On vibrations in thermoelasticity without energy dissipation for micropolar bodies

Transcription

1 Marin and Baleanu Boundary Value Problems : DOI 0.86/s R E S E A R C H Open Access On vibrations in thermoelasticity without energy dissipation for micropolar bodies Marin Marin * and Dumitru Baleanu 2,3 * Correspondence: m.marin@unitbv.ro Department of Mathematics and Computer Science, Transilvania University of Brasov, Brasov, 5008, Romania Full list of author information is available at the end of the article Abstract We consider a micropolar thermoelastic body occupying a prismatic cylinder that is free of loads on lateral surface no body force, no body couple, and no heat supply. On the base of the cylinder are prescribed a time-dependent displacement, a microrotation, and a thermal displacement, which are harmonic in time, and collaborate to induce the motion of the considered body. With the help of a measure associated with the corresponding steady-state vibration and by assuming that the exciting frequency is lower than a certain critical frequency, we will obtain a spatial decay estimate. Keywords: thermoelasticity; micropolar; energy dissipation; vibration Introduction The theory of thermoelastic material behavior without energy dissipation possesses the following properties: the heat flow, in contrast to that in classical thermoelasticity characterized by the Fourier law, does not involve energy dissipation; a constitutive equation for an entropy flux vector is determined by the same potential function as also determines the stress, and it permits the transmission of heat as thermal waves at finite speed. It is well known that in a micropolar continuum the deformation is described not only by the displacement vector but also by an independent rotation vector. This rotation vector specifies the orientation of a triad of director vectors attached to each material particle. A material point can experience a microrotation without undergoing a macrodisplacement. An infinitesimal surface element transmits a force and a couple vector, which give rise to nonsymmetric stress and couple-stress tensors. The former is related to a nonsymmetric strain tensor and the latter to a nonsymmetric curvature tensor, defined as the gradient of the rotation vector. It is believed that this type of the continuum mechanics was originally introduced by Voigt since 887 and the brothers Cosserat since 909. There is a simplified variant of the theory of micropolar bodies, the so-called couple-stress theory, and in this theory the rotation vector is not independent of the displacement vector, but related to it in the same way as in classical continuum mechanics. The motivation for the extension of the classical to micropolar and couple-stress theory was that the classical theory was not able to predict the size effect experimentally observed in problems which had a geometric length scale comparable to the material s microstructural length, such as the grain size in a polycrystalline or granular aggregate. For example, the apparent strength of some materials 206 Marin and Baleanu. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original authors and the source, provide a link to the Creative Commons license, and indicate if changes were made.

2 MarinandBaleanu Boundary Value Problems : Page 2 of 9 with stress concentrators such as holes and notches is higher for smaller grain size; for a given volume fraction of dispersed hard particles, the strengthening of metals is greater for smaller particles; the bending and torsional strengths are higher for very thin beams and wires. There are several hyperbolic theories for describing the heat conduction which are also called theories of second sound. In these theories the flow of heat is modeled with finite propagation speed, in contrast to the classical model based on the Fourier law, leading to infinite propagation speed of heat signals. A review of these theories is presented in the paper by Chandrasekharaiah. First results in the thermoelastic theory without energy dissipation were obtained by Green and Naghdi 2. This thermomechanical theory of deformable media introduces the so-called thermal displacement related to the common temperature. and uses a general entropy balance as postulated in Green and Naghdi 3. By the procedure of Green and Naghdi, the reduced energy equation is regarded as an identity for all thermodynamical processes and places some restrictions on the functional forms of the dependent constitutive variables. The theory is illustrated in detail in the context of the flow of heat in a rigid solid, with particular reference to the propagation of s thermal waves at finite speed. The linear theory of thermoelasticity without energy dissipation for homogeneous and isotropic materials was employed by Nappa 4 to obtain spatial energy bounds and decay estimates for the transient solutions in connection with the problem in which a thermoelastic body is deformed subject to boundary and initial data and body supplies having a compact support, provided positive definiteness assumptions are made upon the constitutive coefficients. Also, in the linear theory of thermoelasticity without energy dissipation Chandrasekharaiah 5 proves the uniqueness of the solutions, Iesan 6 establishes continuous dependence results, while Quintanilla 7 studies the question of existence. In 8 0 we find some results regarding vibrations for magneto-thermoelastic bodies. Other results regarding thermoelasticity of dipolar bodies and of microstretch bodies are presented in 6. Some concrete and practical issues related to porous media can be found in 7and8. Our present study is dedicated to the spatial behavior of the harmonic in time vibrations within the model of the linear thermoelasticity theory without dissipation energy for micropolarbodies. We proveaprioriestimates for the amplitude of a harmonic vibration by means of some auxiliary identities. It provides some estimates describing how the amplitude evolves with respect to the distance to the excited base, provided the frequency of vibration is greater than a certain critical value. The spatial behavior of the harmonic in time vibrations has been studied by Chirita 9 in the theory of classical linear thermoelasticity. Here the author uses a technique developed by Flavin and Knops 20 in the low frequency range. Some differential inequalities are established for the appropriate selected measures which after integration provide exponential estimates for the spatial evolution of the amplitude of vibration, provided the positive definiteness of the constitutive coefficients is assumed. In 2 Ciarletta proposed a theory of micropolar thermoelasticity, which, because it is a theory without energy dissipation, allows propagation of thermal waves at a finite speed. Theplanofthepaperisthefollowing.Wefirstwritedownthemixedinitialboundary value problem within context of micropolar bodies in thermoelasticity without energy dissipation. Then we prove some differential relations for certain cross-sectional integrals.

