Section 4

Behaviour of MaterialsThe section will cover the behaviour of materials by introducing the stress-strain curve. The concepts of elastic and plastic deformation will be covered. This will then lead to a discussion of the micro-structure of materials and a physical explanation of what is happening to a polycrystalline material as it is loaded to failure.

© Loughborough University 2010. This work is licensed under a Creative Commons Attribution 2.0 Licence.

Contents• Introduction• Elasticity• Plasticity• Elastic-Plastic Stress-Strain Relationship• Elastic-Plastic Stress-Strain Curves• Secant and Tangent Modulii• Unloading Modulus and Plastic Strain• True Stress and Strain• True Stress-Strain Curve• Constant Volume Concept• True Stress and Strain Relationships• Ductility Index• Imperfections in Solids• Line Defects or Dislocations• Dislocation Movement and Strain Hardening• Microscopic Interpretation of Elastic-Plastic Stress-Strain Relationship• Hardness• Credits & Notices

Introduction

• Knowledge of a material’s properties, and how it behaves under various loading conditions is essential in design– Materials are selected for specific applications

dependent on their properties and characteristics

• The behaviour of a material under load is markedly different depending on whether the material response is elastic or plastic

Elasticity

• For the tensile bar, the external load has been assumed to be low enough that the bar will resume its initial shape once the external load is removed

• This state of elastic deformation is possible only when the external load is within certain limits

• In the elastic range, the load-displacement or stress-strain curve is linear – loading and unloading follow the same path

Elastic limit

Loading

UnloadingE

Plasticity

• If the loading is increased, it will reach a certain limit whereby elastic deformation would end and plastic deformation would start

• This limit is known as the elastic limit and beyond this point the material is said to have yielded

• The loading is thus beyond the elastic limit– Permanent or irreversible

deformation• Stress corresponding to

yielding is called the yield strength, denoted by y

Elastic limit

Loading Unloading

EPoint of yielding

y

Elastic-Plastic Stress-Strain Relationship

• Linear elastic stress-strain relationship considered thus far– Knowledge of mechanical behaviour in plastic region

also important in structural design / stress analysis– Complex plastic behaviour dependent on nature of

material

• Complete elastic-plastic stress strain relationship can be categorised by two types:– Distinct yielding, i.e. mild steels, low / medium carbon

steels– Less distinct yielding, i.e. aluminium alloys, alloy

steels

Elastic-Plastic Stress-Strain Curves

Low / Medium Carbon Steels Aluminium Alloys and Alloy Steels

ult = Ultimate Strength

YU = Upper yield point

YL = Lower yield pointpr = Proof stress

0.1%

0.2%

0.1% pr

0.2% pr

ult

Limit of Proportionality

YU

YL

ult

E

Strain Hardening

Necking

Secant and Tangent Modulii• Some materials (cast iron,

concrete) do not have a linear elastic portion of the stress-strain curve

• For this nonlinear behaviour, Secant and Tangent Modulii are used

• Secant modulus:– Slope of straight line

between origin and a point on stress-strain curve

• Tangent modulus:– Slope of the stress-strain

curve at a specified level of stress

1

2

secE

tanE

Unloading Modulus and Plastic Strain

• Unloading path is often linear– Slope of an approximate

straight unloading line defined as the unloading modulus

• Plastic or permanent deformation calculated from knowledge of total strain

loadingsecunloading EEE

petot p e

0.1% pr

Esec

Example 4.1

• A steel tensile specimen has a diameter of 10mm and a Young’s modulus of 209GPa. The load corresponding to 0.2% strain limit is 7kN and the maximum load recorded is 10kN with a total strain of 10%. Determine (i) the yield strength, (ii) the ultimate strength, and (iii) the plastic strain at the maximum load.

Answer: (i) y=89 MPa (ii) ult=127 MPa (iii) p=9.94%

True Stress and Strain

• True Stress: a stress defined with respect to the current or true cross-sectional area, Atrue as

• In the plastic range, plastic deformation or permanent reduction in cross-sectional area is significant

• A continuous use of nominal or engineering stress is no longer accurate

true

xtrue A

P

True Stress-Strain Curve

Onset of necking

True Stress-Strain Curve

Corrected for complex stress state in the neck region

Engineering Stress-Strain Curve

Atrue

PxPxPx Px

A

Constant Volume Concept

• A volume change in the elastic range is extremely small and is regarded as negligible

• In the plastic range, plastic deformation occurs through shear and thus no volume change takes place

• Thus we can write:

• This constant volume concept gives the following:

L

L

A

A or LAALVV

truetrue

L

1L

L

A

A

A

A.

A

P

A

P

truetrue

x

true

xtrue

True Stress and Strain Relationships

• True stress-engineering stress relationship:

• True strain defined by:

• True strain-engineering strain relationship:

)1(true

L

dLd true

...432

)1ln(d

dln2

A

Aln

L

Lln

432

truetrue

Ductility Index

• In order to estimate the ability of a metal to deform plastically, the ductility index is used as a measure:

• The ductility index is related to true stress and strain by:

%100A

AAq true

q1

1ln and

q1 truetrue

Example 4.2

• A steel tensile specimen has a diameter of 5mm and a gauge length of 100mm. During loading, a sampling pair of data points was taken at 10kN with a new length of 102mm. Determine (i) the true stresses and true strains at the sampling load, (ii) the corresponding diameter, and (iii) the ductility index at the sampling load.

