Modern inelastic material model formulations rely on the use of tensor-valued internal variables. When inelastic phenomena include softening, simulations of the former are prone to localization. Thus, an accurate regularization of the tensor-valued internal variables is essential to obtain physically correct results. Here, we focus on the regularization of anisotropic damage at finite strains. Thus, a flexible anisotropic damage model with isotropic, kinematic, and distortional hardening is equipped with three gradient-extensions using a full and two reduced regularizations of the damage tensor. Theoretical and numerical comparisons of the three gradient-extensions yield excellent agreement between the full and the reduced regularization based on a volumetric-deviatoric regularization using only two nonlocal degrees of freedom.

翻译：现代非弹性材料模型公式依赖于张量值内部变量的使用。当非弹性现象包含软化时，前者的模拟容易产生局部化。因此，对张量值内部变量进行精确的正则化对于获得物理正确的结果至关重要。本文重点研究有限应变下各向异性损伤的正则化。为此，一种具有各向同性、随动和畸变硬化的柔性各向异性损伤模型配备了三种梯度扩展，采用损伤张量的完整正则化和两种约化正则化。三种梯度扩展的理论和数值比较表明，基于仅使用两个非局部自由度的体积-偏量正则化的完整正则化与约化正则化之间具有极好的一致性。