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Editor-in-Chief
Nikiforov
Vladimir O.
D.Sc., Prof.
Partners
doi: 10.17586/2226-1494-2024-24-4-629-636
Instability of a rectangular CCCC-nanoplate
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Article in Russian
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Abstract
For citation:
Sukhoterin M.V., Sosnovskaya A.A. Instability of a rectangular CCCC-nanoplate. Scientific and Technical Journal of Information Technologies, Mechanics and Optics, 2024, vol. 24, no. 4, pp. 629–636 (in Russian). doi: 10.17586/2226-1494-2024-24-4-629-636
Abstract
The spectrum of critical loads and equilibrium forms of a CCCC-nanoplate (C — clamped edge) at various values of a non–local nanoparameter has been studied. The symmetric solution is represented by two hyperbolo-trigonometric series in two coordinates which obeyed the basic differential equation of the physical state of Eringen. The boundary conditions for the absence of deflections and angles of rotation of the pinched faces were precisely satisfied. As a result, a homogeneous infinite system of linear algebraic equations with respect to unknown coefficients of these series is obtained containing a relative compressive load as the main parameter. After the conversion, the system began to contain only one sequence of coefficients. An iterative process of searching for a non-trivial solution in combination with the method of iterating over the load value is constructed. For each value of the nonlocal parameter, the first four critical loads for symmetric forms of supercritical equilibrium are found and their 3D images are obtained. It was found that critical loads decreased with an increase in the nonlocal parameter. The influence of the number of members held in rows and the number of iterations on the accuracy of the results is investigated. The results obtained can be used in the design of various nanoscale smart structures.
Keywords: rectangular CCCC-nanoplate, Eringen theory, critical load spectrum, Fourier series
References
References
- Eringen A.C. Nonlocal Continuum Field Theories. New York: Springer, 2002, 376 p.
- Wang L.F., Hu H.Y. Flexural wave propagation in single-walled carbon nanotubes. Physical Review B, 2005, vol. 71, no. 19, pp. 195412. https://doi.org/10.1103/PhysRevB.71.195412
- Mikhasev G.I., Avdeichik E.V., Kaplunov Iu.D., Prikazchikov D.A. Study of free longitudinal vibrations of a nano-sized beam from the standpoint of Eringen's two-phase nonlocal elasticity theory. Theoretical and Applied Mechanics: International Scientific and Technical Collection. Vol. 33. Minsk, BNTU, 2018, pp. 72–80. (in Russian)
- Benvenuti E., Simone A. One-dimensional nonlocal and gradient elasticity: Closed-form solution and size effect. Mechanics Research Communications, 2013, vol. 48, pp. 46–51. https://doi.org/10.1016/j.mechrescom.2012.12.001
- Farajpour A., Danesh M., Mohammadi M. Buckling analysis of variable thickness nanoplates using nonlocal continuum mechanics. Physica E: Low-dimensional systems and Nanostructures, 2011, vol. 44, no. 3, pp. 719–727. https://doi.org/10.1016/j.physe.2011.11.022
- Bastami M., Behjat B. Ritz solution of buckling and vibration problem of nanoplates embedded in an elastic medium. Sigma Journal of Engineering and Natural Sciences, 2017, vol. 35, no. 2, pp. 285–302.
- Ebrahimi F., Barati M.R. Buckling analysis of piezoelectrically actuated smart nanoscale plates subjected to magnetic field. Journal of Intelligent Material Systems and Structures, 2017, vol. 28, no. 11, pp. 1472–1490. https://doi.org/10.1177/1045389x16672569
- Wang Z., Xing Y., Sun Q., Yang Y. Highly accurate closed-form solutions for free vibration and eigenbuckling of rectangular nanoplates. Composite Structures, 2019, vol. 210, pp. 822–830. https://doi.org/10.1016/j.compstruct.2018.11.094
- Chwał M., Muc A. Buckling and free vibrations of nanoplates—comparison of nonlocal strain and stress approaches. Applied Sciences, 2019, vol. 9, no. 7, pp. 1409. https://doi.org/10.3390/app9071409
- Wang W., Rong D., Xu C., Zhang J., Xu X., Zhou Z. Accurate buckling analysis of magnetically affected cantilever nanoplates subjected to in‑plane magnetic fields. Journal of Vibration Engineering & Technologies, 2020, vol. 8, no. 4, pp. 505–515. https://doi.org/10.1007/s42417-019-00106-3
- Pradhan S.C., Kumar A. Buckling analysis of single layered graphene sheet under biaxial compression using nonlocal elasticity theory and DQ method. Journal of Computational and Theoretical Nanoscience, 2011, vol. 8, no. 7, pp. 1325–1334. https://doi.org/10.1166/jctn.2011.1818
- Pradhan S.C., Murmu T. Small scale effect on the buckling of single-layered graphene sheets under biaxial compression via nonlocal continuum mechanics. Computational Materials Science, 2009, vol. 47, no. 1, pp. 268–274. https://doi.org/10.1016/j.commatsci.2009.08.001
- Mohammadi M., Goodarzi M., Ghayour M., Alivand S. Small scale effect on the vibration of orthotropic plates embedded in an elastic medium and under biaxial in-plane pre-load via nonlocal elasticity theory. Journal of Solid Mechanics, 2012, vol. 4, no. 2, pp. 128–143.
- Analooei H.R., Azhari M., Heidarpour A. Elastic buckling and vibration analyses of orthotropic nanoplates using nonlocal continuum mechanics and spline finite strip method. Applied Mathematical Modelling, 2013, vol. 37, no. 10–11, pp. 6703–6717. https://doi.org/10.1016/j.apm.2013.01.051
- Karamooz Ravari M.R., Talebi S., Shahidi A.R. Analysis of the buckling of rectangular nanoplates by use of finite-difference method. Meccanica, 2014, vol. 49, no. 6, pp. 1443–1455. https://doi.org/10.1007/s11012-014-9917-x
- Hosseini M., Jamalpoor A., Fath A. Surface effect on the biaxial buckling and free vibration of FGM nanoplate embedded in visco-Pasternak standard linear solid-type of foundation. Meccanica, 2016, vol. 52, no. 6, pp. 1381–1396. https://doi.org/10.1007/s11012-016-0469-0
- Sukhoterin M.V., Potekhina E.V., Annenkov L.V. Etermination of the spectrum of critical loads and forms balance compressed cladding panels hull. Vestnik gosudarstvennogo universiteta morskogo i rechnogo flota imeni admirala S.O. Makarova, 2014, no. 2(24), pp. 44–51. (in Russian)
- Sukhoterin M.V., Rasputina E.I., Pizhurina N.F. Mixed forms of free oscillations of a rectangular CFCF-plate. Scientific and Technical Journal of Information Technologies, Mechanics and Optics, 2023, vol. 23, no. 2, pp. 413–421. (in Russian). https://doi.org/10.17586/2226-1494-2023-23-2-413-421
- Duan W.H., Wang C.M., Zhang Y.Y. Calibration of nonlocal scaling effect parameter for free vibration of carbon nanotubes by molecular dynamics. Journal of Applied Physics, 2007, vol. 101, no. 2, pp. 024305. https://doi.org/10.1063/1.2423140