Answer: (i) true=519 MPa, true=0.0198 (ii) d´=4.95 mm (iii) q=1.93%

Imperfections in Solids

• Metallic materials are made up of millions of crystals with BCC, FCC, or HCP crystal microstructures

• These crystal microstructures are not perfect in their atomic arrangement – they contain inherent structural defects

• Defects exist in metals in the form of line defects or point defects in the crystal structure during the solidification process

• Line defects are commonly known as dislocations

BCC

FCC

HCP

Line Defects or Dislocations

• Dislocations exist in the crystal lattice in the shape of plane (3D space) or line (2D space)

• Dislocations should not be interpreted as an indication of poor material quality

• Two types of dislocation are identified– Edge dislocation– Screw dislocations

Edge Dislocations

Stress

Stress

Stress

Stress

Stress

Stress

Slip plane

Burgess vector, b

Screw Dislocations

• Dislocations that produce the same result (plastic deformation) but the slip plane is parallel to the line of dislocation

• The magnitude and direction of the lattice distortion with a dislocation is expressed in terms of a Burgess vector, b

Burgess Vector

Dislocation Line

Point Defects

• Point defects are localised defects in the crystal microstructure involving one ore more atoms

• Caused by manufacture or heat treatment• Three point defect types: vacancy, interstitial

and substitutional

Vacancy Interstitial Substitutional

Dislocation Movement and Strain Hardening

• Metals and alloys contain thousands of crystal grains and many dislocations

• The existence of dislocations creates local strain or stress fields. – Upon loading, local strain or stress fields intensify

• Local yielding occurs at front of dislocations– Atomic bonds break– Dislocations start moving or slipping across crystal planes –

plastic deformation– Dislocations interact among themselves and with grain

boundaries / point defects– Dense concentration of dislocations occurs – pile-up– Resistance to external load increases – strain hardening

Strain Hardening (cont.)

Tension region

Slip plane

Compression region

Slip plane

(a) Edge dislocations arranged side-by-side

(b) Opposing Edge dislocations

Microscopic Interpretation of Elastic-Plastic Stress-Strain Relationship

• In the OA region: Dislocations are ‘locked’ and stable – linear elastic behaviour

• At point A: local stress sufficient to cause an instability of dislocations

• In the AB region: dislocations start to move freely as their constraints have ‘yielded’ on slip planes

• In the BC region: dislocations piling up during interaction – strain hardening takes place and plastic deformation still uniform

• At point C: slip plane developed into macroscopic fracture surface – material no longer capable of resisting the increasing load – ultimate strength

• Beyond point C: plastic deformation no longer uniform – necking occurs

YU

YL

ult

E

Strain Hardening

Necking

O

C

B

A

Hardness

• Hardness test result used to quickly assess degree of yielding, ductility, strain-hardening, or ultimate strength

• Hardness generally considered as resistance to permanent or plastic deformation

• Hardness test carried out by static indentation– Measured hardness number

• Thickness of specimen must be 10 the indentation produced at the end of the test

• Three major test methods:– Brinell– Vickers (popular in UK)– Rockwell (popular in US)

Brinell Hardness Test

• Hemispherical nose shape tungsten-carbide indenter of 10mm diameter compressed to flat surface of material

• When the load reaches a selected value, held for 30 seconds

• Load removed and diameter of indented area, d, measured

• Hardness number calculated using:

• Following load levels used:– Steels (and cast irons): 3000 kg– Aluminium alloys: 500 kg – 1000 kg

• For many ductile metals: ult 3.4BHN

22 dDDD

P2BHN

D

d

P

Vickers Hardness Test

• Indenter is a square-based diamond pyramid• The angles between the opposite faces of the pyramid

are 136• The surface diagonal of the pyramidal indent is

measured when the load reaches a selected level• Harness number calculated as:

• The selected load levels are:– Steels (and cast irons): 30 kg– Aluminium Alloys: 5 kg

22 L

P854.1

L

P2

136sin2

VHN

d

P

L

L

Rockwell Hardness Test

• A load P is applied to the flat surface of a material.

• Once the load reaches the selected load level, the indentation depth is measured with a dial test indicator (DTI)

• Hardness number is found by using a look-up table– No calculation is required

• For most metals, Rockwell C or Rc is used with the selected load level of 150 kg

Comparison of Hardness Scales

• The table below shows a comparison between the three hardness scales together with the UTS for steels

BHN

(10mm dia. and 300 kg force)

VHNRc

(150 kg Force)

ult

(MPa)

496

465

433

397

360

322

284

247

209

540

500

460

420

380

340

300

260

220

51.7

49.1

46.1

42.7

38.8

34.4

29.8

24.0

-

1792

1655

1517

1379

1241

1110

972

834

696

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Credits