Non-linear mechanical , electrical and thermal phenomena in piezoelectric crystals

Mechanical, electrical and thermal phenomena occurring in piezoelectric crystals were examined by non-linear approximation. For this purpose, use was made of the thermodynamic function of state, which describes an anisotropic body. Considered was the Gibbs function. The calculations included strain tensor εij = f(σkl, En, T ), induction vector Dm = f(σkl, En, T ) and entropy S = f(σkl, En, T ) as function of stress σkl, field strength En and temperature difference T . The equations obtained apply to anisotropic piezoelectric bodies provided that the “forces” σkl, En, T acting on the crystal are known.


Introduction
The present paper provides a thermodynamic description of elastic (mechanical), electrical and thermal properties of crystals in a non-linear approximation.The physical processes and phenomena that occur in a real crystal generally are of a non-linear character, similarly to those in any other physical body which displays a certain degree of nonlinearity.It is conventional to describe non-linear properties (i.e., physical nonlinearity) in terms of higher-order material constants, and such are the factors of proportionality incorporated in the Taylor series expansion of a non-linear relation describing the given effect.In general, the mentioned constants are tensors.
Nonlinearity manifests in the values of the second-order material constants being the functions of the values of the applied forces which act onto the crystal, e.g. the elasticity constants can be a function of the applied stress σ.Thus, the standard Hooke equation may be written as follows: c F.Warkusz, A.Linek where ε denotes strain and s(σ) is an elasticity constant which varies according to the stress σ applied.Hence, s(σ) takes the form where s 1 is a second-order material constant (modulus of elasticity).By linear approximation, equation (1.1) becomes as follows: ε = s 1 σ, where s 1 = ∂ε/∂σ.
Using non-linear approximation, we have where The coefficients s 1 , s 2 , s 3 , . . . of equation (1.2) can be regarded as material constants which are proportional to the stresses (σ) of relevant powers.Hence, they are material constants of second, third, fourth or higher order.Thus, the Hooke equation for non-linear effects can be written as [1,2] With tensor notation for anisotropic bodies, the relation of takes the following form: s = s ijkl + s ijklpq σ pq + s ijklpqrs σ pq σ rs + . . .

and the Hooke equation
in tensor notation, where s ijkl is a fourth-order tensor (second-order material constant), s ijklpq is a sixth-order tensor (third-order material constant), s ijklpqrs is an eight-order tensor (fourth-order material constant), and the coefficients i, j, k, l, p, q, r, s take the values of 1, 2, 3. Summation signs have been omitted [1,3,4].Higher-order material constants are formally derived from thermodynamic functions.The procedure is similar to that for the linear case.If, for example, strain ε is produced simultaneously by stress σ, by electric field E, and temperature variations T , the equation of state for non-linear processes, which includes material constants only up to the third order, can be written as where Apart from second-order material constants, s 1 , d 1 , α 1 , and third-order material constants s 2 , d 2 , α 2 , there are also third-order "mixed" material constants k σE , k σT and k ET .The order of the material constant is defined by the order of the derivative of the thermodynamic function and not by the order of the "force" acting onto the crystal or by the order of the material constant tensor.
The problem will be presented more in detail in section 2.

Thermodynamic relations in crystals and Gibbs functions
According to the first law of thermodynamics, the total energy U of a body is the sum of different energy types.In piezoelectric crystals, energy balance is primarily accounted for by mechanical, electrical and thermal energy.The effect of magnetic energy or gravitational energy on the phenomena occurring in piezoelectric crystals may be neglected.In general, there are eight thermodynamic functions that can be used to describe a piezoelectric phenomenon.The form of the function depends on the choice of the independent variables, selected according to the conditions under which the crystal is to be examined [2].
In the present paper we confine ourselves to thermodynamics, that is, Gibbs function G. Mathematical analysis enables us to define particular material constants, as well as to establish many interesting relations between them, by linear and nonlinear approximation.
Let us consider the differential form dG, which describes the state of the crystal following the application of three different "forces" -stress σ ij , electric field E m , and temperature T .The forces acting onto the crystal are independent variables.Thus, we have By virtue of the first and second law of thermodynamics, where i, j, m = 1, 2, 3. Substituting (2.3) into (2.2),we obtain Strain ε ij , induction D m and entropy S are functions of stress σ kl , electric field E n and temperature T : Differentiating the function G with respect to individual independent variables, and having determined the remaining values, we obtain Comparing the coefficients of (2.4) and (2.5), we can derive the relations that describe particular quantities: The indices in the lower part of the vertical line show the independent variable, which takes a constant value during differentiation.
The function ε ij , D m and S can be expanded into a Taylor series.All derivatives incorporated in the Taylor's series are the material constants.

