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 psi, which is a function of the jump in displacement. In particular, a relaxation is required (and a relaxation formula is known) if psi' (0(+)) not equal infinity. However, formulating a model for dynamic fracture when psi' (0(+)) = infinity is not straightforward, compared to when psi' (0(+)) is finite, and especially compared to when psi is smooth. We therefore formulate a model that is suitable when psi' (0(+)) = infinity 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 psi. We also show that if psi' (0(+)) <infinity, then relaxation is necessary without this constraint, at least for some initial data.