Modeling and Analysis of an Antiplane Piezoelectric Contact Problem

Abstract

We study a mathematical model which describes the antiplane shear deformations of a piezoelectric cylinder in frictional contact with a foundation. The process is static, the material behavior is described with a linearly electro-elastic constitutive law, the contact is frictional and the foundation is assumed to be electrically conductive. Both the friction and the electrical conductivity condition on the contact surface are described with subdi®erential boundary conditions. We derive a variational formulation of the problem which is in the form of a system of two coupled hemivariational inequalities for the displacement and the electric potential¯elds, respectively. Then we prove the existence of a weak solution to the model and, under additional assumptions, its uniqueness. The proofs are based on an abstract result on operator inclusions in Banach spaces. Finally, we present concrete examples of friction laws and electrical conductivity conditions for which our results are valid.

On the solvability of mixed variational problems with solution-dependent sets of Lagrange multipliers

“…These models have already been analysed in the literature; see, for example, [10,15] for a treatment of the model in the frame of quasi-variational inequalities, or [11][12][13] for a treatment in the frame of hemi-variational inequalities. We recall that a slip-dependent frictional contact law is a law in which the friction bound depends on the slip.…”

“…We recall that a slip-dependent frictional contact law is a law in which the friction bound depends on the slip. These models have already been analysed in the literature; see, for example, [10,15] for a treatment of the model in the frame of quasi-variational inequalities, or [11][12][13] for a treatment in the frame of hemi-variational inequalities. Currently, interest lies in a variational approach involving dual Lagrange multipliers, which allows us to apply modern numerical techniques in order to approximate the weak solution (see, for example, [16]).…”

The weak solution of an antiplane contact problem for electro-viscoelastic materials with long-term memory

“…General models for piezoelectric materials can be found in [2], [12], [23]. Static frictional contact problems for elastic and viscoelastic materials were studied in [3], [16], [18], [22], [21], [19], [20], [7], under the assumption that the foundation is insulated. Contact problems with normal compliance for electroviscoelastic materials were investigated in [15], [25].…”