Research Article Weak Solutions in Elasticity of Dipolar Porous Materials

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1 Mathematical Problems in Engineering Volume 2008, Article ID , 8 pages doi: /2008/ Research Article Weak Solutions in Elasticity of Dipolar Porous Materials Marin Marin Department of Mathematics, University of rasov, rasov, Romania Correspondence should be addressed to Marin Marin, m.marin@unitbv.ro Received 25 March 2008; Accepted 17 July 2008 Recommended by K. R. Rajagopal The main aim of our study is to use some general results from the general theory of elliptic equations in order to obtain some qualitative results in a concrete and very applicative situation. In fact, we will prove the existence and uniqueness of the generalized solutions for the boundary value problems in elasticity of initially stressed bodies with voids porous materials. Copyright q 2008 Marin Marin. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 1. Introduction The theories of porous materials represent a material length scale and are quite sufficient for a large number of the solid mechanics applications. In the following, we restrict our attention to the behavior of the porous solids in which the matrix material is elastic and the interstices are voids of material. The intended applications of this theory are to the geological materials, like rocks and soils and to the manufactured porous materials. The plane of the paper is the following one. In the beginning, we write down the basic equations and conditions of the mixed boundary value problem within context of linear theory of initially stressed bodies with voids, as in the papers of 1, 2. Then, we accommodate some general results from the paper 3, and the book 4, in order to obtain the existence and uniqueness of a weak solution of the formulated problem. For convenience, the notations chosen are almost identical to those of 2, asic equations Let be an open region of three-dimensional Euclidean space R 3 occupied by our porous material at time t 0. We assume that the boundary of the domain, denoted by, is a closed, bounded and pice-wise smooth surface which allows us the application of the

2 2 Mathematical Problems in Engineering divergence theorem. A fixed system of rectangular Cartesian axes is used and we adopt the Cartesian tensor notations. The points in are denoted by x i or x. The variable t is the time and t 0,t 0. We will employ the usual summation over repeated subscripts while subscripts preceded by a comma denote the partial differentiation with respect to the spatial argument. We also use a superposed dot to denote the partial differentiation with respect to t. The Latin indices are understood to range over the integers 1, 2, 3. In the following, we designate by n i the components of the outward unit normal to the surface. The closure of domain, denoted by, means. Also, the spatial argument and the time argument of a function will be omitted when there is no likelihood of confusion. The behavior of initially stressed bodies with voids is characterized by the following kinematic variables: u i u i x, t, ϕ jk ϕ jk x, t, σ σ x, t, x, t 0,t 0 ). 2.1 In our study, we analyze an anisotropic and homogeneous initially stressed elastic solid with voids. We restrict our considerations to the Elastostatics, so that the basic equations become as follows. i The equations of equilibrium is as follows: ( ) τij η ij,j ρf i 0, μ ijk,i η jk u j,i M ik ϕ ki M ji ϕ kr,i N ijr ρg jk 0; 2.2 ii the balance of the equilibrated forces is as follows: h i,i g ρl 0; 2.3 iii the constitutive equations are as follows: τ ij u j,k P ki C ijmn ε mn G mnij κ mn F mnrij χ mnr a ij σ d ijk σ,k, η ij ϕ jk M ik ϕ jk,r N rik G ijmn ε mn ijmn κ mn D ijmnr χ mnr b ij σ e ijk σ,k, μ ijk u j,r N irk F ijkmn ε mn D mnijk κ mn A ijkmnr χ mnr c ijk σ f ijkm σ,m, 2.4 h i d mni ε mn e mni κ mn f mnri χ mnr d i σ g ij σ,j, g a mn ε mn b mn κ mn c mnr χ mnr ξσ d i σ,i ; iv the geometric equations are ε ij 1 2( uj,i u i,j ), κij u j,i ϕ ij, χ ijk ϕ jk,i, σ ν ν In the above equations we have used the following notations: i ρ the constant mass density; ii u i the components of the displacement field;

