Dendrites and fronts in a model of dynamical rupture with damage

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Abstract
Inspired by our previous studies [Phys. Rev. Lett. 68, 612 (1992); J. Phys. I 2, 1621 (1992); Phys. Rev. A 45, 8351 (1992)] for a random system of the antiplane model of dynamical rupture controlled by a damage field (also coined ‘‘thermal fuse model’’ in its electric analog), here, we examine mainly the case of two-dimensional ordered systems (square lattices and continuum), in order to highlight the similarities with and also the differences from other growth phenomena. Two situations are studied: (1) the cata- strophic growth of a crack from a nucleus at constant applied stress and (2) the steady-state propagation of a rupture front in a strip under constant applied displacements at the borders. In discrete lattices, numerical simulations of case 1 show a rich ‘‘phase diagram’’ of rupture patterns as a function of the damage exponent m, with four-leaf-clover-shaped cracks for low m and dendriticlike cracks with complex sidebranching for larger values of m. In case 2, the discrete nature of the lattice is at the origin of the observation of many possible coexisting solutions for crack propagation. The case of a one-mesh-thick semi-infinite central crack is solved analytically for its crack tip velocity, which is uniquely determined by a growth criterion involving the history of the damage field at all points ahead of the crack tip. We then present the continuum formulation of the steady-state propagation of a rupture front in a strip under constant applied displacements at the borders, and we find a continuous family of solutions parametrized by the velocity, similar to the Saffman-Taylor problem, which has an infinity of degenerate solutions in the absence of surface tension.