We study some spectral and numerical properties of the solutions to a thermoelastic problem with double porosity. The model includes Cattaneo-type evolution law for the heat flux to remove the physical paradox of infinite propagation speed of the classical Fourier’s law. Firstly, we prove that the operator determined by the considered problem has compact resolvent and generates a -semigroup in an appropriate Hilbert space. We also show that there is a sequence of generalized eigenfunctions of the linear operator that forms a Riesz basis. By a detailed spectral analysis, we obtain the expressions of the spectrum and we deduce that the spectrum determined growth condition holds. Therefore we prove that the energy of the considered problem decays exponentially to a rate determined explicitly by the physical parameters. Finally, some numerical simulations based on Chebyshev spectral method for spatial discretization are given to confirm the exponential stability result and to show the distribution of the eigenvalues and the variables of the problem.
Cowin and Nunziato [9] and Nunziato and Cowin [22] proposed a theory to describe the behaviour of porous solid materials. In these materials there is a skeleton (or material matrix) that is elastic, and the interstices are voids in the material. These kinds of materials have been widely studied in the literature and different applications have been found for instance, in geological materials such as rocks and soils and in manufactured materials such as ceramics and pressed powders. Nowadays, this theory [9, 22] is commonly accepted as one of the nonclassical elasticity theories. There is a huge number of contributions referring to this theory and, hence, it is not possible to cite them all. Nevertheless, some relevant contributions are cited in this paper. See, for example, reference [15]. Later, Ieşan [14] proposed the linear theory of thermoelastic materials with voids. There are several studies related to this subject. Recently, Ieşan and Quintanilla [17] have presented a theory of thermoelasticity with double porosity structure. This model allows for the body to have a double porosity structure: a macro porosity connected to pores in the body and a micro porosity connected to fissures or cracks in the solid skeleton. Straughan pointed out this aspect in [25]. Double porosity materials occur frequently in real life. We point out applications in mathematical biology (biomaterials, bone replacement) [24, 28], geophysics [33], and also in energy production [18, 36]. For a review of the literature on thermoelastic materials with a double porosity structure the reader is referred to [17, 25–27].
The only dissipative mechanism proposed by Ieşan and Quintanilla [17] was the thermal effect; however, no dissipation mechanisms were considered for the porosities. The spatial and the temporal behavior of solutions to the initial boundary value problem as derived by Ieşan and Quintanilla [17] was studied by Arusoaie [5]. For bounded bodies, the author obtained estimates of Saint-Venant type, while for unbounded bodies they deduced some alternatives of Phragmén-Lindelöf type. The temporal behavior of solutions is described using the Cesáro means of various parts of the total energy. The question of exponential stability was tackled for the first time by Bazarra et al. [6]. They considered the same equations derived by Ieşan and Quintanilla [17] by adding dissipation mechanisms for the porosities. They distinguished two cases: first, one dissipative mechanism is considered on every porous structure; and second, the dissipation is considered on one porous structure. They proved exponential decay of the solutions in the first case, while in the second case, the exponential decay holds under sufficient conditions on the coefficients of the considered problem. Recently, Bazarra et al. [7] show the analyticity of solution to elastic problem with double porosity without thermal effects and with dissipation mechanisms in the motion and in both porosities equations. Exponential stability and impossibility of localization of the solutions in time are immediate consequences.
In the present article, we study the spectral and numerical properties of the solutions to a damped thermoelastic problem with double porosity. We have replaced the Fourier’s law for heat conduction by the Cattaneo’s law to remove the physical paradox of infinite propagation speed. This transforms the classical system into hyperbolic one with second sound, in which the thermal disturbances propagate with finite speed. Thus, the resulting model becomes more realistic than the conventional one derived in [17] in dealing with practical problems, but the mathematical analysis much difficult. We want to highlight two issues. The first one is the novelty of the model, and the second one is that our approach is mathematical and numerical. To the best of the authors’ knowledge, the thermoelastic problem with double porosity and second sound has not been considered before in literature. From our point of view, we believe that any theory needs a mathematical and numerical analysis that allows to decide its applications to the real-world situations. Our paper is addressed in this line and our model provides a wider domain of applications and a closer connection to microscopic theories. It is worth noting that our system of equations is composed of four equations coupled between them. The first three are hyperbolic while the two others are parabolic. As a consequence, the problem is characterized by four finite speeds of waves propagation, for the elastic wave, the thermal wave, as well as for volume fraction wave for each porous structure. This problem poses new mathematical difficulties due to its complexity and the presence of second sound. The question proposed in this article is: does the interactions between the coupling determined by these coefficients, the damped terms and the second sound effects bring the thermal dissipation to the mechanical part in a successful way. But the exponential stability issue becomes more difficult due to the second sound effect.
