The questions on the uniqueness of a solutions of the problems are investigated

Vol. 20, No. 2, 2016, 11–23

Explicit Solutions on Same Problems in the Fully Coupled Theory of Elasticity For a Circle with Double Porosity

Ivane Tsagareli^∗ and Lamara Bitsadze

I.Vekua Institute of Applied Mathematics of Iv. Javakhishvili Tbilisi State University 2 University St., 0186, Tbilisi, Georgia;

(Received December 3, 2015; Revised November 2, 2016; Accepted December 5, 2016)

The purpose of this paper is to consider the two-dimensional version of the fully coupled theory of elasticity for solids with double porosity the and to solve explicitly some boundary value problems (BVPs) of statics for an elastic circle. The explicit solutions of this BVPs are constructed by means of absolutely and uniformly convergent series. The questions on the uniqueness of a solutions of the problems are investigated.

Keywords:Double porosity, Explicit solution, Elastic circle.

AMS Subject Classification: 74F10, 74G30, 74G10.

1. Introduction

The poroelasticity is an effective and useful model for deformation-driven bone fluid movement in bone tissue. The suggested double porosity model would consider the bone fluid pressure in the vascular porosity and the bone fluid pressure in the lacunar-canalicular porosity.The extensive review of the results in the theory of bone poroelasticity can be found in the survey article of Cowin [1]. A theory of consolidation for elastic materials with double porosity was presented in [2-4], where the physical and mathematical foundations of this theory were considered.

In these papers the theory of Aifantis unifies a model proposed by Biot [5] for the consolidation of deformable single porosity media with a model proposed by Barenblatt [6] for seepage in undeformable media with two degrees of porosity.

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 by several authors [7,12].

In [13,14] the fully coupled linear theory of elasticity is considered for solids with double porosity. Four spatial cases of the dynamical equations are considered. The fundamental solutions are constructed by means of elementary functions and the basic properties of the fundamental solutions are established.

Porous media theories play an important role in many branches of engineer- ing, including material science, the petroleum industry, chemical engineering, and

∗Corresponding author. Email: [email protected]

ISSN: 1512-0082 print c 2016 Tbilisi University Press

soil mechanics, as well as biomechanics. The problem of elastic bodies with dou- ble porosity was the subject of study for some papers more than fifty years ago.

Many authors have investigated the BVPs of the 2-dimensional and 3-dimensional theories of elasticity for materials with double porosity, that are published in a large number of papers (some of these results can be seen in [15-27] and references therein). There the explicit solutions on some BVPs in the form of series and in quadratures are given in a form useful for engineering practice.

The purpose of this paper is to consider the two-dimensional version of the fully coupled theory of elasticity for solids with double porosity and to solve explicitly some BVPs of statics for an elastic circle. The explicit solutions of this BVPs are constructed by means of absolutely and uniformly convergent series. The questions on the uniqueness of solutions of the problems are investigated.

2. Basic equations and boundary value problems

LetDbe an elastic circle of radiusR with boundaryS,centered at pointO(0,0), and let ¯D =D∪S. Let us assume that the domain D is filled with an isotropic material with double porosity.

The system of homogeneous equations in the full coupled linear equilibrium the- ory of elasticity for materials with double porosity can be written as follows [8]

µ∆u+ (λ+µ)graddivu−grad(β1p1+β2p2) = 0, (1)

(k1∆−γ)p1+ (k12∆ +γ)p2= 0, (k21∆ +γ)p1+ (k2∆−γ)p2= 0,

(2)

whereu = u(u1, u2) is the displacement vector in a solid,p1 and p2 are the pore and fissure fluid pressures respectively. β₁ and β₂ are the effective stress parameters, γ > 0 is the internal transport coefficient and corresponds to fluid transfer rate with respect to the intensity of flow between the pore and fissures, λ, µ, are constitutive coefficients, k_j = κj

µ⁰, j = 1,2, k₁₂ = κ₁₂

µ⁰ , k₂₁ = κ₂₁ µ⁰ . µ⁰ is the fluid viscosity, κ1 and κ2 are the macroscopic intrinsic permeabilities associated with matrix and fissure porosity, respectively,κ₁₂andκ₂₁ are the cross- coupling permeabilities for fluid flow at the interface between the matrix and fissure phases, ∆ is the Laplace operator.

