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 ยฎrst 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 ยฎeld, the socalled strong discontinuities, as a tool for the modeling of these localized eects 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 multidimensional case is assumed in this ยฎrst 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 ยฎrst the exact closedform 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 ยฎnite 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 ยฎnite element simulations that conยฎrm the conclusions drawn from the previous analyses.