Simulation and comparison of the results of the exact analytical solution of the boundary problem of a flow in a thin layer with experimental data

Many technological processes of metal forming include deformation of an elongated workpiece by approaching tools, the contact surfaces of which have shallow grooved cavities. A common special case are stamps or part of their contact surfaces, which are planes: sites for draft and some types of harvesting streams. The article is devoted to mathematical modeling of the flow of a thin metal layer over rough surfaces. The article presents the results of an analytical study of the boundary value problem of the viscoplastic flow of a thin layer of metal located between thin rough plates moving towards each other. Exact solutions based on the classical formulation in the framework of an "ideal fluid" and "viscoplastic fluid" are given. A real experiment is described for assessing the effect of shear stresses averaged over the layer thickness on the kinematics of the flow of a plastic layer. In order to assess the effect of shear stresses averaged over the layer thickness on the flow kinematics, and, in particular, near fixed boundaries, experiments were carried out and the results on settlement between converging rigid plates of a thin plastic sample bounded by fixed walls and initially having the shape of a rectangle in plan are analyzed. . The fixed walls of the tool are located along the long sides of the rectangle. Experimental regularities have been established. Spreading of samples occurs as expected, i.e., along straight streamlines parallel to the long side of the rectangle, or along the rays of a circular sector. It was found that the longitudinal displacements near the immovable walls noticeably lag behind the displacements far from the boundary, and the maximum displacements are observed on the line of symmetry of the sample with respect to the immovable walls. This fact indicates the presence of nonzero shear stresses and (velocities) of deformations in the considered flows of a thin plastic layer. Comparison of the obtained experimental and analytical simulation results allows us to evaluate the correctness of the chosen mathematical model, which can be used to describe the flows.