In this paper, the bending, buckling, and vibration behaviors of bi-directional functionally graded (BDFG) nonuniform micro/nanobeams are investigated. A new Euler–Bernoulli beam model is developed for BDFG tapered micro/nanobeams using Gurtin–Murdoch surface elasticity theory and modified couple stress theory to capture the effects of surface energy and microstructure stiffening, respectively. The present formulation accounts for the physical neutral surface. The material properties of the bulk and surface continuums of the nanobeam are assumed to vary along the thickness and length directions according to power law. Also, the cross section is assumed to vary linearly along the length direction. Hamilton principle is employed to derive the nonclassical equations of motions and boundary conditions. The generalized differential quadrature method (GDQM) is employed to accurately evaluate the variable coefficients of the obtained governing equations. Then after, the Navier’s method is employed for the simply supported BDFG nanobeam for its static bending deflection, critical buckling load, and fundamental frequency. The proposed model is validated by comparing the obtained results with available literature. Effects of different geometrical and material parameters on static and dynamic behaviors of small-scale BDFG nanobeams with the simultaneous effects of microstructure and surface elasticity are comprehensively studied and discussed. Results disclose that the nonuniformity parameters, aspect ratio, dimensionless material length-scale parameter, surface stress, surface elasticity, and gradient indices have a significant effect on the bending, buckling, and free vibration responses of BDFG tapered micro/ nanobeams.

中文翻译：

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具有表面能效应的双向功能梯度非均匀纳米梁静态和动态响应的半解析解

在本文中，研究了双向功能梯度 (BDFG) 非均匀微/纳米梁的弯曲、屈曲和振动行为。使用 Gurtin-Murdoch 表面弹性理论和修正的耦合应力理论，为 BDFG 锥形微/纳米梁开发了一种新的 Euler-Bernoulli 梁模型，以分别捕捉表面能和微结构硬化的影响。本公式说明了物理中性表面。根据幂律，假设纳米束的体积和表面连续体的材料特性沿厚度和长度方向变化。此外，假设横截面沿长度方向线性变化。哈密​​顿原理用于推导运动和边界条件的非经典方程。采用广义微分求积法(GDQM)对得到的控制方程的可变系数进行准确评估。然后，Navier 方法用于简支 BDFG 纳米梁的静态弯曲挠度、临界屈曲载荷和基频。通过将获得的结果与现有文献进行比较来验证所提出的模型。综合研究和讨论了不同几何和材料参数对小尺度 BDFG 纳米梁静态和动态行为的影响以及微观结构和表面弹性的同时影响。结果表明，非均匀性参数、纵横比、无量纲材料长度尺度参数、表面应力、表面弹性和梯度指数对弯曲有显着影响，