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Variational analysis of laminated nanoplates for various boundary conditions

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Abstract

Based on Gurtin–Murdoch’s theory, a new mechanistic laminated nanoplate model composed of nanocomposites under various boundary conditions is proposed, where the top and bottom surfaces of the laminated nanoplate are regarded as two-dimensional films which have independent surface elastic parameters. The influence of surface Lamé constants and residual stress on static bending and vibration properties for the laminated nanoplate is studied. In order to get approximate solutions of bending deflections and first-order frequencies for the laminated nanoplate with complex boundary conditions, the variational method is employed in this paper. The effectiveness of the presented variational method is verified by comparison with available results. Numerical results indicate that the effective bending stiffness of the laminated nanoplate considering surface effects is obviously different from the classical result. Moreover, the influences of surface effects on bending deflections and first-order frequencies are related to the surface elastic parameters and boundary conditions. The presented method and results provide a theoretical benchmark for other numerical and experimental methods to study nano laminates.

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Acknowledgements

This work was supported by the Natural Science Foundation of Hebei Province (A2022203025) and the Science and Technology Project of Hebei Education Department (ZD2021104).

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Authors and Affiliations

  1. Department of Engineering Mechanics, Yanshan University, Qinhuangdao, 066004, China

    Junhua Xiao & Jie Wang

  2. Hebei Key Laboratory of Mechanical Reliability for Heavy Equipments and Large Structures, Yanshan University, Qinhuangdao, 066004, China

    Junhua Xiao & Jie Wang

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  1. Junhua Xiao
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Xiao, J., Wang, J. Variational analysis of laminated nanoplates for various boundary conditions. Acta Mech 233, 4711–4728 (2022). https://doi.org/10.1007/s00707-022-03352-x

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