In this article, we consider the dissipation on one porous structure. Our approach is different from that used in [6] where the exponential decay is achieved using a characterization, going back to Gearhart, Huang and Prüss [19]. Despite the mathematical elegance of this approach; it does not provide the explicit expression of the rate of decay. Our approach is based on the Riesz basis properties to prove that the spectrum determined growth condition holds. As the spectral analysis is an important key in developing the Riesz basis properties of the considered system, we shall present a complete spectral analysis. Thanks to the distribution of the spectrum of this system, we prove that there exists a sequence of (generalized) eigenvectors that forms a Riesz basis for the state space. Then, we show that the spectrum determined growth condition holds, which means that the growth order of the system can be determined by its spectral bound. Consequently, we prove that the solutions decay exponentially at a rate determined explicitly by the physical parameters. Moreover, the explicit approximations of eigenvalues expressions can be useful later to study the controllability problem. The system of equations considered here introduce new mathematical difficulties in order to determine exact explicit forms. As far as the authors know, there are no contributions made in this sense.
The sections of this paper are organized as follows. In Section 2, we recall the basic equations and prove briefly that the considered problem is well-posed. In Section 3, we carry out a complete spectral analysis. We shall prove that the operator has compact resolvent whose spectrum is located in a strip parallel to the imaginary axis. In Section 4, we shall prove that the corresponding eigenvectors form a Riesz basis in an appropriate state space. Then we show that the system satisfies the spectrum determined growth condition and the exponential stability is deduced. In Section 5, some simulations based on Chebyshev spectral method for spatial discretization are given to show the distribution of the eigenvalues and the variables of the problem and to confirm its exponential stability.
Setup of the problem and well-posedness
Following Ieşan and Quintanilla [17], the evolution equations of the theory of thermoelasticity with two porosity in one-dimensional setting and in the absence of supply terms, are given by [6, 17]
together with the heat equation
and the constitutive equations
Here u represents the displacement, φ and ψ are the volume fractions for each porous structure, θ is the difference of temperature with respect to the reference temperature, ρ is the mass density, and are the coefficients of inertia for each porous structure, Υ is the stress, σ and τ the equilibrated stress of each porous structure, R and ϑ the equilibrated body force of each porous structure, η is the entropy, q the heat flux, is the relaxation time describing the time lag in the response of the heat flux to a gradient in the temperature and μ, b, d, β, α, , γ, , , , , , κ and c are the constitutive constants of the material.
If we substitute (2.3) into (2.1) and (2.2), we obtain the system of field equations:
We study the system (2.4) with the following initial data
and the following boundary conditions
Note that we have added the porous dissipations to the fourth and to the fifth equations of (2.3). These are the minimum dissipations allowing to obtain an exponential decay with an explicit decay rate.
It is prescribed by the basic principles that the mass density, the coefficients of inertia, the thermal capacity and the relaxation time are positive. That is:
The internal mechanical energy of the system is given by [6]
where denotes the inner product in .
To guarantee that the internal mechanical energy is positive, the following matrix should be positive definite
For this, we need to assume
We also deduce that
The mechanical dissipation of the system require that
To give an accurate formulation of the problem (2.4)–(2.6), we introduce the following Hilbert spaces,
where is the usual Sobolev space of order k. Let the state space be
endowed with an inner product, for , ,
The corresponding norm in is given by
We note that is nonnegative. It is easy to check that is a Hilbert space. Then, we define the system operator in
with domain
Then the system (2.4)–(2.6) can be rewritten as
We introduce, according to [10], two solution concepts for the Cauchy problem (2.11). Consider a -valued function . If and U satisfies (2.11), then U is said to be a classical solution to (2.11). If with for all and
then U is called a mild solution to (2.11). Assume that generates a strongly continuous in . Due to the definition of , the existence of a classical solutions is equivalent to . For , the unique classical solution to (2.11) is then given by . If, more general, , then the unique mild solution is given again by . The latter statements can be found in Proposition 6.2 and Proposition 6.4 of [10]. Regarding the correspondence of solutions to (2.11) and of solutions to (2.4)–(2.6) see also the discussion in [19].
