Unique solvability of a crack problem with Signorini-type and Tresca friction conditions in a linearized elastodynamic body

TL;DR

This paper proves the unique solvability of a dynamic crack problem in a linear elastic body with Signorini-type contact and Tresca friction conditions, without using viscosity regularization.

Contribution

It establishes the first proof of existence and uniqueness for this dynamic crack model with combined contact and friction laws without additional regularization.

Findings

Existence of a strong solution is proven.

Uniqueness of the solution is established.

No viscosity regularization is needed for the proof.

Abstract

We consider dynamic motion of a linearized elastic body with a crack subject to a modified contact law, which we call the Signorini contact condition of dynamic type, and to the Tresca friction condition. Whereas the modified contact law involves both displacement and velocity, it formally includes the usual non-penetration condition as a special case. We prove that there exists a unique strong solution to this model. It is remarkable that not only existence but also uniqueness is obtained and that no viscosity term that serves as a parabolic regularization is added in our model.