Uniqueness of Solutions in Thermopiezoelectricity of Nonsimple Materials

This article presents the theory of thermopiezoelectricity in which the second displacement gradient and the second electric potential gradient are included in the set of independent constitutive variables. This is achieved by using the entropy production inequality proposed by Green and Laws. At first, appropriate thermodynamic restrictions and constitutive equations are obtained, using the well-established Coleman and Noll procedure. Then, the balance equations and the constitutive equations of linear theory are derived, and the mixed initial-boundary value problem is set. For this problem a uniqueness result is established. Next, the basic equations for the isotropic case are derived. Finally, a set of inequalities is obtained for the constant constitutive coefficients of the isotropic case that, on the basis on the previous theorem, ensure the uniqueness of the solution of the mixed initial-boundary value problem.


Introduction
The classical theory of heat propagation is based on two equations: the energy balance equation and the Fourier law for the heat flux.These equations lead to the classical heat equation that has the unrealistic feature that the velocity of heat propagation is infinite.This classical theory was firstly questioned by Cattaneo [1,2] and Vernotte [3], leading to the rise of new fields of research: heat waves and second sound propagation can be found in the work of Straughan [4] and in extended irreversible thermodynamics (EIT) in the books of Jou et al. [5] and Müller and Ruggeri [6].In [7] Gurtin and Pipkin considered a general theory of heat conduction with finite velocity, based on a heat flux law with memory.For more information on thermoelasticity theories that predict a finite velocity for the propagation of thermal signals, see the the reviews of Chandrasekharaiah [8,9].Furthermore, as presented by Ieşan in [10], the theory proposed by Green and Laws [11] is an alternative way of formulation of the propagation of heat.They make use of an entropy inequality in which a new constitutive function appears (see e.g.[12]).
The origin of the theory of nonsimple elastic materials goes back to the works of Toupin [13,14] and Mindlin [15].The first investigations of nonsimple thermoelastic materials are presented in [16,17].Within the framework of that theory for this materials, the second-order displacement gradient is added to the classical set of independent constitutive variables.The theory of nonsimple thermoelastic materials has been discussed in various papers (see for example [18][19][20][21][22][23][24][25][26]) The problem of the interaction of electromagnetic fields with elastic solids was the subject of important investigations (see e.g.[27][28][29][30][31][32] and the literature cited therein).Certain crystals (for example quartz) when subject to stress, become electrically polarized (piezoelectric effect).Conversely, an external electromagnetic field produces deformation in a piezoelectric crystal.The theory of thermopiezoelectricity has been studied in various works (see e.g.[33][34][35][36][37]).
In this paper, we derive a theory of thermopiezoelectricity of a body in which the second gradient of displacement and the second gradient of electric potential are included in the set of independent constitutive variables.We obtain the appropriate thermodynamic restrictions and constitutive equations, with the help of an entropy production inequality proposed by Green and Laws [11].Next, for both anisotropic and isotropic materials we establish the basic equations of the linear theory and we obtain a uniqueness result for the mixed initial-boundary values problem.

Basic equations
We consider a body that at some instant occupies the region B of the Euclidean three-dimensional space and is bounded by the piecewise smooth surface ∂B.The motion of the body is referred to the reference configuration B and to a fixed system of rectangular Cartesian axes Ox i (i = 1, 2, 3).
We shall employ the usual summation and differentiation conventions: Latin subscripts are understood to range over the integers (1,2,3), summation over repeated subscripts is implied and subscripts preceded by a comma denote partial differentiation with respect to the corresponding Cartesian coordinate.In what follows we use a superposed dot to denote partial differentiation with respect to the time t.Further, we will neglect the issues of regularity, simply understanding a degree of smoothness sufficient to make sense everywhere.
We restrict our attention to the theory of homogeneous piezoelectric solids.Let u i be the displacement vector, D i the electric displacement vector and E i the electric field vector.The equation of motion is while the equations for the quasi-static electric field are [34] where t ij is the stress tensor, f i is the external body force per unit mass, ρ is the reference mass density, g is the density of free charges and ϕ is the electric potential.
Following [14,27,35], we assume that the body is free from initial stress and we postulate an energy balance in the form for every part P of B and for every time.Here e is the internal energy per unit mass, r is the heat supply per unit mass.Moreover, t i is the traction vector, µ ji is the hypertraction tensor, Q i is the generalized surface charge density and q is the heat flux.Following Ieşan [10,35], for the traction vector, the hyperstress tensor µ kji , the heat flux vector q i and the electric quadrupole Q ji we have where n i is the outward unit normal vector to the boundary surface ∂B.By using eqs.( 1) and (3) and the divergence theorem, the balance of energy becomes If we introduce the tensors eqs.
(1) and (2) 1 can be written as and the local form of energy balance becomes As proved by Ieşan [10], the invariance of the energy equation for observers in rotating motion the one with respect to the other, leads to τ ji = τ ij , moreover, without loss of generality, we can suppose given that the skew symmetric part makes no contribution to the rate of work over any closed surface in the body, or over the boundary.Let's set then eq. ( 6) becomes We postulate the entropy production inequality proposed by Green and Laws [11] for every part P of B and every time.Here η is he entropy per unit mass, φ is a new strictly positive thermal variable needing a constitutive equation.Using eq. ( 3) and the divergence theorem, we have in local form or equivalently We now introduce the scalar function σ (electric enthalpy) defined by and we substitute eqs.( 8) and (10) in the inequality (9), we then obtain

