Approximation of the flow of a layer of limited thickness of a multi-connected contour by a collection of flow of single-connected areas

The calculations are based on the one proposed by A.A. Ilyushin. theory of flow in a thin plastic layer, enclosed between two surfaces of tool bodies approaching according to a given law, under the assumption that: the material of the spreading layer is ideally plastic and volumetrically incompressible; elastic deformations of the tool are small and can be neglected; on the contact surfaces of the metal layer with the tool, the classical L. Prandtl flow law is accepted. To study the characteristics of the flow, we use the well-known boundary value problem in the “ideal fluid” formulation, which is described by nonlinear first-order partial differential equations with respect to the contact pressure and components of the flow velocity vector. In the general case, the quantities under study are complex functions that depend on the shape of the deformation zone, the magnitude of the contact pressure at the considered point of the contact surface, the presence, composition and method of applying a lubricant, the roughness of the contacting surfaces and other factors. The paper proposes a method for solving problems of flow of plastic layers in a new formulation. Using the proposed mathematical apparatus, a solution algorithm and a computer model of the flow were constructed.