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On some variational principles in micropolar theories of single-layer thin bodies

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Abstract

The generalized Reissner-type operator of three-dimensional micropolar mechanics of solids is presented, on the basis of which the generalized Reissner-type operator of three-dimensional micropolar mechanics of thin solids with one small size is obtained under the new parameterization of the domains of these bodies. From the last Reissner-type operator, in turn, the generalized Reissner-type variational principle of three-dimensional micropolar mechanics of thin solids with one small size is derived under the new parametrization of the domains of these bodies. It should be noted that the advantage of the new parameterization is that it is experimentally more accessible than other parameterizations (Nikabadze in Development of the method of orthogonal polynomials in the classical and micropolar mechanics of elastic thin bodies, MSU Publishing House, 2014; Contemp Math. Fundam Dir 55:3–194, 2015; J Math Sci 225:1, 2017). Further, applying the method of orthogonal polynomials (expansion of unknown quantities in series in terms of a system of orthogonal polynomials), from the generalized Reissner-type variational principle of three-dimensional micropolar mechanics of thin solids with one small size under the new parameterization of the domains of these bodies, the Reissner variational principle of micropolar mechanics of thin solids with one small size in the moments with respect to the system of Legendre polynomials is derived. In addition, the method is described for obtaining the variational principles of Lagrange and Castigliano of micropolar mechanics of thin solid with one small size under the new parametrization of the domains of these bodies in moments with respect to systems of the first and second kind Chebyshev polynomials. The paper is a continuation of the work “Nikabadze, Ulukhanyan, On some variational principles in the three-dimensional micropolar theories of solid”; therefore, before reading this paper, the authors invite the interested reader to familiarize themselves with the work (Nikabadze and Ulukhanyan in On some variational principles in the three-dimensional micropolar theories of solids, submitted).

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Notes

  1. A three-dimensional body, one size of which is smaller than the others, is called a thin body with one small size, and a solid body, two sizes of which are small compared to the third dimension, is called a thin body with two small dimensions.

  2.  The dependence of the quantities on \(x'\) means their dependence on the curvilinear coordinates \(x^1\) and \(x^2\) of the base surface. The usual rules of tensor calculus used in [2, 3, 21, 24,25,26,27] are applied. The notations and agreements adopted in the work [4] and also in previously published works are preserved [1,2,3,4, 29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47].

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Acknowledgements

The paper was published with the financial support of the Ministry of Education and Science of the Russian Federation as part of the program of the Moscow Center for Fundamental and Applied Mathematics under the Agreement No. 075-15-2019-1621 and of the Shota Rustaveli National Science Foundation (project No FR-21-3926).

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Authors and Affiliations

  1. Moscow Center for Fundamental and Applied Mathematics, Moscow, Russia

    M. Nikabadze & A. Ulukhanyan

  2. Lomonosov Moscow State University, Moscow, Russia

    M. Nikabadze

  3. Bauman Moscow State Technical University, Moscow, Russia

    M. Nikabadze

  4. International Research Center on Mathematics and Mechanics of Complex Systems (M&MoCS), L‘Aquila, Italy

    M. Nikabadze & A. Ulukhanyan

  5. Università degli Studi dell‘Aquila, L‘Aquila, Italy

    A. Ulukhanyan

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Nikabadze, M., Ulukhanyan, A. On some variational principles in micropolar theories of single-layer thin bodies. Continuum Mech. Thermodyn. 35, 1147–1164 (2023). https://doi.org/10.1007/s00161-022-01089-5

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