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Existence and Uniqueness of Weak Solutions to Frictionless-Antiplane Contact Problems

Besma Fadlia, Mohamed Dalah, Delfim F. M. Torres

Abstract

We investigate a quasi-static-antiplane contact problem, examining a thermo-electro-visco-elastic material with a friction law dependent on the slip rate, assuming that the foundation is electrically conductive. The mechanical problem is represented by a system of partial differential equations, and establishing its solution involves several key steps. Initially, we obtain a variational formulation of the model, which comprises three systems: a hemivariational inequality, an elliptic equation, and a parabolic equation. Subsequently, we demonstrate the existence of a unique weak solution to the model. The proof relies on various arguments, including those related to evolutionary inequalities, techniques for decoupling unknowns, and certain results from differential equations.

Existence and Uniqueness of Weak Solutions to Frictionless-Antiplane Contact Problems

Abstract

We investigate a quasi-static-antiplane contact problem, examining a thermo-electro-visco-elastic material with a friction law dependent on the slip rate, assuming that the foundation is electrically conductive. The mechanical problem is represented by a system of partial differential equations, and establishing its solution involves several key steps. Initially, we obtain a variational formulation of the model, which comprises three systems: a hemivariational inequality, an elliptic equation, and a parabolic equation. Subsequently, we demonstrate the existence of a unique weak solution to the model. The proof relies on various arguments, including those related to evolutionary inequalities, techniques for decoupling unknowns, and certain results from differential equations.
Paper Structure (6 sections, 3 theorems, 76 equations)

This paper contains 6 sections, 3 theorems, 76 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Formulation of the Problem
  4. Variational Formulation
  5. Existence and Uniqueness Result
  6. Conclusions

Key Result

Lemma 1

Assume that 45--(49) holds. If $m^{h'} > \alpha^{h'}$, then there exists a unique solution $u^h \in W^{1,2}(0, T, V^h)$ to Auxiliary Problem tp1.

Theorems & Definitions (3)

  • Lemma 1
  • Lemma 2
  • Theorem 1