Use of characteristics in non-stationary problems of plastic flow of a thin layer along planes

The calculations are based on the proposed Ilyushin A.A. [1] theory of flow in a thin plastic layer enclosed between two surfaces of tool bodies approaching each other according to a given law. On contact surfaces, the classical L. Prandtl flow law is adopted [2]. To solve the proposed problem, the well-known boundary value problem in the formulation of the "ideal fluid" is used. This setting is described by nonlinear differential equations in partial derivatives of the first order with respect to the contact pressure and components of the flow velocity vector along the flow plane [3, 4]. In the general case, the desired values are complex functions that depend on the shape of the deformation zone, the value of the contact pressure at the considered point of the contact surface, the presence, composition and degree of lubrication, the roughness of the contacting surfaces, etc. [5, 6]. The paper proposes a method for solving problems of the flow of plastic layers in a new formulation. The analysis of the received result is carried out. Based on the method of characteristics, the problem of the flow of a plastic layer of material between inclined plates in a fixed circular area is solved, when its boundary is formed by grooves in one of the tool bodies, where the plastic material flows freely. Using the proposed solution, it is possible to construct the kinematics of the flow.