3 MarinandBaleanu Boundary Value Problems : Page 3 of 9 Based on these relations we obtain estimates describing how the amplitude evolves with respect to the distance to the excited base. This requires the frequency of vibrations to be greaterthanacertaincriticalvalue. 2 Basic equations Let B be an open set domain of a three-dimensional Euclidian space occupied by the reference configuration of a homogeneous micropolar body. We assume that B is regular and a finite region with boundary B andwe denote the closureof B by B. We use afixedsystem of rectangular Cartesian axes and adopt Cartesian tensor notation. Points in B are denoted by x j and t 0, is the temporal variable. Also, the spatial argument and the time argument of a function will be omitted when there is no likelihood of confusion. A superposed dot denotes the differentiation with respect to time t, and a subscript preceded by a comma denotes the differentiation with respect to the corresponding spatial variable. The governing equations of the theory of anisotropic and homogeneous micropolar thermoelasticity without energy dissipation, as we can find in 2, consist of the equations of motion t ij,j ϱf i = ϱü i, m ij,j ε ijk t jk ϱm i = I ij ϕ j ; and the equation of energy ϱ η = ϱ r q i,i. 2 Equations and2aredefinedforx, t B 0,. When the reference solid has a center of symmetry at each point but is otherwise nonisotropic, the constitutive equations, defined for x, t B 0,, are t ij = A ijmn ε mn B ijmn γ mn D ij θ, m ij = B mnij ε mn C ijmn γ mn E ij θ, ϱη = D ij ε ij E ij γ ij c θ, q i = K ij β j. 3 The deformation tensors ε ij and γ ij used in equations 3 aredefined,in B 0,, by means of the geometric equations ε ij = u j,i ε jik ϕ k, γ ij = ϕ j,i. 4 The system of equations is complete if we add the law of heat flow β i = θ,i 5 for all x, t B 0,.

4 MarinandBaleanu Boundary Value Problems : Page 4 of 9 In the equations above we have used the following notation: u i for the components of displacement vector, ϕ i for the components of microrotatia vector, t ij for the components of stress tensor, m ij for the components of couple-stress tensor, q i for the components of the heat conduction vector, η forthespecificentropyperunitmass,ϱ for the constant reference density mass, for the constant reference temperature, I ij for the components of inertia, β i for the components of the thermal displacement gradient vector, F i for the components of the external body force vector, M i for the components of the external body couple vector, r for the external rate of supply of heat per unit mass, and ε ijk is the alternating symbol. The coefficients from 3, that is, A ijmn, B ijmn, C ijmn, D ij, E ij, c, andk ij are constant constitutive coefficients subject to the following symmetry conditions: A ijmn = A mnij, C ijmn = C mnij, I ij = I ji, K ij = K ji. 6 The free energy, used to obtain the constitutive equations, is given by ϱ = 2 A ijmnε ij ε mn B ijmn ε ij γ mn 2 C ijmnγ ij γ mn D ij ε ij θ E ij γ ij θ c θ 2 c K ij τ,i τ,j Here we denote by τ the thermal displacement related to the temperature variation. The relationship between τ and θ isgiven by τ = θ. 8 Introducing the constitutive equations 3 and the geometric equations 4intheequa- tions of motion and the equation of energy 2, we obtain a system of equations in terms of displacements u i, microrotations ϕ i, and thermal displacements τ as Aijmn u n,m ε mnk ϕ k B ijmn ϕ n,m D ij τ,j ϱf i = ϱü i, Bmnij u n,m ε mnk ϕ k C ijmn ϕ n,m E ij τ,j ε ijk Ajkmn u n,m ε mnk ϕ k B jkmn ϕ n,m D jk τ ϱm i = I ij ϕ j, 9 K ij τ,j,i D ij u j,i ε jik ϕ k E ij ϕ j,i ϱ r = c τ, for any x, t B 0,. 3 Preliminary results Consider a cross-section D of a prismatic cylinder and the boundary of the section, D, assumed to be piecewise continuously differentiable. We choose the system of Cartesian rectangular axis so that its origin is in the center of the cylinder base and the positive x 3 - axis is directed along the cylinder. If we denote by L the length of the cylinder, then the lateral boundary of the cylinder is S = D 0, L. The contents of the prismatic cylinder is a micropolar thermoelastic body which is homogeneous and anisotropic.

