留言板

尊敬的读者、作者、审稿人, 关于本刊的投稿、审稿、编辑和出版的任何问题, 您可以本页添加留言。我们将尽快给您答复。谢谢您的支持!

姓名
邮箱
手机号码
标题
留言内容
验证码

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
    下载: 导出CSV
  • [1] 周剑秋, 黄连军, 王英. 基于应变梯度位错理论的纳晶-无定形态层状复合材料的力学性能研究[J]. 工程力学, 2014, 31(1): 224-228. (ZHOU Jianqiu, HUANG Lianjun, WANG Ying. Mechanical behaviors of crystalline-amorphous nanolaminates based on strain gradient dislocation theory[J]. Engineering Mechanics, 2014, 31(1): 224-228.(in Chinese) doi: 10.6052/j.issn.1000-4750.2012.07.0497
    [2] 王学滨. 考虑应变梯度及刚度劣化的剪切带局部变形分析[J]. 工程力学, 2006, 23(10): 101-106. (WANG Xuebin. Analysis of localized deformation in shear band considering degraded stiffness based on gradient-dependent plasticity[J]. Engineering Mechanics, 2006, 23(10): 101-106.(in Chinese) doi: 10.3969/j.issn.1000-4750.2006.10.020
    [3] LAM D C C, YANG F, CHONG A C M, et al. Experiments and theory in strain gradient elasticity[J]. Journal of the Mechanics and Physics of Solids, 2003, 51(8): 1477-1508. doi: 10.1016/S0022-5096(03)00053-X
    [4] CHEN C Q, SHI Y, ZHANG Y S, et al. Size dependence of Young’s modulus in ZnO nanowires[J]. Physical Review Letters, 2006, 96(7): 075505. doi: 10.1103/PhysRevLett.96.075505
    [5] EOM K, PARK H S, YOON D S, et al. Nanomechanical resonators and their applications in biological/chemical detection: nanomechanics principles[J]. Physics Reports, 2011, 503(4/5): 115-163.
    [6] 秦江, 黄克智, 黄永刚. 采用特征线方法对混合硬化情况下基于变形机制的应变梯度工程塑性理论的研究[J]. 工程力学, 2009, 26(9): 176-185. (QIN Jiang, HWANG Kehchih, HUANG Yonggang. A study on the conventional theory of mechanism-based strain gradient plasticity for mixed hardening by the method of characteristics[J]. Engineering Mechanics, 2009, 26(9): 176-185.(in Chinese)
    [7] ASKES H, AIFANTIS E C. Gradient elasticity in statics and dynamics: an overview of formulations, length scale identification procedures, finite element implementations and new results[J]. International Journal of Solids and Structures, 2011, 48(13): 1962-1990. doi: 10.1016/j.ijsolstr.2011.03.006
    [8] WANG L, HU H. Flexural wave propagation in single-walled carbon nanotubes[J]. Physical Review B, 2005, 71(19): 195412. doi: 10.1103/PhysRevB.71.195412
    [9] WANG L, HU H, GUO W. Validation of the non-local elastic shell model for studying longitudinal waves in single-walled carbon nanotubes[J]. Nanotechnology, 2006, 17(5): 1408-1415. doi: 10.1088/0957-4484/17/5/041
    [10] ANSARI R, SAHMANI S, ARASH B. Nonlocal plate model for free vibrations of single-layered graphene sheets[J]. Physics Letters A, 2010, 375(1): 53-62. doi: 10.1016/j.physleta.2010.10.028
    [11] KHAKALO S, NIIRANEN J. Anisotropic strain gradient thermoelasticity for cellular structures: plate models, homogenization and isogeometric analysis[J]. Journal of the Mechanics and Physics of Solids, 2020, 134: 103728. doi: 10.1016/j.jmps.2019.103728
    [12] 王碧蓉, 邓子辰, 徐晓建, 等. 剪力非局部因子对双壁碳纳米管中弯曲波频散特性的影响[J]. 应用数学和力学, 2014, 35(5): 478-486. (WANG Birong, DENG Zichen, XU Xiaojian, et al. Effects of nonlocal shear factor on flexural wave dispersion in double-walled carbon nanotubes[J]. Applied Mathematics and Mechanics, 2014, 35(5): 478-486.(in Chinese) doi: 10.3879/j.issn.1000-0887.2014.05.002
    [13] 郑晓静, 周又和. 集中载荷作用下弹性地基圆薄板大挠度问题的精确解[J]. 力学学报, 1988, 20(2): 161-172. (ZHENG Xiaojing, ZHOU Youhe. On the exact solution to large deflection problem of circular plate on elastic foundation under a concentrated load[J]. Acta Mechanica Sinica, 1988, 20(2): 161-172.(in Chinese)
    [14] 陈玲玲, 杨旭, 刘洋, 等. 全应变梯度挠曲电纳米梁有限单元法研究[J]. 计算力学学报, 2020, 37(5): 545-552. (CHEN Lingling, YANG Xu, LIU Yang, et al. Research on finite element method of nanobeam considering flexoelectricity based on general strain gradient elasticity theory[J]. Chinese Journal of Computional Mechanics, 2020, 37(5): 545-552.(in Chinese) doi: 10.7511/jslx20191104002
    [15] MINDLIN R D. Micro-structure in linear elasticity[J]. Archive for Rational Mechanics and Analysis, 1964, 16(1): 51-78. doi: 10.1007/BF00248490
    [16] ASKES H, SUIKER A S J, SLUYS L J. A classification of higher-order strain-gradient models-linear analysis[J]. Archive of Applied Mechanics, 2002, 72(2): 171-188.
    [17] POLIZZOTTO C. A second strain gradient elasticity theory with second velocity gradient inertia, part Ⅰ: constitutive equations and quasi-static behavior[J]. International Journal of Solids and Structures, 2013, 50(24): 3749-3765. doi: 10.1016/j.ijsolstr.2013.06.024
    [18] PAPARGYRI-BESKOU S, GIANNAKOPOULOS A E, BESKOS D E. Variational analysis of gradient elastic flexural plates under static loading[J]. International Journal of Solids and Structures, 2010, 47(20): 2755-2766.
    [19] VENTSEL E, KROUTHAMMER T. Thin Plate and Shells[M]. New York: Marcel Dekker, 2001.
    [20] XU X J, DENG Z C. Effects of strain and higher order inertia gradients on wave propagation in single-walled carbon nanotubes[J]. Physica E: Low-Dimensional Systems and Nanostructures, 2015, 72: 101-110. doi: 10.1016/j.physe.2015.04.011
  • 加载中
图(10) / 表(1)
计量
  • 文章访问数:  179
  • HTML全文浏览量:  96
  • PDF下载量:  85
  • 被引次数: 0
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

目录

    /

    返回文章
    返回