The New Solution in Stresses of the Problem for a Thick-Wall Pipe
It has been established that the derivation of the solution of the Lame problem for a thick-walled cylinder is the unjustified extraction of the necessary formulas from a system of equations without proper verification of the remaining ones. The biharmonic equation for the stress functions for the Cartesian and polar coordinate systems for solving two-dimensional boundary problems of elasticity theory allows reducing order to a harmonic equation. In the article, this fact was first established both for the biharmonic equation in Cartesian coordinates, and during the transition to the polar coordinates. The path through Cartesian coordinates is justified by the fact that the traditionally biharmonic equation in polar coordinates for plane problems of the theory of elasticity is derived by changing the coordinates from the biharmonic equation in Cartesian coordinates. Problems in stresses for a circle, a plane with a hole, and a pipe with uniformly distributed stresses at the boundary were solved. From an applied point of view, it was established that solving a problem for a plane with a hole and uniform stresses applied to its edge is equivalent to solving a problem for a pipe with a zero radial component of the stresses at the outer boundary of the pipe. From the point of view of the theory of elasticity, the destruction of a thick-walled pipe must begin not from the inside, but from the outside, and the observed in practice destruction of pipes from the inside is a consequence of corrosion.