A vector-functionU(x) = (u₁, u₂, p₁, p₂) defined in the domainDis called regular if it has integrable continuous second derivatives inD, andU(x) itself and its first order derivatives are continuously extendable at every point of the boundary of D, i.e., U(x)∈C²(D)T

C¹(D); D=DS

S, x∈D, x= (x1, x2).

Note that the system (2) would be considered separately. Further we assume that p_j is known, whenx∈D.We can write the system (2) as

(∆ +λ²₁)∆p_j(x) = 0, j= 1,2.

With the help of this we find the solution of system (2) in the form

p1(x) =ϕ(x) +m1ϕ1(x), p2(x) =ϕ(x) +ϕ1(x), (3) where

∆ϕ= 0, (∆ +λ²₁)ϕ₁ = 0, m₁ = γ−k₁₂λ²₁

γ+k1λ²₁ =−k₂+k₁₂ k1+k21

,

λ₁=i

r γk₀

k₁k₂−k₁₂k₂₁ =iλ₀, k₀=k₁+k₂+k₁₂+k₂₁,

k1 >0, k2 >0, γ >0, µ >0, λ+µ >0, k1k2−k12k21>0.

Substitute the expression β1p1+β2p2 in (1) and search the particular solution of the following nonhomogeneous equation

µ∆u+ (λ+µ)graddivu=grad[(β1+β2)ϕ+ (m1β1+β2)ϕ1]. (4) It is well-known that a general solution of the last equation is presented in the form u(x) =v(x) +v0(x), (5) wherev(x) is a general solution of the equation

µ∆v+ (λ+µ)graddivv= 0. (6)

v₀(x) is a particular solution of the nonhomogeneous equation. We Look for solu- tionv₀(x) in the form [28]

v₀(x) =gradF(x).

Substitutev₀ instead ofu into (4). Now we can find value of the functionF. And finally, for textbf v₀, we obtain

v₀(x) = 1

λ+ 2µgrad

(β₁+β₂)ϕ₀−β₁m₁+β₂ λ²₁ ϕ₁

, (7)

whereϕ0 is a biharmonic function ∆∆ϕ0= 0 and ∆ϕ0=ϕ, ∆ϕ= 0.

So it remains to study the problem of finding the functions p_j(x), j= 1,2.

We consider only the interior boundary value problems. The exterior one can be treated quite similarly.

The basic BVPs in the full coupled linear equilibrium theory of elasticity for materials with double porosity are formulated as follows.

The Dirichlet type BVP problem

Find a regular solution U(u, p₁, p₂) to systems (1) and (2) for x ∈ D satis- fying the following boundary conditions:

u(z) =f(z), p₁(z) =f₃(z), p₂(z) =f₄(z), z∈S. (8) The Neumann type BVP problem

Find a regular solution U(u, p₁, p₂) to systems (1) and (2) for x ∈ D satis- fying the following boundary conditions

P ∂

∂z,n

U(z)=f(z), ∂

∂np1(z) =f3(z), ∂

∂np2(z) =f4(z), z∈S, (9) where f = (f1, f2) and fj(z), j = 3,4, are known functions, n(z) is the external unit normal vector onS at zandP

∂

∂x,n

Uis the stress vector in the considered theory

P ∂

∂x,n

U=T ∂

∂x,n

u−n(β1p1+β2p2), (10)

T ∂

∂x,n

u is the stress vector in the classical theory of elasticity,

T ∂

∂x,n

u(x)=µ ∂

∂nu(x)+λndivu(x)+µ

2

X

i=1

ni(x)gradui(x).

3. The uniqueness theorems

In this section we investigate the question of the uniqueness of solutions of the above-mentioned problems.

