基于偶应力理论微纳米Mindlin板的尺度效应分析  

Size-Dependent Effects of Micro-Nano Mindlin Plates Based on the Couple Stress Theory

作　　者：薛江红 何赞航 夏飞 李泽嵘 金福松 杨鹏 XUE Jianghong;HE Zanhang;XIA Fei;LI Zerong;JIN Fusong;YANG Peng(MOE Key Laboratory of Disaster Forecast and Control in Engineering,School of Mechanics and Construction Engineering,Jinan University,Guangzhou 510632,P.R.China)

机构地区：[1]暨南大学力学与建筑工程学院“重大工程灾害与控制”教育部重点实验室,广州510632

出　　处：《应用数学和力学》2022年第7期740-751,共12页Applied Mathematics and Mechanics

基　　金：广东省自然科学基金(面上项目)(2021A1515012037)。

摘　　要：基于偶应力理论,建立了适用于微纳米结构的Mindlin板理论.考虑横向剪切变形和材料的尺度效应并引入长度尺寸参数,推导了各向同性微纳米Mindlin板的本构方程.根据板的平衡条件,进一步推导出用位移函数和转角函数表示的板的屈曲和振动控制方程.通过对位移和转角变量进行空间和时间域上的分离,得出了四边简支(SSSS)和对边简支、对边固支(SCSC)两种边界情况下微纳米板的屈曲和振动问题的解析解.然后利用MATLAB软件进行算例分析,获得了不同尺寸参数、长宽比、厚长比等情况下板的临界屈曲荷载和固有频率.研究结果与已有文献中的结果以及ABAQUS有限元仿真解进行对比,结果表明,不同参数下的三种方法得到的结果均十分接近.算例分析发现,尺度效应对屈曲载荷和固有频率都有显著影响.A Mindlin plate theory for micro-nano structures was proposed based on the couple stress theory.A length parameter was introduced to consider the size effect,and the constitutive equations for the micro-nano Mindlin plate were derived in view of the transverse shear deformation.The buckling and free vibration governing equations in terms of displacements and the slope functions of the shear deformation micro-nano plate were further deduced with the force equilibrium conditions.The analytical solutions of buckling and free vibration for the shear deformation micro-nano plate were obtained through separation of the displacement and rotation variables in space and time domains.Two scenarios of boundary conditions were analyzed:SSSS(simply supported by 4 edges)and SCSC(2 opposite edges simply supported and other 2 edges clamped).A MATLAB program was developed to compute the critical buckling and natural frequencies with different values of dimensional parameters,aspect ratios and length-to-thickness ratios.The research results,in comparison with those from the ABAQUS finite element analysis and previous literatures,are consistent with the latter ones.The examples show that,the size effects significantly influence the buckling load and the natural frequency.

关 键 词：偶应力 尺度效应 MINDLIN板理论 屈曲 振动 

分 类 号：O345[理学—固体力学]