Applied questions of Il’yushin theory of elastoplastic processes

Based on the analysis of the experimental results of the complex loading processes along the helical deformation trajectories, it is found that the response to the helical deformation trajectory takes a certain form of a limiting regime after the simple loading and after the exhaustion of some trace; in other words, the correspondence of the geometry of the deformation trajectory and the shape of the response takes place. A new form of constitutive equations are considered to study complex loading processes with deformation trajectories of arbitrary geometry and dimension. Some vector constitutive equations and a system of differential equations for the four angles of the stress vector decomposition in the Frenet frame are obtained. It is shown that the stress vector is represented as the sum of the following three components: the rapidly decaying plastic traces of elastic states, the instantaneous responses to deformation processes, and the irreversible stresses accumulated along a deformation trajectory. A new method of mathematical modeling of five-dimensional processes of complex loading is proposed and substantiated for two- and three-dimensional processes.