Crack-like transition wave in lattices

Steady-state dynamic problem is studied for discrete square and triangular lattices whose massless bonds follow a trimeric piecewise linear force-elongation diagram. In a general case, a prestressed or an active lattice is considered. The transition wave is found analytically as that localized between two neighboring lines of the lattice particles. The transition wave itself is accompanied by dissipation waves carrying energy away from the transition front and, in general, away from the transition line. However, in the case of the transition leading to an increased tangent modulus of the bond, there exist non-divergent dissipative waves exponentially localized in a vicinity of the transition line behind the transition front. In particular, where the second branch has zero resistance, the transition wave becomes a crack. In this case, the solution for an infinite lattice includes an energy flux from a remote source, and no prestress is assumed. The Green function corresponding to a general value of the second branch tangent modulus is expressed via the known lattice-with-a-movingcrack fundamental solutions. Mathematically, this allows to study the transition wave in two-dimensional lattices as a one-dimensional problem. The analytical results are compared to the ones obtained for a continuous elastic model and for the lattice-related version of one-dimensional Frenkel-Kontorova model.