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References Dynamic problems of rate-and-state friction in viscoelasticity Elias Pipping Freie Universitt Berlin 10th of December 2014 Dynamic problems of rate-and-state friction in viscoelasticity E. Pipping References Experimental

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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
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SLIDE 2

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)

Dynamic problems of rate-and-state friction in viscoelasticity

  • E. Pipping
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References

Rate-and-state friction

Widely used law µ(V , θ) = µ∗ + a log V V∗ + b log θV∗ L , ˙ θ(θ, V ) =

  • 1 − θV

L

ageing law − θV

L log θV L

slip law Transformation: α = log(θV∗/L) µ(V , α) = µ∗ + a log V V∗ + bα, ˙ α(α, V ) =

  • V∗e−α−V

L

− V

L

  • log V

V∗ + α

  • 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
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References

A typical continuum mechanical problem

ΓN ΓN Ω ΓD ΓC

With prescribed u(0), ˙ u(0), and α(0). σ(u) = Bε(u) + Aε(˙ u) in Ω (linear viscoelasticity) div σ(u) + b = ρ¨ u in Ω (momentum balance) ˙ un = 0

  • n ΓC

(bilateral contact)1 σt = −λ˙ u, λ = |σt| |˙ u| = |sn|µ(|˙ u|, α) |˙ u|

  • n ΓC

with λ = 0 for ˙ u = 0 . . .

  • n ΓN,D

˙ α = ˙ α(|˙ u|, α)

  • n ΓC

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

1Inherited from the rate-and-state friction model

Dynamic problems of rate-and-state friction in viscoelasticity

  • E. Pipping
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References

Weak formulation

We get

  • Ω

ρ¨ u(v − ˙ u) +

  • Ω

Bε(˙ u): ε(v − ˙ u) +

  • Ω

Aε(u): ε(v − ˙ u) +

  • ΓC

φ(v, α) ≥

  • ΓC

φ(˙ u, α) + 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 + C ˙ u + Au + ∂Φ( · , α)(˙ u) − ℓ ⊂ H∗ and ˙ α = ˙ α(|˙ u|, α) a.e. on ΓC

Dynamic problems of rate-and-state friction in viscoelasticity

  • E. Pipping
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References

Time discretisation

Turn 0 ∈ M¨ u + C ˙ u + Au + ∂Φ( · , α)(˙ u) − ℓ, ˙ α = ˙ α(|˙ u|, α) into 0 ∈ M¨ un + C ˙ un + Aun + ∂Φ( · , αn)(˙ un) − ℓn, ˙ α = ˙ α(|˙ un|, α) and then (using a time discretisation scheme/solving the ODEs) 0 ∈ (Mn + C + An)˙ un + ∂Φ( · , αn)(˙ un) − ˜ ℓn αn = Ψ|˙

un|(αn−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
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References

The big picture

|γ(v)| ∈ L2(ΓC) v ∈ H α ∈ L2(ΓC)

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

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

  • 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 nv always converge to a fixed point?

A: Yes/Maybe (depending on the law)

Dynamic problems of rate-and-state friction in viscoelasticity

  • E. Pipping
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References

Application: a simplified subduction zone

x1 x2 x3 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
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References

Numerical stability: Number of fixed point iterations

1775 1780 1785 1790 0.00 0.20 0.40 0.60

time [s]

trench distance [m]

−20 µm −10 µm −5 µm −2.5 µm 0 µm 2.5 µm 5 µm 10 µm 20 µm 40 µm

2 3 4 5 6

iterations

1775 1780 1785 1790 10−3 10−1

time [s] step size [s]

Dynamic problems of rate-and-state friction in viscoelasticity

  • E. Pipping
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References

Comparison with laboratory data

100.5 101 101.5

recurrence time [s]

0.1 0.2 0.3 0.4

rupture width [m]

10−1.5 10−1 10−0.5 experiment ageing law slip law

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
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References

Further reading

  • E. Pipping, O. Sander and R. Kornhuber. “Variational formulation
  • f 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