5 MarinandBaleanu Boundary Value Problems : Page 5 of 9 The cylinder is free of load on the lateral boundary surface, that is, we have a zero body force, couple force and heat supply and zero displacement, microrotations, and thermal displacements. But over the base of cylinder are specified the displacements, microrotations, and thermal displacement, all of which are assumed to be harmonic in time. Therefore, besides the system of equations 9 we can adjoin the following lateral boundary conditions: u i x, t=0, ϕ i x, t=0, τx, t=0, x, t S 0,, 0 and the base boundary conditions u i x, x 2,0,t=ũ i x, x 2 e ιωt, ϕ i x, x 2,0,t= ϕ i x, x 2 e ιωt, τx, x 2,0,t= tx, x 2 e ιωt, x, x 2 D0, t >0, where ũ i x, x 2, ϕ i x, x 2, and tx, x 2 are prescribed smooth functions, ι is the complex unit, and ω is a prescribed positive constant. Loads from induce inside the cylinder some vibrations harmonic in time, having the form u i x, x 2, x 3, t=u i x, x 2, x 3 e ιωt, ϕ i x, x 2, x 3, t= i x, x 2, x 3 e ιωt, τx, x 2, x 3, t=tx, x 2, x 3 e ιωt, x, x 2, x 3, t B 0,. 2 The amplitude U i, i, T of the vibrations satisfies the following system of differential equations: Aijmn U n,m ε mnk k B ijmn n,m ιωd ij T,j ϱω2 U i =0, Bmnij U n,m ε mnk k C ijmn n,m ιωe ij T,j ε ijk Ajkmn U n,m ε mnk k B jkmn n,m ιωd jk T I ij ω 2 j =0, K ij T,j ιωd ij U j,i ε jik k ιωe ij j,i c ω 2 T =0.,i 3 The lateral boundary conditions get the form U i x=0, i x=0, Tx=0, x S, 4 and the base boundary conditions become U i x, x 2,0=Ũ i x, x 2, i x, x 2,0= i x, x 2, Tx, x 2,0= Tx, x 2, x, x 2 D0. 5 In the case of a finite cylinder we will have to prescribe a boundary condition on the superior base of the cylinder, DL. For a forced oscillation the spatial behavior of the amplitude was studied by Chirita 9andCiarletta2,in thecaseofarhombicthermoelasticmate- rial, provided the exciting frequency is less than a certain critical frequency. The main goal

6 MarinandBaleanu Boundary Value Problems : Page 6 of 9 of our study is to estimate how the amplitude evolves with respect to the axial distance to the excited and. In the following we want to prove some estimates on a solution of the system of equations 3, with the lateral boundary conditions 4 and the base boundary conditions 5. We will use the notation U j,i = U j,i ε jik k. In the following theorem we will state and prove four auxiliary identities on which will be based the main result. Theorem Let U i, i, T be a solution of the boundary value problem consisting of equations 3-5. Then the following equalities are satisfied: 2 Aijmn U j,i Ū n,m C ijmn n,m j,i B ijmn U j,i n,m Ū j,i n,m ϱω 2 U i Ū i I ij ω 2 i j ιωdij TU j,i TŪ j,i ιωe ij T i,j T i,j = d 2 A3jmn U n,m B 3jmn n,m ιωd 3j TŪ j A3jmn Ū n,m B 3jmn n,m ιωd 3j TU j B3jmn U n,m C 3jmn n,m ιωe 3j T j B3jmn Ū n,m C 3jmn n,m ιωe 3j T j, 6 ιωdij TU j,i TŪ j,i ιωe ij T i,j T i,j = d A3jmn Ū n,m B 3jmn n,m ιωd 3j TU j d A3jmn U n,m B 3jmn n,m ιωd 3j TŪ j B3jmn Ū n,m C 3jmn n,m ιωe 3j T j d B3jmn U n,m C 3jmn n,m ιωe 3j T j, 7 Kij T,i T,j cω 2 T T ιωe ij j,i T j,i T ιωd ij U j,i T Ū j,i T = d K 33 TT,3 T T,3, 8 ιωd ij U j,i T U j,i T ιωe ij j,i T j,i T = d where z is the notation for the complex conjugate of z. K 3j TT,j T T,j, 9

7 MarinandBaleanu Boundary Value Problems : Page 7 of 9 Proof Considering equations 3 and 3 2 we can prove the following equality: Aijmn U n,m B ijmn n,m ιωd ij T,j ϱω 2 U i Ūi A ijmn Ū n,m B ijmn n,m ιωd ij T,j ϱω 2 Ū i Ui B mnij U n,m C ijmn n,m ιωe ij T,j i ε ijk A jkmn U n,m B jkmn n,m ιωd jk T i I ij ω 2 i j B mnij Ū n,m C ijmn n,m ιωe ij T,j i ε ijk A jkmn Ū n,m B jkmn n,m ιωd jk T i I ij ω 2 i j =0. 20 With some calculations, equality 20canbewrittenintheform 2 A ijmn U j,i Ū n,m C ijmn n,m n,m B ijmn U j,i n,m Ū j,i n,m ϱω 2 U i Ū i I ij ω 2 i j ιωd ij TU j,i TŪ j,i ιωe ij T i,j T i,j = A ijmn U n,m B ijmn n,m ιωd ij TŪ i,j A ijmn Ū n,m B ijmn n,m ιωd ij TU i,j B mnij U n,m C ijmn n,m ιωe ij T i,j B mnij Ū n,m C ijmn n,m ιωe ij T i,j. 2 Integrate equality 2over, apply the divergence theorem, and use the lateral conditions 4; wegettheequality 6. Now, if we again consider equations 3 and 3 2 then it is easy to prove the following equality: Aijmn U n,m B ijmn n,m ιωd ij T,j ϱω 2 U i Ūi A ijmn Ū n,m B ijmn n,m ιωd ij T,j ϱω 2 Ū i Ui B mnij U n,m C ijmn n,m ιωe ij T,j i ε ijk A jkmn U n,m B jkmn n,m ιωd jk T i I ij ω 2 i j B mnij Ū n,m C ijmn n,m ιωe ij T,j i ε ijk A jkmn Ū n,m B jkmn n,m ιωd jk T i I ij ω 2 i j =0. 22 With some calculations, equality 22canbewrittenintheform ιωd ij TU j,i TŪ j,i ιωe ij T i,j T i,j = A ijmn Ū n,m B ijmn n,m ιωd ij TU i,j A ijmn U n,m B ijmn n,m ιωd ij TŪ i,j B mnij Ū n,m C ijmn n,m ιωe ij T i,j B mnij U n,m C ijmn n,m ιωe ij T i,j. 23

