A variational approach to the phase field modeling of brittle and ductile fracture

The modeling of the post-critical behavior of materials is still a scientific challenge. This is especially true when dealing with materials that undergo complex behavior, in which several mechanisms are combined. This physical complexity is reflected in the mathematics and the numerics of this kind of problems. In this work, we study the modeling of brittle and ductile fracture. We adopt regularized kinematics based on a phase field description of the fracture topology. In order to ensure mathematical soundness, we use a rigorous variational framework for dissipative rate-independent materials. This framework allows to introduce several dissipative mechanisms in a straightforward and clear manner. For instance, gradients for both damage and plasticity are introduced. This implies the existence of two internal length scales that control the degree of ductility of the macroscopic fracture mechanism. A finite element discretization allows to test the possibilities of the proposed model to describe different fracture behaviors with several benchmark numerical experiments. In addition, the variational framework naturally leads to a robust staggered algorithm. Despite the simplicity of the numerical solution, different types of fracture processes can be described as particular cases: quasi-brittle, elasto-plastic brittle, and ductile.