Deriving Plasticity from Physics?Sethna, Markus Rauscher, Jean-Philippe Bouchaud, Yor Limkumnerd

• Yield Stress• Work Hardening

• Cell Structures• Pattern Formation

Shock Formation?

Hug

hes

et a

l.

Why?

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

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…

Equations of Motion

/ = h SSijij ij

Scalar Theory• = total dislocation density (includes + and -)• Most general equation of motion allowed by symmetry• Rate independent t→stress • 1st order in Sij=ij–kkij• 2nd order in gradients, • Ignore antidiffusion term• Yields 3D Burgers equation

Tensor Theory• Net dislocation density ij = ti

bj ()

(i = direction, j = Burgers vector)• Dislocations can’t end: i ij=0 → Current Jkl

• Peach-Koehler Force: J=D(4)

• Closure (4)ijkl=½(ikjl+iljk)

• (General law J = D/t = D

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

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 Sijij• 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

Stress-Strain Curves from Symmetry

ij/ = SSij ij G(S, , …)

ij/ = gSSijij (kl

• Assume strain in direction of applied deviatoric stress Sij

• General, nonlinear function G

• Second order in S, constant coefficients, 4th order in gradients, spatial average, one singular term dropped

Looks Good;Needs

4th orderGradient

Stress-Strain: Inset g=(1+S^2/2)

Scalar Theory: Bouchaud, Rauscher

Cell Wall FormationTensor 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.

Stress Free? Cell Walls!Tensor Theory: Yor Limkumnerd

Cell Wall, Grain Boundary:Dislocation Spacing dStress 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 Vinals, 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

Stress Free? Vector Order Parameter!Tensor Theory: Yor Limkumnerd

Dislocation density has six fields:Nine ijminus three: i ij = ki ij = 0

Stress-free dislocation densities have three independent components (Yor):ij(k) = (k) Eij

(k) Eij(k)

Eij

(Eij-kn nim jm)

A=(A1,A2,A3) 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

Cell Wall FormationTensor 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.