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 X v ~ exp(-  H v /k B T)

Vacancy Equilibrium X v ~ exp(-  H v /k B T)

Defect Equilibrium S c = k B ln g c (E) S b = k B ln W b Entropy S s = k B ln W s dF c = dE-TdS c -TdS s, the change in free energy dF c ~ 6 nearest neighbour bond energies (since break on average 1/2 the bonds in the surface) W b =(N+n)!/(N!n!) ~(N+n+1)/(n+1) ~(N+n)/n (If one vacancy added) dS b =k B ln((N+n)/n) For large crystals dS s <<dS b   n ~ N exp –dF c /k B T

Shottky Defect Frenkel Defect Ionic Crystals

Edge Dislocation

Grain Boundaries

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 Young’s Shear Bulk

Mechanical Properties Stress,  xx = F xx /A Shear Stress,  xy = F xy /A Compression Yield Stress –  yield ~Y/10 –  yield ~G/6 (theory -all atoms to move together ) Strain,  =  x/x o Shear Strain,  =  y/x o Volume Strain =  V/V o Brittle Fracture –stress leads to crack –stress concentration at crack tip =2  (l/r) –Vcrack= Vsound

Effect of Structure on Mechanical Properties Elasticity Plastic Deformation Fracture

Elastic Deformation Pulling on a wire decreases its diameter –  l/l o = -  l/R o Poisson’s Ratio,  0.5 (liquid case=0.5) Young’s Modulus –Y(or E)= (F/A)/(  l/l o ) Shear Modulus –G=  /  = Y/(2(1+ )) Bulk Modulus K=-P/(  V/V o ) K=Y/(3(1-2 ))

Microscopic Elastic Deformation Interatomic Forces F T =Tensile Force F C =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

Poly Crystalline Copper

Dislocation Motion due to Shear

Slip Systems in Metals

Plastic Deformation Poly Crystals –by grain boundaries –by slip on slip planes –Engineering Stress, A o –True Stress, A i AoAo AiAi

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

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

Depends on Grain Size

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

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)