含尺度效应的功能梯度三明治微梁振动、弯曲与屈曲特性研究

基于修正的应变梯度理论和精化的高阶剪切变形理论，提出了一种含尺度效应的功能梯度三明治微梁模型。功能梯度材料的等效弹性参数由Mori-Tanaka均匀化方法描述。针对微梁的高阶边值问题，融合微分求积和Gauss-Lobatto求积准则，建立了一种2节点18自由度的微分求积有限元。通过对比性研究，验证了理论及数值模型的有效性。最后，讨论了边界条件、材料尺度参数、功能梯度指数、长细比、各层厚度比等对功能梯度三明治微梁静动态特性的影响。结果表明，功能梯度三明治微梁的静力响应、振动频率、屈曲荷载以及模态均呈现出显著的尺度效应，所得结果有望为微机电系统中承载器件的设计提供数据积累和方法依据。

Research on Vibration,Bending and Buckling Characteristics of Functionally Graded Sandwich Microbeams with Size Effects

This paper proposed a functionally graded sandwich microbeam model considering size effects within the frameworks of the modified strain gradient theory and a refined high-order shear deformation theory. The material distributions were assumed to vary in the thickness direction and estimated through the Mori-Tanaka homogenization scheme. A two-node differential quadrature finite element with 18 degrees of freedom was established by combining the differential quadrature rule and Gauss-Lobatto quadrature rule to solve the high-order boundary value problems.The efficacy of our theoretical model and numerical method was validated by comparing the degenerated results with the reported ones. Finally, the effects of boundary conditions, material length scale parameters, functional gradient index, slenderness ratio, and thickness ratio on the static and dynamic characteristics of functionally graded sandwich microbeams were discussed. It was revealed that the introduction of strain gradients has significant impact on the static responses, vibration frequencies, buckling loads together with the related mode shapes of functionally graded sandwich microbeams. The results obtained were expected to provide data accumulation and methodological basis for the design of load-bearing devices in MEMS.