K. Ranjith

Rate and state dependent friction and the stability of sliding between elastically deformable solids

We study the stability of steady sliding between elastically deformable continua using rate and state dependent friction laws. That is done for both elastically identical and elastically dissimilar solids. The focus is on linearized response to perturbations of steady-state sliding, and on studying how the positive direct effect (instantaneous increase or decrease of shear strength in response to a respective instantaneous increase or decrease of slip rate) of those laws allows the existence of a quasi-static range of response to perturbations at sufficiently low slip rate. We discuss the physical basis of rate and state laws, including the likely basis for the direct effect in thermally activated processes allowing creep slippage at asperity contacts, and estimate activation parameters for quartzite and granite. Also, a class of rate and state laws suitable for variable normal stress is presented. As part of the work, we show that compromises from the rate and state framework for describing velocity-weakening friction lead to paradoxical results, like supersonic propagation of slip perturbations, or to ill-posedness, when applied to sliding between elastically deformable solids. The case of sliding between elastically dissimilar solids has the inherently destabilizing feature that spatially inhomogeneous slip leads to an alteration of normal stress, hence of frictional resistance. We show that the rate and state friction laws nevertheless lead to stability of response to sufficiently short wavelength perturbations, at very slow slip rates. Further, for slow sliding between dissimilar solids, we show that there is a critical amplitude of velocity-strengthening above which there is stability to perturbations of all wavelengths.

Slip dynamics at an interface between dissimilar materials

It has been shown recently that steady frictional sliding along an interface between dissimilar elastic solids with Coulomb friction acting at the interface is ill-posed for a wide range of material parameters and friction coefficients. The ill-posedness is manifest in the unstable growth of interfacial disturbances of all wavelengths, with growth rate inversely proportional to the wavelength. We first establish the connection between the ill-posedness and the existence of a certain interfacial wave in frictionless contact, called the generalized Rayleigh wave. Precisely, it is shown that for material combinations where the generalized Rayleigh wave exists, steady sliding with Coulomb friction is ill-posed for arbitrarily small values of friction. In addition, intersonic unstable modes and supersonic steady-state modes exist for sufficiently large values of the friction coefficient. Secondly, regularization of the problem by an experimentally motivated friction law is studied. We show that a friction law with no instantaneous dependence on normal stress but a simple fading memory of prior history of normal stress makes the problem well-posed.

Stability of quasi-static slip in a single degree of freedom elastic system with rate and state dependent friction

The stability of quasi-static frictional slip of a single degree of freedom elastic system is studied for a Dieterich–Ruina rate and state dependent friction law, showing steady-state velocity weakening, and following the ageing (or slowness) version of the state evolution law. Previous studies have been done for the slip version. Analytically determined phase plane trajectories and Liapunov function methods are used in this work. The stability results have an extremely simple form: (1) When a constant velocity is imposed at the load point, slip motion is always periodic when the elastic stiffness, K, has a critical value, Kcr. Slip is always stable when K > Kcr > 0, with rate approaching the load-point velocity, and unstable (slip rates within the quasi-static model become unbounded) when K Kcr. An implication of this result for slip instabilities in continuum systems is that a critical nucleation size of coherent slip has to be attained before unstable slip can ensue. (2) When the load point is stationary, the system stably evolves towards slip at a monotonically decreasing rate whenever K ⩾ Kcr > 0. However, when K < Kcr, initial conditions leading to stable and unstable slip motion exist. Hence self-driven creep modes of instability exist, but only in the latter case.

Destabilization of long-wavelength Love and Stoneley waves in slow sliding

Abstract Love waves are dispersive interfacial waves that are a mode of response for anti-plane motions of an elastic layer bonded to an elastic half-space. Similarly, Stoneley waves are interfacial waves in bonded contact of dissimilar elastic half-spaces, when the displacements are in the plane of the solids. It is shown that in slow sliding, long-wavelength Love and Stoneley waves are destabilized by friction. Friction is assumed to have a positive instantaneous logarithmic dependence on slip rate and a logarithmic rate weakening behavior at steady-state. Long-wavelength instabilities occur generically in sliding with rate- and state-dependent friction, even when an interfacial wave does not exist. For slip at low rates, such instabilities are quasi-static in nature, i.e., the phase velocity is negligibly small in comparison to a shear wave speed. The existence of an interfacial wave in bonded contact permits an instability to propagate with a speed of the order of a shear wave speed even in slow sliding, indicating that the quasi-static approximation is not valid in such problems.

