This work aims to provide a comprehensive theoretical framework for the nonlinear bending, vibration and buckling of functionally graded (FG) nanobeams resting on an elastic foundation through nonlocal strain gradient theory. To this end, both Timoshenko and Euler–Bernoulli beam theories are considered and the equations of motion are established by Hamilton’s principle. The analytical expressions of the nonlinear deflections, frequencies and critical buckling forces of FG nanobeams are presented with closed-form expressions. Comparing the obtained results with those published in the literature shows the accuracy of the current analysis. Finally, the impacts of some effective parameters are thoroughly investigated and discussed.

这项工作旨在通过非局部应变梯度理论为基于弹性基础的功能梯度（FG）纳米梁的非线性弯曲、振动和屈曲提供一个综合的理论框架。为此，Timoshenko 和 Euler-Bernoulli 梁理论都被考虑，运动方程是由哈密顿原理建立的。FG 纳米梁的非线性挠度、频率和临界屈曲力的解析表达式用闭式表达式表示。将获得的结果与文献中发表的结果进行比较显示了当前分析的准确性。最后，深入研究和讨论了一些有效参数的影响。