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