Kiani Keivan, Żur Krzysztof Kamil
Department of Civil Engineering, K.N. Toosi University of Technology, Valiasr Ave., P.O. Box 15875-4416, Tehran, Iran.
Faculty of Mechanical Engineering, Bialystok University of Technology, Wiejska 45C Street, 15-351 Bialystok, Poland.
Nanomaterials (Basel). 2020 Nov 21;10(11):2306. doi: 10.3390/nano10112306.
Through considering both nonlocality and surface energy effects, this paper suggests suitable mathematical-continuum-based models for free vibration of nanorods with multiple defects acted upon by a bidirectional-transverse magnetic field. By employing both theories of elasticity of Eringen and Gurtin-Murdoch, the equations of motion for the magnetically affected-damaged rod-like nanostructures are derived using the nonlocal-differential-based and the nonlocal-integral-based models. The local defects are modeled by a set of linearly appropriate axial springs at the interface of appropriately divided nanorods. Through constructing the nonlocal-differential equations of motion for sub-divided portions and by imposing the appropriate interface conditions, the natural frequencies as well as the vibrational modes are explicitly obtained for fixed-free and fixed-fixed nanorods with low numbers of defects. The extracted nonlocal-integral governing equations are also solved for natural frequencies using the finite-element technique. For a particular situation, the model's results are successfully verified with those of another work. Subsequently, the effects of nonlocality, surface energy, defect's location, nanorod's diameter, magnetic field strength, and number of defects on the dominant free vibration response of the magnetically defected nanorods with various end conditions are displayed and discussed.
通过考虑非局部性和表面能效应，本文提出了基于数学连续体的合适模型，用于研究双向横向磁场作用下具有多个缺陷的纳米棒的自由振动。利用Eringen弹性理论和Gurtin-Murdoch理论，分别基于非局部微分模型和非局部积分模型，推导了受磁场影响的受损棒状纳米结构的运动方程。通过在适当划分的纳米棒界面处设置一组线性合适的轴向弹簧来模拟局部缺陷。通过构建细分部分的非局部微分运动方程并施加适当的界面条件，明确得到了具有少量缺陷的固支-自由和固支-固支纳米棒的固有频率和振动模式。还使用有限元技术求解了提取的非局部积分控制方程的固有频率。在特定情况下，该模型的结果与另一项研究的结果成功验证。随后，展示并讨论了非局部性、表面能、缺陷位置、纳米棒直径、磁场强度和缺陷数量对具有各种端部条件的磁性缺陷纳米棒主导自由振动响应的影响。
