The article deals with the axisymmetric elastoplastic problem for a hollow thick-walled cylinder (plane deformed state), loaded from the inside and outside by uniform pressures proportional to one parameter. The material is considered to be perfectly plastic, with the elastic modulus and yield strength generally are arbitrary functions of the radius. In addition, the material is considered to be incompressible in both plastic and elastic zones. On the basis of the criteria for the plasticity of Huber - Mises and Tresca - Saint-Venant, the radius at which the first plastic deformations occur is determined. It is shown that, depending on the functions of the inhomogeneity of elastic and plastic parameters and loads, the occurrence of plastic deformations is possible both on the surfaces and on the inner walls of the cylinder.
