The Asymptotic Analysis of the Solution of an Elasticity Theory Problem for a Transversely Isotropic Hollow Cylinder with Mixed Boundary Conditions on the Side Surface
Magomed Farman Mekhtiyev1, Nina Ilyinichna Fomina2, Nazaket Boyukaga Mammadova2
1Department of Applied Mathematics, Institute of Mathematics and Mechanics of National Academy of Sciences of Azerbaijan, Baku, Azerbaijan
2Faculty of Applied Mathematics and Cybernetics, Baku State University, Baku, Azerbaijan
To cite this article:
Magomed Farman Mekhtiyev, Nina Ilyinichna Fomina, Nazaket Boyukaga Mammadova. The Asymptotic Analysis of the Solution of an Elasticity Theory Problem for a Transversely Isotropic Hollow Cylinder with Mixed Boundary Conditions on the Side Surface. Science Journal of Applied Mathematics and Statistics. Vol. 4, No. 6, 2016, pp. 269-275. doi: 10.11648/j.sjams.20160406.14
Received: September 23, 2016; Accepted: October7, 2016; Published:November 3, 2016
Abstract: The problem of elasticity theory for the transversely isotropic hollow cylinder with mixed conditions on the side surface is considered in the paper. Transcendental equations are obtained regarding the eigenvalues of the problem. The roots of the characteristic equations are studied thoroughly. The study of the eigenvalues allowed to establish the essential characteristics of the stress-strain state of an anisotropic shell in comparison with isotropic shells. Homogeneous solutions were built here.
Keywords: Theory of Elasticity, Transversely Isotropic Hollow Cylinder, Side Surface, Mixed Boundary Conditions, Stress-Strain State, Eigenvalues, Transcendental Equation, Anisotropic Shell
The modern theory of shells is deeply developed section of the mechanics of a deformable solid. However, the calculation of shells on the basis of three-dimensional equations of the theory of elasticity is associated with considerable mathematical difficulties. Therefore, it is necessary to apply to a variety of approximate methods to simplify the calculation of shells. Many methods of bringing the three-dimensional problem to a two-dimensional one use small shell thickness compared to its other dimensions in the constructions. Among them a special place is occupied by the asymptotic method. The asymptotic methods of integrating the equations of two-dimensional shell theory obtained the great development in A. L. Goldenveiser’s papers , . V. V. Novozhilov’s  combination of complex transformation of equations of the shell theory with the asymptotic methods is presented in K. F. Chernykh’s works , . With regard to the study of three-dimensional stress-strain state of elastic bodies the development of an asymptotic method belongs to K. Fridrix, L. Dressler , , A.L. Goldenweiser, I.I.Vorovich , , . Further development of the asymptotic method went in two directions. In the first one the solution of the elasticity problem for thin bodies is carried out by means of direct integration of elasticity equations with the help of two iterative processes. This direction is developed in the works of A.L. Goldenveiser, M.I. Huseyn-Zade, A.V. Kolos ,  and L.A. Agalovyan .
The second approach is based on the investigation of a system of uniform solutions. The above-mentioned direction was developed by I.I. Vorovich , , D.C. Aksentyan , , O.S. Malkina , , N.N. Bazarenko , , T.A. Vilenskaya , Y.A. Ustinov , ,  and M.F. Mekhtiev , .
Thus, the asymptotic method developed by A. L. Goldenweiser, I. I. Vorovich, V. B. Lidski  and their followers , , , , , , has made significant contribution to the development of the theory of plates and shells.
2. Statement of the Problem and Its Solution
Let the cylinder occupies a volume
The equilibrium equations in displacements are in the form :
is the radius of the middle surface of the shell,
are dimensionless quantities, are isotropic material constants, are material constant in a plane perpendicular to the plane of isotropy.
The relations of the generalized Hooke's law are :
Suppose that the boundary conditions are given at the cylinder side surface
The nature of the boundary conditions at the ends of the cylinder is not defined yet, but we assume them so that the cylinder is in equilibrium state.
