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Torsional vibration of bi-directional functionally graded nanotubes based on nonlocal elasticity theory

机译：基于非局部弹性理论的双向功能梯度纳米管的扭转振动

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The equation of torsional motion is presented in this paper to investigate the free torsional vibration behaviors of tubes made of a bi-directional functionally graded (FG) material, which is composed of two different materials with continuously varying along the radius and length directions. To incorporate the size effect of long-range forces, the nonlocal elasticity theory is employed to derive the difference equation of torsional motion, which can be reduced to the classical governing equation by simply setting a zero nonlocal parameter. Suppose that the effective material properties of the nanotube vary in the length direction according to an exponential distribute function and in the radius direction according to a power-law function. The closed-form solutions of torsional frequencies and mode shapes are derived. It is shown that the torsional frequencies can be significantly affected by the through-radius and through length gradings of the bi-directional FG nanotubes and hence can be prescribed by tailoring the bidirectional nano-structures of the FG material. The torsional frequencies can be increased with the decreasing nonlocal parameter, whereas the size-dependent behaviors on the mode shape cannot be observed. (C) 2017 Elsevier Ltd. All rights reserved.
机译：本文提出了扭转运动方程，以研究由双向功能梯度（FG）材料制成的管的自由扭转振动行为，该材料由两种不同的材料组成，它们沿半径和长度方向连续变化。为了结合远程力的大小效应，采用非局部弹性理论来导出扭转运动的差分方程，只需设置零非局部参数即可将其简化为经典控制方程。假设纳米管的有效材料特性在长度方向上根据指数分布函数而变化，并且在半径方向上根据幂律函数而变化。推导了扭转频率和振型的闭合解。结果表明，扭转频率会受到双向FG纳米管的穿透半径和长度梯度的显着影响，因此可以通过调整FG材料的双向纳米结构来规定。扭转频率可以随着减小的非局部参数而增加，而不能观察到模态形状上尺寸相关的行为。 （C）2017 Elsevier Ltd.保留所有权利。

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