Abstract
We consider a mathematical model which describes the antiplane shear deformation of a cylinder in frictionless contact with a rigid foundation. The adhesion of the contact surfaces, caused by the glue, is taken into account. The material is assumed to be electro-viscoelastic and the foundation is assumed to be electrically conductive. We derive a variational formulation of the model which is given by a system coupling an evolutionary variational equality for the displacement field, a time-dependent variational equation for the electric potential field and a differential equation for the bonding field. Then we prove the existence of a unique weak solution to the model. The proof is based on arguments of evolution equations with monotone operators and fixed point.
Highlights
Antiplane shear deformations are one of the simplest examples of deformations that solids can undergo: in antiplane shear of a cylindrical body, the displacement is parallel to the generators of the cylinder and is independent of the axial coordinate
The antiplane problems play a useful role as pilot problems, allowing for various aspects of solutions in Solid Mechanics to be examined in a simple setting
This coupling leads to the appearance of electric potential when mechanical stress is present and, mechanical stress is generated when electric
Summary
Antiplane shear deformations are one of the simplest examples of deformations that solids can undergo: in antiplane shear of a cylindrical body, the displacement is parallel to the generators of the cylinder and is independent of the axial coordinate. Static frictional contact problems for electro-elastic materials were studied in [9,10,11,12] and contact problems for electro-viscoelastic materials were considered in [13, 14] In all these references the foundation was assumed to be electrically insulated. In this paper we study an antiplane frictionless contact problem with adhesion for electro-viscoelastic materials, in the framework of the Mathematical Theory of Contact Mechanics, when the foundation is electrically conductive. Our interest is to describe a physical process in which both antiplane shear, contact, adhesion and piezoelectric effect are involved, leading to a well posedness mathematical problem. Taking into account the piezoelectric effect, the conductivity of the foundation and the adhesion in the study of an antiplane problem for viscoelastic materials represents the main novelty of this work. We prove the unique solvability of the intermediate problems, we consider a contraction mapping whose unique fixed point leads us to construct the solution of the original problem
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