三维压电介质界面裂纹的边界积分—微分方程

BOUNDARY INTERGRAL-DIFFERENTIAL EQUATIONS FOR INTERFACIAL CRACKS IN THREE DIMENSIONAL PIEZOELECTRIC MEDIA

基于三维两相横观各向同性压电介质的基本解和压电介质的Somigliana恒等式 ,利用发散积分的有限部理论 ,建立以裂纹面上的不连续位移和不连续电势为基本未知量的三维压电介质界面裂纹问题的超奇异积分—微分方程组 ,其中的积分核具有O(1 r2 )阶的奇异性。当两相材料退化为均质材料或单相材料时 ,方程组中的微分项的系数为零 。

The study on interfacial fracture problem in piezoelectric materials is very important since many smart structures are composite of those materials. Up to now, most of the theoretical researches in this filed are limited to two dimensional problems. In this paper, based on the three dimensional fundamental solutions for two phased transversely isotropic piezoelectric media and the corresponding Somigliana identity, a set of hyper singular integral differential equations is obtained in terms of the displacement discontinuities and the electric potential discontinuity across crack faces. The kernels in the integrals have the singularity of O(1/r 2) . As a special case when the two phased piezoelectric media become homogeneous, the coefficients of the differential parts in these equations equal zero and, therefore, the equations reduce to the hyper singular integral equations. By using of the main part analysis method, the singularity indexes of displacement, electric potential, electric field, electric displacement, strain and stress near the crack tip can be derived.