In this work, a nonlocal strain gradient beam model considering the thickness effect is developed to study the nonlinear vibration response of a functionally graded nanobeam. The governing equation of the functionally graded nanobeam is derived by using the Euler–Bernoulli beam theory with von Kármán’s nonlinear strain-gradient relationship and the Hamilton principle. The expression of the nonlinear frequency for the functionally graded nanobeam with pinned-pinned boundary conditions is obtained with the help of Galerkin technique and the Hamiltonian approach. The obtained results show that the effect of thickness is very important for the size-dependent vibration response of the functionally graded nanobeam; the nonlinear vibration response of the nanobeam depends not only on the material length scale parameter and nonlocal parameter but also on the slenderness ratio. Effects of the slenderness ratio and the power-law index on the vibration response of the functionally graded nanobeam are also investigated and discussed. The numerical results show that the nonlocal parameter reduces the nonlinear frequency of the functionally graded nanobeam, while the material length scale parameter increases the nonlinear frequency of the functionally graded nanobeam. The slenderness ratio leads to an increase in the nonlinear frequency of the functionally graded nanobeam, while the power-law index leads to a decrease in the nonlinear frequency of the functionally graded nanobeam.

中文翻译：

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基于厚度效应的非局部应变梯度理论的功能梯度纳米梁的非线性振动

在这项工作中，开发了一种考虑厚度效应的非局部应变梯度梁模型，以研究功能梯度纳米束的非线性振动响应。利用欧拉-伯努利光束理论，冯·卡尔曼的非线性应变梯度关系和汉密尔顿原理，推导了功能梯度纳米束的控制方程。借助Galerkin技术和哈密顿方法，获得了具有固定销边界条件的功能梯度纳米束的非线性频率表达式。获得的结果表明，厚度的影响对于功能梯度纳米束的尺寸依赖性振动响应非常重要。纳米束的非线性振动响应不仅取决于材料的长度尺度参数和非局部参数，还取决于细长比。还研究和讨论了细长比和幂律指数对功能梯度纳米束振动响应的影响。数值结果表明，非局部参数降低了功能梯度纳米束的非线性频率，而材料长度尺度参数增加了功能梯度纳米束的非线性频率。细长比导致功能梯度纳米束的非线性频率增加，而幂律指数导致功能梯度纳米束的非线性频率减小。还研究和讨论了细长比和幂律指数对功能梯度纳米束振动响应的影响。数值结果表明，非局部参数降低了功能梯度纳米束的非线性频率，而材料长度尺度参数增加了功能梯度纳米束的非线性频率。细长比导致功能梯度纳米束的非线性频率增加，而幂律指数导致功能梯度纳米束的非线性频率减小。还研究和讨论了细长比和幂律指数对功能梯度纳米束振动响应的影响。数值结果表明，非局部参数降低了功能梯度纳米束的非线性频率，而材料长度尺度参数增加了功能梯度纳米束的非线性频率。细长比导致功能梯度纳米束的非线性频率增加，而幂律指数导致功能梯度纳米束的非线性频率减小。而材料长度尺度参数则增加了功能梯度纳米束的非线性频率。细长比导致功能梯度纳米束的非线性频率增加，而幂律指数导致功能梯度纳米束的非线性频率减小。而材料长度尺度参数则增加了功能梯度纳米束的非线性频率。细长比导致功能梯度纳米束的非线性频率增加，而幂律指数导致功能梯度纳米束的非线性频率减小。