In this study, the static bending, buckling, and free vibration behaviors of two-dimensional functionally graded (2D-FG) tapered micro/nanobeams are modelled and analyzed. For the first time, the nonclassical equations of motion and corresponding boundary conditions of 2D-FG beam are simultaneously derived in the framework of the modified couple stress and Gurtin-Murdoch surface elasticity theories in conjunction with the Timoshenko beam theory. Here, all the material properties of the beam are graded along the thickness and length directions according to the power-law. Both thickness and width of the beam are considered to vary linearly along the length direction. Unlike existing Timoshenko beam models, the current formulation accounts for the axial and bending deformations, Poisson’s effect, and exact position of the physical neutral plane. Differential quadrature method is employed to estimate the variable coefficients of the derived equations. The bending deflection, critical buckling load, and free vibration frequency of simply supported 2D-FG Timoshenko nanobeam are obtained employing the Navier's method. A parametric study is accompanied to investigate the effects of the different material and geometrical parameters on the mechanics of 2D-FG micro/nanobeams. It is observed that these parameters are very important in investigating the static and dynamic responses of 2D-FG micro/nanobeam.

在这项研究中，对二维功能梯度 (2D-FG) 锥形微/纳米梁的静态弯曲、屈曲和自由振动行为进行了建模和分析。首次在修正耦合应力和 Gurtin-Murdoch 表面弹性理论的框架下，结合 Timoshenko 梁理论，同时推导了 2D-FG 梁的非经典运动方程和相应的边界条件。此处，梁的所有材料属性均根据幂律沿厚度和长度方向分级。梁的厚度和宽度都被认为沿长度方向线性变化。与现有的 Timoshenko 梁模型不同，当前的公式考虑了轴向和弯曲变形、泊松效应以及物理中性平面的精确位置。采用微分求积法估计导出方程的变量系数。采用 Navier 方法获得简支 2D-FG Timoshenko 纳米梁的弯曲挠度、临界屈曲载荷和自由振动频率。随附参数研究以研究不同材料和几何参数对 2D-FG 微/纳米束力学的影响。据观察，这些参数对于研究 2D-FG 微/纳米束的静态和动态响应非常重要。随附参数研究以研究不同材料和几何参数对 2D-FG 微/纳米束力学的影响。据观察，这些参数对于研究 2D-FG 微/纳米束的静态和动态响应非常重要。随附参数研究以研究不同材料和几何参数对 2D-FG 微/纳米束力学的影响。据观察，这些参数对于研究 2D-FG 微/纳米束的静态和动态响应非常重要。