A problem of harmonic oscillations of a rectangle in the theory of micropolar elasticity: the analytical solution
https://doi.org/10.25587/SVFU.2023.93.57.002
Abstract
We consider the plane problem of natural harmonic oscillations of a rectangle with mixed boundary conditions in the framework of the linear micropolar theory of elasticity. The micropolar or Cosserat model is used for many modern materials with microstructure, when an elementary particle of a continuous medium has six degrees of freedom. A method for solving the original boundary value problem, when it is divided into separate sequences of consistent scalar boundary value problems, including one for rotational component, is proposed. It was revealed that in a micropolar medium there are two «sorts» of natural oscillations of a rectangle, one of which is bounded from below, while in a classical medium there is only one «sort» of natural oscillations and there are no such restrictions. The proposed method can be developed for the case of other boundary conditions and for the three-dimensional case.
About the Authors
Yu. M. Grigor’ev
Ammosov North-Eastern Federal University, Theoretical Physics Department; Academy of Sciences of the Republic of Sakha (Yakutia)
Russian Federation
Russian Federation
Yuriy M. Grigor’ev
58 Belinsky Street, Yakutsk, 677000 Russia
33 Lenin Avenue, Yakutsk 677007, Russia.
A. A. Gavrilieva
Larionov Institute of the Physical-Technical Problems of the North of the Siberian Branch of the RAS, Division of Federal Research Centre "The Yakut Scientific Centre of the Siberian Branch of the Russian Academy of Sciences"
Russian Federation
Russian Federation
Anna A. Gavrilieva
1 Oktyabrskaya street, Yakutsk 677980, Russia
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Review
For citations:
Grigor’ev Yu.M., Gavrilieva A.A. A problem of harmonic oscillations of a rectangle in the theory of micropolar elasticity: the analytical solution. Mathematical notes of NEFU. 2023;30(2):14-29. (In Russ.) https://doi.org/10.25587/SVFU.2023.93.57.002
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