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Some Applications of Eigenvalue Problems for Tensor and Tensor–Block Matrices for Mathematical Modeling of Micropolar Thin Bodies

1
Faculty of Mechanics and Mathematics, Lomonosov Moscow State University, 119991 Moscow, Russia
2
Department of Computational Mathematics and Mathematical Physics, Bauman Moscow State Technical University, 105005 Moscow, Russia
*
Author to whom correspondence should be addressed.
Math. Comput. Appl. 2019, 24(1), 33; https://doi.org/10.3390/mca24010033
Received: 26 January 2019 / Revised: 20 March 2019 / Accepted: 21 March 2019 / Published: 22 March 2019
(This article belongs to the Special Issue Mathematical Modeling in Physical Sciences)
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Abstract

The statement of the eigenvalue problem for a tensor–block matrix (TBM) of any order and of any even rank is formulated, and also some of its special cases are considered. In particular, using the canonical presentation of the TBM of the tensor of elastic modules of the micropolar theory, in the canonical form the specific deformation energy and the constitutive relations are written. With the help of the introduced TBM operator, the equations of motion of a micropolar arbitrarily anisotropic medium are written, and also the boundary conditions are written down by means of the introduced TBM operator of the stress and the couple stress vectors. The formulations of initial-boundary value problems in these terms for an arbitrary anisotropic medium are given. The questions on the decomposition of initial-boundary value problems of elasticity and thin body theory for some anisotropic media are considered. In particular, the initial-boundary problems of the micropolar (classical) theory of elasticity are presented with the help of the introduced TBM operators (tensors–operators). In the case of an isotropic micropolar elastic medium (isotropic and transversely isotropic classical media), the TBM operator (tensors–operators) of cofactors to TBM operators (tensors–tensors) of the initial-boundary value problems are constructed that allow decomposing initial-boundary value problems. We also find the determinant and the tensor of cofactors to the sum of six tensors used for decomposition of initial-boundary value problems. From three-dimensional decomposed initial-boundary value problems, the corresponding decomposed initial-boundary value problems for the theories of thin bodies are obtained. View Full-Text
Keywords: tensor–operator of equations; stress tensor–operator; tensor–operator of stress and couple stress; tensor–block matrix operator; canonical presentation of tensor–block matrix; eigenoperator tensor–operator of equations; stress tensor–operator; tensor–operator of stress and couple stress; tensor–block matrix operator; canonical presentation of tensor–block matrix; eigenoperator
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited (CC BY 4.0).
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Nikabadze, M.; Ulukhanyan, A. Some Applications of Eigenvalue Problems for Tensor and Tensor–Block Matrices for Mathematical Modeling of Micropolar Thin Bodies. Math. Comput. Appl. 2019, 24, 33.

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