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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.