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 Ẽ 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 evolutions 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 Ẽ (such as the dilatation part) are modeled exactly from the single damage variable. Constitutive equations are proposed for the remaining parts of Ẽ (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 Ẽ(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