Qualitative features and discontinuous solutions of ideal plasticity equations

A homotopy of solutions of the well-known Prandtl and Nadai solutions is constructed, i. e., a continuous transformation of one solution into another. This transformation occurs under the influence of a group of continuous transformations allowed by the system of ideal plasticity. In this case, it is possible to observe the evolution of the characteristics of the system, which is determined by the group parameter A. For a = 1, the characteristics of the Prandtl solution are obtained. For a = 0, these are the characteristics of the Nadai solution. At a ≈ 0.47, the characteristics of the first family begin to overlap and voltage gaps occur. In the article, a stress rupture line is constructed. To find the conditions for the occurrence of discontinuities, a qualitative study of the equations of characteristics of the ideal plasticity system was carried out, which made it possible to formulate a condition sufficient for the intersection of characteristics of one family. To illustrate the formulated condition, a program was developed for constructing the characteristics of the Cauchy problem of ideal plasticity equations based on conservation laws. The program was implemented in the Maple environment. The program was tested on the exact solutions of Prandtl and Nadai, the error did not exceed 10–8. Numerical and analytical solutions of boundary value problems were constructed when a sufficient condition on the boundary was not fulfilled. It is shown that in this case the characteristics of the first family intersect and discontinuity lines appear.