A quasistatic electro‐elastic antiplane contact problem with Tresca friction law

Abstract

We study the antiplane frictional contact models for electroelastic materials, both in quasistatic case. The material is assumed to be electro-elastic and the friction is modeled with Tresca's law and the foundation is assumed to be electrically conductive. First, we derive the classical variational formulation of the model which is given by a system coupling an evolutionary variational equality for the displacement field and a time-dependent variational equation for the potential field. Then we prove the existence and uniqueness of weak solution to the model.

Analysis and Study of Weak Solution of Antiplane Electro-Viscoelastic Problem

“…The uniqueness of the solution follows from the uniqueness of the fixed point of the operator Λ. It can also be obtained by using arguments similar as those used in [1,2].…”

“…Such kind of deformation, associated to a displacement field of the form (5), is called an antiplane shear, see for instance [1,2,4] for details. Below in this paper the indices i and j denote components of vectors and tensors and run from 1 to 3, summation over two repeated indices is implied, and the index that follows a comma represents the partial derivative with respect to the corresponding spatial variable; also, a dot above represents the time derivative.…”

“…We consider the case of antiplane shear deformation i.e., the dispalcement is parallel to the generators of the cylinder and is dependent of the axial coordinate. Our interest is to describe a physical process (for more details see [1,2,5]) in which both antiplane shear, contact, state of material with long-term memory and piezoelectric effect are involved, leading to a well posedness mathematical problem. In the variational formulation, this kind of problem leads to an integro-differential inequality.…”

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

“…Several articles have tackled the thermopiezoelectric contact problem with friction [5,[9][10][11][12][13], thermo-elasto-visco-plasticity [14][15][16], and thermo-visco-elasticity [17,18]. These studies are rooted in variations in constitutive laws and contact conditions [19][20][21][22][23][24][25].…”

Existence and uniqueness of the weak solution for a contact problem

“…For this reason, considerable progress has been made in their modeling and analysis, and the engineering literature concerning this topic (see [2,3,6,8,9,11]) is rather extensive. Mathematical and mechanical state of the art on contact mechanics can be found in [1,10,12,13].…”