Variational formulation of rate‐ and state‐dependent friction problems

Abstract

Abstract. We propose a variational formulation of rate-and state-dependent models for the dynamic sliding of a linearly elastic block on a rigid surface in terms of two coupled variational inequalities. Classical DieterichRuina models are covered as special cases. We show existence and uniqueness of solutions for the two spatial subproblems arising from time discretisation. Existence of solutions to the coupled spatial problems is established for Dieterich's state equation through a xed point argument. We conclude with some numerical experiments that suggest mesh independent convergence of the underlying xed point iteration, and illustrate quasiperiodic occurrence of stick/slip events.

On the efficient and reliable numerical solution of rate-and-state friction problems

“…This abstract reformulation of (1) and (10) allows to use convexity in the construction of fast nonlinear multigrid solvers after discretization (Pipping et al 2013;Pipping 2014). Our numerical routine proceeds as follows: We use an implicit Newmark scheme (Newmark 1959) to discretize (12) in time and piecewise affine finite elements to discretize in space.…”

“…Note that this transformation is motivated by mathematical analysis in Pipping et al (2013). Discretization in time and space leaves us with a sequence of spatial problems…”

“…The solver is mathematically consistent in the sense that it resolves the rate-and-state coupling up to any prescribed tolerance. Its main features are: adaptive time stepping based on an implicit Newmark scheme (Newmark 1959) to resolve almost instantaneous slip events; finite elements with local mesh refinement guided by predefined seismogenic zones to resolve complex geometries and spatial heterogeneities; a fast, reliable algebraic solution procedure involving a fixed-point iteration (Pipping et al 2013;Pipping 2014) and a non-smooth Newton multigrid solver (Gräser 2011;Pipping 2014;Gräser & Sander 2016).…”

Existence of long‐time solutions to dynamic problems of viscoelasticity with rate‐and‐state friction

“…Even though this model for rate‐and‐state dependent friction is very popular in geophysics, it does not have an underlying thermodynamical structure allowing for a proper balance of energies through external powers and dissipation. This mathematical drawback was already pointed out in [], where two energies have been used to govern separately the momentum balance for u and the flow rule for the internal variable α, or in [], where an energetic formulation was obtained for a slightly expanded model. The quasistatic rate‐and‐state friction model (i.e.…”

Truncated nonsmooth Newton multigrid methods for block-separable minimization problems

“…Such minimization problems occur in many fields of science; however, we are mainly interested in the case where they result from discretized nonsmooth partial differential equations (PDEs). Examples of such PDEs include contact and friction models in solid mechanics [26,33], certain models of porous media flow [3], but also many variants of Allen-Cahn-type phase-field models [25]. The block-structure is typically either induced by grouping all vector components of a given vector-valued degree of freedom, or by grouping the degrees of freedom of a grid element in a Discontinuous-Galerkin-type discretization.…”