Anisotropic damage state modeling based on harmonic decomposition and discrete simulation of fracture

This study proposes an anisotropic damage state modeling of the effective (damaged) elasticity tensor (E) over tilde as a function of damage based on (i) a discrete element model for quasi-brittle material and (ii) a decomposition of the elasticity tensor in covariants. A procedure is proposed to measure the evolution of effective elasticity tensors computed by a beam-particle model. Various multiaxial damaging loadings allow us to constitute a dataset of around 76 000 effective elasticity tensors (our virtual testing reference set). We then cross-identify the anisotropic/tensorial damage state for the whole dataset. A detailed analysis of the dataset, using the distance to orthotropy as a guideline, justifies representing the induced micro-cracking by a single second-order damage variable D, even in the final stages with strong micro-cracks interaction. To formulate the damage state coupling, we use a reconstruction formula of orthotropic elasticity tensors in terms of invariants and (tensor) covariants. Thanks to this formula, some parts of the effective elasticity tensors (E) over tilde (such as the dilatation part) are modeled exactly from the single damage variable. Constitutive equations are proposed for the remaining parts of (E) over tilde (such as its generalized shear modulus and fourth-order harmonic part) using physical assumptions from micro-mechanics and a sparse data driven approach. The proposed anisotropic damage state coupling (E) over tilde (D) accurately models the damaged elasticity tensors in multiaxial loading, proportional or non-proportional, up to high damages. The present study firstly highlights the need for an anisotropic damage model for quasi-brittle materials and, secondly, offers a methodology to formulate the damage state coupling by explicit formulas introducing at most two dedicated parameters: the (optional) nonlinear shear-damage coupling parameter m and the harmonic prefactor h.