Structural Continuous Dependence in Micropolar Porous Bodies

Transcription

1 Copyright 15 Tech Science Press CMC, vol.45, no., pp.17-15, 15 Structural Continuous Dependence in Micropolar Porous odies M. Marin 1,, A.M. Abd-Alla 3,4, D. Raducanu 1 and S.M. Abo-Dahab 3,5 Abstract: Our study is dedicated to mixed initial boundary value problem for porous micropolar bodies. We prove that the solution of this problem depends continuously on coefficients which couple the micropolar deformation equations with the equations that model the evolution of voids. The evaluation of this dependence is made by using an appropriate measure. Keywords: micropolar bodies, voids, continuous dependence, uncoupled systems, convergence. 1 Introduction The theory of materials with voids or vacuous pores is the simplest extension of the classical theory of elasticity and was first proposed by Nunziato and Cowin in the paper Nunziato and Cowin In this theory the authors introduce an additional degree of freedom in order to develope the mechanical behavior of a body in which the matrix material is elastic and interstices are voids of material. It is worth to recall that porous materials have applications in many fields of engineering such as petroleum industry, material science, biology and so on. The intended applications of the theory are in geological materials like rocks and soil and in manufactured porous materials. The linear theory of elastic materials with voids was developed by Cowin and Nunziato in Cowin and Nunziato Here the uniqueness and weak stability of solutions are also derived. Then, the problem of bodies with voids was approached in a large number of studies, of which we mention only some of the most recent. For example, Iesan 11 considered the problem of Almansi for porous Cosserat elastic solids. The non-linear deformations of porous elastic solids were approached in Iesan and Quintanilla Dept. of Mathematics, Transilvania University of rasov, rasov, Romania. Correspondent author. m.marin@unitbv.ro 3 Dept. of Mathematics, Taif University, Taif, Saudi Arabia. 4 Dept. of Mathematics, Sohag University, Sohag, Egypt. 5 Dept. of Mathematics, Qena Fac. of Sci., SVU, Egypt.

2 18 Copyright 15 Tech Science Press CMC, vol.45, no., pp.17-15, 15 Also, in the paper Mora and Waas 7 the micropolar and Lamè constants for a circular cell polycarbonate honeycomb are calculated by using a finite element representation of the honeycomb microstructure. The known smeared crack approach is revisited in the paper Heinrich and Waas 13 to describe post-peak softening in laminated composite materials. The paper Abbas and Kumar 14 is a study of the plane problem in initially stressed generalized thermoelastic half-space with voids. In the paper Mahmoud and Abd-Alla 14 the equations of elastodynamic problems of the orthotropic hollow sphere in terms of displacement are solved The minimum principle for dipolar materials with strec is considered in the paper Marin 9. For the same category of materials, Marin and Stan 13 give some weak solutions, while the paper Marin et al. 13 a presents some results obtained with the help of Lagrange Identity. A study of temporal behaviour of solutions in Thermoelasticity of porous micropolar bodies is given in Marin and Florea 14 and in Marin et al. 13 b. Some considerations regarding the localization in time of solutions for thermoelastic micropolar materials with voids can be found in the paper Marin et. al 14. Also, the study Sharma and Marin 14 is dedicated to micropolar thermoelastic solids. Continuous dependence of solutions with regards to the coefficients of the equations that govern the deformation of a body is more important than continuous dependence the initial data and on boundary conditions. This is because, in the mathematical modeling of the continuum, there may be errors or disturbances due the idealization of the model. Also, the continuous dependence of solutions with regards to the coefficients of the equations is important in obtaining some numerical approximations of the solutions of the models. Continuous dependence is important both in terms of pure mathematics and in terms of practical applications. Therefore, many studies published in recent years are devoted to this topic. We recall only the fundamental paper of Knops and Payne 1988 and also the papers Chirita and Ciarletta 11, Franchi and Straughan 1996, Iovane and Passarela 4, Green and Naghdi The paper is structured as follows. First, we formulate the mixed intial-boundary value problem in the context of micropolar porous bodies. Then we will prove some preliminaries identities which we will use in order to derive the continous dependence theorems. In the last part of our paper we deduce the convergence of the solution of our mixed problem, when the coupling coefficients tend to zero, by using an appropriate measure, which is specified in advance.

