Solutions for nonlinear lattices

Discrete two-dimensional lattices with bistable bonds are considered. Initially, Hooke's law is valid as the first stable branch of the force-elongation diagram; then, when the elongation becomes critical, the transition to the other branch occurs. This transition is assumed to occur only in a line of the bonds; the bonds outside the transition line are assumed to be in the initial phase all the time. The transition is considered as a localized wave or as a propagating crack. A regular crack corresponds to the second branch having zero resistance, while if the bond breaks at a point of the second branch of a nonzero resistance, the formulation corresponds to the crack with a one-dimensional 'process zone'. The use of the discrete models leads to a closed mathematical formulation of the considered problems. It allows to determine the transition wave speed (or the crack speed), the total speed-dependent dissipation and the dissipation structure. For the piecewise linear trimeric diagram with a step-wise transition, explicit analytical solutions are found. Also such a solution is built for the crack with the process zone. For a general transition path, the problem can be reduced to an integral equation. The considerations are based on the corresponding intactlattice self-equilibrated fundamental solutions expressed in terms of the previously obtained lattice-with-a-moving-crack fundamental solutions.