Elastic Equilibrium of Porous Cosserat Media with Double Porosity

The static equilibrium of porous elastic materials with double porosity is considered in the case of an elastic Cosserat medium.The corresponding three-dimensional system of differential equations is derived. Detailed consideration is given to the case of plane deformation. A two-dimensional system of equations of plane deformation is written in the complex form and its general solution is represented by means of three analytic functions of a complex variable and two solutions of Helmholtz equations. The constructed general solution enables one to solve analytically a sufficiently wide class of plane boundary value problems of the elastic equilibrium of porous Cosserat media with double porosity. A concrete boundary value problem for a concentric ring is solved.


Introduction
A model of elastic equilibrium of porous media with double porosity was constructed in the works [1][2][3].The theory justified in these papers combines the previously proposed model of Barenblatt for media with double porosity [4] and that of Biot for media with ordinary porosity [5].For a detailed account of the development of the theory of porous media and relevant references, see [6].Various issues related to the elastic equilibrium of bodies with double porosities are treated in [7][8][9][10][11][12][13][14][15].
In Section 2 we give the basic three-dimensional equations of the static equilibrium of porous elastic materials with double porosity in the case of an elastic Cosserat medium.In Section 3 we consider the case of a plane deformed state and write the corresponding two-dimensional system of equilibrium equations in the complex form.In Section 4 we construct the general solution of the abovementioned system of equations by means of analytic functions of a complex variable and solutions of Helmholtz equations.The obtained analogues of the Kolosov-Muskhelishvili formulas [35] make it possible to solve analytically plane boundary value problems of the elastic equilibrium of porous Cosserat media with double porosity.Finally, in Section 5 we solve a boundary value problem for a concentric circular ring.

Basic Three-Dimensional Relations
Let an elastic body with double porosity occupy the domain Ω ⊂  3 .Denote by ( 1 ,  2 ,  3 ) a point of the domain Ω in the Cartesian coordinate system.Let the domain Ω be filled with an elastic Cosserat medium having double porosity.The considered solid body is characterized by the displacement vector u = ( 1 ,  2 ,  3 ), rotation vector  = ( 1 ,  2 ,  3 ), and also the fluid pressures  1 ( 1 ,  2 ,  3 ) and  2 ( 1 ,  2 ,  3 ) occurring, respectively, in the pores and fissures of the porous medium.

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Then a homogeneous system of static equilibrium equations is written in the form [25]     = 0, where   are stress tensor components,   are moment stress tensor components, ∈  is the Levi-Civita symbol, and   ≡ /  ; the summation over the recurring index  is assumed to be done from 1 to 3.
Formulas that interrelate the stress and moment stress components, the displacement and rotation vector components, and the pressures  1 ,  2 have the form [13,25] where ,  are the Lamé parameters, , , ],  are the constants characterizing the microstructure of the considered elastic medium,  1 and  2 are the effective stress parameters, and   is the Kronecker delta.
As is well known, for the internal energy to be positive it is necessary that the following conditions be fulfilled [28]: In the stationary case, the values  1 and  2 satisfy the following system of [13] where  1 =  1 /  ,  2 =  2 /  ,  12 =  12 /  , and  21 =  21 /  ,   is fluid viscosity,  1 and  2 are the macroscopic intrinsic permeabilities associated with matrix and fissure porosity,  12 and  21 are the cross-coupling permeabilities for fluid flow at the interface between the matrix and fissure phases,  > 0 is the internal transport coefficient and corresponds to fluid transfer rate with respect to the intensity of flow between the pore and fissures, and Δ ≡  11 +  22 +  33 is the threedimensional Laplace operator.
The three-dimensional system of (1), (2), and (4) describes the static equilibrium of a porous elastic Cosserat medium with double porosity.Substituting relations (2) into (1), we obtain equilibrium equations with respect to the components of the displacement and rotation vectors: (5) If we add boundary conditions on the boundary Ω of the domain Ω, to the system of equilibrium equations, then we can consider various classical boundary value problems.
The following lemma is easy to prove.
Lemma 1.If  > 0,  1  2 −  12  21 > 0, then the system of ( 4) is equivalent to two independent equations: to the Laplace equation with respect to the combination ( 1 +  21 ) 1 + ( 2 +  12 ) 2 and to the Helmholtz equation with respect to the difference where Proof.Adding the first equation of system (4) to the second equation of this system, we immediately obtain (6).Let us write system (4) in the matrix form: By assumption, the determinant of the matrix det The left multiplication of all members of the latter equation by the matrix If from the first equation of the latter system we subtract the second equation of this system, then we obtain (7).Since the transformation determinant is defined as the validity of the lemma is proved.

Corollary 2. If on the boundary 𝜕Ω of the domain
This corollary follows from the fact that the homogeneous Helmholtz equation (7) with zero boundary conditions has only the trivial solution.

