A Global Damage Theory and the Hyperbolicity of the Wave Problem

Abstract

It is well known that wave equation in materials that suffer damage in the course of deformation loses it hyperbolicity when the damage process is described by a continuum damage theory of the local type. Here we develop a global (nonlocal) damage theory by (a) introducing a damage coordinate which is a spatial functional of the strain field in the material domain and (b) stipulating that the rate of evolution of damage is with respect to the damage coordinate. We then derive the axial wave equation for a thin rod and thereby demonstrate that, while the rod experiences softening, the wave speed is given in terms of the secant modulus and the wave equation retains its hyperbolicity. Various other phenomena, such as the onset of inhomogeneous damage in the presence of homogeneous deformation and the tendency of axial specimens under tension to fracture invariably at the center, are also explained.