Energy Decay in Thermoelastic Bodies with Radial Symmetry

In this paper, we consider the energy decay of some problems involving domains with radial symmetry. Three different settings are studied: a strong porous dissipation and heat conduction, a weak porous dissipation and heat conduction and poro-thermoelasticity with microtemperatures. In all the three problems, the exponential energy decay is shown. Moreover, for each of them some finite element simulations are presented to numerically demonstrate this behavior.


Introduction
The study of elastic solids with voids started with the contributions of Cowin and Nunziato [1][2][3]. This kind of materials has deserved much attention in the last forty years [4][5][6][7][8][9][10][11][12][13][14]. Interesting physical applications of the thermoelasticity with voids are directed to the study of solids with small distributed porous. Rocks, soils, woods, ceramics or even biological materials as bones are natural examples. In fact, in recent years multiporosity structure has been considered [15][16][17][18]. This kind of solids can be seen as a particular kind of elastic materials with microstructure. Indeed, an increasing interest has been developed to consider the theories with microstructure along the last and current centuries [19][20][21][22][23][24][25]. A relevant sub-class of these solids corresponds to the case that the microstructure [26] is determined by microtemperatures [27]. The wide applicability of this kind of structures made that many scientists are interested in the study of elastic materials with microtemperatures [28][29][30][31][32][33][34][35].
Since Quintanilla [36] showed the slow decay of the solutions when only one mechanism of porous dissipation is considered, much studies have been done to analyze the decay of the solutions when different types of dissipative mechanisms are introduced in the system [4-7, 9-12, 28, 33, 37]; however, we can say that all the contributions correspond to the onedimensional problem. To the knowledge of the authors the only exception to this statement corresponds to the article Nicaise and Valein [38]. In that contribution, the authors showed the polynomial decay to a certain kind of thermo-porous elastic problem. We can say that it is generally accepted that for different problems porous-thermo-elasticity is not natural to expect exponential decay of the solutions for dimension greater than one. Therefore, one can ask if it is possible to identify solutions in such a way that they decay in an exponential way. In this sense, we believe that the geometry of the body plays a relevant role in this kind of study. The main aim of the present contribution is to identify solutions for different porousthermo-elastic problems in such a way that they decay in this fast way. For this reason, in this paper we restrict our attention to geometries as which are radially symmetric. Examples of radially symmetric domains are the balls centered at the origin as well as the circular crowns. In this case we can see that the radially symmetric solutions decay in an exponential way. To this end, we will follow the energy arguments, which are naturally applied because they are usual in the study of decay of solutions within several thermoelastic theories (see for instance [39,40]).
In the next section we describe the basic equations corresponding to the theories we are going to work with. Section 3 is devoted to the study of the time decay of solutions for porous-thermoelasticity with strong porous dissipation. Later, we consider the case when the porous dissipation is weak, but in this case we need to change the boundary conditions. Last section is devoted to the case of porous thermoelasticity with microtemperatures.

