1. Structural Defects Mechanical Properties of Solids
Lecture 3.0
Structural Defects
Mechanical Properties of Solids

2. Defects in Crystal Structure
Vacancy, Interstitial, Impurity
Schottky Defect
Frenkel Defect
Dislocations – edge dislocation, line, screw
Grain Boundary

3. Substitutional Impurities
Interstitial Impurities

4. Self Interstitial
Vacancy
Xv~ exp(-Hv/kBT)

5. Vacancy Equilibrium
Xv~ exp(-Hv/kBT)

6. 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

7. Ionic Crystals
Shottky Defect
Frenkel Defect

8. Edge Dislocation

9. Grain Boundaries

10. Mechanical Properties of Solids
Elastic deformation
reversible
Young’s Modulus
Shear Modulus
Bulk Modulus
Plastic Deformation
irreversible
change in shape of grains
Rupture/Fracture

11. Modulii
Shear
Young’s
Bulk

12. Mechanical 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

13. Effect of Structure on Mechanical Properties
Elasticity
Plastic Deformation
Fracture

14. 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)

15. Microscopic Elastic Deformation
Interatomic Forces
FT =Tensile Force
FC=Compressive Force
Note F=-d(Energy)/dr

16. Plastic Deformation Single Crystal   by slip on slip planes
Shear Stress

17. Deformation of Whiskers
Without Defects
Rupture
With Defects
generated by high stress

18. Dislocation Motion
due to Shear

19. Slip Systems in Metals

20. Plastic Deformation Poly Crystals Ao by grain boundaries
by slip on slip planes
Engineering Stress, Ao
True Stress, Ai Ai

21. Movement at Edge Dislocation
Slip Plane is the plane on which the dislocation glides
Slip plane is defined by BV and I

22. 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

23. Strain 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

24. Depends on Grain Size

25. Burger’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

26. 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

27. Precipitation 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

28. Cracking 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)