3 Structural Continuous Dependence in Micropolar Porous odies 19 asic equations An anisotropic elastic material is considered. Assume a body of this type that occupies a properly regular region of the three-dimensional Euclidian space R 3 bounded by a piecewise smooth surface and we denote the closure of by. The boundary is smooth enough to apply the divergence theorem. We use a fixed system of rectangular Cartesian axes Ox i,i = 1,,3 and adopt Cartesian tensor notation. A superposed dot stands for the material time derivative while a comma followed by a subscript denotes partial derivatives with respect to the spatial coordinates. Einstein summation convention on repeated indices is used. Also, the spatial argument and the time argument of a function will be omitted when there is no likelihood of confusion. The motion of a micropolar porous body is described by the independent variables u i x,t, the displacement vector field, ϕ i x,t, the microrotation vector field and φ, the change in volum fraction. We consider the mixed problem associated with the theory of elasticity of micropolar bodies with voids on the time interval I. In the absence of supply terms, it is known that the basic equations on I are, [see, for instance Iesan 11] t i j, j = ρü i, m i j, j ε i jk t jk = I i j ϕ j, h i,i g = ρκ φ, 1 Here, the equations 1 are the motion equations and is the balance of the equilibrated forces. Next, we restricte our considerations only to the case where the materials have a center of symmetry. Consequently, the constitutive tensors of odd order must vanish and therefore the constitutive equations become t i j = C i jmn ε mn i jmn γ mn a i j φ, m i j = mni j ε mn C i jmn γ mn b i j φ, h i = A i j φ, j, 3 g = a i j ε i j b i j γ i j ξ φ τ φ, where the strain tensors ε i j and γ i j are defined by means of the kinetic relations ε i j = u j, i ε jik ϕ k, γ i j = ϕ j, i. 4

4 11 Copyright 15 Tech Science Press CMC, vol.45, no., pp.17-15, 15 In the above equations we used the following notations: ρ-the constant mass density; I i j = I ji -the coefficients of microinertia; κ-the equilibrated inertia; u i -the components of displacement vector; ϕ i -the components of microrotation vector; φ-the volume distribution function which in the reference state is φ ; ε i j,γ i j -kinematic characteristics of the strain; t i j -the components of the stress tensor; m i j -the components of the couple stress tensor; h i -the components of the equlibrated stress vector; g-the intrinsic equilibrated force; A i jmn, i jmn, C i jmn, A i j, a i j, b i j, τ and ξ from the constitutive equations are prescribed characteristic functions of the material, and they obey to the symmetry relations A i jmn = A mni j, C i jmn = C mni j, A i j = A ji. 5 As time interval I we take I =, T and consider the mixed boundary-final problem P defined by the system of equations 1-4, the initial conditions in u i x, = u i x, u i x, = u 1 i x, ϕ i x, = ϕ i x, ϕ i x, = ϕ 1 i x, 6 φx, = φ x, φx, = φ 1 x, and the following boundary conditions u i x, t = ũ i x, t on, T, ϕ i x, t = ϕ i x, t on, T, 7 φx, t = φx, t on, T. Introducing the constitutive relations 3 into equations 1 and, we obtain the following system of equations ρü i = A i jmn ε mn i jmn γ mn a i j φ, j, I i j ϕ j = mni j ε mn C i jmn γ mn b i j φ, j ε i jk A jkmn ε mn jkmn γ mn a jk φ, 8 ρκ φ = A i j φ, i, j a i j ε i j b i j γ i j ξ φ τ φ y a solution of the mixed initial-boundary value problem of the theory of thermoelasticity of micropolar bodies with voids in the cylinder Ω =, T we mean an ordered array u i,ϕ i,φ which satisfies the system of equations 8 for all x,t Ω, the boundary conditions 7 and the initial conditions 6. In our subsequent analysis we need to require the following restrictions ρ >, κ >, τ 9

