Mehanika nekovinskih gradiv

1

Center for Experimental Mechanics, University of Ljubljana, Slovenia

Mehanika nekovinskih gradivi

Center for Experimental Mechanics, Faculty of Mechanical Engineering, University of Ljubljana, Ljubljana, Slovenia , and

Institute for Sustainable Innovative Technologies, Ljubljana, Slovenia

Lecture NotesKonstrukcijski polimeri in kompoziti

Center for Experimental Mechanics, University of Ljubljana, Slovenia2

Institute for Sustainable Innovative Technologies ‐ iSiT

iSIT building, October 2008

Contacts:Center for Experimental MechanicsFaculty of Mechanical Engineering, University of LjubljanaPot za Brdom 104, SI‐1125, LjubljanaSLOVENIA

Telephone: (+386‐1) 6207 100Fax: (+386‐1) 6207 110E‐mail: [email protected]‐lj.si


States of matter

There are several different types of states of matter:

Gasses, Liquids, and Solids

that are examples of physical states,which can be treated separately from:

Electrical states, Magnetic states and Optical states.

A given substance will have a physical state, a magnetic state, electrical and optical properties.

Source: http://en.wikipedia.org/wiki/State_of_matter#Crystalline_vs._glassy

Physical States of matter:

2


Classification of polymeric materials

Natural origin Synthetic origin (plastics)

PolisacaridesLatex

ProteinesElastomers Thermoplastics Thermosets

POLYMERS

Thermoplastic Elastomers


Types of Thermoplastic Polymers (WAK)

Engineering

Special, High Performance

General use

Source: http://wak.mv.uni‐kl.de/


New Polymer Characterization

Standards

3


New standards for Testing of Polymers ‐ 1

Standards for polymers testingHow to observe time-dependency?

To observe time-dependent properties of polymers new standards will include tests on

RELAXATION and CREEP

Alternative techniques for determination of time-dependent behavior of polymers

creeprelaxation


New standards for Testing of Polymers ‐ 2

VDI – Richtlinie VDI 3880: Werkstoff- und Bauteildämpfung, 2005Blatt 1: Einteilung und ÜbersichtBlatt 2: Dämpfung in festen WerkstoffenBlatt 3: Dämpfung von BaugruppenBlatt 4: Modelle für gedämpfte StruktureBlatt 5: Berechnungen für Maschinensätz

ASTM – Springer Handbook of Experimental Solid Mechanics, 2008, (http://refworks.springer.com/)


Definition of Strain and Stress

4


1-2 plane

0,iL

1,iL

0,1

0,11,1

11 L

LL

x

u

1

23

12

0,2

0,21,2

22 L

LL

x

v

0,3

0,31,3

33 L

LL

x

w

Thermo‐mechanical (rheological) properties of materialsDefinition of Normal Strain


23

2-3 plane

23

1

23

232

232

1212 x

v

x

u

2323 x

w

x

v

1313 x

w

x

u

Thermo‐mechanical (rheological) properties of materialsDefinition of Normal Strain


3

3

3

2

2

3

3

1

1

3

2

3

3

2

2

2

2

1

1

2

1

3

3

1

1

2

2

1

1

1

333231

232221

131211

,,

)(2

1)(

2

1

)(2

1)(

2

1

)(2

1)(

2

1

)(2

1

x

u

x

u

x

u

x

u

x

ux

u

x

u

x

u

x

u

x

ux

u

x

u

x

u

x

u

x

u

uu

zzzyzx

yzyyyx

xzxyxx

ijjiij

Plain strain:

000

0)(2

1

0)(2

1

000

0

0

000

0

0

)(2

1

2

2

2

1

1

2

1

2

2

1

1

1

2221

1211

,, x

u

x

u

x

ux

u

x

u

x

u

uu yyyx

xyxx

ijjiij

Thermo‐mechanical (rheological) properties of materialsStrain Tensor

5


S2x

33

3S

1x

31

32

3x

A

A

T

dA

dTS

)( 3eT

3e

),,( 321 eeen

),,( )()()( 321 eee TTTF

Thermo‐mechanical (rheological) properties of materialsDefinition of Stress


zzyzx

yzyyx

xzxyx

zzzyzx

yzyyyx

xzxyxx

ij

333231

232221

131211

Plain stress:

z

yyx

xyx

zz

yyyx

xyxx

ij

00

0

0

00

0

0

00

0

0

33

2221

1211

Thermo‐mechanical (rheological) properties of materialsStress Tensor


Thermo‐mechanical (rheological) properties of materialsClasical Mechanics

In classical mechanics, the properties of elastic solids can be described by Hooke’s law, that states that when a tensile stress, , is applied to a material, it undergoes a elongation per unit length, named strain, , proportional to that stress, such as:

= Ein which E is the Young’s modulus, and the stress is independent of the rate of elongation.

The properties of liquids, on the other hand, are described by Newton’s law, in which the tension is independent of the strain but proportional to the rate at which the rate is applied, i.e., the strain rate, as:

where is a measure of the fluid’s resistance to deformation called the viscosity.

dt

d

6


FLUID TYPICAL VISCOSITY (Pa.s)

Solid glass >1020

Liquid glass (500 ºC) ~1012

Solid bitumen 106 – 108

Molten polymers (150-350 ºC) 102 – 106

Caramel syrup 101 – 102

Honey 100 – 101

Glicerol ~100

Olive oil ~10-1

Lubricating oil 10-2 – 10-1

Water 10-3

Air < 10-5

Thermo‐mechanical (rheological) properties of materialsViscosity


Material Functions


Tensile Experiment

Stress response is rate dependent

Definition of the modulus in a classical sense fails

Time-dependent materials are rate dependent: l

1 1 2

2

t

l

l0

v = const.

7


What is Time‐Dependency?

Materilas which mechanical properties, under constant thermo-mechanical loading, are changing with time we will call TIME-DEPENDENT MATERILAS.

0.0001

0.001

0.01

0.1

1

1 10 100 1000 10000 100000 1000000 1E+07 1E+08 1E+09 1E+10

t [sec]

J(t

) [m

m2 /N

]

100

10-1

10-2

10-3

10-4

100 102 104 106 108 1010

master curves:different Tdifferent cw

( , , , , , , , , )J J t T P T P

Def

orm

acija

Čas

Def

orm

acija

ČasTime

Def

orm

atio

n


dV

( )T t

What is time dependency?

( )ij t

Res

pons

e

Time

( )P t

( )ij t ( )ij tRelaxation

( )ij tPhysical aging

( )ij tCreep

( )ij tPhysical aging

Time

Load


Material Functions ‐ Transfer Functions

BOUNDARY CONDITIONS

Temp.T

Press.p

Env.c

Theory of elasticity: H(s)

THERMODYNAMICSYSTEM

H(s)

CAUSE

STRESSij

STRAINij

RESPONSE

STRAINij

STRESSij

E = const.

1/E = const.

Theory of viscoelasticity: H(s)E(t, T, p, c)

D(t, T, p, c)

8


Material Functions – General Concepts

Material Function Process

Spatial Orientation of Loading Type of Loading

CreepRelaxation

DynamicStaticUniaxial Volumetric

Shear / Torsion

Response to the excitation


Basic material functions:

in shear: G(t), J(t), G’(), G”(), J’(), J”()in bulk: K(t), B(t), K’(), K”(), B’(), B”()in extension: E(t), D(t), E’(), E”(), D’(), D”()Poisson’s ratio: (t) '(), "()

Material functions

Fundamental material functions:

G(t) ... Change of shape K(t).... Change of volume


Material functions

Shear Bulk Uniaxial Extension

Relaxation ( )G t ( )K t ( )E t

Creep ( )J t ( )M t ( )D t

Storage ( )G t ( )K t ( )E t

Strain prescribed Loss ( )G t ( )K t ( )E t

Storage ( )J t ( )M t ( )D t

Harmoni c

Stress prescribed Loss ( )J t ( )M t ( )D t

Type of Loading

Mode

Har

mon

icQ

uasi

stat

ic

9


Relaxation(excitation is deformation)

G(t), G*(), G’(), G”()K(t), K*(), K’(), K”()E(t), E*(), E’(), E”()(t), *(), ’(), ”()

Dynamic measurementsG*(), G’(), G”()K*(), K’(), K”()E*(), E’(), E”()*(), ’(), ”()

Static

G(t), K(t), E(t), (t)

In phase (storage modulus)

G’(), K’(), E’(), ‘()

Out of phase(loss modulus)

G”(), K”(), E”(), ”()

Creep(excitation is stress)

J(t), J*(), J’(), J”()B(t), B*(), B’(), B”()D(t), D*(), D’(), D”()(t), *(), ’(), ”()

Dynamic measurementsJ*(), J’(), J”()B*(), B’(), B”()D*(), D’(), D”() *(), ’(), ”()

Static

J(t), B(t), D(t), (t)

In phase (storage modulus)J’(), B’(), D’(), ’()

Out of phase(loss modulus)

J”(), B”(),D”(),”()

Static and dynamic material functions


Creep


Creep Experiment

Specimen is exposed to an instant stressIncreasing strain is measured as function of time;Experiment is performed at constant temperature, pressure, and moisture conditions.

t t

0

)()( 0 thtstep

dtJt

t

0

)()()(

0)()( tJtstep 0

)()(

t

tJ step

dtJtt

ramp 0

)()( tdt

dttramp

)( ?

t0t0 - Rise time

t1t1 - Beginning of measurement

Exp. Window

t1= f(t0)

10


Ramp loading in creep measurements

QUESTION: When the measured values are valid?

t1 = ?

t

0

t0 t1

t

t0 - Rise time t1 - Beginning of measurement

Transient Phenomena

CVELBAR, Robert, EMRI, Igor. Analysis of Transient Phenomenon in Creep Compliance Measurements of Viscoelastic Materials. V: 1994 SEM Spring Conference, Baltimore, Maryland, USA, Jun. 6-8, 1994, pp: 663-668CVELBAR, Robert, EMRI, Igor. Transient Phenomena in Torsional Creep Experiments. V: Ed.: Hatuo Nakamura. International Conference on Materials and Mechanics'97, Ed.: Hatuo Nakamura, Japan Society of Mechanical Engineers, Tokyo International Forum, Tokyo, Japan, 20-22 jul., 1997, pp: 483-485, Invited Keynote LEcture.


0

5

10

15

20

25

30

35

0.0001 0.001 0.01 0.1 1 10 100 1000 10000

t, [s]

Err

, [%

]

t0=0.001s t0=0.01st0=0.1s

t0=1s

t0=10s

Measurements error after the loading

ErrorJ t J t

J tramp

( ) ( )

( )[%]100

Transient Phenomena

CVELBAR, Robert, EMRI, Igor. Analysis of Transient Phenomenon in Creep Compliance Measurements of Viscoelastic Materials. V: 1994 SEM Spring Conference, Baltimore, Maryland, USA, Jun. 6-8, 1994, pp: 663-668CVELBAR, Robert, EMRI, Igor. Transient Phenomena in Torsional Creep Experiments. V: Ed.: Hatuo Nakamura. International Conference on Materials and Mechanics'97, Ed.: Hatuo Nakamura, Japan Society of Mechanical Engineers, Tokyo International Forum, Tokyo, Japan, 20-22 jul., 1997, pp: 483-485, Vabljeno predavanje..


Transient Phenomena Resonance of the measurinf system when t0 0

0

0

Response to step loadingResponse to ramp loading

CVELBAR, Robert, EMRI, Igor. Analysis of Transient Phenomenon in Creep Compliance Measurements of Viscoelastic Materials. V: 1994 SEM Spring Conference, Baltimore, Maryland, USA, Jun. 6-8, 1994, pp: 663-668CVELBAR, Robert, EMRI, Igor. Transient Phenomena in Torsional Creep Experiments. V: Ed.: Hatuo Nakamura. International Conference on Materials and Mechanics'97, Ed.: Hatuo Nakamura, Japan Society of Mechanical Engineers, Tokyo International Forum, Tokyo, Japan, 20-22 jul., 1997, pp: 483-485, Vabljeno predavanje..

