 | | 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 |
|
| << Previous article | Volume 46, Issue 1 / 2011 | Next article >> |
| D.V. Georgievskii, "Asymptotics of Solutions of the Three-Dimensional Elasticity Equations for Compressible and Incompressible Bodies," Mech. Solids. 46 (1), 96-103 (2011) |
| Year |
2011 |
Volume |
46 |
Number |
1 |
Pages |
96-103 |
| DOI |
10.3103/S0025654411010158 |
| Title |
Asymptotics of Solutions of the Three-Dimensional Elasticity Equations for Compressible and Incompressible Bodies |
| Author(s) |
D.V. Georgievskii (Lomonosov Moscow State University, GSP-2, Leninskie Gory, Moscow, 119992 Russia, georgiev@mech.math.msu.su) |
| Abstract |
We analyze the leading terms in the general asymptotic expansions of solutions of the first boundary value problem of three-dimensional elasticity in displacements. The cases of compressible and incompressible bodies, which have substantially different statements, are considered separately. The minimum-to-maximum ratio of characteristic dimensions of the elastic body is a natural small asymptotic parameter. The third dimension can be of any "intermediate" order, including the endpoints. For example, such a geometry is typical of bodies that simultaneously have characteristic macro-, micro-, and nano-dimensions in three coordinate axes.
An asymptotic analysis showed that a necessary condition for the existence and uniqueness of the leading terms of asymptotics of displacements in the interior of a three-dimensional thin body is that the orders (with respect to the small geometric parameter) of the displacement components prescribed on the boundary be related to each other in a certain way. Exact solutions of the leading approximation systems in displacements are written out. |
| Keywords |
Lamé equations, theory of thin bodies, asymptotic method, plate |
| References |
| 1. | V. M. Alexandrov, B. N. Smetanin, and B. V. Sobol',
Thin Stress Concentrators in Elastic Bodies
(Fizmatlit, Moscow, 1993)
[in Russian]. |
| 2. | S. A. Nazarov,
Asymptotic Theory of Thin Plates and Rods
Vol. 1: Dimension Reduction and Integral Estimates
(Nauchnaya Kniga, Novosibirsk, 2002)
[in Russian]. |
| 3. | M. U. Nikabadze,
"Some Issues Concerning a Version of the Theory of Thin Solids Based on
Expansions in a System of Chebyshev Polynomials of the Second Kind,"
Izv. Akad. Nauk. Mekh. Tverd. Tela,
No. 3, 73-106 (2007)
[Mech. Solids (Engl. Transl.)
42 (3), 391-421 (2007)]. |
| 4. | B. E. Pobedrya,
Numerical Methods in the Theory of Elasticity and Plasticity
(Izd-vo MGU, Moscow, 1995)
[in Russian]. |
| 5. | A. L. Gol'denveizer,
"Derivation of an Approximate Theory of Shells by Means of Asymptotic
Integration of the Equations of the Theory of Elasticity,"
Prikl. Mat. Mekh.
27 (4), 503-608 (1963)
[J. Appl. Math. Mech. (Engl. Transl.)
27 (4), 903-924 (1963)]. |
| 6. | M. I. Gusein-Zade,
"Asymptotic Analysis of Three-Dimensional Dynamic Equations of a Thin Plate,"
Prikl. Mat. Mekh.
38 (6), 1072-1078 (1974)
[J. Appl. Math. Mech. (Engl. Transl.)
38 (6), 1017-1023 (1974)]. |
| 7. | I. V. Simonov,
"Asymptotic Analysis of Three-dimensional Dynamic Equations
for Thin Two-Layer Elastic Plates,"
Prikl. Mat. Mekh.
53 (6), 976-982 (1989)
[J. Appl. Math. Mech. (Engl. Transl.)
53 (6), 772-778 (1989)]. |
| 8. | I. V. Simonov,
"Theory of Dynamic Bending of Thin Elastic High Nonhomogeneous Plates,"
Int. J. Solids Struct.
29 (21), 2597-2611 (1992). |
| 9. | A. M. Krivtsov and N. F. Morozov,
"Anomalies in Mechanical Characteristics of Nanometer-Size Objects,"
Dokl. Ross. Akad. Nauk
381 (3), 345-347 (2001)
[Dokl. Phys. (Engl. Transl.)
46 (11), 825-827 (2001)]. |
| 10. | E. A. Ivanova, D. A. Indeitsev, and N. F. Morozov,
"On the Determination of Rigidity Parameters for Nanoobjects,"
Dokl. Ross. Akad. Nauk
410 (6), 754-758 (2006)
[Dokl. Phys. (Engl. Transl.)
51 (10), 569-573 (2006)]. |
| 11. | V. M. Alexandrov,
"Asymptotic Solution of the Contact Problem for a Thin Elastic Layer,"
Prikl. Mat. Mekh.
33 (1), 61-73 (1969)
[J. Appl. Math. Mech. (Engl. Transl.)
33 (1), 49-63 (1969)]. |
| 12. | S. A. Nazarov,
Introduction to Asymptotic Methods of Elasticity Theory
(Izd-vo LGU, Leningrad, 1993)
[in Russian]. |
| 13. | I. I. Argatov,
"Refinement of the Asymptotic Solution Obtained by the Method of Splicing Expansions
in the Contact Problem of Elasticity Theory,"
Zh. Vych. Mat. Mat. Fiz.
40 (4), 623-632 (2000)
[Comput. Math. Math. Phys. (Engl. Transl.)
40 (4), 594-603 (2000)]. |
| 14. | A. D. Bruno,
Power Geometry in Algebraic and Differential Equations
(Fizmatlit, Moscow, 1998; Elsevier, Amsterdam, 2000). |
| 15. | M. V. Fedoryuk,
Saddle-Point Method
(Nauka, Moscow, 1977)
[in Russian]. |
| 16. | V. F. Kravchenko, G. A. Nesenenko, and V. I. Pustovoit,
Poincaré Asymptotics of Solution to Problems of Irregular
Heat and Mass Transfer
(Fizmatlit, Moscow, 2006)
[in Russian]. |
| 17. | D. V. Georgievskii,
"Asymptotics with Respect to a Small Geometric Parameter for Solutions of Three-Dimensional Lamé Equations,"
Russ. J. Math. Phys.
16 (1), 74-80 (2009). |
| 18. | D. V. Georgievskii,
"On Asymptotics with Respect to a Small Geometric Parameter
for Solutions of the First Boundary Value Problem
of Elasticity Theory,"
in Proc. 10th Intern. Sci. Conf. "Modern Problems of
Mathematics, Mechanics, and Informatics"
(Izd-vo TulGU, Nula, 2009),
pp. 152-154
[in Russian]. |
| 19. | D. V. Georgievskii,
"Stress Problems for a Compressible Viscous Fluid (Stokes' Approximation),"
Dokl. Ross. Akad. Nauk
409 (5), 615-618 (2006)
[Dokl. Phys. (Engl. Transl.)
51 (8), 433-436 (2006)]. |
|
| Received |
04 December 2008 |
| Link to Fulltext |
|
| << Previous article | Volume 46, Issue 1 / 2011 | Next article >> |
|
 If you find a misprint on a webpage, please help us correct it promptly - just highlight and press Ctrl+Enter
|
|
Russian 
English