k kc

M ccE E

c M cν

η −=

− η − + (2.35)

40


where E0 is the Young (elasticity) modulus of the matrix material,

M(c)

is the function which takes into account the mutual effect of themicroinclusions, c is the volume fraction of the filler, c

k is the criti-

cal concentration of the filler at which E = E1, i.e. the Young modu-

lus of the filler, η(= E/E0).

The data presented in Table 2.3 show that the values obtained bycalculations using equation (2.35) are in good agreement with theexperimentally determined values for the case in which the volumefraction of the filler is larger than 20%, whereas equation (2.34) issuitable for volume fractions of the filler smaller than 20% and alsofor materials with a small difference of the elasticity moduli of thematrix and the filler.

The experimental results show that the values of the Youngmodulus decrease with increasing volume fraction of the filler. Asthe Young modulus of the binder decreases, the rate of decrease ofthe Young modulus increases. At higher temperatures this relation-ship is sometimes not fulfilled. The experimental results obtained forpowder composites, produced by liquid-phase sintering (iron–copper,tungsten–copper, tungsten–copper–nickel, tungsten-iron–nickel, mo-lybdenum–silver, nickel–silver, etc.) show that the dependence of theelasticity moduli on the volume fraction deviates from the lowervalues determined in accordance with the rules of mixing.

The dispersion-strengthening particles affect the composite whenthey restrict the deformation of the matrix. Usually, the Youngmodulus of such a system is lower than that indicated by equation(2.32). Calculations carried out using equation (2.32) and the ex-perimentally obtained data are compared in Fig. 2.16. For all ex-amined systems, the experiments show positive deviations of theYoung modulus values from the calculated values, indicating theconstriction and limited deformation of the matrix.

In Ref. 20 and 21, the method of finite elements was used todetermine the algorithms for predicting the elasticity moduli of ho-mogeneous isotropic two-phase materials with any geometry of thephases forming the composite. The authors analysed the stress fieldsin the vicinity of the individual particles and groups of the parti-cles in a continuous matrix, and also examples of the application ofthe algorithm for determining the elasticity characteristics of theporous and sintered materials.

The strength of the reinforcing fibres in the composites is oftenvery high but the cross-section of the components may be unsuit-able because it does not ensure the required rigidity and, conse-quently, the stability of the structure. In some cases, the high rela-

41


FFFFFigigigigig.2.16..2.16..2.16..2.16..2.16. Effect of the volume fraction of dispersion-strengthening particles in acomposite on the relative change of experimentally and theoretically determinedelasticity moduli.

tive strength of the composites is not utilised because the energy ofelastic deformation (σ2/E) increases in direct proportion to thesquare of stress or yield limit. All reliability parameters decreasewith increasing energy of elastic deformation.

The parameters affecting the actual service efficiency of materialsgreatly differ from those of real structures. For example, at R

m =

5000 MPa, the energy of elastic deformation decreases by an orderof magnitude and the rigidity parameters are 3–4 times lower thanthe required boundary values. One of the methods of solving theseproblem is to produce alloys where not only the tensile strength butalso Young modulus increases. When retaining the characteristic de-termined by the ratio E/σρn, where ρ is the specific density and1>n>1/3. For materials used in aviation industry with ρ = 1.1–9g ⋅cm–3, n = 0.5. The composites characterised by a high Youngmodulus and low specific density have advantages not only in com-parison with steel but also magnesium and aluminium alloys.

The behaviour of the composites outside the elastic range dependson whether the strengthening particles of fibres are deformed. Solidsurfaces of the inclusions restrict deformation of the softer matrixunder loading. When the hydrostatic component becomes 3–4 times

42


higher than the yield limit of the matrix, failure takes place. Whenthis stress is insufficient for particles to be deformed, failure propa-gates through the matrix.

This description of the elasticity characteristics is also valid forlaminated composites [22]. To solve the boundary problems of theelasticity theory, it is necessary to know the geometrical parametersof the phase, the distribution and cross-section of the fibres. Irregu-lar distribution of the fibres in the cross-section of the componentgreatly complicates the calculations. If the geometry of the phasesand approximation of the stress fields are taken into account, it ispossible to find simple methods of joining the elements of the com-posite. The main problem in determination of the elastic character-istics of the composites with fibres is the evaluation of the moduliby the variance calculation procedures.

The modulus of elasticity along a solid in the direction of rein-forcing fibres can be determined as an additive characteristic,whereas the values in the direction normal to the direction of thestrengthening fibres greatly differ from the values determined fromthe decrease rule. Reinforcement of the matrix with fibres greatlyincreases the Young modulus in the direction normal to the direc-tion of fibres. However, the increase of the Young modulus of thefibre results in a situation in which the increase of the transversemodulus of the composite is not significant and the solution ap-proaches the values determined for infinitely rigid fibres. The ra-tio of the longitudinal and transverse modulus of normal elasticityat higher values of the Young modulus of the fibres is small and thisrestricts the application of fibres with high elasticity moduli in thecomposites. The increase of the rigidity of the system in the trans-verse direction can be achieved by selecting the orientation of fibresin different layers which obviously decreases the rigidity of the com-posite in the axis of the component. A suitable example is a lami-nated composite in which the orientation of the fibres in the indi-vidual layers deviates by the selected angle. The properties of thiscomposite are similar to those with the isotropic characteristics, andthe values of the Young modulus fit in the group of the values de-termined in the longitudinal and transverse direction in relation tothe distribution of the fibres.

The elasticity properties of the component of the composite char-acterise to a certain extent the conditions of failure of componentsmade of composite materials. The fibre-reinforced compositeremains sound if

43


( )2

3.25,

1

F

E dν ν

ησ ≤− µ (2.36)

where η is the proportionality factor, F is the friction factor, µ isthe Poisson number, E

v is the Young modulus, and d

v is the fibre

diameter.The magnitude of the tensile stresses formed in the fibre is low

and, consequently, the critical value of the tensile stresses σ isobtained in the case of long fibres (for example, for boron fibres inand aluminium matrix the length of the fibres is 0.65–0.70 m).Equation (2.36) characterises a simple condition for the process ofsingle-component plastic deformation of the matrix in elastic defor-mation of the fibre.

The complicated nature of the process of failure of fibre-reinforced composites under cyclic loading [22] is shown in Fig.2.17. The composites consist of a matrix made of and aluminiumalloys and strengthening fibres of molybdenum or tungsten, with adifferent volume fraction of the fibres in the component.

Puskar [23] describes the characteristic stages of failure in thelongitudinal loading of components in tension and compression. Ini-tially, transverse cracks form and propagate (Fig. 2.17), and this isfollowed by the formation and propagation of longitudinal cracks(Fig. 2.17b) in the matrix. With loading, the entire cross-sectionfails by fatigue (Fig. 2.17c) or the matrix disintegrates and fall out,initially from the surface and in later stages from the space betweenthe fibres (Fig. 2.17d). The formation of a specific stage is deter-mined mainly by the type and dimensions of the reinforcing fibreand its volume fraction in the component. The author of [23]assumes that the mechanism of failure of composites is based on thesignificant difference of the elasticity moduli of the strengtheningfibres and the matrix material during propagation of an elastoplasticwave in the component.

2.2 MANIFESTATION OF ANELASTICITYElastic deformation is characterised by the complete reversibility ofthe Hooke law which is fulfilled only when the loading rate is verylow and the level of acting stress does not cause any changes in thedensity and distribution of lattice defects or in the distribution ofmagnetic moments.

44


FFFFFigigigigig.2.17..2.17..2.17..2.17..2.17. Stages of damage and failure of a composite material.

2.2.1 Delay of deformation in relation to stressThe Hooke law characterises the time relationship between stress σand strain ε. The total elastic strain of the solid ε

c is the sum of

instantaneous elastic strain εe and additional (quasi-inelastic) strain

εd whose equilibrium value is obtained only after stress σ has been

operating for some time (Fig. 2.18). The time required to obtain theequilibrium value ε

d is determined by the processes associated with

the redistribution of atoms, magnetic moments and temperature inthe material.

The redistribution of the atoms in the interstitial solid solutionsunder loading can be illustrated on an example of the fcc lattice (forexample, α-iron) with interstitial atoms (for example, carbon). Theinterstitial atoms can be displaced to the positions (0, 1/2, 1/2) and(1/2, 1/4,0). In the first case, the atomic spacing is 0.90 nm, in thesecond case it is 0.36 nm. The radius of the carbon atom is 0.8 nm;this results in non-symmetric deformation of the lattice during theformation of a solid solution. The first position is characterised bya potential well whose depth is smaller than that in the second po-sition. The application of the criteria of the minimum deformationenergy for the given positions shows that the first position is sig-nificantly more stable than the second position with respect to the

45


FFFFFigigigigig.2.18. .2.18. .2.18. .2.18. .2.18. Time dependence of the change of strain when loading the solid with constantstress (a) and the change of stress when loading the solid with constant strain (b).

positioning of the interstitial atom. If there is no external stress, theinterstitial atoms can travel to the positions in the direction of thex, y and z axes. Loading with a stress along some axis, for exam-ple z, increases the space in the faces of the cube parallel with thisaxis in comparison with the faces of the cube parallel with the axesx and y. The distribution of the interstitial atoms in the positions ofthe z axis is therefore preferred. This results in tetragonality of thelattice in the z axis. Consequently, additional deformation takesplace and its magnitude increases with loading time.

In fcc lattices, the largest void is found at the point of intersec-tion of the body diagonals. The interstitial atom in this void causescubic stretching of the lattice. Tetragonality maybe the result of thepresence of a pair of interstitial atoms placed in two adjacent lat-tices. Therefore, the redistribution of the atoms in the FCC latticeis observed that lower concentrations of the interstitial atoms incomparison with the FCC lattice. The pair of the interstitial atomsis reoriented in space under the effect of external stress resulting inadditional deformation which increases with time.

The time for the establishment of the new distribution of the in-terstitial atoms (relaxation time τ

r) is the function of the frequency

of transitions of the interstitial atoms from one position to another.In this case, it is determined by diffusion equilibrium.

46


The redistribution of the magnetic characteristics is the reflectionof the relationship between the mechanical and magnetic propertiesof ferromagnetic materials, especially magnetostriction, i.e. thechange of the geometrical dimensions of the ferromagnetic materi-als in the direction of the external magnetic field. The phenomenonis reversible, which means that mechanical loading results in thedisplacement of Bloch walls in these materials. The phenomenon canbe suppressed using a sufficiently strong external magnetic field.

The redistribution of temperatures can be illustrated by, for ex-ample, bending of a beam resting on two supports. When bendingis adiabatic, for example, at a high rate, the flexure is proportionalto the force and is reflected in the compression of the upper fibresand stretching of the lower fibres. The compressed area is heatedwhereas the stretched area is cooled. A temperature gradient formsin the cross-section of the beam and causes additional deformationwhose magnitude depends on time. The time for obtaining the sametemperature is determined by thermal conductivity, heat capacity anddensity. With longitudinal and torsional methods of loading the in-tensity of the phenomenon is low, but for bend loading it can reacha significant value with the scattering of the mechanical energy inthe material.

These examples indicate the occurrence of additional deformation,in addition to primary deformation, especially under repeated load-ing, which is the reason for the lower values of the dynamic elas-ticity moduli in comparison with the values determined by staticmethods. The deformation process of the actual component is linkedwith time by means of additional deformation which may be re-flected immediately after loading or after a certain period of time,and the change of the magnitude of additional deformation is con-trolled by the exponential law and described by the relaxation equa-tion. Movement towards uniform deformation becomes faster withthe increase of the initial deviation of the characteristics. There mayalso be cases in which additional deformation copies the course ofdamped vibrations. Consequently, vibrations may change to reso-nance under the effect of the external force. Additional deformationmay depend on stress by a directly proportional dependence ε

d = cσ,

by means of a certain function (εd = f(σ)) or hysteresis. From the

viewpoint of time, we can determine immediately f(t) = const,determine gradually f(t) = e–(t/γ), or in damped or resonance mannerdetermine f(t) = e–βteiωt, where c is the proportionality constant, σis stress, t is time, τ and β the characteristics of the material andω is circular frequency.

47


These examples of the dependence of additional deformation onstress and time can be combined. Three combinations are importantfor practice: relaxation processes (combination of the relationshipsε

d = cσ and f(t) = e–βteiωt), resonance processes (combination of re-

lationships εd = cσ and f(t) = e–βte (t/γ)), and mechanical hysteresis

(combination of hysteresis and f(t) = const).The relaxation processes are described in Fig. 2.18. At time

t = 0, stress σ is generated in the material and its magnitude ismaintained constant. The deformation, corresponding to this stress,is not detected immediately. Elastic strain ε

c forms immediately but

the total value of strain εc is obtained only after a certain period of

time. The rate of approach to the value εc is

( )1.c ε

σ

ε = ε − ετ

(2.37)

The value τσ is the time required to obtain εc under the effect of

constant stress.The dependences of σ and ε in loading and unloading (Fig. 2.19)

show that during a deformation process the specimen is loaded witha constant stress for some period of time. The tangent of the angleinclination of the ON line is the Young modulus of the material ofthe specimen in the stage in which the total deformation has not yetbeen realised. This modulus corresponds to the adiabatic deforma-tion process and is referred to as the adiabatic or non-relaxed Youngmodulus E

N. The slope of the OR line determines the modulus of

elasticity of the material of the specimen when total deformation hastaken place and relaxation processes have occurred in the specimen.In the conditions with slow deformation (isothermal loading), dur-ing the time longer than the relaxation time we obtain the isother-mal or relaxed Young modulus E

R which is lower than E

N.

A different approach to the phenomenon can also be used. Thestrain ε

c is generated in the specimen at a specific time and is main-

tained constant. It is necessary to decrease the stress, and the rateof this decrease increasea with increase of the difference of thestress and equilibrium values σ

0 (Fig. 2.18b), therefore

( )0

1.

ε

σ = σ − στ (2.38)

48


FFFFFigigigigig.2.19..2.19..2.19..2.19..2.19. Stress–strain dependence under the effect of connstant stress (a) and constantstrain (b).

The values τε is the relaxation time of stress at constant strain.

The dependence between σ and ε is in Fig. 2.19b. It shows that thenon-relaxed and relaxed elasticity moduli must also be differentiatedin this case.

Since σD = E

R ε

c and from equations (2.37) and (2.39) we obtain

the values of σ0 and ε

c, the relaxation equation has the following

form

( ).REε σσ + τ σ = ε + τ ε (2.39)

Solids fulfilling this condition are referred to as standard linearsolids.

The evaluation of additional strain by static methods is difficultbecause these strains are low and the relaxation time is short. It istherefore necessary to use repeated loading ε in which strain lagsbehind applied stress σ and the phase shift is ϕ (Fig. 2.20a). Dur-ing a single loading cycle in the σ−ε coordinates we obtain a char-acteristic hysteresis loop (Fig. 2.20b). Its area corresponds to theenergy scattered in the material during a single loading cycle (∆W).Its axis (line OD) has, however, a different slope in comparisonwith the one corresponding to the non-relaxed or relaxed Youngmodulus. Consequently, the tangent of the angle of the OD line char-acterises the dynamic Young modulus E

D. For the metals, we can

use the equation in the form

49


2 21 ,

1d N

RE E

r = − + ω τ

(2.40)

where R is the degree of dynamic relaxation

,N R

N

E ER

E

−=

ω is the circular loading frequency and τr is the characteristic proc-

ess time.The Young modulus determined by the dynamic method is lower

than that the determined by the static methods, i.e. ED

< ER

. Therelative difference is approximately 1% and is not associated withthe measurement error. The lower value of E

D is caused by the fact

that repeated loading is accompanied by higher elastic deformationin the solid in comparison with the same stress under static load-ing. Another aspect of the phenomenon (E

D < E

R) is that alternated

loading by the same stress as in static loading decreases the defor-mation resistance of the material. The phase shift ϕ is the functionof the loading frequency. At low frequencies (ω→∞) the relaxationprocess can take place and ∆W is low (Fig. 2.21). Consequently,E

D → E

R. At high frequencies (ω → ∞), the relaxation process does

FFFFFigigigigig.2.20.2.20.2.20.2.20.2.20. Delay of strain behind stress (a) and the hysteresis loop (b).

50


not manage to take place, there is no additional deformation andE

D → E

A and ∆W are low (Fig. 2.21). For an intermediate case in

which ωτr = 1 E

D = (E

N + E

R)/2 and ∆W is high (Fig. 2.21).

In cases with the effect of external factors (for example, the mag-nitude of acting stress) or internal factors (for example, the reso-nance of vibrations of dislocations segments with the frequency ofexternal loading) the area of the hysteresis loop ∆W increases withincreasing number of loading cycles (Fig. 2.22). Consequently, thedynamic Young modulus gradually decreases as a result of repeateddeformation and this is reflected in the fact that the selected levelof stress leads to higher deformation of the material.

2.2.2 Internal frictionEvaluation of the scatter of the energy inside metal is often used inthe direct experiments, for example, when measuring the dynamichysteresis loop, internal friction in the region in which it is depend-ent on the strain amplitude, etc. Evaluation of the energy losses ina single loading cycle ∆W in long-term loading characterises thekinetics of fatigue damage cumulation. The temperature, frequency,time and amplitude dependences of internal friction provide a largeamount of important information on the mechanisms of micro-plasticity [24] or elastic characteristics, the defect of the Youngmodulus, the degree of relaxation of the stresses in the examinedmaterial, etc.

FFFFFigigigigig.2.21..2.21..2.21..2.21..2.21. Frequency dependence of the change of the Young modulus and energyscattered during a single cycle.

51


Internal friction is the property of the solid characterising its ca-pacity to scatter irreversibly the energy of mechanical vibrations[25, 26]. In resonance methods, the internal friction of the materialis determined from the width of a peak or depression on the curveof the amplitude of the deviation from the loading frequency at aconstant amplitude of vibrations [27]. The amount of the energyscattered in the material is measured using the quantity

1

0

,3

Q− ∆ω=ω (2.41)

where ∆ω is the half width of the resonance peak at half its height,and ω

0 is the circular resonance frequency of vibrations of the

specimen.Taking equation (2.41) into account, we obtain

12 2

.1

r

r

RQ− ωτ=

+ ω τ (2.42)

At ωτr = 1, internal friction is maximum, Q–1

max = R/2.

Internal friction is also characterised by the relative amount of

FFFFFigigigigig.2.22..2.22..2.22..2.22..2.22. Schematic representation of changes of hysteresis loops.

52


energy scattered in a single load cycle Ψ, which is determined fromthe area of the hysteresis loop ∆W and from the total energy sup-ply to the system W, corresponding to the maximum strain in thecycle in which ∆W was determined, therefore

.W

W

∆ψ = (2.43)

In dynamic measurements

1 .2

Q− ψ=π

(2.44)

The logarithmic decrement of vibrations δ is determined by theequation

1

,n

n

zln

z +δ = (2.45)

where zn and z

n+1 is the amplitude of the n-th and n + 1 cycle of

damped vibrations of the solid. The numerical values δ are equal tothe relative scattering of energy (irreversible change of the energyof vibrations to heat) during a vibration cycle. When δ << 1, the fol-lowing equation is valid

2 .ψ = δ (2.46)

For small phase shift values of σ and ε (Fig. 2.20) tan ϕ ≈ ϕ =(1/2π)(∆W/W). The characteristics Q–1, Ψ, δ, ϕ are linked by theequations

1 tan .2 2

WQ

W− ψ δ ∆= ϕ ≈ ϕ = = =

π π π (2.47)

The amount of energy absorbed by the solid can also be measuredusing other criteria, such as the increment of the vibrations, thecoefficient of absorption of the acoustic wave, etc.

The values of Q–1, ϕ, δ, Ψ etc. can be defined for any stress

53


state. The difference is that in the case of inhomogeneous stress thecharacteristics are identical in the entire volume. For inhomogeneousstress this holds only when Q–1, Ψ, δ, ϕ, etc., are independent of thestrain amplitude. Otherwise, these characteristics represent the meanvalues of these quantities in the entire volume of the solid.

Equation (2.47) is valid only for low values of internal friction.At high values of Q–1 (and, consequently, of Ψ, δ, ϕ, etc.) the in-ternal friction depends on the internal friction mechanisms and mustbe determined separately for every case.

The measurements of ϕ and δ depend not only on the frequencyand amplitude of vibrations but also on the type of loading. Thevalues were determined in transverse and longitudinal vibrations anddepend on, for example, the relaxation mechanism.

The difference of the Young modulus is determined by the equa-tion

( )3

.2 1

G E∆ = ∆+ µ (2.48)

For the majority of metals and alloys µ = 0.27–0.37 which showsthat the changes of ∆G are 10–18% higher than the changes of ∆E.It is therefore more efficient to determine the value of ∆G than ∆E.

2.2.3 Mechanisms of energy scattering in the materialInternal friction is an integral representation of the activity of theindividual components which depend or do not depend on frequency.

The frequency-dependent components include the thermoelasticphenomenon, the movement of valency electrons, the viscosity ofgrain boundaries, the movement of interstitial atoms in interstitialsolid solutions, the movement of interstitial atoms in substitutionalsolid solutions, the change of the orientation of paired defects, anddislocation relaxation.

The thermoelastic phenomenon can be observed when loading sol-ids produced from single crystals or polycrystalline materials. If atest bar is loaded with a sufficiently high rate, the bar is extendedby the value 0a ' (Fig. 2.23). Accurate measurements show that theelongation of the bar is accompanied by its cooling. The extendedand partially cooled bar equalises its temperature to ambient tem-perature and, consequently, it is extended by the value a 'b '. Thisextension is referred to as the direct elastic after-effect and its rate

54


FFFFFigigigigig.2.23..2.23..2.23..2.23..2.23. Diagram illustrating the thermoelastic phenomenon.

is low. As a result of compression the bar is again heated in rela-tion to the environment and a reverse process takes place. In re-peated loading of the material, this phenomenon causes the scatterof supplied energy. Zener calculated that the scatter depends on thedilation coefficient, heat conductivity, specific mass and the Youngmodulus of the examined material. The thermoelastic phenomenon isstrongest in the frequency range from 1 Hz to 1 kHz. The frequencyat which damping is maximum is equal to D/D2

z, where D is the self-

diffusion coefficient and dz is the mean the grain size of the poly-

crystalline material.The movement of valency electrons, associated with external

loading, can be observed especially at low temperatures (severalKelvins) and at a high frequency of changes of the orientation ofloading of the solid. At a relatively high rate of compression of thematerial, for example, in the x direction, the velocity of movementof the valency electrons in this direction increases. This changes theoriginal, for example, the spherical Fermi level to an ellipsoid. Thenew shape of the Fermi level is obtained after a certain period oftime and the stress in compression, required for elastic deformation,gradually decreases. Part of the supplied vibrational energy is con-sumed for these changes.

The viscosity of the grain boundaries is characterised bymicroplastic deformation which already forms under the effect oflow shear stresses and is associated with the displacement of theatoms of the grain boundaries, even though the deformation of thegrains is only elastic. This leads to hysteresis phenomena. When apolycrystal is subjected to repeated loading at low temperature, theenergy scatter is small (Fig. 2.24) because the time available is tooshort for atom displacement. In loading in the medium temperature

55


range there is a significant scatter of the supplied energy explainedby the self diffusion of atoms along the grain boundaries.

The movement of interstitial atoms in the interstitial solid solu-tions can be evaluated most efficiently in α-iron which contains car-bon and nitrogen, already at room temperature and a frequency ofapproximately 1 Hz.

According to Snoek, if steel is not subjected to external loading,the carbon or nitrogen atoms are in octahedral positions, Fig. 2.25.When stress is applied, the atoms are transferred to stretched inter-stitial positions from compressed positions. The length of the jumpin repeated loading is x = (a√2)/2, where a is the lattice spacing.The time t required to travel the distance x is determined by theequation x2 = D⋅t, where D is the coefficient of diffusion in α-iron.

For a diffusion process, time τ1, required for a single jump, is

determined by the equation

1 0 .q

kTe−

τ = τ (2.49)

FFFFFigigigigig.2.24..2.24..2.24..2.24..2.24. Temperature dependence of internal friction for polycrystalline aluminium(1) and single crystal aluminium (2) at a frequency of vibrations of 0.8 Hz.

56


where τ0 is the physical constant, k is the Boltzmann constant, T is

absolute temperature, and q is the size of the energy barrier whichthe carbon atom must overcome when jumping from the interstitialposition to the adjacent one.

When the time t, given by the frequency of external stress, ismany times higher than τ

1 for the given temperature, the carbon

atoms manage to move under the effect of external stress, withoutdeformation lagging behind stress. When t << τ

1, the carbon atoms

remain in their sites and there is no delay (ϕ). Only at t ≈ τ1, the

interstitial atoms exert time-dependent resistance to deformationresulting in a delay and, consequently, damping, Fig. 2.26. In thiscase, x2 = Dτ

1. Consequently

2

1

2.

2

aD= τ

The maximum of internal friction Q–1 is directly proportional to theconcentration of the interstitial element in the solid solution.

The increase of the frequency of vibrations displaces the maxi-

FFFFFigigigigig.2.25..2.25..2.25..2.25..2.25. Positions for the distribution of interstitial atoms in the cubic body-centredlattice: 1) atoms of the main metal; 2) octahedral positions; 3) tetrahedral positions.

57


mum of Q–1 to higher temperatures, without any change of the heightof the peak. Similar maxima of Q–1 were detected for the bcc met-als with interstitial atoms (for example, nitrogen in iron, nitrogenin tantalum, oxygen and nitrogen in vanadium and niobium, etc.).

Snoek friction enables the diffusion coefficient to be measured atroom temperature. It makes it possible to evaluate the change of thecarbon content of the solid solution; this is highly advantageous ininvestigations of precipitation processes.

The movement of interstitial atoms in fcc lattices, being an ana-logue of the previous description, was observed but the internal fric-tion resulting from the movement of these atoms was extremelysmall because the deformation of the lattice by the interstitial at-oms is small. This also applies to vacancies or substitutional atomsplaced in the nodal sites of the lattice.

The change of the orientation of pair defects contributes to theformation of internal friction. The effect of the external forceschanges the orientation of the individual pairs of defects (pairs ofvacancies, substitutional and interstitial atoms, substitutional atomand vacancy, in the close vicinity). It is assumed that a pair ofsubstitutional atoms is situated in the cubic lattice. A pair of de-fects causes deformation of the close vicinity and results in

FFFFFigigigigig.2.26..2.26..2.26..2.26..2.26. Temperature dependence of internal friction for pure iron with 0.027 wt.%of nitrogen.

58


tetragonality of the lattice. If the metal is not subjected to the ef-fect of external forces, the individual pairs of the defects are ori-ented randomly according to the orientation of the individual crys-tals, Fig. 2.27. After applying an external force, the orientation ofthe pair of the defects changes. When the pairs of the defects areformed by atoms larger than the atoms of the main lattice, under theeffect of the external force they are oriented in the direction of theexternal force. This may be used to explain the maximum of damp-ing for Cu–Zn alloy (Zeener), Ag–Zn (Nowick, Fig. 2.28) and otheralloys.

The extent of damping is proportional to the number of defectpairs which can change the orientation.

The relaxation of dislocations (Bordoni phenomenon) is associ-ated with the change of stress in the dislocation line. The decreaseof stress is determined by the movement of small kinks of the dis-locations pinned at impurities or other dislocations (Manson). Thedislocation line can be shortened as a result of vibrations of the dis-locations and depends on the shape of the dislocation line and thelevel of local stress (Seeger). The formation of the maximum of in-ternal friction at a specific temperature depends strongly on the fre-quency of repeated stress because the phenomenon is of the reso-nance nature. Dislocational frequency-dependent friction forms atmegahertz frequencies. In the case of kilohertz frequencies, it is pos-

FFFFFigigigigig.2.27..2.27..2.27..2.27..2.27. Illustration of the change of a pair of defects under the effect of a pairof defects.

59


FFFFFigigigigig.2.28..2.28..2.28..2.28..2.28. Temperature dependence of internal friction of Ag–Zn single crystals withthe (110) orientation containing: 1) 3.7 wt.% Zn; 2) 10.6 wt.% Zn; 3) 16.5 wt.%Zn; 4) 26.2 wt.% Zn; 5) 30.25 wt.% Zn.

sible to ignore the dislocational frequency-dependent friction.Information on the operation of the individual mechanisms of

scattering of mechanical energy in the material at different frequen-cies is presented in Fig. 2.29. The formation of internal frictionpeaks, their height and frequency affiliation are a function of thetype of material, its composition, substructural state and, in particu-lar, temperature during loading.

The frequency-independent processes include the magneto-mechanical phenomenon and the amplitude-dependent vibration ofdislocation segments.

The magnetomechanical phenomenon is the result of magnetichysteresis in ferromagnetic materials and is associated with themovement of Bloch walls. The effect of this component of internalfriction can be greatly suppressed by applying a sufficiently strongexternal field to the loaded solid. For example, a magnetic flux withan intensity of 2⋅104 A⋅m–1 greatly decreases the contribution of thisphenomenon in iron and mild steel.

The amplitude-dependent vibration of the segments of the dislo-cations can be described using the model of vibration of an an-

60


FFFFFigigigigig.2.29..2.29..2.29..2.29..2.29. Change of internal friction in relation to loading frequency the orientationindication of maxima at 20°C for different energy scattering mechanisms: 1) changeof orientation of pair defects; 2) viscosity of grain boundaries; 3) viscosity of twinboundaries; 4) movement of interstitial atoms; 5) thermoelastic phenomenon; 6)exchange of thermal energy between grains.

chored spring, in accordance with the Granato–Lücke theory [28],supplemented by other authors.

We shall assume a random dislocation distribution in a crystal ofa real metal. Some parts of the dislocations L are pinned, for ex-ample, due to the presence of clusters of point defects K at the dis-locations, Fig. 2.30. Selecting a dislocation segment and graduallyincreasing the amplitude of external stress σ

a, after reaching some

value σa the segments with the length L

p will bend (Fig. 2.30b) in

areas where they are not blocked. As a result of tensile loading, thepinning points can move in the dislocation line, depending on theorigin. After reaching the critical value σ

a, when 2r = L

p (2.30c),

the individual segments with the original length Lp can combine in

a single segment with length Ln (Fig. 2.30d).

The increase of stress σ results in a linear increase of strain εdisl

(section ABC in Fig. 2.31) up to the stress resulting in unpinningof the dislocation from the pinning points (section CD). Another in-crease of stress increases the strain in accordance with the DGcurve, up to the formation of a new dislocation from a source withthe distance of the pinning points L

n. After unstressing, the strain

61


σa = 0 increasing σa→

FFFFFigigigigig.2.30..2.30..2.30..2.30..2.30. Diagram describing the vibration of parts of pinned dislocations and theformation of new dislocations.

decreases along the GDA line, i.e. in the direction differing fromthat during loading. In repeated loading the curve passes along thepoints ABEF and FEA. The area defined by these curves is propor-tional to the absorbed scattered energy and its size depends on thenumber of points from which the dislocation became unpinned. Theprocess of unpinning of the dislocation and generation of new dis-locations is shown in Fig. 2.30e. After a further increase of thestress amplitude up to some critical value, the dislocations segmentwith the initial length L

n can therefore generate dislocation kinks

FFFFFigigigigig.2.31..2.31..2.31..2.31..2.31. Dependence of stress on dislocational strain.

62


and renew the activity of the dislocations source (Fig. 2.30f) whichleads in the final analysis to cyclic microplastic deformation.

These processes, starting with the vibration of the segments ofthe dislocations pinned at the points K up to the generation of newdislocations, depends on the magnitude of the stress amplitude σ

a

or the strain amplitude ε and provide different contributions to theextent of internal friction.

Because of the importance of the effect of the stress amplitudeon the extent of internal friction in the examination of substructural,structural and deformation characteristics of materials and the deri-vation of relationships of fatigue damage cumulation in materials,the problem is examined in detail, especially in section 3.2 andchapter 6 of this book.

Hysteresis is observed to various degrees in all materials. Themain reason for it is the irreversible movement of the dislocationsin the stress field, i.e. microplastic deformation. The hysteresiscurve in the ideal case is independent of time and depends only onthe magnitude of stress. Consequently, the Young modulus and in-ternal friction are independent of frequency but depend on the strainamplitude. The slope of the hysteresis loop and its area do notchange in relation to the rate of repetition of loading. The shape ofthe hysteresis loop is influenced mainly by the level of the maximumstress.

Figure 2.32 shows schematically the frequency (A) and amplitude(B) dependence of the Young modulus and internal friction for thecase of relaxation (I), resonance (II) and hysteresis (III).

The third case is of special importance for engineering. Strainamplitude-dependent internal friction, formed at high strain ampli-tudes, makes it possible to suppress the development of dangerousresonance states of components, sections and whole structures.

2.3 Defect of the Young modulusThe above results show that the Young modulus is not a physicalconstant but it is a characteristic influenced by a large number offactors.

In practice, the concept of the ideal Young modulus can be usedonly for an isotropic, single-phase and defect-free metallic material,deformed in the elastic stress range.

When loading a real solid, total strain ε contains the elasticstrain ε

e and additional plastic strain ε

p, which depends on the mag-

nitude of acting stress and also loading rate, Fig. 2.33. The occur-rence of the additional, plastic deformation changes the Young

63


modulus of the material which is characterised in this case by theeffective value, for example, E

ef. In static measurements of E

ef, we

can use the following equation

.efe p

Eσ=

ε + ε (2.50)

When measuring the elastic characteristics of materials we deter-mine their effective values. The difference between the ideal andeffective modulus of elasticity (Young moduus) is small but, in somecases, for example, in measurements of the elasticity moduli bystatic methods at stresses of (0.7–0.8) R

e the difference may reach

40–50% [29]. When evaluating the effective values of the elastic-ity moduli, it is useful to take into account information on the struc-tural sensitivity of the elasticity moduli.

The effective values of the elasticity moduli are useful not onlyfor engineering calculations but they also represent a source of valu-able information on the nature and mechanisms of the additional,plastic component of the total strain. In this case, it is recommended

FFFFFigigigigig.2.32..2.32..2.32..2.32..2.32. Changes of the Young modulus and internal friction in relation to frequency(A) and strain amplitude (B) for relaxation (I), resonance (II) and hysteresis processes(III).

64


FFFFFigigigigig.2.33..2.33..2.33..2.33..2.33. Composition of total strain during loading of material.

to use a dimensionless parameter, the so-called defect of the Youngmodulus, defined by the equation

,efE EE

E E

−∆ = (2.51)

where E is the ideal or initial value of the Young modulus. The re-lationship of the defect of the Young modulus with ε

p depends on

the loading conditions. Figure 2.33 shows that in static loading

.pdE

E d

ε∆ =ε

In repeated loading with the stress σ = σ0 exp (i ω t) the result-

ant strain contains the elastic component of strain εe and the plas-

tic component of strain εp, i.e. ε = ε

e + ε

p. The inelastic, plastic

component of strain can be divided into the following parts: εp =

ε′p – iε″

p. The phase of the first part (ε′

p) is identical with the ex-

ternal loading phase. The phase of the second part (ε″p) is displaced

by 90° in relation to the external loading phase. The occurrence ofε

p is determined by the scatter of the elastic energy of vibrations in

65


the material. Different mechanisms of the transformation of theenergy of elastic deformation to thermal energy are included in theintegral name: internal friction.

Consequently

**1 .p

e

Q− ε=

ε (2.52)

The formation of εp results in a change of the elastic properties

of the material which are characterised in this case by the dynamicYoung modulus

.De p

Eσ=

′ε + ε (2.53)

The difference between the values of the static and dynamic elas-ticity moduli is associated with the fact that the real solid is notcharacterised by ideal elasticity. The value of E

D depends on the

type, nature and mechanisms of anelasticity of the material and,consequently, it is the equivalent of E

ef, determined in static load-

ing. For the case of repeated loading we have

.p

e

E

E

′ε∆ =ε (2.54)

The effective elasticity moduli are the result of the effect of sev-eral factors which can be conventionally divided into two groups.The first group includes the factors which take into account thephase composition, structural state, the level of micro- andmacrostresses, etc. This classification has a number of problemsbecause both groups overlap.

It is useful to distinguish between the nonlinear and nonelasticbehaviour of the material. The anelastic behaviour the material re-sults in the formation of a hysteresis loop. The nonlinear nature ofthe σ–ε dependence results in the formation of E

ef and ∆E/E with the

change of ε. The anelasticity phenomena, associated with the for-mation of the hysteresis loops, indicate the nonlinear behaviour ofthe material.

66


FFFFFigigigigig.2.34.2.34.2.34.2.34.2.34. Form of the hysteresis loop of the material with ∆E/E = 0 (a) and∆E/E ≠ 0 (b)

However, the occurrence of the defect of the Young modulus isnot direct confirmation of the anelastic behaviour of the material.The evaluation of the contour of the hysteresis loop makes it pos-sible to determine the parameter characterising the amount of scat-tered energy, i.e. the area of the hysteresis loop ∆W. The level ofthe defect of the Young modulus is determined by the middle line ofthe hysteresis loop [30].

In the symmetric hysteresis loop (Fig. 2.34a) the middle line isstraight. Here, the defect of the Young modulus is equal to 0. If ∆E/E ≠ 0, the loading curve must be nonlinear, Fig. 2.34b. Conse-quently

2

2,D

WE =

ε (2.55)

where

= σ ε εε

The nonlinear form of the middle line of the hysteresis loop is theresult of the effect of two types of factors. They are factors whoseeffect results only in the formation of the defect of the Young modu-

67


lus, and factors causing both the defect of the Young modulus andanelastic scattering of energy in the material. Additional deforma-tion is immediately evident and depends only on the magnitude ofacting stress.

The type of anelastic processes, related to the second group ofthe factors, is determined by the type of the dependence of the ad-ditional plastic deformation on the acting stress and loading time.

In anelastic processes of the relaxation type, the defect of theYoung modulus, referred to as the degree of relaxation, is deter-mined by the equation

.N R

N

E EE

E E

−∆ = (2.56)

The magnetomechanical mechanism of relaxation in ferromagneticmaterials can cause the formation of the defect of the Young modu-lus depending on saturation. The value ∆E is the tensor of the fifthorder and depends on the mutual orientation of the main axes of thestrain tensor, crystallographic axes and the saturation vector. Theformation of the ∆E effect is associated with the change of the do-main structure of the ferromagnetic material during loading. It isreflected in additional, anelastic deformation. The increase of satu-ration from zero to the value I results in a change [31] of the Youngmodulus in accordance with the equation

( ) ( )0 2

21 ,I

bK K Vf g

a − = + −

(2.57)

where

2

2,

Ff

V

∂=∂

F is the free energy of the ferromagnetic material

,F

gV

∂=∂

a, b are the coefficients in the Bridgman equation

68


26 ,V

aP PV

∆ = +

where P is pressure and V is volume.The change of the Young modulus ∆E of the ferromagnetic ma-

terials

,SE E E∆ = − (2.58)

where ES and E are the elasticity moduli of the material in the satu-

rated and initial conditions.The value of the ∆E phenomenon is high in materials with high

magnetostriction, with moderate magnetic crystallographic anisot-ropy and a low level of internal stresses. The characteristic ∆E/E

S

reaches almost 20%, for example, in the case of annealed nickel.One of the manifestations of the ∆E phenomenon is the difference

between the values of the elasticity moduli measured by dynamicmethods whilst maintaining a constant value of the external mag-netic field (E

H) and saturation (E

S), and

21 ,S H

H

E Ek

E

−= (2.59)

where k1 is the coefficient of the magnetomechanical bond.

When the temperature is increased to the range around Curie tem-perature, the material changes to the paramagnetic state, the ∆Ephenomenon disappears and normal changes of the elasticity moduliin relation to temperature are recorded above T

C (Fig. 2.35).

In the saturated condition, the dependence of the Young moduluson temperature has the conventional form. However, for certain al-loys (for example, Elinvar) the effect of the magnetic field with thesaturated value results in anomalies in the E – T dependence. Ironwith 45 wt.% Ni in a magnetic field with an intensity of 0.8 × 104

A m–1 shows values independent of temperature from 0 to 470°C.The movement of the dislocations under the effect of the exter-

nal stress is the reason for the dislocation or anelasticity of the ma-terials. The existence of a large variety of dislocation structures andof specific features of the interaction of dislocations with point

69


defects and atoms of the alloying elements in alloys requires thedevelopment of various mechanisms describing dislocation anelast-icity. These mechanisms have been described in greater detail inRef. 2, 32.

Additional strain εp, caused by dislocation anelasticity, is propor-

tional to the total length of moving dislocations and the mean valueof displacement of the dislocations from the initial (equilibrium) po-sition. The controlling factor is the change of the local character-istics of interaction of the dislocations and the atoms of the alloy-ing elements situated in the atmospheres around the dislocations inthe solid solution.

In cases in which the dislocation does not interact with other dis-locations, the movement of a single unpinned dislocation in theideal, pure material is of the viscous type. The rheological proper-ties of the elastic–viscous bond are important in this case. The hys-teresis loops of mechanical hysteresis of such a material are in factellipses which grow symmetrically with increasing stress (Fig.2.36a). In actual metallic materials, the movement of the individualdislocations is associated with overcoming barriers of various type.The magnitude of stress, required for the start of movement of thedislocations, is associated with the considerations of the dry fric-

FFFFFigigigigig.2.35..2.35..2.35..2.35..2.35. Temperature dependence of the Young modulus of polycrystalline nickelfor different saturated magnetic fields (104 Am–1): 1) 0; 2) 0.05; 3) 0.08; 4) 0.33;5) 0.85; 6) 4.8 (complete saturation).

70


FFFFFigigigigig.2.36..2.36..2.36..2.36..2.36. Hysteresis loops in operation of different mechanisms of energy scatteringin material.

tion mechanism. The nonlinear form of the σ – ε curve and open-ing of the hysteresis loop takes place only at σ > σ

t, where σ

t is the

stress for dry friction (Fig. 2.36b). Here, the defect of the Youngmodulus and internal friction depend on the amplitude of actingstress. The form of the curve in Fig. 2.36b is identical with thedeformation diagram for the mechanism of dragging the atmospheresof alloying elements by the dislocations temporarily arrested andblocked on a system of the disordered obstacles. σ

t has its own

physical meaning for every case.The mechanisms of dislocation anelasticity are based on the

physical models of two types. They are the models of detachmentof dislocations which are probable a modification of the dry fric-tion model. In these models, friction does not take place in all pointsof the space, but only in microvolumes with the effect of solute at-oms. The hysteresis loop for this case is in Fig. 2.36c. The secondtype of model links the displacement of the elastic stress field,which surrounds the dislocation during its movement, with specialfeatures of the redistribution of the atmosphere of solute atoms.Redistribution is regarded as a diffusion process in a changinginhomogeneous stress field around the dislocation and is describedon the basis of the considerations of the elastic–viscous nature ofbonding.

71


Qualitatively similar results were obtained for the case of inter-action of the dislocations by the mechanism of dislocationanelasticity. They are based on the assumption on the conservationof the dislocation density of the type of dislocation configuration.Hysteresis anelasticity, formed by the movement of dislocations,results in a decrease of the elasticity moduli in the form

. ,dislE dE

E d

ε∆ =σ

(2.60)

where εdisl

is the dislocational strain. Equation (2.60) characterisesthe defect of the Young modulus in the stress range preceding thedevelopment of microplastic deformation.

Taking into account the model of vibrations of a spring accord-ing to Granato and Lücke [33], the defect of the Young modulus isdetermined by the equation

22

6,d

EL

E

∆ Ω= ρπ (2.61)

where Ω is the orientation factor, ρd is the dislocation density, L is

the effective length of the dislocation determined by the equation1/L = 1/L

p + 1/L

n.

The redistribution of dislocations and also dislocation multipli-cation processes contributes to the anelasticity of the material. Forthese processes to take place, it must be σ > σ

t (Fig. 2.36d).

The increase of the proportion of mutual interaction of the dis-locations at a sufficiently high external stress results in a more dis-tinctive nonlinear form of the σ−ε curve and in a significant in-crease of the defect of the Young modulus. In the microplasticrange, the dependence of the defect of the Young modulus on thestress amplitude or strain (e) amplitude has the form

,aEAe

E

∆ = (2.62)

where A and a are the parameters that depend on the type and con-ditions of loading.

The mechanisms of dislocational anelasticity are characterised by

72


a wide spectrum of the relaxation times. All models are character-ised by the directly proportional relationship of the defect of theYoung modulus with the density of moving dislocations. In annealedmaterials, the defect of the Young modulus, based on dislocation be-haviour, is only several tenths of a per cent.

At higher temperatures, the anelasticity mechanism at the grainboundaries is also important. It is associated mainly with the vis-cous slip in the vicinity of the grain boundaries. Calculations of thedefect of the Young modulus for this type of relaxation show thatit is strongly influenced by the Poisson number and may representseveral tens of per cent, as confirmed for alpha brass.

The anelastic nature of deformation of the material duringcycling loading results in the occurrence of two mutually linked phe-nomena, internal friction and the defect of the Young modulus. Inaccordance with the equations (2.52) and (2.54), we can write that

1

,p

p

QM

E

E

−′′

′

ε= =

∆ ε (2.63)

The main source of information on the anelasticity mechanisms andthe values of the parameters controlling these mechanisms is still themeasurement of internal friction. On the basis of a large number ofmeasurements and publications of internal friction it was assumedthat the defect of the Young modulus is a characteristic of second-ary importance which does not provide any further information. Thelatest results show that internal friction and the defect of the Youngmodulus are not identical anelasticity characteristics, because inter-nal friction evaluates the size of the hysteresis loop and the latterthe nonlinearity of the central line of the loop. Under the combinedeffect of a large number of external factors and reactions of thematerial in the form of activation of various anelasticity mecha-nisms, it may be seen that each of them provides a separate contri-bution to the formation of hysteresis loops observed in practice.Detailed information on the behaviour of the material under the ef-fect of external alternating loading may be obtained by combinedmeasurements with evaluation of the internal friction and the defectof the Young modulus.

These characteristics (Q–1, ∆E/E) are measured by two methods,i.e. as the isothermal dependences of the defect of the Young modu-

73


lus on strain amplitude or as the dependence of the defect of theYoung modulus on temperature at a constant stress or strain ampli-tude. In the first case, the defect of the Young modulus is deter-mined from the equation

2 2

2,r x

r

f fEA

E f

−∆ = (2.64)

where fr and f

x is the natural frequency of vibrations of the speci-

men, determined at the minimum stress amplitude and at the selectedstress amplitude. In the second case, the defect of the Young modu-lus is determined from equation (2.51), and E

ef has the meaning of

the dynamic Young modulus at temperature T determined from equa-tion (2.22). The numerical values of the parameters in equation(2.22) are determined by the approximation of the temperature de-pendence of the elasticity moduli from the region of lower tempera-tures where the effect of the evaluated mechanisms of anelasticitycan be ignored.

The relationship between Q–1 and ∆E/E (equation 2.63) is deter-mined by the type of the mechanism of the scattering of mechani-cal energy in the material. For the case of the effect of the relaxa-tion and resonance mechanisms, internal friction has only the tem-perature-frequency dependence. The internal friction, caused by thehysteresis mechanism, depends on the strain amplitude and remainsalmost constant with increasing loading frequency. Taking theresults into account, equation (2.63) changes for different mecha-nisms of scattering of mechanical energy.

When the value of the fraction M in equation (2.63) is 0.5, theinternal friction peak characterises a specific relaxation time, or thepeak forms by the superimposition of several relaxation maxima. Inthe second case, the value of the defect of the Young modulus isdetermined with the accuracy equal to the accuracy of determina-tion of the sum of the degrees of relaxation of every relaxationprocess. The total height of the peak is smaller than the sum of theheights of every separate maximum. In the case of superimpositionof several relaxation maxima, the value of ratio M in equation(2.63) is lower than 0.5. If thermal relaxation takes place in thematerial, the defect of the Young modulus is determined by the dif-ference of the adiabatic and isothermal elasticity moduli.

The effect of the resonance mechanisms of internal scatter of

74


energy can be explained on the example of reverse movement of thedislocations in a viscous medium (Granato–Lücke model). Additionaldeformation occurs by displacement of the dislocations in the exter-nal stress field ε

p = ρ

d bl, where ρ

d is dislocation density, b is the

Burgers vector, and l is the mean displacement of the dislocationsin the slip plane determined from the equation

( )0

1,

pL

p

l l x dlL

= ∫ (2.65)

where Lp is the length of the dislocation segment between the pin-

ning points. The x axis is oriented in the direction of dislocationmovement.

For the model of the elastic string, the equation of movement atkilohertz and megahertz frequencies has the form

2

2 2,i tl l

B bet x

ω∂ ∂+ χ = σ∂ ∂

(2.66)

where B is the characteristic of viscous friction, χ2 is the linear en-

ergy of the dislocations. Consequently

411

2

,d pA B LQ− ω ρ

=χ (2.67)

22 ,d p

EA L

E

∆ = ρ (2.68)

where A1

≈ A2

≈ 1/6 is a coefficient which depends on the value ofχ

2.The equations (2.67) and (2.16) show that the values of Q–1 and

∆E/E for the same model characterise different aspects of the proc-ess. The area of the hysteresis loop and the formation of Q–1 are de-termined by the viscous movement of the dislocations and, conse-quently, depend on the loading frequency and viscous friction coef-ficient B. The defect of the Young modulus is determined by the

75


effective force of linear stretching of the dislocation.For the given resonance mechanism of anelasticity, the ratio M

from equation (2.63) is proportional to

2

2

.pB Lωχ

For copper at room temperature, a frequency of 100 kHz and Lp =

5 × 10–3 mm M = 0.04.Simultaneous measurements of Q–1 and ∆E/E using equations

(2.67) and (2.16) have been described only in a very small numberof cases. The available studies indicate that for alloys the ratio Mfrom equation (2.63) is in the range 0.05–0.1 when the content ofthe atoms of the alloying elements is increased from 0.5 up to 5wt.%, but when the prior strain is increased to 30%, the value ofthe ratio rapidly decreases. Increase of the carbon concentrationresults in a decrease of L

p in proportion to C–0.6, and the value of

Lp is 10–40 times higher than the calculated values of the mean

spacing of the atoms of the alloying elements on the dislocations.Areas of dislocation pinning in concentrated solid solutions are notrepresented by the individual atoms of the alloying elements but bytheir clusters containing several tens of these atoms, as shown forsolid solutions based on aluminium and iron [34].

For the hysteresis type of anelasticity of the material, the rela-tionship between Q–1 and ∆E/E depends on the relationship betweenanelastic strain ε

p and external stress σ. When the anelasticity of

the material is linear under the effect of external stress, both com-ponents ε′

p and ε″

p depend in the same manner on the external stress

σ. Consequently, the ratio determined by equation (2.63) does notdepend on stress amplitude σ nor on loading frequency.

Two mechanisms of dislocational anelasticity have been devel-oped. For the first case, the hysteresis of dislocations is the resultof cyclic break away of the dislocations from the atmospheres of thesolute atoms. The ratio M, determined by equation (2.63), for thiscase is a constant and its value varies from ~ 0.3 (for the theoreti-cal model according to Granato and Lücke) up to ~ 1 for the mecha-nisms of thermal activation of release of dislocations from the at-mospheres.

In these models, the maximum value of dislocational strain is re-stricted by the tensile force in the dislocation. Increase of the ex-

76


ternal stress amplitude increases the magnitude of bowout of the dis-location and the level of dislocational strain is influenced by inter-actions of the dislocations with internal barriers, for example, withstress fields from a dislocation forrest, mosaic boundaries, etc.Subsequently, increase of the stress amplitude should result in adecrease of the ratio determined by equation (2.63).

The second type of mechanisms is based on the evaluation of vis-cous movement of the dislocations in the lattice with uniformly dis-tributed solute atoms. For this approach, the dependence of thespeed of dislocations v

d on the value of acting stress σ has the form

01

,m

dV V σ= σ

(2.69)

where v0, σ

1, m are experimental constants. For this model [35] with

the change of m from 1 to higher values, the ratio determined byequation (2.63) is in the range 0.85–0.93.

In simultaneous measurements of Q–1 and ∆E/E on single crystalsof copper, zinc, and on solid solutions of these elements with thecontent of solute elements to 0.01 wt.%, the ratio determined byequation (2.63) in the range of dependence of Q–1 on strain ampli-tude is 0.5–1.0. For a copper alloy with 35 wt.% Zn, the ratio isfrom 0.06 to 0.25.

Under the effect of higher strain amplitude the material behavesas a nonlinear solid. This is reflected in the value of the ratio de-termined by equation (2.63), because it becomes dependent on thestrain amplitude. Two types of dependence can be observed in thiscase.

In the case of a purely hysteresis mechanism of anelasticity, theincrease of nonlinearity of the σ−ε dependence results in a directlyproportional decrease of the value of the ratio determined by equa-tion (2.63), as shown in Fig. 2.37a. The second type of the depend-ence of M on stress amplitude γ appears when the viscous movementof fresh unpinned dislocations in the crystal lattice dominates atγ < γ

cr2. Increase of the level of microplasticity increases the den-

sity of fresh dislocations and this lead to an increase of the valueof coefficient B which depends on the frequency of intersection ofthe dislocations moving in non-complanar slip planes. In this case,the value of M is proportional to the parameter B and will increasewith increasing ε. This is observed for, for example, copper, Fig.

77


FFFFFigigigigig.2.37..2.37..2.37..2.37..2.37. Dependence of σ, ∆G/G and r on the strain amplitude of annealed materialsat a frequency of 2 Hz.

2.37. At γ < γcr2

M is constant. The change of M with increasing γwith both dependences added up together is shown in Fig. 2.37c.This shows that the M–ε dependence is highly sensitive to thedevelopment of nonlinear anelasticity. Consequently, measurementsof the M–ε dependence can be used to determine the range of lin-ear and microplastic anelasticity.

If microplastic processes take place in the material, it can beseen that the dependence of ∆G/G on γ is irreversible. Evaluationof the defect of the Young modulus with the change of temperatureand at a specific value of the strain amplitude showed that the val-ues of ∆E/E decrease to lower strain amplitude and are associatedwith the processes of thermal activation of dislocation movement.Consequently

,H V

kTEAe

E

−σ−∆ = (2.70)

where A is the proportionality constant, H is the effective value of

78


activation enthalpy, and V is the activation volume.It is obvious that the elasticity characteristics and anelasticity of

the material are influenced by a large number of external and inter-nal factors.

Factors Affecting Anelasticity of Materials

79

3

FACTORS AFFECTING ANELASTICITYOF MATERIALS

The resultant value of the extent of internal friction and the defectof the Young modulus is the integral representation of frequency-dependent and frequency-independent mechanisms of scattering ofmechanical energy in materials and also of many other processes,phenomena and mechanisms which can operate in different mate-rials under the effect of different external influences.

In this chapter, attention will be given to explaining the physi-cal nature of internal friction which is independent of the strainamplitude, the so-called internal friction background Q–1 and alsothe effect of temperature for materials with different structures andsubstructures. Since the problem of cyclic microplasticity is exam-ined in Chapter 6, in this chapter, we shall present only the maindata on the effect of the strain amplitude on internal friction andthe defect of the elasticity (Young) modulus. However, special at-tention will be given to the effect of loading frequency, loading time,the number of load cycles and the effect of the magnetic field inferromagnetic materials.

Each section shows that, in addition to understanding the natureand effects of the individual influences, it is also important to ob-tain information on the accuracy of measurement of Q–1 or ∆E/Eon the change of these factors.

Many of the effects are not unique and often overlap, acttogether or against each other. This is also explained by the dif-ferences in the effect of temperature on internal friction and pos-sibly the defect of the Young modulus, because other mechanismsare determined by the response of the material to repeated load-ing and this reaction is significantly influenced by temperature inloading.


80

3.1. INTERNAL FRICTION BACKGROUNDWhen measuring the internal friction of materials in the area inwhich there are no phase transformations and in the range of thestrain amplitudes in which the internal friction does not depend onthe strain amplitude, the internal friction background is recorded asthe reflection of the integral mechanisms of scattering of mechani-cal energy in the material.