5 Structural Continuous Dependence in Micropolar Porous odies 111 Also, we need to impose the positivity of constitutive tensors from the above relations A i jmn η i j η mn a 1 η, a 1 > i jmn η i j η mn b 1 η, b 1 > 1 C i jmn η i j η mn c 1 η, c 1 > A i j ξ i ξ j a ξ, a > 11 These assumptions are in agreement with the usual restrictions imposed in the mechanics of continua. We can couple these restrictions with the assumption that the Helmholtz free energy is a positive definite quadratic form and the best of their interpretation finds its place in the theory of mechanical stability: Ψ = 1 A i jmnε i j ε mn 1 C i jmnγ i j γ mn 1 A i jφ, i φ, j 1 ξ φ i jmn ε i j γ mn a i j ε i j φ b i j γ i j φ 1 a u v φ a φ, a >. In the above relations a, a 1 and a are appropiate constants. Also, u and v are the vector notations, namely u = u i, v = ϕ i. The functions, together with their domains of definition, used in the above equations of motion, in initial conditions and boundary conditions are supposed to be as smooth as required. 3 Preliminary identities efore tackling the mixed initial-boundary value problem of the theory of thermoelasticity of micropolar bodies with voids we will construct certain auxiliary problems and will prove some bounds for the solutions of these auxiliary problems. The identities that we will obtain are commonly called Rellich identities. Using an idea suggested in the paper Knops and Payne 1988, we will consider the following formal boundary value problem A i j H, j, i =, in H = q, on. We assume that the functions H and q as well as the surface satisfy the regularity conditions required by the theorem of existence see Fichera 197. Therefore, based on this theorem, we ensure the existence of a solution of the problem

6 11 Copyright 15 Tech Science Press CMC, vol.45, no., pp.17-15, 15 Theorem 1. If H is a solution of the problem 13, then the quantities A i j H, i H, j dv, H, H da n admit some bounds in terms of the data function q. Proof. Starting from the obvious equality x k H, k A i j H, j, i dx =. we can derive, without much difficulty integrating by parts, the following sequence of relations = x k, i H, k A i j H, j dv x k H, ki A i j H, j dv n i x k H, k A i j H, j da = A i j H, i H, j dv 1 x k A i j H, i H, j, k dv n i x k H, k A i j H, j da = 1 A i j H, i H, j dv 1 14 x k A i j, k H, i H, j dv n i x k H, k A i j H, j da 1 n i x k A i j H, k H, j da where n i represent the components of the unit normal to the surface. If we denote by s i the components of the tangential vector to the surface, we can write the derivative H, i in the form H, i = n i H n s i s H where / n denotes the derivative in the direction of the normal and s is the tangential derivative. If we denote by a αβ the coefficients of the first fundamental form of the surface, then the tangential derivative can be written in the form s H = x i; α a αβ H ; β where the notation f ; α represents the differentiation of function f with respect to surface variables t α. In what follows we suppose that the surface is star shaped with respect to origin and, furthermore, we have x k n k h, x k s k δ on.

7 Structural Continuous Dependence in Micropolar Porous odies 113 Also, we need the following limitation of the gradients of A i j x k A i j, k a s < µ a, < µ < 1 15 In the above relations h, δ, a s and µ are constants. If we take into account these conditions, 14 can be rewritten in the form 1 µ A i j H, i H, j dv h a H da n 1 x kn k s i s j n i s j s k x k A i j s q da 16 δ α A i j n i n j H n da δ α α where α will be conveniently chosen. Thus, if we choose α of the form α = h a δ a 3 then h a > δ αa 3 where we used the notation a 3 = max n i n j A i j A i j n i n j s q da With these considerations, we can write the inequality 16 in the form 1 µ A i j H, i H, j dv h a H da 4 n [ 1 x kn k s i s j n i s j s k x k δ a ] 3 n i n j A i j s q da 17 h a On the other hand, with the help of the Poincare inequality we are lead to λ 1 H dv H, i H, j dv C 1 H da 18 Inequalities 17 and 18 give us the desired bounds for quantities A i jh, i H, j dv, φ and ϕ n da, these bounds being expressed with the help of the boundary function q. The proof of Theorem 1 is concluded.

