3-dimensional solutions from two variables

In this paper, we consider stationary 3-dimensional equations of ideal plasticity with the Mises flow condition. The material is assumed to be incompressible. The case when all three components of the velocity vector and hydrostatic pressure depend only on two coordinates x, y is studied in detail. For this case, a new name is introduced - 3-dimensional solutions from two variables, to distinguish it from the generally accepted two-dimensional state, when only two components of the velocity vector and hydrostatic pressure differ from zero. It is proved that the system admits, in the sense of S. Lie, a Lie algebra of dimension 10. It is shown that are all 3-dimensional solutions from two variables a superposition of the plane stress state and plastic torsion around the z-axis. Two invariant solutions of the equations describing the 3-dimensional deformed state are constructed. The first solution can be used to describe plastic flows between two rigid plates that approach at different speeds. The second solution is used to describe the stress-strain state of the material inside a flat channel formed by converging plates.