Modeling of elastoplastic deformation of steel 45 along Archimedes spiral type trajectories

This paper addresses the mathematical modeling of complex elastoplastic deformation of steel 45 along the plane trajectory in the Ilyushin’s deviatoric space, which consists of sections of both constant and variable curvature (Archimedes spiral). The constitutive equations of the proposed mathematical model are based on the Ilyushin’s vector representation of strain and stress. An approximate model of the theory of elastoplastic processes is used in mathematical modeling for plane trajectories with approximations of process functionals, which depend on the initial value of the curvature, rather than on the current curvature of the deformation trajectory. The constitutive equations of the mathematical model are reduced to the Cauchy problem, a numerical solution to which is obtained using the fourth order Runge–Kutta method. The validity of the mathematical model for this class of curvilinear strain trajectories was verified by comparing the calculation results with the experimental data obtained on the automated test machine SN-EVM in the mechanical testing laboratory of the Tver State Technical University. The experiment was carried out on thin-walled cylindrical specimens of steel 45 under complex loading (combined tension- compression and torsion). The calculation results and experimental data characterizing the scalar and vector properties of the material are presented graphically. It has been established that the proposed approximate mathematical model is able to capture (both qualitatively and quantitatively) the main effects of complex plastic deformation for the considered class of strain trajectories in the areas of small and medium curvature. More accurate calculation results in the approximations of the plasticity functionals can be obtained by taking into account all complex loading parameters, including the current curvature of the strain trajectory, especially for strain trajectories with large curvature.