The following property of holds.
Letandbe defined as before. Thenis dissipative in.
Using the inner product form together with the divergence theorem and the boundary conditions, we find
In view of assumptions (2.9), we have , the Lemma is proved. □
Letandbe defined as before. Thenis compact onand 0 belongs to the resolvent set of, i.e.,.
As a direct result, the spectrum ofconsists only of isolated eigenvalues with finite multiplicity,i.e.,, and satisfies,.
Since the infinitely differentiable function spaces are always dense in , it is easy to check that is dense in . The proof of the compactness of is divided on two steps.
Step 1: We shall prove that is injective. Let satisfying , that is,
From (2.13)1 we have for all . Moreover, from (2.13)5 and the boundary conditions (2.13)7 we get that for all .
Multiply the second, third and the fourth equation of (2.13) by , and , respectively, and then integrating the obtained identities from 0 to 1 yields
Thus, we have
Since the matrix Λ (defined by (2.7)) is positive definite, then we have
By the discussion above, together with the boundary conditions (2.13)7,8, we obtain that
Therefore, we obtain and therefore is injective.
Step 2: We shall prove that is surjective. To do this, let take so we will show that there exists , satisfying the equation , i.e.,
From (2.16)1 we have and . From (2.16)6 and the boundary conditions (2.16)8, we get
A simple integration gives
Setting , from (2.16)5 and (2.16)1 we have after simple integration
Substituting (2.18) into (2.17) we get
From (2.18) and (2.19) we conclude that and .
Multiplying (2.16)2, (2.16)3 and (2.16)4, respectively, by the conjugates of test functions and and then integrating from 0 to 1 yields
Consequently, the system (2.20) is equivalent to the problem
where
Note that, since the matrix Λ (defined by (2.7)) is positive definite, then satisfies
where is a positive constant. In the other hand, there exists such that the function satisfies
It is clear that is a continuous linear form on and is a bounded bilinear and coercive function. Choosing and , by Lax-Milgram’s Theorem, we obtain an unique solution to equation (2.22).
By taking in (2.22) (or in (2.20)), we obtain
Then we have
The regularity theory then yields that . In addition, for any with , (2.26) is also true. Then we see that
Integration by parts yields in (2.27)
Combining (2.26) with (2.28), we get that by the arbitrariness of . Hence, . Similarly, we can get
Thus, we have obtained the vector such that . Therefore, is surjective in .
By the inverse operator theorem, we deduce that . Furthermore, according to the Sobolev embedding theorem (see [1]), we obtain that is compact on , since is a compact subspace of . Therefore, the spectrum of consists only of isolated eigenvalues with finite multiplicity. Furthermore, we obtain that for all due to the dissipativity of and . The proof is complete. □
By following the previous arguments (step 2), we can prove that the operator is surjective for all . Since is dissipative, then it is maximal dissipative.
A dissipative operator always has a closure, which also is a dissipative operator; in particular, a maximal dissipative operator is a closed operator.
Then, as the operator is closed, the use of the above lemma and the Lummer–Phillips Corollary to the Hille–Yosida Theorem [23] leads to the next theorem.
The operatorgenerates a-semigroupon. Hence, the system (
2.4
)–(
2.6
) is well-posed, i.e., for any, the system (
2.4
)–(
2.6
) has a unique mild solution. Furthermore, if,, becomes the classical solution to (
2.4
)–(
2.6
).
Spectral analysis of
In this section, we shall discuss the distribution of the spectrum of the operator defined by (2.10) (see [2, 4] and references therein). We have obtained from Lemma 2.2 that . Thus, it suffices to consider the distribution of the eigenvalues of . For any , we suppose that is an eigenvector of corresponding to λ. Then
which leads to
and u, φ, ψ, θ and q satisfy the eigenvalue problem
Then we consider the following two cases:
Case 1:. From (3.1)5 and the boundary condition on θ we deduce that and is an eigenvector of the eigenvalue . Thus, we have .
Case 2:. From (3.1)5, we have and . Since , set , and , then system (3.1) is equivalent to the following equations:
Since (see (2.8)5), from (3.2)2,3 we obtain
Obviously, if makes system (3.3) have nonzero solutions, then this λ is an eigenvalue of . In the subsection below, we are going to give the asymptotic fundamental solution matrix to system (3.3) and then based on which, we can calculate the eigenvalues using the boundary conditions.
