Linear tensor properties

In this section we provide a brief review of topics in linear algebra and tensor property relations that are used frequently throughout the book. Nye s book on tensor properties contains a complete overview and is also a valuable resource [6]. 

Piezoelectric materials are materials that exhibit a linear relationship between electric and mechanical variables. Electric polarization is proportional to mechanical stress. The direct piezoelectric effect can be described as the ability of materials to convert mechanical stress into an electric field, and the reverse, to convert an electric field into a mechanical stress. The use of the piezoelectric effect in sensors is based on the latter property. For materials to exhibit the piezoelectric effect, the materials must be anisotropic and electrically poled ie, there must be a spontaneous electric field maintained in a particular direction throughout the material. A key feature of a piezoelectric material involves this spontaneous electric field and its disappearance above the Curie point. Only solids without a center of symmetry show this piezoelectric effect, a third-rank tensor property (14,15). 

In studies on the non-linear electric properties of dielectrics, one gains by introducing a tensor accounting for the variation induced in the electric permittivity by the polarizing field ... 

The linear elasticity of an anistropic solid is a tensor property of rank 4. It... 

Let us now derive phenomenological equations of the kind (5.193) corresponding to the expression (5.205). As has been mentioned before, each flux is a linear function of all thermodynamic forces. However the fluxes and thermodynamic forces that are included in the expression (5.205) for the dissipative function, have different tensor properties. Some fluxes are scalars, others are vectors, and the third one represents a second rank tensor. This means that their components transform in different ways under the coordinate transformations. As a result, it can be proven that if a given material possesses some symmetry, the flux components cannot depend on all components of thermodynamic forces. This fact is known as Curie s symmetry principle. The most widespread and simple medium is isotropic medium, that is, a medium, whose properties in the equilibrium conditions are identical for all directions. For such a medium the fluxes and thermodynamic forces represented by tensors of different ranks, cannot be linearly related to each other. Rather, a vector flux should be linearly expressed only through vectors of thermodynamic forces, a tensor flux can be a liner function only of tensor forces, and a scalar flux - only a scalar function of thermodynamic forces. The said allows us to write phenomenological equations in general form... 

In this paragraph we specialize the results for the nonsimple fluid (3.171)-(3180) on the linear dependence in vectors and tensors i.e., in D, g and h (while the dependence on scalars p, T may be nonlinear) [9, 14, 23, 24, 27, 45]. We denote this model as a linear fluid or fluid with linear transport properties because the results describe the classical Navier-Stokes (Newtonian) and Fourier fluid with linear viscosity and heat conduction at the same time the classical thermodynamic relations (local equilibrium) are valid. 

In this book, we confine ourselves only to the special case of fluids mixture (4.128) which is linear in vector and tensor variables.We denote it as the chemically reacting mixture of fluids with linear transport properties or simply the linear fluid mixture [56, 57, 64, 65]. Then (see Appendix A.2) the scalar, vector and tensor isotropic functions (4.129) linear in vectors and tensors (symmetrical or skew-symmetrical) have the forms ... 

Equation 3 can now be employed to calculate the linear and nonlinear susceptibilities in (1) and (2), revealing very specific tensor properties exhibited by poled polymers. With the foregoing model for the potential energy, the thermodynamic averages are sometimes also expressed in terms of the Langevin functions, which are defined as... 

Liquid crystals are anisotropic materials, hence their linear optical properties are determined by a symmetrical dielectric tensor, e, rather than a scalar refractive index. In certain cases it is possible to prepare uniformly oriented liquid crystal films. In these films e is a constant and light propagation can be described by the well-known laws of crystal optics. ... 

Table 7.3 Structure of the tensor components for some of the non-linear material properties in quartz (32)...

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. 

Here c[-], which will be called the elastic modulus tensor, is a fourth-order linear mapping of its second-order tensor argument, while b[-], which will be called the inelastic modulus tensor, is a linear mapping of k whose form will depend on the specific properties assigned to k. They depend, in general, on and k. For example, if k consists of a single second-order tensor, then in component form... 

Besides the elementary properties of index permutational symmetry considered in eq. (7), and intrinsic point group symmetry of a given tensor accounted for in eqs. (8)-(14), much more powerful group-theoretical tools [6] can be developed to speed up coupled Hartree-Fock (CHF) calculations [7-11] of hyperpolarizabilities, which are nowadays almost routinely periformed in a number of studies dealing with non linear response of molecular systems [12-35], in particular at the self-consistent-field (SCF) level of accuracy. 

Most of the four above-mentioned properties for Raman spectra can be explained by using a simple classical model. When the crystal is subjected to the oscillating electric field = fioc " of the incident electromagnetic radiation, it becomes polarized. In the linear approximation, the induced electric polarization in any specific direction is given by Pj = XjkEk, where Xjk is the susceptibility tensor. As for other physical properties of the crystal, the susceptibility becomes altered because the atoms in the solid are vibrating periodically around equilibrium positions. Thus, for a particular... 

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