Phase field based nonlocal anisotropic damage mechanics model

•Order parameters, microcrack density and diagonal damage tensor are correlated.•Generalized free energy functional to account for anisotropic behavior is proposed.•Set of three coupled Allen–Cahn type PDEs are proposed to show damage evolution.•Two new tensors as interpolation and dissipation functions are proposed.•Application of the formulation to describe damage evolution in 2D is studied.

A nonlocal anisotropic damage theory is developed in this work through the phase field method to address the anisotropic damage evolution in materials. The anisotropic damage is discussed and appropriate nonconserved order parameters in three mutually perpendicular directions are defined to find the growth of the components of a second order diagonal damage tensor corresponding to the principal directions of a general second order damage tensor. In contrast to the previous models, two new tensors are proposed to act as interpolation and potential functions along with the Allen–Cahn equation in order to obtain the evolution of the order parameters, which is the basis of the definition of the damage rate. The tensor formulation of the growth of the components of the damage tensor is proposed for the first time. It is shown that, by introducing a set of material parameters including a length scale parameter due to damage, there is a robust and simplified way to model the nonlocal behavior of damage and predict the corresponding material behavior as components of a second order diagonal damage tensor.