 | | 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: | | 12804 |
| In Russian (Èçâ. ÐÀÍ. ÌÒÒ): | | 8044
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| In English (Mech. Solids): | | 4760 |
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| G. L. Brovko, "On one structural model of the Cosserat medium," Mech. Solids. 37 (1), 60-73 (2002) |
| Year |
2002 |
Volume |
37 |
Number |
1 |
Pages |
60-73 |
| Title |
On one structural model of the Cosserat medium |
| Author(s) |
G. L. Brovko (Moscow) |
| Abstract |
The attempts of scientists of the 19th century in the field
of mechanics to find a generalization of the classical
(Cauchy) model of a continuous medium which would provide
a more detailed description for interactions, coupling, and states
in actual deformable media have been first consistently
implemented by E. Cosserat and F. Cosserat in their classic
work in 1909 [1]. The intensive investigations in non-classical
continuum mechanics that started a half-century later have been aimed
at the creation and development of various approaches
(including Cosserat's approach) to
the mathematical modeling of solids taking into account their
microstructure. The development of moment theories in mechanics
of continuous media (in particular, Cosserat type media, including
the Cosserat pseudocontinuum [2, 12]) was concentrated mostly
on mathematical issues and, as a rule, did not cover physical
examples and specific applications of Cosserat model which
would take into account the structure of the solids, their mass
and inertial characteristics, and the character of force/moment
interactions and loads.
In the present paper, using Il'yushin's method of mechanical (structural)
modeling [13, 14], we show that the properties inherent in Cosserat media
can be discovered and graphically illustrated
in some special mechanical models of structural materials.
The approach to the construction of such models demonstrated in this paper
for a special case of a solid deformable planar structure provides a clear
physical interpretation for the structure of this solid, its motions
and interactions (of Cosserat type) and shows that structurally
inhomogeneous materials of Cosserat type can be implemented technologically.
A detailed description of the model structure is given. Averaged properties
of this structure treated as a Cosserat continuum are established. Tensor
characteristics of state (energy-conjugate strain and stress tensors) are
constructed. The equations of motions for finite deformations in the material
and spatial representations are derived [15]. The general structure
of the constitutive relations is given. For the case of small deformations
and small rotations of the inclusions, we obtain the constitutive relations,
including those for the elastic properties of the medium, and derive
the system of equations of motion in terms of displacements and rotations;
the cases are indicated where this system is hyperbolic.
In the case of arbitrary deformations and rotations, subclasses
of media (for example, moment-free medium or decoupled medium)
and motions are separated for which the appropriate boundary-value
problems are substantially simplified. The model constructed
and the equations derived agree with the general results of [1],
the familiar equations of non-symmetric elasticity (see, e.g., [2-10]),
and the results of [14-18]. |
| References |
| 1. | E. Cosserat and F. Cosserat, Theorie de Corps Deformables,
Hermann, Paris, 1909. |
| 2. | C. Truesdell and R. A. Topin, "The classical field theories,"
in Handbuch der Physics, Vol. 3/1, S. 226-793, Springer, Berlin, 1960. |