The quantification and analysis of the internal friction back-ground has been the subject of special attention in the developmentof new structurally stable creep-resisting materials [25,28]. Conse-quently, it is necessary to evaluate theoretical models and activityof the mechanisms in a wide temperature range [36].

There are three stages of the dependence of the internal fric-tion background Q–1

0 on temperature: 1) slight, approximately linear

dependence Q–10 – T at temperatures lower than (0.5–0.6)T

m; 2)

the experimental dependence of the type Q–10 at temperatures higher

than 0.6Tm; 3) the stage of decrease of the growth of the internal

friction background at temperatures close to the melting point.In the first stage of the dependence Q–1

0 (T) we can use the

theory proposed by Granato and Lucke [28] in the form

1 4 20 ~ and ~

MQ L L

M− ∆ρ ρ (3.1)

where ρ is the density of the vibrating dislocations, and the mean-ing of L was explained in the section 2/2/3. The temperature de-pendence of Q–1

0 is associated with the function of distribution of

dislocations on the basis of their orientation [37] and with the typeof dislocation clusters [13]. The length subdivision of the disloca-tion segments has no significant effect on Q–1

0

[39].

At present, special attention is given to the next two stages ofthe Q–1

0 (T) dependence. The general relationships were evaluated

for the first time by the author of Ref. 40 and then in Ref. 26,where it was shown that the internal friction background correlateswith the creep strength of the materials, i.e. as the creep strengthof the alloy increases, the value of Q–1

0 at the examined tempera-

ture decreases. The results showed a relationship between the flowstress and creep rate with the value of internal friction [41]. Theinternal friction background is influenced by many factors.


81

Fig. 3.1. Dependence of internal friction on temperature for copper filament crystals(a) silicon filament crystals (b). a: 1) initial condition, 2) after plastic deformationat room temperature, 3) polycrystalline specimens. b: 1) initial condition, 2) after0.6% deformation, 3) after 1.1% deformation, 4) after 1.5% deformation, 5) afteranneling at 800°C/1 h.

3.1.1. The substructural and structural state of materialThe intergral value of Q–1

0 is influenced significantly by the mecha-

nisms determined by the presence of dislocations [42,43]. For ex-ample, examination of whisker crystals showed that there is a re-lationship between the value of Q–1

0 and dislocation density, Fig.

3.1. Internal friction reacts sensitively to the increase of disloca-tion density in whisker crystals [44], i.e. Q–1

0 increases with in-

creasing temperature and the second and third stage of the Q–10

dependence is displaced to lower temperatures. Perfect filamentcrystals also show increase of Q–1

0 , in the vicinity of the melting

point [45], i.e. when the vacancy concentration rapidly increases.On the basis of the capacity to stabilise the internal friction

background of iron, the alloying elements can be placed in the fol-lowing sequence: Ni, Co, Cu, Cr, Mo. Increase of the content ofa carbide-forming element in iron (for example, molybdenum) shiftsthe high-temperature section to the range of high temperatures.

Polymorphous transformations are characterised by suddenchanges of the internal friction background. For example, such asudden change is well-known to take place in the transformation ofalpha iron to gamma iron [46,47]. In pure metals in the form of sin-gle or polycrystals at temperatures close to the melting point thereare anomalies of the changes of the internal friction background(Fig. 3.2a) [48], i.e. with increasing melting point, the internal fric-tion background is saturated or decreases. Similar behaviour has


82

Fig. 3.2. Temperature dependence of internal friction. a) aluminium, magnesiumand copper; b) beta tin, where: 1) initial heating; 2) after heating at 310°C/1 h,cooling, 3) initial heating of stabilised beta tin, 4) after heating at 310°C/1 h, cooling,5) heating after holding for 1 h at 410°C.

been observed for many metals (Al, Mg, Cu [48], Ta [49], Pt [49],etc). According to Ref. 49, these phenomena at temperatures closeto the melting point can be interpreted by diffusion mechanisms.

After melting of the metals which have closely packed latticesin the solid state, the liquid state is characterised by the occurrenceof a structural relaxation mechanism of energy absorption [50] sothat it is possible to obtain, for example, for tin, the relationshipbetween the structure in the solid and liquid state. The high-temperature gamma phase is obtained by cooling the melt at 300°C,the stable beta phase is obtained from the melt at 410–450°C andthe ß phase with the nuclei of the gamma phase can be obtainedfrom the melt at 320°C (Fig. 3.2b). The coefficient of absorptionof sound α(T), caused by structural relaxation, has a backgroundnature in molten metals and changes with temperature in accord-ance with the change of the ratio T3/v3, where v is the velocity ofsound. These considerations are justified when the short-rangeorder is retained in melting.

3.1.2. Vacancy mechanismThe models used to explain the value of the internal friction back-ground differ. The vacancy mechanism is based on the considera-tion according to which the intensity of the background is controlledby the equilibrium concentration of the vacancies migrating in the


83

field of repeated loading [51]. The formation of the equilibriumvacancy concentration requires a certain period of time (relaxationtime). The relaxation of vacancy concentration causes nonuniformdeformation leading to the scattering of the energy of mechanicalvibrations, i.e. the internal friction background. This can be ex-pressed by the equation

( )10

U

RTf

Q eT

−− σ= (3.2)

where f(σ) is the function of the magnitude of repeated loading σ,U is the energy required for the formation of 1 mole of vacancies.The most frequently used equation for calculating the activationenergy of the internal friction background according to Schoeck[52] has the form

1

10 const ,

nU

RTQ e

−

− = ω

(3.3)

where ω is the circular frequency of oscillations, n1 is the struc-

tural constant. To determine this constant, it is necessary to obtainthe dependence of internal friction on frequency since

( )( )

10 1

10 2

12

1

ln

.ln

Q

Qn

−

−

ωω

= ωω

(3.4)

The experimentally determined values of n1 for annealed, poly-

crystalline samples of metals are in the range from 0.17 to 0.38[53], and for copper single crystal n

1= 0.65 [49]. At the these val-

ues of n1 we can obtain the activation energy of the internal fric-

tion background U, corresponding to the energy of activation ofself-diffusion in pure metals. The phenomenological description ofthe Schoeck model was published in Ref. 54. Here the internal fric-tion background is treated as the superimposition of relaxationswith a wide time spectrum.


84

3.1.3.Diffusion-viscous mechanismIn real solids, processes determined by directional flows of pointdefects can also take place [55]. The sources and sinks of theseflows can be inclusions and various interfaces. If we use themechanisms of diffusion-conditioned deformation for interpretingthe internal friction background at infra-sound frequencies (ω<<102

s–1), we obtain the asymptotic form of the dependence [56]

1

10 2

const,

U

RT

z

Q ed T

−

− = ω

(3.5)

where dg

is the mean grain size. At high frequencies (ω << 102 s–

1) we can use the equation in the form

1,

21

0const efU

RT

z

Q ed T

−

− = ω

(3.6)

where Uef

is the effective energy of bonding, numerically equal tothe sublimation energy. The dependence of the internal frictionbackground on the grain size (decrease of Q–1

0 with increasing grain

size) has been determined in many studies [57].When characterising the internal friction background, it is also

possible to use the mechanisms of diffusion flow with the movingdislocations. Anelastic strain, caused by the displacement of dis-locations, is ε

n(T) = –bρξ(T), where ρ = 1/dh is the dislocation

density in the crystal, d is the distance between the dislocations inthe dislocation wall. If we obtain the conventional dependence ofthe internal friction background on temperature and loading fre-quency, this dependence is similar to the Schoeck characteristicwith the exponent n

1 = 1 at low and n

1 = 1/2 at high loading fre-

quencies [48]. The activation energy of the internal friction back-ground in models of unlimited dislocation walls and in the model ofblocks for polycrystals is identical at the corresponding loading fre-quencies. Excellent agreement between the theoretical and experi-mental data is obtained for the model of displacement of the dis-location at the grain boundaries [58]. This procedure can be usedto obtain the analytical dependence of frequency of temperature


85

[59]. For example, at a frequency of several hertz and at T > (0.3–0.4)T

m, for the fcc metals, the dependence has the following form

21

0 ~ ,

mU

RTQ e

−

− ω

(3.7)

where U is the activation energy of bulk self-diffusion. The mag-nitude of m is determined by comparing the distribution functionf(N) and the experimentally determined histograms of the distribu-tion of the grain boundaries N(ϑ) on the basis of the angle ofmisorientation ϑ of the crystals [60]. For copper m ~ 0.43–0.78.The model of displacement of the dislocations at the grain bounda-ries for describing the internal friction background has restrictionsat specific temperatures because of the recovery of the structureas a result of the formation of grain boundaries [48].

3.1.4 Dislocation mechanismsIn accordance with the calculations based on the model of theoscillating side, at low frequencies the internal friction backgroundis negligible. However, the experimental results show that the value

Fig. 3.3. Displacement of a dislocation segment in the stress field (a–e), wherethe solid lines indicate the position of the dislocation, broken lines the initial position,1 is the position of the point defect at the start of displacement, 2 is the positionof the point defect in the vibration period T, and a) t = 0, b) t = T/4, c) t = T/2,d) t = 3T/4, e) t = T. The diagram of relaxation of the dislocation in the stressfield (f), where the solid lines represent the positions with maximum energy, brokenlines those with minimum energy, A and B are the areas of pinning of dislocationswith length L, H is the height of the Peierls barrier.


86

of Q–10 in the Hertz frequency range is relatively high [61]. This

phenomenon can be explained of the basis of the assumption thatthere is a possibility of small displacements of pinned impuritiesbehind the moving dislocations (Fig.3.3a–e) [62]. Internal frictionis determined by two components. The first component is thedirectly proportional and the other one indirectly proportional tofrequency. With decreasing loading frequency the significance ofthe first component decreases and that of the second componentincreases. The diffusibility of the blocking areas determines the ex-ponential dependence of the internal friction background on tem-perature in the form

1 00 ~ ,

U

RTDDQ e

RT RT

−− =ω ω

(3.8)

where D is the coefficient diffusion of point defects, U is the ac-tivation energy of diffusion.

The contribution from overcoming Peierls and Nabarro barrierscan be taken into account utilising the model of relaxation of dis-locations [62]. In the Mason model, the dislocations are parallelwith one of the directions occupied most densely by the atoms inthe crystal (Fig. 3.3f) and the length of the initial double kink onthe dislocation is equal to the entire length of the dislocations seg-ment. The scatter of energy, associated with the high-temperaturebackground of internal friction, forms during the thermally activatedseparation of adjacent double kinks from their pinning point. Sincethe number of separations is described by the Boltzmann distribu-tion, the following equation is valid

1/ 201

0

2,

2

Ut ef RTN L

Q eG

−− ντ = π (3.9)

where τ(T) is the critical shear stress required for overcoming aPeierls–Nabarro barrier, v

t is the velocity of sound during trans-

verse vibrations, N0 is the number of atoms in 1 cm3 of materials,

Lef is the effective length of the dislocation segment, U is the en-

ergy of activation of separation from the solute atom. The ther-mally-activated release of the dislocations with the kinks from thepinning points increases their mean length which increases with


87

lateM·q 01 02

)J(U

nim001· 02

)J(U

nim0/U

Ufe

01· 02

)J(n = U

fe/U

dCbPlAgAuAuC

40.108.097.178.111.271.2

94.242.269.482.539.580.6

2.031.022.081.012.081.0

80.227.608.4

––

10.8

71.033.002.0

––

52.0

increasing temperature. The internal friction background increaseswith increase in temperature in proportion to T2 [63].

3.1.5. The relaxation mechanismThe general result, according to which the magnitude of internalfriction is determined by the structural defects in metals, has beenutilised in the relaxation mechanism of energy scattering duringvibrations of solids [64]. The collective excitation of the crystal lat-tice with increasing temperature, the so-called relaxons, are areaswith the disrupted arrangement of the particles which form in theareas of occurrence of structural defects of the crystal lattice.

Table 3.1 gives the values of the internal friction background andrelaxons of some metals at m

min = 7, where U is the actual and

Uef

= nU is the effective activation energy of the internal frictionbackground according to Schoeck [52]. For all evaluated metals,the ratio U

min/U is approximately the same (~0.2, with the excep-

tion of lead). The value of exponent n for cadmium, aluminium andcopper is also close to 0.2. Comparison of these parameters con-firms the possibility of operation of the relaxation mechanism.

The formation of a general theory of the internal friction back-ground is still an open area with many complications.

3.2. EFFECT OF TEMPERATURE ON INTERNALFRICTION

The measurements of internal friction at different temperatures pro-vide a large amount of information because temperature, combinedwith other external factors of the threshold states [1], has varyingeffects in different mechanisms of the scattering of mechanical en-ergy in metallic materials. Many examples have been presented andothers will be given in the following chapters of the book.

Table 3.1 Parameters of the relaxation structure and internal friction backgroundin metals


88

The total relaxation maxima of this type can be found by, for ex-ample, the superimposition of the Snoek maximum, determined bythe presence of carbon and nitrogen in iron. In some cases, therelaxation maxima of different nature can overlap if they are dis-tributed closely to each other on the temperature axis. In bothcases, the evaluation of the individual relaxation mechanisms startswith the division of the overall curve Q–1

0 (T) into the individual in-

ternal friction maxima caused by the effect of various mechanismsof mechanical relaxation. This task is similar to the case of distri-bution of the profile of X-ray diffraction into individual components.Like in X-ray diffraction, satisfactory solutions are obtained onlywhen certain conditions, relating to the distribution of maxima onthe temperature axis and the height of the partial peaks of inter-nal friction, are fulfilled. An advantageous feature is that when di-viding the spectrum of the maxima of internal friction, the specificrelaxation time τ =

τ

0eH/RT holds for the given maximum and the

analytical description of the profile of the relaxation maximum ofinternal friction Q–1

r (T) = Q–1

0 (T)

–

Q–1

t (T) is available in the form

( )1 1max

max

1 1sech ,r

HQ T Q

R T T− −

= −

(3.10)

where Q–1max

is the maximum height of the internal friction maximum(half of the relaxation step of the process) obtained at the maxi-mum temperature T

max.

In the occurrence of independently acting relaxation processes,the experimentally determined internal friction at any temperatureis the sum of the partial contributions of the individual relaxationprocesses in accordance with the equation

( ) ( )1 1 .r rii

Q T Q T− −= ∑ (3.11)

Since the partial relaxation process is characterised by only onerelaxation time, equation (3.11) has the following form

( )1 1max

max

1 1sech ,i

r ii

HQ T Q

R T T− −

= −

∑ (3.12)


89

where summation is carried out for all partial relaxation mecha-nisms involved.

When dividing the overall relationship Q–1 (T) into partial proc-esses, we can use graphical and analytical methods.

The graphical method is used when combining several relaxationprocesses determined by the single relaxation mechanisms. In thiscase, it is necessary to process the Q–1

r (T) curve in a hyperbolic

coordinate system, i.e.

1

1max

1arsech .rQ

fTQ

−

− =

(3.13)

In this system of coordinates, the internal friction curve thatdescribes the single relaxation maximum is a straight line. The pointof intersection of the straight line with the temperature axis indi-cates maximum temperature T

max. The slope of the straight line

makes it possible to determine the activation enthalpy of the proc-ess. In the case of purely relaxation processes, the straight line isshifted when the loading frequency is changed.

The graphical method can be used most efficiently in cases inwhich the maxima are divided far away from each other on thetemperature axis. Consequently, on the overall curve Q–1(T) it ispossible to separate the branches of the maxima in which thecourse of the Q

r–1(T) curves is determined by the effect of only

one relaxation process. The graphical procedure depends on the re-lationship between the heights of the maxima. If the height of onemaximum is many times greater than that of another maximum,then the parameter Q

max– 1 in the equation is replaced by the height

of the total maximum at the maximum point (Fig. 3.4a). Transfor-mation to the hyperbolic coordinate system (Fig. 3.4b) makes itpossible to determine the linear section. If this section is analysed,it is possible to determine the activation parameter of the relaxa-tion process which forms the highest internal friction maximum. Theapplication of the determined parameters makes it possible to ex-press the analytical form of the higher maximum. The smaller maxi-mum of internal friction is determined by subtraction from theevaluated maximum.

In most cases, i t is necessary to use the analytical methodsbased on the approximation of the Q

r–1(T) dependence by the equa-

tions of the type (3.12), using the method of least squares. Thecalculation programmes of the break-up of the internal friction


90

spectrum into individual components are based on the assumptionon the ideal Debye shape of the individual partial relaxationmaxima [65]. The method of dividing the Q–1

r(T) can be greatly

simplified if we use the apriori data on the values of the activa-tion enthalpy of the temperature position of the maxima, eventhough only for a part of the individual relaxation processes.

The width of the examined partial maxima of internal friction,especially the width of these maxima, often exceeds the width ofthe ideal Debye maximum, as indicated by the narrow spectra ofrelaxation times or, in most cases, the specific distribution of theactivation enthalpies of the partial relaxation process. The expan-sion of the observed relaxation maxima β can be evaluated in ac-cordance with Ref. 66. If there is no expansion (β = 0), theDebye maximum has the ideal shape, and after transforming thedata to the reciprocal coordinate values of temperatures (1/T), themaximum is slightly asymmetric. Analysis of the Snoek maximum

Fig. 3.4. Temperature dependence of internal friction of Fe–N–2 at% Mn alloy inconventional (a) and hyperbolic coordinates (b), broken lines show the main peaksof the dependence.


91

by the calculation method, for titanium alloys with 20–40 mol% ofniobium or molybdenum, taking into account the expansion of thepartial maxima, was carried out in Ref. 67. The Snoek maximumwas divided into five partial maxima (Fig. 3.5), associated with theoccurrence of oxygen atoms in the lattice distributed in five differ-ent energy positions.

Using chromium steels as an example, it is possible to evaluatethe thermodynamic activity of carbon in alloyed ferrite wheremicroregions with a different chromium concentration can form atthe nodal points of the crystal lattice [68].

The graphical and analytical division of the partial relaxationmaxima makes it also possible to determine the activation param-eters of the relaxation parameters, especially activation enthalpy H.In some cases, the parameters H , τ

0 and Q–1

max are already ob-

tained when dividing the dependence into partial maxima. In othercases, the activation parameters of the processes are determinedindependently. Usually, it is necessary to use two groups of meth-ods, based on analysis of the shape or temperature at which therelaxation maxima occur.

The temperature at which a relaxation maximum occurs and theactivation parameters of the relaxation process are linked by theequation:

Fig. 3.5. Temperature dependence of internal friction on Ti–33 at.% Mo alloyafter annealing for 1 h at 1200°C. The dotted lines represent the main peaks, thesolid curve is the theoretical curve.


92

0 1,H

eRT

ωτ = (3.14)

Qr–1 which makes it possible to determine experimentally H and τ

0.

If we change the loading frequency of the specimens from ω1 to

ω2, the relaxation maximum Q–1

r (T) is displaced from the tempera-

ture Tmax1

to Tmax2

. Consequently, H and τ0 can be determined from

the equations

max1 max 2 2

max 2 max1 1

ln ,T T

H RT T

ω=− ω (3.15)

( )max1 max 20 1 2

max1 max 2

1ln ln .

2

T TH

R T T

+τ = − − ω ω (3.16)

If it is required to use equation (3.15) with sufficiently high ac-curacy, frequency ω must be reduced by several orders of magni-tude. When it is difficult to fulfil the condition of the change offrequency and the shift of the temperature maximum is 20–40°C,the relative error of determination of the enthalpy of activation inthe sense of equation (3.15) is up to 20%. The accuracy of meas-uring the temperature shift T

max2 – T

max1 can be increased by evalu-

ating the position of the maximum on the basis of transformationof the curves to hyperbolic co-ordinates and by applying linear re-gression to the resultant dependence. Two obstacles may occur inthis procedure: final expansion of the maximum of the dependencesQ

r–1

(T) or the occurrence of the temperature dependence Q–1

max.In

this case, the method gives average values of activation enthalpy.The activation enthalpy of the relaxation processes can be de-

termined efficiently using the equation proposed by Wert and Marxin the form

maxmax maxln ,

kTH RT T S= + ∆

ω (3.17)

where is Planck’s constant, ∆S is activation entropy, ω is loadingfrequency.


93

Activation entropy is 40–20 J ⋅mol–1 K–1. Usually, the values∆S = 10–12 J mol–1 K–1 are used. The equation proposed by Wertand Marx is used mainly owing to the fact that it is very simple.When using this equation, it must be taken into account that theboundary conditions must be fulfilled. When deriving equation(3.17), it was assumed that the value τ

0 in equation (3.14) is the

same for all relaxation processes and equals 10–13 s, which corre-sponds to the frequency of Debye oscillations of a single atom υ

D=

1/τ0. There are a large number of processes in which τ

0 greatly

differs from this value. For example, in the mechanism of dislocationrelaxation τ

0 ≈ 10– 9–10–10 s. Equation (3.17) can be used in cases

in which the relaxation process is associated with the thermal ac-tivation of displacement of the individual atoms to interatomic dis-tances.

The shape of the relaxation maximum can be used in evaluat-ing the activation parameters of relaxation. The activation enthalpycan be determined utilising the low- and high-temperature sectionsof the maximum of internal friction Q –1

r (T). From equation (3.14)

and (3.23) we obtain the following equation for the low-tempera-ture section of the maximum (T < T

max)

( )1H

RTrQ T e

−− ≈ (3.18)

and for the high-temperature section of the maximum (T > Tmax

) theequation

1 .H

RTrQ e− ≈ (3.19)

Equation (3.18) shows that H can be determined from theangle of inclination of the low-temperature and high-temperaturesections of the maximum of the dependence Q–1

r (T) in the co-or-

dinates ln Qr–1 – 1/T. This procedure is suitable only for distinctive

maxima whose broadening is close to zero.Activation enthalpy H can also be determined from the width of

the maximum of the dependence Qr–1 (T). Usually, the width of the

maximum is determined at the position Q–1max

/2. The temperaturescorresponding to this level of Q

r–1 (T) in accordance with equation

(3.23) fulfil the condition


94

02

2 20

1.

41

H

RT

H

RT

e

e

ωτ =+ ω τ

(3.20)

The roots of equation (3.20) fulfil the equation, ωτ0eH/RT= 2 ± √3

which shows that

1 2

2 1

2.63 ,TT

H RT T

=− (3.21)

where T1 and T

2 are temperatures at which Q–1

r = Q–1

max/2.

The value H determined using equation (3.21) is accurate whenthe process is characterised by a single relaxation time. If we havethe spectrum of relaxation times, the effective value of H, obtainedusing equation (3.21), will be lower.

The analysed methods of calculations of the activation enthalpyof relaxation processes show that the most suitable method is theone based on analysis of the temperature shift of the maximum ofthe dependence Q–1

r (T). Other methods can be used under certain

conditions when the process is characterised by a single relaxationtime and there is no expansion of the maximum (β = 0), but whenthe mechanism of the relaxation process is clear.

The individual mechanisms of the relaxation processes are de-scribed in subsequent parts of this chapter.

3.2.1 Mechanisms associated with point defectsIn substitutional solid solutions, there is relaxation associated withthe change of the relative position of the atoms of the alloying el-ement and the main metal under the effect of external load. Themaximum of Q–1 was reported for the first time by Zeener [69] inthe relaxation spectrum of α-brass and later by Nowick [70] in thespectrum of the Ag–Zn alloys (see Fig.2.28). In this case, themaximum is more distinctive because there is a larger differencebetween the radii of the atoms of silver and zinc than between theatoms of copper and zinc. At low concentrations of the alloyingelements the height of the maximum is directly proportional to thesquare of the concentration of the dissolved element. For solid so-lutions with a high content of the added metal, where the atoms ofthe alloying elements form complicated complexes rather than iso-


95

lated pairs, the proposed relaxation mechanism takes into accountthe changes of the short-range order after loading the body.

The kinetics of Zeener’s relaxation is determined by the spatialmigration of the atoms of the solid solution. This results in achange of the number of pairs of atoms with different orientation,or the parameters of short-range order. The same mechanism is thebasis of the diffusion phenomena and, consequently, the kineticcharacteristics of relaxation (relaxation time) are similar to the dif-fusion parameters in appropriate alloys.

Experimental examination of Q–1 in ternary and multicomponentalloys showed that Zeener’s relaxation processes also take placehere [71]. The magnitude of relaxation is several times higher andthe relaxation maximum is 1.5–2 times higher than in the binary al-loys of the components forming the ternary alloy.

The presence of alloying elements in interstitial solid solutionsbased on the cubic space-centred lattice results in a more compli-cated relaxation spectrum of Snoek’s mechanism and in the devel-opment of other relaxation processes [72], as shown in Table 3.2.

Fe–C–metal alloys show high-temperature and also additionallow-temperature maxima.

The increase of the difference of the size of the atoms of ironand the alloying elements at comparable concentrations results inan increase of the temperature difference of the additional maximaand Snoek’s maximum. The magnitude of relaxation of the addi-tional maxima of both types for interstitial solid solutions with car-bon or oxygen does not exceed 1×10–3 – 5×10–3.

The maximum proposed by Finkel’shtein and Rozin is relativelywide and in certain cases is divided into two closely spaced peaks[73]. The height of the maximum increases with increasing concen-tration of the interstitial atoms. At concentrations lower than criti-cal concentrations c

cr the dependence of the height of the maximum

on carbon content is linear. The shape of the concentration depend-ence of the height of the maximum is the same for all alloys inwhich this phenomenon has been observed. The value differs from0.1 to 0.25 wt.% of carbon or nitrogen.

In solid solutions with a face-centred cubic lattice or a hexago-nal close-packed lattice the octahedral and tetrahedral planes, inwhich the interstitial atoms can be distributed, have the same sym-metry as the lattice itself. In these materials, Snoek’s relaxationprocess does not occur. Despite this, the point defects (interstitialatoms) are the reason for a special relaxation phenomenon re-ferred to as the Finkel’shtein and Rozin phenomenon. This maxi-


96

Table 3.2 Characteristics of the additional maximum in alloyed ferrite

mum was detected for the first time in an austenitic chromium–nickel alloy in the vicinity of 300°C at a loading frequency of 1 Hzand later in different austenitic steels and alloys based on the face–centred cubic lattice which also contained nitrogen or carbon.

The principle of Finkel’shtein and Rozin phenomenon is explainedby Werner’s model, based on the diffusion rotation of a pair of in-terstitial atoms of nitrogen or carbon under the effect of externalstress. This pair if interstitial atoms can be stable when it is ori-ented along ⟨110⟩, ⟨112⟩, ⟨130⟩ and distributed in the third to fifth

gniyollAtnemele)%.tw(

gnidaoLycneuqerf

)ZH(

erutarepmeTkeonSfomumixam

)K(

erutarepmeTnoitamroffolanoitiddafo)K(mumixam

noitavitcAfoygrenelanoitidda

mumixamlom.Jk( 1– )

metsySN-eFnM2–5.0

rC5.0oM5.0

V5.0lA0.1

metsySC–eF51.5–94.0

V34.3–20.1

oMoM68.0iS56.1

iS3iS64.3

iT22.0–1.0oC9.0oC35.4

W4.4iN9.3rC9.3

rC8rC51rC02rC52rC03

1

1118.0

0542.0059059

106956.0059059069549079

11111

792

692692692382

383992883093213393

–983983493683693

–––––

082803023843063323

843192374724433134353354263063363374064015535055065

*96*6.18**0.48**0.29**0.59

0.18

5.08

5.5012.290.781.390.3313.101

6.872.779.875.201

–––

431631

Comment: * - calculated from the shift of the frequency of temperature maximum;** - calculated from maximum temperature at τ

0 = 10–11.5 s


97

co-ordination sphere. The pairs result in tetragonal distortion of thelattice. During rotation of the pair in relation to the axis of tensileloading of the specimen this causes mechanical relaxation to takeplace.

The increase of the level of internal stresses by quenching, colddeformation, ageing, etc., causes significant broadening and in-creases the height of the maximum [74]. For the Finkel’shtein–Rozin phenomenon to take place, the material must contain latticedefects. For example, after cold deformation, the high-temperaturepart of the Finkel’shtein–Rozin maximum shows an additional maxi-mum whose height depends on the level of internal stresses andtheir heterogeneity. For a steel with 24 wt.% Ni, 0.5 wt.% Mn and0.5 wt.% C, the maximum height of the additional maximum is ob-tained when plastic deformation results in the formation of a ho-mogeneous dislocation structure. When deformation leads to theformation of a cellular structure, the height of the maximum rap-idly decreases and the additional maximum is not detected in thefine-grained recrystallised material.

3.2.2 Dislocation relaxation mechanismsThe effect of low strain amplitudes characterised by internal fric-tion irrespective of the strain amplitude results in the operation ofdifferent mechanisms of relaxation of dislocation segments andentire dislocation substructures. The activation energy of the proc-esses, controlling the movement of dislocations in the crystal lat-tice, is not higher than 2.4×10–19 J. All processes of dislocationrelaxation under repeated loading from Hertz to kilohertz frequen-cies occur at temperatures lower than 500–600 K, i.e., below thecondensation temperature of Cottrell atmospheres T

C.

At temperature lower than TC, every dislocation in annealed ma-

terial is surrounded by the atmospheres of solute and alloying el-ements. The dislocations blocked in this manner are stationary anddo not contribute to the process of dislocation relaxation. For mo-bile dislocations to appear in this material, the amplitude of re-peated loading must be higher than the microscopic limit of elas-ticity.

In the region in which internal friction is independent of thestrain amplitude, the dislocation relaxation processes can developas a result of the movement of fresh dislocations formed duringprevious plastic deformation of the material, or during phase trans-formations accompanied by strengthening. Since strengthening is acondition for dislocation relaxation, the appropriate maximum on the


98

Qr–1

(T) curve is regarded as the deformation maximum or the

maximum of cold deformation.Bordoni’s relaxation is the main process of dislocation relaxa-

tion in pure metals. At kilohertz frequencies, the Bordoni maximaon the Q

r–1

(T) dependence are found at low temperatures (~100

K). The phenomenon is found in single crystals and polycrystallinematerials, but always after prior plastic deformation.

Efficiently annealed metals do not show Bordoni’s relaxation.The height of the maximum depends on the magnitude of prior plas-tic deformation ε

p. The increase of ε

p to approximately 3% results

in a non-monotonic increase of the maximum. A further increaseof ε

p has no longer any effect on the height of the maximum

(Fig.3.6).Bordoni’s maximum is resistant to annealing and disappears only

after complete recrystallisation (for copper at 500°C). In heattreatment (Fig.3.7) and also when the content of the alloying ele-ments is increased, the height of the maximum decreases, despitethe fact that the temperature position of the maximum does notdepend on the concentration or type of alloying atoms. With thechange of the frequency of oscillations the temperature position ofthe maximum changes, but the change of strain amplitude has theform of a maximum. The activation energy of the relaxation proc-

Fig. 3.6. Bordoni’s peak of polycrystalline copper after plastic deformation of:a) 0.1%, b) 0.5%, c) 2.2%, d) 8.4%. Loading frequency 1.1 kHz.


99

Fig. 3.7. Effect of annealing temperature of polycrystalline copper on the heightof Bordoni’s peak: 1) after 8.4% cold deformation, 2) after annealing for 1 h at180°C, 3) after annealing for 1 h at 350°C. Loading frequency 1.1 kHz.

ess for all evaluated metals varies from 8 to 20 kJ⋅mol–1 and τ0

varies from 10–10 to 10–12 s. A special feature of Bordoni’s relaxa-tion is that the height of the maximum is an order of magnitudehigher that in the case of the relaxation maximum with a singlerelaxation time.

Bordoni’s relaxation is explained by the mechanism of disloca-tion relaxation based on examination of the thermally-activated for-mation of pairs of kinks on screw dislocations situated in the di-rections parallel to the direction of the closest packing, in positionswith the minimum Peierls potential energy.

Analysis shows that the enthalpy of activation of Bordoni’srelaxation is H = 2W

k, where 2W

k is the energy of formation of a

double kink on the dislocation. Although the model is in good agree-ment with the observed phenomena, the width of the maximum isseveral times smaller than that indicated by the model. However,if we consider the displacement of the dislocation in the field ofexternal repeated loading or thermal or geometrical kinks that areon dislocations and are not situated in potential wells, the theory isin good agreement with the observations. The width of the maxi-mum is explained by the distribution of the length l of dislocationsegments N (l). Consequently,


100

( )1 302 2

,1

l

AnQ l N l dl

kT− ωτ=

+ ω τ∫ (3.22)

where n0 is the density of geometrical kinks without the presence

of external stress, τ = 1/πD, where D is the coefficient of diffu-sion of kinks of a longitudinal dislocation, A is the proportionalitycoefficient.

At temperatures lower than the room temperature, fcc metalsshow other relaxation phenomena after plastic deformation. Thesephenomena are explained by the movement of kinks on the dislo-cations [2]. To explain low-temperature maxima that are close toBordoni’s maximum, Hasighuti proposed a concept according towhich in certain cases relaxation can be caused by braking of dif-fusing dislocation kinks by point defects, especially vacancies. Therecorded maxima are characterised by high activation energy incomparison with that for Bordoni’s maximum.

Dislocation relaxation in deformed alloys has many other mani-festations. The internal friction maximum of deformed iron withnitrogen atoms was reported for the first time by Snoek in 1941 ata temperature of 200°C and a loading frequency of 0.2 Hz.

The studies by Snoek, Kêo, Köster and others show that the oc-currence of the maximum at 200°C requires the presence of a smallnumber of atoms of C and N and also prior plastic deformation. InFe–C and Fe–N systems at loading frequencies of 0.2–0.1 Hz isrecorded at 200–250°C and are referred to as the Snoek–Köstermaximum (S–K relaxation).

Later, relaxations identical with S–K relaxation were observedin different metals with the fcc lattice (Ta, V, Mo), when the solidsolution contained the atoms of C, N, O, H [75], and also insubstitutional solutions and interstitial solutions based on the fcclattice [76] and the hcp lattice [77]. This shows that the Snoek–Koster relaxation is a general phenomenon in the case of interac-tion of dislocation with atoms forming atmospheres around the dis-locations that formed during plastic deformation. The activationenergy of the deformation maxima is always higher than the acti-vation energy of the volume diffusion of the appropriate alloyingelement. For example, the activation energy of S–K relaxation inalloys of iron with atoms of C and N is 127–168 kJ⋅mol–1 and thefrequency factor τ−1

0 = 1014 s–1. When the level of prior strain is

increased, the height of the S–K maximum increases by 50–70%


101

to saturation which depends on εp. In unidirectional deformation,

the height of the S–K maximum is Q–1S–K

≈ 10–2εp

1/2, and in thermo-mechanical treatment it is Q–1

S–K ≈ 10–2 ε

p. The increase of the prior

strain changes the temperature position of the S–K maximum. Forexample, for iron with 10–3 mol.% C after ε

p = 2% at room tem-

perature TS–K

= 557 K, but after εp

= 10% it decreases so thatT

S–K = 545 K.

The height of the S–K maximum depends on the type of defor-mation and is determined not only by the density of fresh disloca-tions but also by their type and mutual distribution. The decreaseof temperature during deformation from 200 to –70°C reduces theheight of the S–K maximum. When ε

p = const, the highest height

of the maximum after deformation is recorded at temperaturesaround 200°C.

The effect of the grain size on the height of the S–K maximumand its temperature position is only very slight. However, the heightof the S–K maximum depends strongly on the content of intersti-tial atoms in the solid solution. When ε

p = const, the height of the

S–K maximum increases with increasing content of the interstitialatoms, initially linearly and later rapidly increases to saturationwhose level increases with increasing ε

p. S–K relaxation in iron

with nitrogen is characterised by a certain anomaly because thedeformation maximum is recorded at a low nitrogen content of thesolid solution insufficient for the formation of a measurable Snoek’s

Fig. 3.8. Dependence of the height of the Snoeck–Köster maximum of Armco ironon its heating rate.


102

maximum. The temperature of S–K relaxation initially increaseswith increasing content of the interstitial atoms and, subsequently,after stabilisation of the height of the maximum, it no longerchanges. The concentration, corresponding to the maximum heightof the S–K maximum, for nitrogen atoms is on the whole 9–10times higher than in the case of carbon, and the maximum corre-sponding to the nitrogen is 4–5 times higher than the maximumcorresponding to the Fe–C system.

The height of the S–K maximum is strongly influenced by theheating rate when measuring the changes of internal friction in re-lation to temperature (Fig.3.8).

Sarrak and Golovin generalised the results of evaluation of theeffect of the chromium content of iron on the temperature positionof the S–K maximum and of the mixed maximum formed afterquenching a steel from 1250°C followed by 12% cold deformation(Fig.3.9). The S–K phenomenon forms as a result of phase hard-ening during martensitic transformation. The presence of retainedaustenite has only an indirect effect on the parameters of the maxi-mum because it is manifested in quenching in the form ofFinkel’shtein–Rozin relaxation. Certain special features are observedin interpretation of this phenomenon. It is assumed that the control-

Fig. 3.9. Effect of Cr content on the temperature at which Snoek’s maximum (1)and Snoek–Köster maximum (upper curve) of Fe–C alloy is observed. The temperaturesat which the maxima are detected are indicated (2 Hz).

wt.%


103

ling role in the formation of the S–K maximum in phase hardeningis played by the dislocation density and the local distribution of theclusters of carbon atoms in the vicinity of the dislocation kernelwhere the arrangement process may take place. The interaction ofdislocations with the clusters of carbon atoms is used to explain theS–K relaxation in quenched or deformed steel [78]. The distribu-tion of carbon in the vicinity of structural defects causes hetero-geneity of internal stresses in martensite.

In addition to the maxima, recorded at 200°C in the iron–carbonand iron–nitrogen systems, there are also other maxima resultingfrom the deformation of the material. For example, in the Fe–Nsystem, cold deformation results in the formation of a small maxi-mum at 29°C, i.e. several degrees higher than the Snoek maximum(T

S = 22°C, f = 1 Hz). The maximum is found after quenching

from the temperatures higher than the temperature of the ferrite toaustenite transformation, and also after cooling from temperaturescorresponding to ferrite and subsequent plastic deformation ε

p =

10–4–10–3. When the strain amplitude increases from 2.8 × 10–4 to5.7 × 10–4, this 29°C maximum increases, whereas the Snoek maxi-mum decreases. The phenomenon is explained by the interaction ofdislocations with the nitrogen atoms. A similar maximum was alsofound by Breshers in the Fe–C system. When the carbon contentwas ~5 × 10–6 mol%, a small Snoek maximum is overlapped in thedeformed material by a high deformation maximum which is dis-placed by 7°C to higher temperatures, Fig. 3.10.

The deformation maxima of internal friction recorded for alloys

Fig. 3.10. Snoek maximum (lower curve) and deformation maximum (upper curve)of Fe–C alloy with temperatures at which maxima are recorded (2 Hz).


104

of iron with carbon or nitrogen can also be detected at tempera-tures below 75 and 135°C at a loading frequency of 1 Hz [79]. Themaximum was recorded immediately after deformation during thefirst measurement, but after cooling or during the second measure-ment it was no longer recorded. At temperatures higher than 200–250°C, the mobility of the atoms of carbon and nitrogen is alreadyquite high, the atmospheres around the dislocations are dispersedand the intensity of interaction of the interstitial atoms with the dis-locations is insufficient for the formation of deformation maxima.In fcc metals, such as Ta and Nb with an oxygen impurity, defor-mation maxima are recorded in the temperature range 287–470°Cor 272–452°C at a loading frequency of 1 Hz [75].

The interpretation of the Snoek and Köster relaxation, proposedby Shoeck and improved by Seeger and also Seeger and Hirth, isdescribed in greater detail in Ref. 14.

Further examples of the effect on internal friction are in factpresented in every other chapter of this book.

3.3 EFFECT OF STRAIN AMPLITUDEThe dependence of internal friction on strain amplitude or alternat-ing stress in external loading is interesting. This dependence is theresult of the effect of various mechanisms, such as the relaxationmechanisms associated with point structural defects of the crystallattice, relaxation and resonance dislocation mechanisms, the relaxa-tion mechanisms associated with the boundaries of the blocks andgrains, inelastic phenomena during phase transformations, themagnetomechanical relaxation mechanisms, the mechanical–thermalrelaxation mechanisms, and others.

The overall value of Q–1 is the result of the joint effect of a setof inelasticity mechanisms operating in the given ranges of tem-perature, frequency and amplitude of acting stresses. In manycases, Q–1 = Q

0– 1 + Q

r–1, where Q

0– 1

is the internal friction back-

ground, Qr–1

is the internal friction associated with the effect of the

relaxation mechanisms.A special feature of relaxation in materials is that the relaxa-

tion maximum of internal friction Qr–1 can be described in most

cases by relaxation time τt, using the equation

12 2

,1r

r

EQ

E− ∆ ωτ=

+ ω τ (3.23)


105

where ω is the frequency of external loading, and ∆E/E is 10–3–10–1. An exception is represented by inelastic phenomena associ-ated with phase transformations and magnetomechanical phenom-ena.

3.3.1 The Granato–Lücke spring modelThe studies and investigations of the effect of strain amplitude onthe extent of internal friction are based on the spring model, pro-posed by Granato and Lücke [28]. The initial experiments alreadyshowed good agreement between the theoretical calculations andexperimental data, and further work has confirmed assumptions, al-though the experiments have been carried out under different con-ditions. The original theory is applicable to different metals with aspecific dislocation structure at 0 K. The criticism of the model isassociated with the development of detailed physical considerationsregarding the effect of bows on the displacement of dislocations,with the new results obtained regarding the movement of point de-fects anchoring the dislocations, and with the data on thermal ac-tivation and microplasticity. However, the theory can also be ap-plied to materials with a high content of solute elements [80], attemperatures higher than the temperature of Bordoni relaxation[81], etc.

The model of the vibrating spring (Fig. 2.30), which alreadytakes into account the tensile force in the dislocation, and viscousand inertia forces, was proposed by Köhler [82]. The model alsolinks the scatter of mechanical energy with unpinning of the dis-locations from the pinning points.

Granato and Lücke [28] proposed a quantitative theory of themovement of dislocations based on the dependence of internal fric-tion on strain amplitude. The calculations are based on the follow-ing assumptions:

1. All dislocation segments of the dislocation network have thesame length L

n,

2. Ln > > L

p

3. When stress is zero, the distribution of the dislocation seg-ments with length l is disordered, and is expressed by the Köhlerdistribution in the form

( ) 2exp ,

pp

lN l dl dl

LL

ρ= − (3.24)


106

where N(l) dl is the number of segments with the length from l tol + dl, L

p is the mean length of the segment, ρ is the density of

the dislocations released from the pinning points. Consequently

2

01 11/20 1

Cn

ii

CQ X

−ε−

=

=πε ∑ (3.25)

where

30

1 2n

p

LC C

L

Ω∆ ρ= (3.26)

2 ,p

k bC

L

η= (3.27)

here Ω is the orientation factor, ε0 is the strain amplitude, ∆

0 is:

2

02

8,

Gb

S∆ =

π

G is the shear modulus of elasticity, S is the tensile force in thedislocation, k is the Boltzmann constant, η is the Cottrell param-eter, b is the Burgers vector. Calculations can be carried out un-der the condition that σ

0/r << 1, where σ

0 is the amplitude of the

stress in external loading, r is the stress required for separating thedislocation with the length L

p. Shortcomings, such as, for example,

the large calculated distance of movement of the dislocations af-ter unpinning, are explained by the inaccuracy of the boundary con-ditions.

At low stress amplitudes, equation (3.25) is in good agreementwith the experimental results. The condition L

n >> L

p shows that

theory is not suitable for pure metals and this has been confirmedin a large number of investigations. Theory is in excellent agree-ment with the experiments on commercial purity metals at tempera-tures higher than the condensation temperature of the Cottrell at-mospheres T on the dislocations.


107

Theory is supplemented by the dynamic losses in unpinning ofthe dislocations from the solute atoms [83] by the assumption ac-cording to which the movement of the dislocation, unpinned fromthe pinning points, is determined not only by linear stretching butalso by the interaction of the dislocation with the solute atomspresent in the solute solution, in the area in which the dislocationmoves. Statistical analysis of the process of unpinning of the dis-location from the pinning points with different binding forces formedium stress amplitudes gives Q–1 ≈ σ2 [84]. If equation (3.24)is replaced by the distribution in accordance with the equation

( ) 1/ 2.... iN l c l −=

we obtain the value Q–1 close to that determined using equation(3.24).

3.3.2 Thermal activationIn the field of repeated stresses, structural defects, especially dis-locations, change their position and configuration ‘in a jump’. En-ergy scatter in the formation of steps by thermal activation is de-termined from the following procedure [85]. If N(t) is the numberof defects (dislocations) which have ‘jumped’ from position x toposition y during time t, then

( ) ( ) ( )0 ,ox x x y y y

dN tN N t P N N t P

dt = − ν − + ν (3.28)

where Nox

(Noy

) is the number of defects at position x (or y) at timet, x (ν

y) is the frequency of vibrations of the defects at position x

(or y), Px(P

y) is the probability of jump from position x to position

y (or from position y to position x).The probability of a thermally activated jump

( )exp .U

PkT

σ = −

If we consider the deformation of the crystal εd, we can deter-

mine


108

( ) ( )1

,,

2

l N t dQ

W−

ϕ σ σ=

π∫ (3.29)

where ∆W = ( ) ( ); , 1 .d dd N tε σ ε = ϕ σ∫Thermomechanically activated separation of the dislocation

(Teutonico model) describes the situation of two dislocation seg-ments pinned in the centre between a point defect [86]. The totalenergy of the system consists of the energy of the bowed disloca-tion U

1, the energy of elastic interaction U

2 and the work carried

out by applied stress U3. The diagram of the model is in Fig. 3.11a,

b. The effect of the magnitude of acting stress is shown in Fig.3.11c.

Under the effect of very small (4) or very high amplitudes (4)there is only one energy minimum, which means that there is onlyone stable position of the segments. For the mean values of thestress amplitude (5) we record two stable positions, separated byan energy barrier. Under the effect of low stresses, there is no un-pinning and the increase of stress results in the formation of a sec-ond energy minimum and unpinning may take place. This results inthe disappearance of the first energy minimum and the dislocationis mechanically unpinned from the pinning points.

Fig. 3.11. Model of separation (a), dependence of the energy of the double loopon the distance s between the dislocation and the pinning point (b) and stress(c), where 1 is linear energy, 2 Cottrell energy and 3 is the strain energy; m - lowstress, v - high stress,


109

The stress of thermal–mechanical unpinning can be expressed asthe stress of mechanical unpinning using the equation

1/ 2

1

0

1 ln .kr

CkT

U

′ν σ = σ − ω (3.30)

The model is constructed for a temperature of 0 K. If we con-sider the effect of temperature, it can be seen that the internalfriction is proportional to

0

exp ,kr σ− σ

where σcr

is determined by equation (3.30) and σ0 is the external

stress amplitude.The development of the Teutonico model in Ref. 87 resulted in

the determination of the conditions of catastrophic unpinning of thedislocation pinned by several point defects. The conclusions showthat catastrophic unpinning takes place in a narrow stress range(~σ

t1, where the difference between σ

t1 and σ

kr increases with in-

creasing temperature), and catastrophic repeated pinning also takesplace in a narrow the stress range (~σ

t2, where σ

t2 << σ

t1).

These results support the equations of the Granato–Lückemodel, when σ

kr is replaced by σ

t1.

3.3.3 Internal friction with slight dependence on strainamplitude

Lücke et al. [88] analysed the conditions of thermomechanicalseparation for a dislocation segment pinned in the centre of thelength by a point defect. It is assumed that the value of f changesexponentially from f

0 to the equilibrium value

1

2

1

1 exp .a bU Uf

kT

−

∞ ν −= + ν

(3.31)


110

At low temperatures and high frequencies, f∞(σt2

) and f∞(σt1

)approach 0 and 1. The changes of f∞ are described by a steppedfunction. Figure 3.12 presents the dependences σ – ε

d and σ

d, f∞,

σ in relation to t for the given conditions.The investigations carried out in Ref. 88 shows that the inter-

nal friction, dependent on the stress amplitude, is

1 1

31 3

0 0

exp .r tmQ L− σ σ

= ρ − σ σ (3.32)

Friedel [89] derived equations forming a link between internal fric-tion, frequency and temperature, assuming a certain number ρ ofdislocations with length L

n uniformly pinned by point defects, lo-

cated at the distance Lp, where L

n >> L

p >> b. The change of en-

ergy in relation to the magnitude of stress is expressed by the equa-tion

Fig. 3.12. Diagrams of the dependence of individual quantities in thermomechanicalseparation of the dislocation.


111

( ) ,M pU W b dLσ = − σ (3.33)

where WM

is the total energy of the point defect–dislocation bond,d is the distance of the effective bond which is (1–3) b. Repeatedpinning of the dislocation was ignored. For the given conditions

401 0

2 exp .24

M pnm

p

W b dLL bQ

kTL− − σ ρ ν= − π

(3.34)

The equation is valid when

0 .Mkr

p

W

bdLσ < σ = (3.35)

Unpinning takes place at σ0 ≥ σ

kr. Therefore

231

0

.12

n krm

LQ− ρ σ= π σ

(3.36)

The magnitude of Q–1m at σ

0 < σ

kr changes with the change of tem-

perature and frequency. Unpinning from the first pinning point takesplace at a low frequency in comparison with unpinning from thelast pinning points, so that the following conditions must be satis-fied

exp 1.2

pbdL

kT

σ> (3.37)

210 .pL

b−ε > (3.38)

For a high-purity material, the conditions (3.37) and (3.38) arefulfilled when L

p > 102b, ε > 10–4, and the average concentration

of the solutes is c0 < 10–5 wt.%.

When calculating Qm–1, Koiva and Hasigutti [90] used the follow-


112

ing equation instead of (3.28)

( ) ( ) 00 1 exp .

dN t UN N t

dt kT

− σν = − ν − (3.39)

Approximate calculations were carried out for the case in whichσ

0ν/kT < 1, and they obtained the equation

( ) ( )3

1 00 .

12n

m

LQ N f h

kT− σ ν = α + α π

(3.40)

An interesting result of the solution is that it predicts a maxi-mum on the dependence of internal friction on strain amplitude with

Fig. 3.13. Positions of dislocation lines in the process of overcoming a point defect,where a is the equilibrium position when overcoming the obstacle, b shows approachto the concentrated force, c shows the binding energy for the case of repulsion(w > 0) and attraction (w < 0). 1) stable position in front of the obstacle, 2)unstable position behind the obstacle (saddle of the curve); 3) stable position afterovercoming the obstacle.


113

the change of temperature. The level of stress influences the func-tions f(α), h(α) and the shape of the maximum.

The presented data show that the model proposed by Lücke etal. is suitable for low temperatures and high frequencies, whereasthe model proposed by Friedel is suitable for pure materials, etc.The results are generalised in Ref. 91.

The model proposed by Indenbon and Chernov [92] analyzes thethermal fluctuation overcoming of an elastic field of a point defectby a dislocation, by introducing transitional configuration of the dis-location, prior to the establishment of the equilibrium position. Thedislocation segment is situated on the slip plane x, y (Fig. 3.13),and is pinned at the ends at points (±l, 0, 0). Under the effect ofexternal stress σ

0 and thermal fluctuation, the segment is separated

from the point defect C(0, 0, z0). The equilibrium positions are cal-

culated in a computer. With a probability close to 1, unpinning takesplace at the energy

( )0

0ln ,U kTσν = ξ ν

(3.41)

where ν0 is the frequency of vibrations of the dislocation segment,

ν is the frequency of external loading.The temperature dependence of the segment length L

p fulfils the

equation

( )( )( )0

0ln /.

/p

p

L k

T U Lσ

∂ ν ν = ∂ ∂ ∂ (3.42)

With increasing temperature, the shortest segments are the firstto be unpinned and segments with the length being a multiple of L

p

form at relatively low stresses. The first stage is characterised bythe increase of the number of segments contributing to the dislo-cation hysteresis, and the second stage is reflected in a decreaseof the strength of the effect of each segment.

Consequently, internal friction

( ) ( )min

2

10

0

, ,nm n

L

Q T L N l dl∞

− σσ = σ ∫ (3.43)


114

where 2Ln is the length of the dislocation segment, σ

n is the stress

at which the thermal fluctuation unpinning of the segment from thedefect takes place, σ

0 is the external stress amplitude. The value

Lmin

= LM

(σ0, T).

The effect of thermal activation can also be evaluated by solvingthe equations of vibrations of a solid in analysis of functions foradditional strain ε

d [93]. Into the vibration equation we substitute

the rate of change of the additional strain determined by the move-ment of the dislocation

0 00 1 exp exp exp ,d

U USb F

k kT kT

− σν + σν∆ ε = ∆ ρ ν − − − (3.44)

where ∆F is the area through which the dislocation travels, ∆s isthe change of entropy accompanying the displacement of point de-fects. From the resultant equations it is possible to determine thecharacteristics of internal friction and the defect of the Youngmodulus.

At relatively high strain amplitudes, microplastic deformation cantake place in certain areas of the material; this is a source of sig-nificant scattering of the energy in the material.

3.3.4 Plastic internal frictionIn accordance with the schematic representation in Fig. 2.30, theeffect of a certain amplitude of external stress results in the gen-eration of dislocations, Fig. 2.30, and cyclic microplastic deforma-tion of the material takes place. In order to describe the given proc-ess uniformly and in considerable detail, and take into account theimportance of this response of the material to repeated loading,this part of the dependence Q–1(ε) is discussed in Chapter 6 of thisbook.

3.4 EFFECT OF LOADING FREQUENCYThe Granato–Lücke theory [28] is suitable for examining the ef-fect of frequency on internal friction and also for predicting manyprocesses taking place in solids. However, there are two compli-cating features.

Calculations of the dislocations strain at stresses (10–7–10–8) Gin the MHz frequency range with the value δ = 10–3 and at a dis-location density of ρ = 1010 m–2 indicate that the segments vibratewith an amplitude of 10–7–10–8 mm, which is smaller than the


115

atomic spacing. This small displacement can no longer be describedby the tensile force and the Burgers vector. It has been confirmedthat the dislocations start to move under the effect of the Peierlsstress which is lower than (10–7–10–8)G. The model of the vibrat-ing spring does not take into account the existence of Peierls bar-riers and does not explain the possibility of overcoming these bar-riers in measurement of Q–1 in the range of low strain amplitudes.

The theory of internal friction caused by dislocations does notconsider the possibility of thermally activated release of the dislo-cation segments. It is well-known that already at room temperature,the frequency of unpinning of the dislocation segments as a resultof temperature variations amounts to hundredths of a megahertz.In the MHz frequency range, a change of external stress is ac-companied by a multiple, thermally-activated separation of the seg-ments.

Despite the fact that these complications are important, it isquite surprising that the theory of the vibrating spring describessufficiently the resultant experimental dependences. The explana-tion is based on taking into account other phenomena, for examplethe concept and mechanism of movement of bows on the disloca-tions during overcoming barriers according to Shockley [94]. Thestatistical and dynamic theory of bows assumes that a dislocationin the given slip plane is oriented preferentially in the direction withthe highest density of the atoms and, consequently, causes bows toform. At temperatures higher than 0 K, the dislocation lines are un-stable, form pairs of bows of the reversed sign and are periodicallydisplaced from one potential well to another. The energy barrier,overcome by movement of the bow along the dislocation line in thedirection of dense distribution of the atoms (so-called Peierls stressof the second kind) at stresses of 10–4G and the width of the bowof 100b is 10–7G, which corresponds to the stress amplitudes usedin internal friction measurements.

Evaluation of the strain amplitude during movement of bowsshowed that its value is 1/n

pb times higher than the amplitude used

in the model of the vibrating spring (np is the number of bows on

a dislocation segment with length l). The theory of resonancedislocational internal friction solves the movement of bows underthe effect of repeated external loading if the following conditionsare fulfilled:

–the pinning points of the dislocation line are not located in thesame potential well and, consequently, the dislocations contain geo-metrical bows;


116

–below the temperature corresponding to the Bordoni peaks, np

is constant and does not change during the loading cycle;–the activation energy of movement of the bows along the dis-

location line is relatively small in comparison with the energy offormation of double bows;

– with increasing excitation amplitude used in measurement ofQ–1 the dislocations situated in the same potential well do not con-tribute to anelastic deformation.