8 114 Copyright 15 Tech Science Press CMC, vol.45, no., pp.17-15, 15 Our next considerations are analog to those of Theorem 1 but for a vector version of the inequality 17. y analogy with 13, taking into account the geometric equations 4, we consider U i, F i a solution of the boundary value problem [ ] Ai jmn Uj, i ε jik F k i jmn F n, m =, in [ mni j U n, m ε nmk F k C i jmn F n, m ], j, j ε i jk [ A jkmn U n, m ε nmk F k jkmn F n, m ] =, in 19 U i = ġ i, F i = ḟ i, on. We assume that the functions U i, F i, g i, f i, as well as the surface, satisfy the regularity conditions required by the theorem of existence of Fichera see the paper Fichera 197. ased on this theorem, we ensure the existence of a solution of the problem 19. Theorem. If U i, F i is a solution of the problem 19, then the quantities [ Ai jmn Uj, i ε jik F k Un, m ε nmk F k i jmn Uj, i ε jik F k Fn, m C i jmn F j, i F n, m ]dv, U, F, Ui n Ui da, n Fi n Fi da n admit some bounds in terms of the data functions g i and f i Proof. To obtain an a priori bounds for [ ] Ai jmn Uj,i ε jik F k Un,m ε nmk F k i jmn Uj,i ε jik F k Fn,m C i jmn F j,i F n,m dv Ui Ui n n Ui Ui n n da, da, we can use similar assessments with those in the proof of Theorem 1. To obtain an a priori bounds for U, F we can use the Poincare inequality, and thus we find that λ 1 U i U i dv U i, j U i, j dv C 1 ġ i ġ i da λ F i F i dv F i, j F i, j dv C ḟ i ḟ i da

9 Structural Continuous Dependence in Micropolar Porous odies 115 and these inequalities give a bound for PUP and PFP. W Now, we proceed to derive some estimates for a solution of equations 8. For this purpose, we use the bounds found in Theorem 1 and in Theorem for the solutions of the auxiliary problems 13 and 19. First, we write equations 8 in the form ρü i A i jmn ε mn i jmn γ mn a i j φ, j = I i j ϕ j mni j ε mn C i jmn γ mn b i j φ, j ε i jk A jkmn ε mn jkmn γ mn a jk φ =, 1 ρκ φ a i j ε i j b i j γ i j ξ φ τ φ A i j φ, j, i = the system of equations 1 being defined on the cylinder,t We assume that a solution u i, ϕ i, φ of the system 1 satisfies the initial conditions 6 and boundary conditions 7. Theorem 3. If u i, ϕ i, φ is a solution of the system 1 which satisfies the initial conditions 6 and boundary conditions 7, then the quantities t v ds, t ϕ ds admit some bounds in terms of the data functions u i, u1 i ϕ i, ϕ1 i, φ and φ 1, with v = u i, ϕ i. Proof. Let us denote by H the solution of equation 13 1 which satisfies the boundary condition H = ḣ on. If we take into account equation 1 3 we obtain easily the identity t A i j ϕ, j, i a i j ε i j b i j γ i j ξ ϕ τ ϕ ρκ ϕ, H ϕ ds =. We integrate by parts with respect to s [,t] and with respect to the spatial variable on domain and thus we are lead to the identity 1 ξ ϕ t dv 1 ρκ ϕ t dv 1 = 1 A i j ϕ, i tϕ, j t dv A i j ϕ, iϕ, j dv t t t τ ϕ s dv ds a i j ε i j b i j γ i j ϕs dv ds n i n j A i j h H n da ds 3