Fundamental solution matrix
Since , we can reformulate the system (3.3) in terms of the vector
where
Then we transform as follows
where
It is easy to check that for . The signs of , , and will be discussed later.
Then substituting (3.9) into (3.5) leads to
By setting , and , (3.12) becomes
where
Therefore, we have the following
Let. Then there exists a fundamental matrixsolution to system (
3.12
) such that for all λ with sufficiently large modulus, it haswhere
Following [21], we write in the following formal asymptotic series
where
By Eqs (3.5) and (3.6), assumption 2.1 of [31] on pp. 135 is satisfied and hence Theorem 2.2 of [31] on pp. 134 can be directly applied to our problem (see also [8]), that is to say, there is a fundamental matrix solution to Eq. (3.12) which takes the following form
where is uniformly bounded in λ and x. Following the same arguments used in [2, 4], we arrive at (3.16). □
Asymptotic expressions of the eigenvalues
In this subsection, we shall discuss the asymptotic expression of the spectrum of using the Birkhoff asymptotic technique [21]. Here, we shall use the expression given in (3.17). Setting , we rewrite the boundary conditions (3.3)7,8 as the matrix form
where
Set
The expression of the fundamental solution matrix to (3.12) is given by (3.17).
By using Lemmas 3.1 and 3.2 below, we shall provide an estimation of the asymptotic values of eigenvalues of . Since the first element of is , Lemmas 3.2 and 3.3 only concern the eigenvalues .
The asymptotic expressions of the spectrum of is given by the following.
The spectrum ofis contained in a strip parallel to the imaginary axis, i.e., there exists a sufficiently large positive constantsuch thatWhenandis large enough, we havewhere,,are given by (
3.14
)1−3, and
By Lemma 3.1, we know that if and only if . Since and , a direct computation yields
We shall carry out our estimations by using the following notation . From (3.16) we get
and from (3.17) that
which can be rewritten as follows
By Lemma 3.1 and by a direct calculation follows
which gives the expressions of (3.21) and (3.22).
To prove (3.20), we look for the limit of (3.21) when , we find using (3.10) and (3.11)
From (A.1) and i), we have , , and (see Appendix A), then there exists a positive constant L such that
Then by limit definition we have
where . So for we obtain
Therefore, there exist positive constants , and such that
Now let , we have to show that
We proceed by contradiction by supposing that
which yields, from Lemma 2.2 that . By (3.24), we have
Since , we obtain , then , which is absurd. Then . Therefore, we conclude that (3.20) holds. □
Since the relaxation time takes small values, it clear that the eigenvalue , is contained in the strip of . But this can not included in Lemmas 3.1 and 3.2 since the characterization for excludes the eigenvalue .
Now, we are in position to estimate the asymptotic values of eigenvalues of using Lemmas 3.1 and 3.2.
Letbe the eigenvalues ofwith sufficiently large modulus, then they are simple and have the following asymptotic branches:
Thanks to the asymptotic expansion of in Lemma 3.2, we only need to solve
When is sufficiently large, we have . Hence, the asymptotic eigenvalues can be determined by , . Firstly, we have , that is
Using the Rouché’s Theorem, the roots of Eq. (3.27) can be estimated by those of , which are found explicitly as following
where and are given by (3.11)2 and (3.14)10, respectively. Similarly, by , that is , which leads to
where is given by (3.14)11. By , we have , which leads to
where is given by (3.14)18. Similarly, by , we have , which leads to
where is given by (3.15)4. This completes the proof. □
Letbe defined as before and,, be the eigenvalue ofgiven by (
3.26
). Then, forsufficiently large, we have
By (A.1) and (A.2) (see the Appendix) we have that . Since , , (see Remark 3.1), we infer from (3.26)1 that for sufficiently large.
By a direct computation we get that , then by Remark 3.1 and (3.26)4 we have for sufficiently large.
Since , and, , and (see i) in the Appendix) then, we infer from (3.26)2 that .
Since and (see (A.3) and (A.4) of the Appendix), by (3.26)3 we have . □
Riesz basis propriety and exponential stability
In this section, we will discuss the stability of the system (2.11). First we prove that the system satisfies the spectrum-determined-growth condition, that is, the growth order of the system is determined via its spectral bound. Based on this propriety, we deduce the decay rate in light of the asymptotic distribution of the spectrum obtained in the last section. Finally, we prove the exponential stability of the considered problem.