LetU(u, p1, p2) be a regular solution of equations (1) and (2) inD. Multiply the equation (1) byu, the first equation of (2) byp₁ and the second byp₂.Integrating overDand summing the results, we arrive at Green’s formulas

R

D

[E(u,u)−(β₁p₁+β₂p₂)divu]dx=R

S

uP(∂_y,n)Ud_yS,

Z

D

{γ(p₁−p₂)²+ (k₁₂+k₂₁)(gradp₁·gradp₂)}dx

+ Z

D

k1(gradp1)²+k2(gradp2)² dx= Z

S

pP⁽¹⁾(∂y,n)pdyS,

where

E(u,u) = (λ+µ)(divu)²+µ ∂u₁

∂x₁ −∂u₂

∂x₂ 2

+µ ∂u₂

∂x₁ +∂u₁

∂x₂ 2

,

P⁽¹⁾(∂_x,n)p=





k1 k12

k21 k2





∂p

∂n, p= (p₁, p₂).

Now we prove the following theorems:

Theorem 3.1 : The Dirichlet boundary value problem has at most one regular solution in the domainD.

Proof : Let the Dirichlet BVP have in the domainDtwo regular solutionsU⁽¹⁾(x) andU⁽²⁾(x). Denote U=U⁽¹⁾−U⁽²⁾. Evidently the vector Usatisfies equations (1),(2) and the boundary condition U(z) = 0 on S. Note that if U is a regular solution of equations (1),(2), we have the Green formula and taking into account the fact that the potential energy is positively definite, we conclude thatU(x) =C, for x ∈ D, where C = const. Since U(z) = 0, z ∈ S, we have C = 0 and

U(x) = 0 at every point x∈D.

Theorem 3.2 : Two regular solutions of the Neumann boundary value problem may differ by the vector(u, p1, p2),whereu is a sum of a rigid displacement vector andc₁x,and p₁ =p₂ =c=const.

Proof : Let

P(∂_z,n)U(z) = 0, ∂

∂np₁(z) = 0, ∂

∂np₂(z) = 0, z∈S.

Then applying Green’s Formulas to a regular solution and taking into account the positive definiteness of the potential energy we have

u₁ =α₁−εx₂+c₁x₁, u₂ =α₂+εx₁+c₁x₂, p₁=p₂ =c, f or x∈D, where

c₁ = c(β₁+β₂) 2(λ+µ)

and ε, α₁, α₂, c are arbitrary constants.

4. Explicit solution of the Dirichlet BVP

A solution of the system (2) with boundary conditions p1(z) = f3(z), p2(z) = f₄(z), z ∈ S is sought in the form (3), where the functions ϕ and ϕ₁ are unknown in the circleD.On the basis of boundary conditions we reformulate the problem in question as follows

ϕ(z) =h(z), ϕ₁(z) =h₁(z), z∈S, (11)

where

h= 1 k0

[(k₁+k₂₁)f₃+ (k₂+k₁₂)f₄], h₁ = 1

k₀(k₁+k₂₁)(f₄−f₃), k₀ 6= 0.

(12)

The functionsh(z) and h1 in (12) can be represented in Fourier series. Obviously the functionϕis solution of the equation ∆ϕ= 0 and it is represented in the form of the following series [30]

ϕ(x) =

∞

X

k=0

ρ R

k

(Y_k·ν_k(ψ)), (13)

where

x= (ρ, ψ), ρ²=x²₁+x²₂, Y_k = (A_k, B_k),

νk= (coskψ,sinkψ), Y0 = (A0,0), A0 = 1 2π

2π

Z

0

h(θ)dθ,

A_k = 1 π

2π

Z

0

h(θ) coskθdθ, B_k= 1 π

2π

Z

0

h(θ) sinkθdθ.

The metaharmonic function ϕ1(x) in the circleD can be written as follows [29]

ϕ1(x) =I0(λ0ρ)C0+

∞

X

k=1

Ik(λ0ρ)(Zk·νk(ψ)) (14)

where I_k(λ₀ρ) is the Bessel function of an imaginary argument, Z_k = (C_k, D_k), C0, Ck, Dk are the unknown quantities. Keeping in mind (14) and boundary conditions (11) we obtain the values ofC_k and D_k

C0 = 1 2πI0(λ0R)

2π

Z

0

h1(θ)dθ, Ck = 1 πI_k(λ0R)

2π

Z

0

h1(θ) coskθdθ, (15)

Dk= 1 πI_k(λ0R)

2π

Z

0

h1(θ) sinkθdθ.

If we substitute the values of ϕand ϕ1 into (3), we find the functionsp1(x) and p₂(x) in D.

