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.

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 [

A theory of consolidation with double porosity structure was proposed by Wilson and Aifantis [

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 [

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.

Let

The system of homogeneous equations of the linear equilibrium theory of elasticity for solids with double porosity structure can be written as follows [

Note that BVPs for system (

Obviously, from system (

It is easy to see that the solutions of system (

First, we assume

It is well known that the general solution of (

Let us rewrite equalities (

Formula (

Formula (

Equations (

Expressions (

Equation (

Hooke’s law can be written as follows:

For systems (

Find in domain

Ellipse with double porosity.

Note that when stress components

Clearly, BVPs for (

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 on

Now let us formulate the boundary value problems for semiellipse as follows: find in domain

Semiellipse.

The solutions of (

The components of stress vector (

Let us rewrite conditions (

In (

By substituting (

Diagram of the main matrix of the algebraic equation system.

The dimension of all matrices

Thus, the system is uniquely solvable. Let us substitute the obtained values,

Find regular solutions

The solution of (

Let us assume that harmonic function

For determining

Substituting (

Quite similarly, we obtain the solution, when

When we have the nonhomogeneous boundary conditions, then the coefficients in (

By substituting

The main results of this work can be formulated as follows:

The system of equations of the linear equilibrium theory of elasticity for solids with double porosity structure is written in terms of elliptic coordinates.

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.

Analytical (exact) solutions are obtained for 2D BVPs for the ellipse with double porosity structure.

It is possible to construct explicitly the solutions of basic BVPs for systems (

It is possible to obtain numerical solutions of the boundary value problems.

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.

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.

If

Elliptic coordinate system.

The coordinate curves are ellipses and hyperbolas:

The Laplacian operator in the elliptic coordinates has the form

The operator grad has the form

The Helmholtz equation can be written as

Helmholtz equation in the elliptic coordinateshas the following form:

Let function

Let us assume that

When

When

The modification differential equation of Mathieu has the form

If

If

The double porosity model has received a lot of attention from mathematicians and from engineers. In such a model, there are two pore systems with different permeability. There is first a set of isolated porous blocks of low permeability (sometimes called matrix), surrounded by network of high permeability connected porous medium (usually called fractures network). Pores are pervasive in most of the igneous, metamorphic, and sedimentary rocks in the earth’s crust. In fact, porosity found in the earth may have many shapes and sizes, but two types of porosity are more important. One is the matrix porosity, and the other is fracture or crack porosity which may occupy very little volume, but fluid flow occurs primarily through the fracture network.

In physical terms, the theory of poroelasticity postulates that when a porous material is subjected to stress, the resulting matrix deformation leads to volumetric changes in the pores. The pores are filled with fluid. The presence of the fluid results in the flow of the pore fluid between regions of higher and lower pore pressure.

The physical process of coupled deformation is governed by the following equations:

The equations of motion [

The equations of fluid mass conservation [

The constitutive equations (extending Terzaghi’s effective stress concept to double porosity) [

Darcy’s law for material with double porosity [

Substituting (

The authors declare that they have no competing interests.