【作者】 姚伟岸；

【导师】 钟万勰；

【作者基本信息】 大连理工大学 ， 固体力学， 2004， 博士

【摘要】 辛对偶求解体系近来得到越来越多的关注,它成功地应用于许多传统方法如半逆求解方法难于应用的课题。本文的工作是将辛对偶求解体系方法分别应用于弹性楔体的佯谬分析、弹性薄板的弯曲分析、多层层合板、Reissner板弯曲和电磁弹性固体问题。主要的研究成果如下: 在极坐标辛对偶体系下重新求解了各向同性弹性楔体所有佯谬问题的解析解。本文的工作表明,欧几里得空间下的佯谬解就是辛空间下的约当型解,而且通过标准的数学方法可以直接获得所有佯谬问题的解析解。 基于平面弹性与薄板弯曲的相似性原理,利用比传统求解方法应用更广泛的辛对偶求解体系,给出了薄板弯曲经典理论的另一套基本方程,并通过分离变量及辛本征函数展开方法给出薄板弯曲问题的分析解。 辛对偶求解体系易于同时描述多层层合板层间的位移连续性条件和应力平衡条件,给出平面各向异性多层层合板问题的对偶方程组。然后,求解出零本征值的所有本征解,并通过零本征值辛子空间的展开求解给出了平面各向异性多层层合板圣维南问题的一个解析求解方法。 建立了Reissner板弯曲问题的辛对偶求解体系,并求解出圣维南问题的所有基本解,它们形成一个完备的辛子空间。其意义是为Reissner板弯曲问题的解析求解开拓出一条新路。同时,进一步完善了Reissner板弯曲问题与平面偶应力理论的模拟关系,模拟理论将可为两类问题的求解提供一些新的解析与数值求解方法。 最后,给出了电磁弹性固体三维问题所有变量为自变量的最一般形式的广义变分原理,它涵盖了电磁弹性固体所有的基本方程和边界条件。同时,将电磁弹性固体反平面问题导入辛对偶求解体系,并通过分离变量形成辛本征问题。然后,在对本征解定性分析基础上论述了电磁弹性固体反平面问题的圣维南原理。更多还原

【Abstract】 The symplectic dual solution system is getting more and more concerns recently, which has been proved pretty applicable for a number of cases where traditional approaches, such as semi-inverse technique, are difficult to be applied. In this dissertation, this system is applied to the paradox analysis of elastic wedge, the bending analysis of elastic thin plate, multi-layered composite plate, Reissner plate, and the couple analysis of magnetoelectroelastic solids, respectively. Major contributions are as follows:All paradox problems in the isotropic elastic wedge are re-solved in analytical forms under a symplectic dual polar coordinate system. It is revealed that the paradox solution in Euclidean space is just a solution in symplectic space in the Jordan form which can easily be obtained by conventional mathematical means.By making an analogy to the case of plane elasticity and utilizing the symplectic dual solution system in which more complex problems can be solved in analytical forms by comparison with traditional approaches, a set of new governing equations for classical thin plate bending problems are presented, and solutions are given by using variable separation and eigenfunction-vector expansion techniques.The symplectic dual solution system, facilitating to describe compatibility conditions of displacements and stress at interfaces, is exploited to derive dual equations for plane multi-layered anisotropic composite plates. All eigenfunctions with eigenvalue zero are obtained, and an analytical approach of Saint Venant solutions of plane multi-layered anisotropic composite plates is proposed by expending eigenfunctions in symplectic subspace with eigenvalue zero.A symplectic dual solution system for the bending problem of Reissner plate is established, providing all basic Saint Venant solutions which constitute a perfect symplectic subspace. The establishment of this system gives a new way to solve bending problems of Reissner plate in analytical forms. Furthermore, this work makes the analogue relationship between Reissner plate theory and plane couple-stress theory more perfect, and may bring some new analytical and numerical methods to solve the problems in both fields.Lastly, three-dimensional generalized variational principles in magnetoelectroelastic solids, including all variables, fundamental equations and boundary conditions, are presented. Furthermore, the anti-plane problem of magnetoelectroelastic solids is considered in the symplectic dual solution system, a symplectic eigenvalue problem is formed via a variable separation scheme, and Saint Venant principle for the anti-plane problem of magnetoelectroelastic solids is illustrated by analyzing symplectic eigenfunction vectors qualitatively.更多还原