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

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.


Introduction
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 [1], [2]. V. V. Novozhilov's [3] combination of complex transformation of equations of the shell theory with the asymptotic methods is presented in K. F. Chernykh's works [4], [5]. 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 [6], [7], A.L. Goldenweiser, I.I.Vorovich [8], [9], [10]. 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 [11], [12] and L.A. Agalovyan [13].

Statement of the Problem and Its Solution
Let the cylinder occupies a volume The equilibrium equations in displacements are in the form [32]: 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 ( ) m ξ is subjected to the condition: Substituting (4) into (1) to (5), we obtain the following boundary value problem 1 0 The general solution of (6) has the form of: By satisfying the homogeneous boundary conditions (7), we obtain the characteristic equation where ( ) ( ) ( ) ( ) ( ) The transcendental equation (10) where n C are arbitrary constants.
As for the stresses, they can be determined by the generalized Hooke's law [33], [34].

The Asymptotic Analysis of the Problem
The left side of equation (1.10) as an entire function of the parameter µ 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 ε is a small parameter 0 µ = .
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: Stress state corresponding to zero 0 µ = is equivalent to the principal vector P of stresses directed along the axis of the cylinder.
( ) ( ) Let us prove that the characteristic equation (13)  ,  (16) This shows that the characteristic equation has no other restricted roots besides 0 µ = . Thus, all the roots of the characteristic equation tend to infinity as 0 ε → .
In principle, there could be the following limiting cases: As in [32], we can prove that the cases 1 and 2 are not feasible. In the third case, we seek n µ in the form: 2) The roots of the characteristic equation (9) are multiple.
With regard to the cases 3 and 4, their results are obtained from cases 1 and 2 by a formal replacement of 1 2 , s s into 1 2 , i s i s , and of p into i p . 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:

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 ( ) ( ) where ( ) ( ) ( ) In [32] 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: According to (21) , , , In general, the boundary value problem is reduced to solving systems of linear infinite algebraic equations using Lagrange variational principle.

Conclusion
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 0 1 G = boundary-layer solutions totally coincide with Saint-Venan solution for anisotropic plate. By the same method there were investigated various problems some of which we consider [35], [36], [37], [38], [39], [40].
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 [24], [25].