11


Creep experiment

1 2

1 2

( )( ) ( )( ) n

n

tt tD t

t

rheodictic

arrheodictic

D(t)

De

Dg

t

rheodictic

arrheodictic

D(t)

De

Dg

log t

log D(t)rheodictic

arrheodictic

Log Dg

Log De


Shear Creep Testing – CEM Torziometer

Metlikovič P., Emri I., Journal of Mechanical Engineering, Vol. 35, /7-9/, 1989, pp. 102-108Metlikovič P., Emri I., Journal of Mechanical Engineering, Vol. 35, /4-6/, 1989, pp. 56-58


Physical background

0

ttJ

4 ( )( )

32

d tJ t

mglR

0t

p

M r

I

4

32p

dI

2d r

12


The Creep ExperimentPhysical background

The temperature and mechanical profile of loading in theshear-creep measurement of the extruded LDPE specimens.

0

ttJ

4 ( )( )

32

d tJ t

mglR

0t

p

M r

I

4

32p

dI

2d r


Shear Creep Testing – CEM Torziometer

Cylindrical rods:diameter = 6 mmlength = 40 mm

40 mm

Shear Creep Torziometer (SCT‐CEM) -3.3

-3.1

-2.9

-2.7

-2.5

-2.3

-2.1

-1.9

0 2 4 6 8 10 12 14 16log t [s]

log

J( t

) [1

/MP

a]

Tref = 30°C

PA6

time‐temperature superposition principle

-2.2

-2

-1.8

-1.6

-1.4

-1.2

-1

0 1 2 3 4 5log t [s]

log

J [

1/M

Pa]

33.7°C38.9°C48.6°C58.1°C67.1°C76.7°C

LDPE_263_37


Creep experiment

0,0

0,0

0,0

0,1

1,0

1 10 100 1000 10000

t [ sec]

J(t

) [m

m2/N

]

T[oC]

35.8

34.4

32.3

30.628.426.420.0

100 5 101 5 102 5 103 5 104

100

10-1

10-2

10-3

10-4

Creep measurements on PVAc:

Emri I., Pavšek V., On the Influence of Moisture on the Mechanical Properties of Polymers, Materials Forum, No. 16, 1992, pp. 123-131.

13


Relaxation


Relaxation experiment: Torsion

Specimen is exposed to an instant deformation;Decreasing stress is measured by a torque cell;Experiment is performed at constant temperature, pressure, and moisture conditions

)()( 0 thtstep

dtGt

t

0

)()()(

0)()( tGtstep 0

)()(

t

tG step

dtGtt

ramp 0

)()( tdt

dttramp

)( ?

stre

ss

1t0t

Exp.Window

0t

Response0t

Load


Relaxation Experiment

1 2

1 2

( )( ) ( )( ) n

n

tt tE t

arrheodictic

rheodictic

gE

eE

)(tE

t

arrheodictic

rheodictic

gElog

eElog

)(log tE

tlog

14


Relaxation ExperimentCharacteristic regions of teh relaxation process

log

G(t

)

Complete characterization of the polymeric material requires measurements ower several decades of time

Impossible to measure in one experiment!

Tschoegl, N.W., Knauss, W.G., and Emri, I, The effect of temperature and pressure on the mechanical properties of thermo- and/or piezorheologically simple polymeric materials in thermodynamic equilibrium - a critical review. Mech. time-depend. mater., 6, 53-99, (2002)


Measurement conditions:

• pressure up to 600 MPa (6000bar), ±0.1 MPa

• temperature from –30°C to +120°C, ±0.01°C

CEM Relaxation apparatus

Kralj, A., Prodan T., and Emri, I., J. Rheol., 45, 929-943, (2001).Emri, I., and Prodan, T., Experimental Mechanics, 46, 429-439, (2006)

data acquisition

circulator

measuring inserts

thermal bath

pressurizing system

pressure vessel

carrier amplifier

electromagnet

magnet and motor charger


280

mm

loading device

load cell

specimen

slidermechanism

triggeringmechanism

electric motor

(a)

Relaxometer


Load

Response

15


Assumptions:

• homogeneous, isotropic specimen

• (volume) = 3 (lenght)

260

mm

LVDT

LVDT rod

specimen

Measurement Principle

P, T

Dilatometer



Specimens for the Dilatometer and the Relaxometer

specimen for the relaxometer

specimen for the dilatometer

stainless steel sheet

specimen holder


Measuring Capabilities of the CEM Apparatus

Physical Properties Symbols

Shear moment ( ), ( ), ( )M t M T M P or ( , , )M t T P

Me

as

-ure

d

Specimen length ( ), ( ), ( )L t L T L P or ( , , )L t T P

Shear relaxation modulus ( ), ( ), ( )G t G T G P or ( , , )G t T P

Specific volume ( ), ( ), ( )v t v T v P or ( , , )v t T P

Linear thermal expansion coeff. ( ), ( )T P , or ( , )T P

Volumetric thermal expansion coeff. ( ), ( ), , ,g e fT P , or , , ( , )g e f T PBulk creep compliance ( ), ( ), ( )B t B T B P or ( , , )B t T P

Ca

lcu

late

d f

rom

d

efin

itio

ns

Bulk modulus ( ), ( )K T K P or ( , )K T P

WLF constants 1 2, c c

WLF material parameters 0, f f

FMT constants 1 2 3 4 5 6, , , , , c c c c c c

Ca

lcu

late

d

fro

m m

od

els

FMT material parameters *0( ), ( ), , , , , f e eP f P B K k K k

16


G (t) fo r P V A c a t p = 0 .1 M P a (ra w d a ta )

1

1 0

1 0 0

1 0 0 0

1 E -1 1 E + 0 1 E + 1 1 E + 2 1 E + 3

t [s ]

G(t

) [M

Pa

]

T = 4 0 °C

T = 3 2 °C

T = 3 0 °C

T = 2 6 °C

T = 2 0 °C

Relaxation experiment: Torsion

Measurements of G(t) on PVAc:

Response

Load

Emri, I., and Prodan, T., Experimental Mechanics, (2005), In print


Boltzman Superposition and Rheological Models


Boltzmann Superposition Principle‐I

Any linearity of operation, or relation between an effect and its cause, requires satisfaction of:

Postulate (a): Proportionality with respect to amplitude, and

Postulate (b): Additivity of effects independent of the time sequence, when the corresponding causes are added, regardless of the respective application times.

Basic Postulates:

17


Boltzmann Superposition Principle‐II

1 1 1 1( ) ( ) ( ) ( )t h t t t D t t

1 1 2 2( ) ( ) ( )t h t t h t t

1

t

2

1

1 1 2 2( ) ( ) ( ) ( )N Nt D t t D t t D t t

1 1

( ) ( ) ( )i

i N i N

i i ii i t

t D t t D t t tt

1 00

( ) lim ( ) ( )i

ti N

iNi tt

dt D t t t D t d

t d

0

( ) ( )t d

t D t dd

0

( ) ( )t d

t E t dd


Linear Modeling of Forces

1F k x

2F k x

3F k x

i) Forces proportional to the displacement:

ii) Forces proportional to the velocity:

iii) Forces proportional to the acceleration:.

E

( ) ( )t E t

( ) ( )t t

F m a

F

Force is a secondary physical quantity. It is a result of the interaction between space and energy . Definitions of forces are intuitive and are based on the macro scale observations:

Observed ProcessX(t) y(t)

; ;y k x y k x y k x

i iF Q x


Basic rheological elements

E

( ) ( )t E t ( ) ( )t t

a) Hookean spring b) Newtonian dashpot.

18


Maxwell Model

( ) ( )( )

t tt

E

Constitutive equation for the Maxwell model:

( ) ( ) ( )d st t t

( ) ( ) ( )d st t t

E

( )t

( ) ( )s st E t ( ) ( )d dt t

( ) ( ) ( )d st t t

( )( ) d

d

tt

( )( ) s

s

tt

E


Maxwell Model Stress relaxation

E

( ) ( )( )

t tt

E

0( ) ( )t h t

1 ( ) ( ),

d t t

E dt

Stress Relaxation:

0( ) exp( / )Mt E t

0

( )t

M t0

0 / e

00.631

00.369

0

( )t

M t0

0 / e

00.631

00.369

0 0E

/M E

0

( )( ) exp( / )M

tE t E t

Since ( 0) 0, we findt

( 0)gE E t E

( )H

loglog M

E

Rheodictic material with a single spectrum line


Maxwell Model Creep process

Since ( 0) 0, we findt

E

( ) ( )( )

t tt

E

Creep:

0

( )t

t0

0tan / 0

( )t

t0

0tan / 0( ) ( )t h t

0d dt

00( ) .t t

0

( ) 1( )

t tD t D t

E

0 ( 0)D D t D

Material exhibits elastic-viscous behavior

19


Voigt Model

E

( )t

( ) ( )s st E t

( ) ( )d dt t

Constitutive equation for the Voigt model:

( ) ( ) ( )d st t t

( ) ( ) ( )d st t t

( ) ( ) ( )t t E t

( ) ( ) ( )d st t t


Voigt ModelStress relaxation

0( ) ( )t h t

( )E t const

( ) ( ) ( )t t E t

Stress Relaxation:

0( )t E

0

( )t

t0

0

( )t

t0

E

For t > 0 Material exhibits elastic behavior0Since ( 0) ,and ( 0) 0, we findt t


Voigt Model Creep process

E

( ) ( ) ( )t t E t

Creep:

0( ) ( )t h t

( )t

V t0

(1 1/ e)

0.631

0.369

( )t

V t0

(1 1/ e)

0.631

0.369

0( ) [1 exp( / )]p Vt tE

( ) [1 exp( / )]VD t D t

/V E

( )D t D

0Since ( 0) , we havet

( ) , andh pt ( )L

loglog V

D

Arheodictic material with a single retardation spectrum line

20


Maxwell and Voigt Modells Summary

0

( ) 1( )

t tD t

E

0

( )( ) exp( / )M

tE t E t

/M E

When elements are added in parallel relaxation modulus may be added

When elements are added in series then compliances may be added

E

0

( )( ) [1 exp( / )]V

tD t D t

/V E

( ) ( ) ( )d st t t

( ) ( ) ( )d st t t

0

( )( )

tE t E

E


Standard Linear SolidStress relaxation

Ee

E1

( ) ( ) ( )S Mt t t

0( ) ( ) ( ) ( )S Mt t t h t

0 1 1( ) exp( / );M t E t

0s eE 1 1/ E

0 1 1( ) exp( / )et E E t

1 1( ) exp( / )eE t E E t

Maxwell element:

Spring:

Elements are added in parallel, i.e., relaxation modulus may be added:

log ( )E t

log t1log

log gE

log eE

( )H

log1log

1E

Arheodictic material with a single spectrum line

1( 0)g eE E t E E

00

( )( ) ( ) ( )

tt E t E t


Standard Linear Solid Creep process

0( ) [1 exp( / )];V t tE

E

Eg

E

Eg

( ) ( ) ( )S Vt t t

0( ) ( ) ( ) ( )S Vt t t h t

0 /s gE

/ E Voigt element:

Spring:

Elements are added in series, i.e., compliances may be added:

0

1 1( ) [1 exp( / )]

g

t tE E

( ) [1 exp( / )]gD t D D t

log ( )D t

log tlog

log eD

log gD

( )e gD D t D D

( )L

loglog

D

Arheodictic material with a single retardation spectrum line

00

( )( ) ( ) ( )

tt D t D t

21


Standard Linear Liquid Stress relaxation

0 1( ) ( ) ( )M Mt t t

0 1 0( ) ( ) ( ) ( )M Mt t t h t

0 0 0 0( ) exp( / );M t E t 0 0 0/ E Maxwell element 0:

Maxwell element 1:

Elements are added in parallel, i.e.,relaxation modulus may be added:

E0

E1

1 0 1 1( ) exp( / );M t E t 1 1 1/ E

0 0 1 1( ) exp( / ) exp( / )E t E t E t

0 1

( )t

t0

0 1( 0)gE E t E E

( )H

log1log

1E

Rheodictic material with two spectrum lines

0log

0E


Standard Linear Liquid Creep process

0( ) [1 exp( / )];V t tE

( ) ( ) ( ) ( )S V Dt t t t

0( ) ( ) ( ) ( ) ( )S V Dt t t t h t

0 /s gE

/ E Voigt element:

Spring:

Elements are added in series, i.e., compliances may be added:

f

E

Eg

1 1( ) [1 exp( / )] [1 exp( / )]g f

g f

tD t t D D t t

E E

Dashpot: 0( ) .Df

t t

oe

( )t

t0

0

( )t

0( )t t

( )L

loglog

D

Rheodictic material with a single retardation spectrum line

( 0)g gD D t D

oe gD D D

( ) is a steady-state complianceoe gD D t D D


Wiechert ModelGeneralized Maxwell Model

eE 1E 2E 3E NE

1 2 3 N

Elements are added in parallel, i.e.,relaxation modulus may be added

1

i N

g e ii

E E E

1 1

( ) exp( / ) 1 exp( / )i N i N

e i i g i ii i

E t E E t E E t

The set of relaxation times i , and corresponding iE , , ; 1, 2, ,i iE i N we will call the

discrete relaxation spectrum, which we commonly denote as ( )iH .