3 Marin Marin 3 iii ϕ jk the components of the dipolar displacement field; iv ν the volume distribution function which in the reference state is ν 0 ; v σ a measure of volume change of the bulk material resulting from void compaction or distension; vi τ ij, η ij, μ ij the components of the stress tensors; vii h i the components of the equilibrated stress; viii F i the components of body force per unit mass; ix G jk the components of dipolar body force per unit mass; x L the extrinsic equilibrated body force; xi g the intrinsic equilibrated body force; xii ε ij, κ ij, χ ijk the kinematic characteristics of the strain tensors; xiii C ijmn, ijmn,...,d ijm, E ijm,...,a ij, b ij, c ijk, d i, ξ represent the characteristic functions of the material the constitutive coefficients and they obey to the following symmetry relations C ijmn C mnij C ijnm, ijmn mnij, G ijmn G ijnm, F ijkmn F ijknm, A ijkmnr A mnrijk, a ij a ji, P ij P ji, g ij g ji. 2.6 The physical significances of the functions L and h i are presented in the works 6, 7. The prescribed functions P ij, M ij and N ijk from 2.2 and 2.3 satisfy the following equations: ( ) Pij M ij,j 0, N ijk,i P jk Existence and uniqueness theorems In the main section of our paper, we will accommodate some theoretical results from the theory of elliptic equations in order to derive the existence and the uniqueness of a generalized solution of the mixed boundary-value problem in the context of initially stressed bodies with voids. Throughout this section, we assume that is a Lipschitz region of the Euclidian threedimensional space R 3. We use the following notations: W W 1,2 ] 13, W0 W 1,2 0 ]13, 3.1 with the convention that A 13 A A A, the Cartesian product is considered to be of 13-times. Also, W k,m is the familiar Sobolev space. With other words, W is defined as the space of all u u i,ϕ ij,σ, where u i, ϕ ij, σ W 1,2 with the norm 3 u 2 W σ 2 W 1,2 3 ( 3 ) ui 2 W 1,2 ϕij W 1,2 i 1 For clarity and simplification in presentation, we consider the following regularity hypotheses on the considered functions: j 1 i 1

4 4 Mathematical Problems in Engineering i all the constitutive coefficients are functions of class C 2 on ; ii the body loads F i,g jk,andh are continuous functions on. The ordered array u i,ϕ jk,σ is an admissible process on provided u i, ϕ jk, σ C 1 C 2. Also, the ordered array of functions τ ij,η ij,μ ijk,h i is an admissible system of stress on if τ ij, η ij, μ ijk, h i C 1 C 0 and τ ij,i, η ij,i, μ ijk,k, h k,k, h C 0. Let S u S t C be a disjunct decomposition of, where C is a set of surface measure and S u and S t are either empty or open in. Assume the following boundary conditions: u i ũ i, ϕ jk ϕ jk σ σ on S u, t i ( ) τ ij η ij nj t i, μ jk μ ijk n i μ jk, h h i n i h on S t, 3.3 where the functions ũ i, ϕ jk, σ, t i, μ jk,and h are prescribed, ũ i, ϕ jk, σ W 1,2 S u,and t i, μ jk, h L 2 S t.also,wedefinev as a subspace of the space W of all functions u u i,ϕ ij,σ which satisfy the boundary conditions: u i 0, ϕ ij 0, σ 0 on S u. 3.4 On the product space W W, we consider a bilinear form A v, u, defined by { A v, u Cijmn ε mn u ε ij v G mnij εij v κ mn u ε ij u κ mn v ] F mnrij εij v χ mnr u ε ij u χ mn v ] ijmn κ ij v κ mn u D ijmnr κij v χ mnr u κ ij u χ mn v ] A ijkmnr χ ijk v χ mnr u P ki u j,k v j,i M ik u j,i ψ jk v j,i ϕ jk N rik u j,k ψ jk,r v j,k ϕ jk,r a ij εij v σ ε ij u γ ] b ij κij v σ κ ij u γ ] c ijk χijk v σ χ ij u γ ] d ijk εij v σ,k ε ij u γ,k ] e ijk κij v σ,k κ ij u γ,k ] f ijkm χijk v σ,m χ ijk u γ,m ] 3.5 d i σγ,i γσ,i ] gij σ,i γ,j ξσγ } dv, where u ( u i,ϕ ij,σ ), v ( v i,ψ ij,γ ), ε ij u 1 2( uj,i u i,j ), ε ij v 2( 1 ) vj,i v i,j, κij u u j,i ϕ ij, κ ij v v j,i ψ ij, χ ijk u ϕ jk,i, χ ijk v ψ jk,i. 3.6 We assume that the constitutive coefficients are bounded measurable functions in which satisfy the symmetries 2.6. Then, by using relations 3.5 and 2.6 it is easy to deduce that A v, u A u, v. 3.7

