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A variational method of homogeneous solutions for axisymmetric elasticity problems for semiinfinite and finite cylinders with loaded end faces and free lateral surface has been developed. As examples of application of the proposed approach the problem of bending of the thick disk by concentrated forces applied to its end faces have been considered.

Mathematical models and methods for determination of axisymmetric residual stresses in a finite cylinder are considered. The model of residual stresses is built using the conception of incompatible eigenstrain tensor. Within the frame of this model, a direct problem for residual stresses determination is formulated. A method based on the variational method of homogeneous solutions is developed for solving the direct problem. Using the obtained solution, features of residual stresses, caused by continuous and piece-wise homogeneous distributions of eigenstrain components are studied. A variational formulation of the inverse problem for residual stresses determination on the base of empirical data obtained by a photoelasticity method is suggested. The inverse problem is solved numerically with the use of iterative calculations of values of the criterion functional. The results presented in the paper can be used for the development of methods and means for nondestructive testing and engineering characterization of materials and structural elements.

An axially symmetric problem for a hollow cylinder with unloaded bases is considered. On the inner and outer cylindrical surfaces, the normal and tangential loads are prescribed. The problem is reduced to a biharmonic equation with corresponding boundary conditions. Application of the method of variables separation results in a homogeneous boundary value problem for the ordinary differential equation. Its eigenfunctions have been used to construct an infinite system of homogeneous solutions for the initial biharmonic problem. Its solution, represented as a series expansion in terms of homogeneous solutions, depends on four infinite sequences of real constants. To determine them, the variational method has been applied, in which the subordination of the solution to the boundary conditions, given on cylindrical surfaces, is performed in the norm L 2. It brings to an infinite system of algebraic equations which has been solved by the reduction method. The quantitative studies have confirmed the good convergence of the method.

We develop a variational method of homogeneous solutions for the solution of axisymmetric elasticity problems for a semiinfinite cylinder with free lateral surface. We consider four types of boundary conditions imposed on the end of cylinder, namely, the conditions in stresses, in displacements, and two types of mixed conditions. The solution of the problems with the help of this method is reduced to the solution of infinite systems of linear algebraic equations. We perform the numerical analysis of convergence of the obtained solutions. We also consider an example of application of the proposed approach to the determination of stress concentration near the joint of the end of the cylinder with a perfectly rigid lateral surface.The problems of determination of the stress-strain state of cylindrical bodies appear in various scientific disciplines and, in particular, in the applied theories of the strength of solids [9]. Despite significant advances in the development of the methods aimed at the numerical analysis of the problems of mechanics and a wide choice of certified highly efficient software environments intended for the solution of these problems with the help of the finite-element method [12], the interest in analytic methods still remains fairly high. This can be confirmed, in particular, by the appearance of new scientific publications (see, e.g., [10,11,[13][14][15]) and explained by several factors. One of these factors is connected with the fact that the analytic solution of the direct problem enables one to significantly simplify the process of solution of the corresponding inverse problems.We especially mention the following two well-known approaches to the analytic solution of axisymmetric elasticity problems for a circular cylinder: the method of superposition [2, 10] and the method of homogeneous solutions [3,5].The first method is based on the representation of solutions in the form of a sum of two components: one of which guarantees the validity of boundary conditions on the end face and the other component satisfies the conditions imposed on the lateral surface. Each of these components is represented as an expansion in a complete systems of function. Thus, the systems of hyperbolic-cylindrical and trigonometric-cylindrical functions are used for this purpose in the monograph [2]. The coefficients of these expansions are determined by subordinating the solution to the boundary conditions imposed on the end face of the cylinder and on its lateral surface. As a result, an infinite system of algebraic equations is obtained, which enables one to express the coefficients of expansions via the Fourier and Fourier-Bessel coefficients of the functions appearing in the boundary conditions.The second method is based on the representation of required solutions in the form of expansions in complete systems of functions (i.e., in the so-called homogeneous solutions). The role of systems of homogeneous solutions is played by the eigenfunctions of homogeneous axisymmetric problems of the theory of elasticity...