Variants of theories of plastic flow under combined hardening, widely used in practical calculations of structures, are considered. A comparative analysis of the variants of theories under complex loading along plane and spatial deformation trajectories is carried out, covering a wide range of trajectories from multi-link polylines to curved trajectories of variable curvature and torsion. Trajectories from medium to large curvature and torsion are considered. The analysis of the research results is carried out in the vector space of A.A. Ilyushin. The plane trajectories of deformations in the form of a square and a curved trajectory of variable curvature in the form of an astroid are considered, as well as the spatial trajectory of deformations of variable curvature and torsion in the form of a helix. The results of calculations are compared with the results of experimental studies on stress response trajectories, scalar and vector properties. Variants of the theories are considered: the isotropic hardening model; the Ishlinski–Prager–Kadashevich–Novozhilov model (linear kinematic hardening + isotropic hardening); a model similar to the Ono–Wang model; the Armstrong-Frederick-Kadashevich model; the Shabosh model with three Armstrong–Frederick–Kadashevich evolutionary equations; the Temis model based on the invariant theory of plasticity; the Cooper model with a three-membered structure the evolutionary equation for kinematic hardening. The material parameters (functions), the closing versions of the theories of plasticity are given. Satisfactory compliance with the experiment for all deformation trajectories is achieved by calculations based on the Shabosh and the Bondar models – the difference between the results of calculations and experiments does not exceed 30% with the best correspondence to the experiment of the Bondar model. It should be noted that the Bondar plasticity model is closed by three parameters of anisotropic hardening and one function of isotropic hardening, and the Shabosh model is closed by six parameters and one function.

We revealed some features and differences in isotropic and anisotropic hardening under monotonic and cyclic loads by analyzing the experimental results of the samples made of 12X18H10T stainless steel under a rigid (controlled) deformation process, which includes a sequence of monotonic and cyclic loading modes under uniaxial tension-compression and different temperature levels. To describe these features with the theory of thermoplasticity, which belongs to the class of flow theories for combined hardening, a memory surface is introduced in the space of the plastic strain tensor components that separates the processes of monotonic and cyclic deformations. The main assumptions and equations of the thermoplasticity theory are formulated. To describe the transition from the monotonic to the cyclic and from the cyclic to the monotonic deformations, the evolutionary equations are formulated for the parameters of isotropic and anisotropic hardening. The basic experiment, which determined the material functions, consists of three stages, such as cyclic loading, monotonic loading and the subsequent cyclic up to destruction. The method of identifying the material functions based on the results of the basic experiment is considered. The material functions that close the thermoplasticity theory at different temperature levels are determined for 12X18H10T stainless steel due to the basic experiment and identification method. We considered the results of the computational and experimental studies of the rigid cyclic deformation under isothermal and non-isothermal loadings up to destruction of 12X18H10T stainless steel. The kinetics of the stress range and the average stress during isothermal and non-isothermal cyclic loadings are analyzed. A reliable compliance of the computational and experimental results is obtained.