A numerical assessment of phase-field models for brittle and cohesive fracture: Γ-Convergence and stress oscillations

• A numerical demonstration that in phase-field models for brittle fracture the smeared crack length may not converge.
• A demonstration that the numerical results of the phase-field model for brittle fracture are sensitive to the boundary conditions.
• A proof that the phase-field model for cohesive fracture does not satisfy a two-dimensional patch test.

Recently, phase-field approaches have gained popularity as a versatile tool for simulating fracture in a smeared manner. In this paper we give a numerical assessment of two types of phase-field models. For the case of brittle fracture we focus on the question whether the functional that describes the smeared crack surface approaches the functional for the discrete crack in the limiting case that the internal length scale parameter vanishes. By a one-dimensional example we will show that Γ-convergence is not necessarily attained numerically. Next, we turn attention to cohesive fracture. The necessity to have the crack opening explicitly available as input for the cohesive traction-relative displacement relation requires the independent interpolation of this quantity. The resulting three-field problem can be solved accurately on structured meshes when using a balanced interpolation of the field variables: displacements, phase field, and crack opening. A simple patch test shows that this observation does not necessarily extend to unstructured meshes.