This paper introduces a non-linear non-classical model of nanoscale interaction forces between atomic force microscope and carbon nanotubes in a manipulation process. To accurately model the 3D manipulation process, Prandtl-Tomlinson friction model and the Hertz, JKR, and Lundberg contact mechanics theories were employed, taking into account the CNT geometry and manipulation conditions. The applied forces exerted by both the substrate and the probe on the CNT were extracted. The dynamic characteristics of the nanorods and nanotubes were investigated by employing the non-linear modified couple stress Timoshenko beam. Additionally, the cantilever was modeled using the Mindlin plate theory. The findings showed a good agreement between the experimental data of multi-walled carbon nanotube with a diameter of 100 nm, and the theoretical simulations. In addition to critical force and time, CNT bending results were presented using various Timoshenko theories. In order to validate the results of the maximum bending, the outcomes derived from the computational modeling of a polystyrene nanorod using the Euler-Bernoulli and Timoshenko theories were juxtaposed with the existing classical bending outcomes. Finally, the effects of the size parameter were investigated in non-classical polystyrene models. The findings indicated that considering the size effects is important; For instance, the variance between classical and non-classical models can exceed 57% for / = 0.5.

中文翻译：

---

原子力显微镜与碳纳米管之间纳米级相互作用力的非线性建模

本文介绍了操纵过程中原子力显微镜与碳纳米管之间纳米级相互作用力的非线性非经典模型。为了精确模拟 3D 操纵过程，采用了 Prandtl-Tomlinson 摩擦模型以及 Hertz、JKR 和 Lundberg 接触力学理论，同时考虑了 CNT 几何形状和操纵条件。提取基底和探针施加在 CNT 上的力。采用非线性修正力偶应力 Timoshenko 梁研究了纳米棒和纳米管的动态特性。此外，悬臂梁是使用 Mindlin 板理论建模的。研究结果表明，直径为100 nm的多壁碳纳米管的实验数据与理论模拟具有良好的一致性。除了临界力和时间之外，CNT 弯曲结果还使用各种 Timoshenko 理论给出。为了验证最大弯曲的结果，将使用欧拉-伯努利和铁木辛科理论对聚苯乙烯纳米棒进行计算建模得出的结果与现有的经典弯曲结果进行了比较。最后，在非经典聚苯乙烯模型中研究了尺寸参数的影响。研究结果表明，考虑尺寸效应很重要；例如，当 / = 0.5 时，经典模型和非经典模型之间的方差可能超过 57%。