Further Application Of Symplectic Dual Solution System In Elasticity

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.