Introduction

We present the methods of approximate solution of the 3D basic problem of elasticity for the solids of the special types in this work. The classic formulation of the problem is the following: given the boundary displacements it should be possible to find the displacements in the whole elastic body which satisfy the equilibrium equations. This problem is named the second basic problem of elasticity [7]. There exist the well-known exact solutions of this problem in the symmetrical cases (e.g. the solids of revolution with the symmetrical stresses) [6]. The exact solution for the general case has not been found yet, so the engineers apply the approximate methods (Finite Element Method, Boundary Element Method). The application of these methods for the solids with the certain singularities (e.g. cones in the neighborhood of the vertex) or asymmetrical boundary conditions often fail to be correct.

Note that Kolosov-Muskhelishvili method based on the application of the complex variables and analytic functions yields the exact solutions for the wide range of the plane problems [7]. There were numerous attempts of the generalisation of Kolosov-Muskhelishvili method for the 3 -dimensional solids, for example, by A.F. Tsalik, A. Alexandrov and F.A. Bogashov [14, 1, 3]. But their methods imply either solution of very large systems of symbol equations ([14, 3]) or transform the original problem to the different one (which is not equivalent to the given problem in the general case [1]). We might also recall the quaternion matrix representation method developed in [2], which also implies necessity of some complicated non-commutative calculations.....