A solutionv(x) = (v₁, v₂) of homogeneous equation (6) is sought in the form [17]

v₁(x) = ∂

∂x₁[Φ₁+ Φ₂]−∂Φ3

∂x₂, v2(x) = ∂

∂x2

[Φ1+ Φ2] + ∂Φ3

∂x1

,

(16)

where Φ₁, Φ₂ and Φ₃ are scalar functions,

∆Φ₁ = 0, ∆∆Φ₂ = 0, ∆∆Φ₃ = 0, (λ+ 2µ) ∂

∂x₁∆Φ₂−µ ∂

∂x₂∆Φ₃ = 0, (λ+ 2µ) ∂

∂x2

∆Φ2+µ ∂

∂x1

∆Φ3 = 0.

(17)

Taking into account (5) and boundary conditions (8), we can write

v(z) =Ψ(z), z∈S, (18)

whereΨ(z) =f(z)−v0(z) is the known vector; ϕ(z) and ϕ1(z) are defined by equalities (11). On the basis of equation ∆ϕ₀ =ϕ the functionϕ₀ is represented in the following form

ϕ0(x) = R² 4

∞

X

k=0

1 k+ 1

ρ R

k+2

(Yk·νk(ψ)), (19)

whereY_k is defined by (13).

In view of (17) we can represent the harmonic function Φ₁,biharmonic functions Φ2 and Φ3 in the form

Φ₁ =

∞

P

k=0

ρ R

k

(X_k1·ν_k(ψ)),

Φ2 =

∞

P

k=0

ρ R

k+2

(X_k2·ν_k(ψ)),

Φ₃ = λ+ 2µ µ

∞

X

k=0

ρ R

k+2

(X_k2·s_k(ψ)),

(20)

where X_ki = (X_ki1, X_ki2), k = 1,2 are the unknown two-component vectors, ν_k= (coskψ,sinkψ), s_k= (−sinkψ,coskψ).

Using the formulas

∂

∂x₁ =n1

∂

∂ρ −n2

ρ

∂

∂ψ, ∂

∂x₂ =n2

∂

∂ρ+ n1

ρ

∂

∂ψ,

let us rewrite the boundary conditions (18) in the form

vn(z) = Ψn(z), vs(z) = Ψs(z), z∈S, (21) where v_n and Ψ_n(z) are the normal components of the vectors v = (v₁, v₂) and Ψ = (Ψ1,Ψ2) respectively; vs and Ψs(z) are the tangent components of the vectors v = (v₁, v₂) and Ψ = (Ψ₁,Ψ₂) respectively. Substituting the equalities (16),(20) into (21), we get

vn= ∂

∂ρ(Φ1+ Φ2)−1 ρ

∂

∂ψΦ3, v_s= 1

ρ

∂

∂ψ(Φ₁+ Φ₂) + ∂

∂ρΦ₃,

Ψ_n=n₁Ψ₁+n₂Ψ₂, Ψ_s=−n₂Ψ₁+n₁Ψ₂, n= (n1, n2), s= (−n₂, n1), n1= x1

ρ , n2 = x2

ρ .

(22)

Let us expand the functions Ψ_n and Ψ_s in Fourier series, those Fourier coef- ficients areγk and δk,respectively

γ₀= (γ₀₁,0), γ_k= (γ_k1, γ_k2), δ₀ = (δ₀₁,0), δ_k= (δ_k1, δ_k2),

γ₀₁= 1 2π

2π

Z

0

Ψ_n(θ)dθ, δ₀₁= 1 2π

2π

Z

0

Ψ_s(θ)dθ,

γ_k1 = 1 π

2π

Z

0

Ψ_n(θ) coskθdθ, δ_k1= 1 π

2π

Z

0

Ψ_s(θ) coskθdθ,

γk2 = 1 π

2π

Z

0

Ψs(θ) sinkθdθ, δk2 = 1 π

2π

Z

0

Ψn(θ) sinkθdθ.

(23)

If we substitute (22) into (21), then passing to limit as ρ −→ R, for determining the unknown values we obtain the following system of algebraic equations

2

RX01i = γ_0i

2 , 2(λ+ 2µ)

µR X02i = δ_0i 2 , k

RXk1i+k(λ+ 3µ)

µR Xk2i =γki, i= 1,2, k= 1,2, ..., k

RX_k1i+k(λ+ 3µ) + 2(λ+ 2µ)

µR X_k2i=δ_ki.

