压电材料反平面切口奇性指数分析

Analysis of singularity orders in piezo-electric notches under anti-plane loading

为理解压电材料反平面切口尖端奇异状态,提出了一种切口奇性特征分析法。基于切口根部位移场幂级数渐近展开假设,从应力平衡方程和电荷守恒条件出发,导出了关于压电材料反平面切口奇性指数的特征微分方程组,并将切口的力电学边界条件以及界面协调条件表达为奇性指数和特征角函数的组合。从而,压电材料切口反平面奇性指数的计算被转化为相应边界条件下常微分方程组特征值的求解问题,采用插值矩阵法可以计算出各阶奇性指数和相应的特征角函数。该法既适合裂纹奇性分析,也可用于单、双材料切口的奇性计算,并避免了用迭代法求解超越方程的不足。因而具有适应性强的特点。计算发现,压电材料反平面切口存在两个奇性指数,切口的奇异性程度随着开角的增大而增强。

A new approach is presented to evaluate the singularity orders in the piezoelectric notch tip under anti-plane loading.Firstly,the displacement field at the notch tip is expanded as the power series asymptotic expansions,and the stress equilibrium equation and the charge conservation condition are turned into the characteristic equations with respect to the singularity orders.Then,the boundary conditions on the notch surfaces and the compatibility conditions on the bonded interface are transformed into the combination of the singularity orders and the eigen-functions.Thus,the calculation of the piezoelectric notch singularity orders under anti-plane loading is turned into evaluating the eigen-values of the ordinary differential equations under corresponding boundary conditions.The singularity orders and the corresponding characteristic angle functions can be obtained after the interpolate matrix method is introduced to solving the established ordinary differential equations.The present method can be used to analyze the singularity of the crack,the single material notch and bi-material notch.The accuracy of the present method is high because solving the transcendental equation by the iterative method can be successfully avoided.The numerical results obtained show that there are two singularity orders for the anti-plane piezoelectric notch under the boundary conditions of traction free and electrical insulation.The singularity degrees become strong with the increase of the notch opening angle.