# Explicit Solutions of the Boundary Value Problems for an Ellipse with Double Porosity

Authors: Lamara Bitsadze; Natela Zirakashvili
Journal: Advances in Mathematical Physics
 (2016)

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## Abstract

The basic two-dimensional boundary value problems of the fully coupled linear equilibrium theory of elasticity for solids with double porosity structure are reduced to the solvability of two types of a problem. The first one is the BVPs for the equations of classical elasticity of isotropic bodies, and the other is the BVPs for the equations of pore and fissure fluid pressures. The solutions of these equations are presented by means of elementary (harmonic, metaharmonic, and biharmonic) functions. On the basis of the gained results, we constructed an explicit solution of some basic BVPs for an ellipse in the form of absolutely uniformly convergent series.


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## Body

## 1. Introduction

In a material with two degrees of porosity, there are two pore systems, the primary and the secondary. For example, in a fissured rock (i.e., a mass of porous blocks separated from one another with an interconnected and continuously distributed system of fissures), most of the porosity is provided by the pores of the blocks or primary porosity, while most of permeability is provided by the fissures or secondary porosity. When fluid flow and deformation processes occur simultaneously, three coupled partial differential equations can be derived [1, 2] to describe the relationships governing pressure in the primary and secondary pores (and therefore the mass exchange between them) and the displacement of the solid.A theory of consolidation with double porosity structure was proposed by Wilson and Aifantis [1]. The physical and mathematical foundations of the theory of double porosity were considered in the papers [1–3], where analytical solutions of the relevant equations are also given. This theory unifies a model proposed by Biot for the consolidation of deformable single porosity media with a model proposed by Barenblatt for seepage in indeformable media with two degrees of porosity. The basic results and the historical information on the theory of porous media were summarized by De Boer [4]. However, Aifantis’ quasi-static theory ignored the cross-coupling effect between the volume change of the pores and fissures in the system. The cross-coupled terms were included in the equations of conservation of mass for the pore and fissure fluid and in Darcy’s law for solids with double porosity structure by several authors [5–8].Porous media theories play an important role in many branches of engineering, including material science, petroleum industry, chemical engineering, and soil mechanics, as well as biomechanics. In recent years, many authors investigated the BVPs of the 2D and 3D theories of elasticity for materials with double porosity structure, publishing a large number of papers (some of these results can be seen in [9–17] and references therein). In those works, the explicit solutions on some BVPs in the form of series are given in a form useful for the engineering practice.In the present paper, the basic two-dimensional boundary value problems of the fully coupled linear equilibrium theory of elasticity for solids with double porosity structure are reduced to the solvability of two types of a problem. The first one is similar to the BVPs for the equations of classical elasticity of isotropic bodies, while the second one is the BVPs for the equations of the pore and fissure fluid pressures. The solutions of these equations are presented by means of elementary (harmonic, metaharmonic, and biharmonic) functions. On the basis of the gained results, we constructed an explicit solution of some basic BVPs for an ellipse in the form of absolutely uniformly convergent series.