5 Marin Marin 5 Also, by using symmetries 2.6 into 3.5, itresultsin A u, u Cijmn ε mn u ε ij v 2G mnij ε ij u κ mn u ijmn κ ij u κ mn u 2F mnrij ε ij u χ mnr u 2D ijmnr κ ij u χ mnr u A ijkmnr χ ijk u χ mnr u P ki u j,k u j,i 2M ik u j,i ϕ jk 2N rik u j,i ϕ jk,r 2a ij ε ij u σ 2b ij κ ij u σ 2c ijk χ ijk u σ 3.8 and thus 2d ijk ε ij u σ,k 2e ijk κ ij u σ,k 2f ijkm χ ijk u σ,m 2d i σγ,i g ij σ,i σ,j ξσ 2] dv, A u, u 2 UdV, 3.9 where U ρe is the internal energy density associated to u. We suppose that U is a positive definite quadratic form, that is, there exists a positive constant c such that C ijmn x ij x mn 2G ijmn x ij y mn 2F ijmnr x ij z mnr ijmn y ij y mn 2D ijmnr y ij z mnr A ijkmnr z ijk z mn P ki x ji x jk 2M ik x ji y jk 2N rik x ji z jkr 2a ij x ij w 2b ij y ij w 2c ijk z ij w 2d ijk x ij ω k 2e ijk y ij ω k f ijkm z ijk ω m 2d i wω i g ijk ω i ω j ξw 2 c ( x ij x ij y ij x ij z ijk z ijk ω i ω i w 2), for all x ij, y ij, z ijk, ω i,andw. Now, we introduce the functionals f v and g v by f v ρ ( F i v i G jk ψ jk Lγ ) dv, v W, g v ( t i v i μ jk ψ jk hγ ) da, v W, S t 3.11 where v v i,ψ jk,γ W and ρ, F i, G jk, L L 2. Let v ũ i, ϕ jk, γ W be such that ũ i, ϕ jk, γ on S u may by obtained by means of embedding the space W 1,2 into the space L 2 S u. The element v u i,ϕ jk,σ W is called weak (or generalized) solution of the boundary value problem, if u ũ V, A u, u f u g v 3.12

6 6 Mathematical Problems in Engineering hold for each v V. In the above relations, we used the spaces L 2 and L 2 S u which represent, as it is well known, the space of real functions which are square-integrable on, respectively, on S u. It follows from 3.10 and 3.8 that A v, v 2c εij v ε ij v κ ij v κ ij v χ ijk v χ ijk v γ i γ i γ 2] dv, 3.13 for any v v i,ψ jk,γ W. Let us consider the operators N k v, k 1, 2,...,49, mapping the space W into the space L 2, defined by N i v ε 1i v, N 3 i v ε 2i v, N 6 i v ε 2i v, N 9 i v κ 1i v, N 12 i v κ 1i v, N 15 i v κ 1i v, N 18 i v χ 11i v, N 21 i v χ 12i v, N 24 i v χ 13i v, N 27 i v χ 21i v, N 30 i v χ 22i v, N 33 i v χ 23i v, N 36 i v χ 31i v, N 39 i v χ 32i v, N 42 i v χ 33i v, N 45 i v σ,i v, N 49 v σ v i 1, 2, It is easy to see that, in fact, the operators N k v, k 1, 2,...,49, defined above, have the following general form: m N k v n kr αd α v r, p α, 3.15 r 1 α k r where n kr α are bounded and measurable functions on. Also, we have used the notation D α for the multi-indices derivative, that is, D α x α xα 2 2 xα 3 3 y definition, the operators N k v, k 1, 2,...,49 form a coercive system of operators on Wif for each v W the following inequality takes place: 49 k 1 α N k v 2 13 L 2 v r 2 L 2 c 1 v 2 W, c 1 > r 1 In this inequality, the constant c 1 does not depend on v and the norms L2 and W represent the usual norms in the spaces L 2 and W, respectively. In the following theorem, we indicate a necessary and sufficient condition for a system of operators to be a coercive system. Theorem 3.1. Let n psα be constant for α k s. Then the system of operators N p v is coercive on W if and only if the rank of the matrix ( Nps ξ ) ( ) n psα ξ α 3.18 α k s is equal to m for each ξ C 3, ξ / 0,whereC 3 is the notation for the complex three-dimensional space, and ξ α ξ α 1 1 ξα 2 2 ξα

