A new functionally graded non-classical Timoshenko microbeam model is developed by using a variational formulation. The new model incorporates strain gradient, couple stress (rotation gradient) and velocity gradient effects and explains a power-law variation of two-phase materials through the thickness direction. The new model contains three additional material constants to account for the relative three gradient effects. The static bending, buckling and free vibration problems of the newly developed functionally graded material (FGM) beam are then analytically solved. Numerical results show that the differences between the prediction results of the current model and the classic model are significant when FGM beam thickness is very small, but they are diminishing as the thickness increases.

中文翻译：

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完全依赖微结构的功能梯度材料微梁的弯曲、屈曲和振动分析

通过使用变分公式开发了一种新的功能分级非经典 Timoshenko 微束模型。新模型结合了应变梯度、耦合应力（旋转梯度）和速度梯度效应，并解释了两相材料在厚度方向上的幂律变化。新模型包含三个额外的材料常数来解释相对的三个梯度效应。然后分析解决了新开发的功能梯度材料（FGM）梁的静态弯曲、屈曲和自由振动问题。数值结果表明，当前模型与经典模型的预测结果在 FGM 梁厚度很小时差异显着，但随着厚度的增加，差异逐渐减小。