Harmonic vibrations of nanosized piezoelectric bodies with surface effects

Abstract

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.

Nonlinear finite element modeling of vibration control of plane rod-type structural members with integrated piezoelectric patches

“…Let us note that in the piezoelectricity these matrices may have special properties which can be used for numerics, see, e.g., [96,[98][99][100][101].…”

Three-dimensional bending and vibration analysis of functionally graded nanoplates by a novel differential quadrature-based approach

“…They applied the Rayleigh variational principle to the weak setup of the problems and established certain spectral properties for them. Nasedkin and Eremeyev [47] formulated weak statements for the eigenvalue problems associated with the piezoelectric nanodimensional bodies with accounting for the surface stresses and surface electric charges. A variational minimal principle was constructed for the determination of the harmonic behavior of nanosized piezoelectric bodies.…”

Analysis of Surface Stress Influence on the Effective Properties of Nanoporous Piezocomposites

“…òàêaeå [9, 10, 25−28]). Î÷åâèäíî, ÷òî ýôôåêòèâíûå ïüåçîìîäóëè è äèýëåêòðè÷åñêèå ïðîíèöàåìîñòè äîëaeíû èçìåíÿòüñÿ ãîðàçäî áîëåå ñóùåñòâåííî, åñëè âìåñòî íåñâÿçàííûõ ìåõàíè-÷åñêèõ ïîâåðõíîñòíûõ íàïðÿaeåíèé íà èíòåðôåéñå ó÷èòûâàòü ïîëíîñòüþ èëè ÷àñ-òè÷íî ñâÿçàííûå ýëåêòðîìåõàíè÷åñêèå ïîâåðõíîñòíûå ýôôåêòû ïî ìîäåëÿì [23,24,29] .…”