层状压电陶瓷致动器中力电耦合场奇异性的数值分析

Numerical analysis on the singularities of electromechanical fields in multi-layer piezoelectric actuators

首先推导了不同压电材料界面裂纹尖端处的扇形区域内包含基本方程、裂纹面D-P边界条件和交界面处边界条件的弱形式。通过假设力电耦合位移场(位移和电势)与到裂纹尖端距离的(λ+1)次方成正比,可以得到一个分析压电材料裂纹尖端处力电耦合场奇异性的特殊的一维有限元列式。该一维有限元列式只需对扇形区域在角度方向上离散,最后的总体方程为一个关于λ的二次特征根方程。探讨了层状压电陶瓷致动器中可能出现奇异力电耦合场的部位的裂纹面边界条件及交界面处边界条件,进而将该一维有限元法进行推广,用于研究了这些部位的力电耦合场的奇异性。通过数值算例与相应的精确解的比较表明该方法是正确的,而且仅用很少单元就可以得到非常精确的结果。

By using the weak form of governing equations and local boundary conditions for sectorial bimaterial domains of piezoelectrics and assuming that the displacement fields and electric potential are proportional to the （λ＋ 1）th power of the distance from the crack tips, a second order characteristic matrix equation on λ is derived by a one-dimensional finite element formulation that only discretizes domains circumferentially. After a discussion on the boundary conditions on the interfaces and on the crack lines, the present method is extended to determine the singularities of electromechanical fields in piezoelectric ceramic multi-layer actuators. Validity of the formulation is verified by comparing the computed results with the existing analytical solution. Accurate solutions are yielded by very few elements.