Plastic Deformation Permanent, unrecovered mechanical deformation s = F/A stress Deformation by dislocation motion, “glide” or “slip” Dislocations Edge, screw, mixed Defined by Burger’s vector Form loops, can’t terminate except at crystal surface Slip system Glide plane + Burger’s vector maximum shear stress

Crystallography of Slip Slip system = glide plane + burger’s vector Correspond to close-packed planes + directions Why? Fewest number of broken bonds Cubic close-packed Closest packed planes {1 1 1} 4 independent planes Closest packed directions Face diagonals <1 1 0> 3 per plane (only positive) 12 independent slip systems a1 a2 a3 b = a/2 <1 1 0> | b | = a/2 [1 1 0]

HCP “BCC” b = a/2 <1 1 1> | b | = 3a/2 b = a <1 0 0> Planes {0 0 1} 1 independent plane Directions <1 0 0> 3 per plane (only positive) 3 independent slip systems Planes {1 1 0} 6 independent planes Directions <1 1 1> 2 per plane (only positive) 12 independent slip systems Occasionally also {1 1 2} planes in “BCC” are slip planes Diamond structure type: {1 1 1} and <1 1 0> --- same as CCP, but slip less uncommon

    Why does the number of independent slip systems matter? Are any or all or some of the grains in the proper orientation for slip to occur? s = F/A HCP     CCP maximum shear stress Large # of independent slip systems in CCP  at least one will be active for any particular grain True also for BCC Polycrystalline HCP materials require more stress to induce deformation by dislocation motion

Dislocations in Ionic Crystals like charges touch 2 1 like charges do not touch long burger’s vector compared to metals compare possible slip planes (1) slip causes like charges to touch (2) does not cause like charges to touch

Energy Penalty of Dislocations bonds are compressed E R0 R compression tension Energy / length  |b|2 Thermodynamically unfavorable Strong interactions bonds are under tension attraction  annihilation repulsion  pinning Too many dislocations  become immobile

Summary Now on to elastic deformation Materials often deform by dislocation glide Deforming may be better than breaking Metals CCP and BCC have 12 indep slip systems HCP has only 3, less ductile |bBCC| > |bCCP|  higher energy, lower mobility CCP metals are the most ductile Ionic materials/Ceramics Dislocations have very high electrostatic energy Deformation by dislocation glide atypical Covalent materials/Semiconductors Dislocations extremely rare Now on to elastic deformation

Elastic Deformation Connected to chemical bonding Stretch bonds and then relax back Recall bond-energy curve Difficulty of moving from R0 Curvature at R0 Elastic constants (stress) = (elastic constant) * (strain) stress and strain are tensors  directional the elastic constant being measured depends on which component of stress and of strain  R0 R E

Elastic Constants F A0 l0 s (stress) e (strain) Y: Young’s modulus (sometimes E) F stress = uniaxial, normal stress A0 material elongates: l0  l strain = elongation along force direction l0 observation: s (stress) Y e (strain) material thins/necks: A0  Ai true stress: use Ai; engineering stress: use A0

Elastic Constants Connecting Young’s Modulus to Chemical Bonding R0 R0 R E F = k DR  k*(length) = Y stress*area strain*length Coulombic attraction want k in terms of E, R0 observed within some classes of compounds

Elastic Constants Bulk Modulus, K apply hydrostatic pressure s = -P P = F/A s = -P measure change in volume linear response Can show: analogous to Young’s modulus Coulombic: Useful relationship:

Elastic Constants Poisson’s ratio, n apply uniaxial stress s = F/A F measure e|| - elongation parallel to force measure e - thinning normal to force e|| e Rigidity (Shear) Modulus, G apply shear stress t = F/A A Dl measure shear strain F f = tanf  f l0 y F x

Elastic Constants General Considerations Stress, s: 3  3 symmetric tensor  6 parameters Strain, e: 3  3 symmetric tensor  6 parameters In principle, each and every strain parameter depends on each and every stress parameter  36 elastic constants Some are redundant  21 independent elastic constants in the most general case Material symmetry  some are zero, some are inter-related Isotropic material  only 2 independent elastic constants normal stress  only normal deformation shear stress  only shear deformation Cubic material  G, Y and n are independent