Multiscale ModelingQuestions for the Mathematicians

• For a given continuum law, what can we deduce about the defect laws?

(If time permits):• Fits of emergent theories are often sloppy – the parameters are not well determined by the data. Can we explain the characteristic common features of these sloppy models?

Transitions between ScalesMultiscale Modeling

Microphysics:Atoms, Grains,

Defects…

Numerics:Finite Element/Diff,

Galerkin…

Continuum LawsDefect Dynamics

Match?

For a given continuum law, what can we deduce about the defect laws?

Coupled system: continuum and defects. Defect properties, evolution determined by gradients in continuum fields.

Deducing Defect LawsMore specific formulation

(1) Extracting defect laws (Activated 2D Dislocation Glide):Complete PictureVelocity explicitly calculated from local stress fieldsEnvironmental Impact and Dependence, Functional Forms

(2) Guessing defect laws (2D Crack Growth): Velocity assumed dependent on local stress fieldsSymmetry and analyticity assumptions yield form of law

(3) Laws from the continuum? (Faceting in etched Silicon)Shock evolution law?Viscosity solution disagrees with experiment

In the space of all reasonable microscopic systems (numerical implementations, regularizations) consistent with a given continuum law, what defect laws can emerge?

Extracting LawsDislocation Glide: Nick Bailey

DislocationEdge of Missing Row

Burgers Vector b

Thermally activated glide Glide slides planes of atoms, v×b=0

Barrier ~ midway between equilibriaExternal stress Velocity ~ v0() exp(-EB()/kBT)

How fast will the dislocations move, given an external stress tensor ? What is the barrier EB() and prefactor v0()?

Environmental Impact, DependenceDislocation Glide, Nick Bailey

General solution to continuum theory expandable in multipoles

ui(r) = r n M[n]i

Environmental Impact:• n=0, logs, arctan: Dislocation displacement field b• n=-1: Volume change, elastic dipole due to dislocation• n=-2, … Near-field correctionsControls interaction between defects

Environmental Dependence:• n=1: External stress • n=2, 3, … Boundary conditions, interfaces

Multipole expansions for arbitrary continua?

Finding Functional FormsDislocation Glide, Nick Bailey

EB(xx, yy, xy) = -(a2/2) xy + (a2 c/)

(arcsin(xy/c) + n An (1-(xy/c)2)n+1/2)

Symmetries: Inverting Stress EB(xy) = EB(-xy) – a2 xy Singularities: Saddle-Node Transition EB(xy) = c3/2 (c –xy)3/2+ c5/2 (c –xy)5/2…Physical Model:

Sinusoidal Potential + Corrections

Fit to Physical Functional Form

Taylor Series for c, A1, A2: Nine Parameters Total Fits Entire Range(Nine Measurements or DFT Calculations!)

EB

xy

xx

xy

xx

c

(Ballistic)

Guessing LawsCrack Growth Laws: Jennifer Hodgdon

Solution of Elasticity with Cut: Three terms with r-1/2

Stress Intensity Factors K I,K II, K III

Environmental Dependence

• Mode I: Crack Opening• Mode II: Shearing• Mode III: Twisting

How fast will the crack grow, given an external stress tensor ? What direction will it grow?

Guessing LawsCrack Growth Laws: Jennifer Hodgdon

Ingraffea: FEMGiven current shape,

forceFinds stress intensities

KI, KII, KIII

Wants Direction (or n) and Velocity v of GrowthSymmetry Implies:

dX/dt = v(KI, KII2) n

dn/dt = -f(KI, KII2) KII b

dn/dt: b odd, needs odd KII Doesn’t turn if KII=0

Cotterell and Rice: KII = KI /2exp[f KI /2v) x]

How big is the decay length 2v /f KI? Length set by microscopic

scale of material: grain size, nonlinear zone size, atom size

Crack turns abruptly until KII=0(Principle of Local Symmetry)

2v /f KI

Is Analyticity Guaranteed?Crack Growth Laws: Jennifer Hodgdon

Abraham, Duchaineau, and De La RubiaBillion atoms of copper

Too small to see nonlinear zone!

Landau theory assumes power series: analyticity. Analyticity natural for finite systems, time t<, temp. T>0(Else critical points, bifurcations, power laws)Ductile fracture: large region around crack tip: collective behaviorFatigue fracture: large region, long times, historyBrittle fracture: OK!

Restrictions to exclude ductile fracture would be prudent, acceptable.

Laws from the Continuum?Faceting in etched Silicon

Melissa Hines, Rik Wind, Markus Rauscher

Etching rate has cusps at low-index surfaces

Etching rate jumps are associated with

a faceting transition

First-order: nucleationCACTUS, FFTWCCMR, Microsoft

Which shock evolution law?Faceting in etched Silicon

Melissa Hines, Rik Wind, Markus Rauscher

Continuum law: h / t = [vn = etch rate ()]Forms facets in finite time: how to evolve thereafter?“Viscosity solution” flattens. Experimental facets persist!Energy anisotropy can affect evolution at cusps (Watson)Math: What shock evolution laws can emerge?Experiment: What do we need to measure?Numerics: How do we implement them?