This paper studies the geometrically nonlinear free vibration of a novel nanocomposite circular plate. The matrix of the nanocomposite structure is made of functionally graded (FG) polymer, and the distribution of graphene platelets (GPLs) as the reinforcement is assumed based on the five various linear FG models. Therefore, this novel nanocomposite is called dual-FG. The Voigt rule of mixture and Halpin-Tsai method are employed to homogenize FG polymer, and distribution of the GPLs within the matrix, respectively. Moreover, the whole structure is fixed on a nonlinear Kerr elastic foundation. The displacements of the plate are estimated by means of the first-order shear deformation theory (FSDT). Von-Kármán type of geometrically nonlinear relations expresses the plate's strains. The generalized differential quadrature (GDQ) technique, bilateral iterative displacement control method (BIDCM), and the weighted residual Galerkin procedure are implemented to extract the free vibration characteristics of the structure. The developed structure is validated with those reported in the open literature. A comprehensive parametric study is conducted to illustrate the effect of various parameters on the dynamic response of the dual-FG nanocomposite circular plate. It will be observed that the influence of the volume fraction and distribution model of GPL is more pronounced on the vibration frequency in the FG matrix. It was noted that the volume fraction of graphene platelets (GPL) and the power-law index of the matrix exhibit a direct and inverse relationship, respectively, with the natural frequency of the structure. Furthermore, the V-GPL model exhibits the highest percentage growth in nonlinear frequencies, while the Λ-GPL model shows the least growth. This observation is particularly pronounced for plates with hinged boundary conditions.

中文翻译：

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双边迭代位移控制方法在双FG纳米复合材料圆板非线性自由振动分析中的应用


本文研究了新型纳米复合材料圆板的几何非线性自由振动。纳米复合材料结构的基体由功能梯度（FG）聚合物制成，并且基于五种不同的线性FG模型假设石墨烯片（GPL）作为增强体的分布。因此，这种新型纳米复合材料被称为双FG。分别采用 Voigt 混合物规则和 Halpin-Tsai 方法均匀化 FG 聚合物以及 GPL 在基质内的分布。此外，整个结构固定在非线性克尔弹性基础上。板的位移通过一阶剪切变形理论（FSDT）来估计。冯卡门类型的几何非线性关系表达了​​板的应变。采用广义微分求积（GDQ）技术、双边迭代位移控制方法（BIDCM）和加权残差伽辽金程序来提取结构的自由振动特征。所开发的结构通过公开文献中报道的结构进行了验证。进行了全面的参数研究，以说明各种参数对双 FG 纳米复合材料圆板动态响应的影响。可以看出，GPL 的体积分数和分布模型对 FG 基体振动频率的影响更为明显。值得注意的是，石墨烯片的体积分数（GPL）和基质的幂律指数分别与结构的固有频率呈现正相关和反相关关系。此外，V-GPL 模型在非线性频率方面表现出最高的百分比增长，而 Λ-GPL 模型则表现出最低的增长。 对于具有铰接边界条件的板，这一观察结果尤其明显。