## 2. Basic Equations

Letx
=
x
1
,
x
2 be a point of the Euclidean 2D space R
2. In what follows, we consider an ellipse with a double porosity structure that occupies the region Ω of R
2.The system of homogeneous equations of the linear equilibrium theory of elasticity for solids with double porosity structure can be written as follows [9]:(1)
μ
Δ
u
+
λ
+
μ
grad
⁡
div
⁡
u
=
grad
⁡
β
1
p
1
+
β
2
p
2
(2)
k
1
Δ
-
γ
p
1
+
k
12
Δ
+
γ
p
2
=
0
,
k
21
Δ
+
γ
p
1
+
k
2
Δ
-
γ
p
2
=
0
,where u
=
u
1
x
,
u
2
x
T is the displacement vector in a solid, p
1
x and p
2
x are the pore and fissure fluid pressures, respectively, β
1 and β
2 are the effective stress parameters, γ
>
0 is the internal transport coefficient and corresponds to the fluid transfer rate with respect to the intensity of flow between the pores and fissures, λ
,
μ
,
k
1
,
k
2 are all constitutive coefficients, k
j
=
κ
j
/
μ
′  j
=
1,2, k
12
=
κ
12
/
μ
′, k
21
=
κ
21
/
μ
′, μ
′ is the fluid viscosity, κ
1 and κ
2 are the macroscopic intrinsic permeabilities associated with matrix and fissure porosity, respectively, κ
12 and κ
21 are the cross-coupling permeabilities for fluid flow at the interface between the matrix and fissure phases, and Δ is the two-dimensional Laplace operator. We consider the vectors as column matrices, if necessary. Throughout this paper, it is assumed that β
1
2
+
β
2
2
>
0. Superscript “T” denotes transposition. We will suppose that(3)
k
1
>
0
,
k
2
>
0
,
k
1
k
2
-
k
12
k
21
>
0
,
k
0
=
k
1
+
k
2
+
k
12
+
k
21
>
0
.Note that BVPs for system (2) which contain p
1
x and p
2
x can be investigated separately. Then if supposing p
j
x as known, we can study BVPs for system (1) with respect to u
x. By combining the obtained results, we arrive at explicit solutions of BVPs for systems (1)-(2).Obviously, from system (2), we have the following equations for p
1
x and p
2
x:(4)
Δ
Δ
+
λ
1
2
p
j
=
0
,
j
=
1,2
,where(5)
λ
1
=
i
γ
k
0
k
1
k
2
-
k
12
k
21
,
k
0
=
k
1
+
k
2
+
k
12
+
k
21
.It is easy to see that the solutions of system (2) have the form(6)
p
1
=
ϕ
+
m
1
ϕ
1
,
p
2
=
ϕ
+
ϕ
1
,where(7)
m
1
=
-
k
2
+
k
12
k
1
+
k
21
,
Δ
ϕ
=
0
,
Δ
+
λ
1
2
ϕ
1
=
0
.First, we assumep
j to be known. Then, for u
x, we get the following nonhomogeneous equation:(8)
μ
Δ
u
+
λ
+
μ
grad
⁡
div
⁡
u
=
grad
⁡
β
1
+
β
2
ϕ
+
β
1
m
1
+
β
2
ϕ
1
.It is well known that the general solution of (8) has the form(9)
u
=
v
+
v
0
,where v is a general solution of the equation(10)
μ
Δ
v
+
λ
+
μ
grad
⁡
div
⁡
v
=
0and v
0 is a particular solution of the nonhomogeneous equation(11)
v
0
=
1
λ
+
2
μ
grad
⁡
β
1
+
β
2
φ
0
-
β
1
m
1
+
β
2
λ
1
2
ϕ
1
;here,(12)
Δ
ϕ
=
0
,
Δ
φ
0
=
ϕ
,
Δ
+
λ
1
2
ϕ
1
=
0
.


