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Lecture 3.0. Structural Defects Mechanical Properties of Solids. Defects in Crystal Structure. Vacancy, Interstitial, Impurity Schottky Defect Frenkel Defect Dislocations – edge dislocation, line, screw Grain Boundary. Substitutional Impurities Interstitial Impurities. Self Interstitial

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lecture 3 0

Lecture 3.0

Structural Defects

Mechanical Properties of Solids

defects in crystal structure
Defects in Crystal Structure
  • Vacancy, Interstitial, Impurity
  • Schottky Defect
  • Frenkel Defect
  • Dislocations – edge dislocation, line, screw
  • Grain Boundary

Substitutional Impurities

Interstitial Impurities


Self Interstitial


Xv~ exp(-Hv/kBT)

vacancy equilibrium
Vacancy Equilibrium

Xv~ exp(-Hv/kBT)

defect equilibrium
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)


For large crystals dSs<<dSb


\n ~ N exp –dFc/kBT


Ionic Crystals

Shottky Defect

Frenkel Defect

mechanical properties of solids
Mechanical Properties of Solids
  • Elastic deformation
    • reversible
      • Young’s Modulus
      • Shear Modulus
      • Bulk Modulus
  • Plastic Deformation
    • irreversible
      • change in shape of grains
  • Rupture/Fracture




mechanical properties
Stress, xx= Fxx/A

Shear Stress, xy= Fxy/A


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 Properties
effect of structure on mechanical properties
Effect of Structure on Mechanical Properties
  • Elasticity
  • Plastic Deformation
  • Fracture
elastic deformation
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
Microscopic Elastic Deformation
  • Interatomic Forces
  • FT =Tensile Force
  • FC=Compressive Force
  • Note F=-d(Energy)/dr
plastic deformation
Plastic Deformation

  • Single Crystal
    • by slip on slip planes

Shear Stress

deformation of whiskers
Deformation of Whiskers

Without Defects


With Defects

generated by high stress


Dislocation Motion

due to Shear

plastic deformation1
Plastic Deformation


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


movement at edge dislocation
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
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 work hardening
Dislocation Movement forms dislocation loops

New dislocations created by dislocation movement

Critical shear stress that will activate a dislocation source


G=Shear Modulus

b=Burgers Vector

l=length of dislocation segment

Strain Hardening/Work Hardening

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

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 hardening
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
cracking vs plastic deformation

Poor dislocation motion

stress needed to initiate a crack is low

Ionic Solids

disrupt charges

Covalent Solids

disrupt bonds

Amorphous solids

no dislocations


good dislocation motion

stress needed to initiate slip is low


electrons free to move

Depends on T and P

ductile at high T (and P)

Cracking vs Plastic Deformation