Piezoelectric constitutive equations

The full tensor form of the piezoelectric constitutive equations can be written by adding the linear elastic (Hook s law) and dielectric responses to Equation (16.1) [1] ... 

Actually, D and x can also be used as independent variables. By choosing one of the electric variables (E and D) and one of the elastic variables (X and x) as the two independent variables, different piezoelectric constitutive equations can be written to describe the piezoelectric response under different conditions. In addition to Equation (16.2), there are three more sets of piezoelectric constitutive equations. 

Figure 1 shows the geometrical configuration of the piezoelectric ceramic disk with radius R and thickness h. The piezoelectric ceramic disk is assumed to be thin (R h) and polarized in the thickness direction. If the cylindrical coordinates (r, 0, z) with the origin in the center of the disk are used. The linear piezoelectric constitutive equations of a piezoelectric ceramic with crystal symmetry Camm, can be expressed as [IEEE, 1987] ...

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

The constitutive equations of piezoelectric media in linear range coupling the two are given by... 

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 ... 

Equations for structural dynamics within the solid material, along with the standard equations for electrostatics, are sufficient to model most piezoelectric materials and their applications. The full electromagnetic equations are unnecessary since the mechanical motion propagates at 10 m/s, several orders of magnitude less than the electric field, permitting the so-called quasistatic assumption. For the same reason, magnetic fields are typically weak and may be safely ignored. Still, equations to describe the electrical and mechanical behavior are needed in addition to the constitutive equations for the material. 

The constitutive equation of IPMC based upon small deflections may be derived based on the constitutive equation of piezoelectricity ... 

Here s are the elastic strains, a are the stresses, E are the electric potentials and D are the electrie displacements. [C] represents the adiabatic elastic compliance tensor at constant electric filed, [d] is the adiabatic piezoelectric tensor and [p] is the adiabatie electrie permittivity at constant stress. From these constitutive equations, it is readily known that the pie-zoelectrie element generates eleetrie signals due to mechanical motions and vice versa. 

For instance, consider a piezoelectric crystal. Assume a one-dimensional model of the crystal and let F be the force acting on the crystal, x the mechanical deformation, u the voltage across the crystal, and q its electrical charge for the time instant t. Then a commonly known form of the constitutive equations is... 

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. 

The subsequent characterization of electromechanical coupling covers the various classes of piezoelectric materials. Details with respect to definition and determination of the constants describing these materials have been standardized by the Institute of Electrical and Electronics Engineers [f04]. Stresses flux density D and field strength E on the electrostatic side, may be arbitrarily combined into four forms of coupled constitutive equations ... 

For piezoelectric materials with at least orthotropic behavior and polarization along the ea-direction, the constitutive equation is... 

In the structure of the constitutive equation for the mechanically and electrostatically orthotropic piezoelectric material of Eq. (4.17), the partial coupling needs to be noted. The normal stresses and strains in all three directions are solely connected to flux density and field strength along the polarization direction ... 

Electrostatic held components in parallel with the polarization direction induce normal mode actuation, see Eq. (4.19). Not visible in the constitutive equation, but explainable by a behavior corresponding to Poisson s effect included in the piezoelectric constants, the signs of strains or stresses parallel and transverse to the applied electrostatic fields are opposed. The deformation of a piezoelectric cube subjected to electrostatic fields in the direction of polarization is shown in Figure 4.3. 

For the remaining case of unidirectional electric flux density confined to the component D3, the transverse electric field strength components E, E2 may be expressed in terms of the shear strains 731, 723. Therewith Ei and E2 can be eliminated from the constitutive equations by static condensation. Thus, this modification of Eqs. (4.20) represents a purely mechanical interaction with strengthened shear stiffnesses as the result of the piezoelectric effect ... 

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. 

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 ... 

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