| 3. | E. L. Aero and E. V. Kuvshinskii, "Basic equations of elasticity for
media with rotational interaction of particles,"
Fizika Tverdogo Tela, Vol. 2, No. 7, pp. 1399-1409, 1960. |
| 4. | R. A. Toupin, "Elastic materials with couple-stresses,"
Arch. Rat. Mech. Anal., Vol. 11, No. 5, pp. 385-414, 1962. |
| 5. | R. D. Mindlin and H. F. Tiersten, "Effects of couple-stresses
in linear elasticity,"
Arch. Rat. Mech. Anal., Vol. 11, No. 5, pp. 415-448, 1962. |
| 6. | V. A. Pal'mov, "Basic equations of nonsymmetric elasticity,"
PMM [Applied Mathematics and Mechanics], Vol. 28, No. 3,
pp. 401-408, 1964. |
| 7. | A. E. Green, "Micro-materials and multipolar continuum mechanics,"
Intern. J. Eng. Sci., Vol. 3, No. 5, pp. 533-537, 1965 |
| 8. | V. A. Lomakin, Static Problems in Mechanics of Deformable
Solids [in Russian], Nauka, Moscow, 1970. |
| 9. | A. A. Il'yushin and V. A. Lomakin, "Moment theories
in mechanics of deformable solids," in Strength and Plasticity
[in Russian], pp. 54-60, Nauka, Moscow, 1971. |
| 10. | W. Nowacky, Theory of Elasticity [Russian translation],
Mir, Moscow, 1975. |
| 11. | I. A. Kunin, Theory of Elastic Bodies with Microstructure
[in Russian], Nauka, Moscow, 1975. |
| 12. | A. C. Eringen, "Theory of micropolar elasticity,"
in Fracture [Russian translation], Vol. 2, pp. 646-751, Mir, Moscow, 1975. |
| 13. | A. A. Il'yushin, "Puzzles of mechanics of deformable bodies,"
in Unsolved Problems of Mechanics and Applied Mathematics [in Russian],
pp. 68-73, Izd-vo MGU, Moscow, 1977. |
| 14. | G. L. Brovko and A. A. Il'yushin, "On one planar model
of perforated slabs," Vestnik MGU [Bulletin of Moscow State University],
Ser. 1, Mat., Mekh., No. 2, pp. 83-91, 1993. |
| 15. | G. L. Brovko, "Material and spatial representations for constitutive
relations of deformable media,"
PMM [Applied Mathematics and Mechanics], Vol. 54, No. 5,
pp. 814-824, 1964. |
| 16. | E. D. Martynova, "Determination of static and dynamic averaged
characteristics of periodic elastic frameworks,"
Elasticity and Inelasticity. Part 1 [in Russian], pp. 155-162, Izd-vo MGU,
1993. |
| 17. | A. E. Green and P. M. Naghdi, "A unified procedure for construction
of theories of deformable media," Proc. Roy. Soc. London, Ser. A, Vol. 448,
No. 1934, Part 1: Classical continuum physics, pp. 335-356;
Part 2: Generalized continua, pp. 357-377, 1995. |
| 18. | G. L. Brovko, "Modeling of nonhomogeneous media of complex
structure and Cosserat continuum," Vestnik MGU [Bulletin of Moscow State
University], Ser. 1, Mat., Mekh., No. 5, pp. 55-63, 1996. |
| 19. | A. Love, A Treatise on the Mathematical Theory of Elasticity
[Russian translation], ONTI, Moscow, Leningrad, 1935. |
| 20. | E. P. Popov, Theory and Design of Flexible Elastic Rods
[in Russian], Nauka, Moscow, 1986. |
| 21. | E. N. Berezkin, A Course in Theoretical Mechanics [in Russian],
Izd-vo MGU, Moscow, 1974. |
| 22. | A. A. Il'yushin, Continuum Mechanics [in Russian],
Izd-vo MGU, Moscow, 1990. |
| 23. | I. M. Vinogradov (Editor), Encyclopedia of Mathematics. Volume 3
[in Russian], Sovetskaya Entsiklopediya, Moscow, 1982. |
| 24. | A. A. Il'yushin, "Asymmetry of strain and stress tensors
in continuum mechanics,"
Vestnik MGU [Bulletin of Moscow State University],
Ser. 1, Mat., Mekh., No. 6, pp. 6-14, 1996. |
| 25. | B. E. Pobedrya, "Fundamentals of structural mechanics
of deformable solids," in Mathematical Modeling of Systems and
Processes [in Russian], No. 4, pp. 66-74, PGTU, Perm, 1996. |
| 26. | G. A. Vanin, "Gradient theory of elasticity,"
Izv. AN. MTT [Mechanics of Solids], No. 1, pp. 46-53, 1999. |
|
| Received |
29 December 1999 |
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