/i i iE

22


Kelvin Model Generalized Voigt Model

1E 2E 3E NE

1 2 3 N

fgE

1

i N

e g ii

D D D

o

1

lim ( )i N

e f g it

i

D D t t D D

o

1 1

( ) exp( / ) { } [1 exp( / ] { }i N i N

e i i f g i i fi i

D t D D t t D D t t

The set of retardation times i , and corresponding iD , i.e. , ; 1, 2, ,i iD i N , we will call

the discrete retardation spectrum, denoted as ( )iL .

Elements are in series, i.e.,compliances may be added

1/g gD E1/i iD E /i i iE 1/f f

,

,


Relaxation and Retardation Spectra

( ) ( ) exp( / ) lneE t E H t d

From Generalized Maxwell and Voigt Model we can obtain:

( ) ( )[1 exp( / )] lngD t D L t d

( ) [1 exp( / )]g j jj

D t D L t ( ) exp( / )e j jj

E t E H t


Rheological ModelsHomework ‐ 1

Determine Creep and Relaxation functions for the shown models:

1 21E 2E 1E

2E

1

2

1E

1

2E 23E 3

1 21E 2E 1E

2E

1

2

1E

1

2E 23E 3

1E

1

2E 23E 3

23


Experiments in Uniaxial Extension


New standards for Testing of Polymers

Standards for polymers testingHow to observe time-dependency?

To observe time-dependent properties ofpolymers new standards will include testson RELAXATION and CREEP.

Alternative technique for determinationof time-dependent behavior of thepolymeric material from short (2-3 min)tensile experiments

creeprelaxation

What are the time-dependent properties of material?

Why we want to use tensile experiment?

Equipment for such characterization are notwidely spread and very expensive

Complete characterization takes 2-3 days


The UNIAXIAL TENSION of polymeric materials is one of the most important type of theirdeformation. Extension dominates in production of fibers, films and foils

Tensile machines are widely used and relatively cheap in comparison with apparatuses intendedfor creep and relaxation measurements

Many research groups are developing equipment to observe the behavior of polymers underuniaxial tension. Group of Prof. Maia: extensional rheometer for high viscosity systems

Plenty of tensile experimental data are already available. CAMPUS® (Computer Aided MaterialPreselection by Uniform Standards) is the only plastics database which offers truly comparablematerial data measured according to binding international standards (BASF)

Tensile Experiment. Reasons

Rupture fragile Rupture ductileSchematics of rheometer

24


Physical Background. Time‐Dependent Properties (1/5)

What do we understand under the time-dependency of mechanical properties?

Mechanical properties vary with timeMolecular rearrangements inside the material as a response on applied excitation could be

Infinitely longPurely elastic material

InstantlyPure viscous material

Comparable with the time of experiment

Viscoelastic material

Viscoelastic behavior of polymers is characterized by MATERIAL FUNCTIONS of time, or frequencyExample of molecular rearrangements

in tensile experiment


Time-independent part(material constant)


, lnRR t C F k t d

stat

icdy

nam

ic

G t K t E t t

J t B t D t

G K E

G K E

J B D

J B D

Mathematical representation of materialresponse to the applied excitation –

MATERIAL (RESPONSE) FUNCTIONTime-dependent part

Table 1: Material function of viscoelastic materials Interconversion between relaxation and retardation material functions

Spectral function Kernel function


EXCITATION RESPONSE


t t E tTRANSFER FUNCTION

System, Process

0

tE t

0

t

t E t s dss

Impossible to achieve in reality

Constitutive Equation of Linear Theory of Viscoelasticity (CE LVE)

Step function

Finding E(t) form CE LVE isINVERSE PROBLEM

Standard excitation

Application of nonstandardexcitations

Material

25



Schematics of polymer structureEach color corresponds to certain length of

molecule

Response times

Distribution of response times referred to as SPECTRAL FUNCTION (SPECTRUM)

Mechanical spectrum represents the time-dependentpart of the experimental response function

According to mode of loading it can be relaxation,H(), or retardation, L()

Mechanical spectrum determines all mechanicalproperties of polymeric material

Mechanical spectrum can not be measured directlyfrom the experiment

F(

)



/

1

( ) i

Mt

e g e ii

E t E E E h e

1

1M

ii

h

/

0

( ) lnte g eE t E E E H e d

Continuous Form

(Fredholm integral equation of the first kind)

Discrete Form(more suitable for numerical analysis)

Ee – equilibrium modulusE(t) = Ee, when t ∞

Eg – glassy (instantaneous) modulusE(t) = Eg, when t 0

Finding H() form equations above isINVERSE PROBLEM

Representation of Material Function in terms of Mechanical Spectrum:

log

E(t

), [

Pa

]


Problem Statement

Str

ess

, [

Pa

]

We want to characterize time-dependent propertiesof polymeric material starting from stress-strainexperimental data obtained from tensile experiment

OUTPUT

F(t)

(t) =L0

L

(t) = R·t

F(t)

A0

A0

26


Direct problems

(determination of effects of given cause)

Inverse problems

(determination of causes from the effects)Ill-posed problems (OR):

1. don’t have any solution2. don’t have unique solution3. solution is sensitive to error in input data

Problem Statement. Mathematical Background

,i it ,i it

0

( )( ) ( )

t st E t s ds

s

/

1

( ) i

Mt

e g e ii

E t E E E h e

( )E t , 1,ih i Mexperimental data

1st inverse problem 2nd inverse problem

Relaxation modulus Mechanical spectrum


Structure of the Approach

Str

ess

, [P

a]

log

E(t

), [P

a]lo

g E

(t),

[Pa]

H, [

-]

Algorithm 1

Algorithm 3

Algorithm 2

0

( )( ) ( )

t st E t s ds

s

/

1

( ) i

Mt

e g e ii

E t E E E h e

Time-TemperatureSuperposition Principle


Simultaneous measurements of two materila functions:

Uniaxial relaxation modulus and Poisson coefficient

Relaxation experiment in uniaxial extensionUniaxial extension and Poissins Coefficient

TSCHOEGL, Nicholas W., KNAUSS, Wolfgang G., EMRI, Igor. Poisson's ratio in linear viscoelasticity - a critical review. Mech. time-depend. mater., 2002, vol. 6, no. 1, 3-51.

27


Relaxation ExperimentUniaxial extension and Poissons coefficient

M. Samarin: Naprava za merjenje enoosne relaksacije viskoelastičnih materialov, Magistrska naloga, Fakulteta za strojništvo, Ljubljana, 1999

SKITEK, Tanja, EMRI, Igor, TSCHOEGL, N. W.. An Apparatus for Measuring a Time-Dependent Poisson Ratio. V: 1996 VIII International Congress on Experimental Mechanics, Nashville, Tennessee, USA, Jun. 10-1, 1996, pp: 226-22.EMRI, Igor, SAMARIN, Matjaž, TSCHOEGL, N. W.. The Time-Dependent Poisson's Ratio. An Overview. V: Advanced Technology in Experimental Mechanics, The Japan Society of Mechanical Engineers, Wakayama, Japan, 25-26 jul., 1997, pp: 481-486, Vabljeno predavanje, CEM-97-32


HAAKE C40

ME NU E

45.35ºC

INT-TEMP

08.09.1995 14:30:15

EN TER

2 0

50

1 0015 0

2 00

2 70

HAAKE F6

Borescope

CCD Camera

Heating/Cooling Circulator

Heating / CoolingCylinder

Specimen

Load Cell

Servo Motor

Silicon Oil

Environmental Chamber

Stretching Device

Relaxation experiment in uniaxial extensionCEM Apparatus


The measuring setup

Specimen

Force senzor

Force senzor

M. Samarin: Naprava za merjenje enoosne relaksacije viskoelastičnih materialov, Magistrska naloga, Fakulteta za strojništvo, Ljubljana, 1999SKITEK, Tanja, EMRI, Igor, TSCHOEGL, N. W.. An Apparatus for Measuring a Time-Dependent Poisson Ratio. V: 1996 VIII International Congress on Experimental Mechanics, Nashville, Tennessee, USA, Jun. 10-1, 1996, pp: 226-22.EMRI, Igor, SAMARIN, Matjaž, TSCHOEGL, N. W.. The Time-Dependent Poisson's Ratio. An Overview. V: Advanced Technology in Experimental Mechanics, The Japan Society of Mechanical Engineers, Wakayama, Japan, 25-26 jul., 1997, pp: 481-486, Vabljeno predavanje, CEM-97-32


28


The loading frame The step motor


SKITEK, Tanja, EMRI, Igor, TSCHOEGL, N. W.. An Apparatus for Measuring a Time-Dependent Poisson Ratio. V: 1996 VIII International Congress on Experimental Mechanics, Nashville, Tennessee, USA, Jun. 10-1, 1996, pp: 226-22.EMRI, Igor, SAMARIN, Matjaž, TSCHOEGL, N. W.. The Time-Dependent Poisson's Ratio. An Overview. V: Advanced Technology in Experimental Mechanics, The Japan Society of Mechanical Engineers, Wakayama, Japan, 25-26 jul., 1997, pp: 481-486, Vabljeno predavanje, CEM-97-32



Measuring setup Mesh

Frequency domain

2D FFT

Relaxation ExperimentUniaxial extension and Poissons coefficient



Effect of Temperature and

Pressure

29


Motivation

Shrinkage of the interior trim:

Physical aging

Structural failure due to shrinkage:


Motivation

-1

0

1

2

3

4

-10 -8 -6 -4 -2 0 2 4

log t [s]

log

G [

MP

a]

P0 = 200 Mpa

T0 = 5oC

SBR

NR

-60

-40

-20

0

20

40

60

0 2 4 6 8 10 12 14

Tooth No.

F[N

]


Motivation

The key issue in the field of structural polymers is their durability (long-term stability).

30


Motivation

Δl

glassy

transition

rubbery

Time

log

()

Gt

plateauflow

T = const.

P = const.

Gg

GrSea

ling

forc

e

tcritical


Effect of TemperatureThermal Volume Expansion


Materials and their transitionsGlass transition, melting, crystallization

amorphous material

Semicrystalline material

crystalline material

Tg, Tm, Tc

glass transition (Tg) melting (Tm), crystallization (Tc)

crystal regionamorphous

region

Polymer structure

31


Materials and their transitionsGlass transition, melting, crystallization

100% amorphous semicrystalline100% crystalline

Tm, Tc Tg Tg, Tm, Tc

spe

cific

vol

ume

temperature

spec

ific

volu

me

Vo

Tg

glass

rubber

Property of the crystalline region

Below Tm: Ordered crystalline solid

Above Tm: Disordered melt

A first-order transition

Property of the amorphous region

Below Tg: Disordered amorphous solid with immobile molecules

Above Tg: Disordered amorphous solid in which portions of molecules can wiggle around

A second order transition


First‐ and Second‐Order Transitions

Thermodynamic transitions are classified as being first- or second-order. In a first-order transition there is a transfer of heat between system and surroundings and the system undergoes an abrupt volume change.

In a second-order transition, there is no transfer of heat, but the heat capacity does change.

The volume changes to accommodate the increased motion of the wiggling chains, but it does not change discontinuously..

fFirst order transition

Second order transition


spe

cific

vo

lum

e

temperature

spe

cific

vo

lum

e

Tg

T.

T.

1

2 T1>.

gT = f(T)

Cooling‐ or Heating‐Rate Dependents of Tg

rate-dependent value

When an amorphous polymer is heated, the temperature at which it changes from a glass to the rubbery form is called the glass transition temperature, Tg.

A given polymer sample does not have a unique value of Tg because the glass phase is not at equilibrium.

The measured value of Tg will depend on the molecular weight of the polymer, on its thermal history and age, on the measurement method, and on the rate of heating or cooling.

32


The Glass Transition Tg

The glass transition is a property of only the amorphous polymers and amorphous portion of a semi-crystalline solid. The crystalline portion remains crystalline during the glass transition.

At a low temperature the amorphous regions of a polymer are in the glassy state. In this state the molecules are frozen on place. They may be able to vibrate slightly, but do not have any segmental motion in which portions of the molecule wiggle around, i.e., there are no long-range motion in the polymer chain.

