Theory of inelasticity without loading surface and associated flow rule

Based on the analysis of experimental hysteresis loops (cyclic diagram) of steel 40Х16Н9Г2С three sections are allocated describing the different behavior of the stress, i.e., three types of stresses. For each type of stress the authors have formulated appropriate evolutionary equations characterizing anisotropic hardening. To describe isotropic hardening, evolutionary equation for the parameter saturation stresses of the second type is introduced. In case of additional isotropic hardening under non-proportional cyclic loading we consider the stress saturation parameter of the second type dependent on the measure of disproportionality (complexity) of the loading process. To describe the phenomena of ratcheting under non-symmetric soft cyclic loading we consider the parameter, which is an evolutionary equation for the stress of the first type to be dependent on the accumulated strain. The stress deviator is defined as the sum of stress deviators of three types. To describe the nonlinear process of damage accumulation, the authors introduced the kinetic equation based on the energy principle, where energy equal to the work of stress of the second type on the deformations field is taken as energy spent on creating damage in the material. Kinetic equation for damage caused by stress of the first type on the field deformations is introduced under nonsymmetrical soft cyclic loading in case of ratcheting of the hysteresis loop. It became possible to assign material functions, close theory, formulate the basic experiment and the method of material functions identification. We have presented material functions of steel 40Х16Н9Г2С and the results of verification of the theory under proportional (simple) hard cyclic loading and disproportionate (complex) loading sweep deformations in the form of concentric circles with a common center at the origin of coordinates. Five turns of the trajectory, from the trajectory of the greater curvature to the average trajectory curvature, are considered. There is a reliable agreement between the results of calculations and experiments.