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?