Variational methods for rate- and state-dependent friction problems - PowerPoint PPT Presentation

References Variational methods for rate- and state-dependent friction problems Elias Pipping 1 Oliver Sander 2 Ralf Kornhuber 1 1 Free University Berlin: Institute for Mathematics 2 RWTH Aachen University: Institute for Geometry and Practical Mathematics 28th of November 2013 Variational methods for rate- and state-dependent friction problems E. Pipping, O. Sander, R. Kornhuber

References Experimental background velocity coefficient of friction V 2 µ ss ( V 1 ) V 1 µ ss ( V 2 ) time time Figure : System response to jump in velocity (after steady-state sliding) Variational methods for rate- and state-dependent friction problems E. Pipping, O. Sander, R. Kornhuber

References Phenomenological Law Abstract setting θ = ˙ ˙ µ = µ ( V , θ ) , θ ( θ, V ) Variational methods for rate- and state-dependent friction problems E. Pipping, O. Sander, R. Kornhuber

References Phenomenological Law Abstract setting θ = ˙ ˙ µ = µ ( V , θ ) , θ ( θ, V ) Prominent example µ ( V , θ ) = µ ∗ + a log V θ θ ˙ + b log , θ ( θ, V ) = 1 − L / V ∗ L / V V ∗ Variational methods for rate- and state-dependent friction problems E. Pipping, O. Sander, R. Kornhuber

References Phenomenological Law Abstract setting θ = ˙ ˙ µ = µ ( V , θ ) , θ ( θ, V ) Prominent example µ ( V , θ ) = µ ∗ + a log V θ θ ˙ + b log , θ ( θ, V ) = 1 − L / V ∗ L / V V ∗ Properties: coefficient of friction • ˙ θ = 0 iff θ = θ ss ( V ) := L / V µ ss ( V 1 ) µ ss ( V 2 ) time Variational methods for rate- and state-dependent friction problems E. Pipping, O. Sander, R. Kornhuber

References Phenomenological Law Abstract setting θ = ˙ ˙ µ = µ ( V , θ ) , θ ( θ, V ) Prominent example µ ( V , θ ) = µ ∗ + a log V θ θ ˙ + b log , θ ( θ, V ) = 1 − L / V ∗ L / V V ∗ Properties: coefficient of friction • ˙ θ = 0 iff θ = θ ss ( V ) := L / V • µ ss ( V ) := µ ( V , θ ss ( V )) = µ ∗ + ( a − b ) log V V ∗ µ ss ( V 1 ) µ ss ( V 2 ) time Variational methods for rate- and state-dependent friction problems E. Pipping, O. Sander, R. Kornhuber

References Phenomenological Law Abstract setting θ = ˙ ˙ µ = µ ( V , θ ) , θ ( θ, V ) Prominent example µ ( V , θ ) = µ ∗ + a log V θ θ ˙ + b log , θ ( θ, V ) = 1 − L / V ∗ L / V V ∗ Properties: coefficient of friction • ˙ θ = 0 iff θ = θ ss ( V ) := L / V • µ ss ( V ) := µ ( V , θ ss ( V )) = µ ∗ + ( a − b ) log V V ∗ • µ ss ( V 2 ) − µ ss ( V 1 ) = ( a − b ) log V 2 V 1 µ ss ( V 1 ) µ ss ( V 2 ) time Variational methods for rate- and state-dependent friction problems E. Pipping, O. Sander, R. Kornhuber

References Phenomenological Law Abstract setting θ = ˙ ˙ µ = µ ( V , θ ) , θ ( θ, V ) Prominent example µ ( V , θ ) = µ ∗ + a log V θ θ ˙ + b log , θ ( θ, V ) = 1 − L / V ∗ L / V V ∗ Properties: coefficient of friction • ˙ θ = 0 iff θ = θ ss ( V ) := L / V • µ ss ( V ) := µ ( V , θ ss ( V )) = µ ∗ + ( a − b ) log V V ∗ • µ ss ( V 2 ) − µ ss ( V 1 ) = ( a − b ) log V 2 V 1 µ ss ( V 1 ) Interpretation: µ ss ( V 2 ) • time scale: L / V , regularisation from µ = µ ( V ) . time Variational methods for rate- and state-dependent friction problems E. Pipping, O. Sander, R. Kornhuber

References Opinions What geoscientists think • Widely applicable (e.g. wood/rock, pulverised fault gouge); can even be used for quantitative reproduction of data. • Cumbersome. What mathematicians think • µ ( V , θ ) monotone in V ( � convex energies, etc.) • ˙ θ ( V , θ ) gradient flow for fixed V . • aside: needs slight modifications to be meaningful ( µ ≥ 0): µ � µ s . Variational methods for rate- and state-dependent friction problems E. Pipping, O. Sander, R. Kornhuber

References A problem involving RSD friction Γ D Ω Γ N Γ N Γ C With prescribed u ( 0 ) , ˙ u ( 0 ) , and θ ( 0 ) . σ ( u ) = C ε ( u ) in Ω (linear elasticity) div σ ( u ) + b = ρ ¨ u in Ω (momentum balance) (bilateral contact) 1 , i.e. ˙ u n = 0 ˙ on Γ C u = ˙ u t u | = | s n | µ s ( | ˙ λ = | σ t | u | , θ ) σ t = − λ ˙ with λ = 0 for ˙ u , on Γ C u = 0 | ˙ | ˙ u | . . . on Γ N , D θ = ˙ ˙ θ ( | ˙ u | , θ ) on Γ C (family of ODEs) with s n ≈ σ n , constant in time 1 . 1 Inherited from the RSD friction model Variational methods for rate- and state-dependent friction problems E. Pipping, O. Sander, R. Kornhuber

