Wing Kam Liu* and Wei Chen, Northwestern University ... - SAMSI

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Stochastic Formulation for Multiresolution Continua Solid Mechanics ( v) ⎡ + �( ) � � ⎤( v) ⎡T + S + S ⎤ = −T N f + f g h ⎣ Z⎦ ⎣ Z ⎦ Intrinsic stress ( ) σ · ∇−Σ: ∇⊗∇ + f = � γ ρ ρ ρ S · ∇−s + B = Γ � , ρ = 1, 2, …, N Σ · n = 0 σ · n−Σ: ∇⊗n S ρ Macro double stress Spatial operator Boundary conditions ( ) body force = t · n = T, ρ = 1,2, …, N acceleration Temporal operator ( ( t ) ( t ) ( t ) ) n n n n σβ , , β = f , f , f sym ∇v x, , Z , L x, , Z −L, ∇L x, , Z σ Z( s) x1 β β Random dimension Z x3 Z() i t1 x1 VELOCITY x2 x3 (1) t1 x2 x1 t2 x3 RANDOM SOLUTIONS t1 x2 t2 MICRO-VELOCITY GRADIENT t2 In stochastic collocation, Galerkin projection, random dimension discretized like space and time Constitutive law an implicit function of a denumerable set of random variables
Stochastic Constitutive Theory Denumerable set of random variables are structural descriptors of microstructure Unique stress distributions averaged to provide constitutive behavior MACRO-SCALE SIMULATION Greene, M.S., Y. Liu, et al. (2011). SVE l δ =
Page 1 and 2: Mathematical Science Methodologies
Page 3 and 4: Multiresolution Modeling Scope: Pro
Page 5 and 6: Mathematical Science Foundation, Co
Page 7 and 8: Multiresolution Concept: zoom into
Page 9 and 10: High Strength Alloy Analysis & Unce
Page 11 and 12: Equations of Archetype Multiresolut
Page 13 and 14: More on the Theory: Motion Assumpti
Page 15 and 16: More on the Theory: Variational Sta
Page 17 and 18: Three Invariants (Strain rates, Inh
Page 19 and 20: Illustrative Example: Two-Scale Ela
Page 21 and 22: Two-Scale Concurrent Multiresolutio
Page 23 and 24: Summary: comparison with experiment
Page 25 and 26: Animation of the Fracture Process I
Page 27 and 28: Northwestern University Materials D
Page 29 and 30: Uncertainty Sources and Quantificat
Page 31: Stochastic Mathematical Formulation
Page 35 and 36: Desired Features Multiresolution Mi

Wing Kam Liu* and Wei Chen, Northwestern University ... - SAMSI

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