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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

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References

Variational methods for rate- and state-dependent friction problems

Elias Pipping1 Oliver Sander2 Ralf Kornhuber1

1Free University Berlin: Institute for Mathematics 2RWTH 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
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References

Experimental background

V1 V2 time velocity µss(V2) µss(V1) time coefficient of friction

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
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References

Phenomenological Law

Abstract setting µ = µ(V , θ), ˙ θ = ˙ θ(θ, V )

Variational methods for rate- and state-dependent friction problems

  • E. Pipping, O. Sander, R. Kornhuber
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References

Phenomenological Law

Abstract setting µ = µ(V , θ), ˙ θ = ˙ θ(θ, V ) Prominent example µ(V , θ) = µ∗ + a log V V∗ + b log θ L/V∗ , ˙ θ(θ, V ) = 1 − θ L/V

Variational methods for rate- and state-dependent friction problems

  • E. Pipping, O. Sander, R. Kornhuber
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References

Phenomenological Law

Abstract setting µ = µ(V , θ), ˙ θ = ˙ θ(θ, V ) Prominent example µ(V , θ) = µ∗ + a log V V∗ + b log θ L/V∗ , ˙ θ(θ, V ) = 1 − θ L/V

µss(V2) µss(V1) time coefficient of friction

Properties:

  • ˙

θ = 0 iff θ = θss(V ) := L/V

Variational methods for rate- and state-dependent friction problems

  • E. Pipping, O. Sander, R. Kornhuber
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References

Phenomenological Law

Abstract setting µ = µ(V , θ), ˙ θ = ˙ θ(θ, V ) Prominent example µ(V , θ) = µ∗ + a log V V∗ + b log θ L/V∗ , ˙ θ(θ, V ) = 1 − θ L/V

µss(V2) µss(V1) time coefficient of friction

Properties:

  • ˙

θ = 0 iff θ = θss(V ) := L/V

  • µss(V ) := µ(V , θss(V )) = µ∗ + (a − b) log V

V∗

Variational methods for rate- and state-dependent friction problems

  • E. Pipping, O. Sander, R. Kornhuber
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References

Phenomenological Law

Abstract setting µ = µ(V , θ), ˙ θ = ˙ θ(θ, V ) Prominent example µ(V , θ) = µ∗ + a log V V∗ + b log θ L/V∗ , ˙ θ(θ, V ) = 1 − θ L/V

µss(V2) µss(V1) time coefficient of friction

Properties:

  • ˙

θ = 0 iff θ = θss(V ) := L/V

  • µss(V ) := µ(V , θss(V )) = µ∗ + (a − b) log V

V∗

  • µss(V2) − µss(V1) = (a − b) log V2

V1

Variational methods for rate- and state-dependent friction problems

  • E. Pipping, O. Sander, R. Kornhuber
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References

Phenomenological Law

Abstract setting µ = µ(V , θ), ˙ θ = ˙ θ(θ, V ) Prominent example µ(V , θ) = µ∗ + a log V V∗ + b log θ L/V∗ , ˙ θ(θ, V ) = 1 − θ L/V

µss(V2) µss(V1) time coefficient of friction

Properties:

  • ˙

θ = 0 iff θ = θss(V ) := L/V

  • µss(V ) := µ(V , θss(V )) = µ∗ + (a − b) log V

V∗

  • µss(V2) − µss(V1) = (a − b) log V2

V1

Interpretation:

  • time scale: L/V , regularisation from

µ = µ(V ).

Variational methods for rate- and state-dependent friction problems

  • E. Pipping, O. Sander, R. Kornhuber
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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 µs.

Variational methods for rate- and state-dependent friction problems

  • E. Pipping, O. Sander, R. Kornhuber
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References

A problem involving RSD friction

ΓN ΓN Ω ΓD ΓC

With prescribed u(0), ˙ u(0), and θ(0). σ(u) = Cε(u) in Ω (linear elasticity) div σ(u) + b = ρ¨ u in Ω (momentum balance) ˙ un = 0

  • n ΓC

(bilateral contact)1, i.e. ˙ u = ˙ ut σt = −λ˙ u, λ = |σt| |˙ u| = |sn|µs(|˙ u|, θ) |˙ u|

  • n ΓC

with λ = 0 for ˙ u = 0 . . .

  • n ΓN,D

˙ θ = ˙ θ(|˙ u|, θ)

  • n ΓC

(family of ODEs) with sn ≈ σn, constant in time1.

