A phase field model for mixed-mode cohesive failure through fracture energy dissipation at crack surfaces

A new phase field model for mixed-mode cohesive failure is proposed. The coupled displacement and damage problems are addressed through a thermodynamic framework. The critical energy release rate (CERR) is revisited as the sum of the surface energy and the dissipative fracture energy associated with the mixed-mode ratio. By incorporating this CERR into the displacement-damage thermodynamic framework, the widely used Benzeggagh and Kenane (BK) mixed-mode criterion is directly introduced into the phase field evolution equation. Moreover, the crack angle and damage initiation criteria, consistent with the BK law, are derived through the thermodynamic framework. Four representative examples are conducted to validate the proposed phase field model, including a uniaxial loading test and mixed-mode failure tests on an L-shaped panel, a tension-shear mortar specimen, and an FRP-to-concrete joint. The first example demonstrates that the proposed phase field model can accurately reproduce the target mixed-mode cohesive relationships. In the last three examples, the predictions show very good agreement with experimental data both qualitatively and quantitatively. © 2025 Elsevier Ltd.