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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
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Received 21 June 2010
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