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Olga Markelova
Laboratory of Analysis and Synthesis of Dynamic Systems in Precision Mechanics, Institute of Precision Mechanics and Control, Russian Academy of Sciences, Rabochaya 24, Saratov, Russia, 410028
Astronomy Chemistry Earth & Planetary Sciences Mathematics Physics

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Analytical Solution for Bending and Free Vibrations of an Orthotropic Nanoplate based on the New Modified Couple Stress Theory and the Third-order Plate Theory Marina Barulina; Dmitry Kondratov; Sofia Galkina; Olga Markelova
Journal of Mathematical and Fundamental Sciences Vol. 54 No. 1 (2022)
Publisher : Institute for Research and Community Services (LPPM) ITB

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.5614/j.math.fund.sci.2022.54.1.2

Abstract

In the present work, the equations of motion of a thin orthotropic nanoplate were obtained based on the new modified couple stress theory and the third-order shear deformation plate theory. The nanoplate was considered as a size-dependent orthotropic plate. The governing equations were derived using the dynamic version of Hamilton’s principle and natural boundary conditions were formulated. An analytical solution in the form of a double Fourier series was obtained for a simply supported rectangular nanoplate. The eigenvalue problem was set and solved. It was analytically shown that the displacements of the median surface points in the plane of the plate do not depend on the material length scale parameters in the same directions; these in-plane directional displacements depend on the material length scale parameter in the out-of-plane direction only. On the other hand, the out-of-plane directional displacement depends on the length scale parameter in the plane directions only. The cross-section rotation angles depend on all length scale parameters. It was shown that the size-dependent parameters only have a noticeable effect on the deformed state of the plate if their order is not less than the order (plate height)-1.
Co-Authors Dmitry Kondratov Marina Barulina Sofia Galkina