功能梯度材料平面问题的辛弹性力学解法  

THE SYMPLECTIC METHOD FOR PLANE ELASTICITY PROBLEM OF FUNCTIONALLY GRADED MATERIALS

作　　者：陈伟球[1] 赵莉[1] 

机构地区：[1]浙江大学土木系,杭州310027

出　　处：《力学学报》2009年第4期588-594,共7页Chinese Journal of Theoretical and Applied Mechanics

基　　金：国家自然科学基金(10725210;10432030);教育部高等学校博士点专项基金(20060335107);新世纪优秀人才支持计划(NCET-05-05010)资助项目~~

摘　　要：将辛弹性力学解法推广用于功能梯度材料平面问题的分析,考虑沿长度方向弹性模量为指数函数变化而泊松比为常数的矩形域平面弹性问题,给出了具体的求解步骤.提出了移位Hamilton矩阵的新概念,建立起相应的辛共轭正交关系;导出了对应特殊本征值的本征解,发现材料的非均匀特性使特殊本征解的形式发生明显的变化.There are several typical methods that have been widely employed to analyze static and dynamic behavior of functionally graded material （FGM） structures, such as the simplified analytical methods （based on one or two-dimensional structural theories）, three-dimensional exact elasticity methods, approximate elasticity methods based on laminated models, semi-analytical methods, and the numerical methods. In this paper, the symplectic approach, originally developed for homogeneous or piece-wisely homogeneous materials, is extended to consider the plane elasticity problem with a rectangular domain of FGM. In the present FGM, Young＇s modulus varies exponentially with the axial coordinate, while Poisson ratio remains unaltered. After introducing new stress components, the problem is formulated within the frame of state space, and solved using the method of separation of variables along with the eigenfunction expansion technique. The operator matrix, called shift- Hamiltonian matrix, is not in an exact Hamiltonian form, since the eigenvalues are symmetric with respect to -α/2, rather than zero in the standard Hamilton matrix. In this case, the symplectic adjoint eigenvalue of zero is induced as -α. The Saint-Venant solutions derived in the paper exhibit some unique characteristics, but they can be degenerated to the ones for homogeneous materials after imposing certain rigid motions. The symplectic method enriches the analysis methodology for heterogeneous material. Furthermore, it can indicate the certain physical essence of the problem that can not be revealed by other methods.

关 键 词：功能梯度材料 平面问题 辛弹性力学 移位Hamilton矩阵 辛共轭正交关系 

分 类 号：O343.7[理学—固体力学]