平面偶应力问题的辛求解方法

【作者】 房桂祥；

【导师】 钟万勰；

【作者基本信息】 大连理工大学 ， 固体力学， 2004， 硕士

【摘要】 平面偶应力理论虽然早在上世纪初就出现了，但是其分析求解一直没有得到很好的解决。现有的求解手段主要采用数值方法——如有限元法。而能给出其解析解的只限于某些特殊的偶应力问题。辛方法作为一种崭新的理论求解体系已成功应用于板、梁等弹性力学问题的求解，与经典的弹性力学求解体系相比有着其独特的优越性。本文目的在于把这种解析方法应用到平面偶应力问题的求解。 本文借助于Reissner板与平面偶应力的模拟关系，在平面偶应力问题的类Hellinger-Reissner变分原理的基础上，以应力函数为原变量，部分应变为其对偶变量，推导出力法形式的平面偶应力问题的Hamilton对偶方程组。于是把平面偶应力问题引入到Hamilton体系，从而利用辛空间的分离变量和本征函数向量展开法获得其解。本文讨论了两种典型边界条件——对边自由和对边固支矩形域问题的解析求解。首先求解出由于用应变代替位移作为基本变量而带来的对边自由矩形域问题的所有非齐次特解，这些解均是有特殊物理意义的解。然后，推导出这两类边界条件各自的本征值超越方程，并进一步给出其对应的非零本征值的本征解。从而依据叠加原理，获得这两种典型边界条件问题的解。最后，本文求解了一半无穷矩形域单向拉伸问题，数值结果证明了微尺寸下经典弹性力学的求解方法得出的结果不再适用，由于偶应力的影响，单向拉伸问题在固定端角点处的奇异性消失。 本文将辛方法成功应用于矩形域平面偶应力问题的求解，为这一类问题提供了一条崭新的解决途径。算例结果也很好地证明了辛方法的有效性和优越性。更多还原

【Abstract】 Plane couple stress problem appeared at the beginning of last century, but it has not been well analytically solved before. There are mainly some numerical methods such as FEM etc. at present, and there are only few analytic solutions for some special problems. Symplectic method has been applied in some plate and beam bending problems successfully. Compared with classical method, it has some unique advantages. This paper aims to apply this analytic method in plane couple stress problem.Based on the Pro-Hellinger-Reissner variational principle of plane couple stress problem, the dual PDEs are proposed corresponding to the force method extension. The duality solution methodology is thus extended to plane couple stress problem, and then the method of separation of variables and eigenfunction expansion in the symplectic space is used to find the analytical solutions. A long strip domain plate with both lateral edges free, fixed at one end and under simple tension at the far end, is solved analytically. The solution is composed of the inhomogeneous boundary condition solutions (which are induced from that the stress functions are selected as primary unknowns) and the superposition of eigensolutions of homogeneous lateral boundary conditions. The method of separation of variables is used for the dual PDEs, from which the eigen-root transcendental equation is solved and the corresponding eigenvector functions are obtained. The boundary conditions at the fixed end are derived via’ the variational method. The superposition of these eigensolutions gives the stress distribution at the fixed end. Numerical results show that due to the effect of couple stress, the stress distribution is no longer infinity as given by the classical theory of elasticity at the corner of fixed end.Results improve that in micro scale the couple stress effect should not be neglect, and symplectic method is effective and advantageous to handle with plane couple stress problem in rectangle field.更多还原