电磁弹性固体反平面问题辛求解体系及圣维南原理

在由原变量位移、电势和磁势以及它们的对偶变量——纵向的剪应力、电位移和磁感应强度分量组成的辛几何空间，电磁弹性固体反平面问题被导入哈密顿体系，从而有效的数学物理方法如分离变量法及辛本征向量展开法可以用于该问题的求解．首先，通过理性分析直接求解出矩形域问题所有的本征值及其本征函数向量．然后，在对本征函数向量构成的原问题解的定性分析基础上提出了电磁弹性固体反平面问题的圣维南原理。

Symplectic solution system and Saint-Venant principle on anti-plane problem of magnetoelectroelastic solids

The anti-plane problem of magnetoelectroelastic solids is led into Hamiltonian system in symplectic geometry space, which consists of the original variables, the displacement, electric potential, magnetic potential, and their duality variables, lengthways shearing stress, electric displacement and magnetic induction. Thus the effective methods of mathematical physics such as the separation of variables and symplectic eigenfunction expansion can be employed to solve the problem. At first, all the eigenvalues and their eigenfunction vectors in rectangular domain are solved directly with rational analyses. Then, the Saint-Venant principle on the anti-plane problem of magnetoelectroelastic solids is extracted by analyzing qualitatively the solutions of the original problem corresponding with eigenfunction vectors.