The Plane Deformation Case
From the basic three-dimensional equations we obtain the basic equations for the case of plane deformation.Let Ω be a sufficiently long cylindrical body with generatrix parallel to the  3 -axis.Denote by  the cross section of this cylindrical body; thus  ⊂  2 .In the case of plane deformation  3 = 0,  1 = 0, and  2 = 0, while the functions  1 ,  2 ,  3 ,  1 , and  2 do not depend on the coordinate  3 (see, e.g., [29]).
As follows from formulas (2), in the case of plane deformation Therefore the system of equilibrium equations (1) takes the form Relations (2) are rewritten as where Equations ( 6) and ( 7) take the form where Δ 2 fl  11 +  22 is the Laplace operator in two dimensions.
To write system (12) in the complex form, the second equation of this system is multiplied by  and summed up with the first equation [35][36][37] (see also [38,39] where, by formulas (13), We write ( 14) and (15) as If relations ( 18) are substituted into system (17), then system (16) is written in the complex form:
Equations (19) imply that where  0 fl  1 +  2 +  12 +  21 , () is an arbitrary analytic function of a complex variable  in the domain , and (, ) is an arbitrary solution of the Helmholtz equation From system (21) we easily obtain the expressions for the pressures  1 and  2 : Theorem 3. The general solution of the system of ( 20) is represented as follows: where  = ( + 3)/( + ),  fl 4(( 2 +  12 ) 1 − ( 1 +  21 ) 2 )/( + 2) 2 , () and () are arbitrary analytic functions of a complex variable  in the domain , and (, ) is an arbitrary solution of the Helmholtz equation where Proof.We take the operator   out of the brackets in the lefthand part of the first equation of system (20): Since ( 28) is a system of Cauchy-Riemann equations, we have where () is an arbitrary analytic function of .
A conjugate equation to (29) has the form Summing up ( 29) and ( 30) and taking into account that we obtain If from (29) we subtract (30) and write the expression (   + −    + ), then we have The second equation of system ( 20) is written as Substituting formula (33) into formula (34) we obtain the equation The general solution of ( 35) is written in the form where (, ) is a general solution of the Helmholtz equation The multiplier 2/(]+) has been introduced for convenience in writing our subsequent formulas.
Substituting formulas (32) and ( 36) into ( 29) and taking into account that (, ) is a solution of (37), we obtain From formulas (23) we find the following expression for the combination  1  1 +  2  2 : Substituting the latter formula into (38), integrating over , and using we obtain formula (24) which we are proving: +    (, ) . (41) Thus, if the solution of system ( 20) is sufficiently smooth, then it is represented in the form of ( 24) and (25).Conversely, if expressions (24) and ( 25) are substituted into (20), then this system will be satisfied.
Substituting expressions ( 24) and ( 25) into formulas (18), for combinations of stress tensor components we obtain the following formulas: Thus, the general solution of a two-dimensional system of differential equations that describes the static equilibrium of a porous elastic medium with double porosity is represented by means of three analytic functions of a complex variable and two solutions of the Helmholtz equation.By an appropriate choice of these functions we can satisfy five independent classical boundary conditions.
Let mutually perpendicular unit vectors l and s be such that where e 3 is the unit vector directed along the  3 -axis.The vector l forms the angle  with the positive direction of the  1axis.Then the displacement components   = u ⋅ l,   = u ⋅ s, as well as the stress and moment stress components acting on an area of arbitrary orientation, are expressed by the formulas (44)

A Problem for a Concentric Circular Ring
In this section, we solve a concrete boundary value problem for a concentric circular ring.On the boundary of the considered domain which is free from stresses and moment stresses, the values of pressures  1 and  2 are given.Let a porous elastic body with double porosity occupy the domain  which is bounded by the concentric circumferences  1 and  2 with radii  1 and  2 , respectively ( 1 <  2 ) (Figure 1).
Using the boundary conditions (47), we obtain For the coefficients  1 =  −1 and  1 =  −1 we obtain the following system of equations: whence , Thus where the coefficients The analytic functions   () and   () are represented as series: From the first condition of displacement uniqueness it follows that  = 0.It is also assumed that  0 is a real value; that is,  0 =  0 (see [35]).
The metaharmonic function (, ) is sought as a series where After defining the coefficients  1 ,  −1 , and  −1 from (69), the coefficient  −3 is defined from the first or the second equation of system (67).Since (, ) is a real function, we have All other coefficients are equal to zero.Thus the sought functions   (),   (), and (, ) will have the form By substituting the obtained values of these functions and functions   () and (, ) into formulas (42) we find the values of all stress and moment stress components.Though the domain boundary was free from moment stresses, due to fluid pressures in the pores the considered disturbance promoted their appearance inside the domain.
The procedure of solving a boundary value problem remains the same when stresses, moment stresses, and pressures on the domain boundary are given arbitrarily, but the condition that the principal vector and the principal moment of external forces are equal to zero is fulfilled.

Conclusion
We consider the static equilibrium of porous elastic materials with double porosity for a nonsymmetric elastic Cosserat medium.For the case of plane deformation, a general solution of the corresponding system of differential equations is constructed by means of three analytic functions of a complex variable and two solutions of the Helmholtz equations.The constructed general solution can be applied for solving analytically quite a wide class of boundary value problems.An explicit solution is obtained for a boundary value problem for a concentric circular ring.
In our opinion, problems of double porous elasticity for a nonsymmetric elastic medium may be of interest from theoretical and practical standpoints.

Disclosure
Any idea in this paper is possessed by the author and may not represent the opinion of Shota Rustaveli National Science Foundation itself.