Multiscale Modeling Questions for the Mathematicians • For a given continuum law, what can we deduce about the defect laws? (If time permits): • Fits.
Download ReportTranscript Multiscale Modeling Questions for the Mathematicians • For a given continuum law, what can we deduce about the defect laws? (If time permits): • Fits.
Multiscale Modeling Questions 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 Scales Multiscale Modeling Microphysics: Atoms, Grains, Defects… Match? Numerics: Finite Element/Diff, Galerkin… Continuum Laws Defect Dynamics 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? Deducing Defect Laws More specific formulation In the space of all reasonable microscopic systems (numerical implementations, regularizations) consistent with a given continuum law, what defect laws can emerge? (1) 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 Extracting Laws Dislocation Glide: Nick Bailey 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 Edge of Missing Row Burgers Vector b How fast will the dislocations move, given an external stress tensor s? What is the barrier EB(s) and prefactor v0(s)? Environmental Impact, Dependence Dislocation Glide, Nick Bailey 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? Finding Functional Forms Dislocation Glide, Nick Bailey Symmetries: Inverting Stress EB(sxy) = EB(-sxy) – a2 sxy sxx Singularities: Saddle-Node Transition EB(sxy) = c3/2 (sc –sxy)3/2+ c5/2 (sc –sxy)5/2… Physical Model: Sinusoidal Potential + Corrections Fit to Physical Functional Form EB(sxx, syy, sxy) = -(a2/2) sxy + (a2 sc/p) (arcsin(sxy/sc) + Sn An (1-(sxy/sc)2)n+1/2) Taylor Series for sc, A1, A2: Nine Parameters Total Fits Entire Range (Nine Measurements or DFT Calculations!) EB sxy (Ballistic) sc sxy sxx Guessing Laws Crack Growth Laws: Jennifer Hodgdon 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 How fast will the crack grow, given an external stress tensor s? What direction will it grow? Guessing Laws Crack Growth Laws: Jennifer Hodgdon Ingraffea: FEM Given current shape, force Finds stress intensities KI, KII, KIII Wants Direction q (or n) and Velocity v of Growth Symmetry 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 Dq/2 Dq ~ exp[(-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) Is Analyticity Guaranteed? Crack Growth Laws: Jennifer Hodgdon 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! 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. Laws from the Continuum? Faceting in etched Silicon Melissa Hines, Rik Wind, Markus Rauscher Etching rate jumps Etching rate are associated with has cusps at a faceting low-index transition surfaces First-order: nucleation CACTUS, FFTW CCMR, Microsoft 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?