8 MarinandBaleanu Boundary Value Problems : Page 8 of 9 Integrate equality 23over, apply the divergence theorem; if we use the lateral conditions 4wegettheequality7. With the help of equation 3 3, we can deduce immediately the equality T K ij T,j T K ij T,j ιωd ij U j,i ιωe ij j,i c ω 2 T,i ιωd ij U j,i ιωe ij j,i c ω 2 T,i With some calculations, equality 24canbewrittenintheform = Kij T,i T,j cω 2 T T ιωd ij U j,i T U j,i TιωE ij j,i T j,i T = K ij TT,j T T,j. 25,i Integrate equality 25over, apply the divergence theorem, and if we use the lateral conditions 4wegettheequality8. Finally, we use again equation 3 3 thus we will obtain, in a trivial way, the equality T K ij T,j T K ij T,j ιωd ij U j,i ιωe ij j,i c ω 2 T,i ιωd ij U j,i ιωe ij j,i c ω 2 T,i With some calculations, equality 26canbewrittenintheform =0. 26 ιωd ij U j,i T U j,i TιωE ij j,i T j,i T= K ij TT,j T T,j. 27,i Integrate equality 27over, apply the divergence theorem; if we use the lateral conditions 4wegettheequality9 andthe proofof Theorem is completed. The next theorem is also dedicated to a proof of two auxiliary identities on which will be based the main result. Theorem 2 Let U i, i, T be a solution of the boundary value problem consisting of equations 3-5. Then we have the identities A ijmn U n,m Ū j,i C ijmn i,j n,m 2ιω Bijmn U n,m i,j Ū n,m i,j 3ω 2 ϱu i Ū i I ij i j Dij TŪ j,i TU j,i E ij T j,i T j,i ιω Dij x p T,p Ū j,i T,p U j,i E ij x p T,p j,i T,p j,i

9 MarinandBaleanu Boundary Value Problems : Page 9 of 9 = d d d d d d A3jmn U n,m B 3jmn n,m ιωd 3j Tx p Ū j,p A3jmn Ū n,m B 3jmn n,m ιωd 3j Tx p U j,p B3jmn U n,m C 3jmn n,m ιωe 3j Tx p j,p B3jmn Ū n,m C 3jmn n,m ιωe 3j Tx p j,p x 3 A ijmn Ū n,m Ū j,i C ijmn i,j n,m x 3 Bijmn U n,m i,j Ū n,m i,j ϱω 2 U i Ū i ιωx3 D ij TŪ j,i TU j,i ιωx3 E ij T j,i T j,i x 3 I ij ω 2 i j U i x p n p A iαmβ n α n β C iαmβ n α n β i Kij T,i T,j 3cω 2 T T ιωe ij j,i T,p j,i T,p = d d Ū m B U i m iαmβn α n β m ds, 28 ιωd ij x p U j,i T,p U j,i T,p T T x p n p K αβ n α n β ds xα K 3β T,α T,β T,α T,β x α K 33 T,3 T,α T 3 T,α x 3 K33 T,3 T 3 K αβ T,α T,β cω 2 T T. 29 Proof Considering equations 3 and 3 2 it is easy to prove the following equality: Aijmn U n,m B ijmn n,m ιωd ij T,j ϱω 2 U i xp Ū i,p B mnij U n,m C ijmn n,m ιωe ij T,j x p i,p ε ijk A jkmn U n,m B jkmn n,m ιωd jk Tx p i,p I ij ω 2 x p i,p j A ijmn Ū n,m B ijmn n,m ιωd ij T,j ϱω 2 Ū i xp U i,p B mnij Ū n,m C ijmn n,m ιωe ij T,j x p i,p ε ijk A jkmn Ū n,m B jkmn n,m ιωd jk Tx p i,p I ij ω 2 x p i,p j =0. 30

