Mechanical properties of materials

Is Hooke’s Law true?

How well does Hooke’sLaw describe thebehaviour of realmaterials?

Mechanical properties of materials

Rigorously, we define the Young’smodulus as

E = dσ /dε at zero stress (or strain).

This is the concept of stiffness: thesmall strain response of the material.

σ

ε

Mechanical properties of materials

Limits to IDEAL elastic behaviour?

A simple model suggests thatlattice deformations should beelastic to at least 20 per cent strain

Few materials show such behaviour

σ

ε

The essential feature of elastic behaviour is that the deformation is fully recoverable when the stress is removed. It is this reversibilitywhich is the defining characteristic of an elastic response.

If the behaviour is hookean, then the elastic response is also linear. However an elastic response may also be non-linear: that is to say,the stress is not proportional to the strain.

Mechanical properties of materialsRelation between interatomicpotentials and the elasticproperties

Lennard-Jones and Morsefunctions used to describehow the energy E of two atomsvaries with distance ofseparation r

E0 is minimum at equilibium;dE/dr is the force needed tochange separation

See a separate note on how the elastic modulus is determined at the level of the lattice.

Mechanical properties of materialsAt larger strains, various thingsmay happen:

1 Elastic response maintainedbut no longer linearin stress

σ

ε

Mechanical properties of materialsAt larger strains, various thingsmay happen:

1 Elastic response maintainedbut no longer linearin stress

σ

ε

Mechanical properties of materialsAt larger strains, various thingsmay happen:

2 Material YIELDS:maintains cohesion butdoes not recover initialstate on unloading

Combined ELASTIC and PLASTICresponse

σ

εεp

Mechanical properties of materialsAt larger strains, various thingsmay happen:

3 Material FAILS

σ

ε

!!σf

Mechanical properties of materials

Limits to IDEAL elastic behaviour?

A simple model based suggeststhat lattice deformations should beelastic to at least 20 per cent strain

Few materials show such behaviour

σ

ε

Large strain behaviour

Yield stress

From Callister 6eFig 6.10a

Large strain behaviour

Yield stress

From Callister 6eFig 6.10b

Large strain behaviourTensile strength

From Callister 6eFig 6.11

The internal structure of materialsTHE FUNDAMENTAL LATTICE

The surface ofGYPSUM atatomic resolution

Hall and Bosbach 2001Materials Science of Concretevol 6 pp 101-128

Mechanical properties of materials

Macro-slip in a single crystal

Diagram from Callister 6eFig 7.8

Mechanical properties of materials

Slip in a

From Callister 6eFigs 7.8, 7.9

Slip in a zincsingle crystalElam 1935

Mechanical properties of materials

Slip lines inpolycrystalline copperafter deformation

Brady NBS

From Callister 6eFig 7.10

Mechanical properties of materials

Point defects:vacancy andinterstitial type

From Callister 6eFig 4.1

Mechanical properties of materials

Point defects: impurity atomsin substitutional and interstitialsites

From Callister 6eFig 4.2

Mechanical properties of materials

Edge dislocation

From Callister 6eFig 4.3

Mechanical properties of materialsScrew dislocation

From Callister 6eFig 4.4

Mechanical properties of materials

Dislocations on thesurface of tungsten

From Callister 6eFig 4.6

Mechanical properties of materials

Slip lines inpolycrystalline copperafter deformation

Brady NBS

From Callister 6eFig 7.10

Mechanical properties of materialsEtch pits mark screwdislocations emerging atthe surface of a crystalof lithium fluoride LiF

Johnston GEC

From Callister 6ep 162

Mechanical properties of materialsEdge dislocation movesunder an applied shearstress

From Callister 6eFig 7.1

Mechanical properties of materialsDislocation movement:the caterpillar analogy

From Callister 6eFig 7.3

Mechanical properties

Small strain: Elastic modulus or complianceBulk and shearTensile (Young’s) modulusPoisson ratio

Large strain:Yield stress Tensile strengthCompressive strength

Do these properties capture ALL the things we need to know about the mechanical behaviour of materials?

Do these properties capture ALL the things we need to know about the mechanical behaviour of materials?

Friction?

Do these properties capture ALL the things we need to know about the mechanical behaviour of materials?

Friction?

Hardness?

FRICTION

Amontons’ law F = µ LF

L

µ coefficient of friction

FRICTION

Amontons’ law F = µ LF

L

PTFE teflon 0.04Natural rubber 3.0

µ coefficient of friction

FRICTION

Amontons’ law F = µ LF

L

... but law is not generally true!!

Mechanical properties of materialsHardnessresistance to point load, indentation damage

Mechanical properties of materialsHardness

From Callister 6eTable 6.4

Brinell Hardness

From Callister 6eTable 6.4 - zoom

TOUGHNESS

Complex property

roughly speaking, tough is opposed to brittle

Toughness is linked with plasticity

A simple measure of toughness The area under the stress-strain curve to failure

Materials Science and Engineering

PART II

We shall cut the cake a different way and look in turnat each of the major classes of materials

METALSPOLYMERS CERAMICS

Materials Science and Engineering

PART II

We shall cut the cake a different way and look in turnat each of the major classes of materials

METALSPOLYMERS CERAMICS

COMPOSITES