Phase-field cohesive fracture theory: A unified framework for dissipative systems based on variational inequality of virtual works

This work develops a variational framework with dissipation for deriving governing equations in the phase-field cohesive fracture theory. We show that a phase-field model consistent with cohesive zone theory can be established on only two physical foundations. One of them is the law of energy conservation and the other is a variational inequality of virtual works which serves as a stability condition. The proposed phase-field model can implement a mixed-mode cohesive law of arbitrary form. The mode-dependent fracture toughness and direction-dependent crack driving force in a general cohesive fracture can be taken into account in a unified manner. Since brittle fracture can be viewed as an extreme case of cohesive fracture, the newly proposed method is also applicable to brittle fracture. Three numerical examples are presented to demonstrate the effectiveness of the presented method, including a uniaxial tension test, a mixed-mode fracture of concrete and a compressive failure of rock with flaws. In the first example, the numerical cohesive law is extracted and compared with the target analytical cohesive law. Excellent agreement is observed. In the last two examples, the crack morphology and mechanical response predicted by the model are in excellent agreement with the experimental results.