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 > 2013-3 > pp.317-328

Archive of Issues

Total articles in the database: 11223
In Russian (Èçâ. ÐÀÍ. ÌÒÒ): 8011
In English (Mech. Solids): 3212
<< Previous article | Volume 48, Issue 3 / 2013 | Next article >>
M.M. Kantor, M.U. Nikabadze, and A.R. Ulukhanyan, "Equations of Motion and Boundary Conditions of Physical Meaning of Micropolar Theory of Thin Bodies with Two Small Cuts," Mech. Solids. 48 (3), 317-328 (2013)
Year 2013 Volume 48 Number 3 Pages 317-328
DOI 10.3103/S0025654413030084
Title Equations of Motion and Boundary Conditions of Physical Meaning of Micropolar Theory of Thin Bodies with Two Small Cuts
Author(s) M.M. Kantor (Lomonosov Moscow State University, Leninskie Gory 1, Moscow, 119991 Russia)
M.U. Nikabadze (Lomonosov Moscow State University, Leninskie Gory 1, Moscow, 119991 Russia, munikabadze@yandex.ru)
A.R. Ulukhanyan (Lomonosov Moscow State University, Leninskie Gory 1, Moscow, 119991 Russia, armine_msu@mail.ru)
Abstract Nowadays, microcontinuous mechanics (mechanics of media with microstructure) is being developed very intensively, which is testified by recently published papers [1-14] and by many others, as well as by the symposium dedicated to the hundredth anniversary of the brothers Cosserat monograph [15], held in Paris in 2009. A survey of foreign papers is given in [16], and a special place is occupied by earlier publications of Soviet scientists on micropolar theory of elasticity [17-24]. A brief survey of Cosserat theory of elasticity and an analysis and prospects of such theories in mechanics of rigid deformable bodies is given in [21].

It should be noted that, in a majority of cases, the structure strength calculations are based on the classical theory of elasticity. But there are materials such as animal bones, graphite, several polymers, polyurethane films, porous materials (pumice), various synthetic materials, and materials with inclusions which, under certain conditions, exhibit micropolar properties. There are effects which cannot be prescribed by the classical theory. In statics, nonclassical behavior can be observed in bending of thin films and cantilevers, in torsion of thin and thin-walled rods, and in the case of stress concentration near holes, corner points, cracks, and inclusions. For example, thin specimens are more rigid in bending and torsion as is prescribed by the classical theory [25-27]. The stress concentration near holes decreases, and the concentration factor depends on the radius [28]. The stress concentration near cracks also becomes lower. Conversely, the stress concentration near inclusions is higher than predicted by the classical theory [29-31]. If the material has no center of symmetry of elastic properties, then calculations according to the micropolar theory shows that the specimen is twisted in tension [32]. In dynamical problems, several phenomena also differ from the classical concepts. For example, shear waves propagate with dispersion, microrotation waves arise, and the vibration natural modes differ from the classical ones [2, 7, 11-13, 33]. All these phenomena are used to determine material constants of the micropolar theory of elasticity. There are many methods for determining such constants [2, 34].

Since thin bodies (one-, two-, three-, and multilayer structures) are widely used, it is necessary to create new refined microcontinual theories of thin bodies and advanced methods for their computations. In the present paper, various representations of the system of equations of motion are obtained in the micropolar theory of thin bodies with two small parameters in momenta with respect to a system of Legendre polynomials in the case where an arbitrary line is taken for the base. In this connection, a vector parametric equation of the region of a thin body is given for the parametrization under study, different families of bases (frames) are introduced, and expressions for components of the unit tensor of rank two (UTRT) are obtained. Representations of gradient, tensor divergence, equations of motion, and boundary conditions for the considered parametrization are given. Definitions of (m,n)th-order moment of a variable with respect to an arbitrary system of orthogonal polynomials and a system of Legendre polynomials is given. Expressions for the moments of partial derivatives and several expressions with respect to a system of Legendre polynomials and boundary conditions in moments are obtained.
Keywords thin body with two small dimensions, Legendre polynomials, (m,n)th-order moment, micropolar theory of thin bodies
References
1.  A. A. Il'yushin, "Nonsymmetry of Strain and Stress Tensors in Continuum Mechanics," Vestnik Moskov. Univ. Ser. I Mat. Mekh., No. 5, 6-14 (1996).
2.  A. C. Eringen, Microcontinuum Field Theories. I. Foundation and Solids (Springer, New York, 1998).
3.  S. A. Ambartsumyan, Micropolar Theory of Shells and Plates (Izd-vo NAN Armenii, Erevan, 1999) [in Russian].
4.  M. N. Mutafyan and S. O. Sarkisyan, "Asymptotic Solutions of Boundary-Value Problems for Thin Rectangle according to Nonsymmetric Elasticity," Izv. Akad. Nauk Armyan. SSR Ser. Mat. 57 (1), 41-58 (2004) [J. Contemporary Math. Anal. (Engl. Transl.)].
5.  G. L. Brovko, "Modeling of Inhomogeneous Media of Complex Structure and Cosserat Continuum," Vestnik Moskov. Univ. Ser. I. Mat. Mekh., No. 5, 55-63 (1996).
