Mechanics of Solids (about journal) 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
Russian Russian English English About Journal | Issues | Guidelines | Editorial Board | Contact Us
 
Issues > Archive of Issues > 2014-1 > pp.73-82

Archive of Issues

Total articles in the database: 12854
In Russian (Èçâ. ÐÀÍ. ÌÒÒ): 8044
In English (Mech. Solids): 4810
<< Previous article | Volume 49, Issue 1 / 2014 | Next article >>
V.I. Gorbachev and A.N. Emel'yanov, "Homogenization of the Equations of the Cosserat Theory of Elasticity of Inhomogeneous Bodies," Mech. Solids. 49 (1), 73-82 (2014)
Year 2014 Volume 49 Number 1 Pages 73-82
DOI 10.3103/S0025654414010099
Title Homogenization of the Equations of the Cosserat Theory of Elasticity of Inhomogeneous Bodies
Author(s) V.I. Gorbachev (Lomonosov Moscow State University, Leninskie Gory, Moscow, 119992 Russia, vigorby@mail.ru)
A.N. Emel'yanov (Lomonosov Moscow State University, Leninskie Gory, Moscow, 119992 Russia, emlaldr@gmail.com)
Abstract The paper deals with the homogenization of a boundary value problem for an inhomogeneous body with Cosserat properties, which is referred to as the original problem. The homogenization process is understood as a method for representing the solution of the original problem in terms of the solution of precisely the same problem for a body with homogeneous properties. The problem for a body with homogeneous properties is called the accompanying problem, and the body itself, the accompanying homogeneous body. As a rule, a constructive homogenization procedure includes the following three stages: at the first stage, the properties of the inhomogeneous body are used to find the properties of the accompanying homogeneous body (efficient properties); at the second stage, the boundary value problem is solved for the accompanying body; at the third stage, the solution of the accompanying problem is used to find the solution of the original problem. This approach was implemented in mechanics of composite materials constructed of numerous representative elements. A significant contribution to the development of mechanics of composites is due to Rabotnov [1-3] and his students. Recently, the homogenization method has been widely used to solve problems for composites of regular structure by expanding the solution of the original problem in a power series in a small geometric parameter equal to the ratio of the characteristic dimension of the periodicity cell to the characteristic dimension of the entire body. The papers by Bakhvalov [4-6] and Pobedrya [7] were the first in the field. At present, there are numerous monographs partially or completely dealing with the method of a small geometric parameter [8-14].

Isolated problems for inhomogeneous bodies with nonperiodic dependence of their properties on the coordinates were considered by many authors. Most of such papers published before 1973 are collected in two vast bibliographic indices [15, 16]. General methods were considered, and many specific problems of the theory of elasticity of continuously inhomogeneous bodies were solved in Lomakin's papers and his monograph [17]. The theory of torsion of inhomogeneous anisotropic rods was considered in [18].