Several Theories
In this section we recall the basic equations we will work in this paper.
First problem we consider corresponds to the porous-thermo-elasticity with strong porous dissipation. In this case, the problem is determined by the evolution equations [41]: and the constitutive equations: In this system of equations we have that ρ is the mass density, u i is the displacement vector, t ij is the stress tensor, h i is the equilibrated stress tensor, g is the equilibrated body force, η is the entropy, q i is the heat flux vector, J is the equilibrated inertia, T 0 is the reference temperature at the equilibrium state (assumed uniform and equal to one to simplify the calculations), e ij is the strain tensor which is related to the displacement by the relation φ is the volume fraction, θ is the temperature. Here λ and μ are the usual Lamé parameters, β determines the coupling between the displacement and the temperature, μ 0 is the coupling between the displacement and the porosity, k is the thermal conductivity and c the thermal capacity. Other parameters are usual in the study of the present theory.
We will consider this system with homogeneous Dirichlet boundary conditions. Therefore it is suitable to recall that some certain geometries there exist undamped solutions. The main idea is to consider domains where certain spectral conditions are satisfied. Here, we will assume that u i = 0 on the boundary and homogeneous (Dirichlet or Neumann) boundary conditions for the domain for the other variables. We consider domains that satisfy the condition: Condition D. There exists a nonzero field ψ i ∈ H 1 0 such that ψ i,jj = γ 2 ψ i and ψ i,i = 0, where γ 2 = 0.
We note that in dimension one there are no functions satisfying this condition; however, in dimension greater than one this condition is satisfied for several symmetric domains. For instance, if the domain is a ball there are an infinity of eigenvalues satisfying this condition (see [42]).
When the domain is such that condition (D) is satisfied, we look for solutions of the form (u i , φ, θ) = (exp(ωt)ψ i , 0, 0) whenever ω 2 ρ + γ 2 μ = 0. It is clear that our solutions are undamped and therefore they do not decay; however, the aim of this paper is different and we are looking for solutions that decay in an exponential way. For this reason, we will study the radial solutions which are totally different from the ones considered previously.
Second problem we will study corresponds to the porous-thermo-elasticity with weak porous dissipation. Again we will work with the evolution equations (1)-(3), but we need to change the constitutive equations for the equilibrated stress tensor and the equilibrated body force. They should be meanwhile the other constitutive equations continue in the same form, but k 2 = 0. It is worth noting that the examples of undamped solutions can be considered also in this problem.
Third system we are going to study in this note corresponds to thermoelasticity with voids and microtemperatures. In this case to the system (1)-(3) we have to add the equation: where i is the first heat flux moment tensor, Q i is the microheat flux average, q ij is the first heat flux moment tensor. The constitutive equations are ( [41]): where the new constitutive parameters are the usual ones in the theories with microtemperatures. Again, we can obtain undamped solutions of the form (u i , φ, θ, T i ) = (exp(ωt)ψ i , 0, 0, 0).
Though we have seen examples of undamped solutions for certain domains, our main aim in this paper is to prove the exponential decay of solutions for the problems defined previously whenever we consider radially symmetric solutions. It is relevant to recall that this kind of solutions are only possible in the case that the domain would be radially symmetric as in the case of balls or crowns. For this reason, we want to emphasize how the geometry of the domain allow the existence of exponentially stable solutions which is the main aim of the paper.

First Case: Strong Porous-Dissipation and Heat Conduction
The aim of this section is to study the radial solutions of the problem determined by the system obtained after the substitution of the constitutive equations into the evolution equations: This system corresponds to the case of the porous-thermoelasticity where the dissipative mechanisms are given by the heat conduction and the strong porous viscosity given by the term a * φ ,jj . As we said before, (u i ) is the displacement vector, φ is the volume fraction, θ is the temperature, ρ is the mass density, J is the equilibrated inertia, λ and μ are the Lamé's parameters, k is the thermal conductivity and μ 0 , β, a, a * , η and β 1 are constitutive coefficients for this theory. We will study the problem determined by this system with the Dirichlet boundary conditions: and the initial conditions: It is clear that in the case we assume that the initial conditions have radial symmetry we obtain that the solutions are always of radial type. That is, the solutions can be given in the following way: where r 2 = x 2 1 + x 2 2 for the two dimensional case and r 2 = x 2 1 + x 2 2 + x 2 3 in the threedimensional case. It is worth noting that in this situation we have that u i,j (x, t) = u j,i (x, t). Therefore, the first equation of our system can be written as Our aim in this section is to prove that the radial solutions decay in a exponential decay. Therefore we should recall the assumptions that we need to work. In this section we assume that The sign of the other constitutive coefficients is not a restriction, however we need that β must be different from zero.
The problem described previously has solutions (for instance by means of the semigroup arguments) in the Hilbert space Existence and uniqueness of solutions are guaranteed. Even more, the stability of solutions can be obtained; however, we cannot expect (in the general case) its exponential decay. Now, our aim here is to obtain the exponential decay of the radial solutions to the problem proposed previously. We have the following.
Theorem 1 Let us introduce the functions E 1 , E 2 and E 3 given by and define the energy of the system as Then, this energy decays exponentially, that is, there exist two positive constants M and ω such that Proof First, we have thaṫ where n is the normal vector to the surface ∂B. We note that where is as small as we want and C is a calculable positive constant. If we recall Lemma 4.1 of the book of Jiang and Racke [40], we obtain where σ k is a smooth field defined on B such that σ i = n i on the boundary of B and Recalling again Lemma 4.1 of the book of Jiang and Racke [40] we find that The first integral can be bounded in a similar way as previously to obtain where The other integral can be bounded by If we select a parameter χ large enough we can show that As β is different from zero, we can obtain the estimate and therefore the next inequality can be found: We now consider the functions: We haveḞ On the other hand, since we see thaṫ It follows thatḞ If we select χ * large enough we get At the same time, we also note that there exists a positive K such that We now define the function In view of the previous estimates we see thaṫ We note that, for small enough, we can obtain thaṫ where we recall that In view that E * (t) is equivalent to E(t) we obtain the exponential decay of solutions.

