DIFFERENtIAl & DIFFERENCE EqUAtIONS ANd APPlICAtIONS

SYMMETRIES OF THE PLANE PLASTICITY SYSTEM
WITH A GENERAL YIELD CRITERION
PETR KIRIAKOV AND ALEXANDER YAKHNO
Some classes of invariant solution for the system of two-dimensional plasticity with general
yield criterion are considered. In particular, the system with Coulomb law is investigated
from the point of view of symmetry analysis. Moreover, the classification of the
groups of point transformations admitted by the system with respect to the function of
plasticity is realized. The mechanical sense of obtained invariant solutions is discussed.
Copyright © 2006 P. Kiriakov and A. Yakhno. This is an open access article distributed
under the Creative Commons Attribution License, which permits unrestricted use, distribution,
and reproduction in any medium, provided the original work is properly cited.
1. Introduction
In general, the processes of plastic deformations of materials are expressed by the systems
of nonlinear differential equations. For such systems, the numerical methods are used
widely for resolving of the concrete boundary value problems. But as for the exact solution,
there is a lack of them, because of strong nonlinearity of stress-strain relations. In
this case, the symmetry (group) analysis of differential equations is a powerful method of
the construction of exact solutions. It seems that the first application of these methods to
the plasticity theory was made in [2].
The equations of plane plasticity describe stresses of deformed region, when the plastic
flow is everywhere parallel to a given plane (usually x 1 Ox 2 plane). This system consists of
two equilibrium equations [4],
∂σ x1
∂x 1
+ ∂τ x 1x 2
∂x 2
= 0,
∂σ x2
∂x 2
+ ∂τ x 1x 2
∂x 1
= 0, (1.1)
and the law defining the limit of elasticity under some combination of stresses which is
called the yield criterion p(σ,τ) = 0, where σ xi , τ x1x 2
are the components of stress tensor,
σ is the hydrostatic pressure, and τ is the shear stress:
( )
( ) 2
σx1 + σ x2
σ = , τ 2 σx1 − σ x2
=
+ τx 2 2
4
1x 2
. (1.2)
Hindawi Publishing Corporation
Proceedings of the Conference on Differential & Difference Equations and Applications, pp. 555–564