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一种求三维弹性力学基本解析解的特征方程解法

傅向荣 袁明武 岑松 田歌

傅向荣, 袁明武, 岑松, 田歌. 一种求三维弹性力学基本解析解的特征方程解法[J]. 应用数学和力学, 2012, 33(10): 1170-1179. doi: 10.3879/j.issn.1000-0887.2012.10.003
引用本文: 傅向荣, 袁明武, 岑松, 田歌. 一种求三维弹性力学基本解析解的特征方程解法[J]. 应用数学和力学, 2012, 33(10): 1170-1179. doi: 10.3879/j.issn.1000-0887.2012.10.003
FU Xiang-rong, YUAN Ming-wu, CEN Song, TIAN Ge. A Characteristic Equation Solution Strategy for Deriving the Fundamental Analytical Solutions of 3D Isotropic Elasticity[J]. Applied Mathematics and Mechanics, 2012, 33(10): 1170-1179. doi: 10.3879/j.issn.1000-0887.2012.10.003
Citation: FU Xiang-rong, YUAN Ming-wu, CEN Song, TIAN Ge. A Characteristic Equation Solution Strategy for Deriving the Fundamental Analytical Solutions of 3D Isotropic Elasticity[J]. Applied Mathematics and Mechanics, 2012, 33(10): 1170-1179. doi: 10.3879/j.issn.1000-0887.2012.10.003

一种求三维弹性力学基本解析解的特征方程解法

doi: 10.3879/j.issn.1000-0887.2012.10.003
详细信息
    通讯作者:

    傅向荣(1972—), 男, 湖南人, 副教授, 博士(E-mail: fuxr@cau.edu.cn);岑松(1972—), 男, 福建人, 教授, 博士(联系人. Tel: +86-10-62782452; E-mail: censong@tsinghua.edu.cn).

  • 中图分类号: O343.2;O242.2

A Characteristic Equation Solution Strategy for Deriving the Fundamental Analytical Solutions of 3D Isotropic Elasticity

Funds: 国家自然科学基金资助项目(10872108;10876100);国家重点基础研究发展计划基金资助项目(2010CB731503;2010CB832701);教育部新世纪优秀人才支持计划基金资助项目(NCET-07-0477);中国航天科学基金资助项目(ASFC)
  • 摘要: 提出了一种简单的推导各向同性材料,三维弹性力学问题基本解析解的特征方程解法.应用三维问题控制微分方程的算子矩阵,通过计算其行列式可得到问题特征通解所需满足的特征方程.将满足各种不同简化特征方程的特征通解,代入到微分方程算子矩阵所对应的不同的缩减伴随矩阵,可推导得出相应的三维弹性力学问题的基本解析解,包括B-G解、修正的P-N(P-N-W)解和类胡海昌解.进一步对各类多项式形式的基本解析解的独立性进行了讨论.这些工作为构造数值方法中所需的完备独立的解析试函数奠定了基础.
  • [1] LONG Yu-qiu, CEN Song, LONG Zhi-fei. Advanced Finite Element Method in Structural Engineering[M]. Beijing: Tsinghua University Press; Berlin: Springer-Verlag, 2009.
    [2] 龙驭球, 傅向荣. 基于解析试函数的广义协调四边形厚板元[J]. 工程力学, 2002, 19(3): 10-15. (LONG Yu-qiu, FU Xiang-rong. Two generalized conforming quadrilateral thick plate elements based on analytical trial functions[J]. Engineering Mechanics, 2002, 19(3): 10-15. (in Chinese))
    [3] 傅向荣, 龙驭球. 基于解析试函数的广义协调四边形膜元[J]. 工程力学, 2002, 19(4): 12-16. (FU Xiang-rong, LONG Yu-qiu. Generalized conforming quarilateral plane elements based on analytical trial functions[J]. Engineering Mechanics, 2002, 19(4): 12-16. (in Chinese))
    [4] FU Xiang-rong, CEN Song, LONG Yu-qiu, JIANG Xiu-gen, JU Jin-san. The analytical trial function method (ATFM) for finite element analysis of plane crack/notch problems[J]. Key Engineering Materials, 2008, 385/387: 617-620.
    [5] FU Xiang-rong, CEN Song, LI C F, CHEN Xiao-ming. Analytical trial function method for development of new 8-node plane element based on variational principle containing Airy stress function[J]. Engineering Computations, 2010, 27(4): 442-463.
    [6] CEN Song, FU Xiang-rong, ZHOU Ming-jue. 8- and 12-node plane hybrid stress-function elements immune to severely distorted mesh containing elements with concave shapes[J]. Computer Methods in Applied Mechanics and Engineering, 2011, 200(29/32): 2321-2336.
    [7] CEN Song, FU Xiang-rong, ZHOU Guo-hua, ZHOU Ming-jue, LI Chen-feng. Shape-free finite element method: the plane hybrid stress-function (HS-F) element method for anisotropic materials[J]. Science China Physics, Mechanics & Astronomy, 2011, 54(4): 653-665.
    [8] CEN Song, ZHOU Ming-jue, FU Xiang-rong. A 4-node hybrid stress-function (HS-F) plane element with drilling degrees of freedom less sensitive to severe mesh distortions[J]. Computers & Structures, 2011, 89(5/6): 517-528.
    [9] 王敏中. 高等弹性力学[M].北京: 北京大学出版社, 2002.(WANG Min-zhong.Advanced Elasticity[M]. Beijing: Peking University Press, 2002. (in Chinese))
    [10] WANG Min-zhong, XU Bai-xiang, ZHAO Ying-tao.General representations of polynomial elastic fields[J].Journal of Applied Mechanics, 2012, 79(2): 021017.
    [11] 胡海昌. 胡海昌院士文集[M]. 北京 : 中国宇航出版社, 2008. (HU Hai-chang. Collected Works of Academician HU Hai-chang[M]. Beijing: China Astronautic Publishing House, 2008. (in Chinese))
    [12] 王敏中, 王炜, 武际可. 弹性力学教程[M]. 北京: 北京大学出版社. 2011. (WANG Min-zhong, WANG Wei, WU Ji-ke. Tutorials of Elasticity[M]. Beijing: Peking University Press, 2011. (in Chinese))
    [13] 陆明万, 罗学富. 弹性理论基础
    [14] [ M] . 北京: 清华大学出版社, 1990. (LU Ming-wan, LUO Xue-fu. Fundamental of Elasticity[M]. Beijing: Tsinghua University Press, 1990. (in Chinese))
    [15] 徐芝纶. 弹性力学简明教程[M]. 北京: 高等教育出版社.1983. (XU Zhi-lun. Concise Tutorials on Elasticity[M]. Beijing: Higher Education Press, 1983. (in Chinese))
    [16] 武际可, 王敏中. 弹性力学引论[M]. 北京: 北京大学出版社. 2001. (WU Ji-ke, WANG Min-zhong. Introduction to Elasticity[M]. Beijing: Peking University Press, 2001. (in Chinese))
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出版历程
  • 收稿日期:  2011-08-19
  • 修回日期:  2012-04-27
  • 刊出日期:  2012-10-15

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