The solution (1), (3) will be sought in the form of:
where the function is subjected to the condition:
Substituting (4) into (1) to (5), we obtain the following boundary value problem
The general solution of (6) has the form of:
Here , the functions are linearly independent solutions of the Bessel equation, are the arbitrary constants.
are the roots of a quadratic
By satisfying the homogeneous boundary conditions (7), we obtain the characteristic equation
The transcendental equation (10) defines a countable set of roots , and the corresponding constants are proportional to the cofactors of some row of the determinant of the system. Choosing the cofactors of the elements of the first row as a system of solutions, the solutions to system (1) can be written as:
where are arbitrary constants.
3. The Asymptotic Analysis of the Problem
The left side of equation (1.10) as an entire function of the parameter has a countable set of zeros with the accumulation point at infinity. For effective study of its zeros we’ll assume that the shell is thin-walled.
Let us assume that
We believe that is a small parameter . Substituting (12) into (10) we obtain
Equation (13) has one restricted root. From (11) we find that this corresponds to the root of the following decision:
is an arbitrary constant.
Stress state corresponding to zero is equivalent to the principal vector of stresses directed along the axis of the cylinder.
Let us prove that the characteristic equation (13) at does not have any other restricted roots. For this purpose, we expand in a series in and confine ourselves to the first terms of the expansion. We get
This shows that the characteristic equation has no other restricted roots besides . Thus, all the roots of the characteristic equation tend to infinity as .
In principle, there could be the following limiting cases:
1) at ; 2) at ;
3) at .
As in , we can prove that the cases 1 and 2 are not feasible. In the third case, we seek in the form:
As in the , the following cases are possible here:
2) The roots of the characteristic equation (9) are multiple.
In cases 1 and 2, after substituting (17) into (9) and its transformation using a series expansion in we get:
With regard to the cases 3 and 4, their results are obtained from cases 1 and 2 by a formal replacement of into , and of into . These equations coincide with the equations determining the performance of Saint-Venant’s edge effects in an anisotropic elasticity theory for a layer.
The table 1 shows the values of the coefficients for some materials:
4. Asimptotic Analysis of Stress-Strain State
We now present the first terms of the asymptotic expansions of solutions, co-responding to different groups of roots. For displacements and stresses, in the first approximation, we get two classes of solutions, the first of which corresponds to the zeros
and the second – to the zeros of the function
respectively, we have
Expressions for are obtained from the formulas (21), (22) by the replacement of into and of into - respectively.
Expressions for are obtained from (23) by simply replacing ; are arbitrary constants.
We note that the solution (23) is characteristic only for anisotropic shells. It disappears completely in the transition to an isotropic shell . With regard to the solutions (21) and (22) when they merge into one, and this solution coincides with the Saint Venant’s solution for an isotropic plate.
In  a generalized condition of orthogonality of homogeneous solutions for the transverse isotropic hollow cylinder is proved, which allows to accurately satisfy the boundary conditions at the ends on special conditions of the shell edge bearing.
With the help of generalized orthogonality conditions, we consider the following problem: let the condition (3) satisfy on the side surface of the cylinder and the following boundary conditions be defined at the ends:
is the dimensionless height of the cylinder.
According to (21) can be written as
The summation in the series (24) is taken by the roots located in the upper half-plane . In view of the relations of a generalized orthogonality, the desired constants have the form:
In general, the boundary value problem is reduced to solving systems of linear infinite algebraic equations using Lagrange variational principle.
The main results obtained in the article, the following:
1) There are obtained simple asymptotic formulas allowing to find strain-deformed state of cylindrical shell with given precision;
2) There is distinguished a class of solution (23) which is characteristic only for anisotropic shells and totally disappear on passage to isotropic case;
3) It is shown that stress-strain state of a cylindrical shell is a sum of interior stress-strain state and countable set of boundary-layer solutions which is localized near the shell edge;
4) For boundary-layer solutions totally coincide with Saint-Venan solution for anisotropic plate.
One of our authers (Mekhtiyev M.F.) devoted two monographs to the elaboration of asymptotic method of integrating the equations of anisotropic theory of elasticity for plates and shells of variable thickness , .