## 3. Basic Equations in the Elliptic Coordinate System

Let us rewrite equalities (6), (9), and (10)–(12) in elliptic coordinates system ξ
,
η
0
≤
ξ
<
∞
,
0
≤
η
<
2
π (see Appendix A). We obtain the following.Formula (9) takes the form(13)
u
~
=
v
~
+
v
~
0
.Formula (11) takes the form(14)
v
~
0
=
1
λ
+
2
μ
1
h
∂
Ψ
∂
ξ
e
→
ξ
+
∂
Ψ
∂
η
e
→
η
,where(15)
Ψ
≔
β
1
+
β
2
φ
-
0
-
β
1
m
1
+
β
2
λ
1
2
ϕ
1
¯
,
h
=
sinh
2
⁡
ξ
+
sin
2
⁡
η
,
φ
0
¯
=
φ
0
¯
ξ
,
η
,
ϕ
1
¯
=
ϕ
1
¯
ξ
,
η
.Equations (12) take the form(16)
Δ
ϕ
¯
ξ
,
η
=
0
,
1
h
2
∂
2
ϕ
1
¯
∂
ξ
2
+
∂
2
ϕ
1
¯
∂
η
2
+
λ
1
2
ϕ
1
¯
=
0
,
(17)
1
h
2
∂
2
φ
-
0
∂
ξ
2
+
∂
2
φ
-
0
∂
η
2
=
ϕ
¯
.Expressions (6) can be rewritten in the following form:(18)
p
-
1
=
ϕ
¯
+
m
1
ϕ
¯
1
,
p
-
2
=
ϕ
¯
+
ϕ
¯
1
.Equation (10) in the elliptic coordinates system takes the form [18–21](19)
∂
D
∂
ξ
-
∂
K
∂
η
=
0
,
∂
u
-
∂
ξ
+
∂
v
-
∂
η
=
κ
-
2
κ
μ
′
h
0
2
D
,
∂
D
∂
η
+
∂
K
∂
ξ
=
0
,
∂
v
-
∂
ξ
-
∂
u
-
∂
η
=
1
μ
′
h
0
2
K
,where κ
=
4
1
-
ν
,
μ
=
E
/
2
1
-
ν
,
h
0
:
=
cosh
⁡
2
ξ
⁡
-
cos
⁡
2
η
,
u
-
=
2
h
v
~
1
/
c
2
,
v
-
=
2
h
v
~
2
/
c
2, (in what follows, we assume that c
=
1), v
~
1 and v
~
2 are the components of displacement vector v
~
v
~
1
,
v
~
2 along the normal and the tangent to the curve ξ
=
const, κ
-
2
/
κ
μ
D is divergence of displacement vector v
~, 1
/
μ
K is a rotor of displacement vector v
~, ν is Poisson’s ratio, and E is Young’s modulus.Hooke’s law can be written as follows:(20)
h
0
2
μ
σ
ξ
ξ
=
h
0
2
μ
D
-
2
∂
v
-
∂
η
-
2
h
0
2
sinh
⁡
2
ξ
u
-
-
sin
⁡
2
η
v
-
,
h
0
2
μ
σ
η
η
=
h
0
2
μ
D
-
2
∂
u
-
∂
ξ
+
2
h
0
2
sinh
⁡
2
ξ
u
-
-
sin
⁡
2
η
v
-
,
h
0
2
μ
τ
ξ
η
=
h
0
2
μ
K
+
2
∂
u
-
∂
η
-
2
h
0
2
sin
⁡
2
η
u
-
+
sinh
⁡
2
ξ
v
-
.