10 MarinandBaleanu Boundary Value Problems : Page 0 of 9 With simple calculations, equality 30canbewrittenintheform Aijmn U n,m B ijmn n,m ιωd ij Tx p Ū i,p A ijmn U n,m B ijmn n,m ιωd ij Tx p Ū i,pj ϱω 2 x p U i Ū i,p,j B mnij U n,m C ijmn n,m ιωe ij Tx p i,p,j B mnij U n,m C ijmn n,m ιωe ij Tx p i,pj ε ijk A jkmn U n,m B jkmn n,m ιωd jk Tx p i,p I ij ω 2 x p i,p j A ijmn Ū n,m B ijmn n,m ιωd ij Tx p U i,p,j A ijmn Ū n,m B ijmn n,m ιωd ij Tx p U i,pj ϱω 2 x p Ū i U i,p B mnij Ū n,m C ijmn n,m ιωe ij Tx p i,p,j B mnij Ū n,m C ijmn n,m ιωe ij Tx p i,pj ε ijk A jkmn Ū n,m B jkmn n,m ιωd jk Tx p i,p I ij ω 2 x p i,p j =0. 3 This equality leads to A ijmn U n,m Ū j,i C ijmn i,j n,m B ijmn U n,m i,j Ū n,m i,j 3ω 2 ϱu i Ū i I ij i j 2ιωD ij TŪ j,i TU j,i 2ιωE ij T j,i T j,i ιωd ij x p T,p Ū j,i T,p U j,i ιωe ij x p T,p j,i T,p j,i = A ijmn U n,m B ijmn n,m ιωd ij Tx p Ū i,p,j A ijmn Ū n,m B ijmn n,m ιωd ij Tx p U i,p,j B mnij U n,m C ijmn n,m ιωe ij Tx p i,p,j B mnij Ū n,m C ijmn n,m ιωe ij Tx p i,p,j x k A ijmn U n,m Ū j,i x p C ijmn i,j n,m,p x p B ijmn U n,m i,j Ū n,m i,j x p ϱω 2 U i Ū i,p ιωx p D ij TŪ j,i TU j,i,p ιωx p E ij T j,i T j,i x p I ij ω 2 i j,p. 32 We integrate equality 32and use the lateralboundary condition 4; then we are led to the equality A ijmn Ū n,m Ū j,i C ijmn i,j n,m 2ιω Bijmn U n,m i,j Ū n,m i,j 3ω 2 ϱu i Ū i I ij i j Dij TŪ j,i TU j,i E ij T j,i T j,i

11 MarinandBaleanu Boundary Value Problems : Page of 9 ιω Dij x p T,p Ū j,i T,p U j,i E ij x p T,p j,i T,p j,i = d A3jmn U n,m B 3jmn n,m ιωd 3j Tx p Ū j,p d A3jmn Ū n,m B 3jmn n,m ιωd 3j Tx p U j,p d B3jmn U n,m C 3jmn n,m ιωe 3j Tx p j,p d B3jmn Ū n,m C 3jmn n,m ιωe 3j Tx p j,p x 3 A ijmn Ū n,m Ū j,i C ijmn i,j n,m x 3 Bijmn U n,m i,j Ū n,m i,j ϱω 2 U i Ū i d ιωx3 D ij TŪ j,i TU j,i d ιωx3 E ij T j,i T j,i x 3 I ij ω 2 i j x p Ū s,p A psmn Ū n,m x p U s,p A psmn Ū s,p n p ds x p s,p B psmn U n,m x p s,p B psmn Ū n,m n p ds x p s,p C psmn n,m x p s,p C psmn n,m n p ds x p n p Aijmn Ū n,m U j,i C ijmn j,i n,m B ijmn U j,i n,m Ū j,i n,m ds. 33 If we take into account the lateral boundary condition 4weconcludethat U i,3 =0 on. 34 On the curve D we have U i U i,α = n α τ U i α τ, where τ α are components of the unit vector tangent to D and / τ is the tangential derivative. According to the lateral boundary condition 4 wededuce U i / τ =0on the curve D and hence we obtain U i,α = n α U i on the curve D. 35 Using equations 34and35, the lastintegral in 33becomes

12 MarinandBaleanu Boundary Value Problems : Page 2 of 9 = x p n p A ijmn U j,i Ū n,m B ijmn U j,i n,m C ijmn j,i n,m ds U i x p n p A iαmβ n α n β C iαmβ n α n β i m For the other integrals in 33weobtain Ū m B U i m iαmβn α n β ds. 36 x p Ū s,p A psmn U n,m x p U s,p A psmn Ū n,m n p ds U i Ū m =2 x p n p A iαmβ n α n β ds, x p Ū s,p B psmn n,m x p U s,p C psmn n,m n p ds U i m =2 x p n p B iαmβ n α n β ds, x p s,p C psmn n,m x p s,p C psmn n,m n p ds =2 x p n p C iαmβ n α n β i m ds. 37 If we substitute the results of equations 36 and37 intheequality33, we obtain the first relation of Theorem 2,namelyequation28. To prove equation 29 we start fromthe following equality, which is evident: x p T p K ij T,j ιωd ij U j,i ιωe ij j,i c ω 2 T,i x p T p K ij T,j ιωd ij U j,i ιωe ij j,i c ω 2 T,i After some direct calculations, equality 38 acquires the form =0. 38 ιωd ij x p U j,i T,p U j,i T,p ιωe ij x p j,i T,p j,i T,p c = x p ω 2 T T 2 K ij T,i T,j,p x p K ij T,p T,j T,p T,j This equality can be rewritten as follows: K ij T,i T,j 3c x p K ij T,i T,j,i. 39,p ω 2 T T ιωe ij j,i T,p j,i T,p ιωd ij x p U j,i T,p U j,i T,p c = ω 2 xp T T x p K ij T,p T,j T,p T,j K ij T,i T,j,p,i. 40,p

