Thermodynamic vector linear dependence

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. 

Due to the linearity postulate in irreversible thermodynamics, each vector flux must depend linearly on all the vector forces in the system. We may thus rewrite the fluxes as [18] [19] ... 

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... 

We also note that the vector or tensor responses (3.187), (3.189) depend only on the vector or tensor driving forces respectively. This fact is known in linear irreversible thermodynamics as the Curie principle [36, 80, 88, 89] (cf. discussion in [34, 38]). Present theory shows however, that this property follows from the isotropy of constitutive functions and from the representation theorems of such linear functions, see Appendix A.2, Eqs.(A.ll)-(A.13) and (A.57)-(A.59). But representation theorems for nonlinear isotropic constitutive functions [64, 65] show that the Curie principle is not valid generally. 

P. Curie and J. Curie discovered the piezoelectric effect in 1880. It was found that, when a compressive or a tensile force was applied on some crystals along some special directions (for example, a quartz) electrical charges could be created on the corresponding surfaces of the crystal and the size of the created charge was proportional to the strength of the apphed force. This phenomenon is called the piezoelectric effect . All ferroelectric crystals show a piezoelectric effect. The piezoelectric effect can be described by piezoelectric equations. On the basis of thermodynamic principles, piezoelectric equations can be derived (e.g., see Xu, 1991). These equations describe linear relationships between the four variables stress tensor [T], strain tensor [S], electrical field vector E and electric displacement vector D. The piezoelectric equations can be expressed as four kinds of equations, depended on the variables. Selecting E and Tas variables, we have ... 

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