Dynamic anti-plane sliding of dissimilar anisotropic linear elastic solids

Abstract The stability of steady, dynamic, anti-plane slipping at a planar interface between two dissimilar anisotropic linear elastic solids is studied. The solids are assumed to possess a plane of symmetry normal to the slip direction, so that in-plane displacements and normal stress changes on the slip plane do not occur. Friction at the interface is assumed to follow a rate and state-dependent law with velocity weakening behavior in the steady state. The stability to spatial perturbations of the form exp(i kx 1 ), where k is the wavenumber and x 1 is the coordinate along the interface is studied. The critical wavenumber magnitude, | k | cr , above which there is stability and the corresponding phase velocity, c , of the neutrally stable mode are obtained from the stability analysis. Numerical plots showing the dependence of | k | cr and c on the unperturbed sliding velocity, V o , are provided for various bi-material combinations of practical interest.

Asymptotic fields for dynamic crack growth at a ductile-brittle interface

In this work. the asymptotic fields near a crack tip propagating dynamically under plane strain conditions at a ductile-brittle interface are derived. The ductile material is taken to obey the

Asymptotic and finite element analyses of mode III dynamic crack growth at a ductile-brittle interface

J_2

Spectral formulation of the elastodynamic boundary integral equations for bi-material interfaces

flow theory of plasticity with linear isotropic strain hardening. The asymptotic solution is assumed to be of the variable-separable form with a power singularity in the radial coordinate from the crack tip. Results have been generated for ditferent bi-material combinations over a range of crack speeds. At a given crack speed. two solutions resembling the Mode I and Mode II asymptotic fields in a homogeneous material are obtained for all strain hardening between a lower and upper bound. The effect of mismatch in elastic stiffness and density. as well as strain hardening of the ductile phase on the singularity strength, near-tip mixities and the angular stress and velocity functions are examined. Attention is focused on the influence of crack speed on the above features of the asymptotic solution.

A slip wave solution in antiplane elasticity

In this work, dynamic crack growth along a ductile-brittle interface under anti-plane strain conditions is studied. The ductile solid is taken to obey the J2 flow theory of plasticity with linear isotropic strain hardening, while the substrate is assumed to exhibit linear elastic behavior. Firstly, the asymptotic near-tip stress and velocity fields are derived. These fields are assumed to be variable-separable with a power singularity in the radial coordinate centered at the crack tip. The effects of crack speed, strain hardening of the ductile phase and mismatch in elastic moduli of the two phases on the singularity exponent and the angular functions are studied. Secondly, full-field finite element analyses of the problem under small-scale yielding conditions are performed. The validity of the asymptotic fields and their range of dominance are determined by comparing them with the results of the full-field finite element analyses. Finally, theoretical predictions are made of the variations of the dynamic fracture toughness with crack velocity. The influence of the bi-material parameters on the above variation is investigated.

Instabilities in Dynamic Anti-plane Sliding of an Elastic Layer on a Dissimilar Elastic Half-Space

Abstract A spectral formulation of the plane-strain boundary integral equations for an interface between dissimilar elastic solids is presented. The boundary integral equations can be written in two equivalent forms: (a) the tractions can be written as a space–time convolution of the displacement continuities at the interface ( Budiansky and Rice, 1979 ) (b) the displacement discontinuities can be written as a space–time convolution of the tractions at the interface ( Kostrov, 1966 ). Prior work on spectral formulation of the boundary integral equations has adopted the former as the starting point. The present work has for its basis the latter form based on a space–time convolution of the tractions. Tractions and displacement components are given a spectral representation in the spatial coordinate along the interface. The radiation damping term is then explicitly extracted to avoid singularities in the convolution kernels. With the spectral forms introduced, the space–time convolutions reduce to convolutions in time for each Fourier mode. Due to continuity of tractions at the interface, this leads to a simpler formulation and form of the convolution kernels in comparison to the formulation involving convolutions over the slip and opening history at the bi-material interface. The convolution kernels are validated by studying some model problems to which analytical solutions are known. When coupled with a cohesive law or a friction law at the interface, the formulation proposed here is of wide applicability for studying spontaneous rupture propagation.