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.

A model of elastic equilibrium of porous media with double porosity was constructed in the works [

It should be noted that all the papers mentioned above dealt with a classical (symmetric) medium. But we do not know of any works where problems of double porous elasticity would have been considered for a nonsymmetric elastic Cosserat medium [

In Section

Let an elastic body with double porosity occupy the domain

Then a homogeneous system of static equilibrium equations is written in the form [

Formulas that interrelate the stress and moment stress components, the displacement and rotation vector components, and the pressures

As is well known, for the internal energy to be positive it is necessary that the following conditions be fulfilled [

In the stationary case, the values

The three-dimensional system of (

If we add boundary conditions on the boundary

The following lemma is easy to prove.

If

Adding the first equation of system (

Let us write system (

Since the transformation determinant is defined as

If on the boundary

This corollary follows from the fact that the homogeneous Helmholtz equation (

From the basic three-dimensional equations we obtain the basic equations for the case of plane deformation. Let

As follows from formulas (

Equations (

If relations (

On the plane

To write system (

In this section, we construct the analogues of the Kolosov-Muskhelishvili formulas [

Equations (

The general solution of the system of (

We take the operator

A conjugate equation to (

Substituting formulas (

From formulas (

Substituting the latter formula into (

Thus, if the solution of system (

Substituting expressions (

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

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

Let a porous elastic body with double porosity occupy the domain

The considered circular ring.

We consider the following problem:

Since

Using the boundary conditions (

Let us now satisfy the boundary conditions (

The analytic functions

The metaharmonic function

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.

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.

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

The author declares that there is no conflict of interests.

The designated project has been fulfilled by a financial support of Shota Rustaveli National Science Foundation (Grant SRNSF/FR/358/5-109/14).