Some exact solutions of the nonlinear evolutionary equation for the spreading of a plastic layer on a plane

The problem of free spreading of a plastic layer between parallel approaching planes of tool bodies [1] is considered, which simulates the technological process of stamping thin-walled structural elements. To describe the specified flow of a thin plastic layer, A. Ilyushin proposed an effective two-dimensional mathematical model averaged over the layer thickness, to which the original three-dimensional problem of the flow of an ideally plastic body leads. The transition to a two-dimensional problem was carried out on the basis of special hypotheses proposed as a result of the analysis of the well-known Prandtl solution in the problem of upsetting a flat, in a vertical section, layer of plastic material [2]. On the contact surfaces, the condition of complete slippage of the material is assumed, and the shear stresses reach a maximum value equal to the shear yield strength of the layer material. Within the framework of this model, A. A. Ilyushin formulated a boundary value problem in a region with a moving boundary with respect to three unknown functions - contact pressure and two flow velocity components.