By introducing the $$C^{2}$$ C 2 continuous hexahedral element, the free vibration finite element analysis of the nanobeam and nanoplate structures is reported based on the second strain gradient (SSG) theory and three-dimensional elasticity model. The SSG elasticity presents the powerful higher-order continuum theory which can be efficiently used to capture the size-effect on the nano-objects. The finite element discretization procedure is performed within Hamilton’s principle where the quadratic matrix version of the strain and kinetic energies are derived on the basis of the three-dimensional SSG elasticity model. In the proposed $$C^{2}$$ C 2 continuous hexahedral element, the values of the displacement field and the associated higher-order derivatives are considered as the nodal values to satisfy the continuity conditions. As the case studies, the free vibration of the nanobeams and rectangular nanoplates is investigated. Different results are outlined to show the efficiency and convergence of the present model. The influences of the involved parameters on the natural frequencies of nanobeams and plates are also investigated. It is realized that with the increase in the thickness-to-lattice parameter ratio, the difference of the results related to the SSG theory and classical theory decreases.

中文翻译：

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基于三维弹性理论的振动纳米结构二阶应变梯度有限元分析

通过引入$$C^{2}$$C 2 连续六面体单元，基于第二应变梯度（SSG）理论和三维弹性模型，报道了纳米梁和纳米板结构的自由振动有限元分析。SSG 弹性提出了强大的高阶连续介质理论，可以有效地用于捕捉纳米物体的尺寸效应。有限元离散化程序在哈密顿原理内执行，其中应变和动能的二次矩阵版本是基于三维 SSG 弹性模型导出的。在提出的$$C^{2}$$C 2 连续六面体单元中，位移场的值和相关的高阶导数被视为满足连续性条件的节点值。作为案例研究，研究了纳米梁和矩形纳米板的自由振动。概述了不同的结果以显示本模型的效率和收敛性。还研究了所涉及的参数对纳米梁和板的固有频率的影响。认识到随着厚度与晶格参数比的增加，与SSG理论和经典理论相关的结果的差异减小。