Cohesive phase-field fracture with an explicit strength surface: an eigenstrain-based return-mapping formulation

Abstract: Standard phase-field fracture methods are rooted in brittle fracture theory and therefore do not inherently prescribe a material strength for crack nucleation, while also struggling to capture cohesive fracture behaviour. Recent eigenstrain-based formulations overcome both limitations by introducing fracture eigenstrains that decouple the strength surface from the fracture energy, but their implementation has so far relied on direct energy-minimization frameworks rather than standard finite-element procedures. In this work, we exploit the fact that the eigenstrains require no spatial gradients and reformulate the eigenstrain evolution as a local constitutive model, analogous to those used in plasticity, that is resolved at each integration point. As a result, the cohesive phase-field requires no additional global degrees of freedom beyond those of a standard phase-field formulation and can be readily integrated into existing finite-element codes. Two strength criteria are considered: a non-smooth criterion with independent tensile and shear strengths, and a smooth Drucker-Prager-like criterion that captures pressure-dependent strengthening under compression. Consistent tangent operators are derived for both criteria, ensuring robust convergence of the global Newton-Raphson solver. The framework is validated against three benchmark problems: a plate with a hole under tension and compression, a single-edge notched plate under shear, and a notched plate under dynamic loading. The results demonstrate mesh-independent and phase-field length-scale-independent behaviour, confirm that the fracture energy governs the transition between brittle and cohesive regimes, and show that complex phenomena such as crack branching under dynamic loading are naturally captured. All source codes are openly available.