On the construction of stress discontinuity lines for a two-dimensional plastic region

We consider the plasticity equations in the two-dimensional case and construct stress discontinuity lines in this paper. The construction of stress rupture lines has a fact: the rupture lines are located at the intersection point of lines of the same family (characteristics) and are directed along the angle bisector formed by these characteristics. To find these lines, we have constructed characteristics. Such a task is easier to solve in the case of plastic torsion, at that moment there is only one characteristic, and it is directed along the normal to the outer contour, and it is quite simple to find the sliding lines and their intersection points. Most of the works devoted to the construction of stress rupture lines solve the problem of plastic torsion for isotropic and anisotropic media. For problems of plane deformation of plastic material, this method is not sufficiently developed. This is the complexity of constructing sliding lines for such tasks and the presence of two families of sliding lines. A homotopy of two known exact solutions is constructed: Prandtl and Nadai, that is, a continuous transformation of one solution into another in this article. We obtain the characteristics of the Prandtl solution at a=1. We obtain the characteristics of Nadai's solution at a=0. The characteristics of one family begin to intersect and stress discontinuity lines appear at a = 0,5. These lines are constructed in this work.