Characteristic equation solution strategy for deriving fundamental analytical solutions of 3D isotropic elasticity

doi:10.1007/s10483-012-1619-7

Applied Mathematics and Mechanics (English Edition) ›› 2012, Vol. 33 ›› Issue (10): 1253-1264.doi: https://doi.org/10.1007/s10483-012-1619-7

Characteristic equation solution strategy for deriving fundamental analytical solutions of 3D isotropic elasticity

傅向荣¹, 袁明武², 岑松^3,4, 田歌¹

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

收稿日期:2012-04-27 修回日期:2012-05-27 出版日期:2012-10-10 发布日期:2012-10-10
通讯作者: Song CEN, Professor, Ph.D., E-mail: censong@tsinghua.edu.cn E-mail:censong@tsinghua.edu.cn
基金资助:
Project supported by the National Natural Science Foundation of China (Nos. 10872108 and 10876100), the Program for New Century Excellent Talents in University (No.NCET-07-0477), the National Basic Research Programs of China (Nos. 2010CB731503 and 2010CB832701)

Characteristic equation solution strategy for deriving fundamental analytical solutions of 3D isotropic elasticity

Xiang-rong FU¹, Ming-wu YUAN², Song CEN^3,4, Ge TIAN ¹

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

Received:2012-04-27 Revised:2012-05-27 Online:2012-10-10 Published:2012-10-10
Supported by:
Project supported by the National Natural Science Foundation of China (Nos. 10872108 and 10876100), the Program for New Century Excellent Talents in University (No.NCET-07-0477), the National Basic Research Programs of China (Nos. 2010CB731503 and 2010CB832701)

摘要/Abstract

摘要： A simple characteristic equation solution strategy for deriving the fun- damental analytical solutions of 3D isotropic elasticity is 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 is established. Then, by substitution of the characteristic general solution vectors, which satisfy various reduced characteristic equations, into various reduced ad- joint matrices of the differential operator matrix, the corresponding fundamental analyt- ical solutions for isotropic 3D elasticity, including Boussinesq-Galerkin (B-G) solutions, modified Papkovich-Neuber solutions proposed by Min-zhongWANG (P-N-W), and quasi HU Hai-chang solutions, can be obtained. Furthermore, the independence characters of various fundamental solutions in polynomial form are also discussed in detail. These works provide a basis for constructing complete and independent analytical trial func- tions used in numerical methods.

关键词: characteristic equation solution strategy, fundamental analytical solution, modified Papkovich-Neuber solution, point-ring-couple, torsion problem, elastodynamics, HU Hai-chang solution, analytical trial function

Abstract: A simple characteristic equation solution strategy for deriving the fun- damental analytical solutions of 3D isotropic elasticity is 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 is established. Then, by substitution of the characteristic general solution vectors, which satisfy various reduced characteristic equations, into various reduced ad- joint matrices of the differential operator matrix, the corresponding fundamental analyt- ical solutions for isotropic 3D elasticity, including Boussinesq-Galerkin (B-G) solutions, modified Papkovich-Neuber solutions proposed by Min-zhongWANG (P-N-W), and quasi HU Hai-chang solutions, can be obtained. Furthermore, the independence characters of various fundamental solutions in polynomial form are also discussed in detail. These works provide a basis for constructing complete and independent analytical trial func- tions used in numerical methods.

Key words: point-ring-couple, torsion problem, elastodynamics, characteristic equation solution strategy, fundamental analytical solution, modified Papkovich-Neuber solution, HU Hai-chang solution, analytical trial function

中图分类号:

O343.2

傅向荣;袁明武;岑松;田歌. Characteristic equation solution strategy for deriving fundamental analytical solutions of 3D isotropic elasticity[J]. Applied Mathematics and Mechanics (English Edition), 2012, 33(10): 1253-1264.

Xiang-rong FU;Ming-wu YUAN;Song CEN;Ge TIAN . Characteristic equation solution strategy for deriving fundamental analytical solutions of 3D isotropic elasticity[J]. Applied Mathematics and Mechanics (English Edition), 2012, 33(10): 1253-1264.

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Characteristic equation solution strategy for deriving fundamental analytical solutions of 3D isotropic elasticity

Characteristic equation solution strategy for deriving fundamental analytical solutions of 3D isotropic elasticity

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