When the amorphous regions of a polymer are in the glassy state, it generally will be hard, rigid, and brittle, but the polymer structure is still disordered

If the polymer is heated it eventually will reach its glass transition temperature. Above a transition temperature, Tg, the chain segments recover their rotational mobility and long-range motions, thereby becoming softer and more ductile. We say that polymer enters its rubbery state.


The Glass Transition Tg

Tg depends on molecular weight, heating/cooling speed (and thereby is a kinetic process), measuring method, etc.

The value of Tg depends on the mobility of the polymer chain - the more immobile the chain, the higher the value of Tg.

A polymer chain that can move easily will change from a glass to a rubber at a low temperature. If the polymer chains don't move as easily, then it will require a relatively high temperature to change the compound into a rubbery form.

In particular, anything that restricts rotational motion within the chain should raise Tg (e.g., nano particles should raise Tg).

Plasticizers are low molecular weight compounds added to plastics to increase their flexibility and workability. They weaken the intermolecular forces between the polymer chains and decrease Tg. (Plasticizers are added to the plastic used for automobile upholstery. In older automobiles, the plasticizer may be distilled from the upholstery during hot weather so that it becomes brittle over time.)


Definition of the free volume and the glass transition temperature

0

V

V

gT T

r

g

I.Emri, Rheology of Solid Polymers, Rheology Reviews, 2005

glassy

transition

rubbery

Time

log

()

Gt

plateauflow

f l

f g

33


Model defines free-volume fractionat temperature T as:

- thermal expansion coefficient below Tg

- thermal expansion coefficient above Tg

Free‐volume theory


ln ln / fA BV V

Doolitle, 1951, studied influence of temperature on viscosity of polymer melts. He postulated the relation

where( )g f gf f T T

f f f

f

V V Vf

V V V V

f l

f g ; for T < Tg

; for T Tg

1exp exp exp 1f

f f

V VVA B A B A B

V V f

Doolitle Equation


Thermo‐mechanical (rheological) properties of materialsVolume expansion coefficient

Temperature volume expansion coefficient

34


Time‐Temperature Superposition


Time‐temeprature superposition

0( ) / ( )Ta T T Shift factor is defined as a ratio of a particular relaxation or creep time at temperature T, to that ofa reference temperature T0,

T > T0

Log aT

Log (T) Log (T0) Log t /aT

T T0

Log

G(t)

T > T0

Log aT


T T0

Log

G(t)


Time‐temeprature superposition


LogG

(t)

T5>T4>T3>T2>T1

Experimental Window

35


log

G(t

)

Time‐Temperature Superposition principlePhysical Background

log

G(t

)

Complete characterization of the polymeric materialrequires measurements ower several decades of time

Application of time-temperature (or time-pressure) superposition:

Impossible to measure in one experiment!



Time‐temperature superposition

0

300

600

900

1200

-1 0 1 2 3 4log t [s]

G [

MP

a]

20°C30°C40°C50°C65°C80°C95°C

T ref = 50°CP ref = 0.1 MPa

0

300

600

900

1200

-10 -5 0 5 10 15log t [s]

G [

MP

a]

T ref = 50°C

P ref = 0.1 MPa


G(t) for PVAc at p=0.1 MPa (raw data)

1

10

100

1000

1E-1 1E+0 1E+1 1E+2 1E+3

t [s]

G(t

) [M

Pa

]

T=40°C

T=32°C

T=30°C

T=26°C

T=20°C

G(t) for PVAc at p=0.1 MPa (superposed with Tref=40°C)

1E-5 1E-4 1E-3 1E-2 1E-1 1E+0 1E+1

t/ap [s]

Shift Factors

0

1

2

3

4

5

6

15 35 55

T [°C]

log

aT

Time‐temperature superpositionResults on PVAc

36


Time‐temperature superpositionResults on PA6

-16 -12 -8 -4 0 4log t [s]

20°C30°C40°C50°C65°C80°C

2.0

2.2

2.4

2.6

2.8

3.0

3.2

-1 0 1 2 3log t [s]

log

G [

MP

a]

P 0 = 0.1 MPaT 0 = 80°C

Shift factors

0

5

10

15

20 40 60 80T [°C]

log

aT

PA6.


Time‐temperature superpositionOpen questions

Shifting is usually done “by hand”. This means that different people will obtain different master curves forthe same experimental data.

Based on theTIME-TEMPERATURE SUPERPOSITION PRINCIPLEwe want to construct the master curve at selected reference temperature.

Back

log

E(t

), [

Pa]

log

E(t

), [

Pa]

a1 a2 a3 a4 a5 m1 m20

2

4

6

8

10

12

14

, [

%]

0 1 2 3 42

2.2

2.4

2.6

2.8

3

3.2

log(t), [s]

log

G(t

), [

MP

a]

T

1 = 30

T2 = 40

T3 = 50

T4 = 62

T5 = 75

Set of synthetic data Error in relaxation modulus

IS IT POSSIBLE TO REMOVE THE DRAWBACK OF THE “HAND SHIFTING”?


CFS‐ Close form time‐temperature shifting

We have proposed the unique mathematical (numerical) methodology for shifting of the experimental data in the process of constructing Master Curve at selected reference temperature or pressure, which completely removes ambiguity related to different shifting procedures

M. Gergesova, B. Zupančič, I. Saprunov, and I. Emri, The Closed Form t-T-P Shifting (CFS) Algorithm, Accepted to Journal of Rheology, 2010

37


Isochronal curvesScheme of construction

seconds104 reftt


log aT

T>T0

T0

log t/aT log(T) log(T0)

Williams-Landel-Ferry (WLF) Model, 1955:1 0

2 0

( )log T

C T Ta

C T T

C1 and C2 are material constants, when T0 is Tg, they are nearly the same for most polymers (polymers which they investigated), i.e.,

2 54.6gc 1 17.44gc

Later was shown that these are NOT universal constants!

Modeling the Time‐Temperature SuperpositionHorisontal shifting

Tschoegl, N.W., Knauss, W.G., and Emri, I, The effect of temperature and pressure on themechanical properties of thermo- and/or piezorheologically simple polymeric materials inthermodynamic equilibrium - a critical review. Mech. time-depend. mater., 6, 53-99, (2002)

0 0

0 0

( / 2.303 )( )log

/Tf

B f T Ta

f T T

1 0/ 2.303c B f

0

( )

( )

iT

i

Ta

T

2 0 / fc f


ln ln / fA BV V



f f f

f

V V Vf

V V V V

f l

f g ; for T < Tg

; for T Tg

1exp exp exp 1f

f f

V VVA B A B A B

V V f

Doolitle Equation

38


WLF constants expressed in terms of free‐volume

0 0

0 0

( / 2.303 )( )log

/Tf

B f T Ta

f T T

01 002 0

( )log T

c T Ta

c T T

01 0/ 2.303c B f

0 0 0

( ) ( ) 1 1exp

( ) ( )i

Ti

T Ta B

T T f f

02 0 / fc f

02 2 0r

rc c T T 0 01 2 1 2r rc c c c

WLF constants for another reference temperatureTr:



WLF Equation

01 002 0

( )log T

c T Ta

c T T

-20 -10 2010

-2

-4

2

T-T Csº

c1s

c2s

log (T, T ) PSa s

experimental shiftWLF approximation

log Ta

0( )T T

02c

01c

0

01

( )lim (log )T

T Ta c

00 2( )

lim (log )TT T c

a



Determination of the WLF Constants ‐1

log ( ), ; 1,2,3, ,i ia T T i N

0

log ( ) log ( )i i

ii i

T T TS

a T a T


{ , ; 1, 2,3, , }i iS T i N

-20 -10 2010

-2

-4

2

T-T Csº

c1s

c2s

log (T, T ) PSa s

experimental shift

WLF approximation

01 002 0

( )log T

c T Ta

c T T

00 2

00 01 1

1( )

log T

T T cS T T

a c c

39


Determination of the WLF Constants‐2

0( )T T

Z

02C

0201

C

C

S

0

log ( ) log ( )i i

ii i

T T TS

a T a T


{ , ; 1, 2,3, , }i iS T i N

02

0 01 1

1( )i i i i i

cS S T T

c c

011/a c

0 02 1/b c c

( )i i i i iS S T a T b

( )i i iS a T b 2 2

1 1

[ ( )]N N

i i ii i

S a T b



Minimizing the sum of the squares with respect to a and b we obtain,

1

2 [ ( )] 0N

i i ii

S a T b Ta

The solution of this matrix equation is given by

aDa

D

bDb

D

2

1 1 1

1 1

N N N

i i i ii i i

N N

i ii i

aT T S T

T N Sb

2

1 1

1

N N

i ii i

N

ii

T T

D

T N

1 1

1

N N

i i ii i

a N

ii

S T T

D

S N

2

1 1

1 1

N N

i i ii i

b N N

i ii i

T S T

D

T S



Solving the above determinants we find the constants and to be given as:

2

2

1 101

1 1 1

1

N N

i ii i

N N N

i i ii i i

T N T

ca

N S T S T

2

1 1 1 102

1 1 1

N N N N

i i i i ii i i i

N N N

i i ii i i

T S S T Tb

ca

N S T S T

40


Determination of the WLF ConstantsSummary

log ( ), ; 1,2,3, ,i ia T T i N

0 01 0 0 2

00 0 02 0 1 1

( ) 1log ( )

logTT

c T T T T ca S T T

ac T T c c

0( )T T

Z

02C

020

1

C

C

S

2

2

1 101

1 1 1

N N

i ii i

N N N

i i ii i i

T N T

c

N S T S T

2

1 1 1 102

1 1 1

N N N N

i i i i ii i i i

N N N

i i ii i i

T S S T T

c

N S T S T

0

log ( ) log ( )i i

ii i

T T TS

a T a T


The horizontal shifting is accompanied by a vertical shift, to compensate difference in density due to temperature difference (T-T0) – entropic corrections:

00 0

( ) ( )T

D T D TT

0

01 3 ( )T T

00

0

/( ) ( )

1 3 ( )

T TD T D T

T T

is a linear thermal expansion coefficient,

00 0

( ) ( )T

E T E TT

00

0

/( ) ( )

1 3 ( )

T TE T E T

T T

Vertical Shifting of Creep and Relaxation Curves


3


Time‐Temperature SuperpositionSummary

log

E(t

), [

Pa]

log

E(t

), [

Pa]

TTS implies that the response time function of thematerial at a certain temperature resembles theshape of the same functions of adjacenttemperatures.

Time-temperature superposition is a procedure for widening the time time scale of experiments at a given referene temperature.

log

E(t

)

Restrictions:

t-T shifting may be used only if specimen was atall temperatures in equilibrium state

Material can be brought to equilibriumsufficiently close only well above the glasstransition temperature, i.e., T > Tg+20˚C.

