Simulation of strain localization with gradient enhanced damage models

The incorporation of higher order strain gradients into the constitutive equations of continuum damage mechanics is presented. Thereby, not only scalar-valued isotropic damage models but also anisotropic damage models allowing for directional dependent sti€ness degradation are elaborated. An elegant possibility of describing anisotropic material behavior based on the microplane theory is demonstrated. Its conceptual simplicity originates from the idea of modeling the material behavior through uniaxial stress±strain laws on several individual material planes. For each plane individual damage loading functions are introduced allowing for di€erent failure modes. In order to account for long ranging microstructural mechanisms, second-order gradients of the strains are incorporated in each of these damage loading functions. The overall response can be determined by an integration of the resulting microplane laws over the solid angle. The features of gradient enhanced continuum damage are demonstrated by means of several selected examples. Ó 1999 Elsevier Science B.V. All rights reserved.