Thermodynamics restrictions
Let θ be the difference of the absolute temperature T and the absolute temperature in the reference configuration T 0 , i.e. θ = T − T 0 .
We require constitutive equations for σ, φ, η, τ ij , µ ijk , σ i , Q ij and q i and assume that these are functions of the set of variables where If we replace eqs.( 12) into the inequality (11), we arrive to We have the following conditions and if we suppose that q i = 0, we obtain Consequently, assuming ∂φ/∂ θ = 0, we have the following restrictions and the dissipation inequality is Taking into account eqs.( 10), ( 13), ( 14), the equation of energy (8) reduces to
Using these relations, the constitutive equations ( 19) and ( 20) can be expressed as ijkl e kl + a (12) ijklh κ klh + a In the linear approximation, the energy equation is or, with help of eqs.(21), we arrive to ijk κijk + a The dissipation inequality ( 15) becomes The following quadratic form is defined from the previous dissipation inequality If we consider the symmetric matrix associated to the quadratic form P , its positive semi-definiteness implies, in particular, that The basic equations of linear theory of thermopiezoelectric solids consist of the equations of motion ( 5), the equation of energy (22), the geometrical equations ( 7), (2) 2 , the constitutive equations (21) with the restriction (24), on B × I, where I = [0, t 0 ), where t 0 ≤ +∞.
Following Toupin [14] and Mindlin [15], we consider P i , R i , Λ and H defined in such a way that the total rate of work of the surface forces over the smooth surface ∂P can be expressed in the form Here we used where D ≡ n i ∂/∂x i is the normal derivative and is the surface gradient.Now, we denote with U = (u i , θ, ϕ) the solutions of the mixed initial-boundary value problem Π defined by eqs.( 5), ( 22), ( 7), (2) 2 , (21) and the following initial conditions in B and the following boundary conditions with u 0 i , v 0 i , θ 0 , η 0 , ûi , di , θ, φ, ξ, Pi , Ri , q, Λ and Ĥ are prescribed functions and the surfaces S i and Σ i are such that Si where the closure is relative to ∂B.The (external) data of the mixed initial-boundary value problem in concern are Γ = f i , g, r, u 0 i , v 0 i , θ 0 , η 0 , ûi , di , θ, φ, ξ, Pi , Ri , q, Λ, Ĥ .

A uniqueness result
In this section we establish a uniqueness result for a initial-boundary value problem Π.To this aim, using the constitutive equation (21) we prove that where W is the following quadratic form in the strain measures e ij and κ ijk and F is a quadratic form in the variables On the other hand, taking into account eqs.(2) 2 , (4), ( 5), ( 21) and ( 22), we have Eqs. ( 25) and ( 26) imply It is easy to see that with P defined by (23) and satisfying (24).Taking into account eqs.( 27), (28) and introducing the following quadratic form in the variable we can write where

It easily follows the next theorem
Theorem 1 (Uniqueness).Assume that i) ρ, β, T 0 > 0, ii) the constitutive coefficients satisfy the relations (18), iii) W is a positive semi-definite quadratic form, iv) G is a positive definite quadratic form.
Then, if S 4 is nonempty, the initial-boundary values problem Π has at most one solution.
Proof.Suppose that we have two solutions of the problem Π.Then their difference (u i , θ, ϕ) corresponds to null data.By integrating eq. ( 29), we have The last integral must be a decreasing function with respect to time, but since it is not negative and initially null, it can be deduced that ui = 0, θ + β θ = 0, Taking into account that eqs.(30) 1,2 are linear homogeneous equations with null initial conditions and using eq.( 2) 2 , we conclude that If S 4 = ∅ we obtain the uniqueness result, in fact it is ϕ = 0 on S 4 × I =⇒ ϕ = 0 on B × I.
In the isotropic case, the quadratic form W can be expressed as the sum of two independent quadratic forms, the first one W 1 in the variables e ij and the second one W 2 in the variables κ ijk .
If S 4 is nonempty, the initial-boundary values problem considered has at most one solution.