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IssuesArchive of Issues2011-1pp.96-103

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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
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