BOUNDARY CONDITIONS IN SOLUTIONS OF APPLIED PROBLEMS OF THE THEORY OF PLASTICITY

The processes of metal working with pressure are characterized by unified equations of plasticity theory, but different boundar Generalizations for boundary conditions and solutions of the problem are shown, which are a trigonometric connection of tangential stresses and plastic shear resistance. At the level of the boundary conditions, one can see the possibility of simplifying the closed problem of the theory of plasticity, both with respect to stresses and strain rates. This is due to the complete realization of the plasticity condition in the generalized equation of equilibrium and the revised condition for the continuity of the strain rates. The last two differential equations refer to equations of hyperbolic type, which determines the same approaches to their solution, hence the possibility of formulating and obtaining the final result of the closed problem of the theory of plasticity. The universality of the solution of the problem in the analytical form is shown in the paper. On the basis of the obtained identical mathematical expressions using different conditions of applied problems, working formulas for different processes of metal working with pressure are considered and determined. In this case, processing conditions with asymmetric and symmetrical loading of the deformation center (sediment, rolling) were considered. In contrast to the solutions of the linear problem of plasticity theory, the characteristic of the deformation focus was determined by unified formulas without splitting the treatment zone into several blocks. The obtained approaches make it possible to broaden the class of solved applied problems under different loading conditions. Multivariance and multicomponent problems are solved, taking into account the influence of a significant number of technological factors of production. The reliability of the obtained result is confirmed not only by qualitative and in many respects quantitative coincidence with literary theoretical and experimental data, but also by confirmation of the solution of the kinematic problem, which was noted by many authors as an element of the reliability of the result.