Advances in Mathematical Physics

Volume 2016 (2016), Article ID 3845362, 12 pages

http://dx.doi.org/10.1155/2016/3845362

## The Approximate Solution of Some Plane Boundary Value Problems of the Moment Theory of Elasticity

Ilia Vekua Institute of Applied Mathematics, Ivane Javakhishvili Tbilisi State University, University Street 2, 0186 Tbilisi, Georgia

Received 14 August 2015; Revised 19 January 2016; Accepted 31 January 2016

Academic Editor: Mikhail Panfilov

Copyright © 2016 Roman Janjgava. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

We consider a two-dimensional system of differential equations of the moment theory of elasticity. The general solution of this system is represented by two arbitrary harmonic functions and solution of the Helmholtz equation. Based on the general solution, an algorithm of constructing approximate solutions of boundary value problems is developed. Using the proposed method, the approximate solutions of some problems on stress concentration on the contours of holes are constructed. The values of stress concentration coefficients obtained in the case of moment elasticity and for the classical elastic medium are compared. In the final part of the paper, we construct the approximate solution of a nonlocal problem whose exact solution is already known and compare our approximate solution with the exact one. Supposedly, the proposed method makes it possible to construct approximate solutions of quite a wide class of boundary value problems.

#### 1. Introduction

An elastic medium which at every point is characterized not only by the displacement vector but also by the rotation vector is called the Cosserat medium after the Cosserat brothers who considered the deformation of such a medium for the first time as far back as 1909 [1]. The corresponding theory is often called the moment or asymmetrical theory of elasticity because its deformation and stress tensors are not symmetrical. The development of this theory along various lines attained its peak in the sixties–seventies of the last century [2–12]. Though, in subsequent years, more general models of a micropolar elastic medium were proposed and studied [13–15], the investigation of problems related to the Cosserat theory is still going on [16–36]. The popularity of the theory can possibly be explained by its physical clarity and a relative mathematical simplicity.

In the present paper, we consider the static case of plane deformation of the elastic Cosserat medium. The corresponding basic equilibrium equations are given in Section 2. For constructing approximate solutions of boundary value problems, a semianalytical method is proposed, which is based on using the representation of the general solution by solutions of simpler equations of mathematical physics.

In Section 3 we construct the general solution of the corresponding system of equations using two harmonic functions and a solution of the Helmholtz equation. As is known, the method of representation of the general solution by the well-studied functions of mathematical physics is widely applied to the construction of analytical (exact) solutions of boundary value problems, but one succeeds in obtaining analytical solutions of boundary value problems only for some particular types of boundary conditions when the considered domain has a regular configuration.

Section 4 deals with the application of the obtained general solution to the construction of approximate solutions of boundary value problems, which allows us to consider a wide class of boundary value problems without imposing special restrictions on the domain geometry. When the functions appearing in the general solution are expanded into series in Green’s functions and the boundary conditions are satisfied at the points of the domain boundary, we obtain a system of linear algebraic equations for the unknown coefficients of the series. After solving this system and again using the general representation of the solution, we can write an analytical approximate solution which satisfies the basic system of equations up to the domain boundary. The described algorithm is similar to the method of fundamental solutions [37–39] and the boundary elements method [40–44]. However it is simpler than the method of fundamental solutions for which the integral equation is considered on some contour that encircles the domain [39]. As different from the boundary elements method, the boundary does not have to be approximated by boundary elements and there is no need to use local coordinate systems [40], which also simplifies the construction of an approximate solution. But the main difference of our approach from the above-mentioned two methods is that in our case it becomes possible not to use Green’s functions at all. At the end of Section 4, the harmonic functions and the Helmholtz function in the general solution are represented as finite series obtained as a result of the separation of variables in a polar system of coordinates. Further the Cartesian coordinates of the points on the boundary are substituted in the representations of the functions whose values are given on the boundary and the obtained expressions are equated to the respective values of the boundary conditions at the same points. As a result we obtain a system of linear algebraic equations and after solving it we can easily write an approximate solution of the considered boundary value problem.

In Section 5, we construct by the proposed method the approximate solutions of the boundary value problems of stress concentration in the rectangular domains with circular holes. The coefficients of stress concentration along the hole contour are calculated and we compare the stress concentration values obtained in the case of moment elasticity and for the classical elastic medium. For the classical elastic medium we also construct the approximate solution of the nonlocal problem with the known exact solution and compare our approximate solution with the exact one.

