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On the Theory of Multilayer Thin Bodies

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Abstract

Using the basic recurrence formulas for Chebyshev polynomials of the second kind, the several additional relationships have been obtained which play an important role in the construction of various variants of the theory of thin bodies. The moments of the tensor functions, as well as the moments of their derivatives and the moments of the repeated derivatives are determined, too. The moments of the \(k\)th order of some expressions with respect to Chebyshev polynomials are found. Representations of the equations of motion with respect to the contravariant components of the stress and couple-stress tensors, the heat flow equation, constitutive relations of the micropolar theory, and the Fourier heat conduction law of the \(s\)th order approximation are given. From them it is easy to get the corresponding relations in the moments with respect to the systems of Chebyshev and Legendre polynomials. As a particular case, the zero and first approximations motion equations in moments with respect to the contravariant components of the stress and couple-stress tensors are written out, and also the systems of equations in the displacements of the zero and first approximations in moments for non isothermal processes for any anisotropic material are given.

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Notes

  1. Three-dimensional bodies whose one or two sizes are less than the others are called thin bodies, and also a two-dimensional domain whose one size is less than the other is called a thin domain.

  2. We use the usual rules of tensor calculus [20-22, 43, 44, 50]. We mainly preserve the notation and conventions of the previous works. Under symbols, we write indices denoting the serial numbers of layers . The Greek indices under symbols assume their values according to circumstances, and capital and small Latin indices assume the values 1, 2 and 1, 2, 3, respectively.

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Funding

This work was supported by the Russian Foundation for Basic Research, grants no. 18-29-10085-mk and no. 19-01-00016-A and by 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.

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  1. Lomonosov Moscow State University, 119991, Moscow, Russia

    M. Nikabadze

  2. Bauman Moscow State Technical University, 107005, Moscow, Russia

    M. Nikabadze & A. Ulukhanyan

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  1. M. Nikabadze
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  2. A. Ulukhanyan
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Correspondence to M. Nikabadze or A. Ulukhanyan.

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(Submitted by A. M. Elizarov)

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Nikabadze, M., Ulukhanyan, A. On the Theory of Multilayer Thin Bodies. Lobachevskii J Math 42, 1900–1911 (2021). https://doi.org/10.1134/S1995080221080217

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