Transition waves in bistable structures. II. Analytical solution: wave speed and energy dissipation

We consider dynamics of chains of rigid masses connected by links described by irreversible, piecewise linear constitutive relation: the force-elongation diagram consists of two stable branches with a jump discontinuity at the transition point. The transition from one stable state to the other propagates along the chain and excites a complex system of waves. In the first part of the paper (Cherkaev et al., 2004, Transition waves in bistable structures. I. Delocalization of damage), the branches could be separated by a gap where the tensile force is zero, the transition wave was treated as a wave of partial damage. Here we assume that there is no zero-force gap between the branches. This allows us to obtain steady-state analytical solutions for a general piecewise linear trimeric diagram with parallel and nonparallel branches and an arbitrary jump at the transition. We derive necessary conditions for the existence of the transition waves and compute the speed of the wave. We also determine the energy of dissipation which can be significantly increased in a structure characterized by a nonlinear discontinuous constitutive relation. The considered chain model reveals some phenomena typical for waves of failure or crushing in constructions and materials under collision, waves in a structure specially designed as a dynamic energy absorber and waves of phase transitions in artificial and natural passive and active systems.