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基于简化的应变梯度理论下Kirchhoff板模型边值问题的提法及其应用

徐晓建 邓子辰

徐晓建,邓子辰. 基于简化的应变梯度理论下Kirchhoff板模型边值问题的提法及其应用 [J]. 应用数学和力学,2022,43(4):363-373 doi: 10.21656/1000-0887.420286
引用本文: 徐晓建,邓子辰. 基于简化的应变梯度理论下Kirchhoff板模型边值问题的提法及其应用 [J]. 应用数学和力学,2022,43(4):363-373 doi: 10.21656/1000-0887.420286
XU Xiaojian, DENG Zichen. Boundary Value Problems of a Kirchhoff Type Plate Model Based on the Simplified Strain Gradient Elasticity and the Application[J]. Applied Mathematics and Mechanics, 2022, 43(4): 363-373. doi: 10.21656/1000-0887.420286
Citation: XU Xiaojian, DENG Zichen. Boundary Value Problems of a Kirchhoff Type Plate Model Based on the Simplified Strain Gradient Elasticity and the Application[J]. Applied Mathematics and Mechanics, 2022, 43(4): 363-373. doi: 10.21656/1000-0887.420286

基于简化的应变梯度理论下Kirchhoff板模型边值问题的提法及其应用

doi: 10.21656/1000-0887.420286
基金项目: 国家自然科学基金(12072266);中央高校基本科研业务费(300102219315);陕西省自然科学基础研究计划(2020JQ-337)
详细信息
    作者简介:

    徐晓建(1986—),男,副教授,博士 (E-mail:xuxiaojian@mail.nwpu.edu.cn

    邓子辰(1964—),男,教授,博士生导师 (通讯作者. E-mail:dweifan@nwpu.edu.cn

  • 中图分类号: TB383; O342

Boundary Value Problems of a Kirchhoff Type Plate Model Based on the Simplified Strain Gradient Elasticity and the Application

  • 摘要:

    考虑应变梯度和速度梯度的影响,建立薄板控制微分方程及给出其边值问题的提法,修正了前人给出的薄板角点条件。采用Levy法,给出受分布力作用下简支板的挠度及自由振动频率的解析解。通过与文献中分子动力学数据对比,验证了该文模型的有效性并提出校核材料参数的一种方法。研究结果表明,增大弹性地基和应变梯度参数可以有效提高板的等效刚度,而速度梯度参数则相反。该文提出的板的边值问题为研究薄板在复杂支撑边界及外荷载等条件提供了理论依据。同时,有望为其有限元法、有限差分法和基于能量原理的Galerkin法等数值方法提供理论依据。

  • 图  1  板边界及荷载

    Figure  1.  Boundary conditions and loadings

    图  2  弹性地基上的周边简支矩形板

    Figure  2.  A fully simply supported rectangular plate resting on an elastic foundation

    图  3  周边简支方形板的基频与其边长的关系

    Figure  3.  The fundamental frequency vs. the side length of a simply supported square plate

    图  4  地基刚度系数对周边简支方板位移形状的影响

    Figure  4.  Effects of the foundation stiffness on the displacement of a simply supported square plate for y=a/2

    图  5  地基刚度系数对周边简支方板基频的影响

    Figure  5.  Effects of the foundation stiffness on the fundamental frequency of a simply supported square plate

    图  6  应变梯度参数l2对周边简支方板位移形状的影响

    Figure  6.  Effects of strain gradient parameter l2 on the displacement of a simply supported square plate for y=a/2

    图  7  应变梯度参数l2对周边简支方板基频的影响

    Figure  7.  Effects of strain gradient parameter l2 on the fundamental frequency of a simply supported square plate

    图  8  速度梯度参数l1对周边简支方板位移形状的影响

    Figure  8.  Effects of velocity gradient parameter l1 on the displacement of a simply supported square plate for y=a/2

    图  9  速度应变梯度参数l1对周边简支方板基频的影响

    Figure  9.  Effects of velocity gradient parameter l1 on the fundamental frequency of a simply supported square plate

    A1  坐标系(x, y)和(n, s)及板边界Γ

    A1.  Coordinate systems (x, y) and (n, s) at a piecewise smooth plate boundary Γ

    表  1  矩形薄板3种常见的边界条件

    Table  1.   Three common boundary conditions (BCs) for a rectangular plate

    boundary conditionBC1BC2BC3
    clampedw=0w,x=0Mxxx=0 or w,xx=0
    simply supportw=0M*xx=0Mxxx=0 or w,xx=0
    free$ Q_x^* + \rho hl_1^2{\ddot w_{,x}} = 0 $M*xx=0Mxxx=0 or w,xx=0
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出版历程
  • 收稿日期:  2021-09-16
  • 修回日期:  2021-10-13
  • 网络出版日期:  2022-03-25
  • 刊出日期:  2022-04-01

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