On methods for the solution of integral equations of the theory of plasticity based on the concept of slip

We have proposed a modification of the methods for solving the system of integral equations [M. Ya. Leonov and N. Yu. Shvaiko, “Complex plane deformation,” Dokl. Akad. Nauk SSSR, 159, No. 2, 1007–1010 (1964); N. Yu. Shvaiko, “On the theory of slip with smooth and singular loading surfaces,” Mat. Metody Fiz.-Mekh. Polya, 48, No. 3, 129–137 (2005)]. These equations describe the development of plane plastic deformation for simple and complex loading processes. A characteristic feature of these equations lies in the presence of unknown functions both under the integral sign and in the integration limits. We have written analytical solutions for monotone deformation and in a small neighborhood of an angular point of the loading trajectory. For arbitrary piecewise smooth trajectories, we have reduced this problem to the Cauchy problem for a first-order differential equation with known initial conditions. The results obtained simplify significantly the construction of constitutive equations ${\dot{\sigma }_{mn}} \sim {\dot{\varepsilon }_{mn}}$ and their use in applied problems of the theory of plasticity as compared with [N. Yu. Shvaiko, “On the theory of slip with smooth and singular loading surfaces,” Mat. Metody Fiz.-Mekh. Polya, 48, No. 3, 129–137 (2005); N. Yu. Shvaiko, Complex Loading and Problems of Stability [in Russian], Izd. DGU, Dnepropetrovsk (1989)].