Firstly, we shall show the Riesz basis property of the (generalized) eigenfunctions of the operator .
Letandbe defined as previously mentioned. Then the system of the (generalized) eigenfunctions ofis complete in.
Following [34], we define the following auxiliary operator (which is just the operator with and ) in , that is
with domain . By (2.12) we can check directly that for any
We conclude that
hence
agrees with on . However, as previously (see Lemmas 2.1 and 2.2 and Remark 2.2), one can prove that is maximal dissipative, then it is closed on , it follows that is also closed on . Hence with . Then is a skew-adjoint operator and hence satisfies [23]
Now, we prove the completeness of the (generalized) eigenfunctions of , that is,
where is the Riesz projection corresponding to .
By using the closure of and by following the same arguments used in the proof of Lemma 4.1 of [2] or Lemma 5 of [4] together with (4.1), one can show that Span, which completes the proof. □
In order to obtain the Riesz basis property of the (generalized) eigenvectors of , we recall the following abstract result on Riesz basis generation [35].
Letbe a separable Hilbert space andbe the generator of a-semigroupon. Suppose that
, whereconsists of isolated eigenvalues ofwith finite multiplicity.
, whereandis the Riesz projector associated with.
There exists a real numbersuch that, and.
Then the following statements are true:
There exist two S(t)-invariant closed subspacesandsuch that,,andforms a subspace Riesz basis for.
If, then.
has a decomposition of the topological direct sum,, if and only if
Combining Theorem 3.2 with Lemmas 4.1 and 4.2, we get the following result.
There exists a sequence of (generalized) eigenvectors ofwhich forms a Riesz basis in.
The spectrum-determined-growth condition holds true for the-semigroup, that is,whereandare given by (
3.28
)–(
3.31
) for,is the growth order ofandis the spectrum bound of.
Set
We shall prove the hypothesis (1), (2) et (3) of Lemma 4.2 holds. In fact, from (4.2) we have , where consists of isolated eigenvalues of with finite multiplicity. This implies that the hypothesis (1) holds. Since all eigenvalues are simple, then we have which proves that (2) holds. From (4.2), we have and , then there exists such that
Since if then . Hence, (3) of Lemma 4.2 hold.
Then one can deduce the existence of a sequence of (generalized) eigenvectors of forming a Riesz basis in according to statement (i) of Lemma 4.2. From Lemma 4.1, we have proved that the system of (generalized) eigenvectors is complete in . Therefore by the completeness of the (generalized) eigenvectors in , the sequence is also a Riesz basis in .
From Lemma 2.2 and Theorem 3.2 it is easy to obtain that the multiplicities of the eigenvalues of are uniformly bounded. Therefore, according to Corollary 2.1 of [12], the system (2.11) satisfies the spectrum-determined growth condition, i.e; .
The eigenvalues given by Theorem 3.2, and , , can be written in the form
where for , i.e.
The difference between (4.3)1 and (4.3)4 is given by
After a long calculation using (3.10)1,2, we end up with
where
From (4.5), we obtain
Plugging (4.7) into (4.4), we get
By a direct computation we get that , which means that and have opposite sign. Let us suppose that , then . This contradicts the last supposition so is negative and consequently .
Hence the sign of (4.8) is given by the sign of ,
Now we are going compare and . By (3.26)2 and (3.26)3, we have
After a long calculation, it follows from (3.10)3,4 and (3.11)2, we have,
where
Arguing as above, we can prove that . Hence the sign of (4.10) is given by the sign of ,
The last case of equal speeds of propagation, i.e., , is well known in literature for Timoshenko and Bresse systems. □
Based on Theorem 4.1 and Lemme 3.3, we get the following
Ifthe problem (
2.4
)–(
2.6
) is exponentially stable.
Firstly, we show that this problem is asymptotically stable. According to Lemma 2.2, it is sufficient to show there is no eigenvalue on the imaginary axis due to the Lyubich and Phóng’s Theorem (see [20]). Suppose that there exist one , , such that , and is an eigenvector of corresponding to λ. Then we have
From (2.12), we infer that
which implies for all . Hence u, ψ and θ satisfy the following equations
By (4.14)5, together with the boundary conditions on θ, we obtain that for all . From (4.14)2–4, we obtain the system
Since (4.13) holds, then . Due to the boundary conditions on ψ, we get for all . From this, (4.14)4 and the boundary conditions on u, we get for all .
