 | | 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
|
| In English (Mech. Solids): | | 4760 |
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| << Previous article | Volume 49, Issue 4 / 2014 | Next article >> |
| E.I. Ryzhak, "Direct Coordinate-Free Derivation of the Compatibility Equation for Finite Strains," Mech. Solids. 49 (4), 382-388 (2014) |
| Year |
2014 |
Volume |
49 |
Number |
4 |
Pages |
382-388 |
| DOI |
10.3103/S0025654414040037 |
| Title |
Direct Coordinate-Free Derivation of the Compatibility Equation for Finite Strains |
| Author(s) |
E.I. Ryzhak (Schmidt Institute of Physics of the Earth, Russian Academy of Sciences, ul. B. Gruzinskaya 10, Moscow, 123995 Russia, e_i_ryzhak@mail.ru) |
| Abstract |
The compatibility equation for the Cauchy-Green tensor field (squared tensor of pure extension with respect to the reference configuration) is directly derived from the well-known relation expressing this tensor via the vector field determining the mapping (transformation) of the reference configuration into the actual one. The derivation is based on the use of the apparatus of coordinate-free tensor calculus and does not apply any notions and relations of Riemannian geometry at all.
The method is illustrated by deriving the well-known compatibility equation for small strains. It is shown that when the obtained compatibility equation for finite strains is linearized, it becomes the compatibility equation for small strains which indirectly confirms its correctness. |
| Keywords |
finite strains, small strains, compatibility equation, coordinate-free tensor calculus, tensor isomers |
| References |
| 1. | A. I. Lurie,
Nonlinear Theory of Elasticity
(Nauka, Moscow, 1980)
[in Russian]. |
| 2. | C. Truesdell,
A First Course in Rational Continuum Mechanics
(The Johns Hopkins University Press, Baltimore, Maryland, 1972; Mir, Moscow, 1975). |
| 3. | S. K. Godunov and E. I. Romenskii,
Elements of Continuum Mechanics and Conservation Laws
(Nauchnaya Kniga, Novosibirsk, 1998)
[in Russian]. |
| 4. | E. I. Ryzhak,
Coordinateless Tensor Calculus in Continuum Mechanics
(MFTI, Moscow, 2011)
[in Russian]. |
| 5. | E. I. Ryzhak,
"On the Simplest Localization Potentials,"
Izv. Akad. Nauk SSSR. Mekh. Tverd. Tela,
No. 6, 114-121 (1985)
[Mech. Solids (Engl. Transl.)]. |
|
| Received |
27 December 2013 |
| Link to Fulltext |
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