13 MarinandBaleanu Boundary Value Problems : Page 3 of 9 Now we integrate the equality 4on and, after using the lateral boundary condition 4, we are led to Kij T,i T,j 3cω 2 T T ιωe ij j,i T,p j,i T,p ιωd ij x p U j,i T,p U j,i T,p = d x p K 3j T,p T,j T,p T,j x 3 K ij T,i T,j x 3 cω 2 T T xp n p K ij T,i T,j x p K pj T,p T,j T,p T,j n p ds. 4 As we have already shown in the proof of equality 28, the lateral boundary condition implies T,3 =0, T,α = n α T, on the curve. With these arguments, the equality 4 impliesequation29, therefore the proof of Theorem 2 is completed. The conservation laws which will be proved in the following theorem will be used to derive aprioriestimatesforasolutionofourmixedproblem. Theorem 3 Let U i, i, T be a solution of the boundary value problem consisting of equations 3-5. Then the following two conservation laws are satisfied: d d ω 2 ϱu j Ū j I ij i j c T T c K 33 T,3 T,3 K αβ T,α T,β Ai3m3 U i,3 Ū m,3 B i3m3 U i,3 m,3 Ū i,3 m,3 C i3m3 i,3 m,3 d Aiαmβ U i,α Ū m,β B iαm3 U i,α m,β Ū i,α m,β C iαmβ i,α m,β ιωdiα TŪ i,α TU i,α ιωe iα T i,α T i,α =0, 42 A3jmn Ū n,m B 3jmn n,m ιωd 3j TU j d A3jmn U n,m B 3jmn n,m ιωd 3j TŪ j B3jmn Ū n,m C 3jmn n,m ιωe 3j T j

14 MarinandBaleanu Boundary Value Problems : Page 4 of 9 d = d B3jmn U n,m C 3jmn n,m ιωe 3j T j K 3j TT,j T T,j. 43 Proof To prove equation 42 we start by using equations 3 and 3 2 ; with the help of these we obtain the following equality: Aijmn U n,m B ijmn n,m ιωd ij T,i ϱω 2 U j Ūj,3 B ijmn U n,m C ijmn n,m ιωe ij T,i ε jik A ikmn U n,m B ikmn n,m ιωd ik TI ij ω 2 i j,3 A ijmn Ū n,m B ijmn n,m ιωd ij T,i ϱω 2 Ū j Uj,3 B ijmn Ū n,m C ijmn n,m ιωe ij T,i ε jik A ikmn Ū n,m B ikmn n,m ιωd ik TI ij ω 2 i j,3 =0. 44 Performing direct calculations on equality 44we are led to d ϱω 2 U j Ū j I ij ω 2 i j A i3m3 U i,3 Ū m,3 B i3m3 U i,3 m,3 Ū i,3 m,3 C i3m3 i,3 m,3 A iαmβ U i,α Ū m,β B iαm3 U i,α m,β Ū i,α m,β C iαmβ i,α m,β ιωd iα TŪ i,α TU i,α ιωe iα T i,α T i,α A iαm3 U m,3 Ū i,α B iαm3 U m,3 i,α Ū m,3 i,α C iαm3 m,3 i,α,α ιωd iα TU i,3 TŪ i,3,α ιωe iα T i,3 T i,3,α ιωd ij T,3 U i,j T,3 Ū i,j ιωe ij T,3 i,j T,3 i,j =0. 45 Now integrate equality 45 and use the lateralboundary condition 4; we get d ϱω 2 U j Ū j I ij ω 2 i j A i3m3 U i,3 Ū m,3 B i3m3 U i,3 m,3 Ū i,3 m,3 C i3m3 i,3 m,3 A iαmβ U i,α Ū m,β B iαm3 U i,α m,β Ū i,α m,β C iαmβ i,α m,β ιωd iα TŪ i,α TU i,α ιωe iα T i,α T i,α ιωdij T,3 U i,j T,3 Ū i,j ιωe ij T,3 i,j T,3 i,j =0. 46 Using equation 3 3,itisclearthat T,3 K ij T,ij ιωd ij U i,j E ij i,j c ω 2 T T,3 K ij T,ij ιωd ij Ū i,j E ij i,j c ω 2 T =0. 47

