Common Piezoelectric Continua and Active Piezoelectric Structures

Abstract

In this chapter, applications of the generic piezoelectric shell theories to a number of common piezoelectric continua were presented. A four-step reduction procedure was introduced and it was demonstrated in two geometries. The first case was a piezoelectric plate which includes 1) a thick plate and 2) a thin plate. The derived system equations of the thick piezoelectric plate were completely identical to published results (Tiersten, 1969). The second case was a piezoelectric shell of revolution which represents another class of shell continua e.g., piezoelectric spheres, cylinders, cones, etc., which were discussed in detail. Applications of the generic shell vibration theory to other piezoelectric continua can be further explored. Note that the theory was derived based on a symmetrical hexagonal piezoelectric structure—class ${\text{C}}_{6{\text{v}}} = 6\, {\text{mm}}$.

Appendix: Thick Piezoelectric Plate Equations

In Linear Piezoelectric Plate Vibration (Tiersten, 1969; p. 58), the electromechanical equations of a piezoelectric thick plate are written in a tensor expression:

$$\begin{array}{*{20}l} \begin{aligned} & \text{c}_{11} \text{u}_{1,11} + \left( {\text{c}_{11} + \text{c}_{66} } \right)\text{u}_{2,12} + \left( {\text{c}_{13} + \text{c}_{44} } \right)\text{u}_{3,13} + \text{c}_{66} \text{u}_{1,22} \\ & \quad + \text{c}_{44} \text{u}_{1,33} + \left( {\text{e}_{31} + \text{e}_{15} } \right){ {\varphi } }_{,13} = {\uprho} \ddot{\text{u}}_{1} \\ \end{aligned} \hfill \\ \end{array} ,$$

(A.1)

$$\begin{array}{*{20}l} \begin{aligned} & \text{c}_{66} \text{u}_{2,11} + \left( {\text{c}_{66} + \text{c}_{12} } \right)\text{u}_{1,12} + \text{c}_{11} \text{u}_{2,22} + \left( {\text{c}_{13} + \text{c}_{44} } \right)\text{u}_{3,23} \\ & + \text{c}_{44} \text{u}_{2,33} + \left( {\text{e}_{31} + \text{e}_{15} } \right){ {\varphi } }_{,23} = {\uprho} \ddot{\text{u}}_{2} \\ \end{aligned} \hfill \\ \end{array} ,$$

(A.2)

$$\begin{array}{*{20}l} \begin{aligned} & \text{c}_{44} \text{u}_{3,11} + \left( {\text{c}_{44} + \text{c}_{13} } \right)\text{u}_{1,31} + \text{c}_{44} \text{u}_{3,22} + \left( {\text{c}_{44} + \text{c}_{13} } \right)\text{u}_{2,23} \\ & + \text{c}_{33} \text{u}_{3,33} + \text{e}_{15} { {\varphi } }_{,11} + \text{e}_{15} { {\varphi } }_{,22} + \text{e}_{33} { {\varphi } }_{,33} = {\uprho} \ddot{\text{u}}_{3} \\ \end{aligned} \hfill \\ \end{array} .$$

(A.3)