(24)

From (24) we find

X_01i = γ_0iR

4 , X_02i = δ_0iRµ 4(λ+ 2µ), X_k1i = γ_kiR

k −(δ_ki−γ_ki)R

2k(λ+µ) [(λ+ 3µ)k+ 2µ], Xk2i =µ(δki−γki)R

2(λ+µ) , i= 1,2, k= 1,2, ...,

Thus the solution of the Dirichlet boundary problem is represented by the sum (5) in whichv(x) is defined by means of formula (16),v0(x) by formula (7),ϕ0(x) by formula (19) andϕ1(x) by formulas (14) and (5).

It can be proved that if the functions f and f_j, j = 3,4 satisfy the following conditions onS

f∈C³(S), fj ∈C³(S), j= 3,4,

then the resulting series are absolutely and uniformly convergent.

5. Explicit solution of the Neumann BVP

We sought the solution of the Neumann BVP in the form (3), where the functions ϕand ϕ₁ are unknown in the circle D.Taking into account formulas (3) and (9), the boundary conditions can be rewritten as

∂ϕ(z)

∂R =h(z), ∂ϕ1(z)

∂R =h₁(z), z∈S. (25) h(z) and h₁(z) are given by (12), where f₃ = ∂p₁

∂R, f₄= ∂p₂

∂R, Z

S

h(y)d_yS= 0.

Thus, for the unknown harmonic function ϕ we obtain the Neumann problem, the solution that is represented in the form of series [30]

ϕ(x) =c0+

∞

X

k=1

R k

ρ R

k

(Yk·νk(ψ)), (26)

wherec0 is an arbitrary constant; Yk = (Ak, Bk),

Ak= 1 π

2π

Z

0

h(θ) coskθdθ, Bk= 1 π

2π

Z

0

h(θ) sinkθdθ.

The metaharmonic functionϕ1(x) in the circle D can be written as (14), where Z_k = (C_k, D_k); C₀, C_k, D_k are the unknown quantities. Keeping in mind (12)

and boundary conditions (25), we obtain the values of C₀, C_k and D_k

C0 = 1

2πλ₀I₀⁰(λ₀R)

2π

Z

0

h1(θ)dθ, C_k = 1 πλ₀I_k⁰(λ₀R)

2π

Z

0

h1(θ) coskθdθ, (27)

D_k= 1 πλ₀I_k⁰(λ₀R)

2π

Z

0

h1(θ) sinkθdθ,

where

I_k⁰(ξ) = ∂I_k(ξ)

∂ξ , ∂I_k(λ₀ρ)

∂ρ =λ₀I_k⁰(λ₀ρ) I_k⁰(λ₀R)6= 0, k= 0,1,2, ....

Considering equality (5) and (10), the boundary condition (9), for T(∂_z,n)v(z) can be rewritten as

T(∂_z,n)v(z)(z) =Ω(z), z∈S, (28) where

Ω( z) =f(z) +n(z)[aϕ(z) +bϕ1(z)]−T(∂z,n)v0(z)

is the known vector,Ω= (Ω₁,Ω₂);ϕis defined by (26) andϕ₁ - formulas (14) and (27); a=β1+β2, b=m1β1+β2.

Let us rewrite the boundary conditions (28) in the form

[T(∂z,n)v(z)]_n= Ωn(z), [T(∂z,n)v(z)]_s= Ωs(z), (29) where [T(∂z,n)v(z)]_n and Ωn(z) are the normal components of the vectors T(∂_z,n)textbf vandΩ(z) respectively; [T(∂_z,n)v(z)]_s and Ω_s(z) are the tangent components of the vectorsT(∂_z,n)v(z)) and Ω(z) respectively;

[T(∂_z,n)v(z)]_n= (λ+µ)

∂vn(z)

∂ρ

ρ=R

+ λ R

∂vs(z)

∂ψ , [T(∂z,n)v(z)]_s=µ

∂vs(z)

∂ρ

ρ=R

+ µ R

∂vn(z)

∂ψ ; Ωn(z) =fn(z) +aϕ(z) +bϕ1(z)−[T(∂z,n)v0(z)]_n, Ωs(z) =fs(z)−[T(∂z,n)v0(z)]_s, z∈S.

