Natural vibrations of nanodimensional piezoelectric bodies with contact-type boundary conditions

Homogeneous problems of vibrations of nanodimensional piezoelectric bodies with surface stresses and electric charges taken into account are studied. The boundary conditions modeling the frictionless contact of a body with rigid massive punches and the body coverings by a system of open and grounded electrodes are considered. The weak statements of these problems are given. It is proved that the spectrum is real and discrete and the system of eigenfunctions is complete. Several theorems on the eigenfrequency variations are stated with the surface effects taken into account and in the case of variations in the mechanical and electrical boundary conditions and the material characteristics. The finite element approximations are used to state finite-dimensional generalized eigenvalue problems. The results of finite-element calculations of the model problem, which illustrate the influence of surface effects, are given.