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## Lecture 3.0

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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**Vacancy Xv~ exp(-Hv/kBT)**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**Ionic Crystals**Shottky Defect Frenkel Defect**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**Shear Young’s Bulk**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**• 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**Plastic Deformation** • Single Crystal • by slip on slip planes Shear Stress**Deformation of Whiskers**Without Defects Rupture With Defects generated by high stress**Dislocation Motion**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 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)**• 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**• 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**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