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IssuesArchive of Issues2011-3pp.467-479

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A.R. Ulukhanyan, "Dynamic Equations of the Theory of Thin Prismatic Bodies with Expansion in the System of Legendre Polynomials," Mech. Solids. 46 (3), 467-479 (2011)
Year 2011 Volume 46 Number 3 Pages 467-479
DOI 10.3103/S0025654411030137
Title Dynamic Equations of the Theory of Thin Prismatic Bodies with Expansion in the System of Legendre Polynomials
Author(s) A.R. Ulukhanyan (Lomonosov Moscow State University, GSP-2, Leninskie Gory, Moscow, 119992 Russia, armine_msu@mail.ru)
Abstract For a thin anisotropic body that is inhomogeneous with respect to curvilinear coordinates x1 and x2 and for an arbitrary homogeneous prismatic anisotropic elastic body of variable thickness with one small dimension in the case of the classical parametrization of its domain, we obtain the equations of motion of the Cosserat theory of elasticity in terms of moments with the kinematic boundary conditions of kinematic meaning and with boundary conditions of physical meaning taken into account.

For prismatic bodies of constant thickness, these equations are used to obtain the system of zero-order approximation equations for an isotropic medium. For the zero-order moments of the third components of the displacement and rotation vectors, fourth-order wave equations are obtained. In the classical theory, such equations are obtained in the first-order approximation for the zero- and first-order moments of the first invariant of the plane strain and for the third components of the displacement vector. In contrast to the Timoshenko type wave equation, in the equation for the zero-order moment of the third component of the displacement vector, the shear coefficient is k=1. Moreover, the plate cylindrical rigidity coincides with the rigidity obtained by I. N. Vekua, and for Poisson's ratio equal to 0.5, the coefficient multiplying the acceleration is zero.

In the case of a transversally isotropic medium, in the first- and second-order approximations, the fourth- and sixth-order hyperbolic equations are obtained for the zero-, first-, and second-order moments of the first invariant of the plane strain and of the third component of the displacement vector, respectively. The matrix of velocities of the wave propagation in an infinitely transversally isotropic elastic medium is written in the principal directions, and this matrix shows that the coefficients of these equations can be expressed in terms of these velocities. Most of this paper was published in [1].
Keywords Cosserat theory, prismatic body with one small dimension, system of Legendre polynomials, moments of function, hyperbolic equation
References
1.  A. R. Ulukhanyan, "Dynamical Equations of the Theory of Thin Prismatic Bodies with Expansion in Legendre Polynomials and Representation of Their Solution," Dep. VINITI RAN 15.05.09, No. 316-B2009.
2.  I. N. Vekua, Fundamentals of Tensor Analysis and Theory of Covariants (Nauka, Moscow, 1978) [in Russian].
3.  I. N. Vekua, Some General Methods of Constructing Various Versions of Shells Theory (Nauka, Moscow, 1982) [in Russian].
4.  P. K. Suetin, Classical Orthogonal Polynomials (Nauka, Moscow, 1976) [in Russian].
5.  S. Kaczmarz and H. Steinhaus, Theory of Orthogonal Series, and the review paper by R. S. Guter and P. L. Ul'yanov, Ed. by N. Ya. Vilenkin (Fizmatgiz, Moscow, 1958) [in Russian].
6.  N. N. Lebedev, Special Functions and Their Applications (Fizmatgiz, Moscow-Leningrad, 1963) [in Russian].
7.  M. U. Nikabadze and A. R. Ulukhanyan, "Mathematical Modeling of Elastic Thin Bodies with One Small Dimension by Using a System of Orthogonal Polynomials," Dep. VINITI RAN 21.08.2008, No. 723-B2008.
8.  V. D. Kupradze, T. G. Gegelia, M. O. Basheleishvili, and T. V. Burchuladze, Three-Dimensional Problems of Mathematical Theory of Elasticity and Thermoelasticity (Nauka, Moscow, 1976) [in Russian].
9.  W. Nowacki, Theory of Elasticity (PWN, Warsaw, 1970; Mir, Moscow, 1975).
10.  M. U. Nikabadze and A. R. Ulukhanyan, "Statements of Problems for a Thin Deformable Three-Dimensional Body," Vestnik Moskov. Univ. Ser. Mat. I. Mat. Mekh., No. 6, 43-49 (2005).
11.  M. U. Nikabadze, Several Problems of Tensor Calculus, Part II (TsPI MGU, Moscow, 2007) [in Russian].
12.  A. R. Ulukhanyan, "Wave Propagation under Transverse Vibrations of Beams and Plates," Prikl. Mat. Mekh. 12 (2), 287-300 (1948) [J. Appl. Math. Mech. (Engl. Transl.) 12 (2), 287-300 (1948)].
13.  E. I. Grigolyuk and I. T. Selezov, Nonclassical Theories of Rod, Plate, and Shell Vibrations, in Results in Science and Technology. Mechanics of Deformable Solids, Vol. 5 (VINITI, Moscow, 1973) [in Russian].
14.  E. I. Grigolyuk and G. M. Kulikov, "On the Shear Correction Factor in the Timoshenko Type Shell Theory," Dokl. Ross. Akad. Nauk 381 (1), 47-49 (2001) [Dokl. Phys. (Engl. Transl.) 46 (11), 797-799 (2001)].
15.  R. D. Mindlin, "Influence of Rotatory Inertia and Shear on Flexural Motions of Isotropic Elastic Plates," J. Appl. Mech. 18 (1), 31-38 (1951).
16.  I. T. Selezov, "Hypotheses Underlying Modified Equations of Transverse Vibrations of Plates and Some Singularities of These Equations," Prikl. Mekh. 7 (5), 538-546 (1961).
17.  A. R. Ulukhanyan, "Representation of Solutions to Equations of Hyperbolic Type," Vestnik Moskov. Univ. Ser. I. Mat. Mekh., No. 2, 62-66 (2010) [Moscow Univ. Mech. Bull. (Engl. Transl.) 65 (2), 47-50 (2010)].
18.  A. R. Ulukhanyan, "Dispersion Equations and Wave Propagation Velocities in Cosserat Theory," Dep. VINITI RAN 21.08.2008, No. 723-B2008.
19.  M. A. Kulesh, V. P. Matveenko, and I. N. Shardakov, "Constructing an Analytical Solution for Lamb Waves Using the Cosserat Continuum Approach," Zh. Prikl. Mekh. Tekhn. Fiz. 48 (1), 143-150 (2007) [J. Appl. Mech. Tech. Phys. (Engl. Transl.) 48 (1), 119-125 (2007)].
20.  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)].
Received 14 November 2008
Link to Fulltext http://www.springerlink.com/content/220x7p51w587510p/
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