Vibrations of Linear Piezostructures. Andrew J. Kurdila

1.9 Piezoelectric bender actuators available from PI ceramic®, Source: GmbH & Co. KG owner.

1.3 Outline of the Book

The linear and bending actuators shown in Figure 1.8 and 1.9 are considered in examples throughout this text: they are perhaps the most simple macroscale actuators that are commercially available. In fact, they can also be used to understand the basic physics of piezoelectric sensors too. They illustrate the operating principles of and modeling techniques for piezoelectric composites that are evident in much more complex designs.

      Unfortunately, the theoretical underpinnings of piezoelectric mechanics embraces a wide collection of fields of study that must be synthesized. We begin in Chapter 2. Section 2.1 reviews the fundamentals of vectors, bases, and frames of reference. This section is vital in developing an understanding of how physical vectors such as velocity, acceleration, stress vectors, electric field vectors, and electric displacement vectors are represented. The section culminates in a presentation of rotation matrices and their essential role in constructing change of bases for different representations of vectors. Section 2.2 then extends these results by introducing multilinear operators, or tensors, that act between vector spaces. Since vectors are first order tensors, they are a special case of the tensors presented in Section 2.2. In addition, another collection of physical variables critical to linear piezoelectricity are understood as tensors. These include the stress tensor, linear strain tensor, permittivity tensor, piezoelectric coupling tensor, stiffness tensor, and compliance tensor. The section concludes with a discussion of the role of rotation matrices in the representation and change of basis formula for order tensors. Section 2.3 discusses symmetry properties and geometric properties of tensors and crystals. The discussion begins with an overview of the geometry of crystals in Section 2.3.1. The 14 Bravais lattices and seven crystal systems are defined, as are the 32 crystallographic point groups. This section concludes with examples of symmetry transformations for typical crystal classes, and a discussion of tensor invariance associated with symmetry operations.

Chapter 3 reviews the basics of continuum mechanics that are needed to build a coherent framework for linear piezoelectricity. The definition of the stress tensor in three dimensions is presented in Section 3.1.1, Cauchy's formula is presented in Section 3.1.2, and the equations of equilibrium are discussed in Section 3.1.3. Section 3.2 is dedicated to the study of the linear strain tensor, the mechanical field variable that is complementary to the stress tensor. The general definition of the linear strain tensor in three dimensions is presented, as well as the kinematic relationships for common structures such as the axial rod, Bernoulli–Euler beam, and Kirchoff plate. Strain energy is discussed in Section 3.3. The strain energy density function is introduced, and its role in determining additional symmetry properties of the material stiffness tensor is given. A review of the constitutive laws for linearly elastic materials is summarized in Section 3.4, and the structure of the constitutive relationships for triclinic, orthotropic, and transversely isotropic materials are summarized. The chapter ends with an introduction of the initial‐boundary value problem of linear elasticity in Section 3.5.

      We turn to a discussion of continuum electrodynamics in Chapter 4. Charge and current are introduced in Section 4.1, and the static electric and magnetic fields are discussed in Section 4.2. Maxwell's equations are introduced in Equation 4.10 in SI units. Section 4.3.1 relates the polarization and electric displacement vectors, and relates them to bound and mobile charge, respectively. Magnetization and magnetic field intensity are defined in Section 4.3.2, as are the free, bound, and polarization current densities, respectively. Section 4.3.3 discusses the form of Maxwell's equations in Gaussian units, which prove to be convenient for the derivation of the equations of piezoelectricity.

      Chapter 5 presents the theory of linear piezoelectricity, starting with some one dimensional examples in Section 5.1. Section 5.2 gives the detailed account of how the equations of linear piezoelectricity are derived from Maxwell's equations, and Section 5.2.2 summarizes the initial‐boundary value problem of linear piezoelectricity. Section 5.3 surveys the role of thermodynamics in the construction of various equivalent constitutive laws and their associated thermodynamic invariants. The structure of the constitutive laws generated by crystalline materials having different symmetry operations is described in Section 5.4.

      Chapter 6 focuses on the use of Newton's method to derive the governing equations for linearly piezoelectric composite structures. The axial actuator model, which is a prototype for the linear actuators of the type depicted in Figure 1.8, is treated in Sections 6.1 and 6.2. Section 6.3 presents an analysis of the beam actuator as shown in the introduction in Figure 1.9. Section 6.4 uses Newton's method to derive the governing equations for a simplified model of piezoelectric composite plate bending.

      Chapter 7 introduces powerful variational methods for deriving the governing equations of piezoelectric structures. The chapter begins with a review of variational calculus in Chapter 7.1. Hamilton's principle for mechanical systems is introduced in Section 7.2, and its generalization for linear piezoelectricity is presented in Section 7.3. The strength of these variational techniques is illustrated in Section 7.4, which shows how variational methods for electromechanical systems that consist of piezoelectric structures and attached ideal circuits can be modeled. Various authors have discussed variational methods for electromechanical systems over the years, and Section 7.5 discusses the relationships among some alternative forms of these principles. Section 7.6 illustrates how the electromechanical variational principle can be applied using Lagrangian densities instead of Lagrangian functions .

Chapter 8, the final chapter of this book gives a detailed description of approximation methods for linearly piezoelectric composite structures. The chapter begins in Section 8.1 with a discussion of the differences between classical, strong, and weak forms of the governing equations. As discussed in Section 8.1, the approximation strategies in the text are derived from the weak form of the governing equations. Section 8.2 gives a quick overview of modeling damping and dissipation, with the primary emphasis on viscous damping that is so popular in engineering vibrations texts. Galerkin approximations are introduced in Section 8.3, and example applications for the linear and bending actuators are summarized. Two classes of bases are described for use in the Galerkin approximations. Modal or eigenfunction bases are used in Section 8.3.1, and finite element functions are employed in Section 8.3.2. The chapter finishes with a collection of examples that summarize how transient and steady state solutions are obtained from the Galerkin approximations. Particular emphasis is placed on the derivation of complex frequency response equations and FRFs for the piezoelectric composite structures.

      The Appendix contains three sections that provide supplementary material for the discussions throughout the text. Section S.1 gives a streamlined summary of the basic