23. O.Yu.Zharii, Introduction to Dynamic Electroelasticity, Technischer Bericht des Heinz Nixdorf Institut, Universitaet-GH Paderborn, Mechatronik und Dynamik, 1996, 58 p.

The subject of mechanics of piezoelectric materials, also called electroelasticity, becomes not simply interesting, but an essential element of knowledge for specialists working in various theoretical and applied fields of modern mechanics and acoustics. This short text, in accordance with its title, is an attempt to give a complete formulation of initial boundary-value problems in linear mechanics of piezoelectric solids, where mechanical fields of stresses and deformations should be considered together with internal electric fields as enjoying equal rights components of a coupled or conjugate field.

To get acquainted with a subject, it is desirable to read first a thin book instead of a thick one. Having this in mind, the author tried to embody in this text two ideas: (i) on the base of a few examples to display specific features of conjugate wave fields in piezoelectric solids compared to those in purely elastic bodies, and (ii) to demonstrate ways of application of classic mathematical methods to solutions of boundary-value problems arising in dynamic electroelasticity.

In Section 1 we give a brief exposition of the phenomenological theory of the piezoelectric effect on an example of barium titanate ceramics. Then a general formulation of initial boundary-value problems of mechanics of piezoelectric materials is given. First, we introduce the system of equations describing conjugate wave fields in piezoelectrics. Then, for the specific case of piezoelectrics, namely piezoelectric ceramics, we give the resulting system of equations in the form of one vector and one scalar equations of the second order. After that, mechanical and electrical boundary conditions are introduced. At the end of the section, the general formulation of the Poynting theorem (or the energy conservation law) is derived and the concept of electromechanical coupling factor (EMCF) is introduced.

Simplest cases of one-dimensional deformation of piezoelectric solids are considered in Section 2. We show how to calculate EMCF for two modes of deformation of a piezoceramic layer. Then, using the example of longitudinal vibrations of a thin rod, we introduce the concept of the effective EMCF.

Two-dimensional problems are discussed in Section 3. We study plane wave propagation in an unbounded piezoceramic medium and radial vibrations of a thin disk.

In the last and the largest Section 4 we present the method of eigenfunction expansions in dynamic electroelasticity. Note, that this theory cannot be deduced directly from that developed for purely elastic media. Normal mode expansions are shown to be a natural mathematical and physical means for deeper understanding of specific features of dynamic fields in piezoactive media and for the study of electromechanical energy conversion.

Unlike preceding sections, containing information that belongs to "common knowledge", this material is original, and the section is based on the author's paper [16]. However, the way of derivation of eigenfunction expansions (using Betti's identity) here, is clearer and more straightforward than suggested earlier in [16].

Contents

Preface

Introduction

1 Linear theory of piezoelectricity

1.1 Origin of the piezoelectric effect in ceramic materials

1.2 Complete system of equations of dynamic electroelasticity

1.2.1 Constitutive equations

1.2.2 Equations of continuum mechanics and Maxwell equations

1.2.3 Vector system of equations of motion

1.3 Initial and boundary conditions

1.3.1 Mechanical initial and boundary conditions

1.3.2 Electrical boundary conditions

1.4 Internal energy of piezoelectric medium. Poynting theorem

1.5 Electromechanical coupling factors

2 One-dimensional problems of electroelasticity

2.1 Examples of calculation of EMCF

2.1.1 Thickness deformation of a layer

2.1.2 Shear deformation of a layer

2.2 Vibrations of a longitudinally polarized rod

2.2.1 One-dimensional equations of motion

2.2.2 Steady-state vibrations

2.2.3 Resonance and antiresonance frequencies

3 Two-dimensional problems of electroelasticity

3.1 Propagation of plane waves in an unbounded piezoceramic medium

3.2 Radial vibrations of a piezoceramic disk

3.2.1 Equations of motion

3.2.2 Planar EMCF

3.2.3 Harmonic vibrations of a free disk

4 Normal mode expansions in dynamic electroelasticity and their application

4.1 Preliminary remarks

4.2 Formulation of the problem in tensor notations

4.3 Normal modes and their properties

4.3.1 Betti's identity

4.3.2 Eigenvalue problem

4.3.3 Reality and positiveness of eigenvalues

4.3.4 Orthogonality of eigenfunctions and a series expansion of a vector function

4.4 Eigenfunction expansion of an initial boundary-value problem solution

4.4.1 Derivation of the displacement expansion

4.4.2 Expansion for the electric potential. The uncoupled potential

4.4.3 Energy relations associated with eigenfunction expansions

4.5 Electromechanical energy conversion treatment

4.5.1 Capacitance matrices of an N-electroded dielectric body

4.5.2 Two-electroded transducers: short-circuit eigenfunctions

4.5.3 Two-electroded transducers: open-circuit eigenfunctions

4.5.4 Multielectroded transducers

4.5.5 New use of Mason formula

4.6 Examples

4.6.1 Instantaneous electric loading of a piezoceramic rod

4.6.2 Electric discharge of a rod

4.6.3 Four-electroded disk transducer

References

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