Open AccessAnalytical Solution for Bending and Free Vibrations of an Orthotropic Nanoplate based on the New Modified Couple Stress Theory and the Third-order Plate Theory
Author(s) -
М. А. Барулина,
Dmitry Kondratov,
Sofia Galkina,
O. V. Markelova
Publication year - 2022
Publication title -
journal of mathematical and fundamental sciences/journal of mathematical and fundamental siences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.216
H-Index - 12
eISSN - 2337-5760
pISSN - 2338-5510
DOI - 10.5614/j.math.fund.sci.2022.54.1.2
Subject(s) - orthotropic material , plate theory , bending of plates , length scale , boundary value problem , fourier series , bending , geometry , vibration , materials science , plane (geometry) , mathematics , mathematical analysis , equations of motion , mechanics , classical mechanics , physics , composite material , finite element method , quantum mechanics , thermodynamics
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.