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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


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Substitutional Impurities

Interstitial Impurities


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Self Interstitial

Vacancy

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)

dSb=kBln((N+n)/n) 

For large crystals dSs<<dSb

\

\n ~ N exp –dFc/kBT


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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


Modulii
Modulii

Shear

Young’s

Bulk


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

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

Rupture

With Defects

generated by high stress



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Dislocation Motion

due to Shear



Plastic deformation1
Plastic Deformation

Ao

  • Poly Crystals

    • by grain boundaries

    • by slip on slip planes

    • Engineering Stress, Ao

    • True Stress, Ai

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


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

Strain Hardening/Work Hardening



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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

    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)

    Cracking vs Plastic Deformation