Finite element modeling of dynamic frictional rupture with rate and state friction

Abstract

Numerous laboratory experiments have demonstrated the dependence of the friction coefficient on the interfacial slip rate and the contact history, a behavior generically called rate and state friction. Although numerical models have been widely used for analyzing rate and state friction, in general they consider infinite elastic domains surrounding the sliding interface and rely on boundary integral formulations. Much less work has been dedicated to modeling finite size systems to account for interactions with boundaries. This paper investigates rate and state frictional interfaces in the context of finite size systems with the finite element method in explicit dynamics. It is shown that due to the highly non-linear nature of rate and state friction and its sensitivity to numerical noise, the time integration step to achieve an accurate steady state solution is orders of magnitude smaller compared to the stable time step required in boundary integral formulations. We provide evidence that the noise, which is source of instability in the finite element solution, originates from internal discretization nodes. We then investigate the long term behavior of the sliding interface for two different friction laws: a velocity weakening law, for which the friction monotonously decreases with increasing sliding velocity, and a velocity weakening-strengthening law, for which the friction coefficient first decreases but then increases above a critical velocity. We show that for both friction laws at finite times, that is before wave reflections from the boundaries come back to the sliding interface, a temporary steady state sliding is reached, with a well-defined stress drop at the interface. This stress drop gives rise to a stress concentration and leads to an analogy between friction and fracture. However, at longer times, that is after multiple wave reflections, the stress drop is essentially zero, resulting in losing the analogy with fracture mechanics. Finally, the simulations reveal that velocity weakening is unstable at long time scales, as it results in an acceleration of the sliding blocks. On the other hand, velocity weakening-strengthening reaches a steady state sliding configuration.

Robustness of structural superlubricity beyond rigid models

“…S8 in the ESM). In continuum models, friction between elastic plates or at tip-substrate contacts is usually modeled by constitutive rules (e.g., the Coulomb law) implemented through interaction between surface nodes with chosen damping coefficients [47][48][49], or the cohesive zone model (CZM) with parameters derived from interatomic potential functions [50]. However, evolution of the lattice registry has not been incorporated in these models, and further work is needed to capture the multiscale features we report here.…”

A framework for the analysis of fully coupled normal and tangential contact problems with complex interfaces

“…The natural development of the presented interface element, which by itself stems from [34], includes its extension to three dimensions and possibly the inclusion of other interface phenomena which could be much more difficult to be analyzed using standard FEM or BEM approaches, such as the interplay of friction and adhesion, or friction and plasticity. The presented implementation also allows for the possibility of more complex friction laws to be used, as, for example, the ones employed in [48].…”

Velocity-driven frictional sliding: Coarsening and steady-state pulse trains

“…The domain is discretized into a regular mesh composed of bilinear quadrilateral elements (Q4). The sliding interface between the two elastic bodies is modeled using a node to node contact algorithm, whose details can be found in [69]. Time is integrated using the central difference method and the time step is taken small enough to eliminate the numerical instabilities associated with the explicit finite element modeling of rate-and-state friction, as explained in [69].…”

“…The sliding interface between the two elastic bodies is modeled using a node to node contact algorithm, whose details can be found in [69]. Time is integrated using the central difference method and the time step is taken small enough to eliminate the numerical instabilities associated with the explicit finite element modeling of rate-and-state friction, as explained in [69]. In our simulations, we set the time step to ∆t = α FEM ∆t CFL , where ∆t CFL is set by the Courant-Friedrichs-Lewy condition and α FEM is typically taken to be O(0.01).…”

“…FEM simulations of frictional systems, especially when rate-and-state friction is taken into account, are significantly more challenging and computationally demanding than their BIM counterparts [69]. They are also prone to numerical and physical instabilities (associated with the finite H) that are absent in their BIM counterparts.…”