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Defects in Crystal Structure

- Vacancy, Interstitial, Impurity
- Schottky Defect
- Frenkel Defect
- Dislocations – edge dislocation, line, screw
- Grain Boundary

Interstitial Impurities

Vacancy Equilibrium

Xv~ exp(-Hv/kBT)

Defect Equilibrium

Sc= kBln gc(E)

Sb= kBln Wb Entropy

Ss= 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

Mechanical Properties of Solids

- Elastic deformation
- reversible
- Young’s Modulus
- Shear Modulus
- Bulk Modulus

- reversible
- Plastic Deformation
- irreversible
- change in shape of grains

- irreversible
- Rupture/Fracture

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

Mechanical PropertiesEffect of Structure on Mechanical Properties

- Elasticity
- Plastic Deformation
- Fracture

Elastic Deformation

- 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))

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

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

Microscopic Elastic Deformation

- Interatomic Forces
- FT =Tensile Force
- FC=Compressive Force
- Note F=-d(Energy)/dr

due to Shear

Plastic Deformation

Ao

- Poly Crystals
- by grain boundaries
- by slip on slip planes
- Engineering Stress, Ao
- True Stress, Ai

Ai

Movement at Edge Dislocation

Slip Plane is the plane on which the dislocation glides

Slip plane is defined by BV and I

Plastic Deformation -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

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

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

BV Perpendicular to Dislocation

BV parallel to Dislocation

Solution Hardening (Alloying) Multi-phase alloys - Volume fraction rule

- 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

Precipitation Hardening Methods to Produce

- Fine dispersion of heterogeneity
- impede dislocation motion
- c~2Gb/
- is the distance between particles

- c~2Gb/

- impede dislocation motion
- Particle Properties
- very small and well dispersed
- Hard particles/ soft metal matrix

- Oxidation of a metal
- Add Fibers - Fiber Composites

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)

Cracking vs Plastic Deformation
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