We study crack propagation in a viscoelastic solid. Using simple arguments, we derive equations for the velocity dependence of the crack-tip radius, a (v) , and for the energy per unit area to propagate the crack, G (v) . For a viscoelastic modulus E (omega) which increases as omega(1-s) (0< s< 1) in the transition region between the rubbery region and the glassy region, we find that a (v) approximately G (v) approximately v(alpha) with alpha= (1-s) / (2-s) . The theory is in good agreement with experiment.

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The morphology diagram of possible structures in two-dimensional diffusional growth is given in the parameter space of undercooling ⌬ versus anisotropy of surface tension ⑀. The building block of the dendritic structure is a dendrite with parabolic tip, and the basic element of the seaweed structure is a doublon. The transition between these structures shows a jump in the growth velocity. We also describe the structures and velocities of fractal dendrites and doublons destroyed by noise. We introduce a renormalized capillary length and density of the solid phase and use scaling arguments to describe the fractal dendrites and doublons. The resulting scaling exponents for the growth velocity and the different length scales are expressed in terms of the fractal dimensions for surface and bulk of these fractal structures. All the considered structures are compact on length scales larger than the diffusion length and they show fractal behavior on intermediate length scales between the diffusion length and a small size cutoff which depends on the strength of noise.

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