Consequently, the defect of the Young modulus is

( )

2

022

2

0

1

,

1

GZ

G

ω− ω∆ = ω − + ωτ ω

(3.45)

and internal friction

( )

122

2

0

,

1

Q Z− ωτ= ω − + ωτ ω

(3.46)

where

2 2

4 20 0

8, ,

l

l Gb aBZ

S m

Ωρ= τ =π ω (3.47)

4 220 2

0

sin, .

4 sinl

l n p

aS Gb kTS l n

qm l

π ϕ β γω = = +π ϕ (3.48)

In these equations, Ω is the orientation factor, γ is the size factorequal to 5 at a high angle ϕ and equal to 1 at small angle ϕ. Re-laxation time


117

2

2.

aBl

kTτ =

π γ

Resonance frequency ω0 decreases with decreasing angle ϕ and

approaches a constant value

2

20

.kT

m l

π γ

Stretching S1 contains the quantity

,sin

kT

a

γϕ

where a is the lattice parameter.At low frequencies, ωτ < 1, the above equations have the fol-

lowing form

2 2

4

8,

l

G Gb l

G D

∆ Ωρ=π (3.49)

21 4

6 2

8 1

sinl

GbQ Bl

S− Ωρ= ω

ϕπ (3.50)

and at high frequencies, ωτ > 1, the form of these equations is

2 2

2 2 2

8 sin,lG b SG

G l B

ωπ∆ ϕ=ω

(3.51)

21

2

8 sin,

GbQ

B− Ωπ ϕ=

ωπ (3.52)

where the meaning of the symbols is the same as in the case ofequations (3.45)–(3.48).


118

Fig.3.14. Dependence of the relative friction decrement on relative frequency atdifferent values of the friction constant (B) for the delta distribution of the lengthof dislocation segments.

The internal friction associated with the movement of bows alongthe dislocation lines increases in direct proportion to loading fre-quency and the value l4 when approaching the resonance maximum(ωτ = 1). After passing through resonance, the values start to de-crease in inverse proportion to loading frequency.

The movement of bows on the dislocations can be evaluated byinternal friction measurements in the case of small angles ϕ thatwith the magnitude of acting stresses below 10–6G.

Figure 3.14 shows the dependence of normalised friction on nor-malised frequency for different values of the friction characteris-tic B, plotted using equations (3.45) and (3.46). It can be seen thatfriction is of resonance nature, and the resonance frequency is con-trolled by the values of S

1, B and l, and also by the distribution of

the lengths of the dislocation segments in the material. For thecase shown in Fig. 3.40 we use the delta distribution. The ‘qual-ity’ of the specimen decreases with increasing B, the resonancepeak becomes wider and is displaced to lower frequencies. For thecase of damped resonance (ω/ω

0)2 << 1 we have


119

o

Table 3.3 Resonance frequencies ω0 and position of internal friction maxima ω

max

lateM ϖ0

)zHM( ϖxam

)zHM(

gAlAuAuCiNbPnZeG

764868253826487012685046

0.670.560.970.290.531

5.420.960.15

1max

max

.2

G ZQ

G−∆ = = (3.53)

The frequency position of the maximum

2 20

max 2,lS m

BBl

π ωω = = (3.54)

makes it possible to determine, together with the value of Q–1max

, theparameters ρl2/S

1, S/l2B and ρ/B and specify these values further

when the values of S1 and ρ are known.

Lücke [95] determined the resonance frequency of the disloca-tions ω

0 and the frequency position of the ω

max of internal friction

max for several metals at l = 10–3 mm and B = 5 × 10–4 dyne s/cm2 (Table 3.3). At frequencies lower (ωτ < 1) and higher (ωτ >1) than the peak, internal friction is proportional to l4Bω or ω–1.

Fig.3.15. Displacement of dislocations under the effect of stress: a) displacementat low (1) and high (2) friction; b) the case of high friction with oscillations notin phase (1) or in phase (2); c) change of situation after adding another pinningpoint (as b).


120

In the range of low (sonic) frequencies the dislocation frictionis low, as indicated by, for example, experiments with measurementof Q–1 at 1 Hz and 10 kHz. The value of Q–1 consists of two com-ponents, where the first component is directly proportional and thesecond one indirectly proportional to frequency. With decrease offrequency, the importance of the first and second component in-creases.

The spring theory predicts the anomalous decrease of the veloc-ity of sound after irradiation with neutrons or low plastic deforma-tion [96]. The magnitude of internal friction and of the defect ofthe Young modulus are influenced by various parts of dislocationdeformation that are in phase ε

f or not in phase ε

f0 with the in-

duced stresses, i .e. Q–1 = εf0

/εe, ∆G /G = ε

f /ε

e, where ε

0 is the

elastic strain.The increase of loading frequency increases the viscous friction

force and also the value ε0 (Fig. 3.15), and decreases ε

f. This in-

crease the velocity of sound and friction (normal phenomenon). Inthe limiting case ω→∞, ε

f = 0, the wave propagates as in a me-

dium without dislocations. In the presence of other pinning points,formed as a result of neutron irradiation or low deformation, thevalue of ε

f increases (Fig. 3.15c) and ε

f0 decreases. This results

in a decrease of the velocity of sound and a decrease of friction(anomalous phenomenon). At a large number of pinning points, thedislocation component, which vibrates in the phase with the actingstress, is completely blocked thus increasing the velocity of sound.

The main shortcoming of the spring theory is the small displace-ment, i.e. the amplitude of dislocation vibrations. For example, ata stress of 10–7 G this amplitude is higher than the amplitude ofthermal vibrations of the lattice at room temperature only whenl ≥ 105b, at a purity of the material better than 99.999 at.%. How-ever, if it’s taken into account that the spring model is equivalentto the model of movement of bows with a high density, these short-comings are eliminated. The equivalence of the models was con-firmed by Suzuki and Elbaum [97] at temperatures correspondingto Bordoni’s peak and at a strain amplitude sufficient for the for-mation of a single double bow on a dislocation.

These considerations also show that the mechanisms of internalscattering of energy, regarded as frequency-independent, can beinfluenced by frequency, although not directly but by means of aset of mechanisms that are sensitive to frequency and form, forexample, a large part of the internal friction background.


121

Fig.3.16. Diagram of the effect of forces on the pinned atom: a) length of dislocationsegments A = B; b) A < B.

3.5 EFFECT OF LOADING TIMEThe anelasticity characteristics of the material depend on loadingtime and, consequently, the number of load cycles, in the range ofstrain amplitudes that do not yet cause microplastic reactions.

The change of internal friction and the defect of the Youngmodulus, caused by extending loading time, makes it possible todetermine the parameters of diffusion of atoms along dislocations,the characteristics of stability and long-term strength, the processesof fatigue damage cumulation in materials, etc.

The time dependence of internal friction is the result of diffu-sion of solute atoms in a stress field. The solute atoms form clus-ters and, consequently, dislocation segments are released from thepinning area and the extent of friction increases. In addition to dif-fusion displacement of the atoms along the dislocations, these at-oms also diffuse from the dislocations into the volume of the lat-tice. The type of atoms placement mechanisms that operates in aspecific case depends on the magnitude of acting stress, loadingfrequency, temperature, concentration of the solute atoms at dis-locations, bonds between the solute atoms and the dislocation, andon the mutual effect of the solute atoms between each other.

External loading results in bending of dislocation segments (Fig.3.16). The tensile force in the dislocations results in the formationof forces acting on the solute atom. These forces can be dividedinto mutually normal components. Consequently


122

( ) ( )2 2

2 21 || 1, ,

2 8p p p p

b bF L l F L L

C⊥ + +σ σ= + = − (3.55)

where σ is the acting stress, b is the Burgers vector, C is thestretching of the dislocation line, L

p and L

p+1 are the lengths of the

dislocation segments on each side of the pinned atom. Force F||

isindependent of the orientation of external loading and is alwaysdirected from larger to shorter dislocation segment. Force F⊥changes its sense in accordance with the sense of applied stress.Both forces (F⊥and F

||) can cause movement of the blocking sol-

ute atoms. In the case of small bows of dislocation segments, thefollowing equation can be used

( )1|| ,4p pGb L LF

F C+

⊥

−= (3.56)

It can be seen that F| |

is many times smaller than F⊥. The move-ment of the pinning points in the direction normal to the disloca-tion line should be examined under static loading. With repeatedloading, the redistribution of the pinning atoms depends on the re-lationship between the relaxation time and the frequency ofchanges of external loading.

For the simplest model of the dislocation segment with a singlepinning point, situated between two pinning points, the relaxationtime of movement of a pinned atom together with the dislocationat the stress acting in the direction normal to and parallel to theinitial position is

( ) ( )1 2

|||| 2 2

1 2 1 2

4; ,

L L CD D

C L L L L bkT kT

⊥⊥

τ = τ =+ + σ (3.57)

where D⊥ and D|| are the diffusion coefficients of the atoms of the

solute in the direction normal and parallel to the dislocation line, L1,

L2 are the lengths of the adjacent dislocation segments.The displacement of the pinning atoms is interpreted by the dif-

fusion under the effect of the external force and, consequently, itis possible to determine the diffusion coefficients along the dislo-


123

cations from the internal friction measurements at the start of re-covery of the material. A more efficient procedure is the one us-ing the curves of saturation of changes because the measurementof the course is more accurate, the distance between the solute at-oms is sufficiently large and the interaction of the atoms can be ig-nored. The curve of saturation of the changes Q –1

(τ) can be de-

scribed by the equation

( ) ( ) ( ) ( )1

0

1

2

1 1 1 10 2

1,

1

eQ Q Q Q

e

τ−τ

− − − −τ σ σ − τ

τ

− = + − −

(3.58)

where Q–1(τ)

, Q–1(0)

is the internal friction under the effect of stressσ and without it, τ is time, τ

0 is the duration of saturation of the

changes, and τ1 is the time related to the loading frequency.

The unpinning of the dislocations from the solute atoms and theirnew distribution in the atmosphere is described in detail in theanalysis of the Q–1 – ε dependences. In this section, it is possibleto examine the case in which a solute atom moves together witha dislocation.

Force F⊥ (Fig. 3.16) has the meaning corresponding to themeaning of external loading. The solute atom does not travel far

Fig. 3.17. Region of separation (A), diffusion with activation energy of 0.32 ×10–19 J (B), diffusion with the activation energy of 0.8 × 10–19 J (C) and the positionof pinning (D) of the dislocation in relating to pinning atoms in the stress–temperaturediagram.


124

away from the dislocation. When the applied stress carries outwork higher than the binding energy of the solute atom and the dis-location, the dislocation separates from the pinning atoms. With in-creasing temperatures the jump time of the solute atom τ⊥ de-creases and the solute atom and the dislocation can move together.The σ – T dependence (Fig. 3.17) shows the region of unpinningof the dislocation (A) and the region of displacement of the soluteatoms (B) together with the dislocation. The boundaries of the re-gions are a function of temperature, loading frequency, binding en-ergy of the solute atoms and the dislocations, and the activation en-ergy of diffusion. It should be noted that due to rigid pinning (C)the probability of joint movement of the solute atoms with the dis-locations increases with increasing temperature.

This model explains the temperature dependence of the criticalstrain amplitude ε

cr where the value of ε

cr decreases with increas-

ing temperature. Examination of the Q–1 – γ dependence for cop-per with zinc and germanium shows that the change of the tem-perature dependence of γ

cr occurs at a temperature of ~170°C.

This temperature denotes the start of displacement of pinning at-oms together with the dislocations, with the average displacementbeing 4b. Consequently, D⊥ can be determined.

The joint displacement of the atoms of the solute with the dis-location may not be reflected only in the time dependence of in-ternal friction, because the direction of diffusion of the pinned atomduring transverse movement changes during cycling loading. Dis-placement may also influence the function of the distribution of thesolute atoms along the dislocations, with a decrease of the prob-ability of formation of clusters of the atoms. In addition, the defor-mation, caused by the displacement of dislocations increases, andthere are also losses associated with the frequency of jumps of theatoms of the solute with the frequency of external loading.

Continuous recording has a large number of advantages in com-parison with discrete recording where vibrations must be periodi-cally decreased when the saturation branch forms, and when theunstressing branch appears it is necessary to excite the specimensfor measurement of Q–1. Disregarding the short measurement time,this results in errors which are especially evident at short relaxa-tion times.

Since the diffusion coefficient of the solute atoms along the dis-locations causes a change of Q–1, the time dependence of Q–1 isreflected in certain temperature and frequency ranges. At low tem-perature, the mobility of the solute atoms is low and the long-term


125

Fig. 3.18. Temperature dependence of internal friction (A) for aluminium A999at different frequencies: 1) 1.38 kHz, 2) 2.41 kHz in instantaneous measurements;3 and 4) the same frequency and after saturation lasting 3 min. b) temperaturedependence of friction decrement.

effect of loading does not lead to any significant changes of Q–1.At high temperatures, relaxation takes place already during the pe-riod of increasing loading and it is difficult to determine the timedependence of the changes of Q–1, because the atoms managed toreturn to the initial position at the end of the loading cycle.

Figure 3.18a shows the temperature dependence of Q–1 of purealuminium, recorded at the start of saturation and after loading forthree minutes. Figure 3.18b shows the increase of the friction dec-rement ∆δ on temperature. The temperature ranges in whichQ–1 depends on temperature and also the homologous temperaturesof the boundaries of the ranges for different materials are pre-sented in Table 3.4. It can be seen that for all evaluated materi-als, the temperature ranges with the time dependence of thechanges of Q–1 are on average close to temperature (0.3–0.45)T

t.

In these ranges, the relaxation stability of the metals is reduced.Measurements of the time dependence of the changes of Q–1 can

be utilised efficiently in the quantitative evaluation of the relaxa-

Table 3.4 Temperature ranges of the time dependence of internal friction

lateMfotnetnoC

setulos)%.tw(

erutarepmeT)K(egnar

egnarerutarepmeTgnitlemotdetaler

tniopycneuqerfgnidaoL

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10.010.010.010.0

045–073027–036004–052027–005

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126

r →→→→→

Fig. 3.19. Dependence of the increase of the decrement of oscillations on the radiusof the solute atom.

tion stability and stability of the dimensions of metallic solids op-erating under repeated loading. Measurements should be taken us-ing specimens with the natural frequency, the same as the fre-quency of the source of the vibrations. Static and repeated load-ing results in the redistribution and clustering of the solute atomsas a result of their displacement along the dislocations which is theinitial stage of low-temperature creep, and the dimensions of thesolid change by ~0.1%. The temperature ranges of occurrence ofthis phenomenon should be avoided in service of components. Load-ing frequency also influences the occurrence of temperature rangesin which changes of Q–1 depend on time. The decrease of the load-ing frequency results in a decrease of temperature at which thetime dependence of the changes of internal friction is evident.

The time dependence of the changes of Q–1 is recorded in theregion in which Q–1 is independent and also dependent on the strainamplitude. Long-term loading results in a decrease of the criticalstrain amplitude and increases internal friction. The largest increaseof Q–1 is at stresses corresponding to the critical amplitude, butwith increasing ε the increase of Q–1 becomes smaller. In the en-tire strain amplitude range up to the microplasticity range the in-crease of time increases internal friction (see Chapter 6).

The radius of the solute atoms also influences the increase ofthe logarithmic decrement. The largest radius of the solute atom isrelated to the largest change of ∆δ (Fig. 3.19). Evidently, the spac-ing of the solute atoms on the dislocations is also important in thiscase .

Prior plastic deformation, accompanied by ageing, influences theincrease of the logarithmic decrement when examining the time de-


127

Tab

le 3

.5

Fri

ctio

n m

ech

anis

ms

in f

erro

mag

net

ic m

ater

ials

(σ

–

sp

ecif

ic e

lect

rica

l re

sist

ance

, R

– t

he

rad

ius

of

the

spec

imen

, X

a –

the

init

ial

mag

net

isat

ion

det

erm

ined

by

th

e m

ov

emen

t o

f d

om

ain

bo

un

dar

ies,

XR –

rev

ersi

ble

mag

net

izat

ion

, D

– s

ize

of

do

mai

ns,

σs

– s

tres

s o

f m

agn

etic

ela

stic

sat

ura

tio

n)

retemara

Prof

nosaeR

lanretninoitcirf

lacinahceM

lanretninoitcirf

ycneuqerF

egnargniti

miL

ycneuqerf

foecnednepe

DQ

1–no

ycneuqerF

fedutilp

mA

Fnoitasitenga

MI

dnaepah

Sfo

noisnemid

snemiceps

gnitimi

Lesac

lareneG

ecnedneped

Q1–

a

foegnah

Clareneg

noitasitengam

yddeorcaM

stnerrucf

<<

f 1 f>

>f1

f 1»ρ

4/π

RX

2~

f~

f2/1–

f/f 1

(+1

f+f

1)2

tnednepednIQ

1–a

ta0

=I

=ta

dna0

I=

I s

ecnednepeD

Q1–

nb

elbisreveR

fotne

mEvo

mnia

modseiradnuob

yddeorcaM

stnerrucf

<<

f 0

f>

>f 0

f 0»ρ

69/πX

-a

D2

f~ ~f

1–

f/f 0

(+1

f+f

0)2

tnednepednIQ

1–nb

ta0

=I

=I s

tonseo

Ddneped

Q1–

nb

elbisreveR

fosessecorp

fonoitator

rotceveht

f<

<f 0

>>f

f 0

f 0»ρ

52/πX

-R

D2

f~ ~f

1–

f/f0

(+1

f+f

0)2

Q1–

h

elbisreverrIfo

tnemevo

mnia

modseiradnuob

citengaM

citatscitsale

sisiretsyhot

tnednepednIf

»01

6z

H–

,Fta

Q1–

h0

=ta

F<

<F

ssi

I=I

s/1–F

2ta

F=

F2

–


128

pendence of the changes of Q–1. In this case, the dislocation densityincreases and the number of the solute atoms situated along theunit length of the dislocation in the solid solutions decreases. Thisis reflected in an increase of ∆δ.

Of special importance is the examination of the changes ofQ–1 and ∆E/E in relation to the time of repeated loading in therange of cyclic microplasticity and in the part of the Q–1 – ε de-pendence in which Q–1 is an evident function of the loading time(Chapter 6).

3.6 EFFECT OF MAGNETIC AND ELECTRIC FIELDSIn ferromagnetic materials, a large part of external mechanical en-ergy can be dispersed as a result of changes of general magneti-sation, reversible movement of the domain boundaries, the revers-ible process of movement of the vector of magnetisation in domainsand irreversible displacement of the domains boundaries; eddymicrocurrents or magnetic–elastic static hysteresis are also impor-tant [98].

Table 3.5 summarises the most important effects on the extentof internal friction, with special attention to the effect of the mag-netic field. The scatter of energy, associated with eddy micro- andmacrocurrents, is independent of the strain amplitude and is directlyproportional to loading frequency.

However, magnetomechanical friction is more important; in thiscase, the domain structure of the ferromagnetic materials is char-acterised by different orientation of the vectors of magnetisation ofthe adjacent domains separated by Bloch walls. When the orien-

Fig. 3.20. Deformation curve for ferromagnetic materials.


129

tation of external loading is changed, the magnetisation vector alsochanges and the domain boundaries are displaced. The formationof elastic strain ε

c is accompanied by magnetostriction deformation

λ, caused by the change of the orientation of vectors of local mag-netisation; the total deformation of the material is ε = ε

c + λ .

Unstressing is followed by the deformation of the stepped bound-ary of the domains. The effect of cyclic loading is accompanied bycontinuous displacement of the domain boundaries in two mutuallynormal directions with the frequency identical with the frequencyof external loading. This phenomenon results in additional magneto-mechanical scatter of energy in the ferromagnetic materials, the∆E effect appears and appropriate changes take place in the hys-teresis loops (Fig. 3.20).

When the ferromagnetic material is placed in a saturated mag-netic field (H = H

s), the entire material has the properties of a sin-

gle magnetic domain and the deformation curve is linear. Conse-quently, the friction characteristics of the ferromagnetic material arelow.

The scattering of energy in the ferromagnetic materials dependson the initial condition of the material and test temperature, in re-lation to Curie temperature. The loss of energy in the material isindependent of the frequency of vibrations up to 105 – 106 Hz, i.e.to the frequencies at which the duration of the Barkhausen jumpsis considerably shorter than the period of elastic vibrations [99].

Fig. 3.21. Main shapes of the curves of the dependence of the amount of scatteredenergy on stress amplitude in ferromagnetic materials.


130

The scatter of energy by magnetic–elastic hysteresis is associ-ated with the amplitude of the acting cyclic stress through the equa-tion ∆W = Dσn, where n ≤ 3. Under the effect of a low stress am-plitude n = 3, and with increasing amplitude the value of n de-creases to 2 and at the maximum stress amplitude n = 0 at mag-netic saturation. Similarly, at low stress amplitude, the magneticcomponent of internal scattering of the energy Ψ

h ≈ 6, and the

slope of the Ψh(σ) curve decreases. The function Ψ

h(σ) is char-

acterised by a curve with a maximum (for example, for nickel andalloys of iron with nickel). The dependence of Ψ

h on the stress am-

plitude has the form of an S-shaped curve; this has been confirmedby experiments for iron and carbon steels.

With increasing amplitude of the vibrations the magnetic−elas-tic hysteresis loop of soft ferromagnetic materials graduallychanges from oval to bent shape; this is associated with the for-mation of the ∆G effect [100].

Kekalo [98] classified the main types of the dependence of in-ternal friction on the strain amplitude for ferromagnetic materials,Fig. 3.21. The curves with a high maximum, Fig. 3.21a, are re-corded in cases when τ

m = τ

max << τ < τ

d, where τ

d is the mini-

mum shear stress at which the energy losses, associated with dis-location processes, are recorded. This may be observed in iron,nickel, cobalt, low-carbon and low-alloy steels, and also many fer-rous alloys.

The dependence of internal friction on the stress or strain am-plitude is often different (Fig. 3.21b–d) in comparison with thatobserved in nickel, iron, gadolinium, maraging steels, carbon alloys,and also other alloys. For these cases τ

m = τ

max ≈ τ

d < τ . At

τd > τ

max we obtain the case shown in Fig. 3.21d. The curves with

the form shown in Fig. 3.21d are recorded for slightly different ma-terials, i.e. the magnetomechanical losses are not completely sup-pressed. The form of the dependence shown in Fig. 3.21e is re-corded in two cases, i.e. when τ << τ

max, or in materials with in-

creased magnetic hardness (τmax

≈ τel

). The form of the curve inFig. 3.21f is associated with pinning of domains by different obsta-cles, for example, after stabilisation of the domain structure in theordered condition, and this is reflected in the formation of an ad-ditional maximum.

The effect of the structure on the form of the dependence of in-ternal friction on the stress or strain amplitude from the viewpointof magnetomechanical phenomena is quite clear. All factors,decreasing the speed of movement of the domain boundaries, de-


131

Fig.3.22. Dependence of internal friction on strain amplitude of Co and its alloys:1) Co; 2) 2% Re; 3) 10% Re; 4) 1% Zr; 5) 8% Zr; 6) 1% Nb; 7) 10% Nb; 8) 8%V; 9) 7% Fe; 10) 5.65% Mn; 11) 10% Mn; 12) 5% Cr; 13) 15% Cr; 14) 25% Cr;15) 65% Ni; 16) 10% Mo; 17) 2% Ti (wt.%).

Fig. 3.23. Dependence of internal friction on the strain amplitude of the alloy at23 kHz. 1) without the magnetic field, 2) using the electric field with 3 × 104

V m–1, 3) after application of the magnetic field with an intensity of 1.1 × 104 Vm –1.

crease the height of the maximum and increase the stress τmax

. Theform of the dependences is also influenced by the measurementconditions, such as temperature, application of a constant or vari-able magnetic field, electric current, static loading, etc.

the magnitude of internal friction of cobalt and cobalt alloys issignificantly influence by the ε → α polymorphous transformation.Increasing content of the alloying element decreases the amount ofthe low-temperature ε-phase with a hexagonal close-packed latticeand the temperature of the ε → α transformation increases. Thisis also reflected in the dependence of internal friction on strain


132

amplitude, Fig. 3.22. The magnitude of internal friction of the al-loys under the effect of a specific strain amplitude increases witha decrease of the content of alloying elements. If it is taken intoaccount that the transformation in pure cobalt is not completed, themagnitude of internal friction in a wide strain amplitude range de-pends on the amount of the high-temperature phase present in thealloys. Annealing at temperatures in the vicinity of the martensitictransformation temperature decreases the residual stresses and in-creases the friction capacity of cobalt [101].

Measurements taken on amorphous alloys of iron, chromium andnickel [102], produced in the form of ribbons 0.02 mm thick and 2.5mm wide, by cooling from a melt at a speed of 106 K s–1, with atensile strength of approximately 2400 MPa show, Fig. 3.23, thatthe level of their internal friction is influenced not only by thestrength of the magnetic field but also by the application of a uni-directional electric field. The main relationships recorded for crys-talline materials are also valid in the amorphous materials.

Iron alloys are also sensitive to the strength of the external mag-netic field and, consequently, the magnetomechanical component ofinternal friction contributes significantly to the level of their internalfriction. The overall effect depends not only on the chemical com-position of the alloy (this results in the changes of the amount anddistribution of the ferromagnetic phase in the alloy), but also onheat treatment which may greatly change the fraction of theferromagnetic phases in the alloys of a specific composition. Fur-ther results of measurements in this area are published in Chapter6.

In concluding this chapter, it should be noted that the specificlevel of internal friction and of the defect of the Young modulus isa function of a large number of factors having different effects onthe mechanisms of scattering of mechanical energy in the material.It has also been found that in efficiently prepared experiments,internal friction and the defect of the Young modulus reflect thesubstructure, structure, prior deformation, thermal, deformation–thermal and other treatments. This means that the measurementsof Q–1 or ∆E /E provide additional data on the material and onchanges taking place in it.

133

Measurements of Internal Friction and the Defect of the Young Modulus

4

MEASUREMENTS OF INTERNALFRICTION AND THE DEFECT OF THE

YOUNG MODULUS

The requirements on equipment and devices for measuring internalfriction and elasticity moduli different [1,103]. For example, thedetermination of an independent characteristic, such as the internalscattering of energy, i.e. damping capacity, is carried out with dif-ferent force effects on actual solids. For many actual structuralcomponents it is important to know a wide amplitude and frequencydependence of oscillations so that it is important to use a suitabletest system. On the other hand, the tasks of physical metallurgy andthreshold states require exact experiments in different directions (forexample, in analysis of the relaxation spectrum of internal frictionin solids it is necessary to regulate temperature from the liquid he-lium temperature up to temperatures close to the melting point, dif-ferent frequencies at low strain amplitude).

In the following section, we discuss the fundamentals of experi-mental procedures and equipment for measuring internal friction andthe defect of the Young modulus and also examine the possibilitiesof automating these methods.

4.1 APPARATUS AND EQUIPMENTThe basis of the classification of the methods of measuring inter-nal friction and elasticity moduli is the general approach to the con-struction of scientific devices and metrology (Fig. 4.1).

The main structural feature of apparatus and equipment is the ex-citation section, the measurements section and the section for re-cording vibrations of the specimens or test system. These sectionsare characterised by defined parameters and represent independentelements of testing equipment. Frequency is the criterion used in the

134


FFFFFigigigigig. 4.1.. 4.1.. 4.1.. 4.1.. 4.1. Classification of the test methods, measurements and recording of internalfriction and the defect of the Young modulus.

Vis

ual

Ele

ctro

stat

ic

Ele

ctro

mag

neti

c

Pie

zoel

ectr

ic

Mec

hani

cal

Dig

ital

pri

nt

Pro

gram

min

g

Cod

ing

Com

pute

r

Ana

logu

e x–

yre

c. d

evic

e

Rec

ordi

ng

Rec

ordi

ngm

etho

d

Rec

ordi

ngo

utp

ut

EL

AS

TIC

AN

D A

NE

LA

ST

IC P

RO

PE

RT

IES

Mea

sur-

emen

ts

Mea

sure

dch

arac

ter-

isti

cs

Mea

sure

men

tm

eth

od

Dir

ect

Ind

irec

t

Absolute

Relative

Hysteresis loop

Calorimetric

Oscillation damping

Supplied power

Oscillation incrementResonance curvePhase delay

UltrasoundDifferential

Infr

aso

un

d

So

un

d

Ult

raso

un

d

Hy

per

sou

nd

Sta

nd

ing

Ru

nn

ing

Lo

ng

it-

ud

inal

Tra

nsv

erse

Ro

tati

ng

Ex

tern

alin

dep

end

ent

Sel

f-ex

cita

tio

n

Ex

cita

tio

n

Fre

qu

ency

ran

ge

Ela

stic

wav

es

Typ

e of

osci

llat

ions

Ex

cita

tio

nm

eth

od

135


conventional approach for determination of the group of the exci-tation section. Using frequency, the systems are divided into infra-sonic, sonic, ultrasonic and hypersonic. The frequency range in thetest characterises different structural principles of the constructionof equipment and provides different temperature and time changesof the elasticity effects in metals. The frequency criterion has physi-cal meaning when the appropriate frequency range is determined inrelation to the dimensions of the specimens a and the velocity ofpropagation of the wave v in the material [103]. Consequently, therange of low frequencies is represented by the case in whichω = 2π

f << 2πv/a, the range of medium frequencies ω = 2πv/a, and

the range of high frequencies ω > > 2πv/a.Another important classification criterion for the excitation of vi-

brations in the measurements of internal friction and elasticitymoduli, reflecting the nature of propagation of the elastic waves inthe material, is the nature of elastic waves. These waves may bestanding or running. The standing waves are excited in a systemwhose part is the specimens (for example, a torsion pendulum) ordirectly in the specimens (for example, resonance waveguide). Run-ning waves are excited in the surface of the specimens (for exam-ple, Rayleigh waves) or in its volume (for example, the pulsedmethod). The form of the vibrations of the specimens is determinedby the method of excitation of vibrations. There are longitudinal,transverse and torsional vibrations. The nature of the vibrationsdetermines not only the special features of the structure of equip-ment but also the measured characteristics of the materials (for ex-ample, normal or shear Young modulus). According to the excita-tion method, there are various units (Fig. 4.2), i.e. mechanical, elec-tromagnetic, magnetostriction, piezoelectric, laser, etc. Consequently,the excitation methods can be combined on the basis of the control-ling criterion, i.e. external independent and self-excited (Fig. 4.1).

The classification of the circuit of the measurement unit is de-termined by two factors. It is the aim of measurements and themeasurements method. When measuring the elastic or inelastic char-acteristics of metals, there may be different tasks, e.g. determina-tion of the absolute values or relative changes of the characteristics,associated with the change of the state of the evaluated systems, orwith differences between the determined states. In the first case, thehigh accuracy of measurements is controlling, whereas high sensi-tivity is the controlling factor in the second case.

The classification of the measurement methods is more compli-cated [2, 103, 104]. The direct methods include the evaluation of the

136


FFFFFigigigigig. 4.2.. 4.2.. 4.2.. 4.2.. 4.2. Methods of excitation (V) and recording (P is the receiver) of internaldamping and the defect of the Young modulus (1): a) mechanical (2 – hammer,3 – microscope), b) capacitance (4 – electrode of the condenser), c) magnetostriction(5 – magnetostriction device), d) electromagnetic (6 – magnetic layer), e) piezoelectric(7 – piezocrystal), f) eddy current.

static and dynamic hysteresis loops. The method is based on themeasurement of stress and resultant strain σ in the process ofgradual loading ε and unloading of the specimen. The internal scat-ter of energy is characterised by the equation (2.43).

In repeated loading, it is possible to record simultaneously stress

FFFFFigigigigig. 4.3.. 4.3.. 4.3.. 4.3.. 4.3. Dynamic hysteresis loop: a) away from resonance (ω / ωr < 1), b) in

resonance (ω / ωr = 1).

137


and strain in the form of a hysteresis loop. Figure 4.3 shows exam-ples of a dynamic hysteresis loop away from resonance, Fig. 4.3a,and at resonance, Fig. 4.3b.

The energy measurement method is based on evaluating the dif-ference of the electrical or mechanical output of the excitation de-vice, required for maintaining the selected amplitude of the vibra-tions of the specimen. The relative scatter of energy in the materialof the evaluated specimen is determined from the equation

1 2 ,Z Z

fW

−Ψ = (4.1)

where Z1 is the total output of the excitation device, Z

2 is the out-

put required to overcome resistance in the excitation device with thereplacement of the output scattered in the circuit of the vibratingsystem, f is the frequency of the resultant vibrations, and W is thepotential energy of the deformed specimen, corresponding to theamplitude of the resultant vibrations.

The thermal method is based on the measurement of thermal en-ergy U generated in the specimen during cyclic deformation withfrequency f during time t. Energy ∆W scattered in the material dur-ing a single load cycle is determined by the equation

.U

Wt f

∆ = (4.2)

The method is integral and can be used in the case of the homoge-neous stress state of the specimen, without connected masses.

Indirect methods of internal friction measurements, Fig. 4.1, alsoinclude the method of damping vibrations and other methods. In thismethod, the specimen is the elastic part of the vibrating system.Internal friction is characterised by the logarithmic decrement ofvibrations δ.

Taking into account the decrease of the energy of the system dur-ing a vibration cycle, it is possible to determine the difference ofthe energies corresponding to the adjacent maxima A

t, A

t+∆t. We

introduce the concept of the logarithmic decrement of vibrations inthe form δ = Q–1. Consequently,

138


ln .t

t t

At

A +∆

δ = = β∆ (4.3)

The method of increasing vibrations eliminates the shortcomingof the previous method because the latter requires initial deforma-tion of the specimen with the maximum amplitude which may causechanges in the material and inaccuracies so that the evaluated stateof the specimen is not the initial state. The method of increasingvibrations, i.e. the increment of vibrations ν, utilises resonance andthe constant magnitude of the excitation force. The increment of thevibrations

2 .t t t t

t t t ts

A A A

A A A+∆

+∆

− ∆ν = =+ (4.4)

where Ats is the mean amplitude.

The increment and decrement of the vibrations, corresponding tothe amplitude A

ts, are connected by the equation [104] in the form

00 ,

ts

A

Aδ = δ − ν (4.5)

here δ0 is the decrement of the vibrations, corresponding to the

maximum amplitude at resonance A0. The increment of the vibra-

tions is determined from the recording of the vibrations of the in-creasing resonance vibrations and the values of δ

0, A

0 are deter-

mined from the results of processing the recording of damping of thevibrations. The resonance vibrations are excited in the self-excita-tion regime.

The method of the resonance curve determines the width of theresonant maximum or resonance valley in relation to the frequencydependence of the vibrations, at a constant excitation force (Fig.4.4). Internal friction is determined from the equation

1 10.7 0.5and ,3r r

Q Qf f

− −∆ω ∆ ω= = (4.6)

i.e. from the width of the resonance maximum at the height of 0.7

139


or 0.5 of the maximum, where fr is the internal frequency of vibra-

tions of the system without internal friction. The equations (4.6) arevalid for the determination of internal friction in the area in whichthe internal friction is independent of the strain amplitude. For othermeasurement ranges, the equations must be modified.

Internal friction can also be determined from the width of theresonance ‘well’ [104] using the equation

1 ,r

Q Kf

β−β

∆ω= (4.7)

here Kβ is the coefficient of proportionality. Equation (4.7) can beused for any dependence of Q–1 on the strain amplitude.

Internal friction can also be determined, using the method of theresonance curve, from the following equation [4,105]

1 31 ,db

r

Q K− ∆ω=ω (4.8)

where K1 is the proportionality factor which takes into account the

effective mass of the resonance system, ∆ω3db

is the three decibelldeviation of resonance frequency and ω

r is the resonance frequency

of the system.The phase shift method is based on the time delay between the

acting stress and resultant strain, denoted by angle ϕ. Consequently

FFFFFigigigigig. 4.4.. 4.4.. 4.4.. 4.4.. 4.4. Resonance peak of the frequency dependence of deviation µ0.

140


2

1 tan ,rf

ωΨ = 2π − ϕ

(4.9)

where ω is the frequency of the excitation force, fr is the internal

frequency of the system. Since ϕ ≈ 1, then

2 tan .Ψ = π ϕ (4.10)

For the linear system of the hysteresis type, it is possible to findthe relationship of the decrement of the vibrations with the phaseshift ϕ in the form

2 202

tan ,rr

ff

π δ = − ω ϕ (4.11)

where fr0

is the resonance or internal frequency of the vibrations ofthe system, taking into account the dependence of internal frictionon strain amplitude.

In the ultrasound method, a wave is passed through the specimen.The velocity of the wave and its damping in the material are exam-ined. The wave equation has the following form

0 cos ,x xA A e t−α = − ν

(4.12)

where α is the damping factor, A and A0 is the actual and initial

amplitude of the wave, x is the coordinate in the direction of wavepropagation. The damping factor α is determined from the scatterof energy of the sound wave at its length λ. The value ∆W/W islinked with α by the equation

( )21 ,x x

x

A AWe

W A− αλ+λ−∆ = = − (4.13)

and at low values of α and λ by the equation

141


2 4 .W

W

∆ πν≈ αλ = αω

(4.14)

Elasticity (Young) moduliThe elasticity moduli are measured using bar-shaped specimenswhose length is significantly greater than the transverse dimensions.In the case of low damping, we have

1 ,M −ν = ρ

where M is the characteristic of the Young modulus (E or G). Con-sequently

and ,l t

E Gν = ν =ρ ρ (4.15)

where v1 and v

t are the velocities of propagation of the longitudi-

nal and transverse wave.It is also possible to measure the internal frequency of longitu-

dinal (fl) or torsional (f

t) vibrations of the bar [106]. Consequently

2 2 2

21 ,

2 2l

n E n If

L SL

π µ= − ρ (4.16)

and

,2t

n Gf

L=

ρ

where n is the number of harmonic vibrations, L is the length of thebar, ρ is specific density, µ is Poisson’s number, S is the cross sec-tional area, and I is the moment of inertia of the specimen in rela-tion to the longitudinal axis (for the square cross section I = πS2/6, for the circular cross section with radius r I = πr2/2). The sec-ond term in the brackets in equation (4.16) is the correction coef-ficient according to Rayleigh, which takes into account the effect of

142


transverse vibrations on the frequency of longitudinal vibrations.The correction coefficient for the specimens with L/2r > 3 made ofa cubic lattice material is less than 1% and can be ignored.

In complicated systems with inertia masses (for example, a tor-sional pendulum with a moving weight), the moment of inertia canbe determined from the equations derived in Ref. 107, but in mostcases, it is recommended use the dependences G − f 2 with changesof temperature. Usually, measurements are taken at the internal orresonance frequency which correspond to the main harmonic vibra-tions of the specimen.

The main requirements in measurements of the elasticity moduliare described by the standards [14]. The recommended dynamicmethods (strain rates 103 –104 s–1) for the determination of the elas-ticity moduli are based on the measurements of the resonance fre-quency of induced vibrations of the specimen in the form of a barwith a constant cross section.

The resonance methods enable rapid and relatively accurate (rela-tive error 0.5–0.8%) determination of the Young modulus.

When using the pulsed methods, the elastic strain range is 106 –108 s–1. In these methods, it is necessary to determine the velocityof propagation of the elastic wave through a bar where the wave-length is small in comparison with the dimensions of the specimen.The elasticity moduli are calculated using equation (4.15). Thepulsed methods of measurements of the Young modulus are charac-terised by high accuracy because the measurement error is~0.1%. However, the extent of application of these methods is lim-ited, especially at high temperatures of the material.

The values of the elasticity moduli for many materials and alloysare presented in the literature (for example, Ref. 5,14,108).

Let us return to the diagram in Fig. 4.1. The sections for record-ing the vibrations differ, depending on their construction and themethods of obtaining information. For the most extensively usedexperimental machines, the methods of recording vibrations can bedivided into visual, mechanical, piezoelectric, electromagnetic, elec-trostatic and others. The methods of recording data from the meas-urement sections are discrete or analogue. The results of measure-ments can be recorded in a coordinate-recording device or on stripsof digital recording systems. After transformation, coding andprocessing, the results are entered into a computer for furtherprocessing and analysis.

According to the level and state of automation, the experimentalprocedures of measurements of internal friction and the defect of the

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Young modulus can be classified into four groups: non-automatedsystems, semi-automated systems, automatic systems, where theprocesses of measurement and recording data in the analogue or dig-ital form are automated, and fully automated systems, controlled bya computer, with computer processing of the results. The generalprinciples for the construction of the circuits of the automated sys-tems with continuous recording of internal friction and the defect ofYoung modulus are presented in Ref. 109.

The main requirements on the systems for measuring internalfriction and the defect of the Young modulus is to ensure high ac-curacy and sensitivity of equipment. The relative error of measure-ments of internal friction and the defect of the Young modulus inthe most effective systems is: for the torsional pendulum ±2.5%, forresonance bars ±0.3–0.1%, and for the pulsed methods ±10–0.01%.The construction of sensitive equipment and systems is based ontheir automation and resolution capacity. In the case of the auto-mated systems, the resolution power of the measurements of inter-nal friction is sufficiently high: for the torsional pendulum ±0.03%,for the resonance systems up to 10 %, and for the method with therunning wave up to 10 %.

When comparing characteristics, it is very important to ensurethe high resolution power of the systems for measuring internal fric-tion and the defect of the Young modulus [106].

The design of the devices and systems can differ greatly. The me-chanical section must fulfil special requirements (for example, thesections of connection of the specimen, the section of excitation ofvibrations, etc.). It is always necessary to optimise the test condi-tions in order to eliminate unnecessary scattering of the energy inthe measurement section. The introduction of measurements of in-ternal friction into laboratories in plants requires reliable systemsare reliable and fast measurements. In the case of scientific inves-tigations, special attachments can be used.

4.2 EXPERIMENTAL MEASUREMENTS AND EVALUATIONIn the last 20–30 years, the extent of application of the methods ofmeasuring internal friction and the defect of the Young modulus hasexpanded greatly. A large amount of experience has been accumu-lated in the application of devices and systems in solid-state phys-ics, in evaluation of damage and threshold state of materials (seeFig. 1.1).

We shall use the conventional division of the methods of meas-urement of internal friction and the defect of the Young modulus

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when using the frequency criterion for: infrasound (10–4–102 Hz),sonic (102–104 Hz), ultrasonic (105–108 Hz) and hypersonic (109 –1011 Hz). The conventional nature of the division also results fromthe region of sonic and low frequency oscillations but has a certainphysical substantiation [26].

4.2.1 Infrasound methodsThe methods of the static and dynamic hysteresis loops are used inexamination of the characteristic the internal friction and the defectof the Young modulus of materials at stresses close to or higher thanthe fatigue limit. The sensitivity and accuracy of this method isdetermined by the error of measurement of deformation and by theefficiency of the measuring devices (strain gauges, dynamometers).

The range of possible measurements of energy scattering inefficient test systems of the type Instron and Strainset is greatly lim-ited [110].

The principal diagram of the system for measurement of internalfriction in metals using strain gauges is shown in Fig. 4.5 [111]. Apair of sensors 4 is attached to the specimen, two other pairs ofsensors 5 and 9 are attached to the dynamometer. The signal fromthe sensors 4 is directly proportional to the deformation of the

FFFFFigigigigig. 4.5.. 4.5.. 4.5.. 4.5.. 4.5. Diagram of equipment for measuring internal friction using strain gauges:1) oscilloscope; 2, 12) frequency-selective filters; 3) amplifier; 4,5,9) strain gauges;6) switch; 7,8) resistors; 10) two-channel amplifier of strain gauge data; 11)compensator of the phase shift of the signal.

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specimen, passes through the strain gauge bridge, the amplifier andthe filter to the horizontal axis of the oscilloscope. The signal fromthe sensors 9, proportional to the stress, passes through the straingauge bridge, the reverse change of the phase shift and the filter tothe vertical axis of the oscilloscope. The beam of the electrons onthe screen of the oscilloscope describes the loop in the stress–straincoordinates. The application of accurate strain gauges (nonlinearity2.1 × 10–4 %) makes it possible to increase greatly the sensitivityof measurements. In evaluation of materials with Ψ ≥ 0.001 the er-ror does not exceed 10%, Efficient connection of the amplifiermakes it possible to subtract the elastic strain from the total strainof the specimen and, consequently, increase the sensitivity of thesystem [110].

This system is universal and is utilised in fatigue systems load-ing the specimen by pull–push, bending or torsional loading. Athigher loading frequencies, the method of the dynamic hysteresisloop is used for recording the diagrams of cyclic deformation [111].

The torsional pendulum is used in the low-frequency range andis the most efficient method of measurement of the internal frictiondependent on the strain amplitude, and in measurements of the in-ternal friction dependent on temperature. The shear Young modulusis determined by the square of frequency f 2 of damped vibrations(G ~ M*f, where M* is the moment of inertia of the system). Thereare three methods of using the torsional pendulum: direct, reversed,and combined (Fig. 4.6).

A suitable example of the design of measuring heads in the

FFFFFigigigigig. 4.6.. 4.6.. 4.6.. 4.6.. 4.6. Torsional pendulum systems: a) direct, b) reversed, c) combined (1 –specimen, 2 – rod of the moving clamping jaw, 3) inertia mass for testing thespecimen, 4 – clamping jaws, 5 – rod of the fixed jaw, 6 – suspension, 7 –counterweight).

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arrangement for the measurements of the damping of vibrations andinternal vibrations is the RKM-TPI torsion pendulum. In the firstvariant [112], internal friction and the square of frequency weremeasured visually or semiautomatically using a counter of vibra-tions. The second variant [113] is based on the automatic record-ing of the results of measurements, with continuous drawing of thecurve of the temperature dependence of internal friction and thedependence of internal friction on the strain amplitude with discreterecording of the data. The system makes it possible to use speci-mens with a length of 50–120 mm of circular or right-angled crosssection. The frequency range is 0.2–100 Hz, the temperature rangeup to 2500°C. The strain amplitude range is from 10–6 to 10–3, andthe internal friction background is smaller than 10–4 units of inter-nal friction. Measurements can be taken in a unidirectional magneticfield with an intensity of 4×104 A×m–1. The RKM-TPI system canbe fitted with clamping heads with the section for deformation of thespecimen during measurements [114] by superplastic loading [115],and by an attachment for the effect of corrosive media.

Other systems based on the torsional pendulum of different de-sign are also available [116,117]. Improvement of design is associ-ated with the widening of the working characteristics of systems,such as the range of strain amplitude and frequency of vibrations,temperature [118,119], modification of the measuring heads [120,121], increase of sensitivity for the application of strain (for exam-ple, from 10–11) [122].

Some systems based on the torsional pendulum also make it pos-sible to carry out tests with bending vibrations, in addition to tor-sional vibrations [123]. It is important to note low-frequency com-bined pendulum systems with the original design of the suspension[123].

When using the pendulum, excitation is carried out by the mu-tual effect of the magnetic field of coils distributed on a suspensionsystem, with a permanent magnet [124]. To prevent deformation ofthe specimen as a result of the weight of the rest of the system, acontainer with a damping fluid is used. However, systems not us-ing fluid dampers are also available [124]. Centring of the vibrat-ing system is improved by a system of rollers [126] or using pre-cision dimples [125].

Good results have been obtained with the system in which theferromagnetic cylinder of the suspension is centered in holes of twotoroidal electromagnets with the core in the coils thus forming anorthogonal source of alternating current [127]. The suspension also

147


contains a square magnet which interacts with four electromagnetspowered by direct current. If the voltage to the electromagnet ischanged, the moment of inertia and frequency of vibrations of thesystem change. Compensation of the mass of the suspension systemand the formation of a certain mechanical stress in the specimen canbe achieved using lever or spring systems [128].

In measurements of the characteristics of brittle material andglass, it is necessary to ensure rigid contact of the specimen withthe clamping heads. In one of the interesting solutions, the lowerclamping jaw has the form of a vessel and the upper has the formof a seed (the specimen is pulled from the alloy and this is followedby the measurements), or contact is made by means of eutectics witha low melting point [129].

In the basic models of the torsional pendulum, the change of fre-quencies achieved by moving a weight on the one-sided lever of theinertia system, or by changing the number of weights. Improvementhave been made in the automatic displacement of the weights alonga two-sided lever, by regulating the number of discs in the inertiamass on the vertical shafts, or by changing the number of weights–permanent magnets on a shafts as a result of their contact with elec-tric coils. Another solution is the mutually effect of the magneticfields of the inertia mass in the form of a permanent magnetic witha stationary electric coil [127]. Expansion of the frequency rangeis possible in the case of the torsional pendulum by using the re-gime of forced vibrations [113].

The automation of the low-frequency system for measurements ofinternal friction and the defect of the Young modulus is based onthe application of photocells. The diagram of recording vibrationsfor this principle is shown in Fig. 4.7a. The light beam from themirror, secured to the inertia system, falls on photo resistances andthe signal from the passes, after amplification, to a recording de-vice. The distribution of the photoresistances in a specific sequencemakes it possible to obtain recordings in the logarithmic coordi-nates. Of considerable importance for automation was the conceptof determination of the decrement of vibrations from the decreaseof the speed of vibrations (time periods) during the passage of thebeam (Fig. 4.7b [130] which greatly decrease the duration of a sin-gle measurement in the strain amplitude range 10–4 – 10–2 and thefrequency range 0.2 – 20 Hz. The method of measurement of theduration of passage of the light trace was gradually greatly im-proved as a result of the application of photocells in the goniometrichead [116].

148


FFFFFigigigigig. 4.7. . 4.7. . 4.7. . 4.7. . 4.7. The block diagram of equipment: a) (1 – mirror, 2 – light beam, 3 –the light source, 4 – photoresistance, 5 – amplifier, 6,7 – the recording devicewith attachments); b) (1 – mirror, 2 – light source, 3 – light beam, 4 – grating,5 – photoamplifier, 6 – the device for measuring the transfer of vibrations, 7 –the device for measuring time during the passage of vibrations); c) (1 – mirror,2 – the light source, 3 – the input Schmidt trigger, 4 – the frequency divider, 5– chronometer, 6 – printer, 7- photocells, 8 – the output Schmidt trigger); d) (1– pendulum, 2 – mirror, 3 – the light source, 4 – shut-off attachment, 5 – thesource of the signal for arresting the cycle counter, 6 – the unit for stoppingthe counting of cycles, 7 – the counter of pulses, 8 – the unit for the doublenumber of the pulses, 9 – indicating device, 10-the device for selecting the numberof pulses, 11 – the unit for the double number of the pulses, 12 – photodiodes,13 – triggering photographic equipment, 14 – the source of the starting signal,15 – the unit for starting the counter, 16 – printer).

Accurate measurements of the time periods in the frequency rangeof operation of the pendulum from 0.3 to 3 Hz are taken using sili-con photodiodes [117]. The central part of the illuminating scalecontains photodiodes distributed at a distance of 10 mm from thecentre of the scale [Fig. 4.7c].

During the gradual passage of the beam through the photodiodes,Schmidt triggers convert the electric signal to square pulses whichare transferred to a frequency divider and then to a timer. The datafor determining the logarithmic decrement of friction

0

1ln nt

n t

σ =

(4.17)

149


are automatically recorded in a printer. In equation (4.17), n is thenumber of vibrations, t

0 is the duration of passage of the light beam

through a constant base at the start of counting, tn is the identical

time after n vibrations.Another modification is shown in Fig. 4.7d. The light beam, re-

flected from the mirror, falls on the triggering photograph equip-ment. The signals from the starting and stopping mechanism arechange to square pulses which are then transferred to the indicat-ing unit. The equipment for starting and stopping the counter is con-nected to a chronometer. The pulses are then transferred to the con-trol system, consisting of the selection device for the number of vi-brations (n = 1, 2, 4, 8,...) and from a double counter forming thesignal for stopping the counter in accordance with the selected pro-gramme. The device for stopping the counter issues a signal forstopping the chronometer which records the duration of passage ofthe light beam from the triggering to stopping device after the givennumber of vibrations of the system. Equipment is also fitted witha device for the generation of high-power alternating voltage. Themodified form makes it possible to determine automatically thedependence of internal friction on temperature, strain amplitude ortime, in the form of tables or graphical dependences [131,132].

When applying transverse or bending vibrations, it is possible touse specimens of simple shape and different dimensions, a simpleexcitation method and a simple method of recording vibrations, awide frequency range and also a wide range of external loading ofthe specimens. As in the case of the torsional pendulum, the bend-ing vibrations do not load the entire cross section of the specimenand this impairs the conditions of obtaining the actual internal fric-tion characteristics.

The systems with a cantilever specimen are also used [124, 133].Equipment with or without connected mass is available. The secondmethod is used when it is required to generate various stresses inthe specimen. The principle of this method is shown in Fig. 4.8. Thevibrations are excited by the electromagnetic method. Recordingfrom contact-free shadow observation of the vibrations is carried outusing a photographic sensor [134]. Four forms of vibrations of thespecimen are used. In the case of non-ferromagnetic materials, thinsheets of a ferromagnetic material are secured to the specimen. Thelogarithmic decrement of damping is determined by the method ofdamped vibrations or resonance vibrations.

The systems suspended at the nodes of vibrations are used for themaximum equalisation of the stress along the specimen under the

150


FFFFFigigigigig. 4.8.. 4.8.. 4.8.. 4.8.. 4.8. Block diagram of equipment (a) and the shape of pulsed of the cantileverspecimen (b): 1) amplifier, 2) relay, 3) sound generator, 4) oscilloscope, 5) screw,6) measuring discs, 7) base plate, 8) wedge, 9) variable mass, 10) indicatingdevice, 11) electromagnet, 12) specimen, 13) microscope, 14) spring.

FFFFFigigigigig. 4.9.. 4.9.. 4.9.. 4.9.. 4.9. Diagram of the oscillating system and optical recording system. 1) spring,2) inertia mass, 3) prismatic specimen, 4) mirror, 5) light source, 6) recorder.

effect of bending vibrations. On the basis of the principle shown inFig. 4.9, the Institute of Strength of Materials of the Academy ofSciences of the Ukraine has developed a set of systems [104]. Theprismatic specimen 3 is suspended on thin springs 1. The inertiamass 2 is connected to the specimen. The mass can be connectedusing threads, clamps, wedges, etc. Vibration systems consisting oftwo specimens and inertia mass are also available [135].

151


A system with a U-shaped specimen (‘tuning fork’) utilises themagnetoelectric excitation to resonance at the natural frequency orstationary excitation [136]. The frequency of the exciting vibrationsis changed automatically and the signal, directly proportional to theamplitude of the vibrations, is recorded. The second method is basedon the maintenance of a specific frequency, closed to the resonancefrequency, with the change of the temperature of the specimen [104].

The results show that the method of measuring the suppliedpower is most promising in continuous measurements of internalfriction with the change of temperature, amplitude or loading time[113].

The methods based on longitudinal infrasound vibrations are usedmostly in evaluation of materials with high elasticity [26].

For special purposes, it is possible to use an attachment withcryogenic or higher temperature for the examination of the charac-teristics under static or cyclic deformation, under the effect of X-ray or ion radiation, for testing whisker crystals, thin foils andhighly brittle materials [2,26,104,107]. Experimental results showthat in addition to the measurements of internal friction and Youngmodulus, it is also possible to evaluate other physical properties[137].

