Variational formulation of loading-induced damage anisotropy: Theoretical framework for second-order anisotropic damage and Lip-field regularization

Variational approaches for damage and fracture are now well established. Most of them are restricted to isotropic (scalar) damage, but tensorial internal variables are needed to properly account for loading-induced damage anisotropy. The objective of the present work is to present a theoretical framework for the variational formulation of second-order tensorial damage, in order to exploit the finesse and physical consistency of an anisotropic modeling of damage. Theoretical developments are conducted in order to investigate conceptual locks for such a type of formulation. A regularized anisotropic damage model is developed in a Lip-field framework, using the concept of accumulated damage. This quantity is introduced to define a scalar dissipation potential, whereas the tensorial damage variable appears in the free-energy potential. The proposed constitutive modeling serves both as a proof of concept and as a support for the analysis of the advantages and disadvantages of explicitly accounting for damage anisotropy through a tensorial damage variable. The proposed formulation is implemented in an ad-hoc finite element code in order to perform simple computations.