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基于偶应力理论微纳米Mindlin板的尺度效应分析

薛江红 何赞航 夏飞 李泽嵘 金福松 杨鹏

薛江红,何赞航,夏飞,李泽嵘,金福松,杨鹏. 基于偶应力理论微纳米Mindlin板的尺度效应分析 [J]. 应用数学和力学,2022,43(7):1-12 doi: 10.21656/1000-0887.420171
引用本文: 薛江红,何赞航,夏飞,李泽嵘,金福松,杨鹏. 基于偶应力理论微纳米Mindlin板的尺度效应分析 [J]. 应用数学和力学,2022,43(7):1-12 doi: 10.21656/1000-0887.420171
XUE Jianghong, HE Zanhang, XIA Fei, LI Zerong, JIN Fusong, YANG Peng. Size-Dependent Effect of Micro-Nano Mindlin Plate Based on Couple Stress Theory[J]. Applied Mathematics and Mechanics. doi: 10.21656/1000-0887.420171
Citation: XUE Jianghong, HE Zanhang, XIA Fei, LI Zerong, JIN Fusong, YANG Peng. Size-Dependent Effect of Micro-Nano Mindlin Plate Based on Couple Stress Theory[J]. Applied Mathematics and Mechanics. doi: 10.21656/1000-0887.420171

基于偶应力理论微纳米Mindlin板的尺度效应分析

doi: 10.21656/1000-0887.420171
基金项目: 广东省自然科学基金(面上项目)(2021A1515012037)
详细信息
    作者简介:

    薛江红(1965—),女,教授,博士,博士生导师(通讯作者. E-mail:txuej@jnu.edu.cn

    夏飞(1994—),男,博士生(E-mail:1581058682@qq.com

  • 中图分类号: O345

Size-Dependent Effect of Micro-Nano Mindlin Plate Based on Couple Stress Theory

  • 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。
  • 图  1  微纳米板的位移示意图

    Figure  1.  Schematic diagram of micro-nano plate displacements

    图  2  微元体各面上的应力和偶应力分布状态

    Figure  2.  The distribution of the stress and the couple stress on the surfaces of a cubic element

    图  3  各向同性微纳米板微元体的受力平衡状态

    Figure  3.  The force and moment equilibrium of a differential element of isotropic micro-nano plate

    图  4  矩形板的边界条件

    Figure  4.  Boundary condition of the rectangular plate

    图  5  微纳米板的有限元网格示意图

    Figure  5.  Meshing of the micro-nano plate in ABAQUS software

    图  6  SSSS和SCSC边界下微纳米板的临界屈曲荷载随厚长比的变化

    Figure  6.  Effect of the relative depth on buckling load of micro-nano plate under SSSS and SCSC

    图  7  不同厚长比下,微纳米板临界屈曲荷载随尺寸参数的变化:(a)SSSS;(b)SCSC

    Figure  7.  Effect of the scale ratio on buckling load of micro-nano plate for different thickness-to-length ratios: (a) SSSS; (b) SCSC

    图  8  不同尺寸参数下,SSSS微纳米板固有频率随厚长比的变化

    Figure  8.  Effect of the thickness-to-length ratio on natural frequency of SSSS micro-nano plate for different values of scale parameters

    图  9  不同尺寸参数下,微纳米板固有频率随长宽比的变化:(a) SSSS; (b) SCSC

    Figure  9.  Effect of the aspect ratio on natural frequency of micro-nano plate for different scale parameters: (a) SSSS; (b) SCSC

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出版历程
  • 收稿日期:  2021-06-22
  • 录用日期:  2021-09-17
  • 修回日期:  2021-11-14
  • 网络出版日期:  2022-06-13

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