One-dimensional comparative study of the Lip-field and Phase-field formulations for cohesive fracture

Developing numerical models for structures made of softening materials remains a challenging research topic. Continuum models within the framework of damage mechanics are commonly used for this purpose. However, as the material softens, the solution to the equilibrium problem becomes non-unique, leading to a pathological dependence on the finite element mesh. To address this issue, several "non-local" methods have been proposed in the literature. These methods introduce a characteristic length into the numerical formulation. In the last few decades, the Phase-Field variational approach to fracture [1,2] has gained popularity. In this approach, the solution to the damage mechanics problem is sought by minimizing an incremental energy potential, which is the sum of the elastic energy potential and an energy dissipation potential. Regularization is introduced by adding terms that depend on the gradient of the damage field in the second term. More recently, [3] proposed a new approach to prevent spurious damage localization, the Lipschitz formulation. In this case, the model remains purely local, and regularization is achieved by enforcing this a Lipchitz regularity constraint on the damage field. This constraint naturally incorporates the characteristic length into the formulation. The Lip-Field method can be designed to be non-local within sub-zones, significantly reducing computation time compared to other non-local approaches. Focusing on cohesive fracture [4], in this contribution, this new regularization method is compared with the Phase-Field model, in terms of regularization features, robustness, and computational cost. Several finite element implementations, using alternated minimization solvers, in one dimension are proposed. In particular, after comparing two standard implementations of these two approaches, alternative implementations of both approaches are designed to reduce computation times, decrease the number of iterations required for convergence, and improve overall convergence.