Existence, Energy Identity, and Higher Time Regularity of Solutions to a Dynamic Viscoelastic Cohesive Interface Model

We study the dynamics of viscoelastic materials coupled by a common cohesive interface (or, equivalently, two single domains separated by a prescribed cohesive crack) in the antiplane setting. We consider a general class of traction-separation laws featuring an activation threshold on the normal stress, softening, and elastic unloading. In the strong form, the evolution is described by a system of PDEs coupling momentum balance (in the bulk) with transmission and Karush-Kuhn--Tucker conditions (on the interface). We provide a detailed analysis of the system. We first prove the existence of a weak solution, employing a time discrete approach and a regularization of the initial data. Then, we prove our main results: the energy identity and the existence of solutions with acceleration in L-infinity (0, T; L-2). Thanks to the latter we finally prove the existence of strong solutions satisfying the balance of forces in the bulk and on the interface.