Dynamic cohesive fracture: Models and analysis

Our goal in this paper is to initiate a mathematical study of dynamic cohesive fracture. Mathematical models of static cohesive fracture are quite well understood, and existence of solutions is known to rest on properties of the cohesive energy density ψ, which is a function of the jump in displacement. In particular, a relaxation is required (and a relaxation formula is known) if ψ′(0+) ≠ ∞. However, formulating a model for dynamic fracture when ψ′(0+) = ∞ is not straightforward, compared to when ψ′(0+) is finite, and especially compared to when ψ is smooth. We therefore formulate a model that is suitable when ψ′(0+) = ∞ and also agrees with established models in the more regular case. We then analyze the one-dimensional case and show existence when a finite number of potential fracture points are specified a priori, independent of the regularity of ψ. We also show that if ψ′(0+) < ∞, then relaxation is necessary without this constraint, at least for some initial data.