1. Deriving Plasticity from Physics? Sethna, Markus Rauscher, Jean-Philippe Bouchaud, Yor Limkumnerd Yield Stress Work Hardening Cell Structures Pattern Formation Shock Formation? Hughes et al. Why?

2. What’s Weird about Plasticity? Messy Atomic Scale Physics Messy Dislocation Physics Simple Cell Structures Dislocations Messy Dislocation Tangles Simple Cell Structures Simple at Macro-scale Sharp Yield Stress Yield point rises to previous maximum But

3. Why a Continuum Theory? Microscopic  Continuum Dislocation Junction Formation Too Many Dislocations Want Continuum Theory Smear over Details Explain Why Walls Form! Analogues Hydrodynamics, elasticity Surface growth Crackling noise Rival continuum theories: either Fancy math, no dynamics, or Explicit yield & work hardening, no pattern formation, or Pattern formation, no yield stress Our model: Pattern formation, cells Emergent Yield stress Work hardening Derivation from symmetry Condensed-Matter Approach Scalar now, tensor coming…

4. Equations of Motion S ij  /  = h S ij  i  j  Scalar Theory  = total dislocation density (includes + and -) Most general equation of motion allowed by symmetry Rate independent t→stress  1 st order in S ij =  ij –  kk  ij  2 nd order in gradients,  Ignore antidiffusion term Yields 3D Burgers equation Tensor Theory Net dislocation density  ij =   t i  b j   (  ) (i = direction, j = Burgers vector) Dislocations can’t end:  i  ij =0 → Current J kl Peach-Koehler Force: J=D  (4) Closure  (4) ijkl =½(  ik  jl +  il  jk ) (General law J = D   /  t =  D 

5. How to Get Irreversibility: Shocks! Shocks form at local minima in  Shocks introduce irreversibility On unloading, shocks smear On reloading, reversible until  max Work hardening! Yield stress =  max  x 1D Burgers equation Strain  (t) oscillates: loads  and unloads  Cusps form when   Cusps flatten when  Reversible on reloading t  Cusps Scalar Theory: Bouchaud, Rauscher

6. Shocks in 3D: Cell Walls Hughes et al. Al  =0.6 Perp to stretchParallel to stretch Shocks form walls in 3D Shocks separate cells Figures: contours of S ij  i  j  Like cells in stage III, IV Real cells refine (shrink)  -1/2 Our cells coarsen (grow)  1/2 (1D)  extends into cells? Good Incorrect: fix w/tensor theory? Markus Rauscher: Cactus, FFTW, CTC Windows

7. Stress-Strain Curves from Symmetry S ij  ij /  = S ij G(S ,   ,      …) S ij   ij /   = g  S ij (  k  l     Assume strain in direction of applied deviatoric stress S ij General, nonlinear function G Second order in S, constant coefficients, 4th order in gradients, spatial average, one singular term dropped Looks Good; Needs 4 th order Gradient Stress-Strain: Inset g=(1+S^2/2) Scalar Theory: Bouchaud, Rauscher

8. Cell Wall Formation Tensor Theory: Yor Limkumnerd Yor’s simulation from yesterday! Six components of  ij in a one-dimensional simulation. Still tentative. Higher dimensions, finite elements in progress.

9. Stress Free? Cell Walls! Tensor Theory: Yor Limkumnerd Cell Wall, Grain Boundary: Dislocation Spacing d Stress Confined to region of width d Continuum Dislocations: d ~ b goes to zero: STRESS FREE WALLS LED: Cell Walls “minimize” Stress Energy (D. Kuhlmann-Wilsdorf) Precise reformulation: Plastic deformations in continuum limit confined to zero stress configurations Rickman and Viñals, Linear theory:  ij decays to stress-free state Yor: Any stress-free state writable as (continuous) superposition of flat cell walls Circular cell writable as straight walls

10. Stress Free? Vector Order Parameter! Tensor Theory: Yor Limkumnerd Dislocation density has six fields: Nine  ij  minus three:  i  ij = k i  ij = 0 Stress-free dislocation densities have three independent components (Yor):  ij (k) =    (k) E ij    (k) E ij    (k) E ij  (E ij   -k n  nim  j  m ) A=(A 1,A 2,A 3 ) transforms like a vector field (rotation axis) Vector field A(r) for a cell boundary is a jump Explains variations in cells where no dislocations! Twist Boundary

11. Cell Wall Formation Tensor Theory: Yor Limkumnerd Yor’s simulation from yesterday! Six components of  ij in a one-dimensional simulation. Still tentative. Higher dimensions, finite elements in progress.