界面端奇异行为的数值分析

【作者】 王效贵；

【导师】 许金泉； 郭乙木；

【作者基本信息】 浙江大学 ， 固体力学， 2002， 博士

【摘要】 首先，从多重应力奇异性叠加的奇异应力场出发，通过将奇异性次数分离，提出了一种利用常规的数值分析结果来确定多重应力奇异性(包括振荡应力奇异性)次数的数值分析方法，并且对双材料以及三材料界面端模型的平面变形和三维变形问题进行了数值计算。 其次，基于横观各向同性压电材料空间轴对称变形问题的通解，利用特征展开法，给出了奇异点附近的位移场和奇异应力场。在此基础上，对具有任意界面角和结合角的横观各向同性双压电材料空间轴对称界面端一般模型的轴对称变形问题进行了理论分析，给出了该模型界面端的奇异性特征方程以及界面端附近的位移场、电势、奇异应力场和奇异电位移场。为了检验理论分析的正确性，应用有限元程序对轴对称界面端的奇异性次数进行了数值求解。 最后，从最小势能原理出发，在仅仅考虑奇异性支配区域这一前提下，对于弹性接合材料的平面变形问题和拟平面应变问题，以奇异点为原点分别建立极坐标系和圆柱坐标系，通过分部积分消除厂项，从而使奇异性问题的求解由原来的二维降为一维；对于三维变形问题，以奇异点为原点建立球坐标系，通过分部积分消除γ项，从而使奇异性问题的求解由原来的三维降为二维。然后，采用有限元方法，把降阶后的区域离散，得到了确定奇异性的特征方程。用数值方法求解特征方程，就可以得到满足条件0＜Re(λ)＜1的所有奇异性次数。然后，考虑电学物理量的影响，将上述特殊有限元法应用于压电/压电以及压电/弹性接合材料界面端奇异性的求解。 论文的主要创新点：1、提出了一种利用常规的数值分析结果来确定多重应力奇异性次数的数值分析方法；2、导出了具有任意界面角和结合角的横观各向同性压电双材料轴对称界面端在轴对称变形情况下的奇异性特征方程以及奇异应力场和电位移场；3、提出了一种求解压电/压电以及压电/弹性接合材料在平面变形、拟平面应变以及三维变形情况下界面端奇异性的特殊有限元法。更多还原

【Abstract】 Based on the singular stress field with multiple singularities, numerical methods to determine the orders of the multiple stress singularities are proposed from the numerical solutions obtained by common numerical procedures. Several models of bonded dissimilar materials with interface edges are calculated.Displacement, electrical potential, singular stress fields and singular electrical displacement fields near a singular point are deduced by the eigenfuntion expansion method based on the general solution of the spatial axisymmetric problem of the transversely isotropic piezoelectric media. A generally axisymmetric interface edge of bimaterials with arbitrary interface angle and joining angle is analyzed theoretically by using this method. Eigenequation about singularity, singular stress fields and electrical displacement fields near the interface edge are obtained under axisymmetric distortion.Finally, a special finite element formulation which is based on the principle of minimum potential energy has been developed for determining the orders of the singularity of the singular stress fields around the singular point (interface edge, interface corner and the interface crack) in the bonded dissimilar anisotropic/anisotropic, piezoelectric/piezoelectric as well as piezoelectric/anisotropic materials. The numerical results show that this method is very convenient and efficient.This thesis has the following three innovations: 1. One numerical method to determine the orders of the multiple stress singularities is proposed from the numerical solutions obtained by common numerical procedures. 2. Eigenequation about singularity, singular stress fields and electrical displacement fields near the axisymmetric interface edge of two bonded dissimilar transversely isotropic materials are obtained under axisymmetric distortion. 3. A special finite element formulation has been developed for determining the orders of the singularity in the bonded dissimilar piezoelectric/ piezoelectric as well as piezoelectric/anisotropic materials.更多还原