(30)

v₀ is defined by means of formula (7), where function ϕ₀(x) is the solution of

equation ∆ϕ₀=ϕand is represented in the form [17]

ϕ₀(x) =m₀+R³ 4

∞

X

k=1

1 k(k+ 1)

ρ R

k+2

(Y_k·ν_k(ψ)),

Y_k= (A_k, B_k), A_k and B_k are defined by (26); m0 is an arbitrary constant.

Let us expand the functions Ω_n and Ω_s in Fourier series, those Fourier co- efficients areγ₀ = (γ01,0), γ_k= (γ_k1, γ_k2) and δ0 = (δ01,0), δ_k= (δ_k1, δ_k2), respectively.

Taking into account the formulas (22),(20) and (30), then passing to limit as ρ −→ R, for determining the unknown values we obtain the following system of algebraic equations.

According to uniqueness theorem, we assume that the determinant of the system is not zero. The solution of the system has

X_01i= γ0iR²

4(λ+ 2µ), X_02i = δ0iR² 4(λ+ 2µ), X_k1i = R²

a3

δ_ki− a₄R² a2a3−a1a4

(µγ_ki−a₁δ_ki),

X_k2i = a₃R² a2a3−a1a4

(µγ_ki−a₁δ_ki), where

a₁=µk[2(λ+µ)k−(λ+ 2µ)],

a₂= 2(λ+µ)(λ+ 3µ)k²+ (λ+ 2µ)[(3λ+ 5µ)k+ 2µ], a₃=µk(2k−1), a₄ = (λ+ 3µ)k(2k+ 3) + 2(λ+ 2µ).

We assume that the functionsfandf_j, j = 3,4 satisfies the following conditions onS

f∈C²(S), f_j ∈C²(S), j= 3,4.

Under these conditions the resulting series are absolutely and uniformly convergent.

6. Conclusions

1. The main purpose of this work has been to present some explicit solutions of BVPs in the fully coupled theory of elasticity for solids with double porosity.

Solutions of the considered boundary value problems are obtained in the form of absolutely and uniformly convergent series that is useful to obtain numerical solutions of the boundary value problems. The Green’s formulas are obtained.The uniqueness theorems of the BVPs are proved. The solutions are sought by means

of harmonic, biharmonic and metaharmonic functions, which properties are well known in mathematical physics.

2. The obtained results may be of practical use in micro and nanomechanics, mechanics of materials, engineering mechanics, engineering medicine, biomechan- ics, engineering geology, geomechanics, applied and computing mechanics, in the applied mathematics.

3. Using the above mentioned method gives an opportunity to research the wide class of problems for the systems of equations in the modern linear theories of elasticity, thermoelasticity and poroelasticity for materials with microstructures and construct explicitly the solutions of basic BVPs for a circle, sphere and etc., in a complete version.

References

[1] S.C. Cowin,Bone poroelasticity, Journal of Biomechanics,32(1999), 217-238

[2] R.K. Wilson and Aifantis E.C.,On the theory of consolidation with double porosity-I, International Journal of Engineering Science,20(1982), 1009-1035

[3] D.E. Beskos and E.C. Aifantis,On the theory of consolidation with double porosity-II,International Journal of Engineering Science,24(1986), 1697-1716

[4] M.Y. Khaled, D.E. Beskos and E.C. Aifantis,On the theory of consolidation with double porosity-III, International Journal for Numerical and Analytical Methods in Geomechanics,8, 2 (1984), 101-123 [5] M.A. Biot,General theory of three-dimensional consolidation, J. Appl. Phys.,12(1941), 155-164 [6] G.I. Barenblatt, Yu.P. Zheltov and I.N. Kochina,Basic concepts in theory of seepage of homogeneous

liquids in fissured rocks Russian), Priklad. Mat. Mekh.,24, 5 (1960), 852-864

[7] R. De Boer,Theory of Porous Media: Highlights in the historical development and current state, Springer, Berlin-Heidelberg- New York, 2000

[8] N. Khalili, S. Valliappan,Unified theory of flow and deformation in double porous media, European Journal of Mechanics, A/Solids,15(1996), 321-336

[9] N. Khalili,Coupling effect in double porosity media with deformable matrix, Geophysical Research Letters,30(2003), 21-53

[10] N. Khalili and P.S. Selvadurai,A full coupled constitutive model for thermo-hydro-mechanical analysis in elastic media with double porosity, Geophysical Research Letters,30, SDE 7-1-7-3, (2003).

