 Research
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About finite energy solutions in thermoelasticity of micropolar bodies with voids
 Marin Marin^{1}Email authorView ORCID ID profile,
 Adina Chirila^{1},
 Andreas Öchsner^{2} and
 Sorin Vlase^{3}
 Received: 17 April 2019
 Accepted: 6 May 2019
 Published: 14 May 2019
Abstract
Our study is dedicated to the problem with initial and boundary conditions in the theory of thermoelasticity for micropolar materials with pores. We obtain some results regarding the existence and uniqueness of a finite energy solution of this mixed problem by generalizing the corresponding results obtained by Dafermos in the context of classical elasticity. In some specific conditions we prove some properties regarding the possibility to control the finite energy solution.
Keywords
 Micopolar bodies
 Voids
 Finite energy solution
 Existence of solution
MSC
 74A15
 74F05
 35Q74
 35Q79
1 Introduction
It is known that a material that is under the action of high temperatures is influenced by a heat flow which will involve a temperature distribution, and this will give rise to thermal stresses. The role of the pertinent material properties can affect the magnitude of thermal stress, so it must be well understood and all possible modes of failure must be considered. Taking into account that our study approaches the materials with voids, we can hope that our results can help the specialists in applications in which some materials with voids are involved, such as in the production of granular materials or geological layers. It should be underlined that the granular theory of Goodman and Cowin in [1] gave the start of research on materials with voids. This theory together with the approach of Cowin and Nunziato in [2] has a characteristic feature: the presence of some small pores in a model of a classic solid which are considered by adding to each particle an additional degree of freedom. Considering the two theories of materials with voids, this degree of freedom will be used to characterize the mechanical evolution of granular solids. In this case the pores of the material are interstices and the material of matrix is elastic.
The important implications of this theory are related to manufactured porous objects, such as ceramics, pressed powders, and also to geological layers, such as rocks or soil. In the initial stage, the approaches of Cowin and Nunziato were applied only to solids which are nonconductor of heat (see also [3]). Then, the theory was extended to cover thermoelasticity of the materials with voids. So, the paper of Iesan [4] is a straightforward extension for solids for which the thermal effect is considered.
On the other hand, in our work we take into account the micropolar structure of the materials. This is part of a more general concept, that of the microstructure, which includes a large number of extensions of classical elasticity theory such as micromorphic structure, microstretch structure, Cosserat structure, dipolar structure, microtemperatures, and so on. The start in microstructure theories was given by Eringen (see [5, 6]). Then, the interest in these theories has grown enormously, as evidenced by a huge number of studies published in this context. Here we list a few of them, such as [7–24].
In our study we will extend the result obtained by Fichera in [25] and Dafermos [26] regarding the existence of solutions in the context of linear elasticity theory. Here, the asymptotic stability for the finite energy solution is also approached. We must say, however, that the results of Dafermos, in turn, are based on the results of Visik [27].
The strategy of our work is the following. First, we put down the main equations and conditions characteristic to the mixed problem for the theory of the micropolar thermoelastic materials having voids. Then, we introduce the notion of finite energy solutions and show a way to expand the set of finite energy solutions. The result regarding the uniqueness of this kind of solution is obtained for the general case of nonhomogeneous conditions and boundary data. We first obtain an existence result regarding a solution with finite energy, and this is deduced regarding the null initial data. Then, this result is extended to the general nonhomogeneous initial conditions. In our last result we deduce several estimations useful to control the evolution of the finite energy solution.
2 Basic equations
We assume that our thermoelastic micropolar body with voids occupies at the moment \(t=0\) a regular domain D from the space \(R^{3}\), the usual Euclidian space. Denote by ∂D the border of D and assume it to be a sufficiently regular surface, at least to admit the application of the divergence theorem. The closure of D is denoted by D̄. The motion of our medium will be reported to the rectangular axes \(Ox_{i}\), (\(i=1,2,3\)). The usual vector and tensor notations are adopted. We will use a superposed dot to designate the time derivative of a function, while a partial derivative with respect to a spatial variable is denoted by a subscript preceded by a comma. In the case of repeated indices, the Einstein summation is used. When there is no risk of confusion, the time argument and/or the spatial argument and that of a function will be omitted.

