石墨烯增强多孔功能梯度圆板非线性自由振动
Nonlinear Free Vibration of Graphene-Reinforced Porous Functionally Graded Circular Plates
摘要: 基于Halpin-Tsai微观力学模型、闭单元高斯随机场理论和混合定律得到了石墨烯纳米片(Graphene nanoplates, GPL)增强多孔功能梯度材料(functionally graded materials, FGM)板的有效物性参数，其中考虑了GPL沿厚度方向均匀、X形和O形分布和孔隙沿厚度方向两种梯度分布。然后运用一阶剪切变形理论、Von-Karman变形理论和Hamilton原理推导出多孔GPL增强FGM材料圆板的运动方程组并进行无量纲化。最后通过Kartorovich时间平均法和微分求积法(differential quadrature method, DQM)将运动方程组离散得到一个特征方程组。通过直接迭代法求解出非线性自由振动频率，并通过模型退化和现有文献对比验证，验证了计算的正确性和有效性。讨论了边界条件、GPL重量分数、孔隙率、孔隙分布、GPL分布等参数对多孔GPL圆板非线性固有频率的影响。
Abstract: Based on the Halpin-Tsai micromechanical model, the closed-cell Gaussian random field theory, and the rule of mixtures, the effective material properties of graphene nanoplate (GPL)-reinforced porous functionally graded material (FGM) plates were obtained, considering the uniform, X-shaped, and O-shaped distributions of GPLs along the thickness direction and two types of gradient distributions of porosity along the thickness direction. Subsequently, the equations of motion for porous GPL-reinforced FGM circular plates were derived using the first-order shear deformation theory, the Von-Karman deformation theory, and Hamilton’s principle, and these equations were nondimensionalized. Finally, the equations of motion were discretized into a set of eigenvalue equations using the Karman time-averaging method and the differential quadrature method (DQM). The nonlinear free vibration frequencies were obtained through a direct iteration method. The accuracy and validity of the calculations were verified by model degeneration and comparison with existing literature. The effects of boundary conditions, GPL weight fraction, porosity, porosity distribution, and GPL distribution on the nonlinear natural frequencies of porous GPL circular plates were discussed.
文章引用：王天靖, 雷飞鹏, 滕兆春. 石墨烯增强多孔功能梯度圆板非线性自由振动[J]. 力学研究, 2025, 14(2): 144-156. https://doi.org/10.12677/ijm.2025.142014

参考文献

[1] Novoselov, K.S., Geim, A.K., Morozov, S.V., Jiang, D., Zhang, Y., Dubonos, S.V., et al. (2004) Electric Field Effect in Atomically Thin Carbon Films. Science, 306, 666-669. [Google Scholar] [CrossRef] [PubMed]
[2] Rahmani, O. and Pedram, O. (2014) Analysis and Modeling the Size Effect on Vibration of Functionally Graded Nanobeams Based on Nonlocal Timoshenko Beam Theory. International Journal of Engineering Science, 77, 55-70. [Google Scholar] [CrossRef] 
[3] Arefi, M., Mohammad-Rezaei Bidgoli, E., Dimitri, R. and Tornabene, F. (2018) Free Vibrations of Functionally Graded Polymer Composite Nanoplates Reinforced with Graphene Nanoplatelets. Aerospace Science and Technology, 81, 108-117. [Google Scholar] [CrossRef] 
[4] Song, M., Yang, J., Kitipornchai, S. and Zhu, W. (2017) Buckling and Postbuckling of Biaxially Compressed Functionally Graded Multilayer Graphene Nanoplatelet-Reinforced Polymer Composite Plates. International Journal of Mechanical Sciences, 131, 345-355. [Google Scholar] [CrossRef] 
[5] Feng, C., Kitipornchai, S. and Yang, J. (2017) Nonlinear Free Vibration of Functionally Graded Polymer Composite Beams Reinforced with Graphene Nanoplatelets (GPLs). Engineering Structures, 140, 110-119. [Google Scholar] [CrossRef] 
[6] Feng, C., Kitipornchai, S. and Yang, J. (2017) Nonlinear Bending of Polymer Nanocomposite Beams Reinforced with Non-Uniformly Distributed Graphene Platelets (GPLs). Composites Part B: Engineering, 110, 132-140. [Google Scholar] [CrossRef] 
[7] Thai, C.H., Ferreira, A.J.M. and Phung-Van, P. (2019) Size Dependent Free Vibration Analysis of Multilayer Functionally Graded GPLRC Microplates Based on Modified Strain Gradient Theory. Composites Part B: Engineering, 169, 174-188. [Google Scholar] [CrossRef] 
[8] Gholami, R. and Ansari, R. (2018) Nonlinear Harmonically Excited Vibration of Third-Order Shear Deformable Functionally Graded Graphene Platelet-Reinforced Composite Rectangular Plates. Engineering Structures, 156, 197-209. [Google Scholar] [CrossRef] 
[9] 黄小林, 魏耿忠, 刘思奇, 等. 粘弹性地基上含孔隙的石墨烯增强功能梯度板的振动[J]. 力学季刊, 2020, 41(4): 771-782.
[10] Xu, M., Li, X., Luo, Y., Wang, G., Guo, Y., Liu, T., et al. (2020) Thermal Buckling of Graphene Platelets Toughening Sandwich Functionally Graded Porous Plate with Temperature-Dependent Properties. International Journal of Applied Mechanics, 12, Article ID: 2050089. [Google Scholar] [CrossRef] 
[11] Chen, D., Zheng, S., Wang, Y., Yang, L. and Li, Z. (2020) Nonlinear Free Vibration Analysis of a Rotating Two-Dimensional Functionally Graded Porous Micro-Beam Using Isogeometric Analysis. European Journal of Mechanics—A/Solids, 84, Article ID: 104083. [Google Scholar] [CrossRef] 
[12] Anirudh, B., Ben Zineb, T., Polit, O., Ganapathi, M. and Prateek, G. (2020) Nonlinear Bending of Porous Curved Beams Reinforced by Functionally Graded Nanocomposite Graphene Platelets Applying an Efficient Shear Flexible Finite Element Approach. International Journal of Non-Linear Mechanics, 119, Article ID: 103346. [Google Scholar] [CrossRef] 
[13] Cox, H.L. (1952) The Elasticity and Strength of Paper and Other Fibrous Materials. British Journal of Applied Physics, 3, 72-79. [Google Scholar] [CrossRef] 
[14] Shen, H., Xiang, Y. and Lin, F. (2017) Nonlinear Vibration of Functionally Graded Graphene-Reinforced Composite Laminated Plates in Thermal Environments. Computer Methods in Applied Mechanics and Engineering, 319, 175-193. [Google Scholar] [CrossRef] 
[15] Lal, R. and Ahlawat, N. (2015) Axisymmetric Vibrations and Buckling Analysis of Functionally Graded Circular Plates via Differential Transform Method. European Journal of Mechanics—A/Solids, 52, 85-94. [Google Scholar] [CrossRef] 
[16] Żur, K.K. (2018) Quasi-Green’s Function Approach to Free Vibration Analysis of Elastically Supported Functionally Graded Circular Plates. Composite Structures, 183, 600-610. [Google Scholar] [CrossRef] 

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