 | | 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 |
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| G. N. Chernyshev, "Mutual generalization of elastic and gravitational field equations on the basis of solid mechanics," Mech. Solids. 37 (2), 70-81 (2002) |
| Year |
2002 |
Volume |
37 |
Number |
2 |
Pages |
70-81 |
| Title |
Mutual generalization of elastic and gravitational field equations on the basis of solid mechanics |
| Author(s) |
G. N. Chernyshev (Moscow) |
| Abstract |
By extending the phenomenon of deformation of time known from the general
relativity to the elastic space, we construct refined equations of dynamic
elasticity taking into account the deformation of the time coordinate.
In terms of mechanics, this deformation is interpreted as the dynamic
expansion-compression deformation of the substance similar to the thermal
deformation. This generalization makes it possible to take into account
the force interaction between particles of the medium in the inertial terms
of the equilibrium equations and in the continuity relations. This interaction
is ignored in the classical theory of elasticity. The new model of an elastic
body is validated by means of the comparison of the theoretical solutions
of a number of dynamic problems constructed on the basis of the relations
obtained with experimental results. On the other hand, by extending the
model of an elastic space to the model of the gravitational space, we equip
the familiar equations of the gravitational field in the linear
approximation with the generalized Hooke's law and equilibrium equations.
The material of this paper is presented in the following order.
By generalizing the linearized mathematical model of the gravitational
space to the elastic body we construct the dynamic equations of elasticity
in which the time coordinate is deformed as is the case for the
gravitational field theory. Unlike the gravitational field theory (the
general relativity), when deriving the equations of elasticity we do not
introduce in advance a fixed velocity of propagation of excitations equal
to the velocity of light, sound, or some other velocity determining a
four-dimensional metric of the elastic space. As shown in the present
paper, there is no necessity for introducing such a velocity. Instead, an
arbitrary constant having the dimension of velocity is introduced. This is
necessary for forming a four-dimensional metric utilized when determining
the strain state of an elastic space. The velocities of propagation of
excitations and waves in an elastic body are determined by the elastic
moduli and the density.
Then we validate the obtained equations by comparing theoretical solutions with
experimental results. The confidence in the correctness of these equations
suggested us to perform a reciprocal generalization of the constructed model
of an elastic body to the gravitational space. The deformability
of this space similar to the deformability of an elastic body is in fact
recognized in the general relativity by introducing a metric tensor depending
on the stress state. Therefore, by analogy with the theory of elasticity,
the half-difference of the perturbed and unperturbed metric tensors of the
gravitational space is referred to as the strain tensor. |
| References |
| 1. | L. D. Landau and E. M. Lifshits, Theoretical Physics. Volume 2.
Field Theory [in Russian], Nauka, Moscow, 1988. |
| 2. | L. I. Sedov, Continuum Mechanics. Volume 1
[in Russian], Nauka, Moscow, 1994. |
| 3. | Yu. A. Amen-Zade, Theory of Elasticity [in Russian],
Vysshaya Shkola, Moscow, 1976. |
| 4. | R. M. White, "Surface elastic waves,"
TIIER, Vol. 58, No. 8, pp. 68-110, 1970. |
| 5. | I. A. Viktorov, "Types of surface sound waves in solids.
A review," Akust. Zh., Vol. 25, No. 1, pp. 1-17, 1979. |
| 6. | G. W. Farnell, "Properties of elastic surface waves,"
in Physical Acoustics [Russian translation], No. 6, pp. 139-202,
Mir, Moscow, 1973. |
| 7. | F. W. Voltmer, E. P. Ippen, R. M. White,
T. C. Lim, and G. W. Farnell, "Measured and calculated
surface wave velocities,"
TIIER, Vol. 56, No. 9, pp. 247-248, 1968. |
| 8. | G. W. Kaye and T. H. Laby, Tables of Physical and Chemical
Constants [Russian translation], Fizmatgiz, Moscow, 1962. |
| 9. | O. Andersen and V. Liberman, "Sound speed in rocks
and minerals,"
in Physical Acoustics [Russian translation], No. 4, pp. 382-433,
Mir, Moscow, 1970. |
| 10. | B. N. Ivakin, Methods of Modeling of Seismic Wave Phenomena [in
Russian], Nauka, Moscow, 1969. |
| 11. | J. W. Delly and S. A. Thau, "Observations of stress
wave propagation in a half-plane with boundary loading,"
Intern. J. Solid and Struct., Vol. 3, No. 3, pp. 293-398, 1967. |
| 12. | V. B. Poruchikov, Methods of Dynamic
Theory of Elasticity [in Russian], Nauka, Moscow, 1986. |
| 13. | C. Moller, "Trionfi e limiti della teoria di Einstein della
relativita e gravitazione,"
in Astrofisica e Cosmologia, Gravitazione, Quanti e Relativita.
Sentenario di Einstein, pp. 25-26, Giunti Barbera, Firenze, 1979. |
| 14. | E. Almadi and G. Pizella, "La ricerca delle onde gravitazionali,"
in Astrofisica e Cosmologia, Gravitazione, Quanti e Relativita.
Sentenario di Einstein, pp. 281-458, Giunti Barbera, Firenze, 1979. |
| 15. | J. A. Wheeler, "I quanti e l'universo,"
in Astrofisica e Cosmologia, Gravitazione, Quanti e Relativita.
Sentenario di Einstein, pp. 1137-1164, Giunti Barbera, Firenze, 1979. |
| 16. | A. I. Lur'e, Theory of Elasticity [in Russian],
Nauka, Moscow, 1970. |
|
| Received |
23 December 1999 |
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