41


Time‐Temperature SuperpositionHomework ‐ 2

0

300

600

900

1200

1500

-1 0 1 2 3 4log t [s]

G [

MP

a]

20°C30°C40°C50°C65°C80°C95°C

T ref = 50°CP ref = 0.1 MPa


Time‐Temperature SuperpositionHomework ‐ 2

183,9687220,3847277,9006452,4285708,981903,5732979,54641000

187,6638221,9378289,6285473,7961735,4359922,6379990,364630

189,1633227,8286296,2691494,7886765,586939,132997,4329400

192,8585230,881306,7653515,5669790,6486953,64491007,822251

194,5721235,433315,7086539,4516815,3898967,56851014,034158

198,8028238,8068322,135560,9261837,3998975,54771018,051100

199,8202241,4309330,8641583,7929858,2853988,66791027,36963

202,2301246,1971340,7177606,2315878,3141998,03971031,86740

203,8902249,678350,6784629,848896,68261006,0191036,68725,1

206,7286252,2485361,8708652,9292913,17671016,0331041,77515,8

207,9067257,1218371,4033676,9205925,17221020,4781045,79110

210,9592262,7983384,4166700,9656942,36281028,8331048,9516,3

216,4751266,2257397,4834721,2622955,80441035,7411053,2884

215,7789273,8837411,8353747,3423968,38911040,6681056,2872,51

217,6532279,0783425,9196770,0484980,70631046,8261059,4471,58

218,7243285,2903441,4499792,2726989,38161051,861061,1071

222,955291,8237457,5155812,6761998,91411055,2871064,1060,63

226,061298,6249474,8668833,991007,1611058,4471065,7120,4

229,0599304,3015492,8602852,7871014,1771062,3031067,4260,251

228,0425315,6011511,2824870,1381019,9071064,1771067,8010,158

234,7365323,0985528,0442885,88251021,2451067,6581071,7640,1

G(t)7G(t)6G(t)5G(t)4G(t)3G(t)2G(t)1t [s]

95°C80°C65°C50°C40°C30°C20°CT [oC]

183,9687220,3847277,9006452,4285708,981903,5732979,54641000

187,6638221,9378289,6285473,7961735,4359922,6379990,364630

189,1633227,8286296,2691494,7886765,586939,132997,4329400

192,8585230,881306,7653515,5669790,6486953,64491007,822251

194,5721235,433315,7086539,4516815,3898967,56851014,034158

198,8028238,8068322,135560,9261837,3998975,54771018,051100

199,8202241,4309330,8641583,7929858,2853988,66791027,36963

202,2301246,1971340,7177606,2315878,3141998,03971031,86740

203,8902249,678350,6784629,848896,68261006,0191036,68725,1

206,7286252,2485361,8708652,9292913,17671016,0331041,77515,8

207,9067257,1218371,4033676,9205925,17221020,4781045,79110

210,9592262,7983384,4166700,9656942,36281028,8331048,9516,3

216,4751266,2257397,4834721,2622955,80441035,7411053,2884

215,7789273,8837411,8353747,3423968,38911040,6681056,2872,51

217,6532279,0783425,9196770,0484980,70631046,8261059,4471,58

218,7243285,2903441,4499792,2726989,38161051,861061,1071

222,955291,8237457,5155812,6761998,91411055,2871064,1060,63

226,061298,6249474,8668833,991007,1611058,4471065,7120,4

229,0599304,3015492,8602852,7871014,1771062,3031067,4260,251

228,0425315,6011511,2824870,1381019,9071064,1771067,8010,158

234,7365323,0985528,0442885,88251021,2451067,6581071,7640,1

G(t)7G(t)6G(t)5G(t)4G(t)3G(t)2G(t)1t [s]

95°C80°C65°C50°C40°C30°C20°CT [oC]

Construct Master Curve for Tref = 50 CCalculate c1 c2 for Tref = 50 C and Tref = 20 CDetermine Isochronous temperature dependence at t = 10, 100, and 1000 seconds


Time‐Temperature‐ Pressure Superposition

42


The Effect of Pressure

Similarity of Temperature and Pressure Effect

Polymer Applications


An increase of pressure slows both relaxation and retardation, Fillers-Moonan-Tschoegl, 1977, 1983.

)()(

)]([log

2

1,

pTTpc

pTTca

refrr

refrr

pT

p pref ( )p 0The FMT equation comprises the WLF equation. When and equation reduces to the WLF equation.

N.W. Tschoegl and W.G. Kanuss

The Effect of Pressure (and Temperature)


The Effect of Pressure (and Temperature)

experimentalwindow

T T = 0

log t

P P P1 2 5 < < ... <

P P0 3 =

t1 t2 t3

master curve at P3

P1

P2

P3

P4

P5

(b)

log aP2

Fillers-Moonan-Tschoegl (FMT) Model, 1979

log

( )

( ),ac T T P

c T T PP T

1 0

2 0

06

65

04

43 1

c1ln

1

1ln

)(

)()( 0

Pc

Pc

Pc

Pcc

P

PfP

f

T

c k Pr f3 1 / ( )

c k Kr r4 /

c k Pf5 1 / ( )

c k K6 /

)(/02 Pfc f

c B f1 02 303 / .

43


The effect of constant T and P atlo

g

()

Gt

experimentalwindow

P P = 0

master curve at T3

log tt1 t3

T T T1 2 5< < ... <

T T0 3 = T1

T2

T3

T4

T5

t2

(a)

log aT4

experimentalwindow

T T = 0

log t

P P P1 2 5 < < ... <

P P0 3 =

t1 t2 t3

master curve at P3

P1

P2

P3

P4

P5

(b)

log aP2

.

t


What do we know ?

The State-of-Art on the effect of pressure and temperature is summarized in:

• Tschoegl, N.W., Knauss, W.G., Emri, I., “The Effect of Temperature and Pressure on the Mechanical Properties of thermo- and/or Piezorheologically Simple Polymeric Materials in Thermodynamic Equilibrium – A Critical Review”, Mechanics of Time Dependent Materials Vol.6, 2002, pp.53-99.

• Tschoegl, N.W., Knauss, W.G., Emri, I., “Poisson’s Ratio in Linear Viscoelasticity-A critical Review”, Mechanics of Time Dependent Materials Vol.6, 2002, pp.3-51.


Shear Relaxation Modulus with Pressure Superposition

G(t) for PVAc at T=40°C (raw data)

1

10

100

1000

1E-1 1E+0 1E+1 1E+2 1E+3

t [s]

G [

MP

a]

p=0.1MPap=20 MPap=40 MPap=55 MPap=70 MPa

G(t) for PVAc at T=40°C (superposed with pref=40°C)

1E-5 1E-4 1E-3 1E-2 1E-1 1E+0 1E+1

t/aT [s]

Shift Factors

0

1

2

3

4

5

0 20 40 60 80

p [MPa]

log

ap

44


Shear Relaxation Modulus with Pressure Superposition

-8 -5 -2 1 4log t [s]

0.1 MPa25 MPa50 MPa75 MPa100 MPaP 0 = 0.1 MPa

1.0

1.5

2.0

2.5

3.0

3.5

-1 0 1 2 3log t [s]

log

G( t

) [M

Pa

]

T 0 = 35°C

Shift factors

0

4

8

0 50 100P [MPa]

log

aP


The effect of constant T and P on amorphous polymers at

-8 -4 0 4

log t [s]

T 0 = 35°C

1.0

1.5

2.0

2.5

3.0

3.5

-1 0 1 2 3log t [s]

log

G( t

) [M

Pa

]

15°C20°C25°C30°C35°C

Shift factors

0

2

4

6

15 25 35T [°C]

log

aT

P 0 = 0.1 MPa

-8 -5 -2 1 4log t [s]


1.0

1.5

2.0

2.5

3.0

3.5

-1 0 1 2 3log t [s]

log

G( t

) [M

Pa]

T 0 = 35°C

Shift factors

0

4

8

0 50 100P [MPa]

log

aP

1.0

1.5

2.0

2.5

3.0

3.5

-8 -4 0 4log t [s]

log

G( t

) [M

Pa

]

- master curve - master curve

T 0 = 35°C

P0 = 0.1 MPa

TP

Time‐temperature‐pressure supperposition

Constant temperature and pressure does not change the shape of the spectrum.

t


-1.0

0.0

1.0

2.0

3.0

-9 -6 -3 0 3 6log t [s]

log

G [

MP

a]

- master curve

- master curve

P 0 = 200 MPa

T 0 = 5°C

T

P

SBR

The effect of constant T and P at for SBRt

45


0.5

1.0

1.5

2.0

2.5

-8 -5 -2 1 4 7log t [s]

log

G [

MP

a]

- master curve

- master curve

T 0 = 5°C

P 0 = 200 MPa

T

P

EPDM

The effect of constant T and P at for EPDMt


log

( )

( ),ac T T P

c T T PP T

1 0

2 0

c k Pr f3 1 / ( )

c k Kr r4 /

c k Pf5 1 / ( )

c k K6 /

Fillers‐Moonan‐Tschoegl (FMT) Model

)(/02 Pfc f

c B f1 02 303 / .

02

01 )(log

TTc

TTcaT

Williams‐Landel‐Ferry (WLF) Model

ffc /02

c B f1 02 303 / .

06

65

04

43 1

c1ln

1

1ln

)(

)()( 0

Pc

Pc

Pc

Pcc

P

PfP

f

T

Modeling the Effect of P and T at t


Volume Dependence on Pressure

PVAc Specific Volume Dependence on Pressure (T=40°C)

0.810

0.820

0.830

0.840

0.850

0 20 40 60 80 100

p [MPa]

v [

cm

3/g

]

v [cm3/g]Murnaghan's Fit

46


Thermal Expansivity Dependence on Pressure

PVAc Thermal Expansion Dependence on Pressure

0.825

0.830

0.835

0.840

0.845

0.850

38.5 39.0 39.5 40.0 40.5 41.0 41.5

T [°C]

v [

cm

3/g

]

p=0.1MPa, Beta=0.000645/Kp=50 MPa, Beta=0.000560/K


Effect of Pressure on Tg, and Tm

0

V

V

1p

TgT

2p

3p

13 2p p p

Tm


B(t) for PVAc (T=40°C, p=8.5 MPa 0.1 MPa)

0.00041

0.00043

0.00045

0.00047

0.00049

0.00051

10 100 1000 10000

t [s]

B(t

) =

v(

t)/

p [

/MP

a]

The Effect of Pressure Variation

47


Effect of pressure variation on PVAc

1.00

1.50

2.00

2.50

0 1 2 3 4 5log t [MPa]

B( t

) [ x

10 -

3/M

Pa]

35°C

32°C

30°C

P 0 = 0.1 MPa

P = 10 MPa

4.00

5.00

6.00

7.00

0 1 2 3 4 5log t [s]

B( t

) [ x

10

- 4/M

Pa

]

= 100 MPa

= 75 MPa

= 50 MPa

= 25 Mpa

T 0 = 30°C

P 0 = 0.1 MPa

P 1

P 2

P 3

P 4

Understanding of the effect of pressure variation is still insufficient.There are very few models that allow mathematical description of the simultaneous pressure and temperature variation


dtttKttP

t

kk

0

)()(')('3)()(

t

ctT

dttt

)(),(),()(')('

0

1

)(),(),(

1

303.2)(),(),(log

fctTf

bctT

d

Ttf

t

T

0

)()(

dtMf

tkk

0

)()(

3

1 d

ctf

t

c

0

)()(

3

1

cT ffffctTf 0)(),(),(

Stress‐Strain Relations

The Knauss‐Emri (KE) ModelModeling the eefect of pressure


Measurement conditions:

• pressure up to 600 MPa (6000bar), ±0.1 MPa

• temperature from –50°C to +120°C, ±0.01°C

data acquisition

circulator

measuring inserts

thermal bath

pressurizing system

pressure vessel

carrier amplifier

CEM P‐T‐t Apparatus

Kralj, A., Prodan T., and Emri, I., J. rheol., 45, 929-943, (2001).Emri, I., and Prodan, T., Experimental Mechanics, (2005), In print

48


Assumptions:

• homogeneous, isotropic specimen

• = 3

260

mm

LVDT

LVDT rod

specimen

Measurement Principle

P, T

CEM P‐T‐t Apparatus: Dilatometer


280

mm

loading device

load cell

specimen

slidermechanism

triggeringmechanism

electric motor

(a)

CEM P‐T‐t Apparatus: Relaxometer


Load

Response


Typical geometry of teh specimens

CEM P‐T‐t Apparatus: Specimens

49


P h ys ica l P ro p erties S ym b o ls

S hear m om ent ( ), ( ), ( )M t M T M P o r ( , , )M t T P

Mea

s

-ure

d

S pec im en leng th ( ), ( ), ( )L t L T L P o r ( , , )L t T P

S hear re la xa tion m odu lus ( ), ( ), ( )G t G T G P o r ( , , )G t T P

S pec ific vo lum e ( ), ( ), ( )v t v T v P o r ( , , )v t T P

L inear the rm a l e xpa ns ion coe ff. ( ), ( )T P , o r ( , )T P

V o lum e tric the rm a l expa ns ion coe ff. ( ), ( ), , ,g e fT P , o r , , ( , )g e f T PB u lk c reep com p liance ( ), ( ), ( )B t B T B P o r ( , , )B t T P

Ca

lcu

late

d f

rom

de

fin

itio

ns

B u lk m odu lus ( ), ( )K T K P o r ( , )K T P

W LF cons tan ts 1 2, c c

W LF m ate ria l pa ram ete rs 0, f f

F M T constan ts 1 2 3 4 5 6, , , , , c c c c c c

Ca

lcu

late

d

fro

m m

od

els

F M T m ate ria l pa ram ete rs *0( ), ( ), , , , , f e eP f P B K k K k

CEM P‐T‐t Apparatus: Measuring capabilities


Shear relaxation modulus of PVAcEffect of Temperature

40 0

32 ( )( , , )