15 MarinandBaleanu Boundary Value Problems : Page 5 of 9 After doing some calculations, we can write equation 47intheform d c ω 2 T T K 33 T,3 T,3 K αβ T,α T,β 2 K α3 T,3 T,3 ιωd ij T,3 Ū i,j T,3 U i,j,α ιωe ij T,3 i,j T,3 i,j =0. 48 Now we integrate 48 on and use the lateral boundary condition 4; we arrive at the equality d c ω 2 T T K 33 T,3 T,3 K αβ T,α T,β ιωdij T,3 Ū i,j T,3 U i,j ιωe ij T,3 i,j T,3 i,j =0. 49 By using equations 49 and46 weobtaintheequality42. The conservation law 43 is obtained immediately equaling the right-side members of equality 7 and9. This concludes the proof of Theorem 3. Combining equalities 6-9 oftheorem with equalities oftheorem2 and those of Theorem 3, namely42-43, we obtain various measures associated with the amplitude U i, i, T. With the help of these measures, we will obtain suitable spatial estimates to describe the spatial behavior of the respective amplitude. The next result is a first estimate which describes the spatial behavior of the solution. Theorem 4 Let U i, i, T be a solution of the boundary value problem consisting of equations 3-5. Then the following equality holds: A ijmn U j,i Ū n,m B ijmn U j,i n,m Ū j,i n,m C ijmn j,i n,m ω 2 ϱu i Ū i I ij i j c = d T T c K ij T,i T,j ιωdij TU j,i TŪ j,i ιωe ij T j,i T j,i A3jmn U n,m B 3jmn n,m ιωd 3j TŪ j A3jmn Ū n,m B 3jmn n,m ιωd 3j TU j B3jmn U n,m C 3jmn n,m ιωe 3j T j B3jmn Ū n,m C 3jmn n,m ιωe 3j T j K 33 T T,3 TT,3. 50

16 MarinandBaleanu Boundary Value Problems : Page 6 of 9 Proof By combining equations 6 and8 we obtain immediately the above desired identity 50. Another aprioriestimate will be proved in the next theorem. Theorem 5 If U i, i, T is a solution of the boundary value problem consisting of equations 3-5, then we have A ijmn U j,i Ū n,m B ijmn U j,i n,m Ū j,i n,m C ijmn j,i n,m K ij T,i T,j ω 2 ϱu i Ū i I ij i j T T U i x p n p A iαmβ n α n β m ds Ū m B U i iαmβn α n β m T T x p n p K αβ n α n β ds i C iαmβ n α n β = d A3jmn U n,m B 3jmn n,m ιωd 3j TŪ j x p Ū j,p A 3jmn Ū n,m B 3jmn n,m ιωd 3j TU j x p U j,p B3jmn U n,m C 3jmn n,m ιωe 3j T j x p j,p B 3jmn Ū n,m C 3jmn n,m ιωe 3j T j x p j,p K 33 T T,3 TT,3 x α K3α T,α T,β T,α T,β K 33 T,α T,3 T,α T,3 x 3 Ai3m3 U i,3 Ū m,3 B i3m3 U i,3 m,3 Ū i,3 m,3 C i3m3 i,3 m,3 x 3 Aiαmβ U i,α Ū m,β B iαmβ U i,α m,β Ū i,α m,β C iαmβ i,α m,β x 3 ιω D iα TŪ i,α TU i,α E iα T i,α T i,α x 3 K 33 T,3 T,3 K αβ T,α T,β x 3 ω 2 ϱu i Ū i I ij i j c T T. 5 Proof We arrive at the equality 5ifwecombinetheresultsfromequalities28and29 of Theorem 2 with equation 50ofTheorem4. Theresultofthespatialbehaviorwillbebasedonequality5. For the result to be rigorous, we specify assumptions which are really common in continuum mechanics. Thus, we assume that the tensors of the micropolar thermoelasticity satisfy the strong ellipticity condition, A ijmn x i x m y j y n >0, B ijmn x i x m y j y n > 0, for all non-zero vectors x, x 2, x 3, y, y 2, y 3, C ijmn x i x m y j y n >0. 52

17 MarinandBaleanu Boundary Value Problems : Page 7 of 9 Also,thespecificheatc and the conductivity tensor K ij satisfy the conditions c >0, K ij x i x j > 0, for all non-zero vector x, x 2, x It is clear that from 52 that we can deduce A i3m3 x i x m >0, B i3m3 x i x m > 0, for all non-zero vector x, x 2, x 3, C i3m3 x i x m >0. 54 Since the curve D was presumed regular, we deduce that there is h 0 >0suchthatx p n p h 0 > 0. Then we have the inequalities U i 0 x p n p A iαmβ n α n β i C iαmβ n α n β Ui MC m ds Ū i i wherewehaveusedthenotations Ū m 2B U i m iαmβn α n β i ds, 55 C =A iαmβ A iαmβ 2B iαmβ B iαmβ C iαmβ C iαmβ /2, 56 x 2 M = sup 2 x2. 57 x,x 2 D Also, for the conductivity tensor K ij we have T T MK T T 0 x p n p K αβ n α n β ds ds, 58 where M is defined in 57and K =K αβ K αβ /2. 59 Now we introduce the quantities m 0, m, ω 0,andω by m 0 = max x 3 0,L m = max x 3 0,L We can assume that Ū i i i U i ds U, ω0 iūi i i ds = ϱ MCm 0, 60 T T ds T Tds, ω 0 = c MKm. 6 ω > ω = max ω 0, ω, 62 m 0 m 0, m m, 63

