Analytical strain localization analysis of isotropic and anisotropic damage models for quasi-brittle materials

Abstract

Strain and damage localization are usually precursors of rupture. We present a three-dimensional method dedicated to quasi-brittle materials based on the works of Bigoni & Hueckel, and Jirásek and coworkers, aiming at simplifying the analysis of the localization properties of continuous damage models of a general form, and possibly anisotropic. The method reformulates the localization problem as a two-variable polynomial maximization problem, a strategy commonly used in softening plasticity models, but not so much in Continuum Damage mechanics. The quasi-brittle hypothesis is exploited to render the problem solvable in a fully analytical way, and a post-analysis criterion for the validity of the analysis is also exhibited. In this work, the method is fully established from a theoretical viewpoint, and examples illustrating its use are provided. Multiaxial calculations are performed for four continuous damage models (two isotropic and two anisotropic ones). The method applies to induced anisotropy and constitutive models representing isotropic linear elasticity before damage growth, and remains accurate when models display immediate softening after the elastic limit (and thus to multiaxial tensile cases). The analytical method is, however, entirely general and allows for the calculation of (i) the orientation of a potential localization plane, (ii) the mode angle of the weak discontinuity, and (iii) the validity domain of such a simplified analysis.