 | | 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: | | 12854 |
| In Russian (Èçâ. ÐÀÍ. ÌÒÒ): | | 8044
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| In English (Mech. Solids): | | 4810 |
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| << Previous article | Volume 49, Issue 1 / 2014 | Next article >> |
| D.V. Georgievskii, "Compatibility Equations in Systems Based on Generalized Cauchy Kinematic Relations," Mech. Solids. 49 (1), 99-103 (2014) |
| Year |
2014 |
Volume |
49 |
Number |
1 |
Pages |
99-103 |
| DOI |
10.3103/S0025654414010117 |
| Title |
Compatibility Equations in Systems Based on Generalized Cauchy Kinematic Relations |
| Author(s) |
D.V. Georgievskii (Lomonosov Moscow State University, Leninskie Gory, Moscow, 119991 Russia, georgiev@mech.math.msu.su) |
| Abstract |
We obtain integrability conditions for systems of linear partial differential equations based on generalized Cauchy kinematic relations. The generalization pertains to both the dimension of the Euclidean space and the rank of the object corresponding to displacement vector in the classical case. The integrability conditions, or the compatibility equations, are written out as the conditions of vanishing of all components of either the generalized incompatibility tensor introduced here or the generalized Riemann-Christoffel tensor obtained from the incompatibility tensor by convolutions with the Levi-Civita symbols. The ranks and number of independent components of these tensors are found. |
| Keywords |
compatibility equations, integrability conditions, incompatibility tensor, Riemann-Christoffel tensor, Cauchy relations |
| References |
| 1. | Yu. N. Rabotnov,
Mechanics of Deformable Solids
(Nauka, Moscow, 1988)
[in Russian]. |
| 2. | B. E. Pobedrya,
Numerical Methods in Elasticity and Plasticity Theory
(Izd-vo MGU, Moscow, 1995)
[in Russian]. |
| 3. | B. E. Pobedrya and D. V. Georgievskii,
"Equivalence of Formulations for Problems in Elasticity Theory in Terms of Stresses,"
Rus. J. Math. Phys.
13 (2), 203-209 (2006). |
| 4. | C. Amrouche, P. G. Ciarlet, L. Gratie, and S. Kesavan,
"On the Characterization of Matrix Fields as Linearized Strain Tensor Fields,"
J. Math. Purcs Appl.
86, 116-132 (2006). |
| 5. | P. G. Ciarlet, P. Ciarlet (Jr.), G. Geymonat, and F. Krasucki,
"Characterization of the Kernel of the Operator CURL CURL,"
C. R. Acad. Sci. Paris. Ser. I
344, 305-308 (2007). |
| 6. | D. V. Georgievskii and M. V. Shamolin,
"Levi-Civita Symbols, Generalized Vector Products,
and New Integrable Cases in Mechanics of Multidimensional Bodies,"
J. Math. Sci.
187 (3), 280-299 (2012). |
| 7. | D. V. Georgievskii,
"General Solutions of Weakened Equations in Terms
of Stresses in the Theory of Elasticity,"
"General Solutions of Elasticity Systems in Stresses
not Equivalent to the Classical Elasticity,"
Vestnik Moskov. Univ. Ser. I. Mat. Mekh.
67 (6), 26-32 (2012)
[Moscow Univ. Mech. Bull. (Engl. Transl.)
68 (1), 1-7 (2013)] |
|
| Received |
25 August 2013 |
| Link to Fulltext |
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