About the limit state of deformable bodies

Abstract

The theory of limit state deals with statically determinate condition of solids. In this case the system is closed due to the limit conditions, such properties of matter as viscosity, elasticity, etc. cannot influence the limit state. In other words, being at the limit state the nature of the relationship between stress and strain has no effect on the limit state. The article discusses systems of equations which correspond to the classical theory of plasticity. It is assumed that the components of the velocity vector depend on two spatial coordinates only. The constructed system can be used to describe the torsion of a parallelepiped around the three orthogonal axes. For the constructed system of equations group point symmetries, conservation laws were found. It is shown that the system admits 8-dimensional Lie algebra. On the basis of the symmetry group some classes of invariant solutions of rank 1 were constructed. They depend on the 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 an infinite series of conservation laws.