Thus, we obtain , which contradicts to that is an eigenvector of corresponding to λ. Therefore, for any . The Lyubich and Phóng’s Theorem [20] asserts that the problem (2.4)–(2.6) is asymptotically stable.
Secondly, let us recall that the problem (2.4)–(2.6) is exponentially stable if and only if for all . Due to (3.32) and we have that for all . Then our conclusion follows. □
It is worth noting that, by following a characterization due to Gearhart, Huang and Prüss [19], Bazara et al. [7] prove the exponential decay of the solutions to system (2.4) with if condition (4.13) holds.
In general the exponential stability issue becomes more difficult when the second sound effect is considered. In fact, it has been shown in [11] that the dissipative effects of heat conduction with second sound for a Timoshenko system is usually weaker than those induced when , and the coupling via Cattaneo’s law may cause lack of the exponential decay usually obtained in the case of coupling via Fourier’s law. It seems that this aspect is related to the dynamics of Timoshenko’s systems more than to thermal conduction laws. According to the above remark, the exponential stability for thermoelastic system with double porosity, with or without second sound, is obtained under the same condition (4.13) by two different approaches.
Numerical simulations
Numerical approximation based spectral method
In this section, we start by deriving the dimensionless system. The new variables of the system are denoted by
where and . These new variables make our system non-dimensional. After suppressing the primes, the system (2.4) becomes
where the dimensionless parameters are
We use the Chebyshev spectral method for spatial discretization to solve the system (5.2). So, the approximate solution is imposed to satisfy the exact value at the collocation of Chebyshev-Gauss-Lobatto points. Each component solution of the system (5.2) can be approximate by defined in the domain ,
where is the solution to the system (5.2), are the time dependent expansion spectral coefficients (for more information see [3, 32]) and are the Chebyshev basis polynomials of order n defined by
By the principle of spectral method, the spatial derivative , for , at the Chebyshev points is approached by the derivative of the Lagrange polynomials [30]. Moreover, it can be written by a -Chebyshev differentiation matrix,
The second order spatial derivative is obtained by a simple multiplication . To use this method, we have first to modify the definition domain in the interval . Therefore, by changing the space variables from to , the system (5.2) becomes
We replace each spatial derivative by a multiplication of -Chebyshev differentiation, then we get a matrix system, as follows
The last two equations (5.4) are discretized as follows
where , , , and are the solution to the system (5.2) evaluated at Chebyshev collocation points. In addition, we use the discretisation θ-Method for the mixed derivatives time and space by employing the following form
Furthermore, the rest of time derivatives are discretized by Euler Method.
The main numerical difficulty of this problem is to solve a system coupled by five equations with twenty two coefficients. On the other hand, the numerical computation of the eigenvalues is formulated by a block matrix containing the matrices of differentiation of Chebyshev updated to respect the boundary conditions. The numerical approach is programmed by MATLAB software.
For numerical purpose, we provide in Table 1 the values of the relevant parameters for copper as material [16, 29],
The values of material parameters
We recall that μ, β, c and κ are given by the following:
From (5.3), (5.8) and Table 1 together with the value and , we get the values of the non-dimensional parameters (),
The variation of (left) and (right) in .
The variation of (left) and (right) in .
Using the initial data
and the boundary conditions (2.6), we obtain Figs 1–3.
The variation of in (left) and the energy in (right).
The values of the non-dimensional parameters
Based on the values of the non-dimensional parameters given in Table 2, the condition (4.13) is well satisfied. The Figs 1, 2 and 3 presented in show that each variable of our system decays to zeros as time which confirm again the stability of the system (5.2). Moreover, the optimal decay rate in this case is given by (see Table 3). The explicit expression of the decay rate plays an important role in stabilization and controllability issues. They describe the dependence of the decay rate on the damping and the physical parameters (see (3.26)) leading to the best choice of materials realizing the fastest decay of solutions. This is useful to select the suitable materials in engineering and industrial applications. These results represent a pleasant feature from the physical viewpoint.
The eigenvalues of ,
Numerical approach for the eigenvalue problem
In this subsection, we present a numerical approach to solve the eigenvalue problem based on the D-Chebyshev differentiation matrix as described above:
where is the unitary matrix updated with boundary condition at each block matrix. For example, from (5.10) one can obtain the equations:
Based on the differentiation of Chebyshev matrix, we get
where is the null matrix, I is the identity matrix and . The components of are given by . The operator is given
Using the values of the non-dimensional parameters given in Table 2, we give in Table 3 the numerical values of the real part of the eigenvalue (see Eqs (4.3)) by the above numerical approach.