## 4. Boundary Value Problems

For systems (1) and (2), we pose the following BVPs.Find in domainΩ
=
0
≤
ξ
<
ξ
1
,
0
≤
η
<
2
π a regular solution U
u
,
p
1
,
p
2
∈
C
2
Ω to systems (1) and (2) satisfying one of the following boundary conditions (see Figure 1):(21)
for
ξ
=
ξ
1
:
a
u
-
=
f
11
,
v
-
=
f
12
,
p
-
1
=
f
13
,
p
-
2
=
f
14
,
o
r
b
h
0
2
μ
σ
ξ
ξ
=
g
11
,
h
0
2
μ
τ
ξ
η
=
g
12
,
p
-
1
=
f
13
,
p
-
2
=
f
14
,
o
r
c
u
-
=
f
11
,
v
-
=
f
12
,
∂
p
-
1
∂
η
=
f
23
,
∂
p
-
2
∂
η
=
f
24
,
o
r
d
h
0
2
μ
σ
ξ
ξ
=
g
11
,
h
0
2
μ
τ
ξ
η
=
g
12
,
∂
p
-
1
∂
η
=
f
23
,
∂
p
-
2
∂
η
=
f
24
.Here f
j
k and g
j
k are given functions.Figure 1
Ellipse with double porosity.Note that when stress componentsh
0
2
/
μ
σ
ξ
ξ and h
0
2
/
μ
τ
ξ
η are given on boundary ξ
=
ξ
1, then the BVP is solvable if the principal vector and the principal moment of external stresses are equal to zero.Clearly, BVPs for (2) can be investigated separately (Problem B). Then, by admitting p
1
x and p
2
x as known, we can study BVPs for system (1) with respect to u
x (Problem A). By combining the obtained results, we arrive at the explicit solutions of BVPs for systems (1)-(2) and (21).Problem A.
First we state the BVPs for the semiellipse, when the conditions of uninterrupted continuation of solutions (i.e., the conditions of symmetry or antisymmetry) are given onη
=
0, η
=
π, and ξ
=
0 [22]. Thus, we obtain the solutions of BVPs for the ellipse.Now let us formulate the boundary value problems for semiellipse as follows: find in domainΩ
1
=
0
≤
ξ
<
ξ
1
,
0
≤
η
<
π a regular solution v
~
u
-
,
v
-
∈
C
2
Ω of (19) satisfying the following boundary conditions (see Figure 2):(22)
f
o
r
η
=
0
:
a
v
-
=
0
,
τ
ξ
η
=
0
⟺
v
-
=
0
,
∂
u
-
∂
η
=
0
o
r
b
u
-
=
0
,
σ
η
η
=
0
⟺
u
-
=
0
,
∂
v
-
∂
η
=
0
,
(23)
f
o
r
η
=
π
:
a
v
-
=
0
,
τ
ξ
η
=
0
⟺
v
-
=
0
,
∂
u
-
∂
η
=
0
o
r
b
u
-
=
0
,
σ
η
η
=
0
⟺
u
-
=
0
,
∂
v
-
∂
η
=
0
,
(24)
f
o
r
ξ
=
0
:
a
u
=
0
,
τ
ξ
η
=
0
⟺
u
=
0
,
∂
v
∂
ξ
=
0
o
r
b
v
=
0
,
σ
ξ
ξ
=
0
⟺
v
=
0
,
∂
u
∂
ξ
=
0
,
(25)
f
o
r
ξ
=
ξ
1
:
a
u
-
=
f
11
,
v
-
=
f
12
o
r
b
h
0
2
μ
σ
ξ
ξ
=
g
11
,
h
0
2
μ
τ
ξ
η
=
g
12
.Figure 2
Semiellipse.The solutions of (19) and (22)–(25) are represented by means of two harmonic functions φ
1
ξ
,
η and φ
2
ξ
,
η. Consider(26)
u
-
=
sinh
2
⁡
ξ
1
coth
⁡
ξ
∂
φ
1
∂
η
+
∂
φ
2
∂
ξ
+
κ
-
1
φ
2
sinh
⁡
ξ
cos
⁡
η
-
cosh
2
⁡
ξ
1
tanh
⁡
ξ
∂
φ
1
∂
ξ
-
∂
φ
2
∂
η
-
κ
-
1
φ
1
cosh
⁡
ξ
sin
⁡
η
,
v
-
=
-
cosh
2
⁡
ξ
1
tanh
⁡
ξ
∂
φ
1
∂
η
+
∂
φ
2
∂
ξ
+
κ
-
1
φ
2
cosh
⁡
ξ
sin
⁡
η
-
sinh
2
⁡
ξ
1
coth
⁡
ξ
∂
φ
1
∂
ξ
-
∂
φ
2
∂
η
-
κ
-
1
φ
1
sinh
⁡
ξ
cos
⁡
η
,
D
=
κ
μ
cosh
⁡
2
ξ
-
cos
⁡
2
η
∂
φ
1
∂
ξ
-
∂
φ
2
∂
η
cosh
⁡
ξ
sin
⁡
η
+
∂
φ
1
∂
η
+
∂
φ
2
∂
ξ
sinh
⁡
ξ
cos
⁡
η
,
K
=
κ
μ
c
o
s
h
2
ξ
-
c
o
s
2
η
∂
φ
1
∂
ξ
-
∂
φ
2
∂
η
sinh
⁡
ξ
cos
⁡
η
-
∂
φ
1
∂
η
+
∂
φ
2
∂
ξ
cosh
⁡
ξ
sin
⁡
η
.The components of stress vector (17) can be written as(27)
h
0
2
μ
σ
ξ
ξ
=
2
sinh
2
⁡
ξ
1
∂
∂
η
∂
φ
1
∂
ξ
-
∂
φ
2
∂
η
cosh
⁡
ξ
-
κ
-
2
∂
φ
1
∂
η
-
κ
∂
φ
2
∂
ξ
sinh
⁡
ξ
cos
⁡
η
+
2
cosh
2
⁡
ξ
1
∂
∂
η
∂
φ
1
∂
η
+
∂
φ
2
∂
ξ
sinh
⁡
ξ
+
κ
∂
φ
1
∂
ξ
+
κ