4.2.2 Sonic and ultrasound methodsThe design and application of equipment for this frequency rangehave been described in a number of studies [2,26,106,138]. Thedesign of equipment does not contain any special inertia systemsconnected with the specimen. The main groups of the measuringequipment are:

–equipment with the specimen secured at nodes by the contactmethod [139],

–equipment in which specimens are secured at nodes by a con-tactless method [147],

–equipment with one specimen secured at the node of the vibra-tions [141],

–cantilever-type equipment [142].When the specimens are secured by the contact method, it is not

necessary to use any wires or supports connecting the specimenswith the sensor and receiver of the vibrations in the vicinity of thenode of the vibrations. The receiver and the sensor are far awayfrom the specimen and, consequently, they can be protected from theeffect of temperature, magnetic or other energy field. It should beremembered that additional energy scattering takes place in the area

152


of contact of the specimen with the filament. For example, when us-ing filaments 0.1 mm thick and in the case of the minimum distanceto the nodes of the vibrations, the scattering of energy in the con-tact zone is less than ≤ 3% when the level of the background of theinternal friction of equipment is Q–1 < 10–5. This systematic erroris almost completely eliminated in the systems without contact wherethe correction of the filaments or supports to the specimen is car-ried out exactly at the nodes of the vibrations. For example, in thepresence of bending vibrations, the contact is realised in the formof needles. The displacement of the sensors of excitation or thechange of the method of securing the specimens make it possible toproduce not only basic vibrations but also higher harmonic vibra-tions. A shortcoming of these systems is the low level of the strainamplitude (ε ≤ 10–5) and in the case of contactless connection alsothe proximity of the measuring unit to the zone of external influ-ences. In both the contact and contactless variant of equipment withsingle connection at the node of the vibrations it is possible to ex-cite several types of vibrations. All these systems can be fully au-tomated.

Equipment for cantilever securing of the specimen is used inevaluation of thin sheets, foils, filaments and wires.

A special position is occupied by the vibrators consisting ofstepped and exponential concentrators because they make it possi-ble to apply a wide range of strain amplitudes [105,143]. The meas-urements of vibrations and of their changes using piezoceramicsheets connected by long waveguides with the specimens are carriedout to evaluate the relative change of internal friction becauseenergy is lost in contact areas.

In electrostatic excitation, one of the ends of the bar is subjectedto the effect of a periodic force formed as a result of electrostaticattraction between the conducting end of the bar and the stationaryelectrode (Fig. 4.10). In excitation of the bending vibrations theelectrodes are situated in the area with the maximum amplitude ofthe vibrations. The electrodynamic method of excitation is associ-ated with the mutual effect of the conducting metal strip, depositedon the surface of a non-conducting specimen, with the magneticfield. This results in the formation of an alternating force causingvibrations in the specimen. These methods are suitable for materi-als with low internal friction in a wide frequency range.

The effect of the electrostatic force between capacitor plates,where one of the plates is represented by the part of the surface ofthe specimen, makes it possible to excite longitudinal, bending or

153


torsional vibrations in the specimen. In this excitation method, thebond between the mechanical and electrical system is weak. Conse-quently, the effect of the electrical circuit can be ignored. To elimi-nate aerodynamic losses, the working space is evacuated. Elasticvibrations in the specimen can also form as a result of Coulombforces [144] between the face of the specimen and the electrodewhich is parallel to the face of the specimen. The second electrode(the receiver on the other end of the specimen) changes the mechani-cal vibrations to an electrical signal. The modified method requiresonly one electrode [145]. The frequency modulation of the loadingand measuring signal is utilised in this case.

Equipment with electrostatic excitation and reception of informa-tion on vibrations is being used at the A.F. Joffe Physical-Techni-cal Institute [146].

One of the first automatic systems of this type was built at theAcademy of Sciences of Georgia in the former USSR [147]. Thisequipment makes it possible to carry out continuous measurementof the temperature dependence of internal friction. The equipmentis based on the method of supplied power of the method of damp-ing of vibrations.

The electrostatic method of excitation ensures good acoustic con-tact between the specimen and the electromechanical converter. Thisimpairs the measurement results when using the piezovibrator or thepulsed method, especially at high temperatures. Equipment of thistype has been used for a large number of studies of the propertiesof semiconductors, metallic and dielectric materials [106].

Various methods are used in electromagnetic citation. Thesemethods are based on the excitation of longitudinal, bending or tor-sional vibrations using conventional electromagnetic converters (Fig.4.11). In the first variant, Fig. 4.11a, longitudinal vibrations areexcited using thin sheets connected to the ends of the specimen.Disadvantages of the method are the displacement of the discs in the

FFFFFigigigigig. 4.10.. 4.10.. 4.10.. 4.10.. 4.10. Diagram of electrostatic excitation of longitudinal (a) and bending(b) vibrations in specimen 3: 1) power supply from the oscillator; 2) recordedsignal travels to the receiver, recording and printing.

154


fixing area and the fact that direct heating of the specimens is notpossible. Therefore, a new system was developed, Fig. 4.11b, wherethin wire suspension is used for inducing and receiving electromag-netic vibrations. On the one side, the suspension is connected to themembrane of the electromagnetic converters and on the other sideat the nodes of the vibrations (or the ends of the specimen). Thesuspension is outside the furnace and the entire electrical circuit isat room temperature [139].

The circuit for automatic equipment with electromagnetic con-verters for the continuous recording of internal friction and Youngmodulus at normal or torsional vibrations is shown in Fig. 4.12[139]. The parameters of this variant are as follows: frequencyrange 0.5–3 kHz, the strain amplitude range from 10–7 to 10–4, tem-perature range from –196 to 600°C, vacuum 0.13 N⋅m–2, the back-ground of equipment is ≤10–5 unit of internal friction, the sensitiv-ity of equipment to the relative change of internal friction is10–4 %.

On the basis of the circuit with the electromagnetic excitation ofvibrations and suspension of the specimens on filaments, S.A.Golovin and A.A. Morozyuk (Tula Polytechnic Institute) constructedsystems based on the differential measurement method. This systemmakes it possible to record differences of the signals from the ref-erence specimen and the measured specimen. This greatly improvedthe accuracy of the method. Equipment of company Elastomat isalso used widely. These systems include electromagnetic and piezo-electric exciters in different temperature ranges of measurement ofinternal friction and Young modulus (–190°C to 200°C, or 20 to

FFFFFigigigigig. 4.11.. 4.11.. 4.11.. 4.11.. 4.11. Equipment with electromagnetic excitation of oscillations. a: 1 – specimen,2 – ferromagnetic plates, 3 – emitter, 4 – receiver, 5 – oscillator, 6 – amplifierand oscilloscope; b: 1 – specimen, 2 – elastic suspension, 3 – emitter, 4 – receiver,5 – oscillator, 6 – amplifier and oscilloscope, 7 – furnace or cryogenic unit.

155


FFFFFigigigigig. 4.12.. 4.12.. 4.12.. 4.12.. 4.12. Block diagram of automatic equipment with electromagnetic excitation:1 – heating (cooling) system, 2) specimen, 3) sensor of vibrations, 4) amplifier,5 – filter, 6 – potentiometer, 7 – power amplifier, 8 – exciter of vibrations, 9 –comparison unit, 10 – servoamplifier, 11 – servomotor, 12 – source of stabilisedvoltage, 13 – potentiometer, 14 – recording device, 15 – thermocouple, 16 –recording device, 17 – device for recording data from the potentiometer, 18 –voltmeter, 19 – oscilloscope, 20 – frequency meter, 21 – discriminator, 22 –pulse counter, 23 – oscillator, 24 – amplitude limiter – frequency-dependent link– sensor unit.

1000°C, for the 1.015 model), using longitudinal, transverse andtorsional vibrations on the basic or higher harmonic vibrations.

Automatic systems with the excitation of internal vibrations weredescribed in Ref. 148 and 149.

A large amount of data on the properties of metals and structuralmaterials have been obtained on the systems with electromagneticexcitation of vibrations [2,26,104].

A method of piezoelectric excitation of vibrations in specimensfor determination of the elastic characteristics of materials was pro-posed by S.L. Quimbi. At present, this method is used in the vibra-tor proposed by J. Marx [150]. A specimen whose section is thesame as that of the piezoceramic specimen is connected to one ofthe ends of the piezoceramic specimen (Fig. 4.13). The piezoceramicspecimen is secured at the node of elastic displacements formedafter supplying an alternating electric field to the electrodes. Meas-urement of the resonant frequency, the quality of the piezoceramicmaterial and the piezoceramic–specimen system makes it possible tocalculate the internal friction characteristics and the Young modu-lus. At high temperatures, it is recommended to use a triple vibra-

156


tor consisting of piezoceramics, a ceramic bar, and the specimen.Various types of a compound piezoceramic vibrator are possible: 1)piezoceramics with a specimen of arbitrary length; 2) piezoceramicsloaded simultaneously with two specimens of any or identical length;3) piezoceramics with the specimen whose length corresponds to theresonance frequency of the piezoceramics. The third case is mostuseful because the nodal plane of the vibrator is identical with thenodal plane of the piezoceramics.

The damping decrement of the specimen δ0 is determined by the

decrement and mass m of the elements of the compound vibrator[151]. For a double vibrator

( )10 2 2 1

0

,m

mδ = δ + δ − δ (4.18)

for a triple vibrator

( ) ( )10 3 3 1 3 1

0 0

,emm

m mδ = δ + δ − δ + δ − δ (4.19)

where me is the mass of the reference bar, δ

0 and δ

1 or m

0 and m

1

are the decrement of vibrations or the mass of the specimen and thepiezoceramics, δ

2 is the decrement of the vibrations of the

piezoceramics–bar–specimen system. For the natural frequency ofthe specimen in the vibrator

( )10 2 2 1

0

,m

f f f fm

= + − (4.20)

FFFFFigigigigig. 4.13. 4.13. 4.13. 4.13. 4.13. Diagram of piezoelectric excitation: 1) piezocrystal, 2) specimen, 3)oscillator, 4,5) voltmetres, 6,7) resistances.

157


and for a triple vibrator

( ) ( )10 2 3 1 3

0 0

,ee

mmf f f f f f

m m= + − + − (4.21)

here fe is the resonance frequency of the reference bar.

Crystals of alkali halides have been used successfully in theevaluation of the effect of the amplitude vibrations on internal fric-tion and the defect of the Young modulus [152]. A compound vibra-tor, consisting of the specimen, a duralumin bar and piezoceramics,is placed in a cryostat. The equipment has the following character-istics: temperature range from 4.2 to 300 K, working frequency 50–125 kHz, strain amplitude from 10–7 to 10–4, with the error of meas-urement of the decrement of vibrations of approximately 10%.

A suitable example of the development of acoustic systems withpiezoceramics is the automatic system developed at the Joffe Physi-cal-Technical Institute of the Russian Academy of Sciences. Theproblem was solved on two planes: analogue automation [153], andthe variant using a computer and digital equipment [154]. Examplesof the application of this type of equipment for polycrystalline mag-nesium have been published in Ref. 106.

Step vibrators with magnetostriction or piezoceramic exciters(Fig. 4.14) can be utilised in the quantification of internal friction.

A fully automatic system of measurement of internal friction and

FFFFFigigigigig. 4.14. . 4.14. . 4.14. . 4.14. . 4.14. Vibrators with a ceramic converter and stepped concentrator (a) andwith an exponential concentrator (b): 1) supply of electric signal, 2) earthedelectrode, 3) sensing electrode (dimensions in mm).

158


the defect of the Young modulus was developed at the TechnicalUniversity at Zilina (Slovakia) under the supervision of the authorof the book [505]. The measurement of internal friction Q–1 by themethod of determination of the quality of the resonance system andthe defect of the modulus of elasticity ∆E/E by the method of de-termination of the displacement of the resonance frequency of thesystem is carried out in a resonant system consisting of apiezoceramic converter 1, a titanium attachment 2 and the specimen3 (Fig. 4.15). All parts of the resonant system (1, 2, 3) fulfil theresonance condition according to which their length is half thewavelength of damping in the given material at resonance frequencyf

1.

The internal friction of the entire system Q–11,2,3

or of the systemwithout the specimen Q–1

1,2, is determined by the equation

1 31,2,3

1,2,3

,r dB

r

fQ

f− ∆=

or

1 31,2

1,2

,r dB

r

fQ

f− ∆=

(4.22)

where ∆fr3dB

is the 3 dB deviation of the resonance peak. If voltage

FFFFFigigigigig. 4.15.. 4.15.. 4.15.. 4.15.. 4.15. Diagram of equipment for automatic measurement of internal frictionand defect of the Young modulus (Technical University, Zilina).

159


Un with frequency f

r is supplied to the piezocrystal, a standing wave

forms in the resonant system and the piezocrystal generates, on dif-ferent planes, the voltage recorded as U

s. This voltage is directly

proportional to the strain amplitude in the central part of the speci-men in accordance with the equation

1 ,sk Uε = (4.23)

where k1 is a constant for the given system with the specimen.

The internal friction of the entire resonant system is also deter-mined by the equation

1,2,311,2,3 ,n

s r

U KQ

U f− = (4.24)

where K1,2,3

is the characteristic of the system, determined from theequation

1,2,3 3 .sr dB

n

UK f

U= ∆ (4.25)

The quantity ∆fr3dB

is determined at Us/√2 by increasing the fre-

quency of supplied current gradually until we record Us max

at ft.

Subsequently, the system is detuned to value frx

at which Us =

Us max

/√2. The increase of frequency above fr also results in a de-

crease of Us max

to the value Us max

/√2 at which the frequency of thesystem is f

ry. Consequently

3 .r dB ry rxf f f= − (4.26)

The internal friction of the specimen Q–13 is determined from the

equation

1 1 13 2 1,2,3 3 1,2 ,Q k Q k Q− − −= − (4.27)

where k2 and k

3 are the constants calculated from the dimensions,

shape and properties of the connected specimens and their material(156).

160


The defect of the Young modulus of the specimen ∆E/E is deter-mined from the equation

1,2,3 1

3

,r r

r

m f fE

E m f

−∆ = (4.28)

where m1,2,3

and m1,2

at the calculated effective masses of the entiresystem without the specimen, f

r1 is the resonance frequency of the

entire system, determined using the value of ε differing from thatused in the determination of f

r.

For the currently available range of ε from 5×10–6 to 7×10–4 itis sufficient to measure at least 30 points at different values of εfrom the given range. For every point it is necessary to determineU

n, U

s, f

r. Special attention must be given to at least tenfold rep-

etition of the measurement for the determination of ∆fr3dB

(equation4.26), because this value is included in the calculations of Q–1

1,2,3. In

the manual setting of the values of Un in the range from 100 mV

to 1000 mV on a dial-type electronic voltmeter, with the manual set-ting of the frequency of the resonance peak with the accuracy of 1Hz in the frequency meter, and with subjective determination of thevalue of U

s on a dial-type electronic millivoltmeter with the manual

search for the value of frx

of frt with the accuracy of ±1 Hz, sub-

jective determination of Us at U

s max/√2, taking into account the

manual inputting of the initial data, interoperational values and theoutput values, we obtain that at f

r = 22.5 kHz and Q

3– 1 = 10–4 the

measurement error is approximately 1% and at ∆E/E = 10–3 it is~0.1%. The measurement time of every experimental point is nolonger than 60 seconds, i.e. for a set of 60 measurements, togetherwith the adjustment time of the measurements, it is no less than3600 s, with high concentration of the operator.

Puskar, Palcek and Houba [155] constructed equipment in whichthese disadvantages are removed. The decrease of the measurementtime and improvement of the accuracy of measurement of internalfriction and the defect of the Young modulus are based on the com-plete automation of the selection of power supply, measurement,evaluation and graphical expression of the results of measurementusing a microcomputer. The resonant system 1 + 2 + 3, or 1 + 2+ 7 + 3 (Fig. 4.15) is powered by the oscillator 11, controlled bya programmable generator with the attenuator 10, activated usingthe interface 7 from the microcomputer 4. The magnitude of U

n is

161


monitored using the multi-meter 8. The data on the magnitude of Un

are entered through the interface 7 into the operating memory of thecomputer 4. When a standing wave forms in the resonant system theresonance frequency f

t is measured and the signal is transferred to

through the interface 7 to the operating memory of the microcom-puter 4. Subsequently, voltage U

s is generated in the piezoceramic

material of the converter 1, and the magnitude of this voltage ismeasured with the multimeter 9 and the signal travels through theinterface 7 to the operating memory of the microcomputer 4. Themeasurement algorithm is presented in the flow chart in Fig. 4.16.After completing measurements at a selected point, the microcom-puter 4 issues a signal which, after passing through the interface 7,defines the next measurement point for the programmable generatorwith the alternator 10. The operating memory of the microcomputer4 records data of all measurements. On the basis of the programme,

FFFFFigigigigig. 4.16.. 4.16.. 4.16.. 4.16.. 4.16. Flow chart showing measurement and evaluation of internal frictionand the defect of the Young modulus in equipment VTP-A (Technical University,Zilina).

++

Density of specimen ?: HSize of specimen?:L1,L2,D1,D2Speed of sound?: vFrequency range?:t

1,t

2

εmax

?: εmax

Initial value U?:U0

Printing of results Graph

Interpolation U(i)Calcul. Us max, fr, frx,∆f3dB

Calcul. ε, Q–1, k2, k3, m1, m2

Interpolation U(i)Calcul. U

s max, ε, f

r, Q–1, dE/E

162


it transfers the data to the disk storage 5 and the signal for print-ing the selected data and the graphical dependences in the printer6. For the case of a sandwich converter 13, Fig. 4.15, consisting ofthe piezoceramic plates 14, adhesion bonded to the metallic sections15, the algorithm, Fig. 4.16, is the same. Using the extensionwaveguide 17 with the length nλ/2, where n is an integer and λ isthe wavelength of the waves in the material of the waveguide, speci-men 3 can be placed in a furnace or a cryogenic unit with program-mable temperature change, in a unidirectional magnetic field withthe programmed change of the intensity, in vacuum which can bevaried using a programme, or in the testing space of a nuclear re-actor where the intensity of neutron radiation can be varied in ac-cordance with a programme, or it can be placed in other energyfields which are either constant or can be varied in accordance witha programme. The data on the parameters in the unit 18 are proc-essed by the unit for evaluating the defect 19 and, subsequently, thedata are transferred through the interface 7 into the microcomputer4 which then controls the parameters in unit 18.

The system reduces the measurement time of a single point to4 s which means that 30 points can be measured within 600 s, in-cluding the setting time. The system also increases the accuracy ofmeasurement of U

n, U

s, f

r, ∆f

r3dB, f

r1 at a frequency of 22.5 kHz so

that at Q3– 1

= 10–4 the measurement error is approximately 10 per-

cent. The system described in the invention eliminates the subjec-tive error of measurement caused by the operator. As a result ofeliminating manual calculations and drawing graphical dependencesin the selected coordinates, the time required for obtaining reproduc-ible data on Q

3– 1

and ∆E/E is reduced further. The application of the

programmable generator with the alternator 10 and digitalmultimeters 8 and 9 makes it possible to set more sensitively thevalue of U

n and measure appropriate values, as well as select

Un max

= 1500 V thus expanding the strain amplitude range from1.2×10–7 to 1×10–3. The programmed control of the measurementsenables data to be obtained on the time dependences of Q

3– 1

and

∆E/E from 4 s to the selected time.Magnetostriction vibrators for a frequency of 3–20 kHz using

longitudinal vibrations have also been developed at the Institute ofStrength of Materials of the Academy of Sciences of Ukraine [104,157].

The evaluation of internal friction by the calorimetric method ata frequency of ~ 20 kHz, Fig. 4.17a, can also be carried out usingthe magnetostriction vibrator (1). The magnetostriction phenomenon

163


FFFFFigigigigig. 4.17.. 4.17.. 4.17.. 4.17.. 4.17. Diagram of calorimetric (a) and resonance (b) methods: a: 1) magnetostriction vibrator, 2) concentrator, 3) specimen, 4) calorimeter filling, 5) thermocouple,6) casing of the calorimeter, 7) microscope, 8) mixer, 9) rubber membrane, 10)base plate, 11) collar, 12) cooling of vibrator; b: 1) stabilized power source, 2)capacitance sensor, 3) detector, 4) electronic potentiometer, 5) frequency meter,6) oscillator, 7) amplifier, 9) vibrator, 10) concentrator, 11) specimen.

is based on the observation which shows that the magnetostrictionmaterial (Fe, Co, Ni, and alloys of these metals) changes its dimen-sions with the change of the magnetic field in which it is placed.Concentration of the energy in a small cross section is carried outusing the concentrator 2 of the conical or stepped type whichchanges the magnetic field. The length of the vibrator and of theconcentrator corresponds to half the wavelength of the waves in thematerial of the vibrator and the concentrator at a frequency of f

r.

Specimen 3 is attached to the concentrators; the natural frequencyof the specimen correspond to f

r. The areas where the mechanical

joints of the system are located are not loaded with mechanicalstress. The electric signal from the generator, connected to ampli-fier, is supplied to the winding of the vibrator. The latter must becooled. Consequently, the vibrator is placed in the vessel 12 withrunning water.

The heat dissipated in the material can be measured using thecalorimeter 4 filled with water, containing a thermometer and amixer 8. The relative scattered energy is calculated from the equa-tion

164


20

4,

E m T

V ft

∆Ψ =σ (4.29)

where E is the Young modulus of the material of the specimen, mis the mass of water in the calorimeter, ∆T is the change of thewater temperature in the calorimeter during time t of cycling load-ing, V is the volume of the working section of the specimen, σ

0 is

the stress amplitude in the specimen, and f is the frequency of thevibrations.

The method has been used in evaluation of the fatigue propertiesof structural steels and alloys.

Recording of the resonance curve, Fig. 4.17b, for a cyclically de-formed specimen is based on the excitation of longitudinal vibrationsof the system which contains the specimen 11, concentrator 10 andvibrator 9. The excitation of vibrations with a frequency of 3–10kHz is determined by the coil of the magnetostriction vibrator re-ceiving alternating current from the amplifier 7 and direct currentfor the magnetisation of the vibrator. The system is powered by anultrasound oscillator operating in the independent excitation regime.The amplitude of vibrations of the specimen can be measured usingthe electronic potentiometer 4. The amplitude of the vibrations ofthe specimen is recorded by the capacitance sensor 2 powered by thestabilised source 1. The alternating voltage, whose magnitude isproportional to the amplitude of vibrations of the end of the speci-men, is transferred from the sensor to the amplifier and the detec-tor 3. The voltage is then further amplified and direct voltage ismeasured. The voltage is then transferred to the potentiometer.When the movement of paper in the recording device corresponds tothe frequency and the movement of the pen to the amplitude of thevibrations, we obtain a recording of the resonance curve.

Puskar [158] constructed and tested equipment for the visualisa-tion of hysteresis loops at kilohertz loading frequencies. Conse-quently, it was possible to evaluate the cyclic strain curves of dif-ferent materials.

The pulsed excitation of the vibrations is based on the movingwave generated in the converter, Fig. 4.18. The pulse from the high-frequency generator with the frequency corresponding to the reso-nance frequency of the plate of the transducer based on the principleof the inversed piezoelectric phenomenon then induces vibrations inthe ceramics or a crystal. The vibrations are transferred through athin transition layer to the specimen in the direction normal to its

165


surface. A moving ultrasound wave forms. After interrupting theelectric signal passing into the transducer, the ultrasound waves arereflected many times between the walls of the specimen up to com-plete attenuation of the vibrations. The time to complete attenuationis determined using the transducer, Fig. 4.18, which now operatesas a receiver.

In the transmission method, one side of the specimen contains thetransducer operating as an emitter and the other side the transduceracting as a receiver. The amplitude and analysis of echo signals onthe screen of the oscilloscope which is synchronised with the fre-quency of the pulses makes it possible to determine the internal fric-tion. When the sensitivity of the oscilloscope is sufficiently high, itis possible to determine the delay of various echo signals and, con-sequently, the speed of propagation of elastic or waves in the ma-terial can be determined. The relative decrease of the amplitude ofthe reflected signals enables the coefficient of absorption of ultra-sound in the evaluated specimen to be determined.

The method of connection of the reflected echo signals is basedon the phenomenon according to which a group of reflected signalsenters an oscilloscope synchronised with the frequency of the emit-ted signals from the oscillator. When the repetition frequency τ

s of

the synchronised pulses is matched with the measurement time τ theecho signals, i.e. at τ

s = τ∆n, where ∆n is the difference between the

number of reflections of the combined echo signals, the screen ofthe oscilloscope shows the image of superimposed pulses of differentorders n . The value of τ

s can be determined with high accuracy

using and electronic frequency meter. The main inaccuracy of themethod is associated with the visual placement of the signals oneach other on the screen of the oscilloscope which makes it possi-ble to record the change of acoustic delay from 0.01 to 0.05 µs. Theaccuracy of the method is 0.2–0.5%.

FFFFFigigigigig. 4.18. . 4.18. . 4.18. . 4.18. . 4.18. Placing of the transducer on the specimen. 1,2) silver coatings withthe transducer between them, 3) transition layer, 4) specimen.

166


The pulse-phase method of measurement of ultrasound is basedon the comparison of the phases of ultrasound waves passed throughdifferent paths in the examined material, Fig. 4.19, in accordancewith Ref. 159. The piezoconverter 1 emits the ultrasound signal 3into the specimen 2. At the moment of incidence of the reflectedsignal on the piezoconverter 1 (during 2τ) another reference signalis supplied (4) and, consequently, interference of the signals 3 and4 takes place in the specimen. The complete attenuation of the vi-brations taking place in the profile occurs when

( )2 1 ,2 2

nn n

ϕ πω τ + = − (4.30)

where ωn is the circular frequency at attenuation of vibrations, ϕ is

the phase shift of the n-th reflected wave, n is a positive integer.The meaning of (2n–1) is determined from two levels for ω

n and

ωn+1

. In the vicinity of the resonance frequency of the converterwhere the thickness of the transition layer is smaller than 5 µm, thecompensation condition can be simplified to the following form

( ) 1

2

2 1 2 ,n Zn

Z

ω− = − ∆ω (4.31)

where Z1 and Z

2 are the acoustic resistances of the piezoceramics

and the specimen. Since the time τ is real only at the natural fre-quency of the converter ω

0, we use frequencies ω

n and ω

n+1 closer

to ω0 so that ω

n < ω

0 <

ω

n+1. Consequently

FFFFFigigigigig. 4.19.. 4.19.. 4.19.. 4.19.. 4.19. Diagram of propagation of an acoustic pulse in the examined medium.

167


( ) ( )1 2

1

2 1 2 2 1 2and .

4 4n n

n n

+

− π + πτ = τ =

ω ω (4.32)

Linear interpolation of τ1 and τ

2 to frequency ω

0 makes it possible

to determine the actual time of propagation of the wave in the speci-men τ and, consequently, the phase velocity of ultrasound. The rela-tive error of the method is approximately 0.01%.

The authors of Ref. 160 described ultrasound equipment for themeasurement of the velocity of sound in solids by the pulse-phasemethod. The sensitivity to the change of the velocity of sound is10–6. This method makes it possible to measure the elasticity moduliof the third order from constants characterising the anharmonicproperties of the solid and accompanying phenomena [161], exam-ine anharmonic behaviour of the dislocation structure (162] andother phenomena [163].

The compensation phase method is based on comparison of thephases from two high-frequency signals which have passed throughdifferent media, a specimen and a reference liquid [164]. The sys-tem has the form of a balanced (with respect to 0) acoustic bridge.It can record very small relative changes of the velocity of soundin the specimen (~10–6), caused by the effect of external factors. Themethod with the pulsed, continuous or combined wave emission re-gime can be used. The acoustic bridge can also be of the self-bal-ancing type. The phase difference formed when the balance of thebridge is disrupted results in a change of the amplitude of theshifted ultrasound signals and, consequently, the frequency of theoscillator is automatically adjusted up to complete compensation ofthe phase shift. Similarly, small changes of the velocity of ultra-sound in the specimen are transformed to the change of the fre-quency of the reference oscillator; this change can be carried outwith a high accuracy.

Other methods of the excitation of vibrations, for example, theeddy current method [165], the laser pulse method [166], and oth-ers are also available.

4.2.3 Hypersonic methodsMicrowave ultra-acoustics is a new direction in the solid state

physics. The generation of acoustic waves with a frequency higherthan 109 Hz is difficult in the excitation of higher harmonic vibra-tions of piezoceramic sheets. However, some authors have managedto measure the absorption of hypersound up to 2000 Hz, for exam-

168


ple, Ref. 167. In fact, high-frequency microwaves are similar tothermal phonons, with the exception of the non-coherent nature ofthe phonons. For example, the characteristic frequency of thermalphonons of the lattice at 1 K is approximately 20 GHz.

Using an X-cut of quartz, place in a superhigh-frequency reso-nator, Baranovskii [168] observed the generation of waves with afrequency of 10–20 GHz. The diagram of the experimental setup isshown in Fig. 4.20. A quartz horn with the X-cut (or Y-cut) withparallel faces and the diameter considerably smaller than the lengthof the electromagnetic wave but greater than the length of the acous-tic wave was placed in the superhigh-frequency resonator and sub-jected to the effect of radiowave pulses with a frequency of 10–20GHz; the duration of the effect of the pulses was approximately1 µs. Every elementary volume of the piezocrystal, situated in thezone of the effect of superhigh frequency, generates an acousticwave with the direction normal to the face of the crystal. After leav-ing the resonator, a planar hypersonic wave is generated in the hornand it can be transferred to the examined object. The echo signalis received at the reversed effect of excitation of the resonator bythe acoustic wave. The excitation of shear waves with a frequencyof 1 GHz as a result of the ferromagnetic resonance in a thin layerof nickel deposited on the quartz surface was described in Ref. 169.The application of the piezoelectric phenomenon for the generation

FFFFFigigigigig. 4.20.. 4.20.. 4.20.. 4.20.. 4.20. Principle of the hypersonic method utilising the piezoelectric phenomenon:1) resonator for superhigh frequency; 2) ultrasound pulse with a length of 1 µs;3) crystal; 4) transition layer; 5) specimen; 6) propagation of ultrasound waves;7) shape of the echo signal on the oscilloscope.

169


of hypersound in quartz was discussed in Ref. 170, where a layerof Permalloy was used at the end of the horn to make it operate asa receiving transducer. The horn was placed in the super high-fre-quency resonator and vibrations with a frequency of 8.9 GHz wereproduced. The development of this method was impaired by difficul-ties in producing high-intensity magnetic fields. In most cases,hypersonic measurements are taken at low temperatures in order toreduce the extent of damping of the examined material.

4.3 PROCESSING THE RESULTS OF MEASUREMENTSAND INACCURACY

The resultant values of the internal friction and the defect of theYoung modulus as a basis for evaluating the physical processes tak-ing place in the metals and alloys are influenced by the design oftesting equipment and also by the methods used to determine thecharacteristics.

4.3.1 Inaccuracy caused by the design of equipmentMechanical frictionThe main requirement when designing testing equipment is to ensureminimum losses in the system. The losses maybe caused by me-chanical friction in clamping sections, by the friction of movingcomponents with the environment and by the dispersion of the en-ergy of vibrations in the materials of the components of the system.It is assumed that the losses in the clamping sections are small andcan be ignored, and the mass of the specimen is many times greaterthan the mass in the area of clamping.

Using a simple model of a vibrating system, the authors of Ref.170 determined the ratio of the damping decrement of the entiresystem δ

s to the decrement of the specimen δ

v, i.e. δ

s/δ

v in relation

to the ratio of the inertia moments of the fixing section M1 and the

specimen M2, i.e. M

1/M

2, for three constant values of the elasticity

characteristics (q1/q

2), Fig. 4.21a. The ratio δ

s/δ

v is selected as 103,

which corresponds to the experimental observations. For steel speci-mens q

1/q

2 = 5. The effect of the fixing section on the determined

value internal friction is determined from an example.The effect of the design factors on the optimisation of low-fre-

quency systems of the type of portional pendulum was evaluated inRef. 171. The optimisation criterion was the background of inter-nal friction at room temperature. The factor of the effect is deter-mined by the ratio of the rigidity of securing the specimen and the

170


rigidity of the bar moving the specimen (Kt/K

v, where K = (πd4G)/

(32l), d is diameter, l is length), and by the ratio of the rigidity ofthe suspension wire to the rigidity of the specimen. The tests werecarried out on specimens of aluminium, stainless steel and brass.The diameter of the securing bars was 3, 3.5 or 6 mm. The wireswere produced from tungsten, nickel–chromium alloy or capron ofdifferent diameters. The zero level and the variation range were se-lected for all combinations.

Processing of the results has provided a regression equation in-cluding the coefficient of the mutual paired effect. Verification con-firmed the accuracy of the model. This was followed by construct-ing nomograms for selecting design parts of equipment in order toensure the minimum losses in the system (Fig. 4.21b). Referencemeasurements were taken on specimens of 25KhNMFA steel. Secur-ing sections made of brass and the suspension bars with a diameter5 mm were selected from the nomograms. Measurements of the in-ternal friction background showed a decrease of 25% in compari-son with the case in which the normal parts of the measuring sys-tem were used (clamping sections made of stainless steel, suspen-sion bars with a diameter 3 mm). It should be noted that this im-provement of the efficiency of measuring equipment has made itpossible to obtain information on the interaction of dislocations withsolute atoms.

For the equipment based on the principle of torsional pendulum

FFFFFigigigigig. 4.21. . 4.21. . 4.21. . 4.21. . 4.21. Calculated (1–3) and experimental (4) dependences log (σs, σ

v) on

log (M1/M

2) (a) and nomogram for selecting structural members of equipment

with the minimum level of losses in the system (b).

171


the ratio Kt/K

v ≥ 100 is sufficient for obtaining accurate data on the

internal friction of materials.Aerodynamic losses are a function of the density of the medium

in which measurements are taken, and also of the shape of movingcomponents and the rate of changes of their temperature. Takinginto account the fact that the distance of different parts of the sys-tem from the axis of rotation differs, and the speeds of these partsare also different, it is necessary to solve a problem of the non-stationary movement of a solid in a viscous medium. In accordancewith the calculations carried out by Yu.V. Piguzov [107], the aero-dynamic losses by friction with the environment can be expressedby the equation

2 30

2

8,

3a

S R

K Tν

π αρ ϕδ = (4.33)

here α is the resistance coefficient whose value at low speeds isequal to the value of the Reynolds number (Re), ρ is the density ofthe surrounding environment, S is the cross sectional area of thebar, normal to the direction of the vector of the speed of movementv, R is the half length of the inertia arm, ϕ

0 is the maximum angle

of rotation of the torsional pendulum, Kv is the rigidity of the speci-

men, and T is the period of vibrations.The results of the calculation show that the losses of this type

decrease with increase of the vibration frequency and increase inproportion with the increase of the strain amplitude. This gives thenonlinear dependence of the aerodynamic losses on the strain am-plitude in the specimens [172]. It should be noted that these con-clusions are valid up to a frequency of 100 Hz. The calculations oflosses for high loading frequencies should be supplemented by theeffect of the resistance coefficient α and by the contribution of sol-ids with a complicated shape.

The aerodynamic losses can be significantly suppressed is the ac-tive section of the measuring system is placed in vacuum. The mag-nitude of the losses decreases asymptotically with decrease of thepressure of the environment, down to the linear dependence at apressure of 1.33 N⋅m–2 [107].

172


4.3.2 Inaccuracies of the measurement methodThe magnitude of the error of internal friction measurements de-pends of the selected testing method and the sensitivity of measur-ing equipment.

The method of damped vibrations has a relative error based onthe measurement and expressed by the equation derived in Ref. 107

( )1 1,

lnp k

p

k

Q A AA

A

−∆ = ∆ + ∆

(4.34)

where Ap, A

k is the initial and final amplitude of vibrations, respec-

tively; ∆Ap is the dependence of the magnitude of the relative error

∆Q–1/Q–1 on the value of the ratio Ap/A

k. The magnitude of the er-

ror when determining this ratio is 3–4%. The logarithmic decrementis the measure of internal friction from Q–1 ~ 10–1. A different meas-urement method must be used at higher values.

When evaluating the recording of damped vibrations, the inaccu-racies are determined by the accuracy of determination of the rangeof vibrations (2A

i). At the value of the absolute inaccuracy ±h of

measurements of the i-th range of the vibrations, the relative errorof calculation of internal friction is determined by the equation[104]

1 .

1 12

ii n i

i n

i i i i nin

i n i n i n

Ah h

AQ

A A h hA

A A A

+− +

+

+ + +

−∆ =

+− + +

(4.35)

The required value 2Ai, at which these calculations can be car-

ried out with the error not exceeding the maximum error of thedamped vibrations on photographic material and in determination ofthe range by a slide rule, is h = 0.2–0.3 mm. This means that forinternal friction calculations with the inaccuracy smaller than 5%the required difference of the ranges is 8–12 mm.

The application of automatic discriminators passing signals pro-portional to the amplitude of the damped vibrations in the range A

i

to Ai+n

and also of counting devices for the determination of the

173


value of n in the given range or of timers of time periods for nvibrations enables the accuracy of measurements to be increased upto 2.5%.

The resonance curve, used for the determination of internal fric-tion, is also characterised by some inaccuracies when processing theresults. The inaccuracies are caused by the determination of thefrequency of the half width of the curve ∆ω

r (ω

r is the resonance

frequency of the vibrations) from the equation

( )1 ,rQ −∆ = ∆ ∆ω − ∆ω (4.36)

where ∆(∆ω) is the inaccuracy of determination of the half width ofthe resonance curve, ∆ω

r is the relative inaccuracy of determination

of the resonance frequency of the vibrations. The systematic analysisof the inaccuracies in low-frequency experiments showed [124] thatat strain amplitude lower than 10–6 the relative error of measure-ments as a result of the inaccuracy ∆(∆ω) is ~ 10% and rapidly de-creases with increasing strain amplitude. Figure 4.22b shows the in-accuracy of the measurement of internal friction determined by dif-ferent methods (1 – the method of damped vibrations, 2 – themethod of the resonance curve) in relation to the extent of internalfriction in the specimen.

The resonance method is used in equipment with kilohertz fre-quencies in measurements of the Young modulus. For example, whenusing cylindrical specimens, the inaccuracy of determination of theYoung modulus in tension is 0.8%, and in torsion it is 0.5%.

The inaccuracies of the determination of the Young modulus wereevaluated methodically using the instructions in Ref. 173.

The method of supplied power depends, as regards the measure-ment accuracy, on the accuracy and stability of operation of theautomatic control equipment, the selection of suitable devices formeasurements, the stability of the power source, the adaptability ofthe feedback system, etc.

For the equipment of this type, the relative inaccuracy of inter-nal friction measurements can be determined from the equation

1 ,Q Z C N−∆ = ∆ + ∆ + ∆ (4.37)

where ∆Z is the error caused by the instability of the amplificationfactor of the feedback, ∆C is the calibration error, ∆N = ∆Z + ∆N

e,

174


is the reference voltage. The relative inaccuracy for the equipmentassembled on the basis of the principle of the supplied power is ap-proximately 0.5%. The accuracy of the absolute measurements of in-ternal friction is determined by the accuracy of calibration and thescale of the devices for determination of the data.

Evaluation of the inaccuracies of internal friction measurementsis described in instructions published in the former Soviet Union inRef. 174.

Phase and ultrasound methods require consideration of many spe-cial inaccuracies. For example, when using the echo method, theremay be inaccuracies caused by non-monochromatic pulses, diffrac-tion phenomena, non-parallel form of the ends of the specimen, theeffect of side walls, losses in the transition layer, etc. When ana-lysing inaccuracies, reference should be made to specialised litera-ture [2, 151].

4.3.3 Errors in processing the measurement resultsWhen plotting various dependences of internal friction, for exampleon temperature, there are sources of additional errors caused bythermal instability [107]

12 1

0.4 ,T

Q HRT q

−−

∆∆ = (4.38)

FFFFFigigigigig. 4.22.. 4.22.. 4.22.. 4.22.. 4.22. Dependence of the relative error of measurement of internal frictionon the ratio of the initial and final amplitude (a) and internal friction (b).

175


where H is activation enthalpy, ∆T is the absolute error of deter-mination of the temperature peak, q–1 = Q–1/Q–1

maxQ–1 is the internal

friction at temperature T, Q–1max

is the internal friction at Tmax

, andR is the gas constant.

The overlapping of the relaxation maxima, observed in the ex-periments, may exert different effects on the temperature dependenceof internal friction. In Ref. 171, the authors presented models andresults of experiments carried out in the area of overlapping of themaxima of the relaxation processes, where the following effects arerecorded: 1 – superimposition of two relaxation Snoek peaks in ironwith different types of solutes, 2 – superimposition of three high–temperature diffusion processes in vanadium, molybdenum, or tung-sten, containing solute atoms.

In the first case, it is possible to separate the maxima of thetemperature-dependent internal friction, where T

max = T

max – T

max1,

and the height of the peaks is 1:0.5 (Fig.4.23a). Figure 4.23b showsthe dependence of the resolution of the spectrum for temperature–dependent internal friction (the relationship of the height of thesmaller peak and internal friction in the saddle area Q–1

s) on the

relationship of the calculated maxima. The resolution power of thespectrum becomes lower with increasing difference in the sizes ofthe peaks, for example, those corresponding to the concentration ofthe interstitial impurities in the solid solution. When the ratio of theheights of the peaks is 1:0.8 or 1:1, the general curve shows a pla-teau. For complicated shapes of the maximum, for example, in thehigh-temperature region, the resolution power of the spectrum is

FFFFFigigigigig. 4.23.. 4.23.. 4.23.. 4.23.. 4.23. Temperature dependence of internal friction (Tm1

= 313 K, Tm2

= 343K) for computer-calculated two maxima (a) and the dependence of the resolutionof the spectrum on the ratio of the height of the maxima (b).

Q–1

× 1

0–4

0 40 80 120 0 1 2 3 4 5 6

2.0

1.8

1.6

1.4

1.2

1.0

120

100

80

60

40

20

T, °C

0

1

2

1max

1max

Q

Q

−

−

1m

ax 1s

Q Q

− −

176


considerably lower. This is associated with the superimposition ofthe high-temperature background of internal friction and appropri-ate maxima. The systematic calculations carried out in computersshow that the decrease of the resolution power of the spectrum isoften caused by lower efficiency of the devices and equipment used.

The errors in processing the results of measurements of thedependence of internal friction on strain amplitude were evaluatedin detailed in the Laboratory of Metals Physics and Strength of theTula Polytechnic Institute [175].

Figure 4.24a shows the dependence of the decrement of dampingon the amplitude of the deviation of the arm of a torsional pendu-lum for steel 22KhNMFA, obtained from 25 parallel measurementscarried out in a RKM-TPI torsional pendulum [112]. Figure 4.24bshows the dispersion of determination of the mean range of thevalue of the decrement S2. The scatter of the results of the meas-urement increases with increasing amplitude. The coefficient of vari-ation v of determination of the decrement of vibrations is dividedinto two parts by a horizontal broken line (Fig. 4.24b). Above theline v > 5%, below the line v < 5%. A similar result can also beobtained for the alloys, for example, alloys of copper, iron, etc. Theapplication of semiautomatic and automatic systems almost doublesthe range of amplitude in which the condition of identical measure-ment accuracy is fulfilled.

In many pure materials and alloys, the dependence of internal

FFFFFigigigigig. 4.24.. 4.24.. 4.24.. 4.24.. 4.24. Dependence of the decrement of oscillations δ of annealed 25Kh2NMFAsteel on the amplitude of oscillations (a) and the dependence of the variancefactor v and dispersion S2 on the amplitude of oscillations (b).

177


friction on amplitude is regarded as a linear dependence with theslope determined by the tangent of the angle of slope of the depend-ence of the part of the decrement of the vibrations in relation to theaxis of the amplitude (tan α ≈ α). It is recommended to determinethe parameter α by the least squares method which makes itpossible to determine markedly not only the value of α but also thedispersion and probability range of the appropriate quantity.

When the aim of the experiments is the examination of dampingmechanisms, the linearisation hypothesis cannot be accepted but itshould be carefully evaluated because otherwise it would be possi-ble to loose a large number of valuable information and the evalu-ated values of the dispersion and the probability range will beinaccurate. If measurements are taken in the evaluation of kinetic,time or other processes and if it is required to describe the inten-sity of changes in the dependence of internal friction on the strainamplitude and the visual examination results lead, in the first ap-proximation, to a linear dependence, it is then possible to treat thedependence of internal friction on amplitude as linear.

The verification of whether internal friction depends in a linearmanner on amplitude is necessary when dealing with a new mate-rial for the experimentator and also in cases in which there is nooriginal information on the nature of the dependence of internal fric-tion on strain amplitude. Verification of the linearity hypothesis isbased on comparing the dispersion corresponding to the scatter ofthe mean values of the decrement of the vibrations, in relation to theempirically determined regression line; this makes it possible todetermine whether the model is adequate and the dispersion, deter-mined by the error in repetition of the method, can also be deter-mined. The non-adequacy dispersion S2

n can be efficiently calculated

from the equation

( )

( )

2 2

11 12 2

21

1 1

1,

2

m m

i imi i

n i m mi

ii i

n y n x x y

S n ym

n n x x

= =

=

= =

− = − − − −

∑ ∑∑

∑ ∑ (4.39)

where n is the number of parallel measurements of the dependenceof internal friction on amplitude taken on the same specimen, or thenumber of specimens used for measuring the given dependence in the

178


selected amplitude range, m is the number of measurement points onthe dependence, y

i is the mean value of the decrement of the vibra-

tions at the given amplitude, determined from a series of n repeatedmeasurements, y

i is the value of the decrement of the vibrations at

the given amplitude, calculated from the regression equation, x isthe mean value of the amplitude, x

i is the actual amplitude. The dis-

persion of repeatability is determined by the equation

2

1 1 120

1

,

m n m

ij ii j i

m

ii

y n y

S

n m

−

= = =

=

−=

−

∑∑ ∑

∑ (4.40)

where yi is the change of the decrements selected m times from each

curve at n parallel measurements of the dependence of internal fric-tion on amplitude.

If repeated measurements are not taken (this is often the case inmeasurements of the dependence of internal friction on amplitude),the nonadequacy dispersion S2

n can be calculated. In this case, the

experimentator plots the results graphically and evaluates them visu-ally. Consequently, the dispersion can be determined using equation(4.39), if the value of α is known; in determination of the probabil-ity range, the number of degrees of freedom is selected m = 2. Veri-fication of the linearity hypothesis using this procedure can be car-ried out only if the dispersion of internal friction in all amplituderanges is the same.

When the dispersion of determination of the decrement of the vi-brations depends on the strain amplitude (this is the most frequentcase in the measurements), it is also necessary to consider differ-ences in the accuracy of the measurements [176,177]. Specialgroups of experiments are then prepared for evaluating therepeatability of the method, and the number of parallel measure-ments in each group is more than 20. The input data are initiallyverified whether the accuracy is the same, and further statisticalprocessing is carried out using the amplitude range in which the dis-persions are equal.

The results of measurements of the dependence of internal fric-tion on amplitude, presented in the following part, fulfilled the lin-earity hypothesis.

179


The internal friction background of iron, determined on 25 dif-ferent specimens subjected to tensile deformation of 5% for the for-mation of an unstable structural state, is characterised by a vari-ance factor of 5%. The results indicate that the measurements of thebackground are highly reproducible, despite the fact that in manystudies it has been shown that the results are highly sensitive to thestructural condition. The variance factor in measurements of back-ground on a single specimen with 25 parallel measurements, carriedout without dismantling the specimen, was approximately 3% (v =2.97%). The accuracy of determination of the internal friction back-ground is sufficiently high, and the extent of damping in the firstcase was ~ 28×10–4 and in the second case 8×10–4.

The intensity of damping α after annealing of the specimen wasevaluated by measurements. The measurement of the dependence ofinternal friction on the strain amplitude was taken by two methods.In the first method, internal friction was measured at increasingstrain amplitude, in the range in which the linearity hypothesis wasvalid. The amplitude was always increased by 5 mm and then de-creased by 5 mm. In the second method, 20 parallel measurementswere taken at the extreme values of the amplitude range, alwaysafter 10 measurements at every extremum. The measurement time inboth cases was the same. This was followed by standard statisticalprocessing of the results to determine the dispersion and the vari-ance coefficient within the individual groups. In the first case,S2

0 = 1.747, in the second case S2

0 = 0.903.

Comparison of the dispersion carried out using the F criterionshowed that the difference in the dispersion was negligible so thatthe same accuracy was used in all cases. In the first case, v = 3.4%,in the second case v = 2.4%. Since the material has a linear depend-ence of the decrement on amplitude, the accuracy of determinationof α is 2–4%, i.e. slightly higher than the accuracy of determina-tion of the internal friction background. Similar measurements weretaken on specimens after 5% tensile deformation. Subsequently, v =30% in a batch of 20 specimens. This shows that the variance co-efficient in determination of internal friction increases when thespecimens are in the unstable structural condition.

The critical strain amplitude can be obtained simply if we havethe experimentally determined regression equation, obtained by themethod of least squares, and if we know the level of the internalfriction background and the entire course of the dependence of in-ternal friction on strain amplitude. The background is substitutedinto the regression equation and this gives the value of the first

180


critical strain amplitude εkr1

.The variance factor for ε

kr1, determined from a series of 20 par-

allel measurements, taken on specimens prior to tensile deformation,was approximately 15%. If the specimens were not annealed priorto measurements, the measurement error ε

kr1 was not higher than 5%

and for εkr2

it was 9%.To conclude this chapter, it should be stressed that the efficient

selection of the equipment and devices, careful execution of themeasurements and evaluation of the results in accordance with therequired procedures necessitate a critical review and determinationof the measurement accuracy.

1 8 1

Structural Instability of Alloys

5

STRUCTURAL INSTABILITY OFALLOYS

The thermal and frequency dependence of internal friction as a re-sult of relaxation is associated with the mean time of movement ofatoms in the equilibrium position of the crystal lattice τ and relaxa-tion time τ

r. The relaxation time is determined by the nature of

processes and is a material characteristic. In this chapter, wepresent several mechanisms of the relaxation of internal frictionused in solving the problems of physical metallurgy and thresholdstates of materials.

5.1 DIFFUSION MOBILITY OF ATOMSInternal friction measurements make it possible to determine the dif-fusion characteristics of point defects, their thermal activation, for-mation of different pairs of point defects and their redistributionunder the effect of external stresses.

A point defect produces local elastic strains in the crystal and anelastic dipole, which is a tensor of the second order, forms. On thebasis of the group theory, Nowick and Berry [66] produced tableswhich make it possible to determine inelastic interactions for elas-tic dipoles and the formation of peaks on the Q–1 T or Q–1 f depend-ence. The best known inelastic interaction is the one determined bythe presence of an interstitial atom i in the cubic body-centred lat-tice. This atom forms tetragonal symmetry and can be located inoctahedral or tetrahedral voids. The difference of these tetragonaldipoles can be determined from the absolute value of the shape fac-tor (λ

1 – λ

2) of the deformation ellipsoid [66]. It has been confirmed

that the interstitial atoms in iron are distributed in octahedral voids;this is also confirmed by the interpretation of the Snoek maximum.

In the cubic fcc and hcp lattices, the interstitial atoms do notform these anelastic formations [66]. A single substitutional atom

1 8 2


s or vacancy in fcc, bcc and hcp lattices does not form theanelasticity state, because a defect results in distortion whose sym-metry is the same as that of the crystal. However, in the formationof the s – s or s – v pair, the anelastic phenomenon may occur.Forexample, the s–s pair in the bcc lattice forms a tetragonal and inthe fcc lattice an orthorhombic dipole with the orientation ⟨110⟩, andunder the effect of external stress they can be mutually displaced.Zener relaxation and the associated occurence of a kink on the curveof the internal friction dependence in the single crystal of α-brass,caused by the initial displacement of a pair of atoms, is a generalfeature associated with the dissolution of atoms in the substitutionalsolid solutions.

The temperature of formation of the maximum of internal fric-tion and the activation parameters of the relaxation process areconnected together by the expression ωτ

0 exp (H/RT), which is also

used for the determination of H and τ0. If the frequency is changed

from ω1 to ω

2, the internal friction peak is displaced on the tem-

perature axis from Tmax1

to Tmax2

. Consequently

max1 max 2 2

max 2 max1 1

ln ,T T

H RT T

ω=− ω (5.1)

τ ω ω

= − + −

(5.2)

The calculation of activation enthalpy H with sufficient accuracyrequires the changes of ω by several orders of magnitude. When theshift of the maximum with the change of the frequency of oscilla-tions is 20–40°C, the relative error in determining H is approxi-mately 20%. Determination of τ

0 from equation (5.2) using the ef-

fective value of H requires analysis of the physical nature of the re-laxation process. Otherwise, the physical meaning may be lost.

Wert and Marx proposed a suitable equation for further determi-nation of H in the form

maxmax ln ,

kTH RT S= + ∆

ω (5.3)

1 8 3


where R is the gas constant, k is Boltzmann constant, h is the re-duced Planck constant, ∆S is the activation entropy (~10–12J⋅mol–1⋅K–1).

Equation (5.3) can be used only when the relaxation process isassociated with the thermal activation of displacement of the indi-vidual atoms by the distance equal to the atomic spacing. Even inthis case, it should be expected that there will be a systematic er-ror in determination of H, caused by the difference of the relaxa-tion process and the ideal model according to Debye.

The diffusion coefficient, determined by internal friction measure-ments, does not depend on the thermodynamic factor. Its value canbe determined from the equation

2

,a

D = ατ

(5.4)

where a is the spacing of the adjacent atoms, α is the geometricalfactor with the value 1/24 for the bcc lattice and 1/12 for the fcclattice, where τ = 3/2τ

r for the interstitial solid solution, or τ = τ

r

for the substitutional solid solution. Basically, it is possible toevaluate not only the mobility of individual atoms and pairs but alsoof larger clusters of the atoms capable of migrating in the crystalin different dimensions and with different speeds [2].

Determination of the value of D using equation (5.4) and thevalue of H using equation (5.1) or (5.3) makes it possible to deter-mine the value of D

0 in the equation D = D

0 exp (–H/RT).

When comparing the effective coefficient of diffusion from thedisplacement of the mass, determined from internal friction meas-urements, the thermodynamic activity of the element is characterisedby the correlation factor [2].

5.1.1 Interstitial solid solutionsThe fcc, bcc and hcp lattices contain octahedral voids (OV) and tet-rahedral voids (TV). The dimensions of these voids in the fcc andhcp lattices are 0.41r for (OV) and 0.225r for (TV). The size ofthese voids in the bcc lattice is 0.154r and 0.291r, and in the caseof the OV it is the minimum spacing between two adjacent atomsin the lattice. The void is asymmetric and the spacing of the fouratoms, forming this void, is 0.631r. In evaluation of the internalfriction of interstitial solid solutions attention is given especially tothe orientation polarization of the paraelastic medium. This polari-

1 8 4


zation forms when the introduction of a defect results in disruptionof the symmetry which is considerably lower than the symmetry ofthe lattice. This phenomenon is based on the tendency of the dipoleto orient in the stress field in such a manner that its energy is mini-mum. The symmetry group of the tensor of the dipole is the sub-group of the symmetric group of the crystal, and the following in-equality must be fulfilled

1,kt

t

hn

h= ≥ (5.5)

where hk and h

t are the numbers of operations of symmetry for the

crystal and for the tensor, nt is the number of possible operations

of the dipole in the lattice. The condition mt ≥ 1 is essential but

insufficient for relaxation to take place.This model makes it possible to determine the boundary of ap-

plication of internal friction measurements for a homogeneous solidsolution [178]. The shape factor varies in the range from 0.01 to1.0. If it is taken into account that the sensitivity of internal fric-tion measurements is 10–4 to 10–5, it can be seen that it is possibleto record changes of the solid concentration range from 10–4 to10–3. If the measurement conditions are not favourable, this rangevaries from 0.1% to 1.0%. The shape factor is related with solubil-ity and increases with decreasing solubility. When evaluating therelative change of concentration, the sensitivity of the method, basedon internal friction measurements, is a constant value in a wide con-centration range.

The Snoek relaxation is the best known phenomenon caused bythe diffusion of interstitial atoms in the bcc lattice. Snoek [179] ob-served this phenomenon in iron at a frequency of approximately 1Hz at temperatures close to room temperature. The shape of thepeak is similar to the Debye dependence (Fig. 5.1). The formationof the peak is associated with the migration of interstitial atoms(carbon and nitrogen) and of the effect of the external stress in oc-tahedral positions. In the diffusion of oxygen, nitrogen and carbonin iron, this mechanism has been confirmed by experiments. For themetals with the bcc lattices, the interstitial atoms can migrate in theOV and also TV [708]. These types of migration are also referredto as Snoek’s relaxation.

