Weak Solutions in Elasticity of Dipolar Porous Materials

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 problemwithin 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, 5 .


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, 5 .

Basic equations
Let B 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 B, denoted by ∂B, is a closed, bounded and pice-wise smooth surface which allows us the application of the 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 B 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 ∂B.The closure of domain B, denoted by B, means B B ∪ ∂B.
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: 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: ii the balance of the equilibrated forces is as follows: iii the constitutive equations are as follows:

2.5
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; 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 , B 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

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: N ijk,i P jk 0. 2.7

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 B is a Lipschitz region of the Euclidian threedimensional space R 3 .We use the following notations: 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

3.2
For clarity and simplification in presentation, we consider the following regularity hypotheses on the considered functions: i the constitutive coefficients are functions of class C 2 on B; ii the body loads F i , G jk , and H are continuous functions on B.
The ordered array u i , ϕ jk , σ is an admissible process on Let ∂B S u ∪ S t ∪ C be a disjunct decomposition of ∂B, where C is a set of surface measure and S u and S t are either empty or open in ∂B.Assume the following boundary conditions: where the functions u i , ϕ jk , σ, t i , μ jk , and h are prescribed, u i , ϕ jk , σ ∈ W 1,2 S u , and t i , μ jk , h ∈ L 2 S t .Also, we define V as a subspace of the space W of all functions u u i , ϕ ij , σ which satisfy the boundary conditions: 3.4 On the product space W × W, we consider a bilinear form A v, u , defined by where

3.6
We assume that the constitutive coefficients are bounded measurable functions in B 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
Also, by using symmetries 2.6 into 3.5 , it results in A ijkmnr χ ijk u χ mnr u P ki u j,k u j,i − 2M ik u j,i ϕ jk 2d i σγ ,i g ij σ ,i σ ,j ξσ 2 dV,

and thus
A u, u 2 B 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 B 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

3.10
for all x ij , y ij , z ijk , ω i , and w.Now, we introduce the functionals f v and g v by u i , ϕ jk , γ ∈ W be such that u 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 12 hold each v ∈ V.In the above relations, we used the spaces L 2 B and L 2 S u which represent, as it is well known, the space of real functions which are square-integrable on B, respectively, on S u ⊂ ∂B.
It follows from 3.10 and 3.8 that 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 B , defined by

3.14
It is easy to see that, in fact, the operators N k v, k 1, 2, . . ., 49, defined above, have the following general form: where n k r α are bounded and measurable functions on B. Also, we have used the notation D α for the multi-indices derivative, that is,

3.16
By 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: 3.17 In this inequality, the constant c 1 does not depend on v and the norms |•| L 2 and |•| W represent the usual norms in the spaces L 2 B and W, respectively.
In the following theorem, we indicate a necessary and sufficient condition for a system of operators to be a coercive system. is equal to m for each ξ ∈ C 3 , ξ / 0, where C 3 is the notation for the complex three-dimensional space, and The demonstration of this result can be find in 4 .
In the following, we assume that for each v ∈ W, we have where the constant c 2 does not depend on v.We denote by P the following set:

3.23
In the following theorem, it is indicated a necessary and sufficient condition for the existence of a weak solution of the boundary-value problem.A v, u v, u define a bilinear form for each v, u ∈ W/P, where u ∈ u and v ∈ v.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 0.

3.24
Moreover, the weak solution, u ∈ W, satisfies the following inequality: where c 3 is a real positive constant.Further, one has for each v ∈ W/P.
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.

Mathematical Problems in Engineering
Proof. from 3.13 and 3.14 we immediately obtain 3.20 .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.By 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.Proof.In this case, the boundary value problem P is given by 3.27 , where a i , b i , and c are arbitrary constants such that we can apply, once again, Theorem 3.2 to obtain the above result.

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.

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 by V/P the factor-space of classes v, wherev v p, v ∈ V, p ∈ P ,

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 byB ρF i dV ∂B t i dA 0, B ρε ijk x j F k G jk dV ∂B ε ijk x j t k μ jk dA 0,3.28where ε ijk is the alternating symbol.