Mechanical PropertiesThe adaptability of a material to a particular use is determined by its mechanical properties.Properties are affected by

Bonding typeCrystal StructureImperfectionsProcessing

Learning Objectives

Define engineering stress and engineering strain.State Hooke’s law, and note the conditions under which it is valid.Given an engineering stress–strain diagram, determine (a) the modulus of elasticity, (b) the yield strength (0.002 strain offset), and (c) the tensile strength, and (d) estimate the percent elongation.Name the two most common hardness-testing techniques; note two differences between them.Define the differences between ductile and brittle materials.State the principles of impact, creep and fatigue testing.State the principles of the ductile-brittle transition temperature.

Types of Mechanical TestingSlow application of stress

Allows dislocations to move to equilibrium positionsTensile testing

Rapid application of stressAbility of a material to absorb energy as it fails. Does not allow dislocations to move to equilibrium positions.Impact testing

Fracture ToughnessHow does a material respond to cracks and flaws

FatigueWhat happens when loads are cycled?

High Temperature LoadsCreep

Some DefinitionsTensile stress:Where F: force, normal to the cross-sectional area,

A0: original cross-sectional area0AF

=σ

Shear StressFs: force, parallel to the cross-

sectional area A0: the cross-sectional areaunit of stress:

0AFs=τ

2mN

areaForce

=1Pa = 1 Nm-2;

1MPa = 106Pa; 1GPa=109Pa

Engineering StrainNominal tensile strain (Axial strain)

00

0

ll

lll ∆

=−

=ε

For small strain:

θγ tan=

θγ ≅

Engineering Shear Strain

Poisson’s ratio

z

zz l

l

0

∆=ε

x

xx l

l

0

∆−=ε

z

x

straintensilestrainlateral

εε

−=−=

p

pp

p

V-∆VVV∆

=∆

Nominal lateral strain (transverse strain)

Poisson’s ratio: ν

Dilatation (Volume strain)Under pressure: the volume will change

σ

ε

E

Elastic Behavior of Materials(Hooke’s Law)

When strains are small, most of materials are linear elastic.

Young’s modulus

Tensile: σ = Ε ε

Shear: τ = G γ

Hydrostatic: – p = κ ∆

Shear modulus

Bulk modulus

Modulus of Elasticity - Metals

Modulus of Elasticity - Ceramics

Modulus of Elasticity - Polymers

0.01-0.1Rubbers

1-5Polyesters

2-4Nylon

3-3.4Polystyrene (PS)

0.2-0.7Polyethylene (PE)

Elastic Modulus (GPa)Polymers

Physical Basis of Young’s Modulus

dxdUF =Review: Inter-atomic forces (attractive and repulsive forces)

002

2

0 xxxx dxdF

dxUdS == ==

Define: stiffness

Assume the strain is small,

)(

)(

000

00

rrNSAF

rrSF

−==

−≈

σ

Where N: number of bonds/unit area, N=1/r02

σ σ

Unit area

εεσ ErS

rrr

rS

==−

=0

0

0

0

0

0 )(

Young’s modulus0

0 )(rrr −

=εQo

o

rSE ==

εσ

Stiffness & Young’s Modulus for different bonds

2-40.5-1Van der Waals

8-122-3Hydrogen

60-30015-75Metallic

200-100050-180Covalent (i.e: C-C)

32-968-24Ionic(i.e: NaCl)

E(GPa)S0(Nm-1)Bonding type

Material E (GPa)Metals: 60 ~ 400Ceramics: 10 ~ 1000Polymers: 0.001 ~ 10

Tensile Testing• The sample is pulled slowly• The sample deforms and then fails• The load and the deformation are measured

Standard tensile specimenThe load and deformation are easily transform into engineering stress (σ) and engineering strain (ε)

A curve stress-strain is obtained 0AF

=σ00

0

ll

lll ∆

=−

=ε

Parameters Obtained From Stress Strain CurveStrength Parameters

Modulus of ElasticityYield StrengthUltimate Tensile StrengthFracture StrengthFracture Energy

Ductility ParametersPercent ElongationPercent Reduction of AreaStrain Hardening Parameter

Modulus of ElasticityIt is a measure of material stiffness and relates

stress to strain in the linear elastic range.

12

12

ε−εσ−σ

=δεδσ

=E

Yielding and Yield Strength

Proportionality Limit (P): Departure from linearity of the stress-strain curveYielding Point – Elastic Limit: the turning point which separate the elastic and plastic regions –onset of plastic deformationYield strength: the stress at the yielding point.Offset yielding (proof stress): if it is difficult to determine the yielding point, then draw a parallel line starting from the 0.2% strain, the cross point between the parallel line and the σ−ε curve

Prior to TS, the stress in the specimen is uniformly distributed.After TS, necking occurs with localization of the deformation to the necking area, which will rapidly go to failure.

Tensile Strength (TS)

The stress increases after yielding until a maximum is reached. It is also known as the Ultimate Tensile Strength (UTS), or Maximum Uniform Strength.

Fracture Strength

o

ff AP

=σ

σf

Elastic RecoveryAfter a load is released from a

stress-strain test, some of the total deformation is recovered as elastic deformation.

During unloading, the curve traces a nearly identical straight line path from the unloading point parallel to the initial elastic portion of the curve

The recovered strain is calculated as the strain at unloading minus the strain after the load is totally released.

ResilienceResilience is the capacity of a material to absorb energy when it is deformed

elastically and then, upon unloading, to have this energy recovered.

∫=y dUr

εεσ

0Modulus of resilience UrIf it is in a linear elastic region,

EEU yyyyyr 22

121 2σσσεσ =

==

Ductility

100*)(%0

0

lll f −=EL

100*)(%0

0

AAA f−=AR

Ductility is a measure of the degree of plastic deformation at fracture

expressed as percent elongation

also expressed as percent area reduction

lO and AO are the original gauge length and original cross-section area respectively

lf and Af are length and area at fracture

Percentage elongation and percentage area reduction are UNITLESS

A smaller gauge length will produce a larger overall %elongation due to the contribution from necking. Therefore %elongation should be reported with original gauge length.

%Reduction is not affected by sample size, thus it is a better measure of ductility