Piezoelectric constitutive relation

Piezoelectric solids are characterized by constitutive relations among the stress t, strain rj, entropy s, electric field E, and electric displacement D. When uncoupled solutions are sought, it is convenient to express t and D as functions of t], E, and s. The formulation of nonlinear piezoelectric constitutive relations has been considered by numerous authors (see the list cited in [77G06]), but there is no generally accepted form or notation. With some modification in notation, we adopt the definitions of thermodynamic potentials developed by Thurston [74T01]. This leads to the following constitutive relations ... 

From the piezoelectric constitutive relations it is a sinqile matter to derive the wave equation for piezoelectric media. The piezoelectric wave equation is typi-... 

From a constitutive relation of the form t = t(D, ri) (here t is stress not time) it can be readily shown that, since there is no change in electric displacement in an open-circuit, thick-sample configuration, there are no secondary stresses due to electromechanical coupling. Nevertheless, the wavespeed is that of a piezoelectrically stiffened wave. 

It has been experimentally observed that for small deformations, the strain in a body is linearly proportional to the applied stress. In one dimension this is known as Hooke s law, relating the elongation of a spring or elastic material to the tensile force. A principle such as this, which relates stress to strain, is known as a constitutive relation, and can be generalized to three-dimensional, non-piezoelectric solids [1] ... 

The fundamental considerations of Chapter 3 are independent of the properties of the materials and therefore are not sufficient to describe the behavior of the mechanical or electrostatic system. The missing links are the constitutive relations between stresses and strains on the mechanical side, between flux density and field strength on the electrostatic side, and the connection between the mechanical and electrostatic side. Effects like piezoelectricity couple the mechanical and electrostatic fields. For the subsequent considerations identical material properties at every location of the continuum can be presumed due to its macroscopic homogeneity. 

As examined in detail in Section 4.4, it is possible to simplify the constitutive relation of piezoelectric materials in consideration of loading, electroding, and associated geometry of the structure. The two variants of electroding introduced above have different implications. 

These assumptions can be applied immediately in the directions ei and 2 of the cross-sectional plane of the fibers. As the constitutive relation of the considered piezoelectric materials exhibits only a partial electromechanical coupling, normal modes and each of the shear modes may be treated independently, as illustrated by Eqs. (4.19) to (4.21). In accordance with the considerations of Section 4.4.4, the notation with negated electric field strength and the associated constitutive submatrices E, Gi, and G2 will be utilized. 

Unlike the procedure laid out above, which introduces the factor C3, in the publications of Bent and Hagood [14,15] and Bent [13], the fraction is set to one only for the mechanical fields, while it is retained for the electrostatic fields. The resulting constitutive matrix of Eq. (5.27b) therefore becomes non-symmetric with respect to the piezoelectric coupling coefficients. Since the undermost line of the normal mode constitutive relation is not used there any further, this has no consequences. 

The physical basis for the design of piezoelectric membranes is based on simple combined electrical and mechanical relations (Gauss law and Hooke s law). The relationship between the electrical and mechanical properties of piezoelectrics is governed by the following constitutive equations ... 

Book content is otganized in seven chapters and one Appendix. Chapter 1 is devoted to the fnndamental principles of piezoelectricity and its application including related histoiy of phenomenon discoveiy. A brief description of crystallography and tensor analysis needed for the piezoelectricity forms the content of Chap. 2. Covariant and contravariant formulation of tensor analysis is omitted in the new edition with respect to the old one. Chapter 3 is focused on the definition and basic properties of linear elastic properties of solids. Necessary thermodynamic description of piezoelectricity, definition of coupled field material coefficients and linear constitutive equations are discussed in Chap. 4. Piezoelectricity and its properties, tensor coefficients and their difierent possibilities, ferroelectricity, ferroics and physical models of it are given in Chap. 5. Chapter 6. is substantially enlarged in this new edition and it is focused especially on non-linear phenomena in electroelasticity. Chapter 7. has been also enlarged due to mary new materials and their properties which appeared since the last book edition in 1980. This chapter includes lot of helpful tables with the material data for the most today s applied materials. Finally, Appendix contains material tensor tables for the electromechanical coefficients listed in matrix form for reader s easy use and convenience. 

Theory and Physics of Piezoelectricity. The discussion that follows constitutes a very brief introduction to the theoretical formulation of the physical properties of crystals. If a solid is piezoelectric (and therefore also anisotropic), acoustic displacement and strain will result in electrical polarization of the solid material along certain of its dimensions. The nature and extent of the changes are related to the relationships between the electric field (E) and electric polarization (P). which are treated as vectors, and such elastic factors as stress Tand strain (S), which are treated as tensors. In piezoelectric crystals an applied stress produces an electric polarization. Assuming Ihe dependence is linear, the direct piezoelectric effect can be described by the equation ... 

In this chapter, the transition from voluminous to areal structures, as already prepared in Section 4.4, will be implemented with special regard to laminated composites and adaptive capabilities making use of the piezoelectric effect for actuation as well as sensing. Therefore, a comprehensive constitutive description is developed and appropriate kinematic relations are specified. Afterwards, possibilities of different complexity for the reduction to a less general description are considered in view of specialized application cases. 

We conclude this introduction to piezoelectricity by mentioning that other constitutive equations are available for both the direct and the converse effect, providing several linear combinations of the four relevant physical quantities (y, other relations can be found elsewhere °° ° ° and are not strictly needed for what follows. 

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