-
2
∂
φ
2
∂
η
cosh
⁡
ξ
sin
⁡
η
-
4
sinh
⁡
ξ
1
+
ξ
sinh
⁡
ξ
1
-
ξ
cosh
⁡
2
ξ
-
cos
⁡
2
η
∂
φ
1
∂
ξ
-
∂
φ
2
∂
η
cosh
⁡
ξ
sin
⁡
η
+
∂
φ
1
∂
η
+
∂
φ
2
∂
ξ
sinh
⁡
ξ
cos
⁡
η
,
h
0
2
μ
σ
η
η
=
-
2
sinh
2
⁡
ξ
1
∂
∂
η
∂
φ
1
∂
ξ
-
∂
φ
2
∂
η
cosh
⁡
ξ
-
κ
∂
φ
1
∂
η
-
κ
-
2
∂
φ
2
∂
ξ
sinh
⁡
ξ
cos
⁡
η
-
2
cosh
2
⁡
ξ
1
∂
∂
η
∂
φ
1
∂
η
+
∂
φ
2
∂
ξ
sinh
⁡
ξ
+
κ
-
2
∂
φ
1
∂
ξ
+
κ
∂
φ
2
∂
η
cosh
⁡
ξ
sin
⁡
η
+
4
sinh
⁡
ξ
1
+
ξ
sinh
⁡
ξ
1
-
ξ
cosh
⁡
2
ξ
-
cos
⁡
2
η
∂
φ
1
∂
ξ
-
∂
φ
2
∂
η
cosh
⁡
ξ
sin
⁡
η
+
∂
φ
1
∂
η
+
∂
φ
2
∂
ξ
sinh
⁡
ξ
cos
⁡
η
,
h
0
2
μ
τ
ξ
η
=
-
2
cosh
2
⁡
ξ
1
∂
∂
η
∂
φ
1
∂
ξ
-
∂
φ
2
∂
η
sinh
⁡
ξ
-
κ
-
2
∂
φ
1
∂
η
-
κ
∂
φ
2
∂
ξ
cosh
⁡
ξ
sin
⁡
η
+
2
sinh
2
⁡
ξ
1
∂
∂
η
∂
φ
1
∂
η
+
∂
φ
2
∂
ξ
cosh
⁡
ξ
+
κ
∂
φ
1
∂
ξ
+
κ
-
2
∂
φ
2
∂
η
sinh
⁡
ξ
cos
⁡
η
+
4
sinh
⁡
ξ
1
+
ξ
sinh
⁡
ξ
1
-
ξ
cosh
⁡
2
ξ
-
cos
⁡
2
η
∂
φ
1
∂
ξ
-
∂
φ
2
∂
η
sinh
⁡
ξ
cos
⁡
η
-
∂
φ
1
∂
η
+
∂
φ
2
∂
ξ
cosh
⁡
ξ
sin
⁡
η
.Let us rewrite conditions (25) when ξ
=
ξ
1 in the following equivalent form:(28)
(a)
2
h
0
2
cosh
⁡
ξ
1
sin
⁡
η
u
-
+
sinh
⁡
ξ
1
cos
⁡
η
v
-
=
-
1
2
sinh
⁡
2
ξ
1
∂
φ
1
∂
ξ
-
∂
φ
2
∂
η
+
κ
-
1
φ
1
,
2
h
0
2
sinh
⁡
ξ
1
cos
⁡
η
u
-
-
cosh
⁡
ξ
1
sin
⁡
η
v
-
=
1
2
sinh
⁡
2
ξ
1
∂
φ
1
∂
η
+
∂
φ
2
∂
ξ
+
κ
-
1
φ
2
or
b
2
μ
cosh
⁡
ξ
1
sin
⁡
η
σ
ξ
ξ
+
sinh
⁡
ξ
1
cos
⁡
η
τ
ξ
η
=
sinh
⁡
2
ξ
1
∂
∂
η
∂
φ
1
∂
η
+
∂
φ
2
∂
ξ
+
κ
∂
φ
1
∂
ξ
+
κ
-
2
∂
φ
2
∂
η
,
2
μ
sinh
⁡
ξ
1
cos
⁡
η
σ
ξ
ξ
-
cosh
⁡
ξ
1
sin
⁡
η
τ
ξ
η
=
s
i
n
h
2
ξ
1
∂
∂
η
∂
φ
1
∂
ξ
-
∂
φ
2
∂
η
-
κ
-
2
∂
φ
1
∂
η
+
κ
∂
φ
2
∂
ξ
.In (28), functions φ
1 and φ
2 are harmonic functions, which, by using the method of separation of variables [23, 24], can be presented as follows:(29)
φ
i
=
∑
n
=
1
∞
φ
i
n
,
i
=
1,2
,where(30)
φ
1
n
=
B
1
n
sinh
⁡
n
ξ
cosh
⁡
n
ξ
1
sin
⁡
n
η
,
φ
2
n
=
B
2
n
cosh
⁡
n
ξ
cosh
⁡
n
ξ
1
cos
⁡
n
η
or
φ
1
n
=
B
1
n
cosh
⁡
n
ξ
cosh
⁡
n
ξ
1
cos
⁡
n
η
,
φ
2
n
=
B
2
n
sinh
⁡
n
ξ
cosh
⁡
n
ξ
1
sin
⁡
n
η
.When n
=
0, then φ
10
=
B
10
+
a
02
ξ
+
a
03
η
+
a
04
ξ
η and φ
20
=
B
20
+
b
02
ξ
+
b
03
η
+
b
04
ξ
η, where B
10
,
a
02
,
…
,
b
04 are constants. From the condition of limitation of grad
⁡
φ
i
0
=
1
/
h
∂
φ
i
0
/
∂
ξ
+
∂
φ
i
0
/
∂
α in the focuses of ellipse and condition of periodicity of functions φ
i
0 along η coordinate, there ensues the following: a
0
j
=
0
,
b
0
j
=
0
,
j
=
2,3
,
4. Therefore, φ
10
=
0 and φ
20
=
A
20 or φ
10
=
A
10, and φ
20
=
0
. When on boundary ξ
=
ξ
0 the values of the stress vector are given, then the rigid displacements of the ellipse are defined by the equalities. φ
10
=
0 and φ
20
=
A
20 and φ
10
=
A
10, and φ
20
=
0.By substituting (22)–(24) and (29) into (28), for determining unknown B
1
n and B
2
n, we get the following infinite system of algebraic equations, whose main matrix has a block-diagonal form (see Figure 3).Figure 3
Diagram of the main matrix of the algebraic equation system.The dimension of all matricesD
i
i
=
1,2
,
… is 2
×
2 and det
⁡
D
i
≠
0. When i
→
∞, det
⁡
D
i
→
M, where M
≠
0.Thus, the system is uniquely solvable. Let us substitute the obtained values,B
1
n and B
2
n, into φ
1 and φ
