双材料环扇形薄板弯曲问题的辛本征解  

Symplectic eigensolutions for bi-material circular sector thin plate bending

作　　者：王珊[1] 姚伟岸[1] 

机构地区：[1]大连理工大学工业装备结构分析国家重点实验室,大连116024

出　　处：《应用力学学报》2012年第3期252-257,350,共6页Chinese Journal of Applied Mechanics

基　　金：国家自然科学基金(10772039);973国家重点基础研究计划(2010CB832704)

摘　　要：弹性力学辛对偶求解方法是通过引入原变量的对偶变量将问题导入辛空间,从而使得有效的数学物理方法,如分离变量和辛本征函数展开的方法得以实施并得出问题的解析解。本文通过引入弯矩函数和恰当的变换,首先建立了两侧边边界条件自由的双材料环扇形薄板弯曲问题的辛对偶体系。然后,讨论了弯矩函数表示的非齐次边界条件,并给出了三个有特定物理意义的解,其解在端部的力系是非自相平衡的。对双材料的楔形板而言,这三个解表示的就是在尖端有集中弯矩、集中扭矩、垂直集中力作用的解。最后,讨论了弯矩函数表示的齐次边界条件,并给出了辛本征值的超越方程以及辛本征解,所有这些解在端部的力系都是自相平衡的。本文的工作为相关问题的解析求解以及辛本征解的进一步应用研究奠定了基础。By introducing bending moment functions and appropriate transformations,the bending problem of a bi-material circular sector thin plate with both straight sides free can be solved using a scheme of separation of variables and symplectic eigenexpansion in the symplectic space.Firstly,nonhomogeneous boundary condition represented by the bending moment functions is discussed.And three solutions with physical meanings are given,for which the system of forces at the end is non-self-equilibrium.For a bi-material wedge plate,they are the solutions with a unit bending moment,a unit torsion moment and a unit concentrated vertical force acting at the vertex.Secondly,the transcendental equation for the symplectic eigenvalues and the corresponding symplectic eigensolutions are derived under the homogeneous boundary condition represented by the bending moment functions.The system of forces at the end for these solutions all are self-equilibrium.The present work lays a foundation for analytical solutions of related problems and the further application of symplectic eigensolutions.

关 键 词：环扇形板 双材料 辛空间 解析解 楔形板 

分 类 号：O343[理学—固体力学]