An Antiplane Electro-elastic Problem With The Power-law Friction

In this paper the material used is electro- elastic and the friction and it is modeled with Tresca's law and the foundation is assumed to be electrically conductive. First we derive the well posedness mathematical model. In the second step, we give 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 of a unique weak solution to the model by using the Banach fixed-point Theorem. Keywords— Tresca's friction, electro-elastic material, variational inequality, weak solution, fixed point, antiplane shears deformation. Mathematics Subject Classification— 74G25, 49J40, 74F15, 74M10 1. INTRODUCTION We consider the antiplane contact problem for electro-elastic materials with Tresca friction law. In this new work, we assume that 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, 4, 5, 6, 7, 8]) in which both antiplane shear, contact, state of material with Trescafriction law 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. The main result we provide concerns the existence of a unique weak solution to the model, see for instance [2, 3, 6] for details. The rest of the paper is structured as follows. In Section s:2 we describe the well posedness mathematical model of the frictional contact process between electro-elastic body and a conductive deformable foundation. In Section s:3 we derive the variational formulation. It consists of a variational inequality for the displacement field coupled with a time-dependent variational equation for the electric potential. We state our main result, the existence of a unique weak solution to the model in Theorem 3.1. The Proof of the Theorem is provided in the end of Section s: 4, where it is based on arguments of evolutionary inequalities, and a fixed point Theorem.