M tG t T P

D

0

5

10

15

20

25

30

35

40

0 2 4 6 8 10 12 14 16 18 20t [h]

T( t

) [°

C]

0

0.1

0.2

0.3

0.4

P(t

) [M

Pa]

MeasTP

-8 -4 0 4

log t [s]

T 0 = 35°C

1.0

1.5

2.0

2.5

3.0

3.5

-1 0 1 2 3log t [s]

log

G( t

) [M

Pa]

15°C20°C25°C30°C35°C

Shift factors

0

2

4

6

15 25 35T [°C]

log

aT

P 0 = 0.1 MPa

P-T boundary conditions

1.0

1.5

2.0

2.5

3.0

3.5

15 20 25 30 35T [°C]

log

G(t

) [M

Pa]

0.1 s1 s10 s100 s

P 0 = 0.1 MPa


Shear relaxation modulus of PVAcEffect of Pressure

40 0

32 ( )( , , )

M tG t T P

D


0

5

10

15

20

25

30

35

40

0 2 4 6 8 10 12 14 16 18t [h]

T( t

) [°

C]

-20

0

20

40

60

80

100

120

140

P( t

) [M

Pa]

T

P

Meas

-8 -5 -2 1 4log t [s]


1.0

1.5

2.0

2.5

3.0

3.5

-1 0 1 2 3log t [s]

log

G( t

) [M

Pa]

T 0 = 35°C

Shift factors

0

4

8

0 50 100P [MPa]

log

aP

1.0

1.5

2.0

2.5

3.0

3.5

0 20 40 60 80 100P [MPa]

log

G( t

) [M

Pa

]

0.1 s1 s10 s100 s

T 0 = 35°C

50


Specific Volume v(T,P)

( , , )( , )

V t T Pv T P

m

0

48

12

16

20

2428

32

36

40

0 12 24 36 48 60 72 84 96 108t [h]

T(t

) [°

C]

-4

04

8

12

16

2024

28

32

36

P( t

) [M

Pa

]

T

P

Meas

P-T boundary conditions0,760

0,780

0,800

0,820

0,840

0 10 20 30 40P [MPa]

v(P

) [c

m3 /g

]

32°C

28°C

24°C

20°C

16°C

0,760

0,780

0,800

0,820

0,840

0,860

15 20 25 30 35T [°C]

v( T

) [c

m3 /g

]0.1 MPa8 MPa16 MPa24 MPa32 MPa


Thermal expansion coefficient

0

1 ( , , )( , ) ( , , )

L t T PT P t T P

L T

0

48

12

16

20

2428

32

36

40

0 12 24 36 48 60 72 84 96 108t [h]

T(t

) [°

C]

-4

04

8

12

16

2024

28

32

36

P( t

) [M

Pa]

T

P

Meas

P-T boundary conditions3.00

4.00

5.00

6.00

7.00

8.00

15 20 25 30 35T [°C]

[ x

10 -

4/ °

C]

P 0 = 0.1 MPa

Tg

6.00

6.40

6.80

7.20

7.60

8.00

0 10 20 30 40P [MPa]

P

[ x

10 -

4/°

C]

T 0 = 32°C

Pg


Bulk Creep Compliance B(t,T,P) ‐ I

0 00 0 0 0 0 0 0

0 0

( , , )( , , ) ( , , ) ( , , ) ( ) ( )

t tkk

kk

T PT P t B T P t d B T P t P P P h d

0 0 0 00 0

0

( , , ) ( , , )( , , ) kk T P t V T P t

B T P tP V P

0

5

10

15

20

25

30

35

40

0 6 12 18 24 30 36 42 48 54t [h]

T( t

) [°

C]

-2

0

2

4

6

8

10

12

P [

MP

a]

T

P

Meas

1.00

1.50

2.00

2.50

0 1 2 3 4 5log t [MPa]

B( t

) [x

10 -

3/M

Pa]

35°C

32°C

30°C

P 0 = 0.1 MPa

P = 10 MPa


1.00

1.40

1.80

2.20

2.60

28 30 32 34 36T [°C]

B( t

) [ x

10 -

3/M

Pa]

9 s

90 s

1000 s

P 0 = 0.1 MPa

51


Bulk Creep Compliance B(t,T,P) ‐ II

0 00 0 0 0 0 0 0

0 0

( , , )( , , ) ( , , ) ( , , ) ( ) ( )

t tkk

kk

T PT P t B T P t d B T P t P P P h d

0 0 0 00 0

0

( , , ) ( , , )( , , ) kk T P t V T P t

B T P tP V P


0

5

10

15

20

25

30

35

40

0 2 4 6 8 10 12 14 16 18 20 22t [h]

T( t

) [°

C]

-20

0

20

40

60

80

100

120

P [

MP

a]

T

P

Meas

4.00

5.00

6.00

7.00

0 1 2 3 4 5log t [s]

B( t

) [x

10 -

4/ M

Pa]

= 100 MPa

= 75 MPa

= 50 MPa

= 25 Mpa

T 0 = 30°C

P 0 = 0.1 MPa

P 1

P 2

P 3

P 4

4.00

5.00

6.00

7.00

0 30 60 90 120P [MPa]

B( t

) [ x

10 -

4/M

Pa]

9 s

90 s

1000 s

T 0 = 30°C


Bulk Modulus K(T, P)

( , ) ( , , )K T P K T P t

( , )dP

K T P VdV

350

400

450

500

550

600

650

0 10 20 30 40P [MPa]

K(P

) [M

Pa

]

T 0 = 32°C

300

500

700

900

15 20 25 30 35T [°C]

K(t

) [M

Pa

]

P 0 = 0.1 MPa


Free Volume and WLF Parameters

TABLE 4— WLF CONSTANTS AND PARAMETERS FOR PVAc.

Parameter [Units]

0P

[MPa] 0T

[C]

01c 0

2c

[C]

B f

[ 410 /C] 0f

[ 210 ]

Value 0.1 35 23.37 83.58 1 1.81 1.62

c B f1 02 303 / .02

01 )(log

TTc

TTcaT

ffc /02

52


FMT Parameters and Constants

log

( )

( ),ac T T P

c T T PP T

1 0

2 0

c k Pr f3 1 / ( )

c k Kr r4 /

c k Pf5 1 / ( )

c k K6 /

)(/02 Pfc f

c B f1 02 303 / .

06

65

04

43 1

c1ln

1

1ln

)(

)()( 0

Pc

Pc

Pc

Pcc

P

PfP

f

T

TABLE 5— FMT PARAMETERS FOR PVAc.

Parameter [Units]

0P

[MPa] 0T

[C]

001c 00

2c

[C]

B f

[ 410 /C] 0f

[ 210 ]

*K

[MPa]

k *rK

[MPa] rk

Value 0.1 35 23.37 83.58 6.756 15.02 12.55 1976 11.09 1067 9.11


The Effect of Moisture


0,0001

0,001

0,01

0,1

1

1 10 100 1000 10000 100000 1000000

1E+07 1E+08 1E+09 1E+10

t [sec]

J(t

) [m

m2/N

]

100

10-1

10-2

10-3

10-4100 102 104 106 108 1010

master curves:different Tdifferent cw

0,0001

0,001

0,01

0,1

1

1 10 100 1000 10000

J(t

) [m

m2 /N

]

t [sec]100 5 101 5 102 5 103 5 104

Cw[%]

2.72

1.81

1.291.050.740.00

100

10-1

10-2

10-3

10-4

0,0

0,0

0,0

0,1

1,0

1 10 100 1000 10000

t [ sec]

J(t

) [m

m2 /N

]

T[oC]

35.8

34.4

32.3

30.628.426.420.0

100 5 101 5 102 5 103 5 104

100

10-1

10-2

10-3

10-4

Effect of Moisture

53


Knauss‐Emri model: Modeling teh Effect of Moisture and Solvents

de

tttGtSt ij

ij

0

)()(')('2)(

dtttKt

t

kk

0

)()(')('3)(

t

ctT

dttt

)(),(),()(')('

0

1

)(),(),(

1

303.2)(),(),(log

fctTf

bctT

d

Ttf

t

T

0

)()(

dtMf

tkk

0

)()(

3

1 d

ctf

t

c

0

)()(

3

1


Stress‐Strain Relations


Effect of Moisture on Fracture Properties of PA



54




Dynamic Loading


Dynamic Loading ‐ 1

Creep and stress relaxation tests are convenient for providing information on the material response at long times (minutes to days), but are not useful at shorter times (seconds or less) because of inertial effects (ringing) induced by the step-function input.

In dynamic tests one applies a sinusoidally oscillating stress or strain. Such tests are well-suited for covering the short-time range in the response (high frequencies) but are inconvenient at long times (low frequencies).

55



0( , ) sint t ( 0) 0t 0( , ) cost t

( ) ( ) exp( / ) lneE t E H t d

0

( )( ) ( ) (0) ( )

t ut E t E t u du

u

0

0

( , ) [ ( ) exp( ) ln ]cost

e

t ut E H d u du

0 0

0 0

( , ) cos [ ( ) exp( ) ln ] cost t

e

t ut E u du H d u du

0 0

0

( , ) sin ( )[ exp( )cos ] lnt

e

t ut E t H u du d

Excitation:

Response: Material property:

2 2

0 0 02 2 2 2

2 2

0 2 2

( , ) sin sin ( ) ln cos ( ) ln1 1

+ ( ) exp( / ) ln1

et E t t H d t H d

H t d



( , ) ( )t

2 2

2 2( ) ( ) ln

1eE E H d

2 2

( ) ( ) ln1

E H d

Usually we are interested in steady-state response, i.e., when

2 2

0 0 02 2 2 2

2 2

0 2 2

( , ) sin sin ( ) ln cos ( ) ln1 1

+ ( ) exp( / ) ln1

et E t t H d t H d

H t d

The storage modulus, is a measure of the material's ability to storage energy elastically. The loss modulus is a measure of its ability to dissipate energy through viscous mechanisms, and is a parameter often related to the toughness and impact resistance of the material. In composite materials Coulomb friction between the matrix and the fibers can substantially increase

)(

)()(tan

E

E

22 )()()(~ EEE

)(sin)(~

)( EE

)cos()(~

)( EE )(

~)( 00 E

)(sin)()(sin)()(cos)(sin)()( 022

000 ttEEtEtE


Hysteresis Experiment ‐ 1

Frequency dependent material functions may be conveniently determined with a so called hysteresis experiment.We excite the material, for example, uniaxial with a harmonic strain excitation and measure the correspondingstress, which is phase shifted in respect to strain

0( , ) sint t 0sin ( , ) /t t

2

0 0 0

( , ) ( , ) ( , )cos ( ) 1 sin ( )

( )

t t t

2 2

20 0 0 0

( , ) 2 ( , ) ( , ) cos ( ) ( , )1

( ) sin ( ) ( ) sin ( )

t t t t

)(sincos)(cossin)()(sin)(),( 00 tttt

56


0( ) /t

0( ) /t

1.0

1.0AB

B

A

0.5

Hysteresis Experiment ‐ 2

2 2

20 0 0 0

( , ) 2 ( , ) ( , ) cos ( ) ( , )1

( )sin ( ) ( ) sin ( )

t t t t

( , ) 0

0

( , )sin ( )

( )t

tA

0( , )

0

( , )cos ( )

( )t

tB

( , ) 0

0

( , )sin ( )t

tA

0( , ) ( )

0

( , )cos ( )t

tB

)(

)()(tan

E

E

B

A

B

A


Hysteresis Experiment ‐3

)(sin)(~

)( EE

Wd

2 / 2 /

0 0

( )(loop) ( ) ( ) ( )

d tW t d t t dt

dt

2 /

0 0

0

(loop) ( ) cos sin[ ( )]W t t dt

0( ) sint t 0( ) ( )sin[ ( )]t t

20

(loop)( )

WE

2 2

0 0

(loop) (loop)( )( )

tan ( )

W B W BEE

A A

20

(loop)( )

WD

2 20 0

(loop) (loop)( )( )

tan ( )

W B W BDD

A A

)(sin)()loop( 00 W

)()(sin)(~

)loop( 20

20 EEW

0

0(

22 )()()(~ EEE

)(~

)( 00 E

)(cos)(~

)( EE

2 2

20 0 0 0

( , ) 2 ( , ) ( , ) cos ( ) ( , )1

( ) sin ( ) ( ) sin ( )

t t t t



2 2( ) ( ) ( )E E E

( )tan ( )

( )

E

E

0 0( ) ( ) sin[ ( )] ( ) sin[ ( )]E t t

2 20 0 0( ) ( ) ( ) ( )E E E

The Absolute Modulus:

The Loss Tangent:

0 0( ) ( )sin ( ) cosE t E t

We observe that when a viscoelastic material is subjected to a sinusoidally oscillating strain the resulting stress will bealso sinusoidal, having the same angular frequency, but leading the strain by the phase angle, which is a function of theloading frequency.