Numerical Simulations
In this section, we want to present numerical simulations for the solutions to our problem in the case that we have a two-dimensional ball with radius 1. We also assume that the time interval is fixed as [0, T ], where T denotes the final time, and we note that, in order to simplify the writing, we do not indicate the dependence on the spatial variable (x i ).
Taking into account the boundary conditions, defining the velocity v i =u i and the porosity speed e =φ system (4)-(6) leads to the following variational formulation, for a.e. t ∈ [0, T ], where v i (0) = v 0 i and e(0) = ψ 0 , the variational spaces E and V are given by 2 ; w i = 0 on ∂B for i = 1, 2}, and we denote by (·, ·) the inner product in the space L 2 (B). Here, the displacements and the porosity are defined as Now, we consider a fully discrete approximation of problem (14)- (17). This is done in two steps. First, we assume that the domain B is polyhedral and we denote by T h a regular triangulation in the sense of [43]. Thus, we construct the finite dimensional spaces V h ⊂ V and E h ⊂ E given by where P 1 (T r) represents the space of polynomials of degree less or equal to one in the element T r, i.e. the finite element spaces V h and E h are composed of continuous and piecewise affine functions. Here, h > 0 denotes the spatial discretization parameter. Moreover, we assume that the discrete initial conditions, denoted by u 0h i , v 0h i , φ 0h , e 0h and θ 0h , are given by where P h is the classical finite element interpolation operator over E h (see, e.g., [43]). Secondly, we consider a partition of the time interval [0, T ], denoted by 0 = t 0 < t 1 < · · · < t N = T . In this case, we use a uniform partition with step size τ = T /N and nodes t n = n τ for n = 0, 1, . . . , N. For a continuous function z(t), we use the notation z n = z(t n ) and, for the sequence {z n } N n=0 , we denote by δz n = (z n − z n−1 )/τ its corresponding divided differences.
Therefore, using the backward Euler scheme, the fully discrete approximations are considered as follows.
where the discrete displacements and the discrete porosity are obtained from the relations: The above fully discrete problem leads to a linear system written in terms of a product variable which has a unique solution due to the conditions imposed on the constitutive coefficients.
Our aim now is to show numerically the asymptotic energy behavior. Therefore, we use the following data: T = 30, ρ = 1, μ= 5, λ= 2, μ 0 = 2, β = 2 J = 5, a = 2, a * = 5, η = 5, β 1 = 3, c = 1, k = 0.1, and the initial conditions: Taking the discretization parameter τ = 0.001 and using the finite element mesh shown in Fig. 1, in Fig. 2 we plot the evolution in time, in both natural and semi-log scales, of the discrete energy E hτ n defined as: As can be seen, the discrete energy tends to zero and the theoretical exponential asymptotic behavior seems to be achieved.