2. As a result, we obtain the vector v
~
u
-
,
v
-.Problem B.
Find regular solutionsp
1 and p
2 to (2) in domain Ω
=
0
≤
ξ
<
ξ
1
,
0
≤
η
<
2
π satisfying one of the following boundary conditions (see Figure 1):(31)
when
ξ
=
ξ
1
:
a
p
1
=
f
13
,
p
2
=
f
14
o
r
b
∂
p
1
∂
η
=
f
23
,
∂
p
2
∂
η
=
f
24
.The solution of (2) is (18), where function ϕ
1
¯ is a solution of (16). The solution of (16) by applying the method of separation of variables is reduced to solutions of the Mathieu differential equation or modified Mathieu differential equations (see Appendix B) [25]. These solutions are known as Mathieu functions. Thus, ϕ
1
¯
ξ
,
η
=
R
ξ
·
Φ
2
η, where Φ
2
η is the solution of Mathieu differential equation d
2
Φ
2
η
η
/
d
η
2
+
a
-
2
q
cos
⁡
2
η
Φ
2
η
η
=
0 (see Appendix C), and R is the solution of modified Mathieu differential equations d
2
R
ξ
/
d
ξ
2
-
a
-
2
q
cosh
⁡
2
ξ
R
ξ
=
0 (see Appendix D), where a
≔
c
+
1
/
2
λ
1
2
,
q
≔
1
/
4
λ
1
2
,
c
=
const is an arbitrary constant.Let us assume that harmonic functionϕ
¯ from (18) is represented in the form of series:(32)
ϕ
¯
=
∑
n
=
0
∞
C
n
cosh
⁡
n
ξ
cos
⁡
n
η
+
∑
n
=
1
∞
D
n
sinh
⁡
n
ξ
sin
⁡
n
η
.Taking into account the boundary conditions (31), to determine unknown coefficients C
n and D
n, we obtain the system of infinite algebraic equations, whose main matrix has a block-diagonal form (Figure 3). By solving this system, we find ϕ
¯
ξ
,
η.For determiningφ
-
0 from (17), let us consider the particular cases: keeping in mind the homogeneous boundary conditions, we get C
n
=
0 or D
n
=
0. Let us assume that D
n
=
0.Substituting (32) in (17), we obtain the following equation:(33)
∂
2
φ
-
0
∂
ξ
2
+
∂
2
φ
-
0
∂
η
2
=
∑
k
=
0
∞
C
k
cosh
⁡
k
ξ
cos
⁡
k
η
cosh
⁡
2
ξ
-
cos
⁡
2
η
=
∑
n
=
0
∞
E
n
cosh
⁡
n
ξ
cos
⁡
n
η
.The solution of (33) is sought in the following form:(34)
φ
-
0
=
φ
-
0
g
+
φ
-
0
p
,where φ
-
0
g is a harmonic function and it can be obtained in the same way as ϕ
¯. Thus, we have(35)
Δ
φ
-
0
g
=
0
,
φ
-
0
g
=
∑
n
=
0
∞
T
n
cosh
⁡
n
ξ
cos
⁡
n
η
.
φ
-
0
p is the particular solution of the following equation:(36)
∂
2
φ
-
0
p
∂
ξ
2
+
∂
2
φ
-
0
p
∂
η
2
=
∑
n
=
0
∞
E
n
cosh
⁡
n
ξ
cos
⁡
n
η
.The solution of (36) is sought in the form(37)
φ
-
0
p
=
∑
n
=
0
∞
Q
n
cosh
⁡
n
ξ
cos
⁡
n
η
.Substituting (37) in (36), we obtain the relations between Q
n and E
n and hence it follows that(38)
φ
-
0
=
∑
n
=
0
∞
T
n
cosh
⁡
n
ξ
cos
⁡
n
η
+
∑
n
=
0
∞
Q
n
cosh
⁡
n
ξ
cos
⁡
n
ηor(39)
φ
-
0
=
∑
n
=
0
∞
M
n
cosh
⁡
n
ξ
cos
⁡
n
η
,where M
n
=
T
n
+
Q
n.Quite similarly, we obtain the solution, whenC
n
=
0. By combining the obtained results, we obtain an explicit solution of (17).Remark 1.
When we have the nonhomogeneous boundary conditions, then the coefficients in (32) are different from zero, C
n
≠
0
,
D
n
≠
0, and function φ
-
0 will be obtained by above-mentioned method for each series, separately.
By substitutingφ
-
0 and ϕ
¯
1 in (14), we obtain function v
~
0. Finally, from (13), we obtain displacement u
~ in Ω at arbitrary point. Stress components are defined from (20).


