About torsion of parallelepiped around three axis

The theory of limit state deals with statically determinate condition of solids. In this case the system is closed due to extreme conditions, such properties of matter such as viscosity, elasticity, etc. cannot influence the limit state. In other words, when reaching the limit state the nature of the relationship between stress and strain has no effect on the ulti- mate state. The study of such systems has been consistently pursued by D. D. Ivlev and his coauthors. To the equilib- rium equations they attached two or an equation relating the components of the stress tensor. This led to the closure of the system of equilibrium equations. In the theory of plasticity equations, which are closed with a single yield stress are studied well. The most well-known system describing the ultimate state of deformable bodies are well-studied equations describing the torsion of the plastic bodies, the two-dimensional stationary problem of the theory of plasticity. The arti- cle discusses some other systems of equations which are closed only by one equation of flow, which corresponds to the classical theory of plasticity. It is assumed that the components of the velocity vector depend only on two spatial coor- dinates. In addition, for the component of velocity vector conditions of deformations compatibility are performed identi- cally. The constructed systems can be used to describe the twisting of the parallelepiped around the three orthogonal axes. For the constructed system of equations point group symmetries, conservation laws have been found. It is shown that the system allows 8 -dimensional Lie algebra. On the basis of the symmetry group some classes of invariant solutions of rank 1 have been constructed. They depend on arbitrary functions of one variable. It is shown that these solutions can be used to describe plastic torsion of a parallelepiped around three orthogonal axes. It is shown that the system admits infinite series of conservation laws. The concluding paragraph describes the construction of elastic solutions to the problem. It is shown that it boils down to finding three harmonic functions.