7 Marin Marin 7 The demonstration of this result can be find in 4. In the following, we assume that for each v W, we have 49 A v, v c 2 Nk v 2 L 2, c 2 > 0, 3.20 k 1 where the constant c 2 does not depend on v. We denote by P the following set: { 49 P v V : Nk v } 2 L 2 0, 3.21 k 1 and by V/P the factor-space of classes ṽ, where having the norm ṽ { v p, v V, p P }, ṽ V/P inf p P v p W In the following theorem, it is indicated a necessary and sufficient condition for the existence of a weak solution of the boundary-value problem. Theorem 3.2. Let A v, u ṽ, ũ define a bilinear form for each ṽ, ũ W/P, whereu ũ and v ṽ. If it is supposed that the inequalities 3.17 and 3.20 hold, then a necessary and sufficient condition for the existence of a weak solution of the boundary value problem is p P f p g p Moreover, the weak solution, u W, satisfies the following inequality: where c 3 is a real positive constant. Further, one has for each ṽ W/P. ũ u W/P c ( m 3 W fil2 i 1 )1/2 ( m gil2 S i 1 )1/2], 3.25 A ( ṽ, ṽ ) c 4 ṽ W/P, c 4 > 0, 3.26 For the prove of this result, see 3. In the following, we intend to apply the above two results in order to obtain the existence of a weak solution for the boundary value problem formulated in the context of theory of initially stressed elastic solids with voids. Theorem 3.3. Let P {0}. Then there exists one and only one weak solution u W of our boundaryvalue problem.

8 8 Mathematical Problems in Engineering Proof. Clearly, from 3.13 and 3.14 we immediately obtain The matrix 3.20 has the rank 13 for each ξ C 3, ξ / 0. Thus by Theorem 3.1 we conclude that the system of N k operators, defined in 3.14, is coercive on the space W. According to definition 3.21 of P, we have that ε ij v 0,κ ij v 0, χ ijk v 0, γ i v 0, ψ 0 for each v P, v v i,ψ jk,ψ. So, we deduce that P reduces to P { v v i,ψ jk,γ V : v i a i ε ijk b j x k,ψ jk ε jks b s,γ c }, 3.27 where a i and b i and c are arbitrary constants and ε ijk is the alternating symbol. We will consider two distinct cases. First, we suppose that the set S u is nonempty. Then the set P reduces to P {0}, and therefore, condition 3.24 is satisfied. y using Theorem 3.2, we immediately obtain the desired result. In the second case, we assume that S u is an empty set. Then we have the following result. Theorem 3.4. The necessary and sufficient conditions for the existence of a weak solution u W of the boundary-value problem for elastic dipolar bodies with stretch, are given by ρf i dv t i da 0, 3.28 ( ) ( ) ρε ijk xj F k G jk dv ε ijk xj t k μ jk da 0, where ε ijk is the alternating symbol. Proof. In this case, the boundary value problem P is given by 3.27, where a i, b i,andc are arbitrary constants such that we can apply, once again, Theorem 3.2 to obtain the above result. 4. Conclusion For the considered initial-boundary value problem the basic results still valid. Now, for different particular cases, the solution can be found because it exists and is unique. References 1 D. Ieşan, Thermoelastic stresses in initially stressed bodies with microstructure, Thermal Stresses, vol. 4, no. 3-4, pp , M. Marin, Sur l existence et l unicité danslathermoélasticité des milieux micropolaires, Comptes Rendus de l Académie des Sciences. Serie II, vol. 321, no. 12, pp , I. Hlaváček and J. Nečas, On inequalities of Korn s type. I: boundary-value problems for elliptic system of partial differential equations, Archive for Rational Mechanics and Analysis, vol. 36, no. 4, pp , I. Nečas, Les Méthodes Directes en Théorie des Équations Elliptiques, Academia, Prague, Czech Republic, M. Marin, On the nonlinear theory of micropolar bodies with voids, Applied Mathematics, vol. 2007, Article ID 15745, 11 pages, M. A. Goodman and S. C. Cowin, A continuum theory for granular materials, Archive for Rational Mechanics and Analysis, vol. 44, no. 4, pp , J. W. Nunziato and S. C. Cowin, Linear elastic materials with void, Elasticity, vol. 13, pp , 1983.

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