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Multiscale Modeling

Multiscale Modeling. 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?.

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Multiscale Modeling

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  1. Multiscale Modeling • 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? Questions for the Mathematicians

  2. Transitions between Scales Microphysics: Atoms, Grains, Defects… Numerics: Finite Element/Diff, Galerkin… Match? Continuum Laws Defect Dynamics Multiscale Modeling Coupled system: continuum and defects. Defect properties, evolution determined by gradients in continuum fields. For a given continuum law, what can we deduce about the defect laws?

  3. Deducing Defect Laws In the space of all reasonable microscopic systems (numerical implementations, regularizations) consistent with a given continuum law, what defect laws can emerge? • Extracting defect laws (Activated 2D Dislocation Glide): • Complete Picture • Velocity explicitly calculated from local stress fields • Environmental Impact and Dependence, Functional Forms • (2) Guessing defect laws (2D Crack Growth): • Velocity assumed dependent on local stress fields • Symmetry and analyticity assumptions yield form of law • (3) Laws from the continuum? (Faceting in etched Silicon) • Shock evolution law? • Viscosity solution disagrees with experiment More specific formulation

  4. Extracting Laws Thermally activated glide Glide slides planes of atoms, v×b=0 Barrier ~ midway between equilibria External stress s Velocity ~ v0(s) exp(-EB(s)/kBT) Dislocation Glide: Nick Bailey How fast will the dislocations move, given an external stress tensor s? What is the barrier EB(s) and prefactorv0(s)? Dislocation Edge of Missing Row Burgers Vector b

  5. Environmental Impact, Dependence General solution to continuum theory expandable in multipoles ui(r) =  r n M[n]i(q,f) • Environmental Impact: • n=0, logs, arctan: Dislocation displacement field s b • n=-1: Volume change, elastic dipole due to dislocation • n=-2, … Near-field corrections • Controls interaction between defects • Environmental Dependence: • n=1: External stress s • n=2, 3, … Boundary conditions, interfaces • Multipole expansions for arbitrary continua? Dislocation Glide, Nick Bailey

  6. Finding Functional Forms Symmetries: Inverting Stress EB(sxy) = EB(-sxy) – a2sxy Singularities: Saddle-Node Transition EB(sxy) = c3/2 (sc –sxy)3/2+ c5/2 (sc –sxy)5/2… Physical Model: Sinusoidal Potential + Corrections EB sxx sxy Dislocation Glide, Nick Bailey Fit to Physical Functional Form (Ballistic) EB(sxx, syy, sxy) = -(a2/2) sxy + (a2sc/p) (arcsin(sxy/sc) + Sn An(1-(sxy/sc)2)n+1/2) sc sxy Taylor Series for sc, A1, A2: Nine Parameters Total Fits Entire Range (Nine Measurements or DFT Calculations!) sxx

  7. Guessing Laws Environmental Dependence Solution of Elasticity with Cut: Three terms with s r-1/2 Stress Intensity Factors K I,K II, K III • Mode I: Crack Opening • Mode II: Shearing • Mode III: Twisting Crack Growth Laws: Jennifer Hodgdon How fast will the crack grow, given an external stress tensor s? What direction will it grow?

  8. Guessing Laws Ingraffea: FEM Given current shape, force Finds stress intensities KI, KII, KIII Wants Direction q(or n)and Velocity v of Growth 2v /f KI Crack Growth Laws: Jennifer Hodgdon Symmetry Implies: dX/dt = v(KI, KII2) n dn/dt = -f(KI, KII2) KII b How big is the decay length 2v /f KI? Length set by microscopic scale of material: grain size, nonlinear zone size, atom size dn/dt: bodd, needs odd KII Doesn’t turn if KII=0 Crack turns abruptly until KII=0 (Principle of Local Symmetry) Cotterell and Rice: KII = KIDq/2 Dq ~ exp[(-f KI /2v) x]

  9. Is Analyticity Guaranteed? 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 behavior Fatigue fracture: large region, long times, history Brittle fracture: OK! Crack Growth Laws: Jennifer Hodgdon Abraham, Duchaineau, and De La Rubia Billion atoms of copper Too small to see nonlinear zone! Restrictions to exclude ductile fracture would be prudent, acceptable.

  10. Laws from the Continuum? Etching rate jumps are associated with a faceting transition Etching rate has cusps at low-index surfaces Faceting in etched Silicon Melissa Hines, Rik Wind, Markus Rauscher CACTUS, FFTW CCMR, Microsoft First-order: nucleation

  11. Which shock evolution law? Faceting in etched Silicon Melissa Hines, Rik Wind, Markus Rauscher Continuum law:  h / t = [vn = etch rate (q,f)] 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?

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