One or two meshes for Lip-field and Phase-field damage formulations? Comparison and performance assessment of standard and alternative FE solvers

Continuum models developed within the framework of damage mechanics are commonly used to represent the behavior of brittle and quasi-brittle materials. However, as the material softens, the formulation become ill-posedness and loses objectivity. To address this issue, several non-local formulations have been proposed in the literature. In the Phase-field variational approach, the solution to the damage mechanics problem is obtained by minimizing an incremental energy potential. Regularization is introduced by adding terms that depend on the gradient of the damage field to the potential. More recently, a new variational formulation -the Lip-field formulation -has been proposed to prevent spurious damage localization. In this case, the potential remains purely local, and regularization is achieved by enforcing a Lipschitz regularity constraint on the damage field. This article compares these two regularization methods in terms of regularization properties, robustness, and computational cost. A new spatial discretization is provided for each of these formulations and one-dimensional finite element implementations using alternating minimization solvers are proposed and discussed.