Anisotropic Damage Model Based on a Decomposition of the Elasticity Tensor in Terms of Covariants

Anisotropic damage models usually require complex experimental identification procedure. To avoid this issue, this study aims at deriving an anisotropic damage model from micromechanical simulations. In particular, we will focus on the determination of a relevant second order damage variable. In the first part, a particular-lattice model is used to generate a dataset of elasticity tensors. Such discrete models provide an explicit representation of cracking and its impact on the mechanical behavior. It is used to represent the behavior of a unit cell of a heterogeneous quasi-brittle material. The cell is submitted to a range of mechanical proportional and non-proportional loadings: tension, bi-tension, shear, ... During those loadings, two phases can be highlighted: diffuse micro-cracking appears first, then micro-cracks coalesce into a macro-crack. At each load step of each loading, the elasticity tensor is measured by means of periodic elastic loadings. Those measures provide the evolution of the elasticity tensor during a damaging loading. Then, the harmonic decomposition of 2D elasticity tensor is used to analyze those evolutions. It enables to describe the elasticity tensors by two invariants, one second order and one fourth order covariant tensors. Additionally, distances to isotropy and orthotropy are calculated. The analysis of the distance to isotropy shows that an anisotropic damage model is necessary. Moreover, the analysis of the distance to orthotropy shows that an orthotropic damage model is sufficient. In the second part, we seek to reconstruct the elasticity tensor of a cracked cell from the elasticity tensor of the undamaged cell and a second order tensor. The reconstruction is based on the harmonic decomposition and assumes that the elasticity tensor is orthotropic. The dilation tensor d=tr_12(E) is chosen as the second order variable. This choice leads to an exact reconstruction of the dilatation part. However, models are required to reconstruct isotropic and harmonic parts. Thanks to the orthogonality of the terms of the harmonic decomposition, reconstruction of both parts are independent. Different models, based on various assumptions, are proposed and compared. While building those models, we try to limit the number of parameters as much as possible. Then, the evolutions of elasticity tensors are reconstructed from the models and are compared to the exact measures. The results show that effective elasticity tensors can be satisfactorily reconstructed from a second order variable with few or no parameters. Finally, a damage variable is deduced from the second order tensor variable. In conclusion, a second order damage variable has been obtained from a micromechanical model. This variable enables to retrieve the elasticity tensor using a reconstruction formula based on the harmonic decomposition. A future study will aim at deriving an evolution law for the damage variable. It will lead to the complete formulation of an anisotropic damage model in the framework of thermodynamics of irreversible process. Moreover, the framework laid in this study will also be used to study the transition from diffuse micro-cracking to localized macro-crack.