Second Case: Weak Porous-Dissipation and Heat Conduction
In this section, we consider a similar problem as in the previous one, but in the case that we assume weak porous dissipation. That is, we replace equation (5) by where η * is a positive constant and we assume that k 2 = 0. The analysis in this section follows the lines proposed before; however, as we have seen in the previous section, the boundary conditions proposed there impose that the manipulations are very cumbersome and it is not clear that we can directly extend to the present situation. In order to overcome this difficulty, we change a little bit the boundary conditions to assume that To define a well-posed problem we need to consider the initial conditions proposed in (8). Again, we assume radial solutions of the form proposed in the previous section and equation (4) becomes (10). This problem can be solved in the Hilbert space and H 1 * (B) = H 1 0 (B) ∩ L 2 * (B). Now, our aim is again to obtain the exponential decay of the radial solutions to the problem proposed in this section. So, we have the following.

Theorem 2 Let us consider the functions E i , i = 1, 2, 3, defined in Theorem 1 and the energy function E given by
Then, this energy decays exponentially; i.e., there exist two positive constants M and ω such that Proof We haveĖ We note that, in view of the boundary conditions, the expression of the time derivative of E 3 is easier to manipulate. Using similar arguments to the ones used in the previous section we seeĖ 1 Proceeding as in the proof of Theorem 1 we also need to introduce a couple of functions, but being a little bit different: It follows thaṫ Again, the expression of the time derivative of F 2 (t) is easier due to the boundary conditions we assume. We also note that inequality (13) also holds here. Therefore, we find thaṫ u i,jj u i,kk +ü iüi +φ ,jj φ ,ii +φ ,i φ ,i dv +C * B θ ,i θ ,i +u i,iuj,j +φ ,jφ,j dv.
The function Obviously, if we select small enough it follows thaṫ and, therefore, the exponential decay can be obtained.

Numerical Simulations
In this section, the system obtained replacing equation (5) by (18) is numerically solved. Since the differences with respect to the numerical algorithm shown in Sect. 3.1 are minor, we skip the details for the sake of reading. Therefore, we use the following data: and the same initial conditions than in Sect. 3.1. Taking now the time discretization parameter τ = 0.001 and the finite element mesh shown in Fig. 1, the evolution in time of the discrete energy is shown in Fig. 3 in both natural and semi-log scales. Again, we can observe that the discrete energy tends to zero and the expected asymptotic exponential decay.

Third Case: Heat Conduction and Microheat Conduction
Third case we want to analyze in this paper corresponds to the study of porous-thermoelasticity with microtemperatures. The system of equations is: We pay attention that this system does not impose any kind of porous dissipation and the damping is proposed by the temperature and the microtemperatures. To study this system, we need to impose the initial conditions proposed in (8), but we also need initial conditions for the microtemperatures. That is, We also impose the boundary conditions proposed in (19) as well as Dirichlet homogeneous boundary conditions for the microtemperatures: Apart of the assumptions proposed previously for the different constitutive coefficients in the case that the microtemperatures are present, it is usual to assume that Under these assumptions we can determine the existence and uniqueness of solutions in the Hilbert space: However, we here concentrate our attention to radial solutions. That is, solutions of the form (9) and such that w i (x, t) = x i W (r, t). As we have seen for the displacement, the microtemperatures satisfy w i,j = w j,i and, therefore, equation (23) can be written as where κ * = κ 4 + κ 5 + κ 6 and we will assume, from now on, positive. As we have imposed in the previous sections, we assume that β is different from zero, but in this section we also assume that d is different from zero.

Theorem 3
Let us introduce the functions E 1 , E 2 and E 3 given by Then, this energy decays exponentially, that is, there exist two positive constants M and ω such that
We also consider the functions: and we obtain thaṫ According to inequality (13) we can show that In view of (13) and (25) it follows thaṫ We can define the function which is also equivalent to the function (u i,j u i,j + φ ,j φ ,j + u i,jj u i,kk +ü iüi +φ| 2 )dv.
It is clear that if we choose small enough we obtain thaṫ E * (t) + CE(t) ≤ 0, and, therefore, the exponential decay can be concluded.
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