\(u_{i}(x,t)\)—the displacement vector field from the reference configuration;

\(\varphi _{i}(x,t)\)—the vector for the micopolar displacement;

θ—the difference between the current temperature and \(T_{0}\), that is,$$ \theta =TT_{0}; $$

ϕ—the difference between the current volume fraction temperature and \(\nu _{1}\), that is,$$ \phi =\nu \nu _{1}. $$

the motion equations:$$\begin{aligned}& \begin{gathered} t_{ij,j}+\varrho f_{i}=\varrho \ddot{u}_{i}, \\ m_{ij,j}+\varepsilon _{ijk} t_{jk}+\varrho g_{i}=I_{ij}\ddot{\varphi } _{j}; \end{gathered} \end{aligned}$$(4)

the equations of the equilibrated forces:$$\begin{aligned}& h_{i,i}+\varrho l=\varrho \kappa \ddot{\phi }; \end{aligned}$$(5)

the equation of evolution of energy:$$\begin{aligned}& \varrho T_{0}\dot{\eta }=q_{i,i} +\varrho r. \end{aligned}$$(6)

ϱ—the density of mass;

η—the entropy specific to the body;

\(T_{0}\)—the constant temperature in the initial state;

κ—the balancing inertia;

\(\varepsilon _{ij}\), \(\gamma _{ij}\)—the tensors of strain;

\(t_{ij}\), \(m_{ij}\)—the tensors of stress;

\(h_{i}\)—a stress vector for equilibrium;

\(q_{i}\)—a heat flux vector;

\(f_{i}\), \(g_{i}\), l—body forces;

