Because of fasttechnological development, electrostatic nanoactuator devices like nanosensors, nanoswitches, and nanoresonators are highly considered by scientific community. Thus, this article presents a new solution technique in solving highly nonlinear integro-differential equation governing electrically actuated nanobeams made of functionally graded material. The modified couple stress theory and Gurtin–Murdoch surface elasticity theory are coupled together to capture the size effects of the nanoscale thin beam in the context of Euler–Bernoulli beam theory. For accurate modelling, all the material properties of the bulk and surface continuums of the FG nanoactuator are varied continuously in thickness direction according to power law. The nonlinearity arising from the electrostatic actuation, fringing field, mid-plane stretching effect, axial residual stress, Casimir dispersion, and van der Waals forces are considered in mathematical formulation. The nonlinear nonclassical equilibrium equation of FG nanobeam-based actuators and associated boundary conditions are exactly derived using Hamilton principle. The new solution methodology is combined from three phases. The first phase applies Galerkin method to get an integro-algebraic equation. The second one employs particle swarm optimization method to approximate the integral terms (i.e. electrostatic force, fringing field, and intermolecular forces) to non-integral cubic algebraic equation. Then, solved the system easily in last phase. The resulting algebraic model provides means for obtaining critical deflection, pull-in voltage, detachment length, minimum gap, and freestanding effects. A reasonable agreement is found between the results obtained from the present method and those in the available literature. A parametric study is performed to investigate the effects of the gradient index, material length scale parameter, surface energy, intermolecular forces, initial gap, and beam length on the pull-in response and freestanding phenomena of fully clamped and cantilever FG nanoactuators.

中文翻译：

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驱动功能梯度纳米梁的拉入和独立不稳定性，包括表面和硬化效应

由于技术的快速发展，纳米传感器、纳米开关和纳米谐振器等静电纳米致动器设备受到科学界的高度重视。因此，本文提出了一种新的求解技术，用于求解控制由功能梯度材料制成的电驱动纳米梁的高度非线性积分微分方程。修正偶应力理论和 Gurtin-Murdoch 表面弹性理论耦合在一起，以在 Euler-Bernoulli 梁理论的背景下捕捉纳米级薄梁的尺寸效应。为了精确建模，FG 纳米致动器的体块和表面连续体的所有材料特性都根据幂律在厚度方向上连续变化。由静电驱动、边缘场、中平面拉伸效应引起的非线性，在数学公式中考虑了轴向残余应力、卡西米尔色散和范德华力。基于 FG 纳米梁的执行器的非线性非经典平衡方程和相关的边界条件是使用哈密顿原理精确推导出来的。新的解决方案方法由三个阶段组合而成。第一阶段应用伽辽金方法得到一个积分代数方程。第二种采用粒子群优化方法将积分项（即静电力、边缘场和分子间力）近似为非积分三次代数方程。然后，在最后阶段轻松解决系统。由此产生的代数模型提供了获得临界偏转、拉入电压、分离长度、最小间隙和独立效应的方法。从本方法获得的结果与现有文献中的结果之间存在合理的一致性。进行参数研究以研究梯度指数、材料长度尺度参数、表面能、分子间力、初始间隙和梁长度对完全夹紧和悬臂 FG 纳米致动器的拉入响应和独立现象的影响。