Harmonic vibrations of nanosized piezoelectric bodies with surface effects

We consider the harmonic and eigenvalue problems for piezoelectric nanodimensional bodies with account for surface stresses and surface electric charges. For harmonic problem new mathematical model is suggested, which generalizes the model of the piezoelectric medium with damping properties, boundary conditions of contact type and surface effects. The classical and generalized (weak) statements for harmonic and eigenvalue problems are formulated in the extended and reduced forms. The spectral properties of the eigenvalue problems with account for surface effects are determined. A variational minimal principle is constructed which is similar to the well-known variational principle for problems for pure elastic and piezoelectric media. The discreteness of the spectrum and completeness of the eigenvectors are proved. As a consequence of variational principle, the properties of the natural frequencies increase or decrease are established for changing the mechanical, electric and "surface" boundary conditions and the moduli of piezoelectric body. The finite element approaches are described for determination of the natural frequencies, the resonance and antiresonance frequencies and harmonic behavior of nanosized piezoelectric bodies with account for surface effects.