Gradient-extended anisotropic brittle damage modeling using a second order damage tensor – Theory, implementation and numerical examples

Abstract

The paper presents a general gradient-extended continuum mechanical framework for materials with internal variables based on additional generalized balance equations. The framework is applied to the case of anisotropic brittle damage where damage is modeled by a second order damage tensor. Although using a second order damage tensor the proposed efficient formulation being implemented into finite elements uses only one scalar additional nodal degree of freedom. Based on the damage growth criterion a specific form of the elastic strain energy is proposed for initially isotropic materials such that artificial stiffening effects are excluded a priori. Special focus is placed on the numerical implementation at the integration point level: Within the concept of generalized standard materials a regularized dissipation potential is used to cope with different inequality constraints, leading to the introduction of penalty viscosity parameters which are chosen sufficiently large such that the occurring errors remain negligibly small. Furthermore, a novel additional damage hardening is suggested which ensures that the eigenvalues of the damage tensor do not exceed the value one. By means of several numerical examples it is demonstrated that the model delivers mesh-independent results and is able to represent (i) localized damage (fracture) and (ii) diffuse (distributed) damage. Finally, isotropic damage (which can be shown to be a special case of the model) and anisotropic damage are compared considering two numerical examples where the occurrence of either localized or diffuse damage will be shown to be crucial.