57



0 0( ) ( )sin ( )cosE t E t

*( ) ( ) ( ) ( ) exp[ ( )]E E jE E j 1j

1

1j

j

( )E

( )E

( )E

( )

( )

( )D

( )D

( )D

( ) ( ) cos ( )E E ( ) ( )sin ( )E E


j

1

~( )E

E ' ( )

E"( )

j

1

~( )J

J ' ( )

J "( )

Hysteresis Experiment ‐4

Dynamic Creep and Relaxation Modulus


Dynamic Materila FunctionsHomework ‐ 3

For the three viscoleastic models shown below determine:

E´() and E”() D´(), D”() tan(), andHysteresis energy as function of excitation frequency(loop)W

1E2E1E

1E

1

1 1

1E2E1E

1E

1

1 1

58


Interrelation Between Material

Functions


Interrelation Between Material Functions ‐1

We discuss here the interrelations between the time- and frequency-dependent viscoelastic material functions measured in shear.

The interrelations between other viscoelastic material functions are from the mathematical stand-point of view analogous. Thus, the findings may be readily generalized to other modes of deformation.

ln)(1

)()(ln)(1

)()()(

ln)()(

22

2

dHGdHGG

deHGtG

e

t

e



The theoretical interrelations between the harmonic and the time dependent material functions are given in the formof the (generalized) Fourier transform,

( ) ( ) ( ) ( ); ( ) exp( )G G jG j G t j j G t j t dt

( ) ( ) ( ) ( ); ( ) exp( )J J jJ j J t j j J t j t dt

0

( ) ( ) cosG G t t dt

0

( ) ( ) sinJ J t t dt

0

( ) ( ) cosJ J t t dt

0

( ) ( ) sinG G t t dt

Static Dynamic

59



1 1 ( )( ) ( ) ; exp( )

2

GG t G j t j t d

j

1 1 ( )

( ) ( ) ; exp( )2

JJ t J j t j t d

j

Dynamic Static

0 0

( )2 2 ( ))( ) sin (1 cos )g

g g

G G GG t G t d G t d

0

0

( )2( ) sin

( )2(1 cos ) .

gg f

f

g f

J JJ t J t d t

JJ t d t



0

( )lim lim ( )ft

dJ tJ

dt

0lim ( ) lim ( )gt

G G t G

0

lim ( ) lim ( )g tJ J t J

0 00

( )lim ( ) lim ( ) lim ( ) lim

t

f t t

Gt G u du

0 0 1e eG J 1g gG J 1f f



Time- and Frequency- Dependent Functions Expressed in Terms of Relaxation and Retardation Spectra:

2 2

2 2 2 2

1( ) ( ) ln ( ) ln

1 1e gG G H d G H d

2 2( ) ( ) ln

1G H d

2 2

1( ) ( ) ln

1H d

2 2

( ) ( ) ln1

eGH d

,

0lim ( ) lim ( )et

G G t G

( ) ( ) exp( / ) ln ( ) 1 exp( / ) lne gG t G H t d G H t d

( ) ( ) 1 exp( / ) lnet G t H t d

,

60



Time- and Frequency- Dependent Functions Expressed in Terms of Discrete Relaxation and Retardation Spectra:

2 2

2 2 2 21 1

1( )

1 1

i n i ni

e i g ii ii i

G G H G H

2 2

1

( )1

i ni

ii i

G H

2 21

1( )

1

i n

i ii i

H

2 21

( )1

i ne i

i ii i

GH

1 1

( ) exp( / ) 1 exp( / )i n i n

e i i g i ii i

G t G H t G H t

1

( ) ( ) 1 exp( / )i n

e i i ii

t G t H t

0lim ( ) lim ( )et

G G t G



( ) lng eG G H d

0 ( ) lne gJ J L d

( ) lnf H d

,

1

i n

g e ii

G G H

0

1

i n

e g ii

J J L

1

i n

f i ii

H

Material Constants:



1 t

G t G

Simple (Laun) (1991):Christansen (1982):

Ninomiya in Ferry (1959):

Schwarzl in Struik (1967):

Schwarzl1 (1975):

Schwarzl2 (1975):

1

( ) 0.4 0.4 0.014 10t

G t G G G

1

( ) 0,337 0,323t

G t G G

1

( ) 0.00807 16 0.00719 8

0.00616 4 0.467 2 0.0918

0.0534 2 0.08 4 0.0428 8t

G t G G G

G G G

G G G

1

( ) 0.496 2 0.0651 4 2

0.0731 2 .111 2

0.03 8 16 0.00683 32 64t

G t G G G G

G G G G

G G G G

2 t

G t G

R. Cvelbar: Interkonverzija materialnih funkcij viskoelastičnih materialov, Doktorska disertacija, Fakulteta za strojništvo, Ljubljana, 2005, zagovor.

Approximate Interrelations:

61



An Owerview:


Interrelation Between Material Functions

Determination of Spectra Using Emri‐Tschoegl

Algorithm


Determination of the mechanical spectrumPhysical meaning of the Mehanical Spectrum

Assumption:Positions of the spectrum lines are fixed:

1) Spectrum lines are equally distributed,2) Two lines per decade.

Open questions: How to determine positions?Where to position the first spectrum line?

Response time i corresponds to length of the molecule.Equal distribution is not realistic.

Schematics of polymer structureEach color corresponds to certain length of

molecule

Response times

62


Determination of the mechanical spectrum

( ) ( ) exp( / ) ln ( ) 1 exp( / ) lne gG t G H t d G H t d

Selected papers related to this subject: 1. Cost T.L., Becker E.B.: Int. J. Numerical Methods in Engineering, 2, 207-219,

(1970). 2. Honerkamp J.: Rheol. Acta, 28, 363-371, (1989).3. Baumgaertl M., Winter H.H.: Rheol. Acta, 28, 511-519, (1989).4. Emri I., Tschoegl, N.W.: Rheol. Acta, 32, 311 –321, (1993).5. Tschoegl, N.W., Emri I.: Rheol. Acta, 32, 322 – 327, (1993).6. Tschoegl, N.W., Emri I.: Int. J. of Polymeric Mater., 18, 117-127, (1992). 7. Emri I., Tschoegl, N.W.: Rheol. Acta, 33, 60 – 70, (1994). 8. Emri I., Tschoegl, N.W.: Rheol. Acta, 36, 303- 306, (1997) 9. Emri, I., and Tschoegl, N.W.:Polimeri (Zagreb), 19, 79-85, (1998).10. Winter, H.H.: J. Non-Newton. Fluid Mech., 68, 225-239, (1997).11. Malkin A.Y., Kuznetsov V.V.: Rheol. Acta, 39, 379 – 383, (2000).

)

log

0

300

600

900

1200

1500

-10 -5 0 5 10 15log t [s]

G

[MP

a]

Tref = 50°C

Pref = 0.1 MPa

Log H(

Emri-Tschoeglalgoritem

Emri-Tschoeglalgorithm

Solving the inverse problem

0( ) ( )t h t ( )G t( )H


The first spectrum line does not contribute to value of point A.

We could not use this points, as well as points lying to the right frompoint A, to reconstruct the first spectrum line.

Point B contains information about all spectrum lines, however, itcould not be used for calculation of the first spectrum line if the level ofnoise in point B exceed magnitude of spectrum line, h1.

Such conditions allow qualitative determination of “Window” whichshould be used for calculation of the first spectrum line.

Quantitative determination implies information about the kernelfunction and number of spectrum lines per decade.

Algorithm Windowing Algorithm

/

1

( ) i

Mt

e g e ii

E t E E E h e

0.6

0.4

0.2

0

0.2

0

0.6

0.4

0.2

0

0.4

0.2

0

0.4

0.2

0

0.4

0.2

0

0.6

0.8

1

-4 -3 -2 -1 0 21 3 4

log (t/ )

f1(t)

f2(t)

f3(t)

f4(t)

hi

h4

h3

h2·exp(-t/ )A

B

SPECTRUM

RELAXATION MODULUS

i

/ , 1, 2,3, 4iti if t h e i

log

E(t

), [P

a]

H, [

-]

Main idea of WA:[ 1 0 1 2]


Windowing Algorithm (Emri‐Tschoegl)

Definition of the Window:

11/

1 1

1log( / ) log ln log

(10 1) logk k

l k nk k k

tn e

1/1 1

1/1

10log( / ) log ln log

(10 1) log

nk k

u k nk k k

tn e

( , ) / (1/ ) exp( / )k k kdk t dt t

log ilog t

exp(-t/i)

1

00.99 443.72 10log k

Window

1/1

1

10 nk k

k k

n is a number of spectrum lines per decade

n log /l kt log /u kt n log /l kt log /u kt

1 -0.59 0.41 5 -0.10 0.10 2 -0.27 0.23 6 -0.08 0.08 3 -0.18 0.15 7 -0.07 0.07 4 -0.13 0.12 8 -0.06 0.06

1

( ) exp( / )i N

e i ii

G t G G t

0

1

( ) exp( / )i N

e i i fi

J t J J t t

63



1

ˆ ˆ ˆ( ) ( ) exp( / ) ( )i N

j M i j i ji

g t g t g t t

1

( ) exp( / )i N

e i ii

G t G G t

1

( )ˆ( ) j

jM

G tg t

G G

1

( )ˆ( ) M

MM

G tg t

G G

1

ˆ ii

M

Gg

G G

1

1 1

ˆ ˆ ˆ ˆ ˆ( ) ( ) exp( / ) exp( / ) exp( / ) ( )i k i N

j M i j i k j k i j i ji i k

g t g t g t g t g t t

1

1


j M i j i k j k i j i ji m i k

g t g t g t g t g t t

( 2 1) 1m k n

1

1

ˆ ˆ ˆ ˆ ˆ( ) ( ) ( ) exp( / ) exp( / ) exp( / )i k i N

j j M i j i k j k i j ii m i k

t g t g t g t g t g t

,

,

( )ˆ/ 2 ( ) 0

ˆ

k u

k l

j sj

k k jj s k

tE g t

g

,

,

2( )k u

k l

j s

k jj s

E t

,

,

,

,

( ) exp( / )

ˆ

exp 2( / )

k u

k l

k u

k l

j s

m j j kj s

k j s

j kj s

t t

g

t

1

1

ˆ ˆ ˆ ˆ( ) ( ) ( ) exp( / ) exp( / )i k i N

m j j M i j i i j ii m i k

t g t g t g t g t



0

1

( ) exp( / )i N

e i i fi

J t J J t t

1

ˆ ˆ ˆ( ) ( ) exp( / ) ( )i N

j M i j i ji

j t j t j t t

1

( )ˆ ( )

j f j

jM

J t tj t

J J

1

ˆ( ) MM

M

Jj t

J J

1

ˆ ii

M

Jj

J J

1

1


j M i j i k j k i j i ji m i k

j t j t j t j t j t t

,

,

,

,

( ) exp( / )ˆ

exp 2( / )

k u

k l

k u

k l

j s

c j j kj s

k j s

j kj s

t t

j

t

1

1

ˆ ˆ ˆ ˆ( ) ( ) ( ) exp( / ) exp( / )i k i N

c j j M i j i i j ii m i k

t j t j t j t j t


Physical Aging

64


Aging

AGING

chemical physical

leads to modification of polymer chain(chemical reaction)

chemistry remains unchanged,but the local packing of the chains alters

(dimensional changes)


Effect of temperature

amorphous solids

glassy state rubber state

Tg


temperature

spec

ific

volu

me

TgTo

cooling

Physical aging

Aging occurs in broad temperature rangebelow Tg (temperature range of practicalinterest)

Should be considered for prediction oflong-term behavior

Physical aging is basic feature of solidstate in general

65


coolin

g

coolin

g

coolin

g

coolin

g

coolin

g

coolin

g

coolin

g

coolin

g

coolin

g

coolin

g

coolin

g

Physical aging


Physical Aging of PVAc

0

0.001

0.002

0.003

0.004

0.005

0.001 0.01 0.1 1 10 100 1000

log t [hours]

V

/V0

raw data

Knauss-Emri

A

B

C

D

E

T =27.5 ºC1

T =30 ºC2

T =32.5 ºC3

T =35 ºC4

T =37.5 ºC5

V

TT1T2T3T4 T5 T0

A B C D E

T =40 ºC0

Experimental data (shown in circles) were obtained by Kovacs, 1964.