10 116 Copyright 15 Tech Science Press CMC, vol.45, no., pp.17-15, 15 t n j s i A i j h s ḣ da ds 1 1 ρκϕ 1,ϕ 1 ρκ ϕ, H t t ξ ϕ, ϕ ρκϕ 1,H a i j ε i j b i j γ i j, H ds t t ρκ ϕ,ḣ ds τ ϕ,hds ξ ϕ, H ds For the last five terms on the right-hand side of relation 3 we will use the arithmetic-geometric mean inequality in the form < a,b > 1 µ a 1µ b, µ >. 4 For this reason we use the notation a M = max ai j, bm = max bi j, ρm = max ρ, κ M = max κ, ξ M = max ξ, τ M = max τ Using this notation, we can conveniently choose the constants µ 1, µ, µ 3, µ 4, µ 5 and so by applying the arithmetic-geometric mean inequality 4, from relation 3 we can deduce 1 ξ ϕ t dv 1 t ρκ ϕ t dv τ ϕ s dv ds 1 t A i j ϕ, i tϕ, j t dv a i j ε i j b i j γ i j ϕs dv ds 1 t A i j ϕ, iϕ, j dv n i n j A i j h H da ds n t n j s i A i j h s ḣ da ds 1 ξ ϕ, ϕ ρκϕ 1,H 1 ρκϕ 1,ϕ 1 µ t 1 εi j a t ds M H ds 5 µ 1 µ ξ M µ 3 µ 5 t t t γi j b ds M µ H ds µ 4 t ϕ ds ρ Mκ M µ 5 H ds µ 3 ρκ ϕ, ϕ t t ϕ ds ρ Mκ M µ 4 H Ḣ ds µ 6 t ϕ ds

11 Structural Continuous Dependence in Micropolar Porous odies 117 τ t M H ds µ 6 Terms relating to φ, φ and φ, i from the right-hand side of relation 5 may be bounded using the left-hand side, while the terms relating to H from the righthand side of relation 5 may be bounded by using the estimations found by using Theorem 1 and Theorem. Analog considerations with the above one will now be made relative to the first two equations of system 1 So, we consider the equation 1 1 and if we take into account equation 19 1 we find the identity t [A i jmn U n, m ε nmk F k i jmn F n, m ], j a i j ϕ, j ρü i,u i u i ds = 6 Finally, if we consider the equation 1 and if we take into account equation 19 we find the identity t [ mni j U n, m ε nmk F k C i jmn F n, m ], j b i j ϕ, j I i j φ j,f i φ i ds = 7 Adding together the relations 6 and 7, term by term, then integrating by parts in space and in time and taking into account the geometric equations 4 we get 1 t [ ] d ρ ui u i I i j φ i φ j A i jmn ε i j ε mn i jmn ε i j γ mn C i jmn γ i j γ mn dv ds ds t t t t a i j ε i j b i j γ i j ϕ dv ds = t n m i jmn g n U i, j C i jmn f n F i, j dads a i j U i, j b i j F i, j ϕ dv ds ρ U i u i I i j Ḟ i φ j dv ds n m A i jmn g n U i, j i jmn f n F i, j dads ρui u i I i j F i φ j dv ρui u 1 i I i j F i φ j 1 dv 8 For the last three integrals on the right-hand side of relation 8 we will use the arithmetic-geometric mean inequality in the form 4. Thus we can conveniently choose the positive constants α 1, α,..., α 6 so that 1 ρ ui u i I i j φ i φ j A i jmn ε i j ε mn i jmn ε i j γ mn C i jmn γ i j γ mn dv

