Solution to the first boundary value problem of plane elasticity theory using conservation laws

A huge number of works are devoted to solving boundary value problems for the equations of plane elasticity theory. The largest number of studies in this area are based on the formula found by G. V. Kolosov. He was the first to express the general solution to the problem of plane elastic deforma-tion by finding two independent functions of a complex variable. This made it possible to apply a welldeveloped theory of analytic functions to solving problems of elasticity theory. Later, the solution method based on Kolosov's formula was developed by his student N. I. Muskhelishvili. But the described method also has significant limitations. It is applicable only to those areas that can be conformally mapped onto a circle. Therefore, other methods for solving elasticity theory problems are also needed, since a large number of practically important problems are solved for areas that do not satisfy this condition. The method developed in the work is based on the use of conservation laws that are constructed for equations describing a plane deformable state. The assumptions made in the work make it possible to construct a solution to the first boundary value problem for arbitrary plane areas bounded by a piecewise smooth contour. In this case, finding the components of the stress tensor is reduced to calculating contour integrals along the boundary of the region under consideration. As in the case considered by G. V. Kolosov, the solution to the problem is based on two exact solutions of the Cauchy – Riemann equations, which have singularities at an arbitrary point in the region under consideration.