In 1991, in his Doctoral dissertation, one of the authors of this paper proposed a version of the homogenization method based on an integral formula representing the solution of the original static problem of inhomogeneous elasticity via the solution of the accompanying problem [19, 20]. An integral formula for the dynamic problem of elasticity was published somewhat later [21]. This integral formula was used to develop a constructive method for the homogenization of the dynamic problem of inhomogeneous elasticity, which can be used in the case of both periodic and nonperiodic inhomogeneity of the properties [22]. The integral formula in the case of the Cosserat theory of elasticity was published in [23]. The present paper briefly presents constructive methods for homogenizing the problems of the Cosserat theory of elasticity based on the integral formula.
Keywords elasticity, inhomogeneous body, Cosserat theory, composite, homogenization
References
1.  Yu. N. Rabotnov, Mechanics of Deformable Solids (Nauka, Moscow, 1979) [in Russian].
2.  Yu. N. Rabotnov, Elements of Hereditary Mechanics of Solids (Nauka, Moscow, 1977) [in Russian].
3.  Yu. N. Rabotnov, Creep of Structural Members (Nauka, Moscow, 1966) [in Russian].
4.  N. S. Bakhvalov, "Averaged Characteristics of Bodies with Periodic Structure," Dokl. Akad. Nauk SSSR 218 (5), 1046-1048 (1974) [Soviet Phys. Dokl. (Engl. Transl.) 9 (10), 650-651 (1975)].
5.  N. S. Bakhvalov, "Homogenization of Partial Differential Equations with Rapidly Oscillating Coefficients," Dokl. Akad. Nauk SSSR 221 (3), 516-519 (1975).
6.  N. S. Bakhvalov and G. P. Panasenko, Homogenisation: Averaging Processes in Periodic Media (Nauka, Moscow, 1984) [in Russian].
7.  B. E. Pobedrya, Mechanics of Composite Materials (Izd-vo MGU, Moscow, 1984) [in Russian].
8.  V. L. Berdichevskii, Variational Principles of Continuum Mechanics (Nauka, Moscow, 1983) [in Russian].
9.  E. Sanchez-Palencia, Nonhomogeneous Media and Vibration Theory (Springer, Berlin, 1980; Mir, Moscow, 1984).
10.  I. V. Andrianov, V. A. Lesnichaya, and L. I. Manevich, Averaging Method in Statics and Dynamics of Ribbed Shells (Nauka, Moscow, 1985) [in Russian].
11.  A. L. Kalamkarov, Composite and Reinforced Elements of Construction (Wiley John, Baffins Lane, Chechester, West Sussex PO19, England, 1992).
12.  A. B. Movchan, N. V. Movchan, and C. G. Poulton, Asymptotic Models of Fields in Dilute and Densely Packed Composites (Imperial College Press, London, 2002).
13.  D. I. Bardzokas and A. I. Zobnin, Mathematical Modeling of Physical Processes in Composite Materials of Periodic Structure (Editorial URSS, Moscow, 2003) [in Russian].
14.  V. I. Bolshakov, I. V. Andrianov, and V. V. Danishevskii, Asymptotic Methods for Calculating Composite Materials with Periodic Structure Taken into Account (Porogi, Dnepropetrovsk, 2008) [in Russian].
15.  G. B. Kolchin and E. A. Faverman, Theory of Elasticity of Inhomogeneous Bodies. Bibliographic Index of Domestic and Foreign Literature (Shtiintsa, Kishinev, 1972) [in Russian].
16.  G. B. Kolchin and E. A. Faverman, Theory of Elasticity of Inhomogeneous Bodies. Bibliographic Index of Domestic and Foreign Literature for 1970-1973 (Shtiintsa, Kishinev, 1977) [in Russian].
17.  V. A. Lomakin, Theory of Elasticity of Inhomogeneous Bodies (Izd-vo MGU, Moscow, 1976) [in Russian].
18.  S. G. Lekhnitskii, Torsion of Anisotropic and Inhomogeneous Rods (Nauka, Moscow, 1971) [in Russian].
19.  V. I. Gorbachev, "Green Tensor Method for Solving Boundary Value Problems of the Theory of Elasticity for Inhomogeneous Media," Vych. Mekh. Sploshn. Sred No. 2, 61-76 (1991).
20.  V. I. Gorbachev, A Version of the Averaging Method for Solving Boundary-Value Problems of Inhomogeneous Elasticity, Doctoral Dissertation in Physics and Mathematics (Lomonosov Moscow State Univ., Moscow, 1991).
21.  V. I. Gorbachev and A. S. Kokarev, "Integral Formula in Dynamic Problem of Inhomogeneous Elasticity," Vestnik Moskov. Univ. Ser. I. Mat. Mekh., No. 2, 62-66 (2005).
22.  V. I. Gorbachev, "Dynamic Problems of Composite Mechanics," Izv. Ross. Akad. Nauk. Ser. Fiz. 75 (1), 117-122 (2011) [Bull. Russ. Acad. Sci. Phys. (Engl. Transl.) 75 (1), 110-115 (2011)].
23.  V. I. Gorbachev, "Integral formulas in Symmetric and Asymmetric Elasticity," Vestnik Moskov. Univ. Ser. I. Mat. Mekh. 64 (6), 57-60 (2009) [Moscow Univ. Mech. Bull. (Engl. Transl.) 64 (6), 148-151 (2009)].
24.  W. Nowacki, Dynamic Problems of Thermal Elasticity (Mir, Moscow, 1970) [in Russian].
25.  V. I. Gorbachev and A. N. Emel'yanov, "Homogenization of Problems of Cosserat Theory of Elasticity of Composites," in Additional Materials. Intern. Scientific Symposium in Problems of Mechanics of Deformable Solids Dedicated to A. A. Il'yushin on the Occasion of His 100th Birthday, January 20-21, 2011 (Izd-vo MGU, Moscow, 2012), pp. 81-88 [in Russian].
26.  Z. Hashin and B. W. Rosen, "The Elastic Moduli of Fiver-Reinforced Materials," J. Appl. Mech. 31 (2), 223-232 (1964).
27.  V. I. Gorbachev and L. V. Olekhova, "Effective Properties of a Nonuniform Beam under Torsion," Vestnik Moskov. Univ. Ser. I. Mat. Mekh. 62 (5), 41-48 (2007) [Moscow Univ. Mech. Bull. (Engl. Transl.) 62 (5), 123-130 (2007)].
28.  E. I. Grigolyuk and L. A. Filshtinskii, Perforated Plates and Shells (Nauka, Moscow, 1970) [in Russian].
29.  E. I. Grigolyuk and L. A. Filshtinskii, Regular Piecewise Homogeneous Structure with Defects (Fizmatlit, Moscow, 1994) [in Russian].
30.  G. A. Vanin, Micromechanics of Composite Materials (Naukova Dumka, Kiev, 1985) [in Russian].
31.  L. V. Olekhova, Torsion of Inhomogeneous Anisotropic Rod, Master's Thesis (Lomonosov Moscow State Univ., Moscow, 2009).
Received 05 September 2013
Link to Fulltext
<< Previous article | Volume 49, Issue 1 / 2014 | Next article >>
Orphus SystemIf you find a misprint on a webpage, please help us correct it promptly - just highlight and press Ctrl+Enter
101 Vernadsky Avenue, Bldg 1, Room 246, 119526 Moscow, Russia • (+7 495) 434-3538 • mechsol@ipmnet.ru • https://mtt.ipmnet.ru
Founders: Russian Academy of Sciences, Ishlinsky Institute for Problems in Mechanics RAS 		© Mechanics of Solids
webmaster		 Rambler's Top100