Rayleigh’s Quotient of Linear Piezoelectricity and Its Use in the Approximate Calculation of the Natural Frequencies of Piezoelectric Continua

The paper discusses an extension to the linear theory of piezoelectricity of the Rayleigh quotient used in the analysis of the properties and the approximate calculation of the natural frequencies of elastic continua. It is shown that for a piezoelectric continuum an infinite number of equivalent expressions can be obtained which generalize the classical Rayleigh quotient. The stationarity conditions of any of these quotients under additional constraints imposed by the Gauss equation of electrostatics and the prescribed natural electrical boundary condition are shown to result in the complete set of the governing equations of the free vibration problem of a piezoelectric continuum. The general results discussed in the paper are illustrated by the approximate calculation of the natural frequencies of a piezoelectric rod by the Rayleigh–Ritz method. Unlike in the case of elastic structures, no monotonic convergence of the approximate frequencies is guaranteed for a piezoelectric continuum, the property which can be explained using the introduced Rayleigh quotients.

Impermeable Crack and Permeable Crack Assumptions, Which One is More Realistic?

This paper investigates the applicability and effect of the crack-free electrical boundary conditions in piezoelectric fracture. By treating flaws in a medium as notches with a finite width, the results from different electrical boundary condition assumptions on the crack faces are compared. It is found that the electrically impermeable boundary is a reasonable one for engineering problems. Unless the flaw interior is filled with conductive media, the permeable crack assumption may not be directly applied to the fracture of piezoelectric materials in engineering applications.