By using MATLAB software, we obtain Figs 4 and 5 describing the distribution of the eigenvalues of the spectrum of .
Distribution of , (left) and , (right).
We see clearly that the spectrum is located on the left-hand side of the complex plane and have two vertical lines, which are the asymptotes (see Fig. 4 and Fig. 5). Thus, the numerical results coincide with the asymptotic spectrum obtained in Theorem 3.2. These figures also confirm our mathematical results concerning the asymptotic behavior of the spectrum of .
Distribution of , (left) and , (right).
Footnotes
Acknowledgements
The authors would like to thank the Editor Prof. Alain Miranville and the anonymous reviewers for their critical reviews, helpful and constructive comments that greatly contributed to improving the final version of the paper.
Appendix
(i) Now, we study the sign of and given by (3.11)1,2 to evaluate (3.32). Since μ, β and are all positives, then from (3.11)1, we infer that and have the same sign. On the other hand, we find
where and are defined by (4.6). We have already proved that and consequently
Using the same argument we get
(ii) Let us study the sign of and given by (3.11)3,4. We rewrite (3.11)3 in the following form
Since , and are all positives, then and have the same sign
where and are defined by (4.11). We have already proved that and consequently .
Using the same argument, we obtain from (3.11)4 that
Since , and δ are all positives, then and have the same sign
Since , we conclude that
(iii) Finally we study the sign of . From (3.10)3 and (3.11)2, we obtain after a long calculation
where
By a direct computation we get that
which means that and have opposite sign. Let us suppose that , then . This contradicts the last supposition so is negative and consequently
Using the same argument (3.10)3 and (3.11)2, we obtain
Consequently
References
1.
R.Adams, Sobolev Spaces, Academic Press, New York, 1975.
2.
M.Aouadi and K.Boulehmi, Asymptotic behavior of non-uniform Timoshenko beam acting on shear force with feedback controller, Z. Angew. Math. Mech.97 (2017), 1579–1599. doi:10.1002/zamm.201700028.
3.
M.Aouadi, I.Mahfoudhi and T.Moulahi, Boundary stabilization of a thermoelastic diffusion system of type II, J. Acta Appl Math.169 (2020), 499–522. doi:10.1007/s10440-019-00308-7.
4.
M.Aouadi and T.Moulahi, Spectral analysis of thermoelastic systems under nonclassical thermal models, J. Therm. Stresses40 (2017), 3–24. doi:10.1080/01495739.2016.1257273.
5.
A.Arusoaie, Spatial and temporal behavior in the theory of thermoelasticity for solids with double porosity, J. Therm. Stresses41 (2018), 500–521. doi:10.1080/01495739.2017.1387882.
6.
N.Bazarra, J.R.Fernández, M.C.Leseduarte, A.Magana and R.Quintanilla, On the thermoelasticity with two porosities: Asymptotic behaviour, J. Math. Mech. Solids24 (2019), 2713–2725. doi:10.1177/1081286518783219.
7.
N.Bazarra, J.R.Fernández, M.C.Leseduarte, A.Magana and R.Quintanilla, On the uniqueness and analyticity in viscoelasticity with double porosity, J. Asymptotic Analysis112 (2019), 151–164. doi:10.3233/ASY-181500.
8.
G.Birkhoff and R.Langer, The boundary problems and developments associated with a system of ordinary linear differential equations of the first order, J. Proc. Amer. Acad. Arts Sci.58 (1923), 49–128.
9.
S.C.Cowin and J.W.Nunziato, Linear elastic materials with voids, J. Elast.13 (1983), 125–147. doi:10.1007/BF00041230.
10.
K.J.Engel and R.Rainer Nagel, One-Parameter Semigroups for Linear Evolution Equations, Graduate Texts in Mathematics, Vol. 194, Springer-Verlag, New York, 2000.
11.
H.Fernández Sare and R.Racke, On the stability of damped Timoshenko systems: Cattaneo versus Fourier law, Archive for Rational Mechanics and Analysis194 (2009), 221–251. doi:10.1007/s00205-009-0220-2.
12.
B.Guo and G.Xu, Expansion of solution in terms of generalized eigenfunctions for a hyperbolic system with static boundary condition, J. Funct. Anal.231 (2006), 245–268. doi:10.1016/j.jfa.2005.02.006.