Material constants derived from function ε ij
Second-order elasticity constant Second-order piezoelectric constant Second-order piezocalorific constant Third-order elasticity constant Third-order piezoelectric constant Third-order piezocalorific constant Third-order electrostriction constant Third-order constant Third-order thermal expansion constant

Material constants derived from function D m
Second-order piezoelectric constant Second-order permittivity constant Second-order electric heat constant Third-order piezoelectric constant Third-order electrostriction constant Third-order constant Third-order permittivity constant Third-order electric heat constant Third-order pyroelectric constant

Material constants derived from function S
Second-order piezocalorific constant Second-order electric heat constant Second-order thermal capacity constant Third-order piezocalorific constant Third-order constant Third-order thermal expansion constant Third-order electric heat constant Third-order pyroelectric constant Third-order thermal capacity constant

Equalities between material constants
With the material constants (tensors) defined above, we can rewrite function ε ij , D m and S from Taylor series into form: ) (2.8)

Linear and non-linear effects of the equations of state
Considering, for example, the equation of state derived from the Gibbs function G for non-linear effects, we can easily obtain the equations of state for linear effects by neglecting higher-order tensors.Thus, by virtue of (2.6), (2.7) and (2.8) we have which are valid for linear effects [7,8,9].By virtue of the symmetry of some tensors (tensor components in matrix tables), the relations of (3.1) can be presented in matrix form.
Hooke's law is often expressed in its contracted notation.Then, the equivalence between the components of the compliance fourth-rank tensor s ijkl and the components of the 6 × 6 matrix s is shown to be: Furthermore, the full tensor suffixes of the stresses σ and strains ε are contracted according to the scheme: The relations of (3.1) constitute the starting set of the equations of state, which describe the environment, and they are widely used to solve the problems dealt with in piezoelectricity.Basic effects are described by the material constants (tensors) s E,T ijkl , ε T,σ nm , ξ E,σ that occur at the diagonal of the set of equations, whereas conjugate effects are defined by the remaining constants.On rewriting the material constants of (3.1) in a matrix form, we obtain a symmetrical matrix of linear effects: The equivalence between the components of the compliance sixth-rank tensor s ijklpq and the components of the 6 × 21 matrix is shown to be: The equation can be written in the form Considering equations (2.6), (2.7) and (2.8) we obtain in a non-linear approximation: An analogical matrix equation can be derived for non-linear effects (equations (2.6), (2.7) and (2.8)): where the basic effects are given by the following constants: and the conjugate effects are described by As we can see, the matrix of non-linear effects is also a symmetrical one.

Summary
Using the Gibbs function (G), the equations describing the physical quantities ε ij , D m , S have been derived for the non-linear effects of piezoelectric bodies.Relevant equations are included in section 3. It has been demonstrated that the relations between the material constants, which are tensors of zeroth to sixth order, can be written in non-linear approximation in the form of material equations where the elements of the matrices are the sums of appropriate matrices which represent the tensors (relation of (3.3)).The relations of (3.3) were used to derive the linear effects (relations of (3.2)).
From the rewritten relations of (2.6, 2.7, 2.8) in the matrix notation (table 1) we can easily identify the interrelations of the tensor components, especially for separate physical effects.Further simplifications and decays of some components of higher order tensors may be obtained for appropriate point groups (crystal classes).
In 1952, with the aim to interpret the properties of the tetragonal antiferroelectric crystal ND 4 D 2 PO 4 , Mason [10,11] made use of the elastic Gibbs function G 1 : The equations and formulae describing ε ij , E n and S, which have been derived by Mason [10], depend on σ kl , D n and T .So it is not surprising that they differ from the equations presented in this paper (equations (2.6), (2.7) and (2.8)) both in linear and non-linear approximations.In our calculations we used a Gibbs function which had the form of G: The equations obtained for ε ij , D n and S depend on σ kl , E n and T .There is a definite sense in which our results differ from those achieved by Mason.For example, according to Mason (function G 1 ) [10] we have the following relations for the linear effects: Our results (based on function G) take the form Relevant differences have been underlined.Considering the physical properties of the crystals, the symmetries of the tensor components and the order of differentiation of the thermodynamic function of state as a total differential, we can use a more concise matrix notation, e.g:During transition from tensor to matrix notation we neglected the coefficients that were different for relevant components (those for s ijkl and s ijklpq are listed in section 3).
Equally concise are the matrices of the coefficients (tensor components) in nonlinear expansion.All of the coefficients (tensor components) incorporated in equations (2.6), (2.7) and (2.8), which occur on the right-hand side of table 1, are presented in the form of matrix 10 × 62: Matrix notation is clear and very convenient, and in its external form it is similar for all (eight) thermodynamic functions, which have been used to describe the physical quantities for piezoelectric crystals.Making use of the data reported in the literature [1,3,5,13,14,15,16], we can establish the number of independent components of particular tensors for each point group, as well as define the relations between some elements of these matrices.

Table 1 .
Matrices of the coefficients (tensor components) in non-linear expansion.