一种求三维弹性力学基本解析解的特征方程解法  

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

作　　者：傅向荣[1] 袁明武[2] 岑松[3,4] 田歌[1] 

机构地区：[1]中国农业大学土木水利学院,北京100083 [2]北京大学工学院,北京100871 [3]清华大学航天航空学院工程力学系,北京100084 [4]清华大学航天航空学院教育部应用力学重点实验室高性能数值仿真中心,北京100084

出　　处：《应用数学和力学》2012年第10期1170-1179,共10页Applied Mathematics and Mechanics

基　　金：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.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 Haichang 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.

关 键 词：特征方程解法 基本解析解 修正的P-N(P-N-W)解 胡海昌解 解析试函数 

分 类 号：O343.2[理学—固体力学] O242.2[理学—力学]