Localization analysis of elastic degradation with application to scalar damage

The present paper extends the results of discontinuous bifurcation analysis of elastoplastic solids to materials that exhibit elastic degradation. As a starting point, the concepts of elastic degradation are formulated in terms of a secant relationship, a threshold function, and an elastic stiffness degradation rule. Differentiation of the secant law renders the governing tangent operator for both stress- or strain-based formulations of elastic degradation. Scalar- and tenser-valued proposals in continuum damage mechanics are particular cases of this unified formulation of elastic degradation, when a reduced set of damage variables is considered. The traditional (1-D) approach of scalar damage describes the stiffness degradation by a single variable, whereby all components of the secant stiffness are affected in a self-similar manner. The paper presents general analytic results for distributed and localized failure of elastic-degrading materials, further specialized for the traditional (1-D) scalar damage model, when it is subjected to specific loading scenarios. For illustration, a geometric localization criterion is introduced to highlight the features of localized failure in the Mohr coordinates.