Abstract

Based on the first-order shear deformation (FSD) model and nonlocal elasticity theory, the simultaneous effects of shear and small scale on the nonlinear vibration behavior of functionally graded carbon nanotube-reinforced composite (FG-CNTRC) beams are investigated for the first time. To this end, the governing equations of bending and stretching with von Kármán geometric nonlinearity are decoupled into one fourth-order partial differential equation in terms of transverse deflection. A closed-form solution of the nonlinear natural frequency, which can be used in conceptual design and optimization algorithms of FG- CNTRC beams with different boundary conditions, is developed using a hybrid method of Galerkin and perturbation technique. First, the decoupled equation is reduced to a nonlinear ordinary one with respect to time by implementing the Galerkin method. Next, multiple scales perturbation technique is used to replace this nonlinear ordinary differential equation with a series of linear equations which can be solved analytically. Finally, numerical results are presented to compare with the existing ones in the literature as well as to study the effects of CNTs distribution, boundary conditions and nonlocal parameter on the frequency ratio. It is seen that the influence of CNTs distribution becomes more significant when the nonlocal parameter increases. Also, the difference between results obtained from the classical and nonlocal elasticity theories increases as the amplitude of vibration increases. Therefore, it is concluded that the classical elasticity theory is inadequate to predict the nonlinear vibration behavior of FG-CNTRC beams with large deformation.

摘要

基于一阶剪切变形（FSD）模型和非局部弹性理论，首次研究了剪切和小尺度对功能梯度碳纳米管增强复合材料（FG-CNTRC）梁非线性振动行为的影响。 . 为此，将具有 von Kármán 几何非线性的弯曲和拉伸控制方程解耦为一个关于横向偏转的四阶偏微分方程。非线性固有频率的闭式解，可用于不同边界条件的FG-CNTRC梁的概念设计和优化算法，使用Galerkin和微扰技术的混合方法开发。第一的，通过实施 Galerkin 方法，解耦方程被简化为关于时间的非线性常方程。接下来，采用多尺度摄动技术，将这个非线性常微分方程替换为一系列可以解析求解的线性方程。最后，给出数值结果与文献中已有的结果进行比较，并研究碳纳米管分布、边界条件和非局部参数对频率比的影响。可以看出，当非局部参数增加时，CNTs 分布的影响变得更加显着。此外，从经典弹性理论和非局部弹性理论获得的结果之间的差异随着振动幅度的增加而增加。所以，