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

This paper presents a study of the solutions characteristic of the localize d failures in inelastic solids under general dynamic conditions. The paper is divided into two parts. In the first part, we present a general framewor k for the inclusion of localized dissipative mechanisms in a local continuu m. 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 prese nt in this context a thermodynamically based derivation of the resulting go verning equations along these discontinuities. These developments are then incorporated in the local continuum framework characteristic of typical lar ge-scale structural systems of interest. The general multidimensional 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 pro pagation in the one-dimensional case of a localized softening bar. We obtai n first the exact closed-form solution involving a strong discontinuity wit h a general localized softening law. We consider next the approximate probl em involving the softening response of the material in a zone of finite len gth. Closed-form analytic solutions are obtained for the case of a linear s oftening law. This analysis reveals the properties of the approximation int roduced by the spatial discretization in numerical solutions of the problem . Finally, we present finite element simulations that confirm the conclusio ns drawn from the previous analyses. (C) 2001 Elsevier Science B.V. All rig hts reserved.