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Generalized Reissner-type variational principles in the micropolar theories of multilayer thin bodies with one small size

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Abstract

The effective parametrization of a multilayer thin domain, called a new parametrization, is considered and consists in using, in contrast to the classical approaches, several base surfaces. In addition, the new parameterization is characterized by the fact that it is experimentally more accessible than other parameterizations used in the scientific literature, since the front surfaces are used as basic ones. Also, when obtaining any relation (a system of equations, constitutive relations, boundary and initial conditions, variational principles, etc.) in the moments of the theory of multilayer thin bodies under the new parametrization of the domain of a thin body, it is sufficient in the corresponding relation of the theory of a single-layer thin body under the root letters of the quantities to supply the index \(\alpha \), which denotes the number of the layer \(\alpha \) and gives these index values from 1 to K, where K is the number of layers. Therefore, for the correct statement of the initial-boundary value problems to the equations of motion and the boundary and initial conditions in the moments, it is also necessary to add interlayer contact conditions, which must also be taken into account when writing the variational operators and formulating the variational principles. What has been said above can be called the rule of obtaining the desired relation in the theory of multilayer thin bodies from the corresponding relation in the theory of single-layer thin bodies. Applying this rule, below we give the representation of the generalized Reissner-type operator and formulate the generalized Reissner-type variational principle both in the case of full contact of adjacent layers of a multilayer structure and in the presence of zones of weakened adhesion. The description of obtaining of dual operators and variational principles of Reissner-type, as well as of Lagrangian and Castiglianian and variational principles of Lagrange and Castigliano, is given. In the presence of domains of weakened adhesion at interphase boundaries in a multilayer thin body, one of the main problems is the problem of modeling the interface (interphase boundary). In this paper, the jump-type model (description of the interface by a surface of zero thickness) is presented in comparative detail.

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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 [19, 40,41,42,43,44,45] are applied. The notations and agreements adopted in previously published works (see [22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 46, 47, 48, 49, and others]) are preserved.

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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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  1. Lomonosov Moscow State University, Moscow, Russian Federation

    Mikhail Nikabadze

  2. Bauman Moscow State Technical University, Moscow, Russian Federation

    Mikhail Nikabadze

  3. Moscow Center for Fundamental and Applied Mathematics, Moscow, Russian Federation

    Mikhail Nikabadze & Armine Ulukhanyan

  4. International Research Center on Mathematics and Mechanics of Complex Systems (M&MoCS), Universitá degli Studi dell’Aquila, L’Aquila, Italy

    Mikhail Nikabadze & Armine Ulukhanyan

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Nikabadze, M., Ulukhanyan, A. Generalized Reissner-type variational principles in the micropolar theories of multilayer thin bodies with one small size. Continuum Mech. Thermodyn. 35, 1207–1221 (2023). https://doi.org/10.1007/s00161-022-01091-x

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