This relaxation is controlled by the thermally activated jumps ofatoms. The relaxation time can be determined using the Arrhenius

1 8 5


FFFFFigigigigig. 5.1.. 5.1.. 5.1.. 5.1.. 5.1. Dependence of internal friction on temperature for tantalum with 0.013wt.% C at two loading frequencies.

equation in the form τr = τ

0 exp (H/kT). Internal friction measure-

ments are usually taken at a constant frequency. If we disregardtemperature changes ∆(T) and ∆(T) at ωτ = 1 and T = T

max, we can

use the equation

( )1

max

sech ,2 1 1

HQ T

kT T

−

∆ = −

(5.6)

where ∆ is the degree of relaxation of the elasticity modulus (atωτ = 1 Q–1

max = ∆/2).

In the exact procedure, the experimentally determined Snoekmaximum does not correspond to the position of the peak at ωτ =1 and corresponds to the condition (ωτ)2 = (1 + kT/H)(1 – kT/H),because ∆ ~ 1/T [180]. Therefore,

1 8 6


( ) ( )( )

20 0

1,

c F g

kT M g−

ν δα∆ = α (5.7)

where c0 is the concentration of the interstitial atoms in the solid so-

lution, v0 is the atomic volume, δλ = λ

1 – λ

2 is the resistance of the

dipoles (the shape factor), g is the geometrical factor of the mainstress axis, α = 4/3, M = G, F(g) = g (in torsion), but α = 2/9,M = E, F(g) = (1 – 3) g (in tensile loading).

The quantity τ0 = ω

0 [∆S/kT], where ∆S is the activation en-

tropy, ω0 = 6υ (υ is the Debye frequency). According to Wert and

Marx [181], one can confirm, with good approximation, the linearrelationship of the temperature of occurrence of the Snoek maximumτ

max and activation enthalpy H using the equation

( )0max max max 0ln ln .H T k T S T k

ω = + ∆ = − ωτ ω (5.8)

Table 5.1 shows the generalised data on the Snoek maximum (H,τ

0) in the interstitial solid solutions obtained by measurements of the

temperature dependence of internal friction [180]. The published re-

TTTTTaaaaabbbbble 5.1le 5.1le 5.1le 5.1le 5.1 Parameters of Snoek relaxation in bcc metals and interstitial soluteatoms

metsySH 01· 01

)J(τ

001· 51

)s(T

xamzH1ta

)K(

O–bNN–bNO–aTN–aTO–VN–VN–rCN–oMC–eFC–eFN–eFN–eFN–eFC–bNC–aTC–V

10.0684.130.0025.210.0967.130.0466.2

460.2215.2409.1080.2133.1492.1572.1712.1372.1652.2867.2259.1

8.056.28.022.15.265.80.246.3

69.015.044.1

1177.56.3166.40.8127.580.28.0280.2

224265914516854445924035213213692692892215726344

1 8 7


FFFFFigigigigig. 5.2.. 5.2.. 5.2.. 5.2.. 5.2. Dependence of activation enthalpy on temperature of formation of Snoekrelaxation peak.

sults contain only data on the experiments in which the number ofinterstitial atoms and the resultant dependence of the height of theSnoek peak on the solute content were determined by analysis.

According to Weller [180], the value of the relaxation parameterscorresponds to the experiments with displacement as a result of thechange of frequency (for example, for tantalum H = 1.92 × 10–19 Jand τ

0 = 7 × 10–5 s [182]).

Using equation (5.8) at a low loading frequency (~1 Hz), we cancalculate the temperatures of formation of the Snoek maximum andplot the H(T) graphic dependence shown in Fig. 5.2. After extrapo-lation for the case H = 0, T = 0, we have H

2 = 4.4 × 10–23⋅T

max(J).

If H = 0 at Tmax

= 0, H1 = 4.5 × 10–23⋅T

max– 8.8 × 10–21(J), and this

value corresponds to τ0 = 2.08 × 10–15 s = const.

The solubility of carbon in niobium, tantalum and vanadium isvery low. This complicates experiments because of the need to usesuperpure metals and dope the material with low concentrations ofcarbon [180]. Snoek peaks are recorded at 525 K (ω = 2.1 Hz) inthe Nb – C system, at 642 (2.2 Hz) in the Ta – C system, and at454 K (1.93 Hz) in the V – C system. Details on the temperatureat which the Snoek maximum is recorded are obtained for nitrogenand oxygen in, for example, the V – N system (T

max = 540 K),

V – O (Tmax

= 455 K), Nb – N (Tmax

= 559 K), Nb – O (Tmax

= 418K), Ta – N (T

max = 616 K) and Ta – O (T

max = 415 K).

The data on the Snoek peaks for the metals of group VIa are pre-

1 8 8


TTTTTaaaaabbbbble 5.2le 5.2le 5.2le 5.2le 5.2 Parameters of Snoek relaxation for metals of group VIa

sented in Table 5.2. Difficulties in experiments are formed not onlyby the low solubility of interstitial atoms (C, N, O), but also bydefects of the actual spectrum of the dependence of internal frictionand temperature as a result of the formation of Snoek and Kösterpeaks (production of alloys maybe accompanied by deformation, e.g.annealing in a gas atmosphere followed by rapid cooling).

The possibility of the formation of a Snoek peak as a result ofthe migration of boron atoms in the solid solution of α-iron hasbeen discussed in the technical literature. Experiments yielded peakson the Q–1(T) dependence [183], widening of the initial peak causedby carbon migration (T

max = 30°C at 1 Hz) [184], and also the

formation of unstable peaks (Tmax

= 13°C at 1–5 Hz) [185]. Later,systematic experiments [186] showed that the addition of boron toiron does not result in the formation of a new peak caused by themigration of boron to the interstitial positions in α-iron. No wid-ening of the peak, associated with carbon migration, was detected.For the interstitial solid solutions of metals with the bcc lattice, thediffusion coefficient at T = T

max is determined by the equation

2

,36 r

aD =

τ (5.9)

where

max max

1 1.

2r fτ = =

ω π

Measurements of the diffusibility of atoms with the evaluation ofQ–1 are in agreement with the results obtained by other methods,Fig. 5.3a. The set of points at higher temperatures corresponds to

kaeP retemaraP C–rC O–rC C–oM O–oM C–W O–W N–W

latnemirepxET

xam)K(

f )zH(514-31453.1–7.0

0061–

5741–

356 ÷ 386÷4.0 1

detaluclaCT

xam)K(

H 9101·)J(

71448.1

28575.2

08565.2

88461.2

80721.3

87366.1

054 ÷ 06599.1 ÷ 84.2

1 8 9


FFFFFigigigigig. 5.3.. 5.3.. 5.3.. 5.3.. 5.3. Temperature dependence of the coefficient of carbon diffusion in alphairon. 1) low-temperature range; 2) high-temperature range; 3) experimentallydetermined curve; 4) calculated according to Wert and Marx.

the results obtained in other methods, for medium temperatures itcorresponds to the method of measurement of Q–1 and at low tem-peratures to the elastic loading method. Processing of the resultsyielded the equation for carbon diffusion in α-iron in the followingform

840.02expCD

RT = −

(5.10)

and for nitrogen in α-iron

760.003exp ,ND

RT = −

(5.11)

where activation enthalpy is in kJ⋅mol–1, and the dimension of thediffusion coefficient is cm2 s–1.

Figure 5.3b shows a more accurate determination of the depend-ence D

C (T). One of the curves was obtained on the basis of empiri-

1 9 0


cal dependences, the other one was calculated according to Wert andMarx from the data on internal friction. If the slope of the straightline in Fig. 5.3a is modified in such manner as to describe the low–and high-temperature measurements, the straight line 4 in Fig. 5.3breflects accurately the results of measurements of Snoek’s peak. Thereason for the difference of the data obtained in the high-tempera-ture range was described in detail in Ref. 2 and is based on theoccurrence of an additional activation process, causing a deviationfrom the Arrhenius relationship, and on the effect of vacancies.

The data on the diffusion characteristics in transition metals canbe supplemented by analysis of the results presented in Table 5.1and 5.2. Consequently

The experimental data on the activation entropy ∆Se can be obtained

from the equation D0 = p αδ2υ exp (∆S/R), where p is the number

of equivalent diffusion paths, α is the geometrical factor, δ is themagnitude of the jump of the atom, υ is the frequency of vibrationsof the atoms. Wert confirmed good agreement of the values of ∆S

e

and ∆Stheor

for carbon and nitrogen in α-iron and oxygen and nitro-gen in tantalum, as stated in Ref. 2.

At present, a large number of data is available on the diffusioncharacteristics in interstitial solid solutions with the bcc latticeobtained from the parameters of the Snoek maximum [187].

The strength of the effect of interstitial atoms on the thermody-namic properties and the diffusion coefficient of the interstitial at-oms in the ternary solid solutions based on niobium changes in thefollowing sequence: Ta, Ti, W, V, Hf, Zr, Cr, which corresponds tothe increase of the difference of the dimensions of the interstitialatoms and niobium and the energy of the deformation interaction ofthe s – i atoms [188]. The binding energy of the s – i complexesis higher than that of the i – i complexes. This results in completeabsence of the i – i complexes in the ternary alloys at any tempera-ture.

Details of the Snoek relaxation in the concentrated solid solutionsbased on Ta, niobium and vanadium have been published in Weller’sstudy [180], and the data on the anisotropy of Snoek’s phenomenon

aT rC oM W

D0

mc( 2 s 1– ) 8100.0 900.0 3100.0 410.0

H lomJk( 1– ) 6.401 2.901 0.061 2.281

1 9 1


in niobium with oxygen and nitrogen are in Ref. 189.The relaxation according to Finkelstein and Rozin was observed

in 1952 in an austenitic steel. The height of the peak is directly pro-portional to the carbon content of the solid solution and the acti-vation energy of the peak is close to the activation energy of car-bon diffusion [190]. These relationships were confirmed for nickel,Ni–Al alloys and a large number of austenitic steels [191,192]. Ini-tial studies already showed the extensive possibilities of utilising theFinkelstein–Rozin (FR) phenomenon in the examination of dissocia-tion of carbides in austenitic steels and also in analysis of phasetransformations in austenitic–martensitic steels and metals with thefcc lattice. Kê et al showed that the formation of the FR peak isdetermined by the reversible movement of carbon atoms (Fig. 5.4),because the carbon atom occupies the vacancy V and forms a pairwith the carbon atoms in the position 2 or 2'; possibly, two atomsare situated in the positions 2,2', and the vacancy is in the positionV.

The concentration dependence of the parameters of theFinkelstein–Rozin peak for different alloys is approximately thesame [193]. At low concentrations, the height of the peak increasesin proportion to c2, and at higher concentrations the height of thepeak is proportional to c, where c is concentration. For manganese-containing alloys, the parabolic region of the concentration depend-ence is less significant because the peak is detected only at concen-

FFFFFigigigigig. 5.4. 5.4. 5.4. 5.4. 5.4. Model for explaining the Finkelstein–Rozin phenomenon (A is the ironatom, V the foreign atom, e.g. substitutional or interstitial atom or vacancy).

1 9 2


trations higher than the critical value c0. The transition from the

parabolic to linear dependence is characterised by the concentrationc

k. Its level differs for different alloys (usually c

k > c

0). When ap-

proaching the solubility limit, the concentration dependence startsto deviate from linear. Increase of the amount of the second phasemay result in a decrease of the height of the Finkel’stein–Rozin peakwith increase of the concentration of interstitial atoms. Verner [193]confirmed the absence of any strong effect of substitutional atomsand vacancies on the FR phenomenon. This makes it possible tounderstand the relaxation mechanism associated with the rotation ofa pair of interstitial atoms under the effect of external stress.

Evaluation of the probability of formation of these pairs, formedby the interstitial and substitutional atoms, was carried out usingthe data in Table 5.3 expressed in fractions of interaction in the co-ordination sphere. In all cases, repulsion is detected in the first twoco-ordination spheres. This shows why pairs of the nearest neigh-

sriapfonoitcaretnIsmotafo

erehpsnoitanidrooC

1 2 3 4 5 6

laititsretnIlaititsretnI

lanoitutitsbusdnalanoitutitsbuS

00.1+

00.1+00.1+

01.1+

01.1+08.1+

43.0–

09.0+09.0+

33.0–

06.0+08.0+

54.0–

07.0+02.0+

61.0

00.001.0–

TTTTTaaaaabbbbble 5.3le 5.3le 5.3le 5.3le 5.3 Energy of interaction of pairs of atoms in fcc alloys in relation tothe first coordination sphere

bours do not form. In the case of interstitial solid solutions, pairscan form between the atoms of the third and fifth co-ordinationsphere.

On the basis of the results it may be assumed that relaxation inthe interstitial solid solutions takes place as a result of the forma-tion of pairs of interstitial atoms distributed in the third and fifthco-ordination sphere. Alternating loading results in the redistribu-tion of the pair by the transition of the interstitial atom from oneco-ordination sphere to another. According to Verner [193]:

1 9 3


2

2

max1max

0 maxmax

exp

,

9 24 exp 1

m

d WE c

dN kTQ

WN kT c

k T

−

λ =

+

(5.12)

where

0

1

m

dN

dN

λ

is the change of the distortion of the lattice during rotation of a sin-gle pair from the direction parallel to the loading axis to the direc-tion normal to this axis, N

0 is the number of voids in the unit vol-

ume, W is the energy of interaction of the interstitial atoms. Therelationships for the vacancies are reversed and, in this case, pairscan form around the vacancies in the adjacent co-ordination spheres.In the case of the interstitial and substitutional atoms, attraction cantake place from the sixth co-ordination sphere. However, this pairwill not change its orientation and no phenomenon will be recorded.

When the concentration of interstit ial atoms is low, then24 c exp (W/kT

max) << 1, we obtain that

1 2max

max

6 exp ,W

Q ckT

− = β

where

2

0 max

1 1.

9 m

E d

N dN kT

λβ =

At higher concentrations Q–1max

= β/4c. The transition from the quad-ratic to linear concentration dependence is expressed less accuratelyby the equation 24c

0 exp (W/kT

max) ≈ 1. In substitutional solid so-

lutions, which greatly differ by the atomic size from the solvent, thephenomenon of suppression of the process of formation of the in-

1 9 4


terstitial pairs takes place until the concentration of the interstitialatoms reaches the value at which in the interstitial atoms are dis-tributed in the fifth and lower coordination sphere. The critical con-centration is approximately 0.1–0.25 wt.% C (or N), and the experi-mental value of c

0 is also in this range [178].

Relaxation in the hcp metals, observed on the Q–1 (T) depend-ence, has not been sufficiently quantified. More systematic analy-sis has been carried out in the case of solid solutions α-Ti. Themaximum on the Q–1 (T) dependence in titanium is observed only incases in which the distribution of the substitutional atoms in thesolid solution [194] is such that the individual interstitial atomssituated in the octahedral voids cause distortion with the symmetryidentical with that of the lattice and no dipoles appear. A dipoleforms when the interstit ial atom is in the vicinity of thesubstitutional atom. The height of the peak on the Q–1 (T) depend-ence increases with increase of the difference of the size of the at-oms of titanium and the substitutional atom. At a constant contentof the alloying substitutional element (for example, zirconium), thepeak grows in direct proportion to the increase of the atomic con-centration of the interstitial atoms (oxygen). In the case of a con-stant content of the interstitial atoms, the dependence of the heightof the peak on the concentration of substitutional atoms is not un-ambiguous [178]. For example, in alloys of titanium with zirconium,the height of the peak of internal friction is maximum at low zir-conium content, decreases with increasing zirconium content, passesthrough a minimum at 0.5 at% Zr (for an alloy with 1 at.% oxy-gen). The position of the minimum corresponds to the line situatedin the ternary Ti–Zr–O diagram in the Zr–O relationship, corre-sponding to compound ZrO

2. Measurements of the Q–1 (T) depend-

ence and of the activation energy of the processes which enables theformation of the peak on the Q–1 (T) dependence at different zirco-nium content show in the Ti–Zr–O system the formation of anordered structure, preceding precipitation of the new phase.

Similar phenomena can be detected in the Mg–Cd substitutionalsolid solutions in the formation of the ordered phase Mg

3Cd in them

[195] and also in the Ti–O–Al system for alloys with the ratio ofthe aluminium and oxygen content close to Al

2O

3. The peak of the

dependence Q–1 (T) for these alloys is completely suppressed. Theabsence of data from the measurements of the Q–1 (T) dependenceon single crystals of solid solutions with different concentrationcomplicates the application of the rule of selection for determina-tion of the symmetric of the dipole in the hcp lattice. Despite this

1 9 5


FFFFFigigigigig. 5.5.. 5.5.. 5.5.. 5.5.. 5.5. Temperature dependence of internal friction for brass (f = 620 Hz).

fact, examination of the solid solutions by measurement of theQ–1 (T) dependence is highly promising and useful.

5.1.2 Substitutional and solid solutionsAt comparable concentrations of components of alloys of fcc, DCCand hcp metals and the formation of substitutional solid solutionswe can observe relaxation caused by changes of the relative distri-bution of the atoms in the alloy under the effect of external load-ing. As mentioned previously, Zener detected, for 30% brass, a peakon the Q–1 (T) dependence which was attributed to the change of theorientation of the axes of the isolated pairs of adjacent atoms ofsolutes under shear loading (Fig. 5.5) [196].

At low solute concentration the height of the relaxation peak isdirectly proportional to the square of the concentration of the dis-solved element. The theory of relaxation of the pair loses the mean-ing for higher concentrations of the dissolved element, where theassumption on the separated pair is not justified. For solid solutionswith higher concentration, where the solute atoms form complexes,Le Claire and Lomer [197] proposed a model which takes into ac-count the change of the shot-range order in alloys after loading thematerial (the model of ‘directional shot-range ordering’).

Zener relaxation is caused by the spatial migration of the atoms

1 9 6


of the solution and this results in changes of the number of pairsof atoms with different orientation or of the shot-range order param-eters in the solution. Consequently, it is possible to examine, forexample, the microscopic characteristics of the ordered alloys.

The diffusion characteristics and thermodynamic activity of alarge number of substitutional solid solutions have been measuredby different authors [2]. For example, for the Ag–15.8–30.2 at.%Zn system, these characteristics were determined by Nowick, for theAl–0.01–0.5 at% Mg by Green and Pavlov, for the Ag–32 at% Cdand Au–32 at% Ag systems by Terner and Williams, etc..

The question why the Zener peak broadens has not as yet beenanswered. There are two extreme cases: a) there is a spectrum ofthe relaxation times, with each time depending on the temperaturein accordance with the Arrhenius equation; b) the temperature de-pendence of the individual relaxation times differs from the Arrhe-nius dependence.

Nowick and Berry [66] assumed the formation of spectral relaxa-tion times characterised by the normal Gaussian distribution of theparameters

0 ,H

kT

ββ = β + (5.13)

where β0 and β

H are independent of temperature and determine the

half width of distribution of ln τ0 and H

1. After calculating the dis-

persion of the distribution βc, the fluctuation of concentration in the

alloy with the approximation according to Gorsky, Bragg andWilliams and after relating the quantities β

H and β

c by the equation

βH = (dH/dc

m)β

c, we obtain the equation

( )

1/ 21 1

,2 1H

m m m

dH zW

dc kT c cn

−

β = − + − (5.14)

where n is the number of atoms in the region of concentrational het-erogeneity, W is the energy of the ordered alloy, c

m is the mean con-

centration of one of the components, H is the mean activation en-ergy, z is the co-ordination number. Therefore, the spectrum of therelaxation times is justified by the fluctuation of the concentrationin the alloys.

1 9 7


Zener relaxation is controlled by the kinetics of changes of shot-range ordering in the alloy [198]. The depth of the potential wellis determined by the energy of interaction of the migrating atomwith its closest neighbours. The height of the potential barrier doesnot depend on the atoms surrounding the potential well and is de-termined only by the type of migrating atom. This approach givesequations for the relaxation time of Zener relaxation. The value ofthe activation energy of Zener relaxation is between the values ofthe activation energy of diffusion of the components of the alloys.

Many theoretical studies take into account the kinetics of changesof shot-range ordering for different cases of loading and deforma-tion in fcc crystals [195], changes of shot-range ordering resultingfrom the effect of temperature [200] and also the correlation func-tions characterising the order around the vacancies [201]. The cal-culated and experimental values of the diffusion parameters areclose. However, coefficients, not reflecting the physical nature of theprocess, are quoted in certain cases.

5.2 RELAXATION OF DISLOCATIONSThe Q–1 (T) dependences of the metal than alloys show, even underthe effect of very low stresses, anomalies in the changes of inter-nal friction with changes of temperature. The activation energy ofthe processes controlling the movement of dislocations in the crys-tal for the maturity of material is not higher than 2.4 × 10–19 K. Theprocesses associated with dislocations relaxation in repeated load-ing of the material with the frequency in the hertz and kilohertzrange take place at temperatures lower than 500–600 K, i.e. belowthe condensation temperature of Cottrell atmospheres T

C. This means

that in the range in which the internal friction depends on the stateamplitude, all types of dislocation relaxation can take place only asa result of the movement of fresh, non-pinned dislocations formedduring prior plastic deformation or phase hardening.

The condition of prior plastic deformation from the processes tooccur causes that these peaks are regarded as the strain maximum.

5.2.1 Low-temperature peaksThe main phenomenon of dislocations relaxation in pure metals isthe Bordoni relaxation [202] whose main features were described insection 3.2.2. In loading in the range of kilohertz frequencies theBordoni peaks are recorded at low temperatures (~ 100 K). TheBordoni peak does not change during annealing up to therecrystallisation temperature. The increase of the content of the al-

1 9 8


loying additions increases the height of the peak, even if the tem-perature at which this is recorded does not change. However, thechange of the vibration frequency results in changes of the tempera-ture at which the Bordoni peak is observed, but the shape of thepeak remains unchanged. This peak is also slightly influenced by thestrain amplitude. The activation energy of the relaxation processvaries from 8 to 10 kJ×mole–1 and τ

0 = 10–1 – 10–12 s [203]. The

characteristic feature of Bordoni relaxation is that the height of thepeak is almost an order of magnitude higher than in the relaxationpeaks with a single relaxation time.

At temperatures lower than room temperature, the fcc metals sub-jected to plastic deformation undergo other relaxation phenomena,differing from Bordoni relaxation, even though their occurrence isexplained on the basis of the assumption on the movement of bowsin the dislocations. To explain the occurrence of peaks at low tem-peratures, but higher than the Bordoni peak, the author of [204]proposed an interpretation based on the assumption that in certaincases the relaxation may be the result of breaking of diffusing dis-location bows by point defects, especially vacancies. Different typesof the defects then determine the formation of several peaks.

5.2.2 Snoek and Köster relaxationThe characteristic peak of the Q–1(T) dependence for deformed ironwith the nitrogen content was detected by Snoek in 1941, at a tem-perature of 200 °C and at a loading frequency of the material of 0.2Hz. Further experiment were carried out by Köster and many otherauthors [205, 206]. They investigated the main characteristics of thepeak, the need for the presence of a small amount of nitrogen at-oms in the solid solution and the need for prior deformation ε

p. The

relaxation nature of the process was already justified in this stage,despite the fact that the width of the peak is significantly greaterthan that of the Debye peak with a single relaxation time. The ac-tivation energy of relaxation in iron alloys containing carbon or ni-trogen is 127–168 kJ ⋅mol–1, and the frequency factor is τ ≈1014 s–1 (for iron with 10 at.% C after 2% cold deformation andT

max S–K = 557 K and after 10% cold deformation T

max S–K = 545 K).

The temperature at which the S–K peak is observed in iron alloysat 0.2–1 Hz is approximately 200–250°C.

Later, in subsequent investigations it was shown [207,208] thatthe peak with this characteristic is recorded in different bcc metals(iron, tantalum, vanadium, niobium, molybdenum, tungsten) if theycontain the atoms of carbon, nitrogen, oxygen and hydrogen in the

1 9 9


solid solution. This peak is referred to as the Snoek–Köster maxi-mum or peak (S–K peak).

The main conditions for the formation of the S–K peak is thepresence of interstitial atoms in the solid solution of the material,prior deformation of the material and the application of heat treat-ment under certain conditions [209]. Usually, it is concluded that theheight of the S–K peak (Q–1

S–K) increases with the increase of the

content of interstitial atoms in the solid solution C0 (Fig. 5.6). This

hypothesis should be treated with care because the authors carriedout deformation in the condition after strengthening by heat treat-ment, in order to determine C

0 from the measurement of the Snoek

maximum. It is evident that this was reflected in the redistributionof the interstitial atoms between the precipitates and the atmospheresaround the dislocations (Fig. 5.7a).

The concentration range of the formation of the S–K peak wasdetermined in practice [209] in alloys of iron with carbon using theFibonachi method taking into account the actual state of the alloyprior to deformation. At ε

p ≈ 20%, the dependence Q–1

S–K (C

0) shows

an extremum in the range of the low initial carbon content in thesolid solution (~9 × 10–4 wt.% C). An analytical procedure was usedto determine the optimum value of ε

p for the maximum height of the

S–K peak (for Armco iron εp ≈ 55%, Fig. 5.7b). This shows that the

FFFFFigigigigig. 5.6.. 5.6.. 5.6.. 5.6.. 5.6. Relationship of the height of Snoek and Köster peaks (Q–1SK

) and Snoekpeak (((((Q–1

S ) in ferrous alloys.

2 0 0


FFFFFigigigigig. 5.7.. 5.7.. 5.7.. 5.7.. 5.7. Dependence of Q –1SK

on strain (a) , the duration of prior ageing at200 °C prior to 20% deformation of iron with 0.015 wt.% C (b), on temperaturein heating (c) and on holding time at T = T

max (d) for quenched and deformed

(20%) specimens with 0.015 wt.% C.

dependence Q–1S–K

~ ε1/2p is not valid generally [205]. The maximum

height of the S–K peak for iron with 0.015 wt.% C after 20% de-formation of the specimens is obtained at 220°C after one hour. Itshould be taken into account that it is necessary to determine thecondition of the atmosphere of solutes during the formation of thecarbon S–K peak.

There are differences in the effect of nitrogen and carbon onS–K relaxation. According to Snoek, during measurement of Q–1 inloading with hertz frequency of nitrogen-alloyed iron we obtain ahigh, complete peak. In the case of carburised iron, the height of theS–K peak is initially small and after heating the material to the tem-perature range T ≥ T

max

S–K this difference may increase [209]. Simi-

lar observation can also be made when heating the Fe–C alloy forshort periods of time to 300°C. At the same concentration of thesolid solution, the height of the Q–1

S–K peak is approximately 4–5

times higher than in the case of the Fe–N alloy. The boundary con-centration for obtaining the maximum value of Q–1

S–K in nitrogen-

alloyed iron is approximately an order of magnitude higher, even

0.001 wt.%C

2 0 1


though the effective binding energy of C and N with the dislocationsin ferrite is similar (~1.28 × 10–19 J). The residual peak, observedat zero nitrogen content in the solid solution, was described in Ref.210. Its occurrence is associated with the partial breakdown of ni-trides during plastic deformation [211].

The characteristics of the S–K peak depend on the type of defor-mation and are determined not only by the density of fresh disloca-tions but also by their type and configuration [212].

In 1963, Shoeck [213] proposed the quantitative interpretation ofthe S–K peak. The model is based on the dragging of the atmos-pheres of solute atoms by dislocations moving under the effect ofrepeated external loading. The dislocations are regarded here as themodel of a spring and the interaction of the dislocations with thesolute atoms is interpreted as the unperturbed distribution of the sol-utes in the Cottrell atmospheres characterised by viscous movement.Consequently

( ) ( )5

1 22 2 4

0

,1

S K c

zQ T A l f z dz

z

∞−−

ωτ= ρ+ ω τ∫ (5.15)

where A is the coefficient of proportionality, ρ is the density ofmoving dislocations taking part during the formation of the S–Kpeak, l

c is the mean value of the length of the dislocation segment,

f(z) is the function of distribution of the length of the dislocationsegments l, z = l/l

c, and the value

2

3,d cc lkT

DG bτ = α (5.16)

where α is the proportionality factor, cd

and D are the concentra-tion and diffusion coefficient of interstitial atoms in the vicinity ofthe dislocation kernel.

Equation (5.15) shows that the maximum of internal friction ischaracterised by the spectrum of relaxation times. The distributionof the lengths of the dislocation segments can be described accord-ing to Keller by the equation f(z) = e–z. Consequently, the integralin equation (5.15) in the Shoeck model has the maximum value of2.2 at ωτ = 0.07 (in the Debye process ωτ = 1). The dependenceQ–1(T) in the Shoeck model is characterised by the dependence c

d(T)

2 0 2


and D(T), and D = D0exp(–Hd/kT), where H

d is the activation en-

ergy of the interstitial atom in the vicinity of the dislocation. Thedependence c

d(T) depends on the selected model of the atmosphere

around the dislocation.Several models of the S–K relaxation have been proposed for the

solid solutions with lower and high concentration of solutes, sincethe distribution of the solute atoms in the atmosphere of the dislo-cations is modelled by the Boltzmann, Fermi or Dirac approach[214, 215, 216, 217].

The relaxation in the presence of hydrogen is still the subject ofspecial interest [218]. The experiments show that the presence ofhydrogen suppresses the low-temperature α and β peaks on theQ–1(T) dependence, caused by the formation of a double bow simul-taneously on the mixed and screw dislocations [215], with the for-mation of the S–K peak. S–K relaxation is the confirmed phenom-enon of the Q–1(T) dependence at temperatures of 80–170 K indicat-ing the interaction of the hydrogen atoms with a dislocation. Themean value of the activation enthalpy of the relaxation process isH ≈ 31.8 s–1. The results show that the values are comparable withthe values resulting in the formation of S–K peaks in the Fe–C andFe–N systems.

Wirth [218] stressed the restrictions of the Shoeck model (Fig.5.8), when describing the relaxation interaction of hydrogen with thedislocation under external loading. For example, the activationenthalpy H = 31.8 kJ⋅mol–1 does not correspond to the enthalpy ofbonding of hydrogen with the dislocation (H

H= 58.6 kJ⋅mol–1). An

alternative model is based on the bending of the double bow of thedislocation atmosphere, formed by the hydrogen atoms under exter-nal loading (Fig. 5.8b). The relaxation time of the bent dislocationcan be expressed by the equation

2

2 2

2,

l

B G bτ =

π (5.17)

where l is the length of the dislocation segment, B is the disloca-tion mobility parameter.

When the dislocation travels the distance x < 1, the followingequation applies:

2 0 3


*2exp ,k kbD F

BkT kT

= −

(5.18)

When the dislocation moves through the distance x > 1

*2exp ,k kl D F

BkT kT

= −

(5.19)

where Dk

is the diffusion coefficient of the bow, 2F*k is the free en-

ergy required for the formation of a double bow with the criticalsize. Since in the case of the schema in Fig. 5.8b–3, it is not pos-sible to correlate the experimental data, we can use the schema ofbending (Fig. 5.8, b–4) and obtain the value of the frequency fac-tor irrespective of the value of H

H

0 2 4

2.

l kT

Gbτ =

π ν (5.20)

At T = 120 K υ = 1013 s–1 and, therefore, τ0= 2.6 × 1017l/b, which

is in agreement with the experimental observations, since l/b = 103

to 104 [219]. The model of bending of the double bow with the at-mosphere generated by hydrogen indicates that the absorption of hy-drogen suppresses the first peak (α) and displaces the second peak(β) of internal friction to lower temperatures.

FFFFFigigigigig.5.8..5.8..5.8..5.8..5.8. Models of bending of a dislocation with the atmosphere formed by hydrogenatoms (a) and kinks formed in double bow (b): 1) without stress; 2) under stress;3) 3,4) dislocations with a bow without hydrogen (3) and with hydrogen in dislocations(4); 5) bending of a kind with the formation of a double bow.

2 0 4


FFFFFigigigigig. 5.9. . 5.9. . 5.9. . 5.9. . 5.9. Dependence of the peak formed at 250°C (Q–1max

or Q–1SK

) of quenchedcarbon steels on carbon content (a, b), quenching temperature (c) and subsequenttempering (d) at a carbon content (wt.%C) of: 1) 1.16; 2) 0,92; 3) 0.71; 4) 0.32;5) 0.12 (f = 1 Hz).

5.2.3 Phenomena associated with martensitictransformation in steel

Since martensitic transformation took place in the steel, the depend-ence Q–1(T) shows a peak similar to the S–K peak. The general re-lationships governing the formation of the peak recorded at 250°C,at a loading frequency of ~1 Hz, in the specimens of carbon steelssubjected to quenching and tempering were presented in Ref. 220and 221. The height of the peak increases with the increase of thecarbon content of the steel and with increasing quenching tempera-ture (Fig. 5.9), and T

max is displaced higher temperature. The in-

crease of the tempering temperature of steel decreases the height ofthe peak. Many authors interpret this peak on the basis of modelsused to explain the Snoek and Köster peaks. However, this relaxa-tion process has certain different features.

Alloying elements, influencing the bonding of dislocations and in-terstitial atoms and changing the breakdown of cementite during

wt.%C wt.%C

0.9 wt.%C

2 0 5


heating, also change the parameters of the peak of the Q–1(T)dependence. In steels alloyed with several elements there are alsoother additional peaks [222]. Special attention should be given tothe data on the origin of the peak recorded at 160°C in quenchedFe–Ni–C materials. The authors of Ref. 222 showed that the for-mation of the peak is associated with the movement of twin bounda-ries containing moving carbon atoms, when the material is subjectedto alternating external loading. Twinned martensite makes it possi-ble to detect a peak at a temperature of 160°C on the Q–1

(T) de-

pendence at a frequency of ~1 Hz in alloyed steels, dislocation mar-tensite at 250°C, and mixed martensite shows both peaks on theQ–1

(T) dependence.

The Snoek and Köster phenomenon and its analogy in thequenched carbon steels are only some of several interesting repre-sentations of the interaction of solute atoms with the dislocations.It can be expected that further work in this area will provide moredetailed relationships between the value of internal friction and thediffusion parameters of the atoms, the movement of grain bounda-ries and behaviour of carbon atoms under external loading, espe-cially in alloys with complicated microstructure and structure.

5.2.4 Migration of solute atoms in the region withdislocations

It will be assumed that low temperatures are those at which T < T0=

–H/kln(fτ0) and at these temperatures the solute atom does not move

under the effect of external loading. High temperatures are those atwhich T > T

0, which means that the atom of the solute at time

∆t = 1/f completes a certain number of jumps, i.e. diffuses to a cer-tain distance (f is the frequency of changes of external loading).

In the low-temperature range, all internal friction mechanisms canbe divided into two large groups. The first group of the mechanismsis associated with the analysis of separation of the dislocation fromthe stationary atoms of the solutes forming the atmospheres aroundthe dislocations (unpinning mechanism). The second group of themechanisms is associated with the analysis of the relationships gov-erning the movement of the dislocation in the slip plane in which theatoms of the solutes are distributed ('friction' mechanism).

If the temperature is increased above T0, to the model is men-

tioned previously we can also add another group of mechanisms, inwhich anelasticity is manifested or depends strongly on the proc-esses of diffusion displacement of the solid atoms during the periodof external loading. If the material is subjected to mechanical stress

2 0 6


τm, it can be seen that each pinning point will be subjected to the

effect of the force F(τ) caused by the bending of the adjacent dis-location segments (see also section 3.5). The force can be dividedinto the component F and F

||, i.e. to the direction normal and par-

allel with the initial position of the dislocation line [223], deter-mined by the equations

( )2 2

2 21 2 ,

8

bF l l

m⊥τ= − (5.21)

( )|| 1 2 ,2

bF l l

τ= + (5.22)

where b and m are the Burgers vector and the linear elongation ofthe dislocation, l

1 and l

2 are the lengths of the dislocation segments

of the dislocations in the vicinity of the pinning point. The exist-ence of the forces F

I and F

II is the physical reason for the appear-

ance of many anelastic processes (Table 5.4) at T > T0 determined

by the effect of the normal or parallel component of the force dur-ing movement of the atom of the solute in the region of the dislo-cation kernel.

A special feature of these manifestations is that under the effectof external loading the dislocation lines do not separate from the at-mospheres of the solute atoms, even though the atoms themselvescan move along the dislocation line by the pipe diffusion mecha-nism. Thermally activated separation of the dislocations can takeplace as an independent process supplementing the phenomenacaused by the diffusion displacement of the solute atoms.

The phenomena of dislocation anelasticity, caused by the diffu-sion displacement of the atoms of the solutes in the atmospheresaround the dislocations are controlled by the transverse and longi-tudinal mobility of the atoms of the solutes in the dislocation ker-nel. This model can be used in the quantification of the appropri-ate diffusion characteristics.

A special feature of Table 5.4 is that it includes the mechanismof formation of the thermal–fluctuation relaxation peak of internalfriction (TF relaxation), caused by the separation of the dislocationsfrom the pinnig solid atoms. In this case, the separation of the dis-locations segments is initiated by the thermal–fluctuation jump ofthe atom of the solute which pins the segments in the direction nor-mal to the dislocation line.

2 0 7


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Analysis of the model in which two dislocation segments l1 and

l2 are pinned by the solute atoms under the effect of the defined

force law during displacement of the dislocation [224] indicates thatthe width of the relaxation peak will be greater than the width ofthe Debye peak with a single relaxation time, although the broad-ening of the peak will not be very large. It is also expected that theincrease of the stress amplitude will increase the peak with satura-tion. An anomaly of this dependence is that with the increase ofmechanical stress the peak is displaced to higher temperatures onthe temperature axis of the Q–1

(T) dependence. Analysis also shows

that the increase of the content of the solute changes the degree ofrelaxation and the position on the temperature axis in the samemanner as mechanical stress τ

mo.

At stress amplitudes of τmo

≥ 10–6 G and binding energies H =0.8 – 16 × 10–19 J, the activation parameters of the TF peak can bedetermined approximately from the equation

( )max 3 0ln 2 ,TF mH kT f= − π κ τ (5.23)

where Tmax

is the temperature at which the maximum of internal fric-tion is recorded at the loading frequency f, where κ

3 is the coeffi-

cient of proportionality expressed in units.Many features of the TF peak are similar to those of the Snoek

or Koster peaks; for this peak to form, the material must containdislocations and a solid solution with a relatively low solute con-tent. The temperature range in which the peak is observed is in thetemperature range of recording the time dependence of internal fric-tion.

The relaxation peaks formed as a result of the TF mechanismwere detected in solid solutions based on aluminium (Fig. 5.10) andcopper [2] and in the Fe–C interstitials solid solutions [224]. Theactivation parameters of the peaks H

TF and τ

0 of many solid solu-

tions are presented in Table 5.5. The level of the activation energyof the diffusion of solute atoms along the dislocations H was deter-mined by the method of the time dependence of internal friction.

The value of τ0 varies in the range 10–12 to 10–13 s–1, which cor-

responds to the value 1/υd, where υd is the Debye frequency of thevibrations of a free atom. The value of Q–1

in the process of meas-

urement of the time dependence of Q–1 is the function of tempera-

ture in measurements and reaches the highest value at temperatureswhere the TF is the peak of internal friction.

2 0 9


FFFFFigigigigig.5.10..5.10..5.10..5.10..5.10. Effect of temperature and frequency on internal friction of Al–0.02wt.% C: a) effect of temperature on Q–1 at different frequencies: 1) f = 1.3 Hz,2) 1820 Hz, 3) 2820 Hz, at strain amplitudes during measurement: 1) ε = 2.42× 10–5, 2,3) 4×10–6; b) frequency shift of the temperature at which the peak forms.

TTTTTaaaaabbbbble 5.5le 5.5le 5.5le 5.5le 5.5 Activation characteristics of different peaks for fcc and bcc metals

metsySHτ 01· 91

)J(τ 01· 21

)s(H

L01· 91

)J(Hτ/Hv

uC–lAgM–lA

C–eF

468.0800.1029.1

3.17.062.0

407.0487.0

–

44.035.083.1

The processes of dragging of the atmospheres of the solute ele-ments by the dislocations belong to the main mechanisms control-ling the dynamics of build-up of microplastic deformation of solidsolutions at elevated temperatures. Examination and quantificationof the TF of internal friction makes it possible to determine the ac-tivation energy of cross diffusion and, under the given conditions,also the type of solute atoms inhibiting the movement of the dislo-cations.

For the temperature range below τ0, the authors of Ref.225 pro-

posed a model of dislocation internal friction based on the mecha-nism of the thermally activated unpinning of the dislocation from thepinning solute atoms. Blair, Hutchison and Rogers presented amodel (BHR model) which is a modification of the internal frictionmechanism, proposed by Teutonico, Granato and Lucke. The BHRmodel takes into account the general interaction law between thedislocation and the solute atoms for the frequently encountered case

2 1 0


of the uniform distribution of the pinning centres along the dislo-cation line. Analysis of the replacement of the anelasticity mecha-nisms with increasing temperature or stress amplitude makes it pos-sible to observe, in the T –

τ

m coordinates, several regions with dif-

ferent processes of the unpinning of dislocations from the pinningpoints (Fig. 5.11):

– in the regions A and E, movement of the dislocations takesplace as a result of simultaneous thermally activated unpinning fromseveral solute atoms. In the regions B, C & D, the process of re-lease of the entire dislocation line starts with the separation of thedislocation line from a single solute atom;

– in the regions A & B, only a small part of the dislocations areseparated as a result of the effect of alternating external stress. Athigh temperatures (regions C, D and E), almost all dislocations areseparated in every loading cycle;

– in region D, the activation energy of separation of the dislo-cation is constant.

In other regions, one may expect the dependence of activation en-ergy on stress amplitude. Critical temperature T

k is linked with the

binding energy of the solute atom with the dislocation. The value ofH

v is determined by the equation H

v = kT

k ln (δ

1/ω), where δ

1 is the

effective frequency of oscillations of the dislocation segment.The BHR theory has been used in analysis of the redistribution

of the solute atoms in the atmosphere around the dislocations dur-

FFFFFigigigigig.5.11..5.11..5.11..5.11..5.11. Maps of mechanisms of thermally activated unpinning of dislocationsfrom solute atoms (a) and dislocation anelasticity of beryllium (b). Region 1 –thermally activated unpinning of dislocations from stationary solute atoms, 2)diffusion along dislocations, 3) diffusion dragging of solute atoms in the atmosphereby the dislocation.

2 1 1


ing their vibrations, by the mechanism of cross diffusion. The cur-rently available results indicate that every diffusion process mayresult in the formation of a relaxation peak on the Q–1

(T) depend-

ence. This hypothesis has been used to develop a single model of adislocation pinned at a single point [226]. The process of scatter-ing of energy is associated in this case with the diffusion draggingof the solute atoms by the dislocations, but no attention is given inthe model to thermally activated unpinning of the dislocations.

The displacement of the atom along the dislocation line by thepipe diffusion mechanism results in the formation of relaxation peak(peak α), with the relaxation time

( )2

2 2,c

L

l

Dα

τ =π + α

where

expC

LL L

HD D

kT = −

is the coefficient of diffusion along the dislocation,

2 32 2

0 .4

cG l

m k Tα = τ

If the stress amplitude is increased, the peak α will be displaced tolower temperatures. The degree of relaxation of the peak will in-crease.

The cross diffusion processes also lead to the formation of arelaxation peak (T peak), for which

,2

cT

T

l kT

D mτ =

where

2 1 2


0exp T

T T

HD D

kT =

is the coefficient of cross diffusion of the solute atoms in the dis-location kernel. Generally, the T peak is reflected in the superim-position of many elementary relaxation peaks and must be widerthan the Debye peak with a single relaxation time. It is useful todraw attention to the anomalous dependence of the characteristicsof the T peak on stress amplitude. If τ

m is increased, the position

of the peak on the temperature axis TTmax

does not change. Further in-crease of τ

m already causes the displacement of the peak on the tem-

perature axis to higher values.The development of diffusion processes in the dislocation kernel

results in the formation, in the temperature–stress dependence (Fig.5.11b), of two additional boundaries of regions of new dislocationrelaxation mechanisms (2, 3).

When examining the Q–1 (T) dependence for pure polycrystalline

beryllium, Levin [227] observed three Q–1 maxima (Fig. 5.12).

Analysis of the experiments showed that both high-temperaturepeaks formed as a result of the mechanism of pipe (L) and trans-verse (T) diffusion in the atmosphere of the solute elements. Thepeak P is the result of the thermally activated unpinning of the dis-locations from the stationary solute atoms, in accordance with theBHR model. The calculation of the activation parameters of the L,T and P peaks yielded data on the characteristics of the dislocationcomplexes and solute atoms in beryllium (Table 5.6) and the resultsare used to construct a general temperature–stress diagram (Fig.5.11b), expressing the operation of different dislocation anelasticitymechanisms, such as thermally-activated unpinning of the disloca-tion, the processes of diffusion along and normal to the diffusionline, and dragging of the dislocation by the solute atoms. Theboundaries of the additional regions are determined by the equations2πfTT

max = 1 and 4πfT

maxL = 1.

Three peaks, similar to P, L, T in Fig. 5.12, were also observedin zirconium [228]. Analysis of the data provides information on thetemperature–stress diagram, thermally activated unpinning of thedislocations and diffusion-controlled dragging of the solute atoms(probably oxygen) in single crystals of zirconium and in polycrys-talline zirconium.

2 1 3


FFFFFigigigigig. 5.12.. 5.12.. 5.12.. 5.12.. 5.12. Dependence of internal friction in heating (a) and cooling (b) and onthe square of frequency (c) on temperature, with the strain amplitude τ/G = 4×10–5.

TTTTTaaaaabbbbble 5.6le 5.6le 5.6le 5.6le 5.6 Anelasticity parameters determined for internal friction peaks

retemaraPskaeP

TLP

Hi

01· 91 )J( 422.0270.321.029.280.0690.2

Hβ 01· 91 )J( 690.2

lc

b061

H=ub/F

mb5.1

5.3 RELAXATION AT GRAIN BOUNDARIESMeasurement of internal friction associated with the relaxation atthe grain boundaries of polycrystalline materials provides a largeamount of information on the structure and properties of the grainboundaries. The Q–1

(T) dependence shows a peak which, after

analysis, provides information on the kinetics of changes at thegrain boundaries, taking under the effect of external loading; thesechanges depend on the structural condition of the polycrystalline

2 1 4


material (mean grain size, the presence of a dislocation substructure,the distribution of the group of the grain boundaries on the basis ofthe angle of misorientation or the same orientation, etc.). Thismakes it possible to investigate and evaluate the interaction of thegrain boundaries with other structural defects and also recombina-tion of the grain boundaries under the effect of various internal andexternal influences on the material. A low content of solute atomsalready has a significant effect on the nature of the peak associatedwith grain boundary relaxation due to the formation of the ss peak(the designation of the peak is taken from Ref. 229), which is theresult of the interaction of the grain boundaries with the solid at-oms. Analysis of the ss peak provides information on this interac-tion [230].

The peak, associated with the relaxation at the grain boundariesobserved on the Q–1

(T)dependence, was described and analysed for

the first time by Kê in 1947. The main data obtained in the meas-urement of the Q–1

(T) dependence in polycrystalline aluminium can

be summarised as follows [231]:–at a loading frequency of ~1 Hz and a temperature of 300°C,

the curves of the Q–1 (T) dependence for polycrystalline aluminium

show a peak associated with the presence of the grain boundaries,because this peak does not form when measuring the Q–1

(T) depend-

ence on single crystal aluminium;–the degree of relaxation of the peak is independent of the grain

size dz in the limiting case when the grain size is smaller than the

size of the specimen;–relaxation time τ is directly proportional to the grain size;–the activation enthalpy of the relaxation process is close to the

activation enthalpy of self diffusion in aluminium and τ0

~ 10–14 s;–the relaxation peak is considerably wider than the peak obtained

for the conventional inelastic material.Similar relationships were also recorded for other metals, so that

the observation made by Kê is universal.The interpretation of the relaxation of the grain boundaries and

the associated peak on the Q–1 (T) dependence which, to simplify

concentrations, will be referred to as GBIF, is based on the assump-tion on viscous sliding (accommodation) of the grains in theirboundaries, formulated by Zeener.

The results of a large number of experiments show that the phe-nomenon of grain boundary relaxation is more complicated becauseit depends strongly on the solutes manifested in the formation of thess peak.

2 1 5


5.3.1 Pure metalsThe peaks on the Q–1

(T) dependences for pure silver, gold, and cop-

per are considerably smaller and wider than in the case of alu-minium. The relaxation spectrum is characterised by a complicatedstructure reflected in doubling of the peak after high-temperatureannealing. The first peak corresponds to the Kê peak. The secondpeak is recorded at higher temperatures. The high-temperature maxi-mum is associated with grain boundary relaxation, because it is notfound in single crystals. In copper and nickel there is also a thirdpeak at transition temperatures. This peak is associated with the for-mation of annealing twins [232].

The relationship between ∆ and γ (where γ is the stacking faultenergy) in the case of gold, copper, nickel and aluminium is linear[233]. The width of splitting

2 .24

G br =

πγ

The results show that the activation enthalpy H and quantity rare linked by the dependence similar to the dependence of ∆ on r(Fig. 5.13). These dependences have been confirmed for many met-als with different crystal lattices [66].

However, other data are also available. For example, in the caseof copper [234] the height of the peak increases with a decrease ofthe grain size and the width of the peak decreases. For 99.999%copper, the value of ∆ is independent of the grain size [235]. In thecase of nickel, ∆ ~ d–1

g in a wide range of d

g. Similar dependences

were also recorded in the case of iron and copper of commercialpurity [236]. If ∆ depends on grain size, this is observed in a spe-cific range of d

g and has the form ∆ ~ d–1

g. The particles distributed

at the grain boundaries form internal stresses around them and com-plicate grain boundary sliding. The phenomenon is independent ofthe elasticity coefficient of the particle.

The change of the relaxation time of the peak associated withgrain boundary relaxation and its activation enthalpy are not inagreement with these values determined for the Kê model. For ex-ample, for pure metals τ ~ dn

g, where 1< n < 2 (n = 2 for Fe, n =

1.86 for Al). The activation enthalpy of the peak associated withgrain boundary relaxation for the pure metals is lower than the ac-tivation energy of self diffusion. The results show [237] that thevalue of H with increasing purity of the metal tends to the activa-

2 1 6


tion enthalpy of grain boundary diffusion. Since the relaxation timeof the peak can be described by the Arrhenius equation, τ

0 is simi-

lar to the value of this factor for the relaxation of point defects(~10–14 s). On the other hand, it was observed [236] that in the caseof pure iron the value of τ

0 decreases with decreasing d

g which

means that it depends on the microscopic parameters of the struc-ture. This phenomenon has not been completely explained.

The peak on the Q–1 (T) dependence, associated with grain bound-

ary relaxation, is described by a set of relaxation times. The param-eter of the logarithmico-normal distribution β changes over a widerange, and in some cases β = 7. The value of this parameter de-pends on temperature which means that there is a set of the valuesof τ

0 and H. The peak has several components forming a wide pro-

file which also indicates that the distribution of the relaxation pa-rameters is not in agreement with the logarithmico- normal distri-bution.

5.3.2 Solid solutionsThe interaction of the grain boundaries with the atoms of the sol-utes in the solid solution results in the redistribution of the relaxa-tion spectrum, and the behaviour of the interstitial and substitutionalsolid solutions differs. Pearson and Rotherham [238] examined cop-per and silver with an addition of ~1 wt.% of the solute and con-cluded that in comparison with the pure metals, the peak on theQ–1

(T) dependence (RM peak) is lower and a ss peak is recorded

at high temperature and is caused by the interaction of solute atomswith the grain boundaries. The activation energy of the peak result-ing from the presence of the solutes is close to the activation en-

FFFFFigigigigig.5.13..5.13..5.13..5.13..5.13. Dependence of the degree of relaxation (a) and relative activation enthalpy(b) of different metals on the extent of splitting r (H

0 is the enthalpy of self

diffusion of pure metal).

2 1 7


ergy of diffusion of the solutes [239] and the activation enthalpy ofself diffusion of the solvent. The value of τ

0 was 10–14 to 10–16 s,

as in the case of the RM peak. The relaxation spectrum of the sspeak is wider than the RM peak. When the concentration of thesubstitutional additions in copper is increased, the original peak forrelaxation at the grain boundaries (RM peak) and the peak due tothe presence of the solutes (ss peak) increases. With the decrease ofthe height of the RM peak its activation enthalpy increases. In thebinary and ternary alloys based on chromium and containing La, Ni,Fe and V, the spectrum of the Q–1

(T) dependence at high tempera-

tures contained complicated boundary peaks [240].At present, there are insufficient experimental data on grain

boundary relaxation for the case in which the solid solution is in-terstitial. The Fe–C and Fe–N alloys have been studied in detail[224]. The main difference of the relaxation spectra for thesubstitutional and interstitial solid solutions is that the interstitialadditions already have an effect at low concentrations, i.e. theychange the temperature at which the peak is formed and also theactivation enthalpy of its formation. The interstitial solid solutionsdid not contain the isolated peak corresponding to the presence ofthe solutes. The concentration dependences of the parameters of thepeaks in the Fe–C and Fe–N alloys are shown in Fig. 5.14. Thesaturation of the grain boundaries with the solute atoms is found atc ≅ 10–2 at% in the Fe–C alloys. A large decrease of the height ofthe peak with a further increase of the concentration is associatedwith the precipitation of the particles of the corresponding phase atthe grain boundaries. In the iron alloys, the internal friction, asso-ciated with grain boundary relaxation, decreases with increasinggrain size.

5.3.3 Relaxation modelsRoberts and Barrand [242] were the first authors to propose thequantitative interpretation of the peak on the Q–1

(T) dependence,

associated with grain boundary relaxation. Their interpretation wasbased on the empirically determined relationship between the degreeof relaxation of internal friction and the width of splitting of thedislocations. They assumed that, depending on the stacking faultenergy of the metal, lattice dislocations in the vicinity of the grainboundaries move under the effect of stress either by climb or slip.In the metals such as aluminium or nickel with a high value of γ,the internal friction, caused by the relaxation at the grain bounda-ries, is the result of dislocation climb and the activation enthalpy

2 1 8


of the process is close to the activation enthalpy of self diffusion.In the metals with the low value of γ (for example, Ag), the extentof dislocations splitting is very high and the relaxation process takesplace by dislocation climb at the grain boundaries because non-conservative displacement of the split dislocations is not advanta-geous from the energy viewpoint.

The activation enthalpy of the peak on the Q–1 (T) dependence as

a result of grain boundary relaxation: in the metals with averagevalues of γ (Au, Cd), it is assumed that the dislocations move bya combined mechanism consisting of slip and climb at the grainboundaries. This is also in agreement with the activation of theenthalpy of the process whose value is between the values for dif-fusion at the grain boundaries and in the volume of the metal.

In single crystal aluminum, the Q–1 (T) dependence shows an

internal friction peak at a temperature of 365°C and a frequency of~ 1 Hz. This peak is the result of the process of climb of the splitdislocations under the effect of stress [243]. Other relaxation phe-nomena, associated with the polygonisation process in aluminiumand diluted Al–Cu solid solutions, have been published in Ref. 244and 245.

Good correlation has been found between the formation of thepeak on the Q–1

(T) dependence at 265°C and the changes of the mi-

crohardness of aluminium. This can be used as an additional methodfor the indirect evaluation of the hardness of materials. The spec-trum of the peaks on the Q–1

(T) dependence for complicated sys-

tems, such as fibre-reinforced composite materials, is not a singlesuperimposition of the phenomena taking place in the components ofthe composite at the fibre–matrix interface. The phenomena associ-ated with relaxation at the grain boundaries or sub-boundaries can

wt.% wt.%

FFFFFigigigigig.5.14..5.14..5.14..5.14..5.14. Dependence of the degree of relaxation (a) and activation enthalpy (b)on the content of interstitial elements in alpha iron.

2 1 9


also be detected when the size of grains all subgrains is smaller thanthe spacing between the fibres [246].

