Eburilitu, Alatancang. Eigenfunction Expansion Method of Upper Triangular Operator Matrix and Application to Two-Dimensional Elasticity Problems Based on Stress Formulation[J]. Applied Mathematics and Mechanics, 2012, 33(2): 221-230. doi: 10.3879/j.issn.1000-0887.2012.02.007
 Citation: Eburilitu, Alatancang. Eigenfunction Expansion Method of Upper Triangular Operator Matrix and Application to Two-Dimensional Elasticity Problems Based on Stress Formulation[J]. Applied Mathematics and Mechanics, 2012, 33(2): 221-230.

# Eigenfunction Expansion Method of Upper Triangular Operator Matrix and Application to Two-Dimensional Elasticity Problems Based on Stress Formulation

##### doi: 10.3879/j.issn.1000-0887.2012.02.007
• Rev Recd Date: 2011-12-01
• Publish Date: 2012-02-15
• The eigenfunction expansion method to solve twodimensional (2D) elasticity problems based on stress formulation was studied. The fundamental system of partial differential equations of the 2D problems was rewritten as an upper triangular differential system based on the known results, and then the associated upper triangular operator matrix was obtained. By further researching, the two simpler complete orthogonal systems of eigenfunctions in some space were obtained, which belong to the two block operators arising in the operator matrix. Then a more simple and convenient general solution for the 2D problem was given by the eigenfunction expansion method. Furthermore, it was indicated what boundary conditions for the 2D problem can be solved by this method. Finally, the validity of the obtained results was verified by a specific example.
•  [1] 钟万勰. 分离变量法与哈密尔顿体系[J]. 计算结构力学及其应用, 1991, 8(3): 229-240. (ZHONG Wan-xie. Method of separation of variables and Hamiltonian system[J]. Computational Structural Mechanics and Applications, 1991, 8(3): 229-240. (in Chinese)) [2] 钟万勰. 弹性力学求解新体系[M]. 大连: 大连理工大学出版社, 1995. (ZHONG Wan-xie. A New Systematic Methodology for Theory of Elasticity[M]. Dalian: Dalian University of Technology Press, 1995. (in Chinese)) [3] 姚伟岸, 隋永枫. Reissner 板弯曲的辛求解体系[J]. 应用数学和力学, 2004, 25(2): 159-165.(YAO Wei-an, SUI Yong-feng. Symplectic solution system for Reissner plate bending[J]. Applied Mathematics and Mechanics(English Edition), 2004, 25(2): 178-185.) [4] 姚征, 张洪武, 王晋宝, 钟万勰. 基于界带模型的碳纳米管声子谱的辛分析[J]. 固体力学学报, 2008, 29(1): 13-22. (YAO Zheng, ZHANG Hong-wu, WANG Jin-bao, ZHONG Wan-xie. Symplectic analysis for phonon dispersion of carbon nanotubes based on inter-belt model[J]. Chinese Journal of Solid Mechanics, 2008, 29(1): 13-22. (in Chinese)) [5] 徐新生, 王尕平, 孙发明. 二维矩形域内 Stokes 流问题的辛解析和数值方法[J]. 应用数学和力学, 2008, 29(6): 639-648.(XU Xin-sheng, WANG Ga-ping, SUN Fa-ming. Analytical and numerical methods of symplectic system for Stokes flow in two-dimensional rectangular domain[J]. Applied Mathematics and Mechanics(English Edition), 2008, 29(6): 705-714.) [6] 周建方, 卓家寿, 李庆典. 基于Hamilton体系的分离变量法[J]. 河海大学学报, 2000, 28(6): 27-31. (ZHOU Jian-fang, ZHUO Jia-shou, LI Qin-dian. Variable separation method based on Hamilton system[J]. Journal of Hohai University, 2000, 28(6): 27-31.(in Chinese)) [7] Luo J H, Li Q S, Liu G D. A biorthogonality relationship for three-dimensional couple stress problem[J]. Science in China Series G: Physics, Mechanics & Astronomy, 2009, 52(2): 270-276. [8] Lim C W, Xu X S. Symplectic elasticity: theory and applications[J]. Applied Mechanics Reviews, 2010, 63(5): 050802 (10 pages). [9] 张鸿庆, 阿拉坦仓, 钟万勰. Hamilton 体系与辛正交系的完备性[J]. 应用数学和力学, 1997, 18(3): 217-221.(ZHANG Hong-qing, Alatancang, ZHONG Wan-xie. The Hamiltonian system and completeness of symplectic orthogonal system[J]. Applied Mathematics and Mechanics(English Edition), 1997, 18(3): 237-242.) [10] Eburilitu, Alatancang. On feasibility of variable separation method based on Hamiltonian system for a class of plate bending equations[J]. Communications in Theoretical Physics, 2010, 53(3): 569-574. [11] Alatancang, Wu D Y. Completeness in the sense of Cauchy principal value of the eigenfunction systems of infinite dimensional Hamiltonian operator[J]. Science in China, Series A: Mathematics, 2009, 52(1): 173-180. [12] 阿拉坦仓, 张鸿庆, 钟万勰. 矩阵多元多项式的带余除法及其应用[J]. 应用数学和力学, 2000, 21(7): 661-668.(Alatancang, ZHANG Hong-qing, ZHONG Wan-xie. Pseudo-division algorithm for matrix multi-variable polynomial and its application[J]. Applied Mathematics and Mechanics(English Edition), 2000, 21(7): 733-740.) [13] 黄俊杰, 阿拉坦仓, 王华. 基于应力形式的二维弹性问题的本征展开法[J]. 应用数学和力学, 2010, 31(8): 992-1000.(HUANG Jun-jie, Alatancang, WANG Hua. Eigenfunction expansion method and its application to two-dimensional elasticity problems based on stress formulation[J]. Applied Mathematics and Mechanics(English Edition), 2010, 31(8): 1039-1048.) [14] Timoshenko S P, Goodier J N. Theory of Elasticity[M]. 3rd ed. New York: McGraw-Hill, 1970: 53-56.

### Catalog

###### 通讯作者: 陈斌, bchen63@163.com
• 1.

沈阳化工大学材料科学与工程学院 沈阳 110142