Theory of Five-Dimensional Elastoplastic Processes of Moderate Curvature

A variant of the constitutive equations for describing complex loading processes with deformation trajectories of arbitrary geometry and dimension is considered. The vector constitutive equations and a new method of mathematical modeling the five-dimensional complex loading processes are obtained. This method is validated for two- and three-dimensional processes of constant curvature. The constitutive equations describe the stages of active loading and unloading. Explicit representations of the stress vector in an arbitrary deformation process are obtained. It is shown that the state parameters of the model in the five-dimensional deformation space are the four angles from the representation of the stress director vector in the Frenet frame, not directly, but in the form of four special functions whose form is known. These functions are called the Vasin functions. The process of complex loading along a three-dimensional helical trajectory of deformation is also considered, where, after diving and subsequent additional loading, the equations of the steady-state loading process are established. Similar results are obtained for five-dimensional helical deformation trajectories. Hence, for this class of processes there exists a correspondence between the geometries of the deformation and reaction paths in the form of a loading path.