References Weak formulation We get � � � � ρ ¨ u ( v − ˙ u ) + C ε ( u ): ε ( v − ˙ u ) + φ ( v , θ ) ≥ φ (˙ u , θ ) + ℓ ( v − ˙ u ) Ω Ω Γ C Γ C for every v ∈ H with H = { v ∈ H 1 (Ω) d : v = 0 on Γ D , v n = 0 on Γ C } or briefly 0 ∈ M ¨ u + A u + ∂ Φ(˙ u , θ ) − ℓ ⊂ H ∗ Variational methods for rate- and state-dependent friction problems E. Pipping, O. Sander, R. Kornhuber

References Time discretisation Turn θ = ˙ ˙ 0 ∈ M ¨ u + A u + ∂ Φ(˙ u , θ ) − ℓ , θ ( | ˙ u | , θ ) into θ = ˙ ˙ 0 ∈ M ¨ u n + A u n + ∂ Φ(˙ u n , θ n ) − ℓ n , θ ( | ˙ u n | , θ ) and then (e.g. using the Newmark- β -method) 0 ∈ 2 u n + ∆ t ∆ t M ˙ 2 A ˙ u n + ∂ Φ(˙ u n , θ n ) − ℓ n + . . . θ n = θ n ( | ˙ u n | , . . . ) � convex minimisation problem + a step on each ODE Variational methods for rate- and state-dependent friction problems E. Pipping, O. Sander, R. Kornhuber

References The big picture Lipschitz, compact | γ ( v ) | ∈ L 2 (Γ C )  ( 1 ) (1) trace map + norm   continuous T : H → H (2) solve ODEs v ∈ H ( 2 ) logarithmic growth  (3) convex minimisation  ( 3 ) L ∆ t -Lipschitz θ ∈ L 2 (Γ C ) Variational methods for rate- and state-dependent friction problems E. Pipping, O. Sander, R. Kornhuber

References The big picture Lipschitz, compact | γ ( v ) | ∈ L 2 (Γ C )  ( 1 ) (1) trace map + norm   continuous T : H → H (2) solve ODEs v ∈ H ( 2 ) logarithmic growth  (3) convex minimisation  ( 3 ) L ∆ t -Lipschitz θ ∈ L 2 (Γ C ) • Q : Does T have a fixed point? A : Yes, by Schauder’s theorem Variational methods for rate- and state-dependent friction problems E. Pipping, O. Sander, R. Kornhuber

References The big picture Lipschitz, compact | γ ( v ) | ∈ L 2 (Γ C )  ( 1 ) (1) trace map + norm   continuous T : H → H (2) solve ODEs v ∈ H ( 2 ) logarithmic growth  (3) convex minimisation  ( 3 ) L ∆ t -Lipschitz θ ∈ L 2 (Γ C ) • Q : Does T have a fixed point? A : Yes, by Schauder’s theorem • Q : Is it unique? Variational methods for rate- and state-dependent friction problems E. Pipping, O. Sander, R. Kornhuber

References The big picture Lipschitz, compact | γ ( v ) | ∈ L 2 (Γ C )  ( 1 ) (1) trace map + norm   continuous T : H → H (2) solve ODEs v ∈ H ( 2 ) logarithmic growth  (3) convex minimisation  ( 3 ) L ∆ t -Lipschitz θ ∈ L 2 (Γ C ) • Q : Does T have a fixed point? A : Yes, by Schauder’s theorem • Q : Is it unique? • Q : Does T n v always converge? A : A subsequence does (in norm). Variational methods for rate- and state-dependent friction problems E. Pipping, O. Sander, R. Kornhuber

References The big picture Lipschitz, compact | γ ( v ) | ∈ L 2 (Γ C )  ( 1 ) (1) trace map + norm   continuous T : H → H (2) solve ODEs v ∈ H ( 2 ) logarithmic growth  (3) convex minimisation  ( 3 ) L ∆ t -Lipschitz θ ∈ L 2 (Γ C ) • Q : Does T have a fixed point? A : Yes, by Schauder’s theorem • Q : Is it unique? • Q : Does T n v always converge? A : A subsequence does (in norm). • Q : Does it converge to a fixed point? Variational methods for rate- and state-dependent friction problems E. Pipping, O. Sander, R. Kornhuber

References The big picture Lipschitz, compact | γ ( v ) | ∈ L 2 (Γ C )  ( 1 ) (1) trace map + norm   continuous T : H → H (2) solve ODEs v ∈ H ( 2 ) logarithmic growth  (3) convex minimisation  ( 3 ) L ∆ t -Lipschitz θ ∈ L 2 (Γ C ) • Q : Does T have a fixed point? A : Yes, by Schauder’s theorem • Q : Is it unique? • Q : Does T n v always converge? A : A subsequence does (in norm). • Q : Does it converge to a fixed point? • Q : What about the time-continuous case? Variational methods for rate- and state-dependent friction problems E. Pipping, O. Sander, R. Kornhuber

References Numerical simulation of a sample problem 24 22 20 18 number of FPIs 16 14 maximum 12 average 10 8 6 4 2 0 2 3 4 5 6 7 8 grid refinements Variational methods for rate- and state-dependent friction problems E. Pipping, O. Sander, R. Kornhuber

References Snapshots of the same problem (1/2) Γ D Ω Γ N Γ N Γ C velocity (logarithmic) velocity (linear) time time Variational methods for rate- and state-dependent friction problems E. Pipping, O. Sander, R. Kornhuber

References Snapshots of the same problem (2/2) time velocity horizontal coordinate Variational methods for rate- and state-dependent friction problems E. Pipping, O. Sander, R. Kornhuber