5.4 ANALYTICAL PROCESSING OF THE RESULTS OFMEASUREMENTS

The discussed mechanisms of anelasticity in metallic materials andthe analysis of the mechanisms indicate that the internal frictionmeasurements can be used efficiently in quantification of a largenumber of specific relationships in metals and alloys. Some exam-ples are given in the following part of the book.

5.4.1 Solubility boundariesThe solid interstitials solutions of metals with the bcc lattice arecharacterised by the linear relationship between the height of theSnoek peak Q–1

max (if the background of internal friction is sub-

tracted) and the concentration of interstitial atoms n (wt.%) in theform

1max ,n pQ−= (5.24)

Here p is the coefficient of proportionality whose values is deter-mined by the type of atoms forming the solid solutions and dependson the grain size, texture and other factors [66]. The value of p fordifferent interstitial atoms is not a quantity changing in a simplemanner (for example, for C in polycrystalline α-iron p ≈ 1.28 – 1.3,and for N p ≈ 1.26 – 1.3). In fact, it is not necessary to know thevalue of p when determining of the solubility boundaries.

A suitable example is the analysis of the Fe–C system. Specimenswith different carbon content (to 0.02 wt.%) should be annealed atdifferent temperatures, followed by rapid cooling and measurementsof the Q–1

(T) dependence. After evaluating the height of the peak

and subtracting the internal friction background, with determinationof the activation enthalpy of the peak, it is possible to plot the de-pendence shown in Fig. 5.15. The graph summarises the results ofmeasurements of the solubility of carbon in ferrite obtained by themethod of depleting the solid solution in carbon [247] with the val-ues obtained from the height of the Snoek peak of the specimenscontaining 0.04 wt.% C, grain size > 1 mm [248]. At p = 0.9, thesolubility values are in agreement. If the temperature of the speci-mens with the C content lower than the solubility limit is increased,

2 2 0


the value of Q–1max

rapidly decreases (Fig. 5.15b). The authors of Ref.247 and 248 assume that the concentration of C in the areas ofgrain boundary segregation is close to the eutectoid concentrationand austenitic layering forms at subcritical temperature. This the de-creases the C content in the lattice of residual α-iron. In the speci-mens with 0.4 wt.% C, cooling from temperatures higher than A

c1

decreases the value of Q–1max

as a result of dissolution of carbon inaustenite.

Consequently, it is possible to determine the limiting concentra-tion of the solubility of carbon in α-iron [249], the solubility lim-its in ternary and more complicated systems (for example, Fe–Cr–C) and also evaluate the vacancy–interstitial atom complexes. Simi-lar measurements by the internal friction method can also be takenat low temperatures (to 150°C) when the solubility of carbon in α-iron is very low.

The possibility of evaluating the content and solubility limit ofinterstitial atoms in the interstitial solutions in the metals and al-loys with fcc and hcp lattices is based on the Finkel’stein and Rozinprinciples. The values of n and Q–1

max are linked by the equation

10 max ,n n K Q−− = (5.25)

here n0 is the critical concentration of the interstitial atoms (at

FFFFFigigigigig.5.15. .5.15. .5.15. .5.15. .5.15. Dependence of the height of Snoek peak (QS–1= Q

40–1, f 1 Hz) and carbon

concentration in ferrite (a), dependence of the height of the Snoek peak on temperature(b) for the carbon contents in wt.%. 1) 0.003, 2) 0.010, 3) 0.040, 4) 0.400.

wt.%

2 2 1


n < n0 the dependence of Q–1 on n is quadratic), K is a coefficient

which takes into account the structure of the alloy. The deviationfrom the linear dependence (equation 5.25) is utilised when deter-mining the solubility limit.

The substitutional solid solutions are characterised by Zeener re-laxation. The following equation is valid for many specific systems

2 1max ,n mQ−= (5.26)

where m is the structure-sensitive coefficient of the alloy. Thismethod has been used successfully for evaluating the solubilitycurves of Ag–Al, In–Tl, etc. alloys [26].

5.4.2 Activation energy and diffusion coefficientThe condition for the formation of a relaxation peak in the formωτ

0 exp (H/kT) = 1 enables several methods to be selected for de-

termination of the activation enthalpy. By changing the loading fre-quency, it is possible to determine H from equation (5.1). The ac-curacy in determination of H can be increased by constructing thedependence Q–1

x = Q–1/Q–1

max on T–1, in which the internal friction

background is subtracted from the internal friction peak so that itis possible to decide whether the spectrum has activation energy ornot. If the change of the frequency of external loading causes nochange of the form of the Q–1

x – T–1 curve, the process is character-

ised by a single relaxation time. Otherwise, the spectrum has acti-vation enthalpy H. In most cases, the value of H is determined us-ing the method based on the determination of the shift of the tem-perature at which the peak is recorded on the Q–1

(T) dependence,

utilising the equation derived by Wert and Marx (equation 5.3), tak-ing into account the fact that the contribution T ∆S changes onlyslightly the value of H. Consequently, we can use the equation inthe following form

maxmax

max

ln ,kT

H RT=ω (5.27)

where ωmax

is the frequency of external loading at the temperatureat which the peak appears. The approach proposed by Wert andMarx is limited to the relaxation processes induced by the thermalactivation of atom migration under the effect of external stress so

2 2 2


that the relaxation mechanism must be known in advance.The value of H is determined from the angle of inclination of the

low- and high-temperature part of the peak of the Q–1 (T) depend-

ence in the coordinates ln Q–1 vs 1/T. This method is suitable onlyfor very narrow peaks of the Q–1

(T) dependence.

The value of H can also be determined from the width of thepeak on the Q–1

(T) dependence measured at the height 1/2 Q–1

max. The

results are satisfactory when the relaxation process is characterisedby a single relaxation time. Since the peak is formed by a spectrumof relaxations, the value of H will be lower than the physically sub-stantiated value of H.

The determination of the bulk diffusion coefficient by internalfriction measurements is based on linking the relaxation time τ withthe diffusion coefficient D (equation 5.4). The method was describedin details when discussing the interpretation of the Snoek peak. Itshould be noted that the actual displacement of the point defect (thefrequency of jumps) is f times smaller than the corresponding val-ues obtained within the framework of the model of random experi-ments. The value of the correlation factor depends on the type oflattice and the diffusion mechanism [2,26]. For example, for selfdiffusion with the vacancy mechanism, the value of f changes from0.78 (for fcc lattice) to 0.5 (for rhombic lattice). If the value of fac-tor f is available, it is possible to calculate more accurately the ac-tual diffusion coefficients from the change of the relaxation time.

Internal friction measurements also make it possible to determinethe pipe diffusion coefficient, from the change of the temperature offormation of the peak on the Q–1

(T) dependence and also from the

time dependence of internal friction. The utilisation of the thermal-fluctuation peaks (TF relaxation) for the evaluation of the activa-tion parameters of diffusion of solute atoms along the dislocationsis discussed in section 5.2.4.

The coefficient of diffusion of the solute atoms along the core ofthe edge dislocation can be determined by measuring the timedependence of internal friction in accordance with Ref. 249. Thephenomenon of time dependence of Q–1

(T) is determined by the mi-

gration of solute atoms along the dislocations under the effect of ex-ternal stress in the areas of strong pinning which changes continu-ously the distribution of dislocations segments and results in theformation of large adsorbed atoms of the solutes, free from dislo-cations segments. When the stress σ

k is higher than some critical

value σc, the value of Q–1

becomes time-dependent. If σ is reduced

in such a manner that σ < σc, the Q–1

(T) dependence is character-

2 2 3


ised by a single relaxation time. Consequently, we obtain the de-pendence of τ on the pipe diffusion coefficient D

d in the form

22,c

d

lD =

τ (5.28)

Relaxation time is determined by transformation of the experi-mental data to suitable coordinates, for example

( )1 1

1 10

ln ,TQ Q

stQ Q

− −∞

− −

−= ν

−

where Q–10 and Q–1

∞ are the values obtained under the effect of σ andin its absence, t is time, and the values of l

c are determined by an

independent method (for example, from the curves of the Q–1 (T)

dependence). The method is described in detailed in Ref. 2.

5.4.3 Breakdown of the solid solutionThe kinetics of precipitation of a phase in alloys is characterised bythe exponent n in the equation proposed by Wert and Zeener [250]according to which the fraction of the particles precipitated from thesolid solution q is determined by the equation

1 exp ,n

tq = − − τ

(5.29)

where t is time, τ is the characteristic process time. The value ofn depends on the mechanism of the precipitation process. The effi-ciency of this approach can be seen in, for example, Ref. 251.

When evaluating the precipitation of carbon or nitrogen from thesolid solution during the thermal and deformation breakdown of thesolid solution of α-iron, the Snoek maximum can be described bythe following equation:

1 1max 0 max

1 1max 0 max

1 ,tQ Qq

Q Q

− −

− −∞

−= −

− (5.30)

2 2 4


where Q–1max t

and Q–1max 0

are the heights of the Snoek peak at time tand t = 0, with Q–1

max ∞ being the same, but at t → ∞. The transfor-mation of the results to the coordinates log ln (1/1 – q) vs ln t canbe utilised when determining the precipitation factor n which char-acterises the breakdown mechanism. At the same time, it is possi-ble to calculate the activation enthalpy and the rate of the processof breakdown of the solid solution [252]. The carbon content ofstructural steels is higher than the concentration associated with thesaturation of the solid solution. In principle, 'reversible' dissolutionof the phases, containing carbon and nitrogen, in ageing can beevaluated from ‘reversible’ dissolution during time-limited heatingof the alloy to high-temperature.

Another example is from the area of the evaluation of the devel-opment of low-temperature brittleness observed at 475°C. From thekinetic parameters of the change of the height of the Snoek maxi-mum and from the splitting of the peak in the breakdown stage, itis possible to plot the diagram of breakdown of ferrite with a highchromium content (Fig. 5.16), as described in Ref. 254. In the caseof short-term ageing, the height of the Snoek peak rapidly decreasedand carbon and nitrogen precipitated at the dislocations (n = 0.5 –0.7). In this stage, 50–70% of interstitial atoms in ferrite precipi-tates from the solid solution (region A, Fig. 5.16). In continuingageing of quenched Kh25 steel (the concentration of C and N is0.01–0.2 wt.%) in a narrow temperature range (450–525°C) theSnoek peak is split as a result of the formation of microregions(s–i complexes) with different content of chromium and interstitialatoms into regions in later stages of the process; this is verified byanother, objective method (region C).

It is also possible to examine the processes of distribution of theinterstitial elements during isothermal annealing of supercooledaustenite in the steel on the basis of the Finkel’stein and Rozinpeaks (FR) which follow the redistribution caused by quenching tothe formation of the Snoek maximum and Snoek and Köster maxi-mum [255].

5.4.4 Intercrystalline adsorptionThe peak on the Q–1

(T) dependence, associated with grain bound-

ary relaxation, is sensitive to intercrystalline absorption also at alow solute content.

Piguzov and Glikman [256] observed a case of intercrystallineadsorption of phosphorus in steel. The peak on the Q–1

(T) depend-

ence at 300°C (f ≈ 1 Hz) is caused by the presence of hundredths

2 2 5


of a percent of P in alloyed iron and low-alloy steels. Temper brit-tleness is the result of adsorption enrichment of the grain bounda-ries with phosphorus. Investigations were also carried out on an ironalloy alloyed with Ni and Cr where the enrichment of the regionsin the vicinity of the grain boundaries cannot develop by the non-adsorption mechanism [257].

The development of temper brittleness was evaluated by the au-thors by measuring the notch toughness of specimens fractured at– 196°C and also on the basis of the etchability of the grainboundaries in the saturated aqueous solution of zirconic acid.

The kinetics of the changes of plasticity at low temperature (Fig.5.17a) and of the size of the area below the peak belonging to phos-phorus (Fig. 5.17b) in relation to the embrittling time of alloyediron, containing 0.032 and 0.005 wt.% P, is shown in Fig. 5.17. Forthe alloy with 0.95 wt.% Si, 0.7 wt.% Ni and 0.032 wt.% P, thepeak initially increases and reaches the maximum value Q–1

gb max (the

concentration at the grain boundary cgb

= 0.5) and then decreases,and after tempering for 2 hours completely disappears (c

gb= 1). The

changes of plasticity at low temperature and of the etchability of thegrain boundaries indicate that the phosphorus concentration at thegrain the boundaries increases during the first two hours at 500°Cand then shows no significant changes. In the alloy with 0.95 wt.%Si, 0.7 wt.% Ni and 0.005 wt.% P, examination showed only theincreasing section of the dependence on tempering time; this is as-

FFFFFigigigigig.5.16..5.16..5.16..5.16..5.16. Regions of breakdown of ferrite with high Cr content: A) breakdownof the interstitial solid solution (complete – 1), to ∆HV

max (2), to 50% breakdown

(4); B) formation of s-i complexes at the total C + N content of 0.2 wt.% (1),0.04 wt.% (2) and 0.02 wt.% (3); C) formation of Cr-enriched zones.

2 2 6


sociated with slight embrittlement and decrease of the etchability ofthe grain boundaries [257].

The increase of the carbon content in iron with Ni and Cr resultsin the phenomenon in which the concentration of phosphorus at thegrain boundaries and the susceptibility to temper britt lenessdecrease. At a constant content of carbon and phosphorus, the phos-phorus concentration at the grain boundaries and the susceptibilityto temper brittleness in iron, containing manganese, is higher thanin iron with Ni and Si in cases in which the alloying elements notforming carbides result in the enrichment of the zones in the vicinityof the grain boundaries in carbon thus impairing intercrystallineadsorption of phosphorus and reducing the rate of embrittlement.

The change of the cgb

vs t ratio during intercrystalline adsorptionof P for five alloys with different phosphorus and carbon contentcan be described by the equation

0

2 20

4 2exp ,gbt gb

gb gb zz

c c Dt Dterfc

c c dd∞

− = − − αα

(5.31)

where cgb 0

is the initial concentration at the grain boundaries, cgb t

is the concentration after tempering for time t, cgb ∞

is the equilib-rium concentration at a specific tempering temperature, α is thec

gb ∞/cgb 0

ratio, D is the diffusion coefficient.The experimental points for different alloys are distributed on the

same curve. Analysis of the results indicates that of the possiblemechanisms, the mechanism with the highest rate is the controlling

FFFFFigigigigig.5.17..5.17..5.17..5.17..5.17. Change of the reduction in area determined at –196°C (a), the heightof the P addition peak of internal friction (b) in relation to tempering time at500 °C of alloyed iron with 0.032 wt.% P (solid circles) and the internal frictionpeak (broken lines).

2 2 7


one. According to the authors of [258], the mechanism of inter-crys-talline adsorption is associated with bulk diffusion to the nearestsub-boundary, i.e. dislocations, and is then transferred at a high ratealong the sub-boundaries to the grain boundaries.

Table 5.7 gives, for several alloys, the experimentally determineddata of the concentration at saturation and the temperature at whichthe peak caused by the atoms forms, and also the data on the bind-ing energy of the solute atoms with the grain boundaries.

5.4.5 Transition of the material from ductile to brittle stateIn a specific temperature range T

t, the strain rate or hydrostatic

pressure Pt, the material may show a change from the nature of fail-

ure from ductile to brittle.The aim of several investigations was to determine whether it is

possible to find a correlation between the changes of internal fric-tion and the transition of the material from the ductile to brittlestate (DB transition), for example, [259, 260]. Measurements of theQ–1

(T) dependence were taken, for example, at strain amplitudes of

ε ≅ 10–7, frequencies of f = 107, 105, 103, and 1 Hz, and also atε = 10–7 to 10–5 at frequencies of f = 105, 103 and 1 Hz, and alsoat ε = 10–5 to 10–4 at a frequency of 1 Hz. The investigations werecarried out using brittle polycrystalline metals with different tran-sition temperatures T

t (–50 to 200°C) and pressures P

t (1 × 108–

7 × 108 N m–2): Zn, Bi, Mo, Cr, steel 09G2, steel 35KhGSA and,for comparison, also single crystals of Zn, Bi, Mo, fine-grained Zn,polycrystalline aluminium and iron which are ductile in the exam-ined temperature range. It was expected that the selected non-cubicbrittle polycrystalline materials Zn and Bi would show, in the rangeT

t and P

t, two phenomena: the DB transition and plastic deformation

caused by the anisotropy of expansion and compression. Only theDB transition was expected in the case of the cubic metals. Thechange of the T

t by ±(10 – 40)°C was obtained by annealing, de-

formation and alloying of the material.The brittle materials in the range of T

t and P

p show a peak on

the Q–1 (T) curve. No peak is recorded in the ductile materials. In

order to be able to explain the nature of the peak, it is necessaryto know in detail the measurement conditions characterised by thefrequency of oscillations and strain amplitude. For example, thepeak at ε ≅ 10–7 is recorded at a frequency of 107 Hz in the rangeT

t (Zn, Mo) and P

t (Zn, Bi, Mo, Cr). At a frequency of 105 Hz, the

peak is recorded only in individual cases in the expected range ofT

t (Zn). At a strain amplitude of ε ≥ 3 × 10–7 the peak on the

2 2 8


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Q–1 (T) dependence is recorded at a frequency of 105 – 103 Hz in the

expected range of Tt and P

t (Zn). At a frequency of approximately

1 Hz in the expected range of Tt the peak is not recorded (Zn) and

is recorded at Tt (Zn, Mo, W, steels 09G2 and 35KhGSA) only in

a specific range of strain amplitude, ε = 1 × 10–5 to 1 × 10–4 forZn. No peak is recorded at lower or higher strain amplitude.

With the change of temperature or at high hydrostatic pressure,the peaks of Q–1

are recorded in different ranges. At low strain am-

plitudes in materials with a cubic lattice, there is no hysteresis andin the materials with a non-cubic lattice hysteresis is evident alsoat ε ≥ 10–7.

The peaks on the Q–1 (T) dependence, recorded at different val-

ues of f and ε, can be found in the same temperature and pressurerange corresponding to T

t and P

t. On the other hand, after a specific

increase of ε, resulting in significant changes in the substructuralthe material, changes are recorded in the position of the peak on theQ–1

(T) dependence, denoted by 1 and 2, and the height of the peak

also changes (in Zn).The change of the initial structure of the material results in the

shift of the peak after annealing to higher temperatures (Zn, Mo, W)and pressures, after prior deformation results in the shift to lowertemperatures and pressures (Zn), and after alloying to high or lowtemperatures, or no effect is recorded. The height of the peak doesnot change. Internal friction peaks can be recorded at lower tem-peratures in comparison with the range of changes of microhardnessand toughness, by approximately 10–40°C (Fig. 5.18).

The application of internal friction measurements in quantifica-tion of the temperature of the DB transition has certain generalcharacteristics.

In brittle polycrystalline materials with cubic and non–cubic lat-tices, the range of the values of T

t and P

t maybe characterised by

the formation of an internal friction peak under certain measurementconditions. As the oscillation frequency increases, the strain ampli-tude decreases. This indicates changes of the dislocation substruc-ture at T

t and P

t. Plastic deformation affects the height and position

of the internal friction peak. Consequently, an internal friction peakcan also be recorded after plastic deformation, after prior deforma-tion at low strains.

The height of the internal friction peak is determined by variousreasons which depend on the type of material, the frequency of os-cillations, amplitude of the material, etc. The internal friction peak,recorded at high loading frequency in the materials with a cubic lat-

2 3 0


tice, is associated with the DB transition. The phenomenologicalmodel proposed in Ref. 261 explains this conclusion by the relaxa-tion of the density of mobile dislocations. The position of the peakis determined by T

t and P

t and its height depends on the frequency

of oscillations. The internal friction peak, determined at a high load-ing frequency of materials with a non-cubic lattice, may be associ-ated with the DB transition and the peak determined at a low load-ing frequency and high values of ε may be associated withmicroplastic deformation taking place during the DB transition.

The temperature or pressure position of the peaks in the range ofthe values of T

t and P

t is associated, according to the authors of

Ref. 261, with the start of the development of the DB transition (Ttx,

Ptx). The value T

max tx depends on the structure of the material, like

Tt.To conclude this section, it should the added that the value of T

t

can be obtained from the results of the notch toughness test, thedynamic tensile test, the torsion test, etc. [262]. Every method givesdifferent values of T

t. It is evident that it is necessary to carry out

extensive experiments to determine the correlation of mechanicalmeasurements and internal friction measurements, especially for thecorrect physical interpretation of the observed dependence with spe-cial attention given to the fact that the DB transition is character-istic of the bcc materials with additional interstitial atoms [262].

5.4.6 Relaxation movement of microcracksArmco iron, subjected to thermomechanical and thermomagnetic

FFFFFigigigigig.5.18..5.18..5.18..5.18..5.18. Effect of temperature on the properties of zinc at the DB transition(Q–1 measured at ~1 Hz).

2 3 1


treatment, shows another internal friction peak at –50°C and a fre-quency of ~1 Hz [263]. Measurements taken on 20GS2 high-strengthsteel at ~1 Hz, subjected to long-term strength tests, after differentholding periods in corrosive media, also showed a similar peak (Fig.5.19) [264] in the range 20–50°C under the effect of tensile stresses(0.5–0.8) R

p0.2. The activation energy of the relaxation peak, meas-

ured at a loading frequency change, was ~57 kJ⋅mol–1. The peak onthe Q–1

(T) dependence was recorded at a specific test time. The time

to the appearance of the peak and the rate of its development (in-crease of the height of the peak) depend on the acting stress, forexample, when σ = 0.5R

p0.2 the peak is detected at the time t > 0.5

h, whereas at higher stresses this time is significantly shorter.Specially organised experiments showed that the formation of the

peak is not typical of strengthened steel in the condition in whichmicrocracks can form [264]. For example, after tensile loading thespecimens of 20GS2 strengthened steel up to the formation of a neckin the central part where the microcracks appear, the height of thepeak is almost doubled. The relaxation peak, associated with theformation of microcracks, was described in Ref. 275. Measurements,taken on grey cast iron, show that T

max ≈ 50°C at 0.83 kHz and the

activation enthalpy is 69 + 5 kJ⋅mol–1. Differences in the mobilityof the front of the areas containing graphite are the result of dif-

FFFFFigigigigig.5.19. .5.19. .5.19. .5.19. .5.19. Temperature dependence of internal friction (a) of 20GS2 steel aftertensile loading without necking (1), after necking (2), after long-term strengthtest at σ = 0.6 R

m after 0.25 h (3), 1 h (4) and 2 h (5) loading and the time

dependence of internal friction (b) under the effect of σ: 6) 0.5 Rm, 7) 0.6 R

m,

8) 0.7 Rm, 9) 0.8 R

m for 20GS2 steel.

2 3 2


ferent curvature of the front of these ‘microcracks’. This results inthe broadening of the peak on the side of the internal friction whichdepends on higher temperature. Tests of Cr18Ni9 steel in a solution160 g CuSO

4·5H

2O, 100 ml H

2SO

4 and 1 l of water showed always

an anomalous increase of Q–1 at temperatures of approximately 80°C

(f ≈ 0.9 kHz), which corresponds to the formation of submicroscopiccracks in the material.

The formation of the peak on the Q–1 (T) dependence corresponds

to the time to the formation of mobile microcracks and also thebuild-up of other structural defects, for example, dislocation clus-ters, cavities, etc. After recording the maximum peak of internalfriction, it can be seen that the surface of the specimens containsnetworks of visible cracks, and the modulus of elasticity decreasesby up to 30% of its initial value.

The interaction of external stress at the front of the movingmicrocracks and around clusters of dislocations at the front of thepropagation results in the formation of suitable conditions for theacts of opening and closure of microcracks and scattering of me-chanical energy by relaxation. The process of migration ofmicrocracks under the effect of stress can be divided on the basisof the ductility of the material. In the case of brittle material andat a high stress cracking takes place by the disruption of cohesionbonds at the front of the crack up to complete failure, or until thecrack encounters a barrier (grain boundary, including, particles ofother phases, etc.). In ductile materials, crack migration is a gradualprocess of merger of many microcracks into a main crack.

The plastic relaxation at the tip of the cracks results in intersec-tion of planar dislocation clusters (Fig. 5.20). When individual dis-

FFFFFigigigigig.5.20..5.20..5.20..5.20..5.20. Interaction of the crack with planar dislocation clusters (a) and thermalfluctuation formation of the microcrack (b).

2 3 3


locations approach each other, they merge and form a microcrack(Cottrell mechanism for the fcc metals). The microcracks can formby the thermal fluctuation mechanism also in the absence of exter-nal stress, for example, in planar clusters at the crack tip [266].The equilibrium depth of thermal fluctuation microcracks, even inthe case of a large number of dislocations in a planar cluster, is thevalue equal to several multiples of the Burgers vector, so that themerger of the microcrack with the main crack may or may not takeplace. In the case of cyclic loading, the process of formation ofmicrocracks is interpreted as the emission of a bow of the disloca-tions to the side and subsequent joining of the parallel parts. Theprocess of crack closure can be interpreted as the movement of thebow back to the original dislocation. The energy for opening of thecrack under the effect of the stress, generated at the tip of the maincrack, is several times lower than the energy required for emissionof the bow.

234


6

CYCLIC MICROPLASTICITY

Taking into account the promising nature and prospects of the prac-tical application of the results, and taking into account chapter 3.3,it is useful to pay special attention to the effect of the strain am-plitude on the internal friction of the materials and the defect of theelasticity (Young) modulus in the range characterised by the exten-sive generation, movement and interaction of the dislocations, i.e.in the region of cyclic microplastic response of the materials.

Mott [267] assumed that in the presence of randomly distributeddislocations situated at certain distances from each other, the ran-dom distribution of the stress exerts an effect

1/ 2

.i

Gbρσ =π

The formation of new dislocations requires the critical stress σkr

=σ

i + Gb/l. If a new dislocation, generated by a source, moves by the

distance d and the number of these dislocations is n, the strain in-creases by the value nbd. It is assumed that the sources generatedislocations in an avalanche like manner and that only some sourcesoperate. The avalanche–like nature of generation of the dislocationsis determined by the fact that the activity of the sources depends onthe sum Gb/l and the local value of σ

i and not on their means value.

Local stress fluctuations generate the avalanche–like formation ofdislocations whose intensity increases with the increase of the de-gree of nonuniformity of distribution of stress in the volume. Un-der the effect of the stress with the reverse orientation, new sourcesoperate and generate the dislocations of the opposite sign.

Golovin [268] determined the internal friction in the microplasticregion in the form

235

Cyclic Microplasticity

( )2 21 a

p mt

nb dQ

q− ρ

=ε (6.1)

where ρa is the density of active sources of dislocations, q is the

distance between the slip planes, m, B are constants depending onthe dispersion of the stress distribution curve, ε

t is the strain at

which microplasticity appears.The actual conditions of dispersion of the energy in the material

are complicated by the fact that the dislocation overcomes obstaclesduring its movement.

It is assumed that ρ0 is the number of dislocations with the length

Ln. During displacement, the dislocation changes its shape, for ex-

ample, in accordance with Fig. 6.1. The probability of the jumpfrom the position M to position N is described by the equation

( )0exp iU bdl

kT

− σ − σ −

(6.2)

Consequently, internal friction is

( )4011 0 0

200

1exp ,in D

p

bdlr L b G UUQ

kT kTl C

σ − σρ = − ν σπ (6.3)

Here r ≈ 0.5–0.75, σi is the internal stress caused by the presence

of point defects, UD

= WM

for non–concentrated solutions, and U0

~0.5W

M for concentrated solutions. The value of l/b changes from

FFFFFigigigigig. 6.1.. 6.1.. 6.1.. 6.1.. 6.1. Diagram of displacement of dislocations in the stress field.

236


( )

1/3

0 0

2

i

G

C

σ −σ

for the non–concentrated solutions to

1/33

202 M

Gb

W C

for concentrated solutions.Davidenkov [269] used the structural heterogeneity of the mate-

rial as the basis for scattering of energy in the material. In repeatedloading, some subgrains and grains are subjected to microplasticdeformation. From many equations, some of the equations are usedfor the numerical description of the hysteresis loop in the form

00

,nE af εσ = ε± ε ε

(6.4)

where f(ε/ε0) is the function describing the shape of the hysteresis

loop.This approach was developed further by Pisarenko [270] who

introduced the stress distribution function in the grains and assumedthat the specimens, loaded to a level lower than the yield limit inthe grains, will differ and some of the specimens may show stresseshigher than the yield limit.

The amount of energy scattered in the unit volume of the mate-rial during a single loading cycle is

( )2 ,x

sW NK P d∞

σ

∆ = σ σ∫ (6.5)

whereN is the number of grains in the unit volume of the material,K

s is the mean value of the capacity for absorption of energy in the

microvolume. The integral in equation (6.5) is the quantity that de-termines the number of plastically deformed grains in the unit vol-

237


ume of the material at a specific mean which depends on the yieldlimit of the grain, the mean stress distribution in the grains and onthe shape of the size distribution curve of the grains. The stress dis-tribution in the grains has the form of a Gaussian dependence. Con-sequently

02 ,m

ss

k

W NK A B σ ∆ = + σ

(6.6)

where m, B, A are constants whose value depends on the dispersionof the stress distribution in the grains. After further transformations,the following equation is obtained for the determination of internalfriction

1 2,msp m

k

E N K BQ− −= − σ

πσ

(6.7)

where σk is the yield limit. The equation describes satisfactorily the

distribution of internal friction in relation to stress amplitude σ.The problem of the effect of the magnitude of the stress ampli-

tude on internal friction and the defect of the modulus of elastic-ity at higher values of the repeated loading is directly linked withthe problems of cyclic microplastic deformation and quantificationof the cyclic plastic response of materials.

Taking into account the attempts for the quantification of the re-lationship between the values of the strain amplitude of the inten-sity of changes of Q–1, in relation to strain amplitude (α =d log Q–1/d log ε

ac) and the change of the response of the material

to loading, it is useful to introduce a convention in the designationof the critical amplitudes (ε

kr) and the intensity of changes of char-

acteristics (α).

6.1 CRITICAL STRAIN AMPLITUDES AND INTENSITY OFCHANGES OF CHARACTERISTICS

According to Puškár [24] and the latest experimental results, thedependence Q–1

– ε or δ – γ can be conventionally divided into four

sections (Fig. 6.2).

238


Under the effect of strain amplitudes with the value to εkr1

(thefirst critical strain amplitudes), the increase of ε

ac is accompanied

by internal friction independent of the strain amplitudes. This is thebackground of internal friction Q–1

0 (section I, where the intensity

of changes of Q–1 is α

0 = 0). Under the effect of strain amplitude

higher than εkr1

but lower than another critical value εac

, there is aslight increase of the magnitude of internal friction, and with the in-crease of ε

ac recording shows a large increase of internal friction,

which is the manifestation of cyclic microplasticity. This region ischaracterised by the occurrence of plastic internal friction Q–1

p (sec-

tion III, where α2

> α1). With a further increase of the strain am-

plitude it is possible to reach εkr3

(third critical strain amplitude),and at values above this amplitude

internal friction depends strongly

not only on the strain amplitude but also the loading time Qt–1 (sec-

tion IV, where α3

> α2).

This convention and the occurrence of all or several strain criti-cal amplitudes

and the intensity of the changes of the characteris-

tics are typical of many materials and different test and measure-ment conditions, and in most cases, some manifestations of thechanges of the dependence Q–1

– ε are not recorded. This may be

caused by the low sensitivity of measurements but in most cases bythe fact that the acting physical mechanisms in heterogeneous ma-

FFFFFigigigigig. 6.2.. 6.2.. 6.2.. 6.2.. 6.2. Changes of internal friction with the change of strain amplitudes indicatingcritical strain amplitude and intensity of changes of the characteristics.

239


terials operate do not operate in different subgrains, grains andphases under the effect of different strain amplitudes.

Therefore, in detailed investigations in this area, in addition todetermining the dependences Q–1

– ε or ∆E/E – ε, it is useful to de-

termine reproducible values of Q–1, εkr1

, εkr2

, εkr3

, but also α1, α2

andα

3. The interpretation of the intensity of the changes of the char-

acteristics (α) is only in the initial state of its development. Usu-ally, it is necessary to use two groups of methods, based on analysisof the shape or temperature at which the relaxation maxima occur.

6.1.1 Physical nature of the critical strain amplitudeThe main problem in the interpretation of internal friction that de-pends on the strain amplitude is the behaviour of dislocations. Itwas assumed for a long period of time that the dislocations movein a reversible way in accordance with external alternating loading.The difference in the nature of the interaction of dislocations withthe lattice and lattice defects is restricted by the second criticalstrain ε

kr2. Consequently, in the models used for explaining the ef-

fect of the strain amplitude on internal friction it was assumed thatin the range ε

ac < ε

cr2 the dislocations structure does not change and

that no displacement of the dislocations will be recorded afterunstressing. The mechanical hysteresis loop will be closed. This typeof the dependence of internal friction on strain amplitude is referredto as hysteresis internal friction Q–1

r (ε).

Under the effect of strain amplitude higher than εkr2

the increaseof internal friction is associated with the irreversible movement ofthe dislocations, accompanied by the change of the characteristicsof the dislocation structure. Since residual deformation is recordedin the material after the effect of strain amplitudes higher than ε

kr2,

attention was given to the value of εkr2

which was regarded as theboundary indicating the start of microplasticity, and the internalfriction in this range of the strain amplitude was referred to as plas-tic internal friction Q–1

p.

The division of the internal friction curve in relation to strainamplitude into the hysteresis and microplastic region is conven-tional. Examination by transmission electron microscopy and inter-nal friction measurements, simultaneous evaluation of the depend-ences Q–1(ε) and the curves of microdeformation in single crystalsand polycrystalline materials such as Cu, Al, Ni [271] and also Cu–Al, Cu–Si, Cu–Sn and Fe–Co alloys [272] showed that the multi-plication and irreversible movement of the dislocations already takeplace at amplitudes lower than ε

kr2.

240


Initially, the processes of microplastic deformation and internalfriction were studied theoretically and experimentally in separate in-vestigations and the physical mechanisms proposed for these proc-esses were differentiated. The dependence Q–1(ε) was interpreted asthe result of reversible movement of the dislocations around theinitial positions which do not change during measurements. Theprocess of microplasticity is associated with the generation and sub-sequent displacement of the dislocations. After finding that in acertain region of the dependence Q–1(ε) some materials showed thegeneration and irreversible movement of the dislocations, studies ap-peared in which both mechanisms were combined into a singlemechanism [273]. The linear section of the curves Q–1(ε) was de-noted as the first stage of microplasticity caused by the slip of dis-locations in the individual grains. The second, parabolic stage ofmicroplasticity was associated with the occurrence of plastic inter-nal friction. The boundary, separating the linear and parabolicstages of microplasticity, is close to the value of the second criti-cal strain amplitude.

The mechanisms of release of the initial dislocations, blocked bythe atoms of the solute elements, and of the generation of disloca-tions have different physical nature. In the former case, the proc-ess is associated with overcoming the segments of dislocations of theshort–range stress fields formed by the atoms of the solute elements.As a result of the rapid decrease of the forces of dislocation–solute interaction, the distance between the equilibrium positions,corresponding to the blocked and unblocked states, is equal to sev-eral lattice spacings. The time required by the moving dislocationto travel this distance is comparable with the duration of existenceof the thermal fluctuation increase of the energy of the ‘dislocation–saolute atom’ complex so that the release of the dislocation from theatmosphere of the solutes is a thermally activated process. Theenthalpy of activation of the processes close to the value of H

B and

the activation volume is V*a ≈ L

pb2.

The activity of the dislocation sources is accompanied by the dis-placement of dislocations to large distances L

n. The external stress

overcomes the interaction forces of the dislocations with the atomsof the solute elements, and the reversible force of the linear stretch-ing of the dislocations and of the load–range field acting over alarge distance where the wavelength is comparable with the lengthof the dislocation segments. In this case, the process of thermal ac-tivation is of lesser importance and the activation volume increasesseveral tens of times in comparison with value V*

a. These differences

241


are even larger when the dislocations generate grain boundarieswhere fresh dislocations form as a result of movement of special ob-jects — dislocations in the active part of the grain boundaries. Ina real material subjected to repeated loading, the mechanisms ofhysteresis and plastic internal friction operate simultaneously.

The magnitude of internal friction, independent of strain ampli-tude Q–1

0 , can be interpreted using one or several of the already men-

tioned mechanisms (section 2.2.3). The strain (stress) amplitude istoo low to enable dislocations segments to vibrate (Fig. 2.30a) andscatter the mechanical energy.

Under the effect of stress amplitude with the value from εkr1

toε

kr2 where Q

m– 1 is recorded, the latter can be expressed by the equa-

tion proposed by Granato and Lücke [28]

1 ,B

m

AQ e

−− ε=ε

(6.8)

here ε is the strain amplitude

3

1 2,n

dp

LA k

L= ρ (6.9)

1 ,4

Fk

s aE

Ω=π (6.10)

2 .p

aB k

L= η (6.11)

In these equations, ρd is the dislocation density, Ω is the orien-

tation factor, s is the factor of shear stress in the slip plane, E isthe Young modulus, k

2 is the constant of proportionality, and η is

the difference of the atom sizes of the main metal and thesolutes. The value of F can be determined from the stress corre-sponding to the amplitude causing bending of the dislocation seg-ment, i.e. ε

kr1. The increase of internal friction in this part of the

Q–1 – ε dependence is interpreted by the non–elastic interaction ofthe dislocation segments with energy barriers. The dislocation seg-

242


ments vibrate in a quasiviscous environment formed by the pointdefects (in steels, these are the atoms of C and N).

The critical strain εkr2

corresponds to the stress causing the gen-eration of dislocations, i.e. ε

kr2 = Gb/L

n, where b is the Burgers vec-

tor of the dislocations. Therefore,

2 .krn

b

Lε = (6.12)

Processing of the results of measurements of internal friction thatdepends slightly on the stress amplitude, using these equations,givesthe quantitative data on the dislocation structure of the mate-rial.

The binding energy between the dislocation and the solute atoms(r) is expressed by the equation [274]

1 3krrc

b Eε = (6.13)

where c is the concentration of solute atoms around the dislocation.For temperatures other than T = 0 K, for which the equation (6.13)was derived, it is necessary to take into account the temperature de-pendence of the solute concentration in the region of the dislocation[275]

0 ,G

kTc c e∆−

= (6.14)

where ∆G is the difference of free enthalpies, k is the Boltzmannconstant, T is absolute temperature. Therefore,

1 03 ,S r

k kTkr

r T Sc e e

b E

∆−− ∆ε = (6.15)

where ∆S is the change of entropy, and c0 is the average composi-

tion of the alloy, in at.%.The entropy change can be determined using the following equa-

tions [69]

243


∆ ∆

= β

(6.16)

0 ,

E

ES n H

T

∂

∆ = ∆∂

(6.17)

where

0

0

,

E

E

TT

∂

β =

∂

E and E0 are the elasticity moduli at experimental temperature and

at absolute zero, ∆H is the activation energy of self diffusion. Thequantity ∂(E/E

0) and coefficient β are determined from the tempera-

ture dependence of the square of frequency (f 2) of the vibrations.The activation entropy can be determined by processing the experi-mental results and using the above equations. Coefficient n must bedetermined by evaluating the entropy of self diffusion and sublima-tion temperature (n = 0.5–0.6).

From experimental measurements, taken on Armco iron (0.03wt.%C + N) after different heat treatments and deformation (ε) andon Mo (99.98 wt.%), using a torsion pendulum with determinationof γ

kr1 and γ

kr2 and other characteristics, the authors of Ref. 276

obtained the experimental data presented in Table 6.1. The increaseof strain is accompanied by increase of the dislocation density andthe length of the dislocation segments decreases.

The interpretation of the results of the Granato–Lücke theory,taking into account the atomic structure of the crystal and thermallyactivated unpinning of the dislocations from the pinning points,makes it possible to take into account the data on bows on dislo-cation lines [277].

Under the effect of strain amplitude higher than εkr2

(section IIIin Fig. 6.2) we determined the internal friction Q–1

p which depends

strongly on the strain amplitude. For the case in which dislocations

244


TTTTTaaaaabbbbble 6.1le 6.1le 6.1le 6.1le 6.1 Change of critical amplitudes, parameters of dislocation structure andbinding energy of the dislocations with solute atoms for materials subjected todifferent treatment

tnemtaertdnalairetaMT

]Cº[ γ1rk

γ2rk

Lp

]mm[L

p

]mm[ β r01·J[ 91 ]

ρm[ 2– ]

noriocmrAh1/Cº039tadelaenna

retfa e %8.2=retfa e %0.7=retfa e %2.11=

Cº059morfdehcneuqmunedbyloM

retfa e %06=tanoitasillatsyrcerretfa

h4/Cº0021

001001001001001

053

053

01·8 5–

01– 4–

01·2 4–

01·4.2 4–

01– 4–

01·1.1 4–

01·8.7 5–

01·2.3 4–

01·1.6 4–

01·0.8 4–

01– 3–

01·9.5 4–

01·2 3–

01·6 4–

01– 3–

01·5.4 4–

01·5.3 4–

01·8.2 4–

01·7.4 4–

01·8.1 4–

01·0.5 4–

01·8.9 5–

01·7.6 5–

01·0.6 5–

01·4.5 5–

01·0.8 5–

01·7.1 4–

01·8.2 4–

4104.03624.01154.02174.00634.0

55.0

555.0

882.0492.0633.0263.0023.0

614.0–

614.0–

01·2 11

01·1 21

01·7 21

01·3 31

01·4 21

01·1 51

01·6 31

overcome obstacles by a thermally activated process [278], plasticinternal friction is characterised by the equation

( )1 11 ,Dp

CQ e

hε−ε− =

ε (6.18)

where

12

and .Q

kTb VGC e D

f kT

−ρ ν= = απ

In the equations, h is a constant with the value 0.5–1 [278], ε is thestrain amplitude, ε

i is the strain amplitude required for overcoming

barriers [278], υ is the frequency of oscillations of the dislocations,v is the area occupied by the dislocation during a oscillation, f isthe frequency of changes of loading, Q is the activation energy, αis the orientation factor (~0.5), V is the activation volume, G is theshear modulus of elasticity.

Another interpretation [279] assumes that at εac

> εkr2

dislocationsources start to generate dislocations. In the range from ε

kr2 to 2ε

kr2

unstable dislocation loops interact together and with a dislocationforrest, but they do not manage to move to the barriers of the typeof grain boundaries. The movement of dislocations to the grainboundaries is possible only at ε

ac ≥ 2ε

kr2 and dislocation clusters

form in the vicinity of the obstacles. When this boundary is ex-

245


ceeded, plastic internal friction, determined by the equation

1 ,mpQ X− = ε (6.19)

takes place, where X and m are the material and experimental con-stants.

Therefore, when εkr2

or 2εkr2

is exceeded, microplastic deforma-tion takes place. This microplastic deformation is of saturation na-ture, i.e. after a specific number of repeated loading cycles, Q–1

p sta-

bilises at a certain value and it is assumed that the density and dis-tribution of dislocations no longer changes significantly. The rep-etition of the strain amplitudes higher than ε

kr3 (region IV in Fig.

6.2) results in the process of fatigue damage cumulation, i.e. inter-nal friction Q–1

p is a function of the number of load cycles. Depend-

ing on the value of ε, fatigue failure of the component is recordedafter a certain number of load cycles. The experimental data and in-terpretation of these data for describing region IV have been pub-lished only in a small number of cases, also owing o the fact thatthe method of measuring internal friction in evaluation of the fatigueprocess is being supplemented by other procedures. The form of theQ–1–ε dependence, Fig. 6.2, is a function of many substructural andstructural factors, the type and composition of the material, and themethod of processing the material. Depending on the sensitivity ofmeasurement of Q–1 and overlapping of different partial mechanismsof scattering of mechanical energy, all three critical strain amplitudemay or may not be recorded.

Therefore, the dependence of internal friction on strain amplitudecan be characterised by the equation

( ) ( ) ( ) ( )1 1 1 1 10 , , , , ,m b p p t pQ Q Q Q Q N− − − − −ε = + ε ρ + ε ρ + ε ρ (6.20)

where ρb is the density of pinned (stationary) dislocations with

ρb ≥ ρ

0, where ρ

0 is the density of dislocations in the annealed ma-

terial, and ρp is the density of mobile dislocations, N is the number

of load cycles.

6.1.2 Methods of evaluating critical amplitudesIn the evaluation and classification of the processes, taking place inthe material during its loading in the region in which the internalfriction depends strongly on strain amplitude, it is necessary to use

246


reproducible and substantiated (from the viewpoint of physical met-allurgy) procedures of determination of critical strain amplitudes.Despite the significance of ε

kr1 and ε

kr3, in determination of these

amplitude there are a number of disputes and unexplained features.Several methods and procedures are still used for the determinationof the second critical strain amplitude.

The value εkr2

is determined as the strain amplitude which causesrapid rising of the Q–1 curve [280] − procedure I, and as the strainamplitude which causes the Q–1/ε – ε dependence change from acurve to a straight line [2] — procedure II. Other studies indicatethat ε

kr2 can also be determined using other methods in which the

effect of εkr2

and higher strain amplitude results in a sharp increaseof the effect of the Young modulus (in the ∆E/E – ε dependence),i.e. procedure III [281]. The large increase of ∆E/E above ε

kr2 is

caused by the increase of the density of mobile dislocations. Thisis a typical sign of plastic deformation.

The experimental studies, concerned with the evaluation of the ef-fect of the prior strain amplitude ε

d on the extent of the change of

background Q–10 and the magnitude of the second critical strain am-

plitude, Fig. 6.3, show [281] that the evaluation of results of themeasurements taken using the procedures I, II and III, gives differ-ent values of ε

kr2. In a specific range of ε

d, the procedures II and

FFFFFigigigigig. 6.3.. 6.3.. 6.3.. 6.3.. 6.3. Dependence of the first and second critical strain amplitude (solidlines) and internal friction background (broken line).

247


III give similar values when determining εkr2

, whereas procedure Iproved to be unreliable, because in comparison with the data pub-lished in Ref. 2 and 282, it gives physically unjustified low valuesof ε

kr2.

In addition to these methods, there are also methods which de-termine ε

kr2 as the strain amplitude whose effect results in the for-

mation of the first slip lines on the surface of the specimen [273],or as the strain amplitude resulting in irreversible changes of theinternal friction background [2, 283]. The principle of the secondmethod is based on determining the background Q–1

0 at ε

0 < ε

kr1. This

is followed by loading with a strain amplitude higher than εkr1

andby repeated measurements of the background Q–1

0 at ε

0. With gradual

increase of ε and measurement of Q–1, after exceeding a certainvalue of ε, the recorded internal friction is already higher thanQ–1

0 . This value then belongs to the second critical strain amplitude.Experiments with the automated measurements of internal friction

carried out in equipment VTP–A (VSDS) showed the possibility ofdetermination of ε

kr2 by another procedure [284]. Automatic record-

ing of the dependence of internal friction on the loading time andof the defect of the Young modulus on loading time at 7×10–7 ≥ ε≥ 5×10–4 at every experimental point, i.e. under the effect of theselected values of ε, showed that the form of the Q–1 – t depend-ence is complicated and is at present very difficult to interpret.However, the ∆E/E – t dependences are of two shapes. The firstshape, at low values of ε, shows a decrease of ∆E/E with increas-ing loading time, and the decrease of ∆E/E with increasing loadingtime becomes smaller with increasing strain amplitude up to thestate in which it no longer changes with time under the effect of aspecific value of ε. The second shape, at the values close to ε

kr2 and

higher than εkr2

, ∆E/E increases with increasing loading time. Thisshape no longer changes at high values of ε. Since at values higherthan ε

kr2 the ∆E/E – t curves are characterised by the saturation

nature of the changes ∆E/E, which is a manifestation of the genera-tion of dislocations, i t may be assumed that ε

ac at which the

∆E/E – t dependence changes from decreasing to increasing has thevalue ε

kr2.

It is useful to note that different materials are characterised bydifferent form of the Q–1 – ε dependences and, consequently, in ex-perimental investigations, it is efficient to select the methods fordetermining ε

kr2 as the important characteristic of microplasticity of

the material. Some other physically substantiated method for deter-mination of ε

kr2 are presented in section 6.2.

248


6.2 CYCLIC MICROPLASTIC RESPONSE OF MATERIALSThe above results show that the Q–1 – ε, ∆/E – ε dependences andalso Q–1 – t or ∆E/E – t dependences at certain strain amplitudes,provide, in addition to the specific values of the critical strain am-plitude and their sensitivity to the history of manufacture and load-ing, a large number of valuable physical–metallurgical and engineer-ing data. A number of specific examples of the measurement and ap-plication of these dependences will be discussed later.

6.2.1 Dislocation density and the activation volume ofmicroplasticity

The plastic strain rate is a function of the speed of dislocations vand the density of mobile dislocations ρ

p, with the Burgers vector

b, in accordance with the equation ρp = ρ

pvb; it is well known that

the total dislocation density ρ = ρ0 + ρ

p, where zero is the initial

density of the dislocations, and ρp = Aεn

p , where A, n are material

characteristics. In experiments with the materials having the struc-ture of fcc substitutional solid solutions at the microplastic strain(ε

p < 0.1), where the flow stress and plastic strain ε

p are linked by

a parabolic relationship, n = 1–2. The value of n for the initialstage of microplastic deformation (ε

p << 0.1) is not available.

It is difficult to determine ρp in the process of continuing defor-

mation because all experimental methods for evaluating the dislo-cation density are basically static, or the dislocation density is de-termined by indirect methods on the basis of certain assumptions,or it is assumed that ρ

p in this range of microplasticity is a vari-

able quantity. The experiments show that dislocations in, for exam-ple, annealed material (ρ

0) do not contribute to microplastic defor-

mation and also that the forest of these dislocations is not a sourceof new dislocations. Fresh dislocations generate on the surface ofthe specimens, at the boundaries of grains and twins, at the secondphase particles and in areas with heterogenities in the structure ofthe material.

Measurements and analysis of the internal friction mechanismsQ–1 with the change of the strain amplitude ε

ac provide a large

amount of information on the dynamics of changes of the disloca-tions structure of the material.

No quantitative relationships have been determined for describ-ing the changes of ρ

p in the strain range 10–6 – 10–4. Measurements

and analysis of the internal friction mechanisms, such as the dy-namic method, can provide valuable information assuming we usea suitable model which would take into account the type of dislo-

249


cations sources, the nature and dynamics of movement of the freshdislocations and also the nature of dislocations structures, examinedby transmission electron microscopy. Cu–Al alloys are highly suit-able from this viewpoint; the generation of dislocations in these al-loys takes place in sources at the grain boundaries and the dislo-cations structure after microdeformation has the nature of planarrows of the dislocations with the same sign [285].

For Cu–Al alloy with an aluminium content of 9.2 and 13.8 at%after annealing at 600°C for 1 hour measurements were taken of theQ–1(ε) curve on each specimens (diameter 1 mm, length 100 mm,pressure 0.5 Pa, frequency 2–2.5 Hz in equipment RKM–TPI) ini-tially with the value of ε gradually increasing from ε = 0 to ε = ε

m,

where εm > ε

kr2. The curve was obtained from 25 measurements

taken on new specimens and denoted by Q–1(↑ε) (Fig.6.4). Afterobtaining the selected values of ε

m, we immediately measured the

internal friction for the gradually decreasing value of εac

, i.e. fromε

ac = ε

m to ε

ac = ε

0, and the curve was denoted Q–1(↓ε). The experi-

ments show that at a specific value of εac

Q–1(↓ε) > Q–1(↑ε) [286].The difference Q–1(↓ε) – Q–1(↑ε) was denoted by ∆Q–1. This is thecontribution of the scattering power of the material by mobile dis-locations with the density ρ

p which formed as a result of loading the

material with the selected value of εm. The phenomenon in which

Q–1(↓ε) ≠ Q–1(↑ε) at a specific value of ε is universal. It does notdepend on the loading method (tension–compression, torsion, bend-

FFFFFigigigigig. 6.4.. 6.4.. 6.4.. 6.4.. 6.4. Diagram for deriving the increment of internal friction as a result ofincreasing dislocation density.

εkr1 ε

kr2

250


ing) nor on loading frequency (for example, 1 Hz or 23 kHz) asshown in other investigations [282, 287].

Analysis of the measurement results shows [282] that with in-creasing value ε, when ε > ε

kr2, the density of mobile dislocations

ρp in the material increases. With decreasing value ε from the se-

lected value εm the density of mobile dislocations remain unchanged

but with increase of ε it may increase in the microplasticity rangeby several orders of magnitude but the measurable change of the to-tal dislocation density is recorded only after exceeding the stresscorresponding to the cyclic yield limit of the material. For example,in Cu +13.8 at.% Al alloy, the relative increase of dislocation den-sity

( )0

0

,ρ ε −ρ

ρ

is 0.1 only when εm

≈ 0.2% [289]. Therefore, when evaluating the

changes of the dislocation density during microplastic deformation,i.e. when ε

m < 7×10–4, it can be assumed that the total dislocation

density ρ and also ρn (the density of stationary dislocations) do not

change. This shows that in the examined range ε (to 7×10–4) thevalue of Q–1 is an inversed function of strain. Consequently,

( ) ( ) ( )1 1 1, , , .m p p m p pQ Q Q− − − ∆ ε ε = ε ρ ε − ε ρ ε (6.21)

Generally, the magnitude of ∆Q–1 depends on the time required toreach ε

m, with the gradually increasing amplitude ε

ac. The time de-

pendence of Q–1 is important for pure metals, but in the Cu–Alalloys with an aluminium content of 1–17.3 at.% it is not evident,because the curves Q–1(↓ε) and the subsequently plotted curvesQ–1(↑ε), when ε

ac < ε

kr2, are more or less identical.

Golovin and Levin [290] quantified microplastic internal frictionfor the case in which the following conditions are fulfilled: a) onlydislocation sources at the grain boundaries operate duringmicroplastic deformation and deformation is accompanied by theformation of planar rows of dislocations with the same sign; b) thedynamics of movement of the dislocations is controlled by the forcesof Newtonian viscous friction. The first condition is fulfilled in thecase of copper–aluminium alloys. The second condition requires a

251


comment. The materials examined were homogeneous solid solutionswith short-range ordering. The movement of the first dislocation ofthe planar row of the dislocations is associated with overcoming theresistance of the lattice whose value in the slip plane is influencedby the short-range ordering of the solid solution. However, aftermovement of several dislocations in the slip plane, the solid solu-tion is no longer ordered and the movement of further dislocationsof the planar row is controlled by the viscous friction mechanisms.After the arrest of the first moving dislocation in front of an obsta-cle of the type of sub–boundaries, dislocation clusters, etc., a pla-nar row of dislocations with length 2L starts to form behind thedislocation. Alternating loading does not change this arrangement ofthe dislocations, because the latter do not return to the sources.Local vibrations of the segments of the moving dislocations in theplanar row of the dislocations then contribute to microplastic inter-nal friction.

This model shows that

( ) ( ) ( )21 1

2

1.24, ,p

p p

B L LQ Q

G− − ω ρ

↑ ε = ε ρ ε = ε (6.22)

( ) ( ) ( ) ( )21 1

2 2, 1.178 1 ... ,24

p p mp p m

m m

B L LQ Q

G− − ω ρ ρ ε

↓ ε = ε ρ ε = + − ε ε (6.23)

where G is the shear modulus of elasticity, ω is the frequency ofexternal loading, B is the viscous factor of movement of the dislo-cations. The low values of ε made it possible to simplify the formof equations, because at ε/ε

m << 1; this is fulfilled under the given

conditions. The series on the right–hand side of equation (6.23) canbe replaced with sufficient accuracy by its first term. Consequently,at low strain amplitudes

( )21

2

3.

