 | | Journal of Applied
Mathematics and Mechanics
Russian Academy of Sciences | | Founded
in January 1936
(Translated from 1958)
Issued 6 times a year
ISSN 0021-8928
(print version) |
Archive of Issues
| Total articles in the database: | | 10512 |
| In Russian (ΟΜΜ): | | 9713
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| In English (J. Appl. Math. Mech.): | | 799 |
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| K.M. Zingerman and V.A. Levin, "Some qualitative effects in the exact solutions of the Lamé problem for large deformations," J. Appl. Math. Mech. 76 (2), 205-219 (2012) |
| Year |
2012 |
Volume |
76 |
Issue |
2 |
Pages |
205-219 |
| Title |
Some qualitative effects in the exact solutions of the Lamé problem for large deformations |
| Author(s) |
K.M. Zingerman (Tver, Russia, zingerman@rambler.ru)
V.A. Levin (Moscow, Russia, v.a.levin@mtu-net.ru) |
| Abstract |
Qualitative effects in the solution of a number of radially symmetric and plane axisymmetric problems for bodies made of non-linearly elastic incompressible materials are analysed for large deformations. In the case of problems of the axisymmetric plane deformation of cylindrical bodies, the lack of uniqueness of the solution for a given follower load in the case of a Bartenev-Khazanovich material and the existence of a limiting load in the case of a Treloar (neo-Hookian) material have been studied in detail and the dependences of the limiting load on the ratio of the external and internal radii of a hollow cylinder in the undeformed state have been presented. A similar study has been carried out for constitutive relations of a special form that well describe the properties of rubber. For this material, the lack of uniqueness of the solution is revealed for fairly high loads. The axisymmetric problem of the plane stress state of a circular ring made of a Bartenev-Khazanovich material has been solved and a lack of uniqueness of the solution for a given follower load was discovered in the case when the dimensions of the ring are given in the undeformed state. Similar studies have been carried out for Chernykh and Treloar materials in the case of the problem of the radially symmetric deformation of a spherical shell. It was established that, in the case of a Chernykh material, the lack of uniqueness of the solution depends considerably on the constant characterizing the physical non-linearity. The limit case of the deformation of a spherical cavity in an infinitely extended body has been investigated. The effect of an unbounded increase in the boundary stresses is observed for finite external loads, that appears in the case of the problem of the plane axisymmetric deformation of a cylindrical cavity in an infinitely extended body made of a Bartenev-Khazanovich material and in the case of the problem of the radially symmetric deformation of an infinitely extended body made of a Chernykh material with a spherical cavity. |
| Received |
28 November 2010 |
| Link to Fulltext |
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