Theory of vibrations of coated, thermopiezoelectric laminae

Abstract

This study presents a theory for dynamic problems of coated laminae in which there is coupling between mechanical and electrical as well as thermal fields. The laminae is coated completely with perfectly conducting electrodes on both its faces, and it may comprise any number of bonded layers, each with a distinct but uniform thickness, curvature and electromechanical properties. First, a generalized variational theorem is derived so as to describe the complete set of the fundamental equations of thermopiezoelectricity. Next, by the use of this theorem, a system of two-dimensional, approximate governing equations of the coated laminae is constructed for the case when the mechanical displacement, electric potential, and temperature fields vary linearly across the laminae thickness. The effects of elastic stiffnesses of, and the interactions between, layers of the laminae and its electrodes are all taken into account. Also, the uniqueness of the governing equations is examined, and a theorem which includes the conditions sufficient for the uniqueness is given.

A uniqueness theorem in Biot's poroelasticity theory

“…By virtue of the conservation of energy together with the initial conditions (10,11) and the boundary conditions (8,9), the integrand of the total energy U of equation (19) and the integrand of the boundary quantity B of equation (24) vanish for the difference set of solutions Λ of equation (13). Accordingly, the quantities U and B which are calculated by integration from the difference set of solutions for the poroelastic region Ω+∂Ω have the same properties in the absence of discontinuities and singularities of any type.…”

“…Since the condition (25) requires the first term, the total energy of poroelastic region, to vanish and the boundary conditions (8,9) require the last term to vanish, then it follows from equation (34) thaẗ…”

“…In linear elasticity, the classical energy argument, that is, the positive-definiteness of material elasticities, is widely used in examining the uniqueness of solutions (e.g., [8]). By use of the energy argument, the first attempt at establishing the uniqueness of solutions in elastostatics is credited to Kirchoff [9] and that in elastodynamics to Neumann [10].…”

Shell Finite Elements for the Analysis of Multifield Problems in Multilayered Composite Structures

“…The first Finite Element (FE) model that relied on these formulations was proposed by [9]. [10], while [11] proposed coupled variational principles for a vibration analysis of thermo-piezoelectric plates.…”

Modeling a Piezothermoelastic Laminated Plate-Membrane

“…Xin and Xueye, as well as Utku, presented an interesting method of controlling a flexible mirror with a PVDF actuator by using wavefront Zernike polynomial decomposition (Xin and Xueye, 1994;Utku et al, 1995). Analytic governing systems of equations have been developed by various authors, but none directly solved (Do¨kmeci, 1978;Sun, 1993;Tzou and Howard, 1994). Solutions to date rely mainly on approximation methods, such as finite element methods, to produce the weak solution (Berdichevsky, 1979;Tzou and Tseng, 1990).…”