A unified regularized variational cohesive fracture theory with directional energy decomposition

In this paper, the cohesive fracture is revisited as an energy minimization problem. The energy functional is rigorously derived based on a pair of integral transformations and a novel directional energy decomposition method. The resulting description of the regularized crack is perfectly suitable for cohesive fracture. A well-defined crack direction is associated with each material point. The resulting constitutive relation shows that a damage-induced material orthotropy within the regularized crack is introduced in this theory, which is suitable for cohesive fracture. The proposed method can implement an arbitrarily given mixed-mode cohesive law. The tensile and shear fracture modes are covered in a unified manner and controlled by the tensile-to-shear strength ratio. These theoretical results are presented with detailed proofs and verified numerically by some representative examples involving tension, shear, and compression.