Elastic-plastic problem in the case of inhomogeneous plasticity under complex shear conditions

In this research, the authors solved a two-dimensional elastic-plastic problem of the stress state under complex shear conditions in the body weakened by a hole that is bounded by a piecewise smooth contour. The stress state of a complex shear occurs in a cylindrical body of infinite length under the action of loads directed along the cylinder generators and constant along the generators. At the same time, with a sufficiently large load, both elastic and plastic zones appear in the body. As in any problem of this kind, it is necessary to find a previously unknown boundary separating the elastic and plastic zones. Finding such a boundary is not an easy task, but the specificity of elastic-plastic problems of complex shear is that solving such problems is easier than solving similar elastic problems. Apparently, for the first time this fact was noted by G. P. Cherepanov. A lot of research is devoted to elastic-plastic problems of complex shear in the case of homogeneous and isotropic plasticity. All articles that solve complex shear problems essentially use the representation of stresses and displacements in the elastic zone in a complex form. In this research, the problems of complex shear are solved using conservation laws. It is assumed that the yield strength is a function of the coordinates of the point where the stress state is being studied. It is known that the elastic properties of structural materials can be homogeneous and isotropic, while their yield point and strength are inhomogeneous. This situation is observed, for example, in the case of neutron bombardment of structural materials. This research will examine exactly this situation. The article presents conservation laws for equations describing a complex shear. It was assumed that the components of the conserved current depend on the components of the stress tensor and coordinates. The components of the stress tensor are included in them linearly. The problem of finding the components of the conserved current was reduced to the Cauchy-Riemann system. The solution of this system allowed us to reduce the calculations of the stress tensor components to a curvilinear integral along the contour of the hole and thus find the boundary between the elastic and plastic areas.