A domain of influence in the Moore–Gibson–Thompson theory of dipolar bodies

We establish a domain of influence theorem for the mixed initial-boundary value problem in the context of the Moore–Gibson–Thompson theory of thermoelasticity for dipolar bodies. Based on the data of the mixed problem, we define, for a finite time t>0, a bounded domain and prove that the displacements and the temperature decrease to zero, outside of the domain . The main result is obtained with the help of two auxiliary results, namely two integral inequalities. We managed to prove that this type of influence domain can be built even if it is considered in a much more complex context. Thus, compared to the classical context in which this concept appeared, we took into account the heat conduction principle from the Moore–Gibson–Thompson theory, we considered the thermal effect and we analyzed the effect of the dipolar structure of the environment.


Introduction
Some researchers consider that type III heat conduction violates the principle of causality. This was the reason for considering the Choudhuri's theory given [1]. This is also the reason why the Moore-Gibson-Thompson theory appeared (see [2], for instance). This theory was developed starting from a third-order differential equation, built in the context of some considerations related to fluid mechanics. Subsequently the equation was considered as a heat conduction equation because it has been obtained by considering a relaxation parameter into the type III heat conduction. Since the advent of the Moore-Gibson-Thompson theory, the number of dedicated studies to this theory has increased considerably. We mention some of them, a part developing the theoretical aspects of the theory, such as [3][4][5][6], others highlighting the practical applicability of this theory [6][7][8][9][10][11][12][13]. For our main result, we will approach the heat conduction of the Moore-Gibson-Thompson type in the case that the dissipation condition holds. We must emphasize that we have considered the Moore-Gibson-Thompson theory in the context of the thermoelastic dipolar bodies, starting from the consideration that these media are very current and have applicability to concrete materials. We can give the following examples, as very convincing: granular media having large molecules (concrete: polymers), animal bones, or human bones, and graphite.
If we consider the extensive number of published papers which are dedicated to the media with dipolar structure, we can conclude that this kind of structure is very suitable to model a large number of media in continuum mechanics [14][15][16][17].