8p m

m

B LQ

G− ρ επ ω∆ =

ε (6.24)

Equation (6.24) makes it possible to characterised the functionρ

p(ε

m). Taking into account the given model, it can be seen that the

experimental points are distributed on straight lines whose tangent

252


is approximately 3. This shows that

3 ,p mAρ = ε (6.25)

which indicates that the density of moving dislocations in the ini-tial stage of microplasticity of the Cu–Al alloys increases with thecube of the plastic strain. The intensity of generation of dislocationsduring the microplasticity phenomena has two values n in the equa-tion ρ

p = Aεn

p. In the first part of the deformation curve, where the

strain is a linear function of stress, the plastic strain εp is propor-

tional to the maximum strain εm so that the density of moving dis-

locations is expressed by the equation ρp = Cεn

p, where n = 3. In

the second part of the deformation curve, where strain ε is a para-bolic function of stress, ε

p ~ ε0.5

m. Consequently, ρ

p = Dε1.5

p, where

n = 3/2.These data should be included in the information on the behav-

iour of an ensemble of the dislocations, on the annihilation of dis-locations of the same time with the reverse sign and on the estab-lishment of the saturated value of dislocation density [24, 291].

Analysis of the dependences Q–1(ε) makes it possible to quantifycertain characteristics of the dislocation network, the stress condi-tions of the start of microplasticity, the dynamics of changes of themicrostructure, differentiate the mechanisms of scattering ofmechanical energy in the material or also determine, for example,the size of the activation volume of microplastic deformation.

In the range εkr1

≤ ε < εkr2

, the scattering of the mechanicalenergy in the material can be interpreted by the spring model inaccordance with Granato and Lücke [28], using equation (6.8).

Interpretation of the factors A, B makes it possible to quantifycertain parameters of the dislocations structure and also the char-acteristics of the interaction of dislocations with point defects, es-pecially interstitial atoms. Therefore, it can be expected that in thisrange of ε we obtain straight lines when the experimental data areplotted in the coordinates ln Q–1

m vs ε–1 with the straight lines hav-

ing the slope B.Several models have been proposed for the range ε

kr2 ≤ ε ≤ ε

kr3.

Peguin, Perez and Gobin [278] assumed that the dislocations over-come obstacles in their movement by a thermally activated process.The appropriate activation energy depends on the level of actingmechanical stress. Consequently, plastic internal friction

253


Q–1p

(= Q–1ε – Q

2–1 , where Q–1

ε1 or Q–1

2 is internal friction at ε > ε

kr2, or

at ε = εkr2

) is determined by equation (6.18).Burdett [280] assumed that the activation volume depends on the

level of acting stress in accordance with the equation

( ),x

i

FV =

σ − σ (6.26)

where F and x are constants. According to Spitzig [292], x = 0.5.After substituting and transforming equation (6.18), where the givenmodel is realistic, it may be expected that in the range of cyclicmicroplasticity in the coordinates ln (Q–1ε) vs. (ε – ε

i)1/2, the experi-

mentally determined points will fit straight lines with the slope cor-responding to D.

Jon, Mason and Beshers [279] derived equation (6.19). If thismodelling assumption is valid, we can expect linearisation of the ex-perimental measurements, when the results are plotted in the coor-dinates log Q–1 – log ε.

These hypotheses have been verified in Ref. 293.In the first part of the experiments, the authors used pure

copper (99.994 wt.%), denoted Cu, iron with 0.03 wt.% C, denotedFe, and an alloy of iron with titanium (0.043 wt.% Fe, 0.03 wt.%Ti), denoted Fe–Ti. The internal friction of Cu, Fe and Fe–Ti in re-lation to the strain amplitude was measured in low–frequency (ap-proximately 1 Hz) vacuum equipment using RKM–TPI torsion pen-dulum, on specimens of Fe and Fe–Ti in the presence of a magneticfield with an intensity of 1.9×104 A⋅m–1, at a temperature of 23°C.

In the second part, the experiments are carried out on iron with0.03 wt.% C (Fe) and on CSN 412013 steel (0.07 wt.% C, 0.27wt.% Mn, 0.03 wt.% Si, 0.013 wt.% P, 0.08 wt.% S, 0.07 wt.% Cr,and 0.006 wt.% N) after heat treatment; the ferrite grain size of Fewas 0.032 mm, that of the steel 0.022 mm. Experiments were car-ried out in RKM–TPI equipment in the presence of a magnetic fieldand at the same temperature as in the first part of the experiments.Measurements were taken at a frequency of 1 Hz in the laboratoryof the Tula Polytechnic Institute, Russia.

The ferrite grain size of CSN 412013 steel after heat treatmentwas d

1 = 0.022 + 0.004 mm, d

2 = 0.29 + 0.045 mm, d

3 = 0.620 +

0.085 mm. The specimens of CSN 412013 steel, diameter 3 mm,were subjected to stabilisation annealing at 220°C for 0.5 hour and

254


chemically polished. The dependence of internal friction on thestrain amplitude in tension–compression loading was measured in amodified Mason system at a loading frequency of 23 kHz in thepresence of a magnetic field with an intensity of 1.9×10 A⋅m–1, ata temperature of 23°C. The experiments were carried out at thelaboratories of VSDS Technical University in Zilina, Slovak Repub-lic.

The first modelling assumption, characterised by equation (6.18),brings linearisation only if we use Burdett’s interpretation. For othermaterials (Cu, Fe – Ti), the ln Q–1

pε vs. (ε – ε

i)1/2 dependence is

characterised by curves, which may indicate that this model does notcharacterise efficiently the interaction of materials with repeatedmechanical loading. Appropriate atmospheres of C and N formed iniron with 0.03 wt.% C t temperatures lower than the condensationtemperature of the atmospheres of solute elements. In the Fe–Ti al-loy, the dislocations are freed from C and N atoms and only veryweak interaction of the solute substitutional atoms with the dislo-cations is observed in copper.

The second modelling assumption, characterised by equation(6.19) results in linearisation in the case of copper whenε > 6.3×10–5, or Fe – Ti, when ε > 1.6×10–4. The authors of themodelling assumptions [279] prepared this model for materials suchas bronze, copper, etc., i.e. the materials characterised by a veryweak effect of the substitutional solute atoms of the dislocations.

The first part of the experiments showed that in the case of thematerials characterised by high–intensity interaction of the soluteswith the dislocations, it is convenient to use Burdett’s approxima-tion [280] also for the quantification of other parameters of theactivation of cyclic microplasticity.

The results of measurements of the internal friction of materialswith high–intensity interaction of the solutes with the dislocationsin relation to the strain amplitude at a frequency of 1 Hz and 23kHz are shown in Fig. 6.5, where each curve is the average of 5measurements. The magnitude of the first critical strain amplitude,determined by the method presented in Ref. 281 for the examinedmaterials, is shown in Table 6.2, and if we use equation (6.8), theresults can be plotted in Fig. 6.6 and the graph can be used to de-termine the values of ε

i, presented in Table 6.2.

The processing of the results of measurements in accordance withthe first model [278], supplemented by Burdett [280], again con-firms the linearisation of ln Q–1

p ε vs (ε – ε

i)1/2 for the region of cy-

clic microplasticity (Fig. 6.7). This enables the equations for cal-

255


FFFFFigigigigig. 6.5.. 6.5.. 6.5.. 6.5.. 6.5. Dependence of internal friction on strain amplitude for different materialsloaded with a frequency of 1 Hz and 23 kHz.

FFFFFigigigigig.6.6. .6.6. .6.6. .6.6. .6.6. Results of measurements for the case of validity of equation (6.8) anddifferent materials loaded with a frequency of 1 Hz and 23 kHz

256


TTTTTaaaaabbbbble 6.2le 6.2le 6.2le 6.2le 6.2 Activation parameters of cyclic microplasticity (b is Burgers vector)

lairetaMezisniarG

]mm[gnidaoLycneuqerf ε

101· 4 ε

i01· 4 V mm[ 3] V/b3

30.0+eFC%.tw31021

230.0

220.0

zH1

zH1

4.1

0.2

0.4

4.401·52.1 81– 35

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zHk32zHk32zHk32

3.10.17.0

2.29.13.1

01·51.9 91– 93

FFFFFigigigigig.6.7..6.7..6.7..6.7..6.7. Results of measurements in Fig.6.7 for the case of validity of modifiedequation (6.18) and different materials loaded with a frequency of 1 Hz and 23kHz

culating D1

(6.18) and (6.26) to be used for the determination of theactivation volume of the microplasticity process.

For the selected value (σ – σi) = 14.7 N ⋅mm–2, the activation

volume at a frequency of 1 Hz is 1.25×10–18 mm3, and at a fre-quency of 23 kHz it is 9.15×10–19 mm3.

In Ref. 292, the activation volume under the effect of the sameeffective stress (σ – σ

i) = 15 N ⋅mm–2) and at a frequency of ap-

proximately 1 Hz was 1.2×10–18 mm3. For iron with a grain size of0.048; 0.057 and 0.076 mm, the activation volume according toBurdett [280] is 1.3×10–18 mm3.

The difference in the activation volume of the valid materials at

257


a loading frequency of 1 Hz and 23 kHz is relatively small, Table6.2.

The experiments showed that the activation volume of themicroplasticity of the examined steels at a temperature of 23°C andan effective stress of 14.7 N⋅mm–2 is approximately 10–18 mm3.

The results of the experiments indicate that in the case of the ma-terials with significant interaction of the solute atoms with the dis-locations in the range of cyclic microplasticity it is possible to usethe interpretation proposed by Peguin, Perez and Gobin [278], af-ter supplementing the approximation according to Burdett [280].

The mechanism of cyclic microplasticity is determined by the in-tensity of the interaction of the atoms of the solutes with the dis-locations and is not influenced by the frequency of loading changes.

The importance of measurements of internal friction and theYoung modulus defect with the change of strain amplitude will beshown on other examples.

6.2.2 Condensation temperature of the atmospheres ofsolute elements

The transition from low–density (Maxwell) to dense (Cottrell) at-mospheres of the solute interstitial elements on the dislocations con-trols the microplastic behaviour of the material also under repeatedloading. Reproducible values of condensation temperature T

c can be

determined by internal friction measurements under different ther-mal and amplitude conditions [294].

Low–carbon unalloyed steel with composition: 0.07% C, 0.27%mn, 0.03% Si, 0.013% P, 0.018% Cr, 0.07% Cr and 0.006% N(CSN 412013) was annealed during the measurements of internalpressure (dependent on temperature and strain amplitude), using themethod of the torsional pendulum [92] with a vibrational frequencyof ~1 Hz in the strain amplitude range γ from 5×10–6 to 7×10–4.Some of the measurements were taken under the effect of a constantmagnetic field with the intensity H = 1.7×10–7 A ⋅m–1, and othermeasurements were taken at H = 0. The experiments were carriedout at the laboratories of the Tula Polytechnic Institute in Tula,Russia.

Bars with a diameter of 20 mm were also used to producetestpieces similar to tensile bars with a diameter in the central partof 3 mm [281]. Internal friction Q–1 and ∆E/E at a frequency of 23kHz were measured by the method described by Mason [282]. Thespecimens welded with symmetric tension–compression at a tempera-

258


ture of 23°C in the strain rate range ε from 1.5×10–6 to 6×10–4.Some of the measurements were taken under the effect of a constantmagnetic field with the intensity H = 1.9×10–4 A⋅m–1, or at H = 0.The experiments were conducted at the laboratories of the VSDSTechnical University in Zilina.

Prior to the measurements, the specimens in both experimentswere initially annealed at 720°C for 0.5 hr and cooled at a rate of100°C/h in a furnace; this was followed by chemical etching of thesurface of the specimens. The ferrite grain size of the specimenswas 0.022 ± 0.004 mm in both cases.

The measurements of internal friction at a loading frequency of~1 Hz in the temperature range 20 – 440°C on specimens of thelow–carbon unalloyed steel after annealing (Fig. 6.8, curve 2) andalso after rapid cooling from 725°C (Fig. 6.8, curve 1) show thatin both conditions of the material, the maximum on the Q–1 – Tcurve is recorded at 40°C (Snoek maximum). The cabin content ofthe solid solution of α-iron, calculated from the Snoek maximum,was 0.003 wt.%% after annealing and 0.014 wt.%% after rapidcooling from 725°C. The activation energy of the maximum on theQ–1 – T curve, determined from the Wert–Marx equation, was ~80.8kJ ⋅mol–1, which corresponds to the activation energy of carbon dif-fusion in the solid solution of iron.

FFFFFigigigigig. 6.8.. 6.8.. 6.8.. 6.8.. 6.8. Temperature dependence of internal friction for mild steel after rapidcooling from 725°C (curve 1) and after annealing (curve 2) at a loading frequencyof ~~~~~1 Hz.

259


FFFFFigigigigig. 6.9. . 6.9. . 6.9. . 6.9. . 6.9. Dependence of internal friction on strain amplitude for different temperaturesat a loading frequency of ~1 Hz (arrows indicate the magnitude of τ

kr1).

Figure 6.8 shows that the low value of the internal friction of theannual material does not change when temperature is increased from100 to 400°C. When 400°C is exceeded, the characteristic increases.Measurements were taken at γ ≅ 5×10–6 and under the effect of amagnetic field with intensity H = 1.7×104 Am–1.

The data on the unstable pinning of the dislocations by the at-mospheres of the interstitial elements at higher temperatures confirmthe results of internal friction measurements (internal friction de-pends on strain amplitude), at different temperatures in the rangefrom 20 to 550°C, as indicated in Fig. 6.9.

Internal friction measurements at different strain amplitudechanging with a frequency of ~1 Hz, under the effect of a magneticfield with an intensity of 1.7×104 A⋅m–1, show that the form of theQ–1 – dependence in the temperature range 100 – 400°C does notchange greatly. The strain amplitude indicating the start of internalfriction, dependent on the strain amplitude, is denoted γ

kr1 and its

value is ~7×10–5. At temperatures of 425°C and higher tempera-tures, the authors detected a significant increase of the magnitudeof internal friction, and a large decrease of the strain amplitude am-plitude the start of the region of internal friction, dependent onstrain amplitude (γ

kr1), was recorded. At γ > γ

kr1, the magnitude of

260


FFFFFigigigigig. 6.10.. 6.10.. 6.10.. 6.10.. 6.10. Temperature dependence of the first critical strain amplitude duringloading with a frequency of ~1 Hz.

increase of internal friction per unit increase of the strain amplitudealso changes.

Figure 6.10 shows that up to 400°C γkr1

≅ 7×10–5, and with in-crease of temperature this value decreases exponentially so that at550°C γ

kr1 = 10–5.

The experimental dependences of the internal friction on thestrain amplitude, determined on specimens of mild steel with thesame structure at a frequency of ~1 Hz and 23 kHz under the ef-fect of a constant magnetic field or in its absence (Fig. 6.11 and6.12) indicate that regardless of the different nature of loading andthe accuracy of the methods used, the curves Q–1 – γ and Q–1 – εare similar.

The Q–1 – ε curves, plotted at a loading frequency of 23 kHz(Fig. 6.12) and ε <

ε

kr1 shows the strain amplitude ε. Consequently,

in the range from ε to εkr1

we can see a slightly higher value of in-ternal friction in comparison with that obtained at a loading fre-quency of ~1 Hz and the corresponding method of determination ofinternal friction in this region. The value was reproducible and ex-ceeded the extent of scatter of internal friction values.

At γ > γkr1

or ε > ε

kr1 (γ

kr1 = 2×10–5 or ε

kr1 = 1.2×10–5 for a load-

ing frequency of ~1 Hz or 23 kHz at H = 0 and γkr1

= 7×10–5, orε

kr1 = 6.4×10–5 at a loading frequency of ~1 Hz or 23 kHz at

H = 1.7×10–4 or 1.9×10–4 A⋅m–1) the curves show a strong depend-ence of internal friction on strain amplitude. The position of the

261


FFFFFigigigigig. 6.11.. 6.11.. 6.11.. 6.11.. 6.11. Q–1 – τ dependence at a loading frequency of ~1 Hz without the effectof the magnetic field (solid lines) and with the effect of the field (broken fields)indicating the Q–1 – τ dependence and also changes of the internal friction backgroundat the second critical amplitude.

FFFFFigigigigig. 6.12.. 6.12.. 6.12.. 6.12.. 6.12. The Q–1– ε and ∆E/E – dependences at a loading frequency of 23kHz without (solid lines) and with the effect of the magnetic field (broken lines).

critical strain amplitude εkr1

can be recorded quite efficiently by themeasurements of ∆E/E at a loading frequency of 23 kHz, becausethe first values of ∆E/E with a gradually increasing strain ampli-tude are obtained at ε

kr1.

262


The modified methods of internal friction measurements in therange of the dependence of internal friction on strain amplitude (af-ter determining the value of internal friction at we determined in-ternal friction at ε = 5×10–6, and this was followed by internal fric-tion measurements at ε

x1 ≤ ε

kr1 and again by internal friction meas-

urements at ε = 5×10–6, etc.), we determined the strain amplitude atwhich the internal friction background (Q–1

0) increases irreversibly.

These strain amplitudes were denoted γkr2

or εkr2

; they represent thestrain amplitudes in the material. At a loading frequency of ~1 Hzor 23 kHz, ε

kr2 = 3.4×10–4 at H = 1.7×10–4, or 1.9×10–4 A⋅m–1.

The critical strain amplitude γkr2

was inspected at a loading fre-quency of up of 1 Hz also by the method of ‘zero shift’ on the scaleof the measuring device. Another method was applied at a loadingfrequency of ~1 Hz as well as at 23 Hz Q–1/ε – ε using Golovin’sprocedure. The results are in the top righthand corner of Fig.6.11.

At a loading frequency of 23 kHz the investigations were supple-mented by microscopic examination of the specimens. The resultsshow that at ε > ε

kr2 the first slip lines appear on the surface of the

specimens [282]. These methods, characterising the second criticalstrain amplitude, give, at both frequencies, the values in the rangefrom 2×10–4 to 3×10–4.

The application of a sufficiently strong magnetic field to suppressthe magnetomechanical component of internal friction in theferromagnetic material for both loading frequencies decreases theinternal friction background, increases the critical strain amplitudeγ

kr1 or ε

kr1, decreases the width of the range between γ

kr1 and γ

kr2, or

εkr1

or εkr2

, and increases the value of γkr2

or εkr2

. The position of thecritical amplitude characterising the microplastic strain γ

kr2 or ε

kr2

can be efficiently recorded by all the given methods with the appli-cation of the magnetic field to the loaded material.

In the independent experiments with the measurements of inter-nal friction in mild steel, the authors of Ref. 249 determined the cy-clic strain amplitude range characterised by the operation of differ-ent energy scattering mechanisms at a frequency of ~1 Hz and 23kHz, and showed that the evaluation criteria for the results are com-parable.

Several methods were used to determine the cyclic strain ampli-tudes at which microplastic deformation starts. On the basis of theagreement of results it can be concluded that in cyclic loading ofmild steel in the annealed condition, without the magnetic field, thecyclic strain amplitude is (3.5–6)×10–4.

263


The condensation temperature of the atmospheres of the solute el-ements in the evaluated steel is ~425°C.

The activation energy of occurrence of γkr1

or γkr2

at a loadingfrequency of ~1 Hz is 24.1 or 16.7 kJ⋅mol–1, and the activation en-ergy of occurrence of ε

kr1 or ε

kr2 at a loading frequency of 23 kHz

is 16.2 or 13.6 kJ⋅mol–1. The differences in the activation energiesof the occurrence of the critical amplitudes were interpreted by theauthors from the viewpoint of the differences in the shape and mo-bility of the dislocation segments at the given loading frequencies.

The cyclic plastic response of the materials is significantlyinfluenced by many factors of the history of the specimens, espe-cially by prior cyclic plastic deformation. Its effects can be effi-ciently evaluated by measuring the internal friction and the defectof the Young modulus [295].

6.2.3 Deformation historyIn order to investigate the effect of prior cyclic microplastic defor-mation on internal friction and changes of the dislocation density,the author of this book carried out [281] experiments on mildunalloyed steel (0.07 wt.% C, 0.006 wt.% N, 0.27 wt.% Mn, 0.03wt.% Si, 0.013 wt.% P, 0.018 wt.% S, 0.07 wt.% Cr), the ferritegrain size was 0.022 ± 0.004 mm.

In the preparation of the specimens with a diameter of 3 mm inthe central part, the specimens were annealed for 30 minutes at200°C in a shielding atmosphere. This was followed by chemicalpolishing of the surface of the specimens. Symmetric cyclic load-ing with a frequency of 23 kHz of the pull–push type with the strainamplitude in the central part of specimens of 1 × 10–6 – 7 × 10–4

was carried out for 45 seconds, i.e. 1×106 load cycles. This wasaccompanied by the measurements of internal friction Q–1 and rela-tive changes of the Young modulus, the so–called defect of theYoung modulus ∆E/E. The experiments were carried out at a tem-perature of 21°C, using a constant magnetic field with a strength of1.9×10 A⋅m–1 to eliminate the magnetomechanical component of theinternal friction in the ferromagnetic material. At a gradually in-creasing strain amplitude in a series of specimens, the maindependences Q–1 – ε or ∆E/E – ε (Fig. 6.13, curves 1) were initiallydetermined. After evaluating the critical strain amplitudes, otherbatches of the specimens were initially loaded with strain amplitudesof ε

d = 2.6×10, ε

d = 3.5×10–4, ε

d ≥ 5.1×10–4, ε

d = 6.5×10–4 for 120

seconds, representing 2.8×106 cycles; this was followed immediatelyby gradual loading of the specimens from a strain amplitude of

264


FFFFFigigigigig. 6.13.. 6.13.. 6.13.. 6.13.. 6.13. Dependence of Q–1 for prestrained steel: 1) εd = 0; 2) 2.6 × 10–4, 3)

3.5 × 10–4; 4) 5.1 × 10–4; 5) 6.5 × 10–4. The arrows pointing upwards indicateε

kr2 and those downwards ε

kr2, where ε

kr1 and ε

kr2 were determined from the occurrence

and changes of ∆E/E in relation to ε.

1×10–6 to 7×10–4, with simultaneous measurements of Q–1, ∆E/E, al-ways for 45 seconds. This gave the corresponding dependencesQ–1 – ε or ∆E/E (Fig. 6.13, curves 2, 3, 4, 5).

The dependence of internal friction on strain amplitude for thespecimen series subjected to different cyclic prior deformation (Fig.6.13) indicates that the form of the curves 1–4 is identical in a widestrain amplitude range and the magnitude of prior cyclic deforma-tion changes the rate of increase of Q–1 only at ε ≥ 1 – 5 × 10–4.Significantly different Q–1–ε dependences were obtained for thespecimens subjected to prior deformation at ε

d = 6.5 × 10–4 (curve

5). The main value of internal friction slightly increases with in-creasing ε

d, but after the application of ε

d = 6.5 × 10–4 the increase

of Q –10 is significant. Valuable information on the activation of

microplastic processes is provided by the measurements of thedefect of the Young modulus ∆E/E. The first reproducibly deter-mined values of ∆E/E were obtained at relatively low values of ε

kr1.

With the increase of ε the defect of the Young modulus slowly in-creases, and after the application of certain strain amplitudes andthe increase of these amplitudes there was a significant increase of∆E/E. Recording of the first values of ∆E/E detects the strain am-plitude at which the dislocation segments start to oscillate in the

265


stress field around the equilibrium positions (the first critical strainamplitude ε

kr1). The absolute values of internal friction and the

defect of the Young modulus are higher for the specimens subjectedto preliminary cyclic deformation at a higher strain amplitude.

In the case of mild steel in accordance with the experimental re-sults [283] we can use the modelling assumptions of the vibrationof a spring for the vibration of dislocation segments around theirequilibrium positions in the quasiviscous environment. In this mod-elling assumption, the ratio of the square of the defect of the Youngmodulus and internal friction determines the instantaneous disloca-tion density in the material using the equation [296]

2

1 ,

E

E kQ−

∆ = ρ (6.27)

where k is a proportionality factor. In accordance with a largenumber of experimental results, it is assumed that in low–carbonunalloyed steel after annealing ρ = 1012 m–2. Cyclic prior deforma-tion changes the dislocation density only when ε

d > ε

kr2. The incre-

ment of the strain amplitude ∆ε = ε – εkr2

in the microplastic regionof cyclic loading results in the increase of the dislocation densityby the value ∆ρ = ρ – ρ

2, where ε are the values of the strain am-

plitude and the dislocation density in the range ε > εkr2

, or ρ > ρ2,

and the values εkr2

, ρ2 are the second critical strain amplitude and

the initial dislocation density in the material. In the coordinateslog ∆ρ – log ∆ε the results were plotted in the form of straight lineswhich can be described by the equation

,ma∆ρ = ∆ε (6.28)

and the values of the exponent m are also a function of the mag-nitude of prior cyclic deformation of the material. The validity ofequation (6.28) indicates the process of irreversible changes in thestructure of the steel by the increase of dislocation density withincreasing strain amplitude.

In the range of the start of microplasticity (ε > εkr2

) the increaseof ε is accompanied by an increase of the dislocation density andof the magnitude of ‘plastic’ internal friction Q–1

p, indicating the

266


elastic–plastic interaction of mechanical loading with structuralchanges of the material, i.e. the existence and effect of the plasticstrain amplitude ε

p. To a first approximation, it will be assumed that

2 .p krε = ∆ε = ε − ε (6.29)

The cyclic strain curve, stress amplitude σa vs. the plastic strain

amplitude εp, characterising the plastic response of the material to

cyclic loading, has the form

,na pσ = χ ε (6.30)

where χ is a constant of the material that is sensitive to the struc-ture and experimental conditions, and n is the cyclic strain harden-ing exponent.

Using the equations (6.28), (6.29) and (6.30), at ε > εkr2

, weobtain the relationship between the stress amplitude and the increaseof the dislocation density in the form

,n

maσ = λ ∆σ (6.31)

where λ = χ a –(n/m). Substituting equation (6.27) into equation(6.31), using the given symbols, we obtain the relationship betweenstress amplitude σ

a, the value of the effect of the Young modulus

∆E/E and the magnitude of internal friction Q–1.These equations and the values of exponent m, determined from

the experiments at the same value of the exponent n for mild steel[297], with the change of the amplitude of prior strain ε

d, yield

some qualitative and also quantitative relationships characterisingthe cyclic plastic response of the material. At the selected value of∆ε, the increase of the dislocation density ε

d is more marked (equa-

tion 6.31, values m). These qualitative relationships indicate thesoftening effect of cyclic microplastic deformation with strainamplitudes from 2.6 × 10–4 to 6.5 × 10–4 in the case of mild steelin the region of the start of microplasticity; this has also been con-firmed indirectly by other studies [279, 296].

The experimentally determined values of Q–1 (and also Q–10, Q–1

p),

∆E/E, εkr1

and εkr2

with the increase of the strain amplitude, afterprior cyclic loading of the steel, are the results of more or less ex-

267


tensive release of the dislocation segments from the area of weakpinning by the atmospheres of C and N at the given prior strain am-plitude. The processes taking place under the effect of ε ≥ ε

kr2 can

be interpreted in the sense of extensive microplastic deformation onthe basis of statistical considerations regarding the structural het-erogeneity of the material.

The cyclic strain curve and its parameters are important for thequantification of the cyclic plastic response of materials [298].Measurements of internal friction and the defect of Young moduluswith the change of the strain amplitude also provide important in-formation in this area.

6.2.4 Cyclic strain curveIt has been shown in many studies that the plastic strain amplitudeis the controlling factor of fatigue damage and the formation andpropagation of static cracks. This amplitude can be characterised ashalf width of the hysteresis loop of the material. In loading withfrequencies of up to 300 Hz the hysteresis loops are recorded by thetesting machine, but at ultrasound frequencies (for example,approximately 20 kHz) direct measurements and recording is notpossible at the current state of measuring devices. Since the meas-urement of internal friction and of the defect of the Young modu-lus provides information on the integral representation of themicroplasticity process, these measurements can also be utilised inthe quantification of the plastic strain amplitude at ultrasound load-ing frequencies.

Low–carbon, unalloyed steel CSN 412013 (0.07 wt.% C, 0.27wt.% Mn, 0.03 wt.% Si, 0.013 wt.% P, 0.08 wt.% S, 0.07 wt.% Crand 0.006 wt.% N) after annealing at 720°C ± 20 hours for 4.25and 108 hours with slow cooling in the furnace was characterisedby the ferrite grain size of d

z1 = 0.022 ± 0.044 mm or d

z2 =

0.29 ± 0.045 mm, or dz3

= 0.620 ± 0.085 mm. After completion ofheat treatment, in order to carry out measurements in VTP equip-ment, the specimens were annealed in Ar at 200°C for 330 min. andcooled in the furnace at a rate of 100°C/h and, in the final stage,subjected to chemical polishing.

Symmetric cyclic pull–push loading with a frequency of 23 kHzin the central part of the specimens was applied by a resonant sys-tem described in a previous study [282]. Equipment makes it pos-sible to generate the total strain amplitude ε

ac in the evaluated cross

section of the specimen in the range from 6×10–6 to 4×10–4, with areproducibility better than 1.5×10–7. The total strain amplitude ε

ac

268


was measured with strain gauges fixed to the essential part of thespecimens and evaluated using a Wheatstone bridge with a selectivenanovoltmeter. The calibration of the proportionality factor of thestrain gauges was measured by the deviation of the free end of thesystem with an accuracy of ±1 µm, using TW5/2A sensors andVibrometer AG equipment with a frequency range from 0 to 100kHz. The proportionality factor of the strain gauges at a loadingfrequency of 23 kHz was 1.8 2%, whereas at the usual frequenciesthe proportionality factor is 1.95 + 2%.

The stress amplitude in the central part of the specimen wasevaluated by approximation in accordance with Fig. 6.14. In theelastic loading range σ = E ε

a, and the value E (= tan α) is related

with the main resonance frequency fr. In the elastic–plastic regionof loading (for example, point x in Fig. 6.14) σ

ax = E

x (ε

aex + ε

apx)

= Ex

εacx

, and the value Ex (= tan α

x) corresponds to the resonance

frequency of the system frx. Consequently, E

x = E – ∆E

x, where ∆E

x

is the change of the Young modulus associated with the microplasticdeformation of the specimen. These considerations show that thestress amplitude, for example, at point x

FFFFFigigigigig. 6.14.. 6.14.. 6.14.. 6.14.. 6.14. Determination of relationship between stress and strain.

269


( ) 1 xax acx x acx

EE E E

E

∆ σ = ε − ∆ = ε − (6.32)

and the amplitude of the plastic component of strain

.ax xapx acx acx acx acx

E

E E

σ ∆ε = ε − ε = ε − = ε (6.33)

The advantage of experimental equipment is that it makes it pos-sible, at the selected value ε

ac, to measure not only the magnitude

of internal friction Q–1 but also evaluate frx and, in accordance with

Mason [299], determine directly

2,x xE M fr fr

E M frν

∆ −= (6.34)

where M and Mv the effective mass of the entire system and of the

specimen. The measuring equipment and corrections made it possibleto evaluate ε

ac with the accuracy of ±1×10–6 and the changes of ε

ac

with a scatter of ±1.5 × 10–7, the relative change of the Youngmodulus with the accuracy of 6 × 10–4 and the changes are alsomade by connecting strain gauges to the reduced section of the ti-tanium attachment of the system at the distance of λ/4 from the endof the attachment, in which no microplastic deformation took placeup to ε

ac = 4 × 10–3 [299]. After evaluating the changes of the value

of the amplification factor in the central part of the specimens, theauthor found that the agreement of the results determined by this ap-proximation and by the direct measurement of the stress amplitudeis better than 98%.

The experiments were carried out at a temperature of 21°C un-der the effect of a constant magnetic field with an intensity of 1.9× 104 A m–1, parallel with the axis of the specimen. Series of thespecimens with the grain sizes of d

z1, d

z2, and d

z3, were loaded

gradually with increasing strain amplitudes εac

. At every selectedvalue of ε

ac, the author determined internal friction Q–1, the relative

change of the Young modulus ∆E/E and, using equations (6.32) and(6.33), also the stress amplitude σ

a and plastic strain amplitude ε

ap.

In calibration, it was observed that the loading of the specimensfor 1 hour with ε

ac = 6 × 10–4 does not cause any measurable

changes of the temperature of the specimen so that there cannot be

270


any changes of the Young modulus of the evaluated material as aresult of the change of the temperature of the specimen.

Measurement of every experimental point at εac

= const was car-ried out over a period of 100 s, which represents 2.30 × 106 ofloading, sufficient for the stabilisation of the changes of the prop-erties of the material in the evaluated strain amplitude range.

The characteristic course of the changes of internal friction in re-lation to the total strain amplitude (Fig. 6.15) showed the effect ofthe ferrite grain size on the internal friction background, determinedat a low value ε

ac, and the differences in the form of the Q–1 –

ε

ac

dependence in the region of the dependence of Q–1 on strain ampli-tude.

The relative change of the Young modulus ∆E/E is recorded af-ter obtaining certain values of ε

ac, and rapidly increases with in-

crease of the total strain amplitude of the specimen. Conventionally,the value ε

ac resulting in a sharp increase of ∆E/E, can be denoted

as the second critical strain amplitude εkr2

[294]. The values of εkr2

are the function of the size of the ferrite grain, i.e. the critical strainamplitude decreases with increasing grain size (Table 6.3).

The intensity of the changes of ∆E/E in relation to εac

in therange ε

ac > ε

kr2 is higher in the case of steels with larger ferrite

grains. In the coordinates log ∆E/E – log εac

, Fig. 6.15, the experi-mental points fit straight lines so that the experimentally obtainedpoints can be expressed by the equation

FFFFFigigigigig.6.15..6.15..6.15..6.15..6.15. Q–1– ε (solid lines) and ∆E/E – ε dependences (broken lines) for differentgrain sizes: 1,1') 0.022 mm; 2,2') 0.290 mm; 3,3') 0.620 mm.

271


,aac

EA

E

∆ = ε (6.35)

where A is the constant that depends on the structure and experi-mental conditions, a is the exponent with the values given in Table6.3, with both factors dependent on the ferrite grain size.

The experimentally determined dependence of stress amplitude σa

on the total strain amplitude εac

(Fig. 6.16) shows that the devia-tion of the dependence from the straight line, determined by theequation σ = E ε

ac, starts in the case of specimens with different

ferrite grain sizes at different values of εac

. The ‘stress amplitudevs. plastic strain amplitude’ cyclic curves, shown in Fig. 6.16, in-dicate that the ferrite grain size has a significant effect on the formof these dependences. The cyclic strain curves can be characterisedby equation (6.30). Figure 6.17 shows that the equation of thecyclic strain curve is also fulfilled for a loading frequency of 23kHz, with the values of the factor χ, given in Table 6.3. With theincrease of the ferrite grain size, the value of χ decreases from 0.41to 0.32, with a simultaneous decrease of the value of the factor χfrom 9700 to 2700 MPa. The relationship between the stress ampli-tude and the total strain amplitude has the form

FFFFFigigigigig.6.16. .6.16. .6.16. .6.16. .6.16. σa – ε

ac (solid lines) and σ

a – ε

ap (broken lines) dependences for steels

with different ferrite grain size: 1,1') 0.022 mm; 2,2') 0.290 mm; 3,3') 0.620mm.

272


TTTTTaaaaabbbbble 6.3le 6.3le 6.3le 6.3le 6.3 Experimentally determined exponents and factors of microplastic responseof steel with different ferrite grain size

lairetaMezisniarG

]mm[ εrc 2

σa

= χpa

n

n χ ]aPM[hdcba

leetS

31021

20.092.026.0

01·3.1 4–

01·0.1 4–

01·3.7 5–

0079014.00044753.00072023.0

062.3290.103.2313.1016.3097.006.259.0166.1719.3237.079.2769.1

iToMbN

01·1.1 3–

01·3.5 4–

01·8.7 4–

0075752.000652803.0

0081942.0

91.8342.120.3601.396.4177.060.3360.247.5847.086.3309.1

,ba acBσ = ε (6.36)

where B (= –χAn), b [= (a +1)n] are constants that depend on struc-ture and experimental conditions, the value of n is the same for allferrite grain sizes examined (Table 6.3). The validity of equation(6.36) is shown in Fig. 6.17.

The plastic strain amplitude in the range εac

> εkr2

can be deter-mined from the equation

,cap acCε = ε (6.37)

where C = (B/χ)1/n, c = b/n, or C = A, c = a + 1. The values C, c

FFFFFigigigigig. 6.17.. 6.17.. 6.17.. 6.17.. 6.17. Experimentally determined dependences of the microplasticity characteristics,determined at a frequency of 23 kHZ for steel with different ferrite grain size:solid lines 0.022 mm, broken lines 0.290 mm, dot-and-dash lines 0.620 mm.

273


are constants depend on the structure and experimental conditions,and the values of exponent c are presented in Table 6.3. To a firstapproximation, the reciprocal value of exponent c is comparablewith the value of the coefficient of cyclic strain hardening for theevaluated ferrites grain sizes. The validity of equation (6.37) isconfirmed in Fig. 6.17, which also shows that the application theselected value of ε

ac results in a higher plastic strain amplitude in

a material with larger ferrite grains.Internal friction in the strain amplitude range resulting in

microplastic deformation (εac

> εkr2

), referred to as plastic internalfriction Q–1

p = Q–1

εc –Q–1

εk, where Q–1

εc is the internal friction at the

selected value of εac

, Q is the internal friction under the effect ofthe critical strain amplitude, depends on ε

ap (Fig. 6.17) in accord-

ance with the equation

1 ,dp apQ D− = ε (6.38)

where the values of the factors D, d depend on the ferrite grain size,and the values of the exponent n are presented in Table 6.3. The re-sultant dependences indicate the strong sensitivity of Q–1

p on the fer-

rite grain size, because the same value of εap

results in more exten-sive scattering of mechanical energy in the material with a largerferrite grain. This scattering is associated with the movement andgeneration of the dislocations.

Internal friction Q–1p can be expressed as the ratio of the energy,

scattered in a single load cycle ∆W = Fσεap

[300] (where F is thecharacteristic of the shape of the hysteresis loop), to the total sup-ply of energy W = 1/2 E ε2

ac. Consequently

12 ,

2p a ap

pac

FWQ

W E− δ σ ε∆= = =

π π π ε (6.39)

where δp is the friction decrement. If the equation for the cyclic

strain curve is substituted into equation (6.39], the plastic compo-nent of the total strain can also be expressed by the equation in theform

1 11 1 1

2 2 .n np p

ap ac ac

Q E E

F F

− + + π δ ε = ε = ε χ χ

(6.40)

274


In equation (6.40), the values Q–1p , δ

p, F, σ, or χ depend on the

magnitude of εac

and, consequently, on εap

.If the equations (6.32), (6.33) are substituted into equation

(6.39), by measurements of plastic internal friction and the relativechange of the Young modulus it is possible to determine, for theselected values of ε

ac, the factor characterising the shape of the hys-

teresis loop by the equation

1

2

2

.pQF

E E

E E

−π=

∆ ∆−

(6.41)

The area of the hysteresis loop in cyclic loading increases withincreasing total strain amplitude ε

ac, i.e., with increasing ε

ap. If we

use equation ∆W = F σεp and equation (6.35), (6.37), we can ob-

tain the equation

,hacW H∆ = ε (6.42)

where the values of the factors H, h increase with increase of theferrite grain size, as indicated in Table 6.3 using the value of theexponent h as an example. The validity of equation (6.42) is illus-trated in Fig. 6.17. The equation show that H = FχAn+1, h =(a + 1) (n + 1). If in derivation of equation (6.42) we use the equa-tion (6.36), then H = Fχ–1/nB(n+1)/n, h = b(1 + 1/n).

Detailed evaluation of the experimental results show that none ofthe factors and exponents in these equations changes its magnitudein accordance with the Hall–Petch equation, i.e. in the region ofcyclic microplasticity of mild steel it is not possible to obtain thelinear dependence of any of these quantities on d

z1/2 .

The increase of the main value of internal friction with increas-ing grain size of the material in a wide range of strain amplitudewas also detected and interpreted by several authors [279, 280], likethe decrease of the values of the critical strain amplitude with in-crease of the grain size. The activation of oscillations of the dis-location segments, release of the dislocations from pinning areas,and the generation of new dislocations and their movement afterexceeding the value ε

ac > ε

kr2 require lower values of ε

ac for the

specimens with larger ferrite grains. In the evaluated range of the

275


cyclic strain amplitude there is no large–scale movement of the dis-locations to the grain boundaries, and this is reflected in the factthat none of the factors, characterising the microplastic response ofthe material to cyclic loading with a frequency of 23 kHz, fulfilsthe Hall–Petch equation. The validity of this equation is confirmedin the majority of processes characterised by higher plastic strains,for example, in the static tensile test or in fatigue test where thespecimens are loaded with stress amplitudes close to or higher thanthe fatigue limit.

For identical steel loaded at a frequency of 70 Hz, χ = 835 MPa,n = 0.156 [297]; these values are lower than those obtained at aloading frequency of 23 kHz. The values of the microplastic straindiffer at the high loading frequency. The experimentally determinedand derived analytical dependences are exponential which increasesthe importance of the amplitude of plastic strain ε

ap in the fatigue

damage cumulation process.When loading a material with a high frequency (for example, 23

kHz), the maximum value of εap

is 3.5% εac

in the evaluated strainamplitude range. This is also consistent with the results of experi-ments carried out on copper at a loading frequency of 21 kHz [301].At usual loading frequencies, for example, 70 Hz [297], the frac-tion of ε

ap in the value ε

ac is considerably higher, which means that

the same total strain amplitude εac

results in far more intensivemicroplastic processes in the material that in loading at a high load-ing frequency.

The difference in the fraction of εap

in the value of εac

may beassociated with the greatly restricted time available for the move-ment of dislocations over larger distances under the effect of thegiven stress amplitude, with the fact that the stress field cannot berelaxed by movement of dislocations over large distances, and withthe significant localisation of microplastic strains in loading thematerial with a high frequency [281].

Equation (6.40) forms a link between the highly sensitivemicroplastic characteristic Q–1

p or δ

p, the shape of the hysteresis loop

F and the value of the ratio εap

/ε2ac

, or the significant characteris-tic of the material in cycling loading (n), which is included in thegeneral equation for the determination of the fatigue life of mate-rials.

The results obtained in Ref. 302 indicate that the given approxi-mation (equations 6.32, 6.33) provides the appropriate factors of cy-clic microplasticity also for titanium (0.39 wt.% Al, 0.04 wt.% Fe,0.04 wt.% Si and 0.0 wt.% Cr), molybdenum (0.09 wt.% Fe, 0.03

276


wt.% Si and trace amounts of Cu, Nb, Ni, Ca, Mg) and also forniobium (0.25 wt.% Mg, 0.18 wt.% Ca, 0.07 wt.% Fe, 0.04 wt.%Si and trace amounts of Ba, Mn), as presented in Table 6.3.

The cyclic strain curve and hardening of materials under repeatedloading are significantly influenced by temperature.

6.2.5 Temperature and cyclic microplasticityMeasurements of internal friction and the effect of the Young modu-lus with the increase of strain amplitude provide important informa-tion on the change of the response of the material to increasing tem-perature, as shown in Ref. 303 by Puškár and Letko using a tita-nium alloy.

Titanium alloy VT 3–1 (according to Russian standard GOST) isa two–phase creep–resisting martensitic alloy with the followingchemical composition (in wt.%): 6.49 Al, 2.36 Mo, 1.47 Cr, 0.42Fe, 0.24 Si, 0.03 C, balance titanium. Artificial ageing was carriedout at a temperature of 550°C for 5 hours. After removing from thefurnace, the specimens were cooled in air. The microstructure con-sisted of the grains of the α–phase and β–phase at a surface arearatio of 1:1 in the form of equiaxed of globular formations, with-out any distinctive boundaries of the initial β–phase. After thistreatment, the properties of the material at 20°C were: R

m = 1200

MPa, Rp 0.2 = 12 000 MPa, A

5 = 14.2%, Z = 52.3%.

Internal friction Q–1 and effect of the Young modulus ∆E/E of theVT 3–1 alloy were determined in equipment VTP, described in de-tail in Ref. 304. For the experiments, equipment was fitted with afurnace and facilities for supplying power to the furnace, and formeasuring and recording temperature. The VTP equipment operateson the resonance principle; therefore, cylindrical specimens with twoheads and the cylindrical central part must also fulfil the resonancecondition. The measurements were taken at temperatures of 20, 200,300, 400 and 550°C, with a scatter of ±1%. A change of the tem-perature of the VT3–1 alloy results in the change of the velocity ofpropagation of sound (v) and also the dynamic Young modulus (E

d)

at a loading frequency of 23 kHz (Table 6.4). Therefore, the lengthof the specimens was changed, whilst maintaining the given shapeof the specimens.

Experimental equipment VTP makes it possible to measure inter-nal friction and the defects of the Young modulus in the total strainamplitude range ε

ac from 5 × 10–7 to 3 × 10–3, also at elevated tem-

peratures, where the dimensions of the specimens are adapted in re-lation to the resonance condition.

277


TTTTTaaaaabbbbble 6.4le 6.4le 6.4le 6.4le 6.4 Experimental values of VT3-1 alloy at 23 kHz and different test temperatures

In the measurements, the experiments were carried out with thelowest suitable value of ε

ac to higher values in such a manner that

measurements at every experimental point was carried out within 50seconds, i.e. 1.15 × 106 load cycles.

The results of the measurements show that the characteristics ofthe functional dependence Q–1 vs. ε

ac do not change with increasing

temperature, Fig. 6.18. The curves in the range of εac

from 3 ×

FFFFFigigigigig.6.18..6.18..6.18..6.18..6.18. Dependence of internal friction on the strain amplitude of VT3-1 alloyat 23 kHz and different temperatures.

ytitnauQCº,erutarepmeT

02 002 003 004 055

v sm[ 1– ]E

daPM,

D2'

d'D

2

Aa

χ ]aPM[R

860501·41.1 5

01·90.1 3–

731.001·353 3–

74.342777.101·65.1 4

063.0

509401·70.1 5

01·46.1 3–

081.001·2.5 3–

50.034267.101·71.1 4

163.0

057401·10.1 5

01·93.2 3–

691.001·624 3–

66.937977.101·4.9 3

063.0

046401·59.0 5

01·50.1 3–

360.001·3 3–

00.258397.101·2.8 3

653.0

044401·78.0 5

01·849.0 3–

770.001·4 3–

19.23412.101·7.1 3

844.0

278


10–5 to 2.5 × 10–3 show that the value Q–1 is independent of εac

upto ε

kr1. This is the internal friction background Q–1

0.

The values of εkr1

can be determined as the magnitude of εac

atwhich the defect of the Young modulus ∆E/E is recorded for thefirst time [304]. With the increase of ε

ac to ε

kr1 and in further meas-

urements of Q–1 at εac

<< εkr1

, for example at εac

= 3 × 10–5, the in-ternal friction background does not change up to the value ε

ac = ε

kr2,

and then the value of Q–10 starts to change [281]. This method was

used to determine the values of εkr2

for all temperatures in the tests.Between the values of ε

kr1 and ε

kr2 there is a slight dependence

of Q–1 on εac

and at values above εkr2

the internal friction dependsin a significant manner on the value ε

ac.

In the temperature range 20–400°C, the internal friction back-ground of VT3–1 alloy gradually increases (Fig. 6.19), but at a tem-perature of 550°C there is an anomaly in the dependence of Q–1 onT.

When test temperature is increased, the magnitude of the firstcritical strain amplitude ε

kr1 decreases (Fig. 6.19) in accordance with

the equation

ε ε

= − (6.43)

FFFFFigigigigig. 6.19. 6.19. 6.19. 6.19. 6.19. Change of internal friction background and the first and second criticalstrain amplitude in relation to temperature for VT3-1 alloy at 23 kHz.

279


where

4 7 11 4.1 10 , 4.23 10 C .x J− − −ε = ⋅ = ⋅ °

The increase of test temperature results in a faster decrease ofthe second critical strain amplitude ε

kr2 than in the case of ε

kr1 (Fig.

6.19). The decrease of εkr2

with increasing temperature can be ex-pressed by the equation

2 2 ,kr x KTε = ε − (6.44)

where

3 6 12 1.14 10 , 1.13 10 C .x K− − −ε = ⋅ = ⋅ °

Figures 6.18 and 6.19 show that increasing temperature decreasesthe width of the range between ε

kr1 and ε

kr2. At a test temperature

of 550°C, the authors of Ref. 303 recorded anomalies in the tem-perature dependence of ε

kr1 and ε

kr2.

In the range of εac

from εkr1

to εkr2

but also above εkr2

(to 2.5 ×10–3), internal friction can be expressed analytically by the equation

12 ,d

acQ D− = ε (6.45)

here D ′2, d', or D′

2, d (for the range of ε

ac from ε

kr1 to ε

kr2, or ε

kr2

and higher) are the experimentally determined coefficients or expo-nents, presented in Table 6.4.

With the increase of εac

at a specific value of this amplitude, theresponse of the material changes from elastic to elastic–plastic,reversible (ε

kr1) up to the region of microplastic deformation (ε

kr2).

This is reflected in the change of the resonance frequency of theVTP system. After evaluating this change, it is possible to determinethe defect of the Young modulus of the experimental material ∆E/E [304] which reflects the integral cyclic microplasticity in the ex-amined volume of the material.

The results of measurements of ∆E/E with increasing εac

at theselected temperatures are presented in Fig. 6.20. Measurementsmade it possible to determine more accurately the value of ε

kr1, be-

cause ∆E/E is recorded only when this value is reached or exceeded.The curves can be expressed analytically by equation (6.35). The

280


FFFFFigigigigig. 6.20.. 6.20.. 6.20.. 6.20.. 6.20. Change of the defect of the Young modulus on the total strain amplitudeof VT3-1 alloy at 23 kHz and different temperatures (arrows indicate the valuesof ε

kr1).

values of the factors of this equation are presented in Table 6.4.Whilst in the temperature range from 20 to 400°C, the value of ais approximately 1.7, at a temperature of 550°C it is already 1.2,which reflects the anomaly in the behaviour of the initial materialat this temperature.

For every experimental point it is possible to determine εac

and∆E/E. Using equations (6.32) and (6.33), we can determine the val-ues of stress amplitude σ

a and the corresponding values of ε

ap. The

results are shown graphically in Fig. 6.21. The cyclic strain curvescan be expressed in the form σ

a = χεn

apεn

ap, where χ or n is the

coefficient of proportionality of the exponent of the cyclic straincurve. The values of these quantities for different temperatures arepresented in Table 6.4.

When the temperature is increased from 20 to 400°C, the valueof χ decreases whereas n does not change, and at 550°C the valuesof χ and n greatly differ from their temperature dependence, ob-tained in the given temperature range.

The results of the measurements show that increasing tempera-

281


ture results in a gradual increase of the internal friction backgroundof VT3–1 alloy in loading with a frequency of 23 kHz. The increaseof temperature during mechanical loading activates the process ofmicroplastic deformation also in the case of VT3–1 titanium alloy,as indicated by the change of the values of ε

kr1, ε

kr2, which decrease

when the temperature is increased in the range 20–400°C.The anomalous behaviour of the VT 3–1 alloy during tests at

550°C is the result of changes of the microstructure of the alloywhich is stable up to 450°C. After exposure of the alloy at 550°C,the structure shows a significant heterogeneity in the shape of theα–and also β–phase. This is also associated with the fact that thetemperature of 550°C is the processing temperature for the processof ageing of this alloy and high–intensity ultrasound greatly short-ens the ageing nd coarsening time of the phases [305].

If we consider the accuracy of the measurements and interpreta-tion of the results, Table 6.4 shows that the value of n does notchange when the temperature is increased in the range 20–400°C.

FFFFFigigigigig. 6.21. 6.21. 6.21. 6.21. 6.21. Cyclic strain curves of VT3-1 alloy at 23 kHz and different temperatures.

282


This is in agreement with the results which shows that at frequen-cies of up to 300 Hz the increase of temperature in the range ofstructural stability of the alloy results in no significant changes inthe shape and nature of the cyclic strain curve, but with the increaseof temperature the curve is displaced along the stress amplitude axis[298].

The measurements of internal friction and the defect of the Youngmodulus make it possible to evaluate the important microplasticcharacteristics of the materials under different service conditions. Itis important to stress the fact that the results of measurements ofthe microplasticity factors may be significantly influenced or evenoverlapped by, for example, the magnetomechanical component ofinternal friction when testing ferromagnetic materials.

6.2.6 Magnetic field and microplasticity parametersTaking into account section 3.6, i t may be noted that thedislocational internal friction which depends on the strain amplitude,being a frequency–independent component, is the basis of a usefulindirect method used in many investigations for evaluating thecyclic plastic response of the materials, whereas the second,frequency-independent component, i.e. magnetomechanical damping,in ferromagnetic materials may influence these results from both thequalitative and quantitative viewpoint. Taking into account the linkbetween the extent of internal friction and the defect of the Youngmodulus, it may be expected that magnetomechanical friction willalso influence the response of the material, characterised by the de-fect of the Young modulus.

The magnetomechanical component of internal friction inferromagnetic materials can be suppressed by placing the componentor specimen in a unidirectional magnetic field with the intensitycausing the magnetically saturated state in the ferromagnetic mate-rial. Consequently, the magnetomechanical component of internalfriction and the defect of the Young modulus can be separated fromthe dislocational friction that depends on the strain amplitude. Tak-ing into account different response of different alloys with differ-ent fractions of the ferromagnetic phases, the minimum strength ofthe external magnetic field required for obtaining magnetic satura-tion differs.

Puškár [306] carried out a number of experiments to determineexperimentally the conditions for magnetic saturation and evaluatethe effect of the magnetic field on the extent of internal friction andthe defect of the modulus of elasticity of the CSN 412032.1 steel

283


in loading with a frequency of 22.9 kHz at different values of thetotal strain amplitude.

Specimens of the shape and dimensions shown in Fig. 6.22, wereproduced from 41 2032.1 steel with the following chemical compo-sition, wt.%: 0.3 C, 1.2 Mn, 0.8 Cr, 0.15 V. The steel was in thecondition after normalizing annealing. After completing preparation,the specimens were annealed at 500°C, for 1 hour, in vacuum.

Internal friction Q–1 and the defect of the Young modulus ∆E/E in relation to the total strain amplitude ε

ac, at ε

ac = const, dur-

ing measurements at a specific point, were determined in automaticequipment for internal friction measurements VTP–A (VSDS Tech-nical University), described in Ref. 155. The experiments were car-ried out at a temperature of 22°C. The specimens were loaded insymmetric pull–push loading at a frequency of approximately 22.9kHz.

In contrast to the measurements taken on non-ferromagneticmaterials, specimens of 12032.1 steel were placed in a coil with theshape and dimensions shown in Fig. 6.22. The coil contained 1000turns of copper wire, diameter 0.5 mm. When measuring the depend-ence of the intensity of the magnetic field H of the coil in relationto the intensity of direct current I

j, the author used the Hall probe,

placed in the centre of a central circular hole in the coil, when thespecimen was not in the coil. The dependence of H on I

s is shown

in Fig. 6.23. To determine the conditions of saturation of specimensof 12032.1 steel, after placing the specimen in the hole in the coil,the author used the circuit shown in Fig. 6.22. The stabilised sourceof direct current Z with the possibility of regulating the intensity ofdirect current (transformer 220 V/0.7 V) was used. The ohmic re-sistance of 1 Ω is characterised by a voltage loss, and the measur-

FFFFFigigigigig. 6.22.. 6.22.. 6.22.. 6.22.. 6.22. Circuit of connection of the coil, shape and dimensions of the specimensand the shape and dimensions of the coil used in the experiments.

284


ing equipment gives the intensity of alternating current Is. Another

measuring equipment controls the intensity of alternating voltage Us.

The circuit makes it possible to determine the dependence of directcurrent I

j on the value of the ratio of alternating voltage U

s and al-

ternating current Is. With the gradual increase of I

j, the ratio U

s/I

s

does not initially change. In a specific range of Ij (from 0.2 A to

0.5 A), the ratio decreases. At values higher than Ij ≥ 0.5 A,

increase of Ij to 2 A no longer causes any changes in this ratio.

Measurements of the magnetic field show that Ij > 0.5 A in the

given coil with the inserted ferromagnetic core of the given shapeand dimensions causes complete magnetic saturation (Fig. 6.23).Therefore, for the given arrangement, I

j = 0.5 A and H = 8 000

A⋅m–1 is sufficient for the magnetic saturation of the specimens ofthe given shape, dimensions and material.

