Spectral Properties of Nanodimensional Piezoelectric Bodies with Voids and Surface Effects

Abstract

This chapter considers the eigenvalue problems for nanodimensional piezoelectric bodies with voids and with account for uncoupled mechanical and electric surface effect. The piezoelectric body is examined in frictionless contact with massive rigid plane punches and covered by the system of open-circuited and short-circuited electrodes. The linear theory of piezoelectric materials with voids for porosity change properties according to Cowin-Nunziato model is used. For modelling the nanodimensional effects the theory of uncoupled surface stresses and dielectric films is applied. The weak statements for considered eigenvalue problem are given in the extended and reduced forms. By using methods of functional analysis, the discreteness of the spectrum, completeness of the eigenfunctions and orthogonality relations are proved. A minimax principle for natural frequencies is constructed which has the properties of minimality, similar to the well-known minimax principle for problems with pure elastic media. As a consequence of the general principles, the properties of an increase or a decrease in the natural frequencies, when the mechanical, electric and “porous” boundary conditions and the moduli of piezoelectric body with voids change, are established. All of these results have been determined for both problems with and without account for surface effects.