1Inherited from the RSD friction model

Variational methods for rate- and state-dependent friction problems

  • E. Pipping, O. Sander, R. Kornhuber
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References

Weak formulation

We get

  • Ω

ρ¨ u(v − ˙ u) +

  • Ω

Cε(u): ε(v − ˙ u) +

  • ΓC

φ(v, θ) ≥

  • ΓC

φ(˙ u, θ) + ℓ(v − ˙ u) for every v ∈ H with H = {v ∈ H1(Ω)d : v = 0 on ΓD, vn = 0 on ΓC}

  • r briefly

0 ∈ M¨ u + Au + ∂Φ(˙ u, θ) − ℓ ⊂ H∗

Variational methods for rate- and state-dependent friction problems

  • E. Pipping, O. Sander, R. Kornhuber
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Time discretisation

Turn 0 ∈ M¨ u + Au + ∂Φ(˙ u, θ) − ℓ, ˙ θ = ˙ θ(|˙ u|, θ) into 0 ∈ M¨ un + Aun + ∂Φ(˙ un, θn) − ℓn, ˙ θ = ˙ θ(|˙ un|, θ) and then (e.g. using the Newmark-β-method) 0 ∈ 2 ∆t M ˙ un + ∆t 2 A˙ un + ∂Φ(˙ un, θn) − ℓn + . . . θn = θn(|˙ un|, . . . ) convex minimisation problem + a step on each ODE

Variational methods for rate- and state-dependent friction problems

  • E. Pipping, O. Sander, R. Kornhuber
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References

The big picture

|γ(v)| ∈ L2(ΓC) v ∈ H θ ∈ L2(ΓC)

(2) continuous logarithmic growth (1) Lipschitz, compact (3)

L ∆t -Lipschitz

T : H → H      (1) trace map + norm (2) solve ODEs (3) convex minimisation

Variational methods for rate- and state-dependent friction problems

  • E. Pipping, O. Sander, R. Kornhuber
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References

The big picture

|γ(v)| ∈ L2(ΓC) v ∈ H θ ∈ L2(ΓC)

(2) continuous logarithmic growth (1) Lipschitz, compact (3)

L ∆t -Lipschitz

T : H → H      (1) trace map + norm (2) solve ODEs (3) convex minimisation

  • 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
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References

The big picture

|γ(v)| ∈ L2(ΓC) v ∈ H θ ∈ L2(ΓC)

(2) continuous logarithmic growth (1) Lipschitz, compact (3)

L ∆t -Lipschitz

T : H → H      (1) trace map + norm (2) solve ODEs (3) convex minimisation

  • 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
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References

The big picture

|γ(v)| ∈ L2(ΓC) v ∈ H θ ∈ L2(ΓC)

(2) continuous logarithmic growth (1) Lipschitz, compact (3)

L ∆t -Lipschitz

T : H → H      (1) trace map + norm (2) solve ODEs (3) convex minimisation

  • Q: Does T have a fixed point? A: Yes, by Schauder’s theorem
  • Q: Is it unique?
  • Q: Does T nv always converge? A: A subsequence does (in norm).

Variational methods for rate- and state-dependent friction problems

  • E. Pipping, O. Sander, R. Kornhuber
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References

The big picture

|γ(v)| ∈ L2(ΓC) v ∈ H θ ∈ L2(ΓC)

(2) continuous logarithmic growth (1) Lipschitz, compact (3)

L ∆t -Lipschitz

T : H → H      (1) trace map + norm (2) solve ODEs (3) convex minimisation

  • Q: Does T have a fixed point? A: Yes, by Schauder’s theorem
  • Q: Is it unique?
  • Q: Does T nv 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
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References

The big picture

|γ(v)| ∈ L2(ΓC) v ∈ H θ ∈ L2(ΓC)

(2) continuous logarithmic growth (1) Lipschitz, compact (3)

L ∆t -Lipschitz

T : H → H      (1) trace map + norm (2) solve ODEs (3) convex minimisation

  • Q: Does T have a fixed point? A: Yes, by Schauder’s theorem
  • Q: Is it unique?
  • Q: Does T nv 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
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References

Numerical simulation of a sample problem

2 3 4 5 6 7 8 2 4 6 8 10 12 14 16 18 20 22 24 grid refinements number of FPIs maximum average

Variational methods for rate- and state-dependent friction problems

  • E. Pipping, O. Sander, R. Kornhuber
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References

Snapshots of the same problem (1/2)

ΓN ΓN Ω ΓD ΓC time velocity (linear) time velocity (logarithmic)

Variational methods for rate- and state-dependent friction problems

  • E. Pipping, O. Sander, R. Kornhuber
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References

Snapshots of the same problem (2/2)

horizontal coordinate time

velocity

Variational methods for rate- and state-dependent friction problems

  • E. Pipping, O. Sander, R. Kornhuber
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Further reading

  • P. Bastian et al. “A Generic Grid Interface for Parallel and

Adaptive Scientific Computing Part II: Implementation and Tests in DUNE”. In: Computing 82.2–3 (2 2008), pp. 121–138. ISSN: 0010-485X. DOI: 10.1007/s00607-008-0004-9.

  • C. Gräser and R. Kornhuber. “Multigrid Methods for Obstacle

Problems”. In: J. Comp. Math. 27.1 (2009), pp. 1–44.

  • E. Pipping, O. Sander and R. Kornhuber. “Variational Formulation
  • f Rate- and State-dependent Friction Problems”. Preprint; to

appear in ZAMM. URL: ftp://ftp.math.fu- berlin.de/pub/math/publ/pre/2013/Ab-A-13-03.html.

Variational methods for rate- and state-dependent friction problems

  • E. Pipping, O. Sander, R. Kornhuber