12 118 Copyright 15 Tech Science Press CMC, vol.45, no., pp.17-15, 15 1 t t t α α 4 a i j ε i j b i j γ i j ϕ dv ds 1 ρ u 1 i u 1 i I i j φ i 1 φ 1 j dv Ai jmn εi jεmn i jmn εi jγmn C i jmn γi jγ mn dv U i n m A i jmn i jmn g n n j n s j s ġ i dads F i n m mni j C i jmn f n n j n s j s ḟ i dads 9 ρui u 1 i I i j F i φ j 1 dv α 1 ϕ ds a M U i, j ds α 1 ϕ ds b t M F i, j α 3 ds ρ u i u i dv 1 ρu i U i dv α α 3 I i j φ i φ j dv 1 I i j F i F j dv α t 5 ρ u i u i dv ds α 4 t ρ U i U i dv ds α t 6 I i j φ i φ j dv 1 t I i j Ḟ i Ḟ j dv α 6 t 1 α 5 where ε i j = u j, i ε jikϕ k, γ i j = ϕ j, i. y adding the relations 5 and 9 we are lead to 1 ρ ui u i I i j φ i φ j A i jmn ε i j ε mn i jmn ε i j γ mn C i jmn γ i j γ mn dv 1 ξ ϕ, ϕ 1 t ρκ ϕ, ϕ τ ϕ, ϕds 1 A i j ϕ, i ϕ, j dv a i j ε i j b i j γ i j ϕ dv µ t 1 ε i j µ t ds γ i j ds 3 t µ 3 α 1 α ϕ ds µ 4 ρκ ϕ, ϕ µ 5 µ 6 ϕ ds α 3 α t 5 ρ ui u i I i j φ i φ j dv ρ ui u i I i j φ i φ j dv ds R Here we have denoted by R the other terms resulting from the gathered formulas 5 and 9. The remaining R contains either terms that involve direct data either terms that can be estimated in terms of data based on Theorem 1 and Theorem, generically denoted by M. With these considerations, we can deduce that the lefthand side of inequality 3 is a positive definite measure. Also, if we use as energy measure the function E, given by Et = 1 [ ρ ui t u i t I i j φ i t φ j t ρκ ϕ t ] dv t t t

13 Structural Continuous Dependence in Micropolar Porous odies [A i jmn ε i j tε mn t i jmn ε i j tγ mn t C i jmn γ i j tγ mn t]dv [a i j ε i j t b i j γ i j t]ϕtdv 1 ξ ϕt,ϕt 1 A i j ϕ, i tϕ, j tdv 31 then from 3 we can deduce the following inequality Et K t Esds M, 3 in which the positive constant K is conveniently chosen. Now we multiply both members of inequality 3 by e Kt and after the resulting inequality is integrated over [,t] we are led to the result t Esds ekt K M. It is clear that this inequality provides a priori estimates for quantities t v ds, t ϕ ds which concludes the proof of Theorem Main result In this section we will prove the continuous dependence of solutions of the mixed initial-boundary value problem P 1 consisting of equations 8, the initial conditions 6 and the boundary conditions 7 with respect to coupling coefficients a i j and b i j To this end we consider a solution u i,ϕ i,φ of our problem P 1 and v i,ψ i, χ a solution of problem P which is similar to the problem P 1 namely, it has the same initial data, the same boundary data, but different coupling coefficients. The coupling coefficients of problem P 1 are a i j, b i j and of problem P are α i j, β i j. We assume that the other characteristics coefficients of the material are the same for both problems P 1 and P, namely A i jmn, i jmn, C i jmn, A i j, ρ, κ, τ and ξ Let us denote by w i,ϖ i,θ the difference between the two solutions and by c i j, d i j the difference between the coupling coefficients, that is w i = u i v i, ϖ i = ϕ i ψ i, θ = φ χ, c i j = a i j α i j, d i j = b i j β i j 34 Due to linearity, the difference w i,ϖ i,θ satisfies a system of equation similar to that of 8: ρẅ i A i jmn ε mn i jmn γ mn, j α i j θ, j = c i j φ, j

