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

FU Xiang-rong; YUAN Ming-wu; CEN Song; TIAN Ge

doi:10.3879/j.issn.1000-0887.2012.10.003

Volume 33 Issue 10

Turn off MathJax

Article Contents

Article Navigation > Applied Mathematics and Mechanics > 2012 > 33(10): 1170-1179

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:

PDF( 376 KB)

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

doi: 10.3879/j.issn.1000-0887.2012.10.003

1.
¹College of Water Conservancy & Civil Engineering, China Agricultural University, Beijing 100083, P.R.China;²School of Engineeing, Peking University, Beijing 100871, P.R.China;³Department of Engineering Mechanics,School of Aerospace, Tsinghua University, Beijing 100084, P.R.China;⁴Key Laboratory of Applied Mechanics & High Performance Computing Center, School of Aerospace, Tsinghua University, Beijing 100084, P.R.China

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）

Received Date: 2011-08-19
Rev Recd Date: 2012-04-27
Publish Date: 2012-10-15

Abstract

Abstract

A simple characteristic equation solution strategy for deriving the fundamental analytical solutions of 3D isotropic elasticity was proposed. By calculating the determinant of the differential operator matrix obtained from the governing equations of 3D elasticity, the characteristic equation which the characteristic general solution vectors must satisfy was established. Then, by substitution of the characteristic general solution vectors, which satisfied various reduced characteristic equations, into various reduced adjoint matrices of the differential operator matrix, the corresponding fundamental analytical solutions for isotropic 3D elasticity, including B-G solutions, modified P-N (P-N-W) solutions, and quasi HU Hai-chang solutions, could be obtained. Furthermore, the independence characters of various fundamental solutions in polynomial form were also discussed in details. These works provide a basis for constructing complete and independent analytical trial functions used in numerical methods.
- characteristic equation solution strategy,
- fundamental analytical solution,
- modified P-N (P-N-W) solution,
- HU Hai-chang solution,
- analytical trial function

FullText(HTML)

References(16)

References

[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))