This work studies a mathematical model describing the static process of contact between a piezoelectric body and a thermally-electrically conductive foundation. The behavior of the material is modeled with a thermo-electro-elastic constitutive law. The contact is described by Signorini's conditions and Tresca's friction law including the electrical and thermal conductivity conditions. A variational formulation of the model in the form of a coupled system for displacements, electric potential, and temperature is derived. Existence and uniqueness of the solution are proved using the results of variational inequalities and a fixed point theorem.

In this paper, we study a quasi‐static contact problem between a thermo‐electro‐viscoelastic body and an electrically and thermally conductive foundation. The contact is modeled using a normal compliance contact condition with Coulomb's friction law, a regularized electrical contact condition and a heat flux contact condition taking into account the frictional heating effects. After introducing the model, the functional framework and the assumptions about the data, we derive the variational formulation of the model and prove its solvability. The proof is based on results of variational inequalities and fixed point argument. Finally, we investigate a fully discrete approximation using, respectively, Euler scheme and finite element method for the spatial variable and the time derivatives. Some error estimates are then derived, leading to convergence results under suitable additional regularity conditions.

In this paper, we consider the penalty method to solve the unilateral contact with friction between an electro-elastic body and a conductive foundation. Mathematical properties, such as the existence of a solution to the penalty problem and its convergence to the solution of the original problem, are reported. Then, we present a finite elements approximation for the penalised problem and prove its convergence. Finally, we propose an iterative method to solve the resulting finite element system and establish its convergence.

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Abstract. We consider a mathematical model which describes the frictional unilateral contact between a thermo-piezoelectric body and a rigid electrically conductive foundation. The thermo-piezoelectric constitutive law is assumed to be nonlinear and the contact is modeled with the Signorini condition, the nonlocal Coulomb friction law with slip dependent friction coefficient and the regularized electrical and thermal conductivity conditions. The variational form of this problem is a coupled system which consists of a nonlinear variational inequality for the displacement field and two nonlinear variational equations for the electric potential and the temperature. The existence of a unique weak solution to the problem is proved by using abstract results for elliptic variational inequalities and fixed point arguments.