The measurement procedure used in VTP–A equipment is basedon the measurement of Q–1 and ∆E/E at 30 different values of ε

ac

from the total strain amplitude range from 1.1 × 10–6 to 7 × 10–4,by applying, at each measurement point, the selected value of ε

ac

during 300 s, followed by selection of a higher value of εac

. Thespecimen was placed in the coil (Fig.6.22) through which the regu-lated direct current I

j passed; the intensity of the current was such

that the intensity of the magnetic field in the coil without the speci-men was H = 0, 800, 1600, 2400, 3200, 4000, 4800, 6400, 9600,12800, 16000 and 19200 A⋅m–1, at a temperature of 22°C.

FFFFFigigigigig. 6.23. . 6.23. . 6.23. . 6.23. . 6.23. Saturation characteristic of the coil, determined by the dependenceU

s / I

s (solid line) and the dependence of the intensity of the magnetic field H

on the intensity of direct current supplied to the coil Ij (broken line).

US/I

S

IJ, A

H×

10–3

, A

m–1

285


The results of measurements show that the internal friction back-ground Q–1

0 changes only slightly with the change of the intensity of

the magnetic field H: increasing H decreases Q–10 (Fig. 6.24). At

εac

> 10–5, the value of Q–1 at H = 0 rapidly increases. At εac

=2 × 10–4 and it reaches the maximum and then slowly decreases withincreasing ε

ac. This qualitative description is similar for the values

of H of up to 6400 A⋅m–1. For higher intensities of the magneticfield H (from 9600 to 19200 A⋅m–1), the form of the dependenceQ–1 vs ε

ac is identical, i.e. the value of Q–1 continuously increases

with increasing εac

. The slow decrease of Q–1 after reaching themaximum value, in the log–log coordinates, is probably the indica-tion that the tested ferromagnetic material is not examined in themagnetically saturated condition.

The dependence of Q–1 on H (Fig. 6.25) shows that with increaseof H the value of Q–1, determined at different values of ε

ac, initially

decreases, and the magnitude of the decrease is a function of thevalue of ε

ac. At H ≥ 9600 A⋅m–1, the value of Q–1 no longer changes

and, consequently, this phenomenon is independent of the value ofε

ac. The results show that the contribution of magnetomechanical

friction to the total value of internal friction depends on the totalstrain amplitude and increases with increasing ε

ac. The experimen-

FFFFFigigigigig. 6.24. . 6.24. . 6.24. . 6.24. . 6.24. Dependence Q–1 – ε of 12 032.1 steel under the effect of magneticfield of different intensity.

286


tal results also show that the component of magnetomechanical fric-tion can be suppressed in cases in which the intensity of the mag-netic field in the coil with the given characteristics is H = 9600A⋅m–1, for the specimens of the given shape and dimensions andmade of 12032.1 steel. For practical purposes, the value almost100% higher is used, i.e. the value of H of approximately 20 000A⋅m–1.

Figure 6.26 shows that the defect of the Young modulus is thefunction of not only the total strain amplitude but also of the inten-sity of the magnetic field. With increasing H ≥ 9600 A ⋅m–1, theincrease of the intensity of the magnetic field has no longer anyeffect on this dependence.

In a number of investigations, the criterion for the determinationof the second critical strain amplitude ε

kr2 is represented by the

value εac

at which the dependence ∆E/E vs. εac

rapidly increases[302]. For the purposes of this chapter of the book, this character-istic will be denoted by ‘ε

2’ ,because in measurements on

ferromagnetic materials this characteristic is not exclusively asso-ciated with cyclic microplastic deformation but also with themagnetomechanical response of the material. This conclusion resultsfrom the comparison which shows that the value of "ε

2" increases

FFFFFigigigigig. 6.25.. 6.25.. 6.25.. 6.25.. 6.25. Dependence of internal friction on the intensity of the magnetic fieldof 12 032.1 steel at different total strain amplitudes.

287


with the increase of the intensity of the magnetic field H up to H= 9600 A⋅m–1, and from the values of H ≥ 9600 A⋅m–1 the increaseof H has no longer any effect of the value of "ε

2", Table 6.5.

The ∆E/E – H dependence, Fig. 6.26, indicates that the value ofthe defect of the Young modulus at a specific value of H is higherat higher values of ε

ac. With increasing H the value of ∆E/E initially

decreases, until H reaches approximately 9600 A⋅m–1. With a fur-ther increase of H the value of ∆E/E no longer changes, in the en-tire range of 0 current of the defect of the Young modulus.

In the log–log representation, using the intensity of the magneticfield lower than H = 9600 A ⋅m–1, the experimental dependences∆E/E – ε

ac have the form of curves, but at H ≥ 9600 A ⋅m–1, the

straight lines overlap (Fig. 6.26). If it is assumed that the curves∆E/E – σ

ac at H < 9600 A⋅m–1 are replaced by the straight lines, the

experimental dependences can be expressed by the equation (6.35).The determined characteristics a, presented in Table 6.5, show

that with increase of the intensity of the magnetic field the value ofa increases up to H ≥ 9600 A⋅m–1, and it then remains constant. Thecomment on the conventional notation "ε

2" also relates to the val-

ues of A, a, with the exception of the case in which the steel is al-ready in the magnetically saturated condition, i .e. H ≥ 9600A⋅m–1.

FFFFFigigigigig. 6.26. 6.26. 6.26. 6.26. 6.26. Dependence of the defect of the Young modulus on the intensity ofthe magnetic field for 12 032.1 steel under the effect of the magnetic field ofdifferent intensity H × 10–3 (A m–1).

288


TTTTTaaaaabbbbble 6.5le 6.5le 6.5le 6.5le 6.5 Change of characteristics of 12 032.1 steel with change of the intensityof the magnetic field

In accordance with the approximation for the determination of theplastic strain amplitude ε

ac and stress amplitude σ

a in high–

frequency loading [302], these quantities can be quantified by theequations (6.32) and (6.33). The cyclic strain curve is defined byequation (6.30). Comparison of the experimental data gives the val-ues of χ, n (Table 6.5). It can be seen that the increase of the in-tensity of the magnetic field H results in a decrease of the value ofn from 0.60 for H = 0 to 0.3 at H ≥ 9600 A⋅m–1, and the value ofχ also decreases.

Figure 6.23 and also 6.24 and 6.26 show that magnetic satura-tion of the given specimen of 12032.1 steel in the coil, used in theinvestigations, takes place up to the intensity of passing direct cur-rent I

j = 0.5 A, which corresponds to H = 8 000 A⋅m–1 in the coil

without the ferromagnetic core. The form of the curves in Fig. 6.25and 6.27 shows that at H > 9600 A⋅m–1 the value of Q–1 or ∆E/Eshows no longer any measurable changes with the increase of H atdifferent values of ε

ac. The conventionally quoted intensity of the

magnetic field for suppressing the magnetomechanical component offriction of 20000 A⋅m–1 [2] is consequently a safe value of H forobtaining guaranteed saturation in coils of different shapes, with dif-ferent technically required air gaps, for the specimens with differ-ent cross sections and dimensions, and also for steels with differ-ent fractions of the ferromagnetic phases. When evaluating theengineering properties of the materials with the ferromagnetic phase,the intensity of internal friction is significantly higher than in thecase in which the magnetomechanical component is suppressed by

H 01· 3–

m.A( 1– )"ε

2" .A 01 3– a χ )aPM( n

08.06.14.22.30.48.44.6

8.21;6.92.91;61

01·7.3 6–

01·3.5 6–

01·3.1 5–

01·1.2 5–

01·50.3 5–

01·0.4 5–

01·8.5 5–

01·1 4–

01·4.2 4–

3154902

032902244012422012212

4637.01857.06818.01558.07588.05909.01449.00000.23501.2

37838316312355870288624105070112508228757666497601

9306.09885.06945.00025.02694.04474.06644.06614.06814.0

289


FFFFFigigigigig.6.27..6.27..6.27..6.27..6.27. Dependence of the defect of the Young modulus on the intensity ofthe magnetic field for 12 032.1 steel at different total strain amplitudes.

the external direct magnetic field. Consequently, it is then possibleto evaluate the cyclic plastic response of the material on the basisof the component of dislocational friction which depends on thestrain amplitude. On the other side, heating of the material is pro-portional to the area of the hysteresis loop, and its formation isdetermined not only by the dislocational friction component, whichdepends on the strain amplitude Q–1

ε, but also the component ofmagnetomechanical friction Q–1

m. The total internal friction is Q–1

c =

Q–1ε + Q

m–1 . Measurements show [306] that Q

m–1 also depends on the

magnitude of εac

and, consequently, it can significantly overlap thevalues of Q–1

ε in a wide range of εac

. The magnetomechanical com-ponent of friction, associated with the direction of the orientationof magnetisation in domains and with the movement of Bloch wallsunder repeated loading of ferromagnetic materials, represents by itscontribution a significant part of the total internal friction of steelQ–1

c. For example, at ε

ac = 3 × 10–4, Q

c–1 = 1.6 × 10–3, but Q–1

ε =4 × 10–4, which means that the contribution from the magneto-mechanical component of friction is Q

m–1 = 1.2 × 10–3.

The measurements also show that the defect of the Young modu-lus in ferromagnetic materials has two components, i.e. (∆E/E)

c =

(∆E/E)ε + (∆E/E)m. One of these components is associated with the

magnetomechanical processes in the material (∆E/E)m and the other

290


one with the generation and interaction of the dislocations in thematerial, i.e., with the cyclic microplastic deformation (∆E/E). Be-low ε

ac = 2.4 × 10–4, the material shows mainly the component (∆E/

E)m and its value is approximately an exponential function of the

value of εac

. The conventionally denoted characteristic "ε2" can be

regarded as the threshold strain amplitude at which themagnetomechanical mechanism of interaction of repeated loadingwith the ferromagnetic material at different intensities of the exter-nal magnetic field is activated (6.5).

Also, at εac

> 2.4 × 10–4, the (∆E/E)m component is significantly

higher than the component (∆E/E)ε. For example, at εac

> 3 × 10–4

the value of (∆E/E)m at H = 0 is 2.04 × 10–3, and the value of (∆E/

E)ε at H = 19200 A⋅m–1 is 1.4 × 10–4.These considerations show that the determination of the actual

second critical strain amplitude εkr2

at which the process of cyclicmicroplastic deformation starts, is determined by fulfilling the con-dition of complete suppression of the magnetomechanical mechanismof scattering of energy in the ferromagnetic material.

Using this comment in characterisation of the cyclic strain curve(equations 6.32, 6.33 and 6.30) shows that after complete magneticsaturation, i.e. when H ≥ 9600 A⋅m–1, n = 0.30 at a loading fre-quency of 22.9 kHz. For steels with a similar content of carbon andother elements, at a loading frequency of approximately 100 Hz,n = 0.10–0.20 [307]. The results of our measurements and process-ing of results should be critically reviewed especially from the view-point of the time required to measure every experimental point,because it is likely that changes of the properties in relation to theloading time are of the saturation nature and loading for 300 s istoo short to stabilise the changes of the properties at a loading fre-quency of approximately 2.9 kHz.

6.2.7 Saturation of cyclic microplasticityChanges of the dislocation structure and mechanical properties ofthe materials under repeated mechanical loading with stress ampli-tude lower than the yield limit are of the saturation nature, whichmeans that after a certain number load cycles they no longer changewith their increase at a specific stress amplitude [298]. The satu-ration characteristics, expressing a certain part of the cyclic plas-tic response of the material in relation to the number load cycles,at conventional loading frequencies are characterised in many stud-ies [307], and it has been shown that they differ for different groupsof the materials [298, 307]. Measurements and evaluation of the

291


coefficient of cyclic strain hardening, present in the equation of thecyclic strain curve, also depend on the assumption that the responseof the material will be evaluated only after stabilising the changesof the properties in relation to the number of cycles.

These results can be verified by measurements of internal fric-tion and the defect of the Young modulus in relation to the loadingtime, as carried out in a study by Puškár [284].

Bars made of electrically conducting copper, purity 99.98%,R

p0.2 = 37 MPa, R

m = 220 MPa, were annealed at 600°C/h, 850°C/

h or 1050°C/h in vacuum. The resultant grain size of the specimenswas z

1 = 0.062 mm, the dynamic modulus of elasticity E = 1.387

× 105 MPa, the grain size z2 = 0.250 mm, E = 1.274 × 105 MPa,

or the grain size z3 = 0.707 mm, E = 1.263 × 105 MPa.

Experiments were carried out using equipment for measuring in-ternal friction and the defect of the Young modulus VTP–A (VSDSTechnical University, Zilina) with completely automatic control,measurements and processing of the measurement results [155].Equipment loads the specimens with symmetric pull–push loadingwith a frequency of approximately 22.9 kHz (R = –1), with the con-trolled total strain amplitude ε

ac which was 2 × 10–7 to 3 × 10–4 in

the given case. The advantage of completely automatic equipment isthat the measurement of a single point can be carried out in a veryshort time of 6 s and, consequently, it is also possible to evaluatethe time dependence of the values of internal friction Q–1 and thedefect of the Young modulus ∆E/E at the selected value ε

ac. The ac-

curacy of measurement in this equipment at Q–1 = 10–3 is 10–2 per-cent, and at ∆E/E = 10–3 is 10–4 %.

The first critical strain amplitude εkr1

was determined as εac

atwhich the defect of the Young modulus of approximately 10–4 isdetected for the first time [302], since the high accuracy of meas-urements gives non–systematic changes of the quantity at ∆E/E <10–4.

The value of the second critical stress amplitude εkr2

, which canbe determined by the currently available methods (see section 6.1.2),was the determined by a new method described later, since the cur-rently available methods are suitable in most cases for materials inwhich the dislocations are strongly pinned by the solute atoms; thisis not typical of copper [293]. All the measurements were taken dur-ing loading of the specimens in air at a temperature of 22 ± 1°C.

The experimental dependences, presented in Fig. 6.28, show thatthe form of the curves Q–1 – ε

ac and ∆E/E – ε

ac for the given grain

size of copper is very similar. The internal friction background, i.e.

292


FFFFFigigigigig.6.28..6.28..6.28..6.28..6.28. Q–1– ε (solid lines) and ∆E/E – ε (broken lines) dependences for copperwith different grain sizes, where triangles refer to ε

kr1 and squares to ε

kr2.

Q–1 at low values of εac

, for example 2 × 10–7, decreases with in-creasing grain size of copper. The values of the first critical strainamplitude ε

kr1 increase with increasing grain size of copper (Fig.

6.28 and Table 6.6). The second critical strain amplitude εkr2

(Fig.6.28 and Table 6.6) also increases with increasing grain size ofcopper. The form of the ∆E/E – ε

ac curve indicates that the dynam-

ics of increase of ∆E/E in relation to εac

is steeper in the case ofthe copper specimens with the larger grain size. The curves shownin Fig. 6.28 were obtained after loading the specimens for 500 s,which represents approximately 1.2 × 107 cycles, i.e. after the satu-ration of the changes of the properties.

At every experimental point (1 – 11 in Fig. 6.29) we recorded thedependence of Q–1 or ∆E/E on loading time τ. The schematic rep-resentation of these dependences in the graph, Fig. 6.29, shows thatthe dependence of Q–1 on loading time (solid lines) is not system-atic and, consequently, is difficult to interpret in this stage of in-vestigations. The dependence of ∆E/E on loading time (brokenlines), which is more important from the viewpoint of cyclic micro-plasticity, is, up to a specific value of ε

ac negative or decreases with

loading time.However, from a specific value of ε

ac the form of the ∆E/E – τ

293


FFFFFigigigigig.6.29..6.29..6.29..6.29..6.29. Q–1– ε (solid lines), ∆E/E – ε (broken lines), Q–1– τ and ∆E/E – τ dependences(inserts 1–11) dependences for copper with a grain size of 0.250 mm in loadingwith a frequency of 23 kHz.

curves is identical (from point 4 and higher) with the course of thechanges at the saturation of the characteristics. This amplitude ofthe total strain is denoted as the second critical strain amplitude ε

kr2.

The physical meaning of this characteristic is such that at εac

≤ εkr2

the cyclic microplastic deformation of copper starts and the changesof the mechanical characteristics (∆E/E) are saturated in the exam-ined range.

The tangent to the origin of the saturation curve (dependence 5in Fig. 6.29) intersects with the horizontal to the saturation value∆E/E at the point which determines the time τ

1. The triple value of

τ1 determines the saturation time τ

s (analogy with magnetic satura-

tion). Consequently, for each value of εac

> εkr2

we obtained a setof data on the saturation time τ

s, and, consequently, on the number

of cycles of loading in saturation Ns (= τ

sf, were f is the resonance

frequency in measurement of the selected point) determined at theselected values of ε

ac, and also the values (∆E/E )

z at the beginning

of saturation (τ = 8 s) and at the end of saturation (∆E/E)k. This

294


shows that it is possible to determine δ(∆E/E)s = (∆E/E)

k – (∆E/E)

z.

Evaluation of the relationships between the change of the defectof the Young modulus in the relationship between the change of thedefect of the Young modulus in relation to time τ of the number ofload cycles N at the selected values of ε

ac can be expressed analyti-

cally by equation (6.5). Evaluation of the value of the parameter ashowed that its magnitude depends on the value of ε

ac. The depend-

ence can described by the equation in the form

,baca B= ε (6.46)

where B, b are the experimentally determined parameters. The val-ues of parameter b are presented in Table 6.6. It can be seen thatthe increase of the grain size increases the rate of increase of thedefect of the Young modulus in relation to the number of load cy-cles. It can also be seen that the increase of the value of ε

ac in-

creases the rate of increase of ∆E/E in relation to the number ofload cycles.

The magnitude of the increase of the defect of the Young modulusδ(∆E/E)

s in relation to the number of load cycles at saturation N

s

at εac

> εkr2

can be expressed in the form

,Cs

s

EC N

E

∆ δ = (6.47)

where the values of C, c for different grain size of copper are pre-sented in Table 6.6. The increase of the defect of the Young modu-lus increases with increasing number of load cycles at saturation.

The number of load cycles resulting in the saturation of thechanges of the properties of copper with different grain size N

s de-

pends on the total strain amplitude. The analytical form of the de-pendence is

3 ,ds acN D= ε (6.48)

where D3, d are the characteristics presented in Table 6.6. The ex-

perimental results show that the increase of the total strain ampli-tude increases the number of load cycles resulting in the saturationof the changes of the properties of copper, and this increase be-

295


comes smaller with increasing grain size of copper.From equations (6.47) and (6.48) we obtain a quantitative cor-

relation between the increase of the defect of the modulus of elas-ticity during saturation and the total strain amplitude used in load-ing. The equation has the following form

,hac

s

EH

E

∆ = ε (6.49)

where H = CDc and h = cd, with the values presented in Table 6.6.From the approximation for the calculation of the stress ampli-

tude σa and the plastic strain amplitude ε

ap, which uses the results

of measurements of the defect of the Young modulus at specificvalues of ε

ac indicates that the equations (6.32) and (6.33) are valid

in this case.The experimental measurements show that the values of ∆E/E at

a certain value of εac

differ at the start of the loading process (de-noted by z) and after saturation of the changes of the properties (de-noted by s).

Processing of the results of measurements using equations (6.32)and (6.33) shows that as a result of the processes taking place dur-ing saturation, the cyclic strain curves is displaced to the right, i.e.to higher values of ε

ap at a specific value of σ

a. The dependence can

be expressed by the equation (6.30). The results show that the val-

TTTTTaaaaabbbbble 6.6le 6.6le 6.6le 6.6le 6.6 Characteristics of the cyclic microplasticity of copper with differentgrain sizes

ezisniarG]mm[ ε

rk 1ε

rk 2b C c D

3d

z1

260.0=z

2052.0=

z3

707.0=

01·70.8 7–

01·67.1 6–

01·09.2 6–

01·1.5 6–

01·9.5 6–

01·8.9 6–

354.0995.0238.0

01·5.2 51

01·5.8 41

01·7.9 72

76.171.261.3

01·0.1 01

01·9.6 9

01·14.1 9

057.0266.0284.0

ezisniarG]mm[ Η h

χz

]aPM[n

z χs

]aPM[ ns

z1

260.0=z

2052.0=

z3

707.0=

01·55.4 4–

01·2.2 8

5.323

252.1634.1325.1

054350576309842

495.0765.0455.0

0263545239202622

685.0655.0945.0

296


ues of χ, n for the start of loading and after saturation differ (Ta-ble 6.6). Increasing loading time decreases the cyclic strain hard-ening coefficient. Copper with the larger grain size is characterisedby a lower value of the cyclic hardening coefficient, both at thestart of loading or at saturation of the changes of the properties.

In order to obtain saturation in the examined strain amplituderange, the number of load cycles at a loading frequency of 22.9 kHzis approximately 1 × 107.

The process of cyclic microplastic deformation is not concen-trated only at the grain boundaries but is also associated with thegeneration and interaction of the dislocations inside the grain. Thisexplains why the values of ε

kr1, ε

kr2 increase with increasing grain

size. The behaviour of the internal friction background is reversed;this may be associated with the relaxation processes taking place atthe grain boundaries which contribute to the extent of internal fric-tion. The dependence of the changes of the properties of the mate-rial on the number of load cycles is the same as under conventionalloading frequencies [307], although the detailed examination andanalytical description provide new information. During saturation,with increasing grain size of copper, i.e. with increasing space inwhich dislocation interaction takes place, the change of the defectof the Young modulus becomes larger with increasing number ofload cycles; this associated with the fact that the magnitude of∆E/E is the manifestation of the integral cyclic microplasticity inthe elementary volumes of the material. The increase of the numberof load cycles with the saturation of the changes of the propertiesresults in increase of the difference between the value of the defectof the Young modulus at the beginning and end of loading; this re-lationship is associated with the result which shows that increas-ing total strain amplitude requires a larger number of load cyclesfor obtaining saturation.

The coefficient of cyclic strain hardening of copper is, accord-ing to Ref. 308, the same at frequencies of 100 Hz and 20 kHz, i.e.n = 0.205, and according to Ref. 309 it is n = 0.209, with the val-ues of n determined at the stress amplitude higher than the fatiguelimit of copper. Our experiments and approximation indicate that n

z

or ns decreases with increasing grain size, and the values of n de-

crease with increase of the number of load cycles; it should also beadded that these characteristics were obtained at stress amplitudeslower than the fatigue limit of copper.

The number of load cycles for obtaining the saturated state of thechanges of the properties in the examined strain amplitude range is

297


approximately 1 × 107 cycles which, at the usual loading frequen-cies, is approximately 1 × 106 cycles [307]. This difference iscaused by the fact that the ratio ε

ap/ε

ac at the high loading frequency

is very low in comparison with loading at the conventional loadingfrequencies and the damaging defect is exerted mainly by the plas-tic strain amplitude.

6.3 FATIGUE DAMAGE CUMULATIONWhen processing the results obtained in this chapter, it can be as-sumed that the measurement of internal friction and the defect of theYoung modulus with increasing total strain amplitude in the rangeabove ε

kr2 provide new possibilities for finding the relationship be-

tween the critical strain amplitude and the rate of changes of thecharacteristics of the materials and their macroplastic behaviourduring fatigue loading.

6.3.1 Hypothesis on the relationship of Q–1 – εεεεε andσσσσσa – Nf dependences

When examining the limiting state of the materials and components[1], it is also useful to include a new concept. The degradationprocess in the material or a component is a time–dependent proc-ess resulting in a change (often for the worse) of the applied prop-erties of the material as a result of the occurrence of internalchanges and due to the effect of external factors or their synergiceffect [310]. A suitable example of the limiting state of the mate-rial is fatigue of the material under mechanical loading, and a goodexample of the degradation process is the internal response of thematerial to external loading in the region of inelasticity and alsomicroplasticity in this region.

The inelastic behaviour of the material is associated with manyprocesses causing that Hooke’s law is not fulfil led in thesubmicroscopic dimensions or in the entire volume of the solid, i.e.the magnitude of deformation is not directly proportional to themagnitude of acting stress. The quantification of the representationof inelasticity under static and quasistatic loading, taking into ac-count the fact that the relaxation time is short in comparison withthe loading time, is demanding from the experimental viewpoint butcan be accomplished by direct methods.

In cyclic or repeated loading, depending on the ratio of relaxa-tion time to the loading time in the same direction, a situation mayarise in which direct examination is not yet possible. Consequently,it is necessary to use indirect methods and also appropriate models

298


for the interpretation of the behaviour of the material.For mild steel, Ivanova [311] proposed to divide the process of

fatigue damage cumulation up to fatigue failure in the high–cyclefatigue region to several stages. Puškár [24] included this proposalin the general model for explaining the fatigue curve. Range a(6.30b) characterises the incubation period of the fatigue process,range b is the region of nucleation of submicroscopic cracks andtheir propagation, range c is the range of propagation of the fatiguecrack, and range d the region of increase of the extent of the finalfracture of the specimen. The fatigue diagram also shows the linescorresponding to the fatigue limit σ

C, the limit of cyclic sensitivity

σCc

and the limit of cyclic elasticity σCe

[311].If the fatigue curve (Fig. 6.30b) is shown together with the de-

pendence of internal friction and the defect of the Young moduluson the total strain amplitude (Fig. 6.30a), it is possible to determinea certain phenomenological relationship between the values of thecritical strain amplitudes and the given fatigue characteristics.

The physical metallurgical similarity of the interpretation of theranges up to and above ε

kr1, ε

kr2, ε

kr3 and the ranges of σ, after con-

version of the values of εkr2

and εkr3

to εapkr2

and εapkr3

using equa-tion (6.33), enabled the author of this book to formulate the hypoth-esis on the mutual relationship of the characteristics using the fol-

FFFFFigigigigig.6.30..6.30..6.30..6.30..6.30. Dependence of change of Q–1, ∆E/E on total strain amplitude (a) andpart of the Wöhler curve.

299


lowing equations:

1,Ce d keEσ = ε (6.50)

2 ,nCc apkrσ = χ ε (6.51)

3,nC apkrσ = χ ε (6.52)

where χ is the coefficient of proportionality in the equation for thecyclic strain curve χ and n is the exponent of cyclic strain harden-ing of the material (equation (6.30)). If it is assumed that the char-acteristics of the cyclic deformation curve in loading below and atthe fatigue limit are the same, the values of χ and n in equations(6.51) and (6.52) are the same. According to some approaches, inloading below and at the fatigue limit the cyclic strain curves dif-fer. Therefore, in equations (6.51) we have χ

1, n

1, and in equation

(6.52) χ2, n

2, and χ

1 ≠

χ

2, n

1 ≠

n

2. In equation (6.50) E

d is the dy-

namic modulus of elasticity of the material.The application of the experimental data, obtained on electrically

conducting copper (section 6.2.6) for different grain sizes, showsthat for the grain sizes of 0.962 mm, 0.250 mm and 0.707 mm thelimit of cyclic sensitivity σ

Cc is 35.8; 33.5 and 30.0 MPa, whereas

the limit of cyclic elasticity σCe

is 0.11; 0.22 and 0.35 MPa.Further information can be obtained from the experiments carried

out on steel CSN 412032.1 (section 6.2.5), with the application ofa magnetic field with the intensity H = 19.2 × 103 A⋅m–1 in meas-urement of internal friction and the defect of the modulus of theelasticity in relation to the total stress amplitude. The limit of cy-clic sensitivity σ

Cc is 198 MPa, and the limit of cyclic elasticity is

σCe

18.4 MPa.The values of the third critical strain amplitude were not deter-

mined in the measurement of the dependences Q–1 vs. ε or ∆E/E vs.ε because of the experimental difficulties determined by the smallrange of the applicable strain amplitudes in equipment VTP–A(VSDS). The fatigue limit of electrically conducting copper is, how-ever, 80 MPa and the fatigue limit of steel 12032 is 225 MPa [307].

It can be assumed that for certain types of materials, differingmainly in the type of structural lattice and also other morphologi-cal features, the ratio of ε

kr2/ε

kr3 will change in accordance with a

specific dependence. Whilst maintaining the internal and external

300


factors, affecting the fatigue limit, it is possible to determine theapproximate value of the fatigue limit by measurements of internalfriction and the defect of the Young modulus in relation to the to-tal strain amplitude. This procedure can then represent a type of theshortened fatigue test.

At present, the author of the book is carrying out extensive ex-periment to verify this hypothesis.

6.3.2 Deformation and energy criterion of fatigue lifeThe evaluation of the conditions in which the material fails by fa-tigue fracture is still the subject of discussions. Its solution, in ad-dition to the considerable importance for deeper understanding of thefatigue process, will also provide a basis for the development ofshortened fatigue tests. The main question is: what causes fatiguefailure at a specific number of load cycles: is it the the limitingamplitude of plastic deformation, at the deformation criterion, or isit the limiting value of the energy scattered irreversibly by the ma-terial when using the energy criterion? [300,307,335].

For some materials some of these questions have already beenpartially answered in tests carried out with the frequency of changesof mechanical loading from 1 to 100 Hz, especially in the low–cycle fatigue range [300,307]. This was carried out using the veri-fied methods of evaluating the plastic strain amplitude ε

ap from the

total strain amplitude εac

for the deformation criterion or the veri-fied methods of determining the area of the hysteresis loop ∆W forthe application of the energy criterion.

Despite the gradual increase of the number of investigations car-ried out using high-frequency loading (approximately 20 kHz), noinvestigations have as yet been carried out in which the applicabilityof the deformation and energy criterion in the quantification of theconditions of formation of fatigue fracture would have been evalu-ated, as also indicated by the results of international conferences inthe USA in 1981 [312] and in the former USSR [313]. Problems arecaused mainly by the determination of ε

ap or ∆W at high frequen-

cies.This problem was solved in Ref. 314 by Puškár and Durmis. The

investigated unalloyed steels differed in the carbon content: steelCSN 412013 0.07 wt.% C, CSN 412040 0.37 wt.% C, CSN 4120600.56 wt.% C.

The internal friction and the defect of the modulus of elasticityof the evaluated materials were determined on three specimens forevery steel in the equipment described previously at a frequency of

301


7 × 10–6 – 2 × 10–3 in loading with symmetric tension and compres-sion and at a frequency of 22 kHz at a temperature of 22°C. Themeasurement procedure was described in Ref. 315.

Fatigue tests were carried out in resonance equipment describedin Ref. 316 and 317 in loading with symmetric tension and compres-sion at a frequency of 22 kHz, always on 25 specimens for everysteel. The temperature of the specimens during loading was main-tained at 25°C by spraying temperature-controlled water on thespecimens. On each of the three stress level the specimens wereloaded by a stress 10 MPa lower than the stress level at which thespecimens failed at a number of cycles to fracture (N

f) of approxi-

mately 107 cycles.Figure 6.31 shows the dependence of internal friction Q–1 and the

defect of the Young modulus ∆E/E in relation to the total strainamplitude ε

ac. The first critical strain amplitude ε

kr2 is determined

as the value εac

at which the measurable value of ∆E/E is recordedfor the first time. The second critical strain amplitude ε

kr2 is deter-

mined as the value of εac

at which there are irreversible changes ofthe internal friction background, i.e. the value Q

0–1 is determined at

εac

≤ 10–5. The specific values of εkr1

and εkr2

are presented in Ta-ble 6.7.

As a result of processing the data obtained in measurements us-ing equations (6.32), (6.33) and (6.35), the authors obtained thedata presented in Table 6.7. The fatigue curves of the examined

FFFFFigigigigig.6.31..6.31..6.31..6.31..6.31. Internal friction (solid lines) and defect of the Young modulus (brokenlines) in relation to strain amplitude of the steel at a loading frequency of 22kHz.

302


steels, Fig. 6.32, were plotted on the basis of the results of detailedfatigue tests. The fatigue life curves can be described by the equa-tion in the form

,ba f fN′σ = σ (6.53)

where σf is the fatigue strength coefficient, b is the fatigue life ex-

ponent. Equation (6.53) is the stress criterion of fatigue life. Thespecific values of σ′

f , a, b for the evaluated steels are presented in

Table 6.7, together with the fatigue limit values σc determined at σ

a

conventional number of cycles 2 × 108.The ratio σ

a /σ

f (Table 6.7) shows that this ratio differs from the

value 1 for Nf = 1, and σ

f = F

f /S

f, where F

f and S

f are the values

of the force and the smallest cross section at the moment of frac-ture in the static tensile test.

The transformation of the fatigue curves from the coordinatesσ

a – N

f to the coordinates ε

ap – N

f was carried out using the cyclic

deformation curves of the evaluated materials (equation 6.30). Thevalues of χ, n are the material and experimental characteristics ofthe materials presented in Table 6.7. The values of ε

ap were calcu-

lated using equation (6.33).The fatigue curves in the Manson–Coffin representation (the

strain criterion of fatigue life) are presented in Fig. 6.33. Thecurves can be described by the equation in the form

citsiretcarahC 31021leetS 04021leetS 06021leetS

εrc 1

εrc 2

Aa

·E 01 5- )aPM(σ'

f)aPM(

bσ

C)aPM(

(σa

σ/ 'f

1=N)χ 01· 4- )aPM(nε'

f

cε

pa/ε'

f

01·4.4 5–

01·8.2 4–

0.82614.12370.2

947970.0–

58157.031.1593.001·40.1 3–

002.0–01·8 4–

01·3.6 5–

01·3.5 4–

2.9503.1

0270.2795

750.0–51284.011.3524.001·31.9 4–

431.0–01·3.1 4–

01·1.9 5–

01·5.6 4–

1.840.1

2770.2164

830.0–03293.024.7094.001·31.3 5–

770.0–01·2.5 5–

TTTTTaaaaabbbbble 6.7le 6.7le 6.7le 6.7le 6.7 Characteristics of carbon steels and factors in equations (6.30), (6.35)and (6.54)

303


,cap f fN′ε = ε (6.54)

where εf is the fatigue ductility coefficient and c is the exponent of

fatigue life with the values presented in Table 6.7.The values presented in Table 6.7 indicate that the ratio ε

ap/ε

f,

where εf is the true strain in the area of fracture in the static ten-

sile test, greatly differs for different materials.In the strain amplitude range above ε

kr2 plastic internal friction

is recorded Qp–1 = Q–1 – Q

2–1 where Q–1 and Q

2–1 are the values of

internal friction at εac

> εkr2

or at εac

= εkr2

(Fig. 6.31, Table 6.7).Using the equations (6.39) and (6.41), gives the equation in the

following form

1

2.p f f b c

f

QW N

E E

E E

−+′ ′π σ ε

∆ =∆ ∆ −

(6.55)

The total energy consumed by the material up to the formation offracture (the energy criterion of fatigue life) is

FFFFFigigigigig.6.32..6.32..6.32..6.32..6.32. Fatigue curves of examined steels at a loading frequency of 22 kHz.

304


FFFFFigigigigig.6.33..6.33..6.33..6.33..6.33. Fatigue curves in Manson and Coffin representation for the examinedsteels at a loading frequency of 22 kHz.

11

2.p f f b c

f f f

QW WN N

E E

E E

−+ +′ ′π σ ε

= ∆ =∆ ∆ −

(6.56)

Using equation (6.53) and substituting into equation (6.32), we ob-tain the equation

1

1 .b

acf

f

E EN

E

ε ∆ = − ′σ (6.57)

From the equations (6.56) and (6.57) we obtain the functional de-pendence for the energy consumed by the material up to fracture inthe form

1 2 31

21

bp f f ac

ff

Q E EW

EE E

E E

+ +− ′ ′π σ ε ε ∆ = − ′σ ∆ ∆ −

(6.58)

305


FFFFFigigigigig.6.34..6.34..6.34..6.34..6.34. Dependence of the energy absorbed to fracture on stress amplitude forthe examined steels at a frequency of 22 kHz.

The experimental data processed using equation (6.58) are shownin Fig.6.34. It appears possible to describe the resultant dependencesby the equation in the form

,ga fGWσ = (6.59)

here G, g are the factors with the values given in Table 6.8.If the energy consumed by the material up to the formation of

fracture is expressed in relation to the number of load cycles tofracture, using equations (6.53) and (6.59) we obtain the depend-ence of the energy absorbed to fracture, Fig. 6.35. The curves canbe expressed by the equation in the form

,hfW H N= (6.60)

for the material listed in Table 6.8.The experiments carried out in Ref. 314 indicate that the values

of the first and, in particular, second critical strain amplitude (Table6.7) increased with increasing carbon content of the steel. This phe-nomenon is be determined mainly by the braking and blocking de-fect of the interphase boundaries which controlled the activity of themechanisms of cyclic microplasticity above ε

kr1 and ε

kr2 [294].

The fatigue life exponent b at a loading frequency of 22 kHz islower for the examined steels (Table 6.7) than for the steels of the

306


TTTTTaaaaabbbbble 6.8le 6.8le 6.8le 6.8le 6.8. Characteristics in equations (6.59) and (6.60) for the examined steels

leetS G gH

]aPM[h b +1(/ b+c)

310214NSC040214NSC060214NSC

176164193

061.0–860.0–540.0–

753.0222.0520.0

547.0948.0848.0

901.0–070.0–340.0–

average slope ≈≈≈≈≈0.81

FFFFFigigigigig.6.35..6.35..6.35..6.35..6.35. Dependence of the energy absorbed to fracture on the number of cyclesto fracture for the examined steels at a loading frequency of 22 kHz.

grade 11, 13 and 15 (according to the former Czechoslovak stand-ards) at a loading frequency of 7–100 Hz [318].

The cyclic strain curves for the examined steels at a loading fre-quency of 7100 Hz are characterised by the exponent of cyclicstrain hardening n with the values in the range from 0.06 to 0.15[297], whereas for a loading frequency of 22 kHz the values are inthe range 0.395–0.490 (Table 6.7), and the value of n at both load-ing frequencies increases with increasing carbon content of the steel.

The fatigue curves in the Manson–Coffin representation (equation(6.54)) are characterised mainly by exponent c whose value at a fre-quency of 22 kHz is significantly lower (Table 6.7) that the valueat a loading frequency of 7–100 kHz, where c = –0.75 (for thesteels CSN 412013 and CSN 412060), and at a loading frequencyof 22 kHz, the values of c decrease with increasing carbon contentin the steel.

When converting the fatigue stress limit σc (Table 6.7) to the fa-

tigue strain limit εapC

using the cycling deformation curves, we ob-tain ε

apC = 3.02 × 10–5, ε

apC = 8.25 × 10–6 or ε

apC = 7.56 × 10–6 for

the steels CSN 412013, CSN 412040, or CSN 412060. At a load-

307


ing frequency of 7–100 Hz, εapC

= 4 × 10–5 for the examined steels[318].

The total energy dissipated by the material up to the fatigue frac-ture increases with increasing number of load cycles in the samemanner at a loading frequency of 7–100 Hz as at 22 kHz.

For low-frequency loading with a frequency of 70–100 Hz,1 + b + c = 0.35 [307], whereas for loading with a frequency of 22kHz 1 + b + c = 0.81.

Comparison of the results obtained for the evaluated steels showsthat to induce fatigue fracture in CSN 412013 steel, the materialdissipates energy 3–4 times more than the steels 412040 and 41262.Discussion of the pseudoelastic behaviour of the CSN 412013 steelwas published in Ref. 307.

For low–cycle fatigue at a frequency of 70–100 Hz b/(1 + b +c) = – 0.25, whereas at a loading frequency of 22 kHz the averagevalue of the fraction is –0.0 72. Consequently, b = 0.971 and notb = nc, as at a loading frequency of 7 – 100 Hz. Approximation ofthe relationship between n, b, c at a loading frequency of 22 kHzis not as simple as in the case of loading with a frequency of70–100 Hz, as shown by our experiments. The mutual relationshipbetween the results obtained in a loading with a frequency of 22kHz, using the deformation and energy criteria, is therefore verycomplicated.

The evaluation of the fatigue process at a frequency of 22 kHzcan be discussed more efficiently on the basis of the deformationcriterion, as implicitly concluded in the studies carried out at aloading frequency of 70–100 Hz [300,307].

The applicability of the deformation and energy criteria of fatiguelife at elevated temperatures has been described by Puškár andLetko [319]. They based their conclusions on the results which showthat mechanical loading at elevated temperatures, acting on the ma-terial, are not in a simple addition correlation, which means that itis not possible to carry out fatigue tests at, for example, 20°C, andtake into account analytically the changes of the properties with in-creasing temperature. The experiments were carried out with VT3–1 two–phase titanium alloy (Russian GOST standards), with the fol-lowing chemical composition, wt.%: 6.39 Al, 2.36 Mo, 1.47 Cr,0.42 Fe, 0.24 Si, 0.03C, balance–titanium. The experiments carriedout in Ref. 319 where similar to those conducted in Ref. 303 whichmeans that the heat treatment of the material and the parameters ofits cyclic microplasticity were published in section 6.2.4.

The experimental results were processed using the procedure de-

308


scribed in Ref. 314 and 320.The Wöhler curves of quenched and tempered VT3-1 alloy,

obtained at different temperatures, are presented in Fig. 6.36. At20°C, σ

C = 650 MPa, at 400°C σ

C = 392 MPa, and at 550°C

σC = 225 MPa, with the scatter of the experimental results of ±7

MPa, with the reference number of load cycles being 2 × 108.Examination of the microstructure of the specimens, loaded with

a stress amplitude of σa = 1.2 σ

c, shown that at 20°C, there are no

significant changes in comparison with the initial condition. Simi-lar agreement was also found in the case of the microstructure af-ter exposure of the specimens to an appropriate stress amplitude ata temperature of 400°C. The microstructure of the specimens,loaded at a temperature of 550°C, is different. It is heterogeneous,with signs of spheroidisation of α–phase.

Figure 6.36 shows that the experimental dependence σa – N

f can

be described by equation (6.53), where σ′f and b for different tem-

peratures are presented in Table 6.9.In the Manson–Coffin representation, Fig.6.37, the curves can be

FFFFFigigigigig.6.36..6.36..6.36..6.36..6.36. Fatigue curves of VT3-1 alloy at different temperatures.

cycles

309


TTTTTaaaaabbbbble 6.9le 6.9le 6.9le 6.9le 6.9 Effect of temperature on the fatigue characteristics of quenched andtempered VT3-1 alloy

T Cº,σ'

f

]aPM[b σ'

f01· 4 c

χ 01· 3

]aPM[n

H]aPM[

h

02004055

54.250123.20749.325

720.0–130.0–050.0–

13.55.5149.3

270.0–411.0–901.0–

06.7171.408.81

473.072.0754.0

415.0300.0520.0

88.078.068.0

described by equation (6.54), where the values of ε′f and c are pre-

sented in Table 6.9.The cyclic strain curves can be described by equation (6.30),

where n = b/c and χ = f/f n.The derivation in [302] and measurements of plastic internal fric-

tion Qp–1 for VT3-1 alloy at selected temperatures [321] indicate that

the amount of energy consumed by the specimen to fatigue fracturecan be described by equation (6.60).

The evaluation of the tests from the viewpoint of the energy cri-terion using equation (6.56) gives the graphical dependence shownin Fig.6.38. The dependences can be described by equation (6.60),where H and h are the experimentally determined characteristicswith the values presented in Table 6.9.

The evaluation of the values of εap

and εac

at Nf = 108 cycles for

FFFFFigigigigig.6.37..6.37..6.37..6.37..6.37. Manson–Coffin curves of quenched and tempered VT3-1 alloy at differenttemperatures.

cycles

310


the experimental results show that at 20°C εap

= 0.022 εac

, at 400°Cε

ap = 0.046 ε

ac and at 550°C ε

ap = 0.024 ε

ac.

The values of the fatigue limit, obtained by the authors ofRef.319 at different temperatures and the reference number of cy-cles of 2 × 108, can be compared with the fatigue limit values ob-tained in loading at a frequency of 100 Hz and a reference numberof cycles of 1 × 107, where the microstructure of VT3–1 alloy wassimilar to that used in this work. Kalachev et al. [322] carried outtests at a temperature of 20°C and the fatigue limit was in the range550–620 MPa, whereas in Ref. 323 it was 620 MPa. In fatigueloading at a temperature 400°C, the fatigue limit was 480–500 MPa[323], and in Ref. 324 it was 330 MPa. No values of the fatiguelimit have been published in the technical literature for a tempera-ture of 550°C. With increase of temperature from 20°C to 400–550°C, the strain fatigue limit decreases from 1.85 × 10–4 to 1.43× 10–5 or even 6.1 × 10–5 and is therefore close to the value of10–5, as generalised for different types of steel [297].

Evaluation of the microstructure prior to and after fatigue load-ing at different temperatures indicate that in the temperature range20–400°C the structure of the VT3–1 alloy is stable and, conse-quently, the changes of the properties with increasing temperatureare controlled by the conventional mechanism. However, loading at

FFFFFigigigigig.6.38..6.38..6.38..6.38..6.38. Application of the energy criterion of fatigue life of VT3-1 alloy atdifferent temperatures.

cycles

311


550°C resulted in significant changes in the microstructure, reflectedin the changes of the exponent c and n in relation to temperature(Table 6.9). The values of χ and n, determined at a testing tempera-ture of 20°C and a loading frequency of 22.5 kHz for the VT3–1alloy are in correlation with the similar characteristics and test con-ditions for the low-alloyed titanium alloy [302].

Increasing temperature decreases the plastic deformation resist-ance of the material. This general conclusion is also reflected infatigue loading at a high frequency where the exponent of the cy-clic strain curve n decreases from 0.374 to 0.275, when the tem-perature is increased from 20 to 400°C. The change of the responseof the material to alternating loading with increasing temperature isalso evident from the ε

ap / ε

ac ratio which is more than doubled with

the temperature increasing from 20 to 400°C, with the number ofload cycles to fracture being 108.

However, if there are significant changes in the microstructureduring loading at 550°C, the response of the material differs. Thevalues of χ and n of the material are higher than those expected inthe case of multiple changes of the properties with increasing tem-perature.

The experimental results and evaluation show that in the analyti-cal evaluation of the results it is possible to use the deformation andenergy criteria of fatigue life also at high loading frequencies andelevated temperatures. The total amount of energy required for theformation of fatigue fracture is indirectly proportional to tempera-ture.

6.3.3 Effect of loading frequency on fatigue limitTaking into account section 3.4, it is useful to note that the load-ing frequency at high total stress amplitudes may have a significanteffect on the fatigue characteristics of materials [143].

The results of the experiments carried out by different authorshave resulted in a conclusion [325] according to which the fatiguelimit at high loading frequency (for example, 20 kHz) is 1.3–1.4times higher than the fatigue limit determined at a loading frequencyof approximately 70 Hz, especially in the case of bcc metals. How-ever, the results are loaded with large errors under the experimen-tal conditions at the compared frequencies. Of special importance istemperature because at a high loading frequency, depending on theextent of internal friction and the rate of its increase with increas-ing strain amplitude of different materials, the rate of heating of thematerials differs.

312


The strictly organised experiments carried out on CSN 415313steel with strictly controlled identical characteristics of the materialin the fatigue tests at 25–45 Hz and 20 kHz showed [326] that whenthe fatigue limit at 20 kHz is determined for 2 × 108 cycles, itsvalue is 270 MPa, and when the fatigue limit is determined at25–45 Hz, at 107 cycles, it is 260 MPa. The increase of the load-ing frequency from 25–45 Hz to 20 kHz results in a significant shiftof the fatigue curve to the higher number of load cycles to fractureN

f, even though the time required to cause fracture at the high load-

ing frequency is significantly shorter [326].Evaluation of the rate of fatigue crack propagation at a loading

frequency of 120 Hz and 20 kHz indicates that the rate (µm ⋅cycle–1) at 20 kHz is up to 100 times lower than in loading at a fre-quency of 120 Hz, whereas the rate of propagation of fatigue cracks(µm⋅s–1) at a loading frequency of 20 kHz is 20 times higher thanat 120 Hz. The basic threshold amplitude of the stress intensityfactor K

ath at a loading frequency of 20 kHz is higher than at a

loading frequency of 120 Hz (4.56 MPa⋅m1/2, 3.8 MPa⋅m1/2).These results and further experiments [327] show that the fatigue

process at high loading frequency is characterised by the same maincharacteristics, stages and relationships in comparison with loadingat low frequency (for example, at 100 Hz). However, there are cer-tain modifications affecting the physical–metallurgical and engineer-ing characteristics of the materials at the higher loading frequencies.These modifications are the result of the effect of various factorswhose influence on the applied characteristics has not as yet beenquantified.

To understand the problem, it is possible to introduce conven-tional symbols. The phenomenon increasing the cyclic deformationresistance of the material will be denoted as the (+) phenomenon,the phenomenon increasing this resistance will be denoted as the(–) phenomenon and the phenomenon having a mixed effect in dif-ferent stages of the process will be denoted by (±). Loading at highfrequency is characterised by the preferential absorption of oscilla-tions at lattice defects resulting in a local increase of temperaturearound these defects (–). The amplitude of the deviation of the dis-location segment at a high loading frequency and a specific shearstress amplitude, used at an ‘arc’ loading frequency is lower be-cause its faster bending is inhibited in a viscous manner by the sol-ute atoms, especially interstitial elements (+). When loading at highfrequency, the time for the relaxation of stress concentration is in-sufficient (+) and, at the same time, there are less suitable condi-

313


tions for the removal of heat from the area in which heat is gener-ated in a single cycle (–). The temperature difference may result ina gradient of internal stresses (±). The bcc metals with interstitialelements significantly increase the deformation resistance with in-creasing strain rate (+). At high loading frequency, the time avail-able is insufficient for general corrosion processes (+) to take place.The fraction of the plastic strain amplitude in the total strain am-plitude at the selected value of the total strain amplitude and at thehigh loading frequency is smaller than at the conventional loadingfrequency (+). At high loading frequency, the slip lines are narrowerand are found in a smaller number of grains, with a smaller surfacerelief in comparison with the conventional loading frequency (+).The width of the plastic zone around the fatigue crack at the highloading frequency is smaller than at the conventional loading fre-quency (–). The shift of the front of the fatigue crack into the ma-terial at high frequency requires a significantly larger number ofload cycles than at the conventional loading frequency (+). The sizeof the activation volume at high and conventional loading frequen-cies is approximately the same.

Each of these reasons is characterised by different intensity ofthe effect on the appropriate fatigue characteristic. Some of themare mutually connected and other combinations mutually excludeeach other. For these reasons, all attempts for the analytical expres-sion of the effect of frequency on the fatigue limit or K

ath are of the

empirical nature with limited validity.To conclude this chapter, it should be noted that important and

valuable information and interpretation can be found in the previ-ously mentioned monographs and also in a compilation edited byGorczyca and Magalas [328].

314


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325

References

Index

A

activation energy of diffusion 86activation enthalpy 78activation volume 78aerodynamic losses 171allotropic transformations 22amplitude–frequency spectrum 3anelastic strain 84anelasticity 1, 28atomic density 12atomic spacing 13

B

background of internal friction 79Barkhausen jumps 129BHR theory 210Blair, Hutchison and Rogers

model 209Bloch walls 46, 59, 128, 289Boltzmann constant 56Boltzmann distribution 86Bordoni maxima 98Bordoni peak 116Bordoni phenomenon 58Bordoni relaxation 105, 197Bordoni’s maximum 100Bordoni’s relaxation 98, 99bowout of the dislocation 76Bridgman equation 67bulk self-diffusion 85Burgers vector 74, 115

C

characteristic process time 49circular resonance frequency 51clustering 126coefficient diffusion of point

defects 86coefficient of absorption of sound 82coefficient of anisotropy 10

compensation phase method 167composite materials 37condensation temperature 257Cottrell atmospheres 97, 106, 197Cottrell parameter 106Coulomb 7Coulomb forces 153critical shear stress 86crystallographic system 7Curie temperature 19, 36, 68cyclic microplasticity 79, 276cyclic strain curve 267cyclic strain hardening 296

D

d-electrons 22damping capacity 133damping decrement 156damping factor 140DB transition 230Debye frequency 186Debye maximum 90Debye oscillations 93Debye peak 208, 212Debye shape 90Debye temperature 11, 14defect of the elasticity modulus 1degree of dynamic relaxation 49diffusibility of atoms 188dilation coefficient 54dimensional factor 22dislocation anelasticity 69, 70dislocation clusters 80dislocation configuration 71dislocation kernel 201dislocation multiplication process 71dislocation segment 99dislocational strain 71dispersion strengthening 38

326


dry friction mechanism 69dynamic methods 142

E

ε-carbide 32, 34effective energy of bonding 84Einstein temperature 15elastic–viscous bond 69elasticity 1elasticity characteristics 1elasticity constant 5elasticity modulus 1electron factor 19, 22, 27electrostatic excitation 153Elinvar 68excitation force 138

F

fatigue stress limit 306Fermi level 54Finkel’shtein and Rozin

phenomenon 95Finkel’shtein–Rozin relaxation 102Finkelstein–Rozin peak 97, 191, 192Finkel’stein–Rozin peak 192Frenkel 16frequency-independent processes 59friction factor 43

G

gas constant 16Granato–Lücke spring model 74, 105Granato–Lücke theory 60, 114

H

magnetostriction 68Hall–Petch equation 273Hertz frequency range 86Hooke law 43, 44hypersonic methods 167hysteresis 1hysteresis anelasticity 71hysteresis loop 72

I

infrasound methods 144inhomogeneous stress 53instantaneous elastic strain 44intercrystalline adsorption 226internal friction background 84interstitial atoms 193Invar alloy 15isotropic pressure 8

K

Köhler distribution 105Kurnakov temperature 29

L

lattice spacing 20linear stretching of the dislocation

75loading frequency 74logarithmic decrement of vibrations

52

M

M–ε dependence 77magnetic hysteresis 76magnetic moment 43magnetomechanical phenomenon 76magnetomechanical bond 68magnetomechanical component 281magnetomechanical phenomenon 59magnetostriction 46magnetostriction vibrator 162Manson–Coffin representation 305Mason model 86maximum of cold deformation 98mechanical hysteresis 69mechanical hysteresis loop 239mechanical relaxation 88microplastic anelasticity 77microplasticity 76microstrain 18molar heat capacity 11

327

References

σ–ε curve 71, 76saturation of cyclic microplasticity

290scattering of mechanical energy

1, 73Schmidt trigger 148Schoeck characteristic 84Schoeck model 83self-diffusion 16self-diffusion coefficient 54Shockley 115Shoeck model 201Snoek 55Snoek and Köster maximum 224Snoek and Köster phenomenon 205Snoek and Köster relaxation 198Snoek maximum

88, 90, 91, 103, 181, 186Snoek peak 219Snoek relaxation 184Snoek’s mechanism 95solute atom 124splitting 215stacking fault energy 215steady-state creep 16strain amplitude 104strain hardening coefficient 16stress sensor 5sublimation energy 84sublimation temperature 11substitutional solid solution 94superlattice 36

T

tensors of the second order 6tetragonality 58Teutonico model 108, 109thermal activation 77thermal–fluctuation relaxation peak

206torsional pendulum 143transmission method 165triclinic 7

U

ultrasound methods 151

N

Nabarro barriers 86natural frequency of vibrations 73Newtonian viscous friction 250non-complanar slip plane 76

O

orientation factor 71

P

paramagnetic state 68Peierls barriers 86Peierls potential energy 99Peierls stress 115Peierls–Nabarro barrier 86phase shift 49pinning points 60pipe diffusion coefficient 223Planck’s constant 92Poisson number 7, 12, 43pulse-phase method 166pulsed methods 12

Q

quasi-inelastic strain 44

R

Rayleigh waves 135relaxation maxima 88relaxation mechanism 87relaxation time 45, 48, 181relaxed Young modulus 47relaxons 87resonance mechanism of

anelasticity 75resonance methods 12, 142resonance peak 51Reynolds number 171rigidity 1RM peak 217

S

S–K maximum 100, 101, 102S–K peak 200S–K relaxation 200

Index

328


unpinned dislocation 76

V

velocity of propagation 8viscous friction 74viscous friction coefficient 74

W

Weert’s equation 10

Werner’s model 96whisker crystals 81Wöhler curve 297

Z

Zener 54Zeener relaxation 95, 182, 195, 197

Internal Friction of Materials Anton Puskar

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