INVESTIGATION OF THE STRESSED STATE OF THE ROLLING PROCESS WITH USING HARMONIC FUNCTIONS

On the basis of the method of harmonic functions, a general mathematical model for the different loading of the deformation focus in the conditions of the plane problem of the theory of plasticity is developed. The problem is posed and solved in a closed form. It is assumed that the solution is of a universal nature, both from the point of view of finding a given class of functions, as well as methods of force action on the deformation center. Under the conditions of the proposed method, for the first time a solution of the applied problem connected with asymmetric loading was obtained. In this case, a complex solution for the entire deformation zone shows that the zones of different metal flow exert a significant influence on each other, which was not taken into account earlier when the deformation center was examined separately. From the point of view of boundary conditions, a region of acceptable solutions was determined under asymmetric loading conditions. Calculations showed that the proposed method can be used not only in solving problems of metal working with pressure with different external effects on the deformation center, in particular during rolling, but also for analyzing the stress state of the metal at each point of the deformation center under the conditions of new technical developments. There is a multifactor effect of different rolling conditions on the power parameters of the process. Values such as coefficient of friction, shape factor, angle of capture change the magnitude and distribution of normal and tangential stresses along the length of the deformation zone. It should be noted that the ratio of the parameters coefficient of friction - angle of capture is the determining factor of the entire rolling process. The process can lose stability, on the verge of slipping, at the same time the circuit of contact stresses changes dramatically, from convex it turns into a concave or partially concave-tuyu. It is determined that the model adequately reacts to changes in technological parameters of the forming process. The distribution of contact normal stresses is characterized by a significant unevenness in length for the thin and medium strips. For medium and high poles, there is a decrease in the effect of contact friction, a more uniform distribution of stress along the length of the arc of contact. It is shown that the stress state of a metal is determined by unified expressions for the entire focus of plastic flow. ssions for the entire focus of plastic flow.