On the dissipation due to wave ringing in nonelliptic elastic materials

Abstract

Initial-boundary value problems describing the mechanics of nonelliptic elastic materials give rise to solutions that involve phase boundaries, the motion of which can dissipate mechanical energy. We investigate whether this dissipation, acting alone, can drive such a system toward equilibrium. Moving phase boundaries are regarded as a localized dissipative mechanism, and we consider a model which specifically excludes dissipation away from a phase boundary (such as that due to viscoelastic damping). In the problem under consideration, wave packets reverberate between the fixed external boundary and a single internal phase boundary. The phase boundary remains stationary unless it is acted upon by one of these wave packets, and each such interaction dissipates a finite amount of energy while causing the initiating wave packet to split into a reflected wave packet and a transmitted wave packet. Consequently, the number of wave packets increases in a geometric fashion. Each individual interaction of a wave packet with the phase boundary is, in a certain sense, mechanically underdetermined, and we augment the mechanical theory with two alternative energy criteria, each of which determines a different interaction dynamics. These alternative energy criteria are motivated by considerations of maximizing the energy dissipation in the system. We treat a system that is perturbed out of an initial minimum energy equilibrium state by a disturbance at the external boundary. A framework is developed for treating the resulting wave reverberations and calculating the energy dissipation for large time. Numerical computation indicates that the total energy dissipated in both versions of the dynamical problem is that which is necessary to settle into a new energy-minimal equilibrium state. We then establish the same result analytically for a meaningful limit involving a vanishingly small dynamical perturbation.