 | | 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
|
| In English (Mech. Solids): | | 3212 |
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| P. A. Belov, A. G. Gorshkov, and S. A. Lurie, "Variational model of nonholonomic 4D-media," Mech. Solids. 41 (6), 22-35 (2006) |
| Year |
2006 |
Volume |
41 |
Number |
6 |
Pages |
22-35 |
| Title |
Variational model of nonholonomic 4D-media |
| Author(s) |
P. A. Belov (Moscow)
A. G. Gorshkov (Moscow)
S. A. Lurie (Moscow) |
| Abstract |
To describe models of nonholonomic media, we use a variational approach
based on the formulation of the kinematic constraints in the medium under
study and on the construction of the corresponding variational forms using
the Lagrange multipliers (a "kinematic" variation principle).
To state the model, we use a formal generalization of the continual model
of mechanics of deformable solids in which the four-dimensional (4D) space
of events with a four-dimensional vector of generalized displacements of
the medium is introduced. Time is included in the generalized
four-dimensional coordinate system and is treated as one of independent
coordinates. For the kinematic constraints we suggest to use generalized
Cauchy relations determining generalized 4D-strains from generalized
4D-displacements. We find a sufficiently general form of the constitutive
equations for nonholonomic media; they are determined with the help of the
nonintegrability conditions for the possible work of internal forces. We
write out the variational equation for nonholonomic linear media and pose
the corresponding boundary (initial-boundary) value problem. We consider a
special case of the model. In the framework of the special model, we obtain
a generalized heat equation and an interpretation of the Fourier and
Duhamel-Neumann hypotheses. |
| References |
| 1. | L. I. Sedov, On Fundamental Principles of Continuum Mechanics [in
Russian], Izs-vo MGU, Moscow, 1961. |
| 2. | L. I. Sedov,
"On fundamental concepts of continuum mechanics,"
in Several Problems of Mathematics and Mechanics [in Russian], pp. 227-235,
Izd-vo SO AN SSSR, Novosibirsk, 1961. |
| 3. | L. I. Sedov and M. E. Eglit,
"Construction of nonholonomic models of continuum mechanics
with the finiteness of strains and several physical and mathematical effects
taken into account,"
Doklady AN SSSR, Vol. 142, No. 1, pp. 54-59, 1962. |
| 4. | V. L. Berdichevskii,
"Variational methods for constructing models of continuous media,"
PMM [Applied Mathematics and Mechanics], Vol. 30, No. 6,
pp. 1081-1086, 1966. |
| 5. | V. L. Berdichevskii,
"Construction of models of continuous media by using
the variation principle,"
PMM [Applied Mathematics and Mechanics], Vol. 30, No. 3,
pp. 510-530, 1966. |
| 6. | L. I. Sedov, "On the energy-momentum tensor and on macroscopic
internal interactions in the gravitational field and in material media,"
Dokl. AN SSSR, Vol. 164, No. 3, pp. 519-522, 1965. |
| 7. | I. F. Obraztsov, S. A. Lurie, P. A. Belov, and Yu. G. Yanovskii,
"On a model of cohesion interactions in continuous media,"
Izv. Vyssh. Uchebn. Zaved. Sev.-Kavk. Region. Estestv. Nauki,
No. 3, pp. 110-118, 2000. |
| 8. | I. I. Goldenblatt, Nonlinear Problems of Elasticity [in Russian],
Moscow, 1969. |
| 9. | W. Nowacki, Dynamic Problems of Thermoelasticity [Russian
translation], Mir, Moscow, 1970. |
|
| Received |
18 July 2006 |
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