Lecture 3.0Lecture 3.0

Structural Defects

Mechanical Properties of Solids

Defects in Crystal StructureDefects in Crystal Structure

Vacancy, Interstitial, ImpuritySchottky DefectFrenkel DefectDislocations – edge dislocation, line,

screwGrain Boundary

Substitutional ImpuritiesInterstitial Impurities

Self InterstitialVacancy

Xv~ exp(-Hv/kBT)

Vacancy EquilibriumVacancy Equilibrium

Xv~ exp(-Hv/kBT)

Defect EquilibriumDefect EquilibriumSc= kBln gc(E)

Sb= kBln Wb EntropySs= kBln Ws

dFc = dE-TdSc-TdSs, the change in free energy

dFc ~ 6 nearest neighbour bond energies (since break on average 1/2 the bonds in the

surface)

Wb=(N+n)!/(N!n!) ~(N+n+1)/(n+1) ~(N+n)/n (If one vacancy added)

dSb=kBln((N+n)/n) 

For large crystals dSs<<dSb

  n ~ N exp –dFc/kBT

Shottky DefectFrenkel Defect

Ionic Crystals

Edge Dislocation

Grain BoundariesGrain Boundaries

Mechanical Properties of SolidsMechanical Properties of Solids

Elastic deformation– reversible

• Young’s Modulus• Shear Modulus• Bulk Modulus

Plastic Deformation– irreversible

• change in shape of grains

Rupture/Fracture

ModuliiModulii

Young’s

Shear

Bulk

Mechanical PropertiesMechanical Properties

Stress, xx= Fxx/A

Shear Stress, xy= Fxy/A

Compression

Yield Stress yield ~Y/10

yield~G/6 (theory-all

atoms to move together)

Strain, =x/xo

Shear Strain, =y/xo

Volume Strain = V/Vo

Brittle Fracture– stress leads to crack– stress concentration at crack tip

=2(l/r)– Vcrack= Vsound

Effect of Structure on Effect of Structure on Mechanical PropertiesMechanical PropertiesElasticityPlastic DeformationFracture

Strain

Str

ess Plastic

Deformation

Fracture

Elastic DeformationElastic Deformation

Pulling on a wire decreases its diameter l/lo= -l/Ro

Poisson’s Ratio, 0.5 (liquid case=0.5)

Young’s Modulus– Y(or E)= (F/A)/(l/lo)

Shear Modulus– G=/= Y/(2(1+))

Bulk Modulus• K=-P/(V/Vo)

• K=Y/(3(1-2))

Microscopic Elastic DeformationMicroscopic Elastic Deformation

Interatomic Forces FT =Tensile Force

FC=Compressive Force

Note F=-d(Energy)/dr

Forc

e

0

FC

FT

r

ao

Repulsion

Attraction

Plastic DeformationPlastic Deformation

Single Crystal– by slip on slip

planes

30/

)cos(cos

coscoscos/

cos

max

Gstressshearyield

Yielding

a

A

o

oyield

Shear Stress

Deformation of WhiskersDeformation of Whiskers

Without DefectsRupture

With Defectsgenerated by high stress

Poly Crystalline CopperPoly Crystalline Copper

Dislocation Motiondue to Shear

Slip Systems in MetalsSlip Systems in Metals

CrystalStructure

Slip Planes SlipDirections

Number ofSlipSystems

Examples

fcc {111} <1-10> 12 Al, Cu, Nibcc {110}

{211}{321}

<-111><-111><-111>

121224

Fe,Ta,W

hcp {0001}{10-10}{10-11}

<11-20><11-20><11-20>

336

Be, Mg,Zn,Ti, Zr, Re

Plastic DeformationPlastic Deformation

Poly Crystals– by grain boundaries

– by slip on slip planes

– Engineering Stress, Ao

– True Stress, Ai

ooii

oyield

lAlA

sizegraind

dk

Ao

Ai

Movement at Edge DislocationMovement at Edge Dislocation

Slip Plane is the plane on which the dislocation glides

Slip plane is defined by BV and I

Plastic DeformationPlastic Deformation-Polycrystalline sample-Polycrystalline sample

Many slip planes – large amount of

slip (elongation)

Strain hardening– Increased difficulty of

dislocation motion due to dislocation density

– Shear Stress to Maintain plastic flow, =o+Gb

• dislocation density,

Strain Hardening

Strain HardeningStrain Hardening/Work Hardening/Work Hardening

Dislocation Movement forms dislocation loops– New dislocations

created by dislocation movement

Critical shear stress that will activate a dislocation source

c~2Gb/l– G=Shear Modulus

– b=Burgers Vector

– l=length of dislocation segment

Depends on Grain SizeDepends on Grain Size

Burger’s Vector-Burger’s Vector-Dislocations are characterised by their Dislocations are characterised by their Burger's vectors.Burger's vectors.  These   These represent the 'represent the 'failure closure'failure closure' in a Burger's circuit in imperfect (top) in a Burger's circuit in imperfect (top) and perfect (bottom) crystal.and perfect (bottom) crystal.

BV Perpendicular to DislocationBV parallel to Dislocation

Solution Hardening (Alloying)Solution Hardening (Alloying)

Solid Solutions• Solute atoms segregate to dislocations =

reduces dislocation mobility• higher required to move dislocation

– Solute Properties• larger cation size=large lattice strain• large effective elastic modulus, Y

Multi-phase alloys - Volume fraction rule

Precipitation HardeningPrecipitation Hardening

Fine dispersion of heterogeneity• impede dislocation motion

c~2Gb/ is the distance between particles

– Particle Properties• very small and well dispersed• Hard particles/ soft metal matrix

Methods to Produce– Oxidation of a metal– Add Fibers - Fiber Composites

Cracking vs Plastic DeformationCracking vs Plastic Deformation

Brittle• Poor dislocation motion• stress needed to initiate

a crack is low

– Ionic Solids• disrupt charges

– Covalent Solids• disrupt bonds

– Amorphous solids• no dislocations

Ductile• good dislocation motion

• stress needed to initiate slip is low

– Metals• electrons free to move

Depends on T and P– ductile at high T (and P)