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For the first time the conservation laws for von Mises plasticity equations in three dimensions as well as for plane stress equations are given. In the plane case conservation laws are used to construct characteristics for the Cauchy problem. For the system of the plane strain, the conservation laws are used to solve the free boundary problem for any convex smooth contour loaded with constant normal and zero tangential stresses.

Conservation laws are becoming one of the most important tools for studying and solving differential equations. Such significance was gained after the article by Noether [

Later it was shown that such a relationship exists only for the equations derived from the variational principle. For other equations the so-called operator of universal linearization should be constructed. And then it is necessary to consider the kernel of its (formally) adjoint operator [

Let us give some basic definitions of the theory of conservation laws due to [

A

There follows (see [

Let system

Let us recall [

Let us consider [

System (

Transformations (

System (

It is easy to verify that (

Let us note that the von Mises yield criterion (

For system (

The direct verification of (

For generator

For

Generator

It is easy to see that the linear combination

Let us note that generator

For Navier-Stokes equations the complete set of conservation laws was found in [

A plane stress state is approximately achieved in a thin lamina deformed under the action of forces which lie in its median plane. Equations for stresses in a Cartesian plane

To construct the conservation laws let us use the method proposed in [

After some manipulations (

Let us note that it is possible to determine other solutions of (

Relations (

Finally, the following theorem is valid.

The system of plane stress state admits infinite series of conservation laws.

System (

Then system (

Finally, for the components of conserved current we have the following linear equations:

Let functions

The Cauchy problem.

Let us determine the coordinates of point

For the second coordinate of

The solution of the aforementioned problems for conserved currents gives the solutions of the Cauchy problem for the initial system, because at point

In such a way it is possible to determine the families of characteristic curves for the system of the plane stress. These curves do not coincide with the so-called slip lines (lines where the tangent stress achieved its maximum value) as in the case of the plane strain system considered later. But principal directions of the stress tensor bisect the angles between characteristic curves. That is why it is possible to reconstruct slip lines from the known characteristic curves field.

Let us consider the system of perfect plane plasticity with Tresca-Saint-Venant-Mises yield condition [

Quasilinear system (

The complete set of conservation laws of (

Let us consider the problem of a free boundary. Let the constant normal

Substituting (

Applying the above method we define the coordinates of

As a particular example, let us consider contour

Free boundary for the contour

For all point symmetries of a three-dimensional perfect plasticity system with the von Mises yield criterion the corresponding conservation laws are calculated. Some of them (conservation of the mass and of the impulse) are in the basis of this system; however, there are new conservation laws.

Recently a rigid plasticity constitutive model with the linear kinematic hardening has been analyzed in [

How two linear systems for the components of conserved currents for the system of plane stress state can be used to determine its characteristic curves is shown.

For the perfect strain plane plasticity with Tresca-Saint-Venant-Mises yield condition, the problem of a free boundary for arbitrary convex cavity situated in the infinite plastic deformed medium is solved with the use of the corresponding conservation laws.

The work was supported by the Ministry of Education and Science of the Russian Federation (Project 1.3720.2011) to S. I. Senashov and by PRO-SNI 2013-UDG to A. Yakhno.