Existence, energy identity and higher time regularity of solutions to a dynamic visco-elastic cohesive interface model

TL;DR

This paper analyzes a complex PDE system modeling visco-elastic materials with a cohesive interface, establishing existence, energy conservation, and higher regularity of solutions under realistic physical laws.

Contribution

It introduces a comprehensive analysis of a coupled PDE system with a cohesive interface, proving existence, energy identity, and higher time regularity of solutions.

Findings

Existence of weak solutions via time discretization and regularization.

Proof of energy identity for the system.

Solutions with bounded acceleration in $L^ abla(0,T; L^2)$.

Abstract

We study the dynamics of visco-elastic materials coupled by a common cohesive interface (or, equivalently, {two single domains separated by} a prescribed cohesive crack) in the anti-plane setting. We consider a general class of traction-separation laws featuring an activation threshold on the normal stress, softening and elastic unloading. In 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 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 $L^\infty (0,T; L^2)$.