r—the supply of heat;
3 Main results
We assume that our micropolar porous material occupies a properly regular and bounded domain D from \(R^{3}\), which is the usual Euclidean space.
The set of all scalar functions defined on the domain D, which have a derivative of order n in all points of D and this derivative is a continuous function on the space D̄, are denoted by \(C^{n} (\bar{D} )\).
The class of all functions \(f: [0,t_{0} )\to B\) having the time derivatives up to nth order, the last derivative being continuous on \([0,t_{0} )\), is denoted by \(C^{n} ( [0,t _{0} ); D )\), in which B is the notation for a Banach space.
Analog definitions can be given for the spaces \(L_{1} ( (0,t _{0} ); D )\) and \(L_{2} ( (0,t_{0} ); D )\) due to the clear meaning of \(L_{1}\) and \(L_{2}\) notations.
Definition 1
Clearly, inequality (15) can be generalized for \((\mathbf{v}, \boldsymbol{\psi },\chi ),(\mathbf{v},\boldsymbol{\psi },\chi ) \in {\hat{\mathbf{W}}}_{1}(D)\) and inequality (16) can be generalized for \(\theta \in {\hat{W}}_{1}(D)\).
With the help of Sobolev’s embedding theorem and by using Schwarz’s inequality, we can extend by continuity the functional \(F_{3}(y,w)\) on the product space \({\breve{U}} (Q_{0} )\times V (Q _{0} )\) and, analogously, \(F_{4}(z,\delta )\) can be an extended functional on the space \({\breve{U}} (Q_{0} )\times L_{1} ( (0, t_{0} ); G(D) )\).
Similarly, we can extend \(F_{5}(\delta , w)\) so that it can exist for \(\delta \in H_{0}(D)\), \(w\in {\breve{U}} (Q_{0} )\).
Below, we can define the concept of solution, having a finite energy, in the case of the mixed problem defined in our context.
Definition 2
The result that follows shows a way to find a new finite energy solution, starting from a given solution.
Theorem 1
Consider y to be a solution with finite energy, \(y\in V (Q_{0} )\), which verifies Eq. (11) and satisfies the conditions to the limit (10), defined on the cylinder \(Q_{0}\), and which corresponds to the source \(z= (f_{i}, g _{i},l,r)\) and to the initial conditions \(\delta \equiv (u _{i}^{0},\varphi _{i}^{0},\phi ^{0},\theta ^{0} )\).
Proof
The result can be obtained by using the same procedure as in Dafermos [26]. □
Also, by using a procedure similar to the one used in Dafermos [26], we can prove the following uniqueness result with regard to the finite energy solution of the mixed problem.
Theorem 2
In thermoelasticity of initially stressed bodies with voids, the mixed problem consisting of system (11) and the data to the limit (10) admits only one finite energy solution which corresponds to some given initial conditions and some given sources.
Let us restrict the considerations to the more simple case of null initial data, the sources belonging to \(L_{2} ( (0,t_{0} ); G(D) )\). In this case, we have the open path to demonstrate the existence result for a solution with finite energy as well as an estimate of this solution. These results are included in the following theorem.
Theorem 3
Now, we want to extend the result of the existence for the solution with finite energy from the particular case of the null initial data from Theorem 2 to the case of some arbitrary initial data. To this aim we need the following definitions.
Definition 3
In the following theorem we characterize the solution defined by means of the map \(\mathcal{S}(z)\) in Theorem 3. Also, the smoothness of this kind solution is investigated, taking into account some hypotheses on the initial data and some specific sources. This is an auxiliary step to obtain the existence of the solution in the general case of the initial conditions.
Theorem 4
For any \(z\in H_{0}(D)\), \(\mathcal{S}(z)\) is a welldefined and onetoone map so that the inverse map \(\mathcal{S} ^{1}(z)\) is defined for all z from the codomain of \(\mathcal{S}(z) \subset H(D)\) to \(H_{0}(D)\).
Proof
To prove our result, we can use the same procedure as Fischera in the paper [25]. So, we must take into account that the functional \(F_{1}\) satisfies inequality (15). Also, we must consider that the functional \(F_{1}(v,v)\) is coercive in the space \({\hat{\mathbf{W}}}_{1}(D)\) with respect to the norm \({\\cdot\}_{ \mathbf{W}_{1}(D)}\), which follows from inequality (17). □
Based on these new considerations, we can address the problem of the existence of a finite energy solution in the general case of some nonhomogeneous initial data.
Theorem 5
Proof
The result can be obtained by using the same procedure as Fischera in the paper [25]. □
Using the procedure of Dafermos [26] and Fichera [25], we can deduce the following three properties regarding the controllability of the finite energy solution from Theorem 5.
Theorem 6
Remark
It is worth noting that in the simple case of null loads, \(z\equiv 0\), inequality (23) turns into equality.
4 Conclusions
We first put down the main equations and basic conditions for the mixed problem in the theory of micropolar thermoelastic materials with pores. Then we extended the procedure proposed by Dafermos in [26] and Fichera in [25] in order to obtain some uniqueness and existence results. The first result is the uniqueness of the finite energy solution. Then we prove the existence result for a finite energy solution in the particular case of null initial data. In Theorem 5 this result is extended in order to cover the general case of nonhomogeneous initial conditions. The last theorem of our study is dedicated to some properties regarding the controllability of the finite energy solution. We need to emphasize the validity of the results regarding the uniqueness and existence for the finite energy solution is as in the classical theory, even if the context in our study is much more complicated, because we considered the effect of the micropolar structure, the effect of thermal treatment, and the effect of voids.
Declarations