14 1 Copyright 15 Tech Science Press CMC, vol.45, no., pp.17-15, 15 I i j ψ j mni j ε mn C i jmn γ mn, j β i j θ, j ε i jk A jkmn ε mn jkmn γ mn α jk θ = d i j φ, j, 35 ρκ θ α i j ε i j β i j γ i j ξ θ τ θ A i j θ, j, i = c i j ε i j d i j γ i j 36 where we used the notations ε i j = w j, i ε jik ϖ k, γ i j = ϖ j, i Equations are satisfied on the cylinder,t Also, due to linearity, the difference w i,ϖ i,θ satisfies the initial conditions in their homogeneous form w i x, =, ẇ i x, =, ϖ i x, =, ϖ i x, =, θx, =, θx, = 37 and, also, the boundary conditions in their homogeneous form w i x,t =, ϖ i x,t =, θx,t =, x,t,t 38 The main result regarding the continuous dependence with respect to coupling coefficients is proved in the following theorem. Theorem 4. The solutions of the mixed initial-boundary value problem consists of equations 8, the initial conditions 6 and the boundary conditions 7 depend continuously with respect to coupling coefficients a i j and b i j Proof. We multiply equation 35 1 by ẇ i and then integrate over to find [ ] d 1 ρẇ i ẇ i dv A i jmn ε mn i jmn γ mn w i, j dv dt α i j θ ẇ i, j dv = c i j ϕ, j ẇ i dv 39 Similarly, we multiply equation 35 by ϖ i and then integrate over to find [ d 1 I i j ϖ i ϖ j dv mni j ε mn C i jmn γ mn ϖ i, j dv dt ε i jk A jkmn ε mn jkmn γ mn α jk θ ϖ i dv β i j θ ϖ i, j dv = d i j ϕ, j ϖ i dv ] 4

15 Structural Continuous Dependence in Micropolar Porous odies 11 Now, we add equations 39 and 4 and take into account the geometric equations 6 to see that [ d 1 ρẇ i ẇ i I i j ϖ i ϖ j dv dt A i jmn ε i j ε mn i jmn ε i j γ mn C i jmn γ i j γ mn dv αi j ε i j β i j γ i j θdv = [c i j ϕ, j ẇ i d i j ϕ, j ϖ i ] dv ] 41 We will multiply now equation 36 by θ and then integrate over so that we find [ d 1 ρκ θ, θ ] A i j θ, i θ, j dv ξ θ,θ τ θ, θ dt α i j ε i j β i j γ i j θdv = c i j ε i j d i j γ i j θdv 4 If we add the equations 41 and 4, term by term, we deduce that { [ d 1 ρẇ i ẇ i I i j ϖ i ϖ j dv ξ θ,θ ρκ θ, θ dt A i jmn ε i j ε mn i jmn ε i j γ mn C i jmn γ i j γ mn dv 43 = ] A i j θ,i θ, j dv [c i j ϕ, j ẇ i d i j ϕ, j ϖ i ] dv We will use the notation c = c i j c i j, c = c i j, j c ik,k, d = d i j d i j, d = d i j, j d ik,k, ρ m = min } α i j ε i j β i j γ i j θdv τ θ, θ = ρ, κ m = min c i j ε i j d i j γ i j θdv κ, I m = min I i j and we apply the Schwarz s inequality in relation 43 and taking into account the hypotheses 1 we find that { [ d 1 ρẇ i ẇ i I i j ϖ i ϖ j dv ξ θ,θ ρκ θ, θ dt A i jmn ε i j ε mn i jmn ε i j γ mn C i jmn γ i j γ mn dv

