Abstract The problem of pull-in instability of a cantilever micro- or nano-switch under electrostatic forces has attracted considerable attention in the literature, given its importance in designing micro- and nano-electromechanical systems (MEMS and NEMS). The non-linear nature of the problem supports the typical approach that relies on numerical or semi-analytical tools to approximate the solution. By contrast, we determine fully analytical upper and lower bounds to the pull-in instability phenomenon for a cantilever beam under the action of electrostatic, van der Waals or Casimir forces. In particular, the novel contribution of this works consists in accounting for size effects analytically, in the spirit of surface elasticity, which adds considerable complication to the problem, allowing for a nonconvex beam deflection. Surface energy effects are generally ignored in classical elasticity, although they are known to become relevant for very small structures and especially at the nano-scale, owing to their large surface/volume ratio. Closed form lower and upper bounds are given for the pull-in characteristics, that allow to discuss the role of several tuneable parameters. Indeed, the evolution of the cantilever tip deflection is presented as a function of the applied voltage up to the occurrence of pull-in and the contribution of van der Waals and Casimir intermolecular interactions is discussed. It is found that intermolecular forces strongly decrease the pull-in voltage, while surface elasticity works in the opposite direction and stabilizes the system. The accuracy of the bounding solutions, measured in terms of the difference between upper and lower analytical bounds, is generally very good, although it rapidly deteriorates as the effect of surface elasticity becomes more pronounced. Finally, approximated closed-form relations are developed to yield simple and accurate design formulae: in particular, they provide estimates for the minimum theoretical gap and for the maximum operable length for a free-standing cantilever in the presence of the effects of surface elasticity and intermolecular interactions. Results may be especially useful for designing and optimizing NEMS switches.

中文翻译：

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具有表面弹性的 MEMS/NEMS 光束的拉入电压的界限

摘要 悬臂微或纳米开关在静电力作用下的牵引不稳定性问题在文献中引起了相当大的关注，因为它在设计微和纳米机电系统（MEMS 和 NEMS）中很重要。问题的非线性特性支持依赖数值或半分析工具来近似解的典型方法。相比之下，我们确定了悬臂梁在静电、范德华力或卡西米尔力作用下的拉入不稳定性现象的完全解析上界和下界。特别是，这项工作的新贡献在于，本着表面弹性的精神，分析地解释了尺寸效应，这使问题变得相当复杂，允许非凸梁偏转。表面能效应在经典弹性中通常被忽略，尽管众所周知它们与非常小的结构相关，尤其是在纳米级，因为它们的表面积/体积比很大。给出了引入特性的封闭形式下限和上限，允许讨论几个可调参数的作用。事实上，悬臂尖端偏转的演变被呈现为施加电压的函数，直到发生拉入，并讨论了范德华和卡西米尔分子间相互作用的贡献。发现分子间力强烈降低吸合电压，而表面弹性在相反的方向起作用并使系统稳定。边界解的准确性，用分析上限和下限之间的差异衡量，通常非常好，尽管随着表面弹性的影响变得更加明显，它会迅速恶化。最后，开发了近似的闭合形式关系以产生简单而准确的设计公式：特别是，它们提供了在存在表面弹性和表面弹性影响的情况下独立悬臂的最小理论间隙和最大可操作长度的估计值。分子间相互作用。结果对于设计和优化 NEMS 开关可能特别有用。它们提供了在存在表面弹性和分子间相互作用影响的情况下独立悬臂的最小理论间隙和最大可操作长度的估计值。结果对于设计和优化 NEMS 开关可能特别有用。它们提供了在存在表面弹性和分子间相互作用影响的情况下独立悬臂的最小理论间隙和最大可操作长度的估计值。结果对于设计和优化 NEMS 开关可能特别有用。