Closed problem of plasticity theory
A plane closed problem of theory of plasticity has been formulated and solved. Determining expressions for the entire deformation zone in analytical form are obtained. Methods of functions of a complex variable and argument functions are used. Generalizing factors of the problem solution are the argument functions for which differential dependencies are obtained in a form of Cauchy-Riemann and Laplace’s equations. It is shown that differential equations with different purposes and with different physical quantities have the same solution formats, which allows using them to establish the connection between the mechanical characteristics of the process. Closed solution allows determining this connection. A multicomponent model of a plastic medium, depending on integral characteristics of a deformed state of the medium and its temperature, i.e. on thermomechanical processing parameters, is presented in the analytical form. Stress state calculations have been carried out for various methods of metal forming with pressure. Their comparability with a real distribution of contact stresses under symmetrical and asymmetric loading, determined by technological factors of production is shown.