On the characterization of localized solutions in inelastic solids: an analysis of wave propagation in a softening bar

Abstract

This paper presents a study of the solutions characteristic of the localized failures in inelastic solids under general dynamic conditions. The paper is divided into two parts. In the first part, we present a general framework for the inclusion of localized dissipative mechanisms in a local continuum. This is accomplished by the consideration locally of discontinuities in the displacement field, the so-called strong discontinuities, as a tool for the modeling of these localized effects of the material response. We present in this context a thermodynamically based derivation of the resulting governing equations along these discontinuities. These developments are then incorporated in the local continuum framework characteristic of typical large-scale structural systems of interest. The general multi-dimensional case is assumed in this first part of the paper. In the second part, we present in the context furnished by the previous discussion a study of the wave propagation in the one-dimensional case of a localized softening bar. We obtain first the exact closed-form solution involving a strong discontinuity with a general localized softening law. We consider next the approximate problem involving the softening response of the material in a zone of finite length. Closed-form analytic solutions are obtained for the case of a linear softening law. This analysis reveals the properties of the approximation introduced by the spatial discretization in numerical solutions of the problem. Finally, we present finite element simulations that confirm the conclusions drawn from the previous analyses.