16 1 Copyright 15 Tech Science Press CMC, vol.45, no., pp.17-15, 15 ] } A i j θ,i θ, j dv α i j ε i j β i j γ i j θdv τ θ, θ 44 1 ρẇ i ẇ i dv 1 I i j ϖ i ϖ j dv 1 c d ϕ,k ϕ,k dv ρ m I m 1 1 a 1 A i jmn ε i j ε mn 1 b 1 i jmn ε i j γ mn 1 c 1 C i jmn γ i j γ mn c d ϕ dv 1 κ m ρ m θ Now we introduce the measure M by M = 1 [ ρẇ i ẇ i I i j ϖ i ϖ j dv ξ θ,θ ρκ θ, θ dv A i jmn ε i j ε mn i jmn ε i j γ mn C i jmn γ i j γ mn dv 45 ] A i j θ,i θ, j dv α i j ε i j β i j γ i j θdv With the help of notations c { M = max c d }, c M = max { c d } {, C = max, ρ m 1 ρ m κ m from 45 we deduce that [ dm 1 CM c M A i jmn ε i j ε mn 1 i jmn ε i j γ mn 1 C i jmn γ i j γ mn dv dt a 1 b 1 c 1 1 ] ϕ,k ϕ,k dv 1 c M ϕ After we integrate this inequality over [,t], we are led to t Mt C Msds [ t 1 c M A i jmn ε i j ε mn 1 i jmn ε i j γ mn 1 C i jmn γ i j γ mn dv ds 46 a 1 b 1 c 1 1 t ] ϕ,k ϕ,k dv ds 1 t c M ϕ ds }

17 Structural Continuous Dependence in Micropolar Porous odies 13 Using the a priori estimate 33 we deduce that the last term of inequality 46 is bounded. Then we can write, formally, inequality 46 in the following form Mt C t Msds c MD 1 c M D 47 Here D 1 and D 1 are some terms that depend on data. After integration over the interval [,t], inequality 47 leads to t Msds c MD 1 c 1 M D e CT 1 48 C Inequality 47 proves that the solution of the mixed initial-boundary value problem consists of the system of equations 8, the initial conditions 6 and the boundary conditions 7 which depend continuously on coupling coefficients a i j and b i j. The evaluation of this dependence is made by means of measure t. On the other hand, if we substitute the integral t from 48 in 47, we obtain the following inequality Mt c MD 1 c M D e CT This inequality proves that the solution depends continuously on coupling coefficients c i j and d i j from 34. This time the evaluation of this dependence is made by means of measure Mt. With this, the proof of Theorem 4 is complete. 5 Conclusions In the first part of the study, we attach to our mixed problem certain auxiliary problems and prove some bounds for solutions of these problems. The second part of the study is devoted to obtain a priori estimates for the gradient of displacement u i and microrotation ϕ i and the gradient of the volume distribution function φ. In the main result of the study we prove that the solution of this problem depends continuously on coefficients which couple the micropolar deformation equations the equations that model the evolution of voids. The evaluation of this dependence is made by using an appropriate measure. Acknowledgement: The authors are deeply grateful to the reviewers, whose very pertinent suggestions have helped improve the paper. References Abbas, I. A.; Kumar, R. 14: Response of Thermal Source in Initially Stressed Generalized Thermoelastic Half-Space with Voids. J. Comput. Theor. Nanos., vol. 11, no. 6, pp

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19 Structural Continuous Dependence in Micropolar Porous odies 15 Thermoelastic ody with Two Temperatures. Abstract and Applied Analysis, vol. 13 Art. no Marin, M.; Stan, G. 13: Weak solutions in Elasticity of dipolar bodies with stretch. Carpathian J. Math., vol. 9, no. 1, pp Marin, M.; Florea, O. 14: On temporal behaviour of solutions in Thermoelas ticity of porous micropolar bodies. An. St. Univ. Ovidius Constanta, Serie Mathematics, vol., no. 1, pp Marin, M.; Agarwal, R. P.; Othman, M. 14: Localization in time of solutions for thermoelastic micropolar materials with voids. CMC-Comput Mater Con, vol. 4, no. 1, pp Mora, R. J.; Waas, A. M. 7: Evaluation of the Micropolar elasticity constants for honeycombs. Acta Mechanica, vol. 19, no. 1-4, pp Nunziato, J. W.; Cowin, S. C. 1979: A nonlinear theory of elastic materials with voids. Arch. Rat. Mech. Anal., vol. 7, no., pp Sharma, K.; Marin, M. 14: Reflection and transmission of waves from imperfect boundary between two heat conducting micropolar thermoelastic solids. An. St. Univ. Ovidius Constanta, Serie Mathematics, vol., no., pp

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