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IssuesArchive of Issues2008-1pp.18-30

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G. L. Brovko and O. A. Ivanova, "Modeling of properties and motions of an inhomogeneous one-dimensional continuum of a complicated Cosserat-type microstructure," Mech. Solids. 43 (1), 18-30 (2008)
Year 2008 Volume 43 Number 1 Pages 18-30
DOI 10.3103/S0025654408010032
Title Modeling of properties and motions of an inhomogeneous one-dimensional continuum of a complicated Cosserat-type microstructure
Author(s) G. L. Brovko (Lomonosov Moscow State University, GSP-2, Leninskie Gory, Moscow, 119992, Russia, gkb@mech.math.msu.su)
O. A. Ivanova (Lomonosov Moscow State University, GSP-2, Leninskie Gory, Moscow, 119992, Russia, o.ivanova@tochka.ru)
Abstract We consider an approach to modeling the properties of the one-dimensional Cosserat continuum [1] by using the mechanical modeling method proposed by Il’yushin in [2] and applied in [3]. In this method, elements (blocks, cells) of special form are used to develop a discrete model of the structure so that the average properties of the model reproduced the properties of the continuum under study. The rigged rod model, which is an elastic structure in the form of a thin rod with massive inclusions (pulleys) fixed by elastic hinges on its elastic line and connected by elastic belt transmissions, is taken to be the original discrete model of the Cosserat continuum. The complete system of equations describing the mechanical properties and the dynamical equilibrium of the rigged rod in arbitrary plane motions is derived. These equations are averaged in the case of a sufficiently smooth variation in the parameters of motion along the rod (the long-wave approximation). It was found that the average equations exactly coincide with the equations for the one-dimensional Cosserat medium [1] and, in some specific cases, with the classical equations of motion of an elastic rod [4-6]. We study the plane motions of the one-dimensional continuum model thus constructed. The equations characterizing the continuum properties and motions are linearized by using several assumptions that the kinematic parameters are small. We solve the problem of natural vibrations with homogeneous boundary conditions and establish that each value of the parameter distinguishing the natural vibration modes is associated with exactly two distinct vibration mode shapes (in the same mode), each of which has its own frequency value.
References
1.  E. Cosserat and F. Cosserat, Theorie des Corps Deformables (Hermann, Paris, 1909).
2.  G. L. Brovko and A. A. Il’yushin, "On a Plane Model of Perforated plates," Vestnik Moskov. Univ. Ser. I. Mat. Mekh., No. 2, 83-91 (1993) [Moscow Univ. Math. Bull. (Engl. Transl.)].
3.  G. L. Brovko, "On one Structural Model of the Cosserat Medium," Izv. Akad. Nauk. Mekh. Tverd. Tela, No. 1, 75-91 (2002) [Mech. Solids (Engl. Transl.) 37 (1), 60-73 (2002)].
4.  A. A. Il’yushin and V. S. Lenskii, Strength of Materials (Fizmatgiz, Moscow, 1959) [in Russian].
5.  E. P. Popov, Theory and Calculation of Flexible Elastic Rods (Nauka, Moscow, 1986) [in Russian].
6.  A. E. H. Love, A Treatise on the Mathematical Theory of Elasticity, 4th ed. (Cambridge Univ. Press, Cambridge, 1927; ONTI, Moscow, 1935).
7.  A. A. Il’yushin, Continuum Mechanics (Izd-vo MGU, Moscow, 1990) [in Russian].
8.  E. N. Berezkin, Course of Theoretical Mechanics (Izd-vo MGU, Moscow, 1974) [in Russian].
9.  A. A. Il’yushin, "Functionals and Invertibility Measures on Sets of Processes in Continuum Mechanics," Dokl. Ross. Akad. Nauk 337(1), 48-50 (1994). [Russian Acad. Sci. Dokl. Math. (Engl. Transl.)]
10.  C. Truesdell, A First Course in Rational Continuum Mechanics (John Hopkins Univ., Baltimore, Maryland, 1972; Mir, Moscow, 1975).
11.  W. Noll, "A Mathematical Theory of the Mechanical Behavior of Continuous Media," Arch. Rat. Mech. Analysis 2(3), 197-226 (1958).
12.  G. L. Brovko, "Material and Spatial Representations of the Constitutive Relations of Deformable Media," Prikl. Mat. Mekh. 54(5), 814-824 (1990) [J. Appl. Math. Mech. (Engl. Transl.) 54 (5), 668-677 (1990)].
13.  A. A. Atoyan and S. O. Sarkisyan, "Study of Free Vibrations of Micropolar Elastic Thin Plates," Dokl. Akad. Nauk Armen. 104(4), 287-294 (2004).
Received 11 March 2005
Link to Fulltext http://www.springerlink.com/content/xp9434615339v676
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