[11] N. Khalili and P.S. Selvadurai,On the constitutive modelling of thermo-hydro-mechanical coupling in elastic media with double porosity, Elsevier Geo-Engineering Book Series,2(2004), 559-564 [12] J.G. Berryman and H.F. Wang,Elastic wave propagation and attenuation in a double porosity dual-

permiability medium, International Journal of Rock Mechanics and Mining Sciences,37(2000), 63-78 [13] M. Svanadze and S. De Cicco,Fundamental solutions in the full coupled theory of elasticity for solids

with double porosity, Arch. Mech.,65, 5 (2013), 367-390

[14] M. Svanadze,Fundamental solutions in the Linear Theory of Consolidation for Elastic Solids with Double Porosity, Journal of Mathematical Sciences,195(2013), 258-268

[15] B. Straughan,Stability and uniqueness in double porosity elasticity, Int. J. of Engineering Science,6 (2013), 1-8

[16] I. Tsagareli and M.M. Svanadze,Explicit solution of the boundary value problems of the theory of elasticity for solids with double porosity, PAMM -Proc. Appl. Math. Mech.,10(2010), 337-338 [17] I. Tsagareli and M.M. Svanadze,Explicit solution of the problems of elastostatics for an elastic circle

with double porosity, Elsevier, Mechanics Research Communications,46(2012), 76-80

[18] L. Bitsadze and I. Tsagareli,The solution of the Dirichlet BVP in the fully coupled theory for spherical layer with double porosity, Meccanica, DOI: 10.1007/s11012-015-0312-z, 2015

[19] L. Bitsadze and I. Tsagareli, Solutions of BVPs in the fully Coupled Theory of Elasticity for the Space with Double Porosity and Spherical Cavity, Mathematical Methods in the Applied Science, DOI: 10.1002/mma.3629, 2015

[20] I. Tsagareli and L. Bitsadze,Explicit Solution of one Boundary Value Problem in the Full Coupled Theory of Elasticity for Solids with Double Porosity, Springer-Verlag Wien, Acta Mechanica,226, 5 (2015), 1409-1418

[21] I. Tsagareli and L. Bitsadze,Solutions of BVPs in the fully Coupled Theory of Elasticity for a Sphere with Double Porosity, Bulletin of TICMI,19, 1 (2015), 26-36

[22] L. Bitsadze, Fundamental solution in the fully coupled theory of elasticity for solids with double porosity, Seminar of I.Vekua Institute of Applied Mathematics, Reports,41(2015), 21-30

[23] L. Bitsadze,Fundamental solution in the theory of poroelasticity of steady vibrations for solids with double porosity, Proc. of I. Vekua Inst. of Appl. Math.,64(2014), 3-12

[24] L. Bitsadze,The boundary value problems of the fully coupled theory of elasticity for solids with double porosity for Half-plane, Seminar of I. Vekua Institute of Applied Mathematics, Reports,41 (2015), 10-20

[25] M. Basheleishvili and L. Bitsadze,Explicit solution of the BVP of the theory of consolidation with double porosity for half-plane, Georgian Mathematical Journal,19, 1 (2012), 41-49.

[26] M. Basheleishvili and L. Bitsadze,Explicit solutions of the BVPs of the theory of consolidation with double porosity for the half-space, Bulletin of TICMI,14(2010), 9-15

[27] E. Scarpetta,On the fundamental solutions in micropolare elasticity with voids, Acta Mechanica,82 (1990), 151-158

[28] W. Nowacki,Theory of Elasticity (Russian), Moscow, Mir, 1975

[29] I. Vekua,On metaharmonic functions(Russian), Tr. Mat. Inst. Akad. Nauk GSSR,12(1943), 105-174 [30] S. Mikchlin,A Course of Mathematical Physic(Russian), Moscow, Nauka, 1968