#### 2. Basic Relations of the Plane Moment Theory of Elasticity

Let be the Cartesian coordinate system with unit vectors . Consider the plane deformed state of the moment-elastic medium which is parallel to the -plane. In that case, the displacement vector and the rotation vector at the points of the medium have the formHomogeneous equations of static equilibrium in terms of displacement and rotation components are written as [10]where are the Lamé constants; are the constants characterizing the microstructure of the considered medium; and , , and

Relations of Hooke’s law for the moment-elastic medium have the formwhere are the stress tensor components and are the moment stress tensor components. All other stress tensor and moment stress tensor components are assumed in the case of plane deformation to be equal to zero.

#### 3. The General Solution of System (2)

The general solution of system (2) can be represented by two arbitrary harmonic functions and an arbitrary solution of the Helmholtz equation.

We introduce the following notation:Using (4), the first two equations of system (2) can be written as follows:From system (5) we see that and are the self-conjugate harmonic functionsand represent them in the formwhere and are arbitrary harmonic functions and is any nonzero real constant.

*Remark*. In representations (7) only one harmonic function could have been used, but we have introduced two harmonic functions to obtain the general solution which is more convenient for our purpose.

With (7) taken into consideration, from relations (4) we have Let us substitute the second formula (8) in the third equation of system (2). After simple transformations we obtain the equationIn view of the fact that is a harmonic function, the general solution of (9) can be written in the form where is an arbitrary solution of the Helmholtz equationand is any nonzero real constant.

Using representation (10) and keeping in mind that is the solution of (11), we write system (8) as follows:The second equation (12) is satisfied identically if we takeBy substituting formulas (13) in the first equation of system (12), we obtain the Poisson equation satisfied by function : Since and are harmonic functions, we can easily write the general solution of the latter equation:where is an arbitrary harmonic function.

The substitution of expression (15) in formula (13) giveswhereObviously, and are arbitrary harmonic functions. For simplicity, we denote them by and

It is convenient to fix arbitrary constants and as follows: Then formulas (16) and (10) take the formSystem (19) is the representation of the general solution of system (2) by two arbitrary harmonic functions and and an arbitrary solution of the Helmholtz equation.

Let us now express stresses and moment stresses through functions , and . To this end, formulas (19) are substituted in relations (3) to obtain Let us assume that and are mutually normal unit vectors such thatand is the angle between vector and the positive direction of the -axis. Then, by virtue of the transformation formulas of first- and second-rank tensor components, the following formulas hold true: Substituting formulas (19) and (20) in relations (22) we can express normal and tangent displacements as well as stresses and moment stresses acting on the area element of arbitrary orientation through functions , , and

#### 4. Use of the General Solution for the Construction of Approximate Solutions of Boundary Value Problems

In this section it is shown how the general solution derived in the preceding section can be used for constructing approximate solutions of boundary value problems. The technique to be described here may be called a semianalytical method in view of the fact that approximate solutions are written in analytical form. In this sense, it is analogous to the method of fundamental solutions [37–39] and the boundary elements method [40–44], but there are differences which have been mentioned in Introduction.

So, we start the description of the algorithm of constructing the approximate solution of a boundary value problem when Green’s functions are used.

In the general solutions (19) and (20), the harmonic functions and are taken for each index ( is some natural number) in the following manner: and solution of the Helmholtz equation is represented aswhere is a modified Bessel function of second kind or a Macdonald function (of zeroth order) which tends to infinity at point and vanishes at infinity [45] and and , , are the real coefficients to be defined.

By formula (23) the first-order partial derivatives of the harmonic functions and have the formand the partial derivatives of the metaharmonic functions are written as The second derivatives of these functions appear in formulas (20):If we substitute the corresponding expressions (23)–(26) in formulas (19), then for each index we obtain solution of system (2) satisfying this system everywhere except the origin where it has a singularity:The values of stresses and moment stresses are obtained if representations (25)–(27) are substituted in the respective formulas (20):

After that, each th function of (28) and (29) is shifted by value (Figure 1). For this, in formulas (28) and (29) and also in the expressions for stresses and moment stresses variables and are replaced, respectively, by values and . Then functions , as well as also functions , have singularities at points