References Dynamic problems of rate-and-state friction in viscoelasticity Elias Pipping Freie Universität Berlin 10th of December 2014 Dynamic problems of rate-and-state friction in viscoelasticity E. Pipping
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) Dynamic problems of rate-and-state friction in viscoelasticity E. Pipping
References Rate-and-state friction Widely used law � 1 − θ V + b log θ V ∗ ageing law µ ( V , θ ) = µ ∗ + a log V ˙ L L , θ ( θ, V ) = V ∗ − θ V L log θ V slip law L Transformation: α = log ( θ V ∗ / L ) � V ∗ e − α − V µ ( V , α ) = µ ∗ + a log V L + b α , α ( α, V ) = ˙ − V log V V ∗ � � V ∗ + α L General setting • µ is monotone in V for fixed α • µ is Lipschitz with respect to α (but not θ ) • (unlike θ ), α follows a gradient flow for fixed V . • (ideally): ˙ α is Lipschitz with respect to V . Dynamic problems of rate-and-state friction in viscoelasticity E. Pipping
References A typical continuum mechanical problem Γ D Ω Γ N Γ N Γ C With prescribed u ( 0 ) , ˙ u ( 0 ) , and α ( 0 ) . σ ( u ) = B ε ( u ) + A ε (˙ u ) in Ω (linear viscoelasticity) div σ ( u ) + b = ρ ¨ u in Ω (momentum balance) (bilateral contact) 1 u n = 0 ˙ on Γ C λ = | σ t | u | = | s n | µ ( | ˙ u | , α ) σ t = − λ ˙ u , on Γ C with λ = 0 for ˙ u = 0 | ˙ | ˙ u | . . . on Γ N , D α = ˙ ˙ α ( | ˙ u | , α ) on Γ C (family of ODEs) with s n ≈ σ n , constant in time 1 . 1 Inherited from the rate-and-state friction model Dynamic problems of rate-and-state friction in viscoelasticity E. Pipping
References Weak formulation We get � � � � ρ ¨ u ( v − ˙ u ) + B ε (˙ u ): ε ( v − ˙ u ) + A ε ( u ): ε ( v − ˙ u ) + φ ( v , α ) Ω Ω Ω Γ C � ≥ φ (˙ u , α ) + ℓ ( v − ˙ u ) 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 u ) − ℓ ⊂ H ∗ 0 ∈ M ¨ u + C ˙ u + A u + ∂ Φ( · , α )(˙ and α = ˙ ˙ α ( | ˙ u | , α ) a.e. on Γ C Dynamic problems of rate-and-state friction in viscoelasticity E. Pipping
References Time discretisation Turn 0 ∈ M ¨ u + C ˙ u + A u + ∂ Φ( · , α )(˙ u ) − ℓ , α = ˙ ˙ α ( | ˙ u | , α ) into 0 ∈ M ¨ u n + C ˙ u n + A u n + ∂ Φ( · , α n )(˙ α = ˙ ˙ α ( | ˙ u n ) − ℓ n , u n | , α ) and then (using a time discretisation scheme/solving the ODEs) u n ) − ˜ 0 ∈ ( M n + C + A n )˙ u n + ∂ Φ( · , α n )(˙ ℓ n α n = Ψ | ˙ u n | ( α n − 1 ) � A coupling of 1 a convex minimisation problem 2 a family of ordinary differential equations (one-dimensional gradient flows) Dynamic problems of rate-and-state friction in viscoelasticity E. Pipping
References The big picture Lipschitz, compact | γ ( v ) | ∈ L 2 (Γ C ) (1) trace map + norm ( 1 ) continuous T : H → H (2) solve ODEs v ∈ H ( 2 ) sublinear growth (3) convex minimisation ( 3 ) Lipschitz α ∈ L 2 (Γ C ) • Q : Does T have a fixed point? A : Yes, by Banach’s/Schauder’s fixed point theorem theorem • Q : Is it unique? A : Yes/Maybe (depending on the law) • Q : Does T n v always converge to a fixed point? A : Yes/Maybe (depending on the law) Dynamic problems of rate-and-state friction in viscoelasticity E. Pipping
References Application: a simplified subduction zone x 3 x 2 x 1 The lower plate moves at a prescribed velocity while the right end of the wedge is held fixed. Dynamic problems of rate-and-state friction in viscoelasticity E. Pipping
References Numerical stability: Number of fixed point iterations − 20 µm trench distance [ m ] 0 . 60 − 10 µm − 5 µm − 2 . 5 µm 0 . 40 0 µm 2 . 5 µm 0 . 20 5 µm 10 µm 20 µm 0 . 00 1775 1780 1785 1790 40 µm time [ s ] iterations 6 5 4 step size [ s ] 3 2 10 − 1 10 − 3 1775 1780 1785 1790 time [ s ] Dynamic problems of rate-and-state friction in viscoelasticity E. Pipping
References Comparison with laboratory data slip law ageing law experiment 10 0 . 5 10 1 10 1 . 5 0 0 . 1 0 . 2 0 . 3 0 . 4 10 − 1 . 5 10 − 1 10 − 0 . 5 rupture width [ m ] recurrence time [ s ] peak slip [ mm ] Recurrence time and rupture width are well reproduced. Peak slip is off by a factor of approximately 6. The error thus lies within an order of magnitude. Dynamic problems of rate-and-state friction in viscoelasticity E. Pipping
References Further reading E. Pipping, O. Sander and R. Kornhuber. “Variational formulation of rate- and state-dependent friction problems”. In: Zeitschrift für Angewandte Mathematik und Mechanik. Journal of Applied Mathematics and Mechanics (2013). ISSN: 1521-4001. DOI: 10.1002/zamm.201300062 . Dynamic problems of rate-and-state friction in viscoelasticity E. Pipping
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