The onset of frictional instabilities, e.g. earthquakes nucleation, is intimately related to velocity-weakening friction, in which the frictional resistance of interfaces decreases with increasing slip velocity. While this frictional response has been studied extensively, less attention has been given to steady-state velocity-strengthening friction, in spite of its potential importance for various aspects of frictional phenomena such as the propagation speed of interfacial rupture fronts and the amount of stored energy released by them. In this note we suggest that a crossover from steady-state velocity-weakening friction at small slip velocities to steady-state velocity-strengthening friction at higher velocities might be a generic feature of dry friction. We further argue that while thermally activated rheology naturally gives rise to logarithmic steady-state velocity-strengthening friction, a crossover to stronger-than-logarithmic strengthening might take place at higher slip velocities, possibly accompanied by a change in the dominant dissipation mechanism. We sketch a few physical mechanisms that may account for the crossover to stronger-than-logarithmic steady-state velocity-strengthening and compile a rather extensive set of experimental data available in the literature, lending support to these ideas.Comment: Updated to published version: 2 Figures and a section adde

We investigate the ability of frame-invariant amplitude equations ͓G. H. Gunaratne, Q. Ouyang, and H. Swinney, Phys. Rev. E 50, 2802 ͑1994͔͒ to describe quantitatively the evolution of polycrystalline microstructures and we extend this approach to include the interaction between composition and stress. Validations for elemental materials include studies of the Asaro-Tiller-Grinfeld morphological instability of a stressed crystal surface, polycrystalline growth from the melt, grain-boundary energies over a wide range of misorientation, and grain-boundary motion coupled to shear deformation. Amplitude equations with accelerated strain relaxation in the solid are shown to model accurately the Asaro-Tiller-Grinfeld instability. Polycrystalline growth is also well described. However, the survey of grain-boundary energies shows that the approach is only valid for a restricted range of misorientations as a direct consequence of an amplitude expansion. This range covers approximately half the complete range allowed by crystal symmetry for some fixed reference set of density waves used in the expansion. Over this range, coupled motion to shear is well described by known geometrical rules and a transition from coupling to sliding motion is also reproduced. Amplitude equations for alloys are derived phenomenologically in a Ginzburg-Landau spirit. Vegard's law is shown to be naturally described by seeking a gauge-invariant form of those equations under a transformation that corresponds to a lattice expansion and deviations from Vegard's law can be easily incorporated. Those equations realistically describe the dilute alloy limit and have the same flexibility as conventional phase-field models for incorporating arbitrary free-energy/composition curves. As a test of this approach, we recover known analytical expressions for open-system elastic constants ͓F.

We present a continuum theory which predicts the steady state propagation of cracks. The theory overcomes the usual problem of a finite time cusp singularity of the Grinfeld instability by the inclusion of elastodynamic effects which restore selection of the steady state tip radius and velocity. We developed a phase field model for elastically induced phase transitions; in the limit of small or vanishing elastic coefficients in the new phase, fracture can be studied. The simulations confirm analytical predictions for fast crack propagation.PACS numbers: 62.20. Mk, 46.50.+a, 47.54.+r Understanding the day-to-day phenomenon of fracture is a major challenge for solid state physics and materials science. Starting with the early idea of Griffith [1], who realized that crack growth is a competition between a release of elastic energy and an increase of surface energy, various approaches have been developed to describe the striking features of cracks [2]. Usually, the motion of cracks is understood on the level of breaking bonds at sharp tips, and obviously theoretical predictions depend sensitively on the underlying empirical models of the atomic properties (see for example [3]). Plastic effects, however, lead to extended crack tips (finite tip radius r 0 ), and it is conceivable that for example fracture in gels can be described macroscopically. Then a full modeling of fracture should not only determine the crack speed but also the crack shape self-consistently.Recent phase field models go beyond the microscopic limit of discrete models with broken rotational symmetry, and encompass much of the expected behavior of cracks [4,5]; these models are close in spirit though different in details with respect to earlier approaches [6]. However, the scale of the appearing patterns is always dictated by the phase field interface width, and thus these models have problems in the sharp interface limit. Other descriptions are based on macroscopic equations of motion but suffer from inherent finite time singularities which do not allow steady state crack growth unless the tip radius is limited by the phase field interface width [7]. Numerical approaches which are not based on a phase field provide a selection mechanism by the introduction of complicated nonlinear terms in the elastic energy for high strains in the tip region [8], requiring additional parameters.It is therefore highly desirable to look for minimal models of fracture which are free from microscopic details and which are based on well established thermodynamical concepts. This is also motivated by experimental results showing that many features of crack growth are rather generic [9]; among them is the saturation of the steady state velocity appreciably below the Rayleigh speed and a tip splitting for high applied tension.Already in our previous publication [10] we emphasized a connection between fracture mechanics and elastically induced surface diffusion processes: the AsaroTiller-Grinfeld (ATG) instability [11] appears to be a good starting point for the quest f...

The description of surface-diffusion controlled dynamics via the phase-field method is less trivial than it appears at first sight. A seemingly straightforward approach from the literature is shown to fail to produce the correct asymptotics, albeit in a subtle manner. Two models are constructed that approximate known sharpinterface equations without adding undesired constraints. Numerical simulations of the standard and a more sophisticated model from the literature as well as of our two models are performed to assess the relative merits of each approach. The results suggest superior performance of the models in at least some situations.

We present a continuum theory to describe elastically induced phase transitions between coherent solid phases. In the limit of vanishing elastic constants in one of the phases, the model can be used to describe fracture on the basis of the late stage of the Asaro-Tiller-Grinfeld instability. Starting from a sharp interface formulation we derive the elastic equations and the dissipative interface kinetics. We develop a phase field model to simulate these processes numerically; in the sharp interface limit, it reproduces the desired equations of motion and boundary conditions. We perform large scale simulations of fracture processes to eliminate finite-size effects and compare the results to a recently developed sharp interface method. Details of the numerical simulations are explained, and the generalization to multiphase simulations is presented.