Abstract This article presents a numerical model based on shear deformation theory in conjunction with Eringen’s nonlocal theory to investigate the thermo-elastic vibration behavior of bi-directional functionally graded nonuniform thick nanobeams supported on Pasternak's foundation. These beams are subjected to nonlinear temperature field along the thickness and uniform in-plane loading. Two directional variations in material properties are described by a power-law model across the thickness and as per the exponential function along the length. Employing Hamilton's energy principle, the governing equations are derived and then solved for frequency parameter via the generalized differential quadrature method for clamped, simply-supported, and combination of both the boundary conditions. A parametric study is carried out for understanding the vibration characteristic of considered beams, which shows that the combination of in-plane and thermal loadings with the foundation parameter is the most critical combination for bi-directional functionally graded nanobeams. Results are compared with the published work and found in good accordance. The results will be of great use in designing aerospace shuttles and scanning probe microscopes where nonuniform embedded nanobeams are subjected to compression in the thermal environment.

中文翻译：

---

面内载荷和非线性热场对嵌入式双向功能梯度锥形厚纳米梁动态响应的综合影响

摘要 本文提出了一个基于剪切变形理论的数值模型，结合 Eringen 的非局部理论，研究了支撑在 Pasternak 基础上的双向功能梯度非均匀厚纳米梁的热弹性振动行为。这些梁承受沿厚度方向的非线性温度场和均匀的面内载荷。材料特性的两个方向变化由跨厚度的幂律模型和沿长度的指数函数描述。利用哈密顿能量原理，推导出控制方程，然后通过广义微分求积法求解频率参数，适用于钳制、简支和两者结合的边界条件。进行参数研究以了解所考虑梁的振动特性，这表明面内载荷和热载荷与基础参数的组合是双向功能梯度纳米梁的最关键组合。结果与已发表的工作进行了比较，发现符合良好。结果将在设计航天飞机和扫描探针显微镜时非常有用，在这些显微镜中，非均匀嵌入的纳米束在热环境中受到压缩。