13.
Z.Han, G.Xu and X.Tang, Stability analysis of a thermo-elastic system of type II with boundary viscoelastic damping, Z. Angew. Math. Phys63 (2012), 675–689. doi:10.1007/s00033-011-0184-6.
14.
D.Ieşan, A theory of thermoelastic materials with voids, J. Acta Mech.60 (1986), 67–89. doi:10.1007/BF01302942.
15.
D.Ieşan, Thermoelastic Models of Continua, Springer Science & Business Media, 2004.
16.
D.Ieşan and R.Quintanilla, A fully coupled constitutive model for thermohydro-mechanical analysis in elastic media with double porosity, J. Geoph. Res. Letters,vol.30 (2003), 22–68.
17.
D.Ieşan and R.Quintanilla, On a theory of thermoelastic materials with a double porosity structure, J Therm. Stresses37 (2014), 1017–1036. doi:10.1080/01495739.2014.914776.
18.
R.Kumar, R.Vohra and M.G.Gorla, Some considerations of fundamental solution in micropolar thermoelastic materials with double porosity, J. Arch. Mech.68 (2016), 263–284.
19.
Z.Liu and S.Zheng, Semigroups Associated with Dissipative Systems, Chapman & Hall/CRC Research Notes in Mathematics, Vol. 398, Chapman & and Hall/CRC, Boca Raton, FL, 1999.
20.
Y.I.Lyubich and V.Q.Phóng, Asymptotic stability of linear differential equations in Banach spaces, J. Studia Math.88 (1998), 37–42. doi:10.4064/sm-88-1-37-42.
J.Nunziato and S.Cowin, A nonlinear theory of elastic materials with voids, J. Arch Rat Mech Anal.72 (1979), 175–201. doi:10.1007/BF00249363.
23.
A.Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, Berlin, 1983.
24.
E.Scarpetta, M.Svanadze and V.Zampoli, Fundamental solutions in the theory of thermoelasticity for solids with double porosity, J. Therm. Stresses37 (2014), 727–748. doi:10.1080/01495739.2014.885337.
25.
B.Straughan, Stability and uniqueness in double porosity elasticity, Int. J. Eng. Sci.65 (2013), 1–8. doi:10.1016/j.ijengsci.2013.01.001.
26.
M.Svanadze, Plane waves and boundary value problems in the theory of elasticity for solids with double porosity, Acta Applicandae Mathematicae122 (2012), 461–471.
27.
M.Svanadze, Boundary value problems of steady vibrations in the theory of thermoelasticity for materials with a double porosity structure, J. Arch. Mech.69 (2017), 347–370.
28.
M.Svanadze and A.Scalia, Mathematical problems in the coupled linear theory of bone poroelasticity, J. Comput. Math. Appl.66 (2013), 1554–1566. doi:10.1016/j.camwa.2013.01.046.
29.
L.Thomas, Fundamentals of Heat Transfer, Prentice-Hall, Inc., Englewood Cliffs, New Jersey, 1980.
C.Tretter, Spectral problems for systems of differential equations with λ-polynomial boundary conditions, J. Math. Nachr.214 (2000), 129–172. doi:10.1002/1522-2616(200006)214:1<129::AID-MANA129>3.0.CO;2-X.
32.
J.A.C.Weideman and S.C.Reddy, A MATLAB differentiation matrix suite, J. ACM Trans. Math. Softw.26 (2000), 465–519. doi:10.1145/365723.365727.
33.
R.Wilson and E.Aifantis, On the theory of consolidation with double porosity, Int. J. Eng. Sci.20 (1982), 1009–1035. doi:10.1016/0020-7225(82)90036-2.
34.
G.Q.Xu and B.Z.Guo, Riesz basis property of evolution equations in Hilbert spaces and application to a coupled string equation, SIAM J. Cont. and Optim.42 (2003), 966–984. doi:10.1137/S0363012901400081.
35.
G.Q.Xu and S.P.Yung, The expansion of semigroup and criterion of Riesz basis, J. Diff. Equations210 (2005), 1–24. doi:10.1016/j.jde.2004.09.015.
36.
Y.Zhao, L.Zhang, J.Zhao, J.Luo and B.Zhang, Triple porosity modelling of transient well test and rate decline analysis for multi-fractured horizontal well in shale gas reservoirs, J. Pet. Sci. Eng.110 (2013), 253–262. doi:10.1016/j.petrol.2013.09.006.
