variational method of homogeneous solutions

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

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 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.