Basic equations
We will approach a bounded domain D in the usual space R 3 which is filled by an thermoelastic dipolar material. We denote by denoted byD the closure of the domain D, so thatD = ∂D ∪ D, ∂D being the frontier of the set D and it is assumed to be smooth enough to apply the theorem of divergence. Also, n = (n i ) is the outward unit normal to the border ∂D. It will be understood that a vector v has the components v i and a tensor w has the components w ij . For a function f = f (t, x) we will denote byḟ its derivative with respect the variable t and by f ,i its derivative with respect to the space variable x i . Summation over repeated subscripts is implied.
The spatial argument and the time argument of a function will be omitted when there is no likelihood of confusion. We refer the motion of the body to a fixed system of rectangular Cartesian axes Ox i , i = 1, 2, 3.
Let us define the problem of heat conduction of the Moore-Gibson-Thompson type, denoted by P, which consists of the differential equation relative to the temperature ϑ: Here we used the notation α for a positive parameter to designate the thermal relaxation. Also, c(x) is notation for the thermal capacity, and the tensors κ ij and κ * ij represent the thermal conductivity, and respectively, the thermal conductivity tensor in connection with the thermal relaxation parameter α. Both tensors κ ij and κ * ij are assumed to be symmetric.
Our aim is to obtain a mixed initial-boundary value problem for ϑ. For this, we consider the following boundary condition: and the following initial conditions: All functions used in our considerations are assumed to be bounded. Furthermore, the thermal capacity c(x) is a positive function, that is: Let us introduce the tensor K ij by means of the notation K ij = κ ij − ακ * ij . We will suppose that K ij is a positive definite tensor: there exists a positive constant k 0 so that Now, we will introduce the mixed initial-boundary value problem for the Moore-Gibson-Thompson theory of thermoelastic bodies with dipolar structure.
To describe the deformation of this type of bodies, we will use the following variables: The notation u = (u i ) is used for the displacement vector, ϕ = ϕ ij for the dipolar displacement tensor and ϑ is the variation of the temperature from the reference temperature T 0 , that we have already introduced previously.
We will use three strain tensors ε ij , γ ij and χ ijk which are defined by means of the geometric equations: Our considerations will be made in the context of a linear theory, such that it is natural to consider that the internal energy is a quadratic form in its variables, namely, of the following form (see [3]): where C ijmn , G ijmn , . . . , A ijkmnr are the elasticity tensors, whereas a ij , b ij and c ijk are the coupling tensors and K ij is the thermal conductivity tensor, defined above. By using an appropriate procedure, starting by the form of E from (8), we can introduce the tensors of stress, having the components denoted by τ ij , σ ij and μ ijk , with the help of the constitutive equations (see [19]): (9) We have added in (9) the expressions for entropy S and for the vector of heat flux having the components q i . Also, the main equations, namely the motion equations, are obtained in the form (see [22]): where the notation I ij = I ji is used for the tensor of microinertia, is the notation for the constant mass density in the reference state, f = (f i ) is the body forces vector and g = g ij is the dipolar body forces tensor.
We will use the energy equation in the form that follows (see [7]): where we denoted by r the heat supply.
We must suppose that the elasticity and coupling tensors used in the previous equations satisfy in the domain D the following symmetry relations: We substitute the constitutive equations (9) and the geometric equations (7) in the motion equations (10) such that we obtain a system of differential equations of the form: (13) In order to complete the mixed problem for the Moore-Gibson-Thompson theory of thermoelasticity for bodies with dipolar structures, we add the following initial conditions: where Also, we add the boundary conditions: whereũ i ,t i ,φ ij ,m ij ,θ andq are given functions. In (15) we used the notation Also, the surfaces ∂D 1 , ∂D 2 , ∂D 3 and their corresponding complements ∂D c 1 , ∂D c 2 , ∂D c 3 are subsets of the surface ∂D, defined such that We will denote by P the mixed initial-boundary value problem which consists of system the equations (13), the boundary conditions (15) and the initial conditions (14). The ordered array (u i , ϕ ij , ϑ) is a solution of the problem P, if it satisfies the equations (13), the initial conditions (14) and the boundary conditions (15).

Main result
This section has three distinct parts. In the first part, we introduce the concept of the domain of influence. The second part is dedicated to integral inequalities, which are auxiliary results for the main result of our study. In the last part, we prove the theorem of existence of the domain of influence, using the helping results from the second part. First, we define the function U ε (x), for a sufficient small ε > 0, reminiscent of the step function of Heaviside, having the expression: which is a smooth enough and nondecreasing function on the entire real axis. We fix an arbitrary point x 0 ∈ D and use the notation d = |x − x 0 |. Then we can define the function where U ε is the function above defined and the positive constants D and t are arbitrarily fixed. Also, the positive constant v is of the nature of a speed and will be determined in the following. By S(x 0 , ρ) we will denote a usual sphere of centre x 0 and radius ρ, namely and this is used in the following definition Now, we can see that the function W(s, x) from (16) becomes zero outside set S. Clearly, W(s, x) is a function smooth enough for any points from [0, t] × D.
The results that we will obtain in the following are based on the following three basic assumptions, which, in fact, are common in Continuum Mechanics: (i) the conductivity tensor K ij is positive definite, i.e.
there exists a positive constant k such that (ii) > 0, I ij > 0, T 0 > 0, c > 0; (iii) the internal energy is positive definite, i.e. there exists a positive constant C such that for all x ij = x ji , y ij , z ijk , ω i , ω, we have C ijmn x ij x mn + 2G ijmn x ij y mn + 2F ijmnr x ij z mnr + B ijmn y ij y mn + 2D ijmnr y ij z mnr + A ijkmnr z ijk z mnr + c (αω +ω) 2 − a ij x ij + b ij y ij + c ijk z ijk (αω +ω) + K ij ω i ω j ≥ C x ij x ij + y ij y ij + z ijk z ijk + ω i ω i + ω 2 .
The following inequality is useful for the main result. For any solution (u i , ϕ ij , ϑ) of the problem P, we have the following inequality:

Proposition 3.1:
Proof: It is easy to see that the result is obtained directly just by using the above hypotheses (i) and (iii). (19) is the potential energy, which we will note with P, that is P = u iui + I jk ϕ jr ϕ kr + cϑ 2 + C ijmn ε ij ε mn

Remark: Note that the function on the left-hand side of the inequality
Also, the function on the right-hand side of the inequality (19) is the kinetic energy, which we will note with K, that is With these notations, inequality (19) receives on a much simpler form, as follows The inequality that we will approach in next theorem is the basis for the demonstration of our main result. The following notations, for a disk and its border, are common: Proof: In the beginning we use Equation (13) 1 , multiplied by Wu i , so that we obtain Similarly, using Equation (13) Finally, we proceed in the same way with Equation (11) that we multiply by Wϑ to obtain Now we add member with member the identities (24)- (26) and obtain the equality: which, clearly, can be restated as follows and this identity can be written in a shorter form, if we use the potential energy P, defined in (20): Integrating both members of identity (27) over [0, t] × D, after that making use of the divergence theorem and taking into account the boundary conditions (15), we obtain On the other hand, if we take into account the definition (16), we notice that we can replace the function W with the function U ε , so we obtain: For the inequality of the end we used several times the inequality of the means, between the arithmetic mean and the geometric mean, in the form xy ≤ 1 2 choosing every time the suitable parameters p. Finally, we used the definition (21) of kinetic energy K.
Based on (29) we deduce the following inequality the last inequality being a consequence of the inequality (22). With the help of inequalities (29) and (30), from the identity (28) we deduce the following useful inequality: Now we pass to the limit in the inequality (31), with ε tends to 0, and we find the function W has as limit just the characteristic function of S, this being the set above defined. So, we obtained the desired inequality (23), which ends the proof of the theorem.
Now, we will define the domain of influence. For a time instant t we will consider D(t) as a domain which contains the points x ∈D, so that: With the help of the set D(t), the domain of influence is defined by: where the sphere S(y, vt) is defined in (17) and ∅ is the empty set. Our important result is addressed in the following theorem.

Theorem 3.2:
If hypotheses (i)-(iii) are satisfied, then for any solution (u i , ϕ ij , ϑ) of the problem P, we have the following characterization: Proof: For an arbitrarily x 0 ∈ {D \ D t } and s ∈ [0, t], we will write the inequality (23) with t = s and ρ = v(t − s) so that we deduce Because x 0 ∈D \ D t , we deduce that x ∈ B(x 0 , vt), as such x / ∈ D(t). Therefore, we have: Clearly, we have the inclusion B(x 0 , v(t − τ )) ⊂ B(x 0 , vt). Thus, we deduce Based on (35) and (36), from (34) we deduce the inequality that follows: so that, considering the hypothesis (i), we deduce and then, considering (22), the more we have the inequality Now, we take into account the definition (21) of the kinetic energy K so that from (37) we deducė But, we have that's why the final conclusion is which concludes the proof of Theorem 3.2.

Conclusion
Some researchers consider that a result regarding the domain of influence, is, in fact, is a more relaxed form of of the known principle of Saint-Venant from elementary elasticity. So, our result is a generalization of this principle to arrive in the context of the theory of Moore-Gibson-Thompson thermoelasticity for bodies with dipolar structure. Then, we managed to prove that this type of influence domain can be built even if it is considered in a much more complex context. Thus, compared to the classical context in which this concept appeared, we took into account the heat conduction principle from the Moore-Gibson-Thompson theory, we considered the thermal effect and we analysed the effect of the dipolar structure of the environment. Although we have these complexities, we managed to identify a domain, outside which neither the displacement nor the temperature generates disturbance. Our domain of influence theorem is based on two auxiliary results, namely two integral inequalities.

Disclosure statement
No potential conflict of interest was reported by the author(s).