Acknowledgements
We want to thank the reviewers who have read the manuscript carefully and have proposed pertinent corrections that have led to an improvement in our manuscript.
Availability of data and materials
Data sharing not applicable to this article as no datasets were generated or analysed during the current study.
Funding
Not applicable.
Authors’ contributions
All authors contributed equally to the writing of this paper. All authors read and approved the final form of the manuscript.
Competing interests
The authors declare that they have no competing interests.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
References
 Goodman, M.A., Cowin, S.C.: A continuum theory of granular material. Arch. Ration. Mech. Anal. 44, 249–266 (1972) MathSciNetView ArticleGoogle Scholar
 Cowin, S.C., Nunziato, J.W.: Linear elastic materials with voids. J. Elast. 13, 125–147 (1983) View ArticleGoogle Scholar
 Nunziato, J.W., Cowin, S.C.: A nonlinear theory of materials with voids. Arch. Ration. Mech. Anal. 72, 175–201 (1979) MathSciNetView ArticleGoogle Scholar
 Iesan, D.: A theory of thermoelastic material with voids. Acta Mech. 60, 67–89 (1986) View ArticleGoogle Scholar
 Eringen, A.C.: Theory of thermomicrostretch elastic solids. Int. J. Eng. Sci. 28, 1291–1301 (1990) MathSciNetView ArticleGoogle Scholar
 Eringen, A.C.: Nonlocal Continuum Field Theories. Springer, New York (2002) MATHGoogle Scholar
 Iesan, D., Quintanilla, R.: On the thermoelastic bodies with inner structure and microtemperatures. J. Math. Anal. Appl. 354, 12–23 (2009) MathSciNetView ArticleGoogle Scholar
 Svanadze, M.: On the linear equilibrium theory of elasticity for materials with triple voids. Q. J. Mech. Appl. Math. 71(3), 329–348 (2018) MathSciNetView ArticleGoogle Scholar
 Marin, M.: Cesaro means in thermoelasticity of dipolar bodies. Acta Mech. 122(1–4), 155–168 (1997) MathSciNetView ArticleGoogle Scholar
 Abbas, I.A.: A GN model based upon twotemperature generalized thermoelastic theory in an unbounded medium with a spherical cavity. Appl. Math. Comput. 245, 108–115 (2014) MathSciNetMATHGoogle Scholar
 Abbas, I.A.: Eigenvalue approach for an unbounded medium with a spherical cavity based upon twotemperature generalized thermoelastic theory. J. Mech. Sci. Technol. 28(10), 4193–4198 (2014) View ArticleGoogle Scholar
 Abbas, I.A., AboDahab, S.M.: On the numerical solution of thermal shock problem for generalized magnetothermoelasticity for an infinitely long annular cylinder with variable thermal conductivity. J. Comput. Theor. Nanosci. 11(3), 607–618 (2014) View ArticleGoogle Scholar
 Othman, M.I.A.: State space approach to generalized thermoelasticity plane waves with two relaxation times under the dependence of the modulus of elasticity on reference temperature. Can. J. Phys. 81(12), 1403–1418 (2003) View ArticleGoogle Scholar
 Sharma, J.N., Othman, M.I.A.: Effect of rotation on generalized thermoviscoelastic Rayleigh–Lamb waves. Int. J. Solids Struct. 44(13), 4243–4255 (2007) View ArticleGoogle Scholar
 Othman, M.I.A., Hasona, W.M., AbdElaziz, E.M.: Effect of rotation on micropolar generalized thermoelasticity with twotemperatures using a dualphaselag model. Can. J. Phys. 92(2), 149–158 (2014) View ArticleGoogle Scholar
 Marin, M., Öchsner, A.: The effect of a dipolar structure on the Holder stability in Green–Naghdi thermoelasticity. Contin. Mech. Thermodyn. 29(6), 1365–1374 (2017) MathSciNetView ArticleGoogle Scholar
 Hassan, M., Marin, M., Alsharif, A., Ellahi, R.: Convective heat transfer flow of nanofluid in a porous medium over wavy surface. Phys. Lett. A 382(38), 2749–2753 (2018) MathSciNetView ArticleGoogle Scholar
 Marin, M., Nicaise, S.: Existence and stability results for thermoelastic dipolar bodies with double porosity. Contin. Mech. Thermodyn. 28(6), 1645–1657 (2016) MathSciNetView ArticleGoogle Scholar
 Marin, M., Vlase, S., Carstea, C.: A dipolar structure in the heatflux dependent thermoelasticity. AIP Adv. 8, 035220 (2018) View ArticleGoogle Scholar
 Modrea, A., Vlase, S., et al.: The influence of dimensional and structural shifts of the elastic constant values in cylinder fiber composites. J. Optoelectron. Adv. Mater. 15(3–4), 278–283 (2013) Google Scholar
 Niculita, C., Vlase, S., et al.: Optimum stacking in a multiply laminate used for the skin of adaptive wings. J. Optoelectron. Adv. Mater. 5(11), 1233–1236 (2011) Google Scholar
 Craciun, E.M., Barbu, L.: Compact closed form solution of the incremental plane states in a prestressed elastic composite with an elliptical hole. Z. Angew. Math. Mech. 95(2), 193–199 (2015) MathSciNetView ArticleGoogle Scholar
 Marin, M., Craciun, E.M., Pop, N.: Considerations on mixed initialboundary value problems for micropolar porous bodies. Dyn. Syst. Appl. 25(1–2), 175–196 (2016) MATHGoogle Scholar
 Marin, M., Radulescu, V.: A variational approach for the mixed problem in the elastostatics of bodies with dipolar structure. Mediterr. J. Math. 15(6), 221 (2018) MathSciNetView ArticleGoogle Scholar
 Fichera, G.: Existence theorems in elasticity. In: Truesdell, C. (ed.) Linear Theories of Elasticity and Thermoelasticity, vol. VI a/2, pp. 347–424. Springer, Berlin (1973) View ArticleGoogle Scholar
 Dafermos, C.M.: On the existence and the asymptotic stability of solutions to the equations of linear thermoelasticity. Arch. Ration. Mech. Anal. 29(4), 241–271 (1968) MathSciNetView ArticleGoogle Scholar
 Visik, M.I.: Quasilinear strongly elliptic systems of differential equations in divergence form. Trans. Mosc. Math. Soc. 12, 140–208 (1963) Google Scholar
 Carbonaro, B., Russo, R.: Energy inequalities in classical elastodynamics. J. Elast. 14, 163–174 (1984) View ArticleGoogle Scholar
 Marin, M., Öchsner, A.: Essentials of Partial Differential Equations with Applications. Springer, Cham (2019) View ArticleGoogle Scholar