Crack dynamics in a nonlinear lattice

A discrete two-dimensional square-cell lattice with a steady propagating crack is considered. The lattice particles are connected by massless bonds, which obey a piecewise-linear double- humped stress-strain relation. Initially, Hooke's law is valid as the first stable branch of the force-elongation diagram; then, as the elongation becomes critical, the transition to the other branch occurs. Further, when the strain reaches the next critical value, the bond breaks. This transition is assumed to occur only in a line of the breaking bonds; the bonds outside the crack line are assumed to be in the initial branch all the time. The formulation relates to the crack propagation with a 'damage zone' in front of the crack. An analytical solution is presented that allows to determine the crack speed as a function of the far-field energy release rate, to find the total speed-dependent dissipation, and to estimate the role of the damage zone. The analytical formulation and the solution present a development of the previous ones for the crack and localized phase transition dynamics in linear and bistable-bond lattices.