Some symmetry group aspects of a perfect plane plasticity system

Abstract

In this paper, all the known classical solutions of plane perfect plasticity system under Saint Venant -Tresca -von Mises yield criterion are associated with some group of point symmetries. The equations of slip-line families for all solutions are constructed, which permits to determine explicitly boundaries of plastic areas.It is shown, how one can determine the compatible velocity solution for known stresses, considering symmetries. Some invariant solutions of velocities for Prandtl stresses are constructed. The mechanical sense of obtained velocity fields is discussed.

Invariant analytical solutions for the motion of an elastic string with electric current in a static magnetic field

“…The modern group theory analysis of differential equations is a powerful tool to find analytical solutions for complicated systems of differential equations that describe natural phenomena. As recent advances in this field, we refer to the works on gas dynamics, 1 fluid mechanics, 2,3 epidemiology, 4 economy sciences, 5,6 plasticity, 7 nonlinear equations of Korteweg‐de Vries type, 8,9 variable‐coefficient Burgers equations, 10 and generalized Benney system 11 …”

Solution of Boundary Value Problems of Plasticity with the Use of Conservation Laws

“…But the results significantly differ: conservation laws allow solving boundary value problems, and at that fairly effectively. This is illustrated in a series of papers on solving of boundary value problems for plasticity equations and other continuum mechanics equations [6][7][8][9][10].…”