F

L

F

L


Effect of pressure variation on PVAc

1.00

1.50

2.00

2.50

0 1 2 3 4 5log t [MPa]

B( t

) [ x

10

- 3/M

Pa

]

35°C

32°C

30°C

P 0 = 0.1 MPa

P = 10 MPa

4.00

5.00

6.00

7.00

0 1 2 3 4 5log t [s]

B( t

) [ x

10

- 4/M

Pa

]

= 100 MPa

= 75 MPa

= 50 MPa

= 25 Mpa

T 0 = 30°C

P 0 = 0.1 MPa

P 1

P 2

P 3

P 4

Understanding of the effect of pressure variation is still insufficient.There are very few models that allow mathematical description of the simultaneous pressure and temperature variation.

66


Model defines free-volume fractionat temperature T as:

- thermal expansion coefficient below Tg

- thermal expansion coefficient above Tg

Free‐volume theory


ln ln / fA BV V



f f f

f

V V Vf

V V V V

f l

f g ; for T < Tg

; for T Tg

1exp exp exp 1f

f f

V VVA B A B A B

V V f

Doolitle Equation


de

tttGtSt ij

ij

0

)()(')('2)(

dtttKt

t

kk

0

)()(')('3)(

t

ctT

dttt

)(),(),()(')('

0

1

)(),(),(

1

303.2)(),(),(log

fctTf

bctT

d

Ttf

t

T

0

)()(

dtMf

tkk

0

)()(

3

1 d

ctf

t

c

0

)()(

3

1


Stress-Strain Relations:

Knauss‐Emri Model

67


Modeling Physical Aging with Knauss‐Emri Model

0

0.001

0.002

0.003

0.004

0.005

0.001 0.01 0.1 1 10 100 1000

log t [hours]

V

/V0

raw data

Knauss-Emri

A

B

C

D

E

T =27.5 ºC1

T =30 ºC2

T =32.5 ºC3

T =35 ºC4

T =37.5 ºC5

V

TT1T2T3T4 T5 T0

A B C D E

T =40 ºC0

Experimental data (shown in circles) were obtained by Kovacs, 1964.


Center for Experimental Mechanics, University of Ljubljana, Slovenia


Using Creep as a “Spectroscopic Method”

68


Example I: Extrusion of LDPE

Using Creep measurements for the analysis ofdurability of LDPE exposed to different thermo-mechanical boundary conditions during teh extrusion prtocess. s.


LDPE used in this investigation

Physical Characteristics of Low Density Polyethylene OKITEN® 245 S (Dioki)a

Material propertiesb ISO Standard Value Unit

Density 1183 0,924 g/cm3

Melt Flow Rate 1133 2,3 g/10min

VICAT softening temperature 306/A 94 ºC

Tensile strength at yield 527/2 11 MPa

Tensile strength at break 527/2 14 MPa

Elongation at break 527/2 535 %

Hardness, Shore 868 47 Scale D

Melting point (DSC – air) 11357 114 ºC

Hazec 14782 4 %

Friction coefficientc: static/dynamic 3295 ≤0.11 -

a The granulated LDPE contains the following additives: flow additive and thermal stabilizerb Test samples obtained by direct press mouldingc Film thickness 0.025 mm exstruded on a laboratory extruder under standard conditions


Specimen preparation: The equipment

The die fitting with the glass tube tool.

Insulation system for slow cooling of glass tubes with extruded polymer.

69


Thermo‐mechanical boundary conditions

Torque, M, depending on the screw revolutions, n, and the set temperature, T, along the extruder barrel.


Thermo‐mechanical boundary conditions

Diagram 4: Pressure, p, depending on the screw revolutions, n, and the set temperature, T, along the extruder barrel.


Selected thermo‐mechanical boundary conditions

Specimen nameProcessing parameters

Set quantities Measured quantities

Tcylinder1

(°C)n2

(min-1)Tmelt

3

(°C)pmelt

4

(bar)M5

(Nm)

LDPE_168_23 160 25 168 23 30

LDPE_168_77 160 200 168 77 105

LDPE_263_20 250 25 263 20 28

LDPE_263_37 250 250 263 37 75

1 The temperature set on the ten heaters along the extruder barrel2 The number of screw revolutions3 The temperature of the melt measured at the outlet from the extruder barrel4 Pressure measured at the outlet from the extruder barrel5 Torque measured on the screws

70


The geometry of specimens

Cylindrical specimen of extruded LDPE, glued to metal holders and prepared for shear creep measurement.


The Creep Experiment – the long term behavior

The temperature and mechanical profile of loading in theshear-creep measurement of the extruded LDPE specimens.


The CEM Creep Apparatus

The Shear Creep Torsiometer: A - specimen, B - upper grip, C- guide, D - lower grip, E - principal bearing, F-weight, G - measuring cord, H - measuring cord bearing, I-loading wheel, J - friction wheel, K - friction rod, L - measuring bearing, M – inductive displacement meter, N - counterweight, O – loading weight, P – loading device.

71


Creep measurements in shear

-2.2

-2

-1.8

-1.6

-1.4

-1.2

-1

0 1 2 3 4 5log t [s]

log

J [

1/M

Pa]

38.9°C

48.6°C

58.1°C

67.4°C

76.8°C

LDPE_168_23Specimen name

Processing parameters


Tcylinder1

(°C)n2

(min-1)Tmelt

3

(°C)pmelt

4

(bar)M5

(Nm)

LDPE_168_23 160 25 168 23 30

LDPE_168_77 160 200 168 77 105

LDPE_263_20 250 25 263 20 28

LDPE_263_37 250 250 263 37 75


-2.2

-2

-1.8

-1.6

-1.4

-1.2

-1

0 1 2 3 4 5log t [s]

log

J [

1/M

Pa]

38.9°C48.6°C48.1°C57.5°C66.9°C76.9°C




Tcylinder1

(°C)n2

(min-1)Tmelt

3

(°C)pmelt

4

(bar)M5

(Nm)

LDPE_168_23 160 25 168 23 30

LDPE_168_77 160 200 168 77 105

LDPE_263_20 250 25 263 20 28

LDPE_263_37 250 250 263 37 75




-2.2

-2

-1.8

-1.6

-1.4

-1.2

-1

0 1 2 3 4 5log t [s]

log

J [

1/M

Pa]

31.9°C39.1°C48.5°C58.1°C67.4°C76.9°C




Tcylinder1

(°C)n2

(min-1)Tmelt

3

(°C)pmelt

4

(bar)M5

(Nm)

LDPE_168_23 160 25 168 23 30

LDPE_168_77 160 200 168 77 105

LDPE_263_20 250 25 263 20 28

LDPE_263_37 250 250 263 37 75

72



-2.2

-2

-1.8

-1.6

-1.4

-1.2

-1

0 1 2 3 4 5log t [s]

log

J [

1/M

Pa]

33.7°C38.9°C48.6°C58.1°C67.1°C76.7°C




Tcylinder1

(°C)n2

(min-1)Tmelt

3

(°C)pmelt

4

(bar)M5

(Nm)

LDPE_168_23 160 25 168 23 30

LDPE_168_77 160 200 168 77 105

LDPE_263_20 250 25 263 20 28

LDPE_263_37 250 250 263 37 75


Master creep compliance

-2.2

-2

-1.8

-1.6

-1.4

-1.2

-1

0 2 4 6 8 10 12log t [s]

log

Jre

f [1

/MP

a]

LDPE_168_23LDPE_168_77LDPE_263_20LDPE_263_37

T ref = 39°C

10

več kot 106

22 h 12 dni 3170 let

J kr =10-1.46

10-2.18



Tcylinder1

(°C)n2

(min-1)Tmelt

3

(°C)pmelt

4

(bar)M5

(Nm)

LDPE_168_23 160 25 168 23 30

LDPE_168_77 160 200 168 77 105

LDPE_263_20 250 25 263 20 28

LDPE_263_37 250 250 263 37 75


Isochronal creep compliance

0

0.02

0.04

0.06

0.08

30 40 50 60 70 80 90T [°C]

J [

1/M

Pa]

t=10 st=100 st=1000 st=10000 s

LDPE_168_23

∆J (T )

∆J (t)



Tcylinder1

(°C)n2

(min-1)Tmelt

3

(°C)pmelt

4

(bar)M5

(Nm)

LDPE_168_23 160 25 168 23 30

LDPE_168_77 160 200 168 77 105

LDPE_263_20 250 25 263 20 28

LDPE_263_37 250 250 263 37 75

73



0

0.02

0.04

0.06

0.08

30 40 50 60 70 80 90T [°C]

J [

1/M

Pa]

t=10 s

t=100 s

t=1000 s

t=10000 s

LDPE_168_77

∆J (T )

∆J (t )



Tcylinder1

(°C)n2

(min-1)Tmelt

3

(°C)pmelt

4

(bar)M5

(Nm)

LDPE_168_23 160 25 168 23 30

LDPE_168_77 160 200 168 77 105

LDPE_263_20 250 25 263 20 28

LDPE_263_37 250 250 263 37 75



0

0.02

0.04

0.06

0.08

30 40 50 60 70 80 90T [°C]

J [

1/M

Pa]

t=10 s

t=100 s

t=1000 st=10000 s

LDPE_263_20

∆J (T )

∆J (t )



Tcylinder1

(°C)n2

(min-1)Tmelt

3

(°C)pmelt

4

(bar)M5

(Nm)

LDPE_168_23 160 25 168 23 30

LDPE_168_77 160 200 168 77 105

LDPE_263_20 250 25 263 20 28

LDPE_263_37 250 250 263 37 75



0

0.02

0.04

0.06

0.08

30 40 50 60 70 80 90T [°C]

J [

1/M

Pa]

t=10 s

t=100 s

t=1000 s

t=10000 s

LDPE_263_37

∆J (T )

∆J (t )



Tcylinder1

(°C)n2

(min-1)Tmelt

3

(°C)pmelt

4

(bar)M5

(Nm)

LDPE_168_23 160 25 168 23 30

LDPE_168_77 160 200 168 77 105

LDPE_263_20 250 25 263 20 28

LDPE_263_37 250 250 263 37 75

74


Isochronal creep compliance at t=10000s

0

0.02

0.04

0.06

0.08

30 40 50 60 70 80 90T [°C]

J [

1/M

Pa]

LDPE168_23

LDPE168_77

LDPE263_20

LDPE263_37

t=10000sSpecimen name



Tcylinder1

(°C)n2

(min-1)Tmelt

3

(°C)pmelt

4

(bar)M5

(Nm)

LDPE_168_23 160 25 168 23 30

LDPE_168_77 160 200 168 77 105

LDPE_263_20 250 25 263 20 28

LDPE_263_37 250 250 263 37 75

compliance



0

0.005

0.01

0.015

0.02

0.025

LDPE_168_23 LDPE_168_77 LDPE_263_20 LDPE_263_37

∆J

(t)

[1/M

Pa]



Tcylinder1

(°C)n2

(min-1)Tmelt

3

(°C)pmelt

4

(bar)M5

(Nm)

LDPE_168_23 160 25 168 23 30

LDPE_168_77 160 200 168 77 105

LDPE_263_20 250 25 263 20 28

LDPE_263_37 250 250 263 37 75

0

0.02

0.04

0.06

0.08

30 40 50 60 70 80 90T [°C]

J [

1/M

Pa

]

t=10 s

t=100 s

t=1000 s

t=10000 s

LDPE_263_37

∆J (T )

∆J (t )



0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

LDPE_168_23 LDPE_168_77 LDPE_263_20 LDPE_263_37

∆J

(T)[

1/M

Pa]

0

0.02

0.04

0.06

0.08

30 40 50 60 70 80 90T [°C]

J [

1/M

Pa

]

t=10 s

t=100 s

t=1000 s

t=10000 s

LDPE_263_37

∆J (T )

∆J (t )



Tcylinder1

(°C)n2

(min-1)Tmelt

3

(°C)pmelt

4

(bar)M5

(Nm)

LDPE_168_23 160 25 168 23 30

LDPE_168_77 160 200 168 77 105

LDPE_263_20 250 25 263 20 28

LDPE_263_37 250 250 263 37 75

75


Mehanika nekovinskih gradiv

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