6.  G. L. Brovko and O. A. Ivanova, "Modeling of Properties and Motions of an Inhomogeneous One-Dimensional Continuum of a Complicated Cosserat-Type Microstructure," Izv. Akad. Nauk. Mekh. Tverd. Tela, No. 1, 22-36 (2008) [Mech. Solids (Engl. Transl.) 43 (1), 18-30 (2008)].
7.  V. I. Erofeev, Wave Processes in Solids with Microstructure (Izd-vo MGU, Moscow, 1999) [in Russian].
8.  B. E. Pobedrya, "Versions of Modeling in Mechanics of Deformable Body," in Fundamental and Applied Problems of Mechanics. Proc. Intern. Sci. Conf., Vol. 1 (Izd-vo KhGTU, Khabarovsk, 2003), pp. 20-29 [in Russian].
9.  B. E. Pobedrya, "Static Problem of the Nonsymmetric Elasticity Theory for an Isotropic Medium," Vestnik Moskov. Univ. Ser. I. Mat. Mekh., No. 1, 54-59 (2005) [Moscow Univ. Math. Bull. (Engl. Transl.) 60 (1), 14-19 (2005)].
10.  B. E. Pobedrya and S. E. Omarov, "Constitutive Relations of Couple Stress Theory of Elasticity," Vestnik Moskov. Univ. Ser. I. Mat. Mekh., No. 3, 56-58 (2007).
11.  M. A. Kulesh, V. P. Matveenko, and I. N. Shardakov, "Construction and Analysis of an Analytical Solution for the Surface Rayleigh Wave within the Framework of the Cosserat Continuum," Zh. Prikl. Mekh. Tekhn. Fiz. 46 (4), 116-124 (2005) [J. Appl. Mech. Tech. Phys. (Engl. Transl.) 46 (4), 556-563 (2005)].
12.  M. A. Kulesh, V. P. Matveenko, and I. N. Shardakov, "Propagation of Surface Elastic Waves in the Cosserat Medium," Akust. Zh. 52 (2), 227-235 (2006) [Acoust. Phys. (Engl. Transl.) 52 (2), 186-193 (2006)].
13.  M. A. Kulesh, V. P. Matveenko, and I. N. Shardakov, "Dispersion and Polarization of Surface Rayleigh Waves for the Cosserat Continuum," Izv. Akad. Nauk. Mekh. Tverd. Tela, No. 4, 100-113 (2007) [Mech. Solids (Engl. Transl.) 42 (4), 583-594 (2007)].
14.  M. U. Nikabadze, M. M. Kantor, and A.R. Ulukhanyan, "To Mathematical Modeling of Elastic Thin Bodies and Numerical Realization of Several Problems about a Strip," Dep. VINITI RAN 29.04.11, No. 204-B2011.
15.  E. Cosserat and F. Cosserat, Théorie des Corps Déformables (Hermann, Paris, 1909).
16.  M. Ostoja-Starzewski and I. Jasiuk, "Stress Invariance in Planar Cosserat Elasticity," Proc. Roy. Soc. London. Ser. A 451 (1942), 453-470 (1995).
17.  E. L. Aero and E. V. Kuvshinskii, "Fundamental Equations of the Theory of Elastic Media With Rotationally Interacting Particles," Fiz. Tverd. Tela 2 (7), 1399-1409 (1960) [Sov. Phys. Solid State (Engl. Transl.) 2, 1272-1281 (1960)].
18.  E. V. Kuvshinskii and E. L. Aero, "Continual Theory of Axisymmetric Elasticity. Taking Account of "Internal" Rotation," Fiz. Tverd. Tela 5 (9), 2591-2598 (1963) [Sov. Phys. Solid State (Engl. Transl.)].
19.  V. A. Pal’mov, "Fundamental Equations of the Theory of Asymmetric Elasticity," Prikl. Mat. Mekh. 28 (3), 401-408 (1964) [J. Appl. Math. Mech. (Engl. Transl.) 28 (3), 496-505 (1964)].
20.  G. N. Savin, Foundations of the Plane Problem of the Couple Stress Theory of Elasticity (Izdat. Kiev Gos. Univ., Kiev, 1965) [in Russian].
21.  A. A. Il’yushin and V. A. Lomakin, Moment Theories in Mechanics of Rigid Deformable Bodies (Nauka, Moscow, 1971) [in Russian].
22.  A. I. Kalandiya, Mathematical Methods of Two-Dimensional Elasticity (Nauka, Moscow, 1973) [in Russian].
23.  V. D. Kupradze, T. G. Gegelia, M. O. Vesheleishvili, and T. V. Burchuladze, Three-Dimensional Problems of Mathematical Theory of Elasticity and Thermoelasticity (Nauka, Moscow, 1976) [in Russian].
24.  I. A. Kunin, Elastic Media with Microstructure (Nauka, Moscow, 1975) [in Russian].
25.  R. D. Gaudhier and W. E. Jahsman, "A Quest for Micropolar Elastic Constants," Trans. ASME J. Appl. Mech. 42 (2), 369-374 (1975).
