An Introduction to the Theory of Piezoelectricity

In this chapter we develop the nonlinear theory of electroelasticity for large deformations and strong electric fields. Readers who are only interested in the linear theory of piezoelectricity may skip this chapter and begin with Chap. 2, Sect. 2.​2. This chapter uses the two-point Cartesian tensor notation, the summation convention for repeated tensor indices, and the convention that a comma followed by an index denotes partial differentiation with respect to the coordinate associated with the index.

In this chapter we specialize on the nonlinear equations in Chap. 1 for large deformations and strong fields to the case of infinitesimal deformations and weak fields, which results in the linear theory of piezoelectricity. A few theoretical aspects of the linear theory are also discussed.

In this chapter, some static solutions to the equations of linear piezoelectricity are presented. A few simple deformations useful in piezoelectric devices are discussed. The concept of electromechanical coupling factor is introduced. In Particular, Sects. 3.3, 3.4, 3.8, 3.9, 3.10, 3.11, 3.12, and 3.13 are on antiplane deformations of polarized ceramics [1].

This chapter is on waves in regions unbounded in at least one direction. These waves can be propagating or stationary waves. They are nontrivial solutions of homogeneous differential equations and boundary conditions. Sections 4.2, 4.3, 4.4, 4.5, 4.6, 4.7, 4.8, 4.9, 4.10, 4.11, 4.12, and 4.13 are on antiplane problems of polarized ceramics for which the notation in Sect. 2.9 is followed.

In this chapter we study vibrations of finite piezoelectric bodies. In some cases, for example, thickness vibrations of unbounded plates, although the in-plane dimensions of the plates are infinite, what matters is the finite plate thickness. Sects. 5.2 and 5.7 are on antiplane problems of polarized ceramics for which the notation in Sect. 2.9 is followed. The solutions in Sects. 5.1, 5.2, 5.3, 5.4, 5.5, 5.6, and 5.7 are exact. Those in Sects. 5.8, 5.9, and 5.10 are approximate.

The theory of linear piezoelectricity assumes infinitesimal deviations from an ideal reference state of the material in which there are no pre-existing mechanical and/or electrical fields (initial or biasing fields). The presence of biasing fields makes a material apparently behave like a different one, and renders the linear theory of piezoelectricity invalid. The behavior of electroelastic bodies under biasing fields can be described by the theory for infinitesimal incremental fields superposed on finite biasing fields, which is a consequence of the nonlinear theory of electroelasticity when it is linearized around the bias. Knowledge of the behavior of electroelastic bodies under biasing fields is important in many applications including the buckling of thin electroelastic structures, frequency stability of piezoelectric resonators, acoustic wave sensors based on frequency shifts due to biasing fields, characterization of nonlinear electroelastic materials by propagation of small-amplitude waves in electroelastic bodies under biasing fields, and electrostrictive ceramics which operate under a biasing electric field. This chapter presents the linear theory for small fields superposed on finite biasing fields in an electroelastic body.

In this chapter we discuss a few effects that are external to the quasistatic theory of electroelasticity developed in Chap. 1 which is valid through Chap. 6. These include the effects of heat conduction; mechanical and electrical dissipations due to viscosity, dielectric loss and semiconduction; nonlocal and gradients effects; as well as electromagnetic effects.

This chapter presents theoretical analyses on a few piezoelectric devices. Sections 8.1 through 8.3 are based on the exact theory of linear piezoelectricity. In particular, Sect. 8.2 is an antiplane problem for which the notation of Section 2.9 is followed. Sections 8.4 and 8.5 employ approximate one-dimensional equations for thin rods. Sections 8.6 and 8.7 are on elastic rods and beams with piezoelectric elements in extension and bending, respectively.