  Mechanics of Solids A Journal of Russian Academy of Sciences   Founded
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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), 467479 (2011) 
Year 
2011 
Volume 
46 
Number 
3 
Pages 
467479 
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, GSP2, Leninskie Gory, Moscow, 119992 Russia, armine_msu@mail.ru) 
Abstract 
For a thin anisotropic body that is inhomogeneous with respect to curvilinear coordinates x^{1} and x^{2} 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 zeroorder approximation equations for an isotropic medium. For the zeroorder moments of the third components of the displacement and rotation vectors, fourthorder wave equations are obtained. In the classical theory, such equations are obtained in the firstorder approximation for the zero and firstorder 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 zeroorder 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 secondorder approximations, the fourth and sixthorder hyperbolic equations are obtained for the zero, first, and secondorder 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 
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Received 
14 November 2008 
Link to Fulltext 
http://www.springerlink.com/content/220x7p51w587510p/ 
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