Strength of material

• 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 deformati

Section 4

Behaviour of Materials

The 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

• 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

• 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

• 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

• 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

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

Elastic-Plastic Stress-Strain Curves

Low / Medium Carbon Steels Aluminium Alloys and Alloy

Steels

ult = Ultimate Strength

YU = Upper yield point

YL = Lower yield point pr = Proof stress

ult

Limit of Proportionality

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

Unloading Modulus and Plastic Strain

• Unloading path is often

linear

– Slope of an approximate

straight unloading line

defined as the unloading

p e

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

• 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

x true 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

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

LA

A

or LA

ALV

LA

AA

A.A

PA

P

true true

x true

x true

True Stress and Strain Relationships

• True stress-engineering stress relationship:

• True strain defined by:

• True strain-engineering strain relationship:

)1

3 2

) 1

ln(

d

d ln

2 A

A ln

Ductility Index

• In order to estimate the ability of a metal to

deform plastically, the ductility index is used as

1 ln

and

q

true

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

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

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

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.)

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 –

O

C

B A

• 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

D D

P

2 BHN

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

2

P 854

1 L

P 2

136 sin

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

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

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