Modal analysis of piezoelectric bodies with voids. I. Mathematical approaches

The paper is concerned with the eigenvalue problems for piezoelectric bodies with voids in contact with massive rigid plane punches and coved 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. The generalized statements for eigenvalue problem are obtained in the extended and reduced forms. A variational principle is constructed which has the properties of minimality, similar to the well-known variational principle for problems with pure elastic media. The discreteness of the spectrum and completeness of the eigenfunctions are proved. The orthogonality relations for eigenvectors are obtained in different forms. As a consequence of variational 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 solid with voids change, are established.