## 5. Conclusions

The main results of this work can be formulated as follows:(1)
The system of equations of the linear equilibrium theory of elasticity for solids with double porosity structure is written in terms of elliptic coordinates.
(2)
The problems are reduced to the solvability of two types of problem. The first one is similar to the BVPs for the equations of classical elasticity of isotropic bodies, while the second one is the BVPs for the equations of the porous and fissure fluid pressures.
(3)
Analytical (exact) solutions are obtained for 2D BVPs for the ellipse with double porosity structure.By using the above-mentioned method, the following is possible:(4)
It is possible to construct explicitly the solutions of basic BVPs for systems (1) and (2) for simple cases of 2D domains (circle, plane with circular hole.) in the form of absolutely and uniformly convergent series that are useful in the engineering practice.
(5)
It is possible to obtain numerical solutions of the boundary value problems.
(6)
It is possible to construct explicitly the solutions of basic BVPs of the systems of equations in the modern linear theories of elasticity, thermoelasticity, and poroelasticity for materials with microstructures and for elastic materials with double porosity for a circle, and so forth.
(7)
In practice, such BVPs are quite common in many areas of science. The potential users of the obtained results will be the scientists and engineers working on the problems of solid mechanics, micromechanics and nanomechanics, mechanics of materials, engineering mechanics, engineering medicine, biomechanics, engineering geology, geomechanics, hydroengineering, applied and computing mechanics, and applied mathematics.



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