26.  R. D. Gaudhier and W. E. Jahsman, "Bending of a Curved Bar of Micropolar Elastic Material," Trans. ASME J. Appl. Mech. 43 (3), 502-503 (1976).
27.  R. D. Gaudhier, "Experimental Investigations on Micropolar Media," in Mechanics of Micropolar Media, Ed. by R. K. Brulin and T. Hsieh (World Scient., Singapore, 1982), pp. 395-463.
28.  R. D. Mindlin, "Influence of Couple Stresses on Stress Concentration," Experim. Mech. 3 (1), 1-7 (1963).
29.  S. Itou, "The Effect of Couple-Stresses on the Stress Concentration around an Elliptic Hole," Acta Mech. 16 (3-4), 289-296 (1973).
30.  B. S. Kim and A. C. Eringen, "Stress Distribution around an Elliptic Hole in an Infinite Micropolar Elastic Plate," Lett. Appl. Engng Sci. 1 (4), 381-390 (1973).
31.  S. Nakamura, R. Benedict, and R. S. Lakes, "Finite Element Method for Orthotropic Micropolar Elasticity," Int. J. Engng Sci. 22 (3), 319-330 (1984).
32.  R. S. Lakes and R. Benedict, "Noncentrosymmetry in Micropolar Elasticity," Int. J. Engng Sci. 29 (10), 1161-1167 (1982).
33.  A. C. Eringen, "Theory of Micropolar Elasticity," in Fracture, Vol. 1, Ed. by H. Liebowitz (Acad. Press, New York, 1968), pp. 621-729.
34.  R. S. Lakes, "Experimental Methods for Study of Cosserat Elastic Solids and Other Generalized Continua," in Continuum Media for Materials with Microstructure, Ch. 1, Ed. by I. Muhlhaus (Wiley, New York, 1995), pp. 1-22.
35.  I. N. Vekua, Fundamentals of Tensor Analysis and Theory of Covariants (Nauka, Moscow, 1978) [in Russian].
36.  A. I. Lurie, Nonlinear Theory of Elasticity (Nauka, Moscow, 1980) [in Russian].
37.  B. E. Pobedrya, Lectures on Tensor Analysis (Izd-vo MGU, Moscow, 1986) [in Russian].
38.  Yu. I. Dimitrienko, Tensor calculus (Vysshaya Shkola, Moscow, 2001) [in Russian].
39.  A. V. Pogorelov, Geometry (Nauka, Moscow, 1983) [in Russian].
40.  W. Nowacki, Theory of Elasticity (PWN, Warsaw, 1970; Mir, Moscow, 1975).
41.  M. U. Nikabadze, "On Some Problems of Tensor Calculus. I," in Contemporary Mathematics and Its Applications, Vol. 62 (VINITI, Moscow, 2009), pp. 67-95 [in Russian].
42.  M. U. Nikabadze, "On Some Problems of Tensor Calculus. II," in Contemporary Mathematics and Its Applications, Vol. 62 (VINITI, Moscow, 2009), pp. 96-130 [in Russian].
43.  M. U. Nikabadze, "On Some Problems of Tensor Calculus. I," J. Math. Sci., 161 (5), 668-697 (2009).
44.  M. U. Nikabadze, "On Some Problems of Tensor Calculus. II," J. Math. Sci., 161 (5), 698-733 (2009).
45.  P. K. Suetin, Classical Orthogonal Polynomials (Nauka, Moscow, 1976) [in Russian].
46.  M. U. Nikabadze, "A Version of the System of Equations of the Theory of Thin Solids," Vestnik Moskov. Univ. Ser. I. Mat. Mekh., No. 1, 30-35 (2006).
47.  M. U. Nikabadze, "Use of Classical Orthogonal Polynomials for Constructing the Theory of Thin Solids," in Elasticity and Inelasticity, Materials of Intern. Sci. Symp. in Problems of Mechanics of Deformable Bodies Dedicated to A. A. Il'yushin on the Occasion of this 95th Birthday (LENAND, Moscow, 2006), pp. 218-228 [in Russian].
48.  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)].
49.  M. U. Nikabadze, "Application of Chebyshev Polynomials to the Theory of Thin Bodies," Vestnik Moskov. Univ. Ser. I. Mat. Mekh., 62 (5), 56-63 (2007) [Moscow Univ. Mech. Bull. (Engl. Transl.) 62 (5), 141-148 (2007)].
50.  M. U. Nikabadze, "Application of a System of Legendre and Chebyshev Polynomials in Modeling of Elastic Thin Bodies with One Small Dimension," Dep. VINITI RAN 21.08.08, No. 720-B2008.
51.  M. U. Nikabadze, "Mathematical Modeling of Elastic Thin Bodies with Two Small Dimensions by Using Systems of Orthogonal Polynomials," Dep. VINITI RAN 21.08.08, No. 722-B2008.
52.  I. N. Vekua, Some General Methods of Constructing Various Versions of Shells (Nauka, Moscow, 1982) [in Russian].
Received 21 June 2010
Link to Fulltext
<< Previous article | Volume 48, Issue 3 / 2013 | 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