Background: This paper is concerned with a nonlinear semi-continuum model for an ultrathin structure. The basic equations of the theoretical model for silicon micro/nanosheets are derived, and the geometric nonlinearity is introduced in the model. Methods: From two different approaches including the new strain energy and the new external potential energy, we establish the nonlinear semi-continuum theoretical model of silicon micro/nanosheets, respectively. A new dimensionless nonlinear semi-continuum parameter is defined. Based on the theoretical model, the characteristics of bending deformation and free vibration are revealed. Results: The relationships between bending deflection and atomic layers in thickness direction as well as the relaxation coefficient between atomic layers are analyzed. The resonance frequencies of free vibration and their relationship with atomic layers are calculated. By introducing the specific property parameters of silicon micro/nanomaterials, several numerical calculations have been carried out. Conclusion: The theoretical results are compared with other studies in the literature, such as nonlinear finite element method (FEM), experimental and classical results, to validate the semi-continuum model established in the present research. This work can provide new ideas for the mechanical analyses of micro/nanomaterials and structures, and the results could be foundations for the design and application of silicon micro/nanosheets.

中文翻译：

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硅微/纳米片的非线性半连续模型及其在弯曲和振动中的应用

背景：本文关注的是超薄结构的非线性半连续模型。推导了硅微/纳米片理论模型的基本方程，并在模型中引入了几何非线性。方法：从新应变能和新外势能两种不同的途径，分别建立了硅微/纳米片的非线性半连续理论模型。定义了一个新的无量纲非线性半连续参数。在理论模型的基础上，揭示了弯曲变形和自由振动的特性。结果：分析了厚度方向弯曲变形与原子层的关系以及原子层间的弛豫系数。计算了自由振动的共振频率及其与原子层的关系。通过介绍硅微/纳米材料的具体性能参数，进行了多项数值计算。结论：将理论结果与文献中的其他研究，如非线性有限元法（FEM）、实验结果和经典结果进行比较，以验证本研究建立的半连续模型。该工作可为微/纳米材料和结构的力学分析提供新思路，研究结果可为硅微/纳米片的设计和应用奠定基础。结论：将理论结果与文献中的其他研究，如非线性有限元法（FEM）、实验结果和经典结果进行比较，以验证本研究建立的半连续模型。该工作可为微/纳米材料和结构的力学分析提供新思路，研究结果可为硅微/纳米片的设计和应用奠定基础。结论：将理论结果与文献中的其他研究，如非线性有限元法（FEM）、实验结果和经典结果进行比较，以验证本研究建立的半连续模型。该工作可为微/纳米材料和结构的力学分析提供新思路，研究结果可为硅微/纳米片的设计和应用奠定基础。