 | | Mechanics of Solids
A Journal of Russian Academy of Sciences | | Founded
in January 1966
Issued 6 times a year
Print ISSN 0025-6544
Online ISSN 1934-7936 |
Archive of Issues
| Total articles in the database: | | 11223 |
| In Russian (Èçâ. ÐÀÍ. ÌÒÒ): | | 8011
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| In English (Mech. Solids): | | 3212 |
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| S. I. Senashov, "Transformations of the Prandtl solution under the action of a symmetry group," Mech. Solids. 40 (5), 129-133 (2005) |
| Year |
2005 |
Volume |
40 |
Number |
5 |
Pages |
129-133 |
| Title |
Transformations of the Prandtl solution under the action of a symmetry group |
| Author(s) |
S. I. Senashov (Krasnoyarsk) |
| Abstract |
We show how point symmetries transform the known Prandtl solution of the
perfect plasticity equations in the two-dimensional case. As a result of
these transformations, we obtain new classes of exact solutions of the same
equations. Solutions that can be used to describe plane flows arising in a
plastic layer compressed between rigid plates are considered in detail. For
these solutions, slip lines are constructed. These slip lines differ from
the cycloids obtained by Prandtl. Some new properties of the
characteristics of the plane ideal plasticity problem are given.
Symmetry theory is widely used when studying and solving differential
equations. We only note the possibility of finding exact solutions of
differential equations and their classification with the use of symmetry
groups. These and other results can be found in [1].
Symmetries admitted by systems of differential equations possess the following property: they take
solutions of a system to solutions of the same system. This property permits one to construct new
solutions by merely subjecting some known solutions of the system to group transformations rather
than by integrating the original system. Numerous interesting results for various differential
equations were obtained by this method. Note that this method is effective only if there is a
sufficiently wide symmetry group. In the present paper, we apply symmetry transformations to the
well-known Prandtl solution. This gives a number of exact solutions, which are yet to be studied.
Here we single out only solutions that are bounded along the axis OY and hence can be used in the
description of plastic flows arising in a material compressed between rigid parallel plates. |
| References |
| 1. | P. P. Kir'yakov, S. I. Senashov, and A. N. Yakhno, Application of Symmetries and
Conservation Laws to the Solution of Differential Equations [in Russian], Izd-vo SO RAN,
Novosibirsk, 2001. |
| 2. | D. D. Ivlev, Theory of Perfect Plasticity [in Russian], Nauka,
Moscow, 1966. |
|
| Received |
03 April 2003 |
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