Existence results for unilateral contact problem with friction of thermo-electro-elasticity

Abstract

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.

Variational and Numerical Analysis of a Static Thermo-Electro-Elastic Problem with Friction

“…Here (15) is the normal compliance power law and (16) is a variant of Coulomb's friction law. Furthermore, the thermoelectric contact is described with the following regularized conditions (see [13,14]):…”

“…The reason lies in the considerable difficulties of the nonlinear evolutionary inequalities modeling the static contact problems present in the variational analysis. Existence and uniqueness results in the study of static contact problems can be found, for instance, in [14][15][16].…”

Variational and Numerical Analysis for Frictional Contact Problem with Normal Compliance in Thermo-Electro-Viscoelasticity

“…Mindlin [9] was the first to propose the thermopiezoelectric model. e mathematical model which describes the frictional contact between a thermo-piezoelectric body and a conductive foundation is already addressed in the static case in [10,11].…”

Penalty and regularization method for variational‐hemivariational inequalities with application to frictional contact

“…For instance, Fillep et al., Hüeber et al., Kunisch and Stadler, and Laborde and Renard who studied these conditions as a simpler alternative and an approximation of the Signorini condition with the Coulomb friction law, Essoufi et al . where a decomposition method in electro‐elastostatics was treated, Benaissa et al . who investigated these laws including the electric and thermal conductivity conditions.…”

“…Note that contact problems in mechanical engineering with small deformations are often described by the Signorini condition with the Coulomb friction law. For instance, Fillep et al, [9] Hüeber et al, [13] Kunisch and Stadler, [14] and Laborde and Renard [15] who studied these conditions as a simpler alternative and an approximation of the Signorini condition with the Coulomb friction law, Essoufi et al [8] where a decomposition method in electro-elastostatics was treated, Benaissa et al [1] who investigated these laws including the electric and thermal conductivity conditions. Schröder et al [28] and Slimane et al [29] who developed a mixed finite element approximation for the Signorini problem with the Tresca friction.…”