18 MarinandBaleanu Boundary Value Problems : Page 8 of 9 where m 0 = max m = max T H 0 D U i Ū i i i ds U, 64 iūi i i ds T T ds T Tds. 65 Here the maximum from m 0 is calculated for U i H0 D, i H0 D, where H 0 Disthe usual Sobolev space. In this way we obtain an explicit critical value for the frequency of the vibration, namely ω = max ϱ MCm 0, c MKm. Combining the results from equations 5, 55, 58, and 62 we obtain the following estimate of the spatial behavior of the amplitude U i, i, T: d A3jmn U n,m B 3jmn n,m ιωd 3j TŪ j x p Ū j,p A 3jmn Ū n,m B 3jmn n,m ιωd 3j TU j x p U j,p B3jmn U n,m C 3jmn n,m ιωe 3j T j x p j,p B 3jmn Ū n,m C 3jmn n,m ιωe 3j T j x p j,p x α K3α T,α T,β T,α T,β K 33 T,α T,3 T,α T,3 K 33 T T,3 TT,3 x 3 ω 2 ϱu i Ū i I ij i j T T x 3 Ai3m3 U i,3 Ū m,3 B i3m3 U i,3 m,3 Ū i,3 m,3 C i3m3 i,3 m,3 x 3 Aiαmβ U i,α Ū m,β B iαmβ U i,α m,β Ū i,α m,β C iαmβ i,α m,β x 3 ιω D iα TŪ i,α TU i,α E iα T i,α T i,α x 3 K 33 T,3 T,3 K αβ T,α T,β x 3 ω 2 ϱu i Ū i I ij i j c T T A ijmn U j,i Ū n,m B ijmn U j,i n,m Ū j,i n,m C ijmn j,i n,m K ij T,i T,j. 66 With this the proof of Theorem 5 is complete. Conclusion It is appropriatetonote that the differential inequality 66 is different from those used for a deduction of the estimates of Saint-Venant type.

19 MarinandBaleanu Boundary Value Problems : Page 9 of 9 To deduce these estimates we used only the strong ellipticity assumptions for the thermoelastic coefficients. Therefore, these results can be applied to a large scale of materials. Competing interests The authors declare that they have no competing interests. Authors contributions All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript. Author details Department of Mathematics and Computer Science, Transilvania University of Brasov, Brasov, 5008, Romania. 2 Department of Mathematics and Computer Science, Cankaya University, Ankara, Turkey. 3 Institute of Space Sciences, Magurele, Bucharest, Romania. Received: March 206 Accepted: June 206 References. Chandrasekharaiah, DS: Hyperbolic thermoelasticity: a review of recent literature. Appl. Mech. Rev. 5, Green, AE, Naghdi, PM: Thermoelasticity without energy dissipation. J. Elast. 3, Green, AE, Naghdi, PM: On thermodynamics and the nature of the second law. Proc. R. Soc. Lond. A 357, Nappa, L: Spatial decay estimates for the evolution equations of thermoelasticity without energy dissipation. J. Therm. Stresses 2, Chandrasekharaiah, DS: A note on the uniqueness of solution in the linear theory of thermoelasticity without energy dissipation. J. Elast. 43, Iesan, D: On the theory of thermoelasticity without energy dissipation. J. Therm. Stresses 2, Quintanilla, R: On existence in thermoelasticity without energy dissipation. J. Therm. Stresses 25, Abbas, IA, Zenkour, AM: The effect of magnetic field on thermal shock problem for a fiber-reinforced anisotropic half-space using Green-Naghdi s theory. J. Comput. Theor. Nanosci. 23, Abbas, IA: Generalized magneto-thermoelastic interaction in a fiber-reinforced anisotropic hollow cylinder. Int. J. Thermophys. 333, Zenkour, AM, Abbas, IA: Magneto-thermoelastic response of an infinite functionally graded cylinder using the finite element method. J. Vib. Control 202, Marin, M: A temporally evolutionary equation in elasticity of micropolar bodies with voids. Sci. Bull. Politeh. Univ. Buchar., Ser. A, Appl. Math. Phys , Marin, M: On the minimum principle for dipolar materials with stretch. Nonlinear Anal., Real World Appl. 03, Marin, M, Agarwal, RP, Mahmoud, SR: Non-simple material problems addressed by the Lagrange s identity. Bound. Value Probl. 203,ArticleID doi:0.86/ Marin, M: On existence and uniqueness in thermoelasticity of micropolar bodies. C. R. Math. Acad. Sci. Paris 322, Marin, M: An evolutionary equation in thermoelasticity of dipolar bodies. J. Math. Phys. 403, Marin, M: A partition of energy in thermoelasticity of microstretch bodies. Nonlinear Anal., Real World Appl. 4, Anwar Beg, O, Zueco, J, Takhar, HS, Beg, TA: Network numerical simulation of impulsively-started transient radiation-convection heat and mass transfer in a saturated Darcy-Forchheimer porous medium. Nonlinear Anal., Model. Control 33, Mahapatra, TR, Pal, D, Mondal, S: Influence of thermal radiation on non-darcian natural convection in a square cavity filled with fluid saturated porous medium of uniform porosity. Nonlinear Anal., Model. Control 72, Chirita, S: Spatial decay estimates for solutions describing harmonic vibrations in a thermoelastic cylinder. J. Therm. Stresses 8, Flavin, JN, Knops, RJ: Some spatial decay estimates in continuum dynamics. J. Elast. 7, Ciarletta, M: A theory of micropolar thermoelasticity without energy dissipation. J. Therm. Stresses 22,