压电材料平面问题的尖端场和应力强度因子的求解

Crack Tip Field and Stress Intensity Factors in Planar Piezoelectric Problem

利用Lekhnitskii理论和Stroh理论的相互联系,把已知的基于Lekhnitskii理论平面应变结果转化为Stroh理论形式的结果,直接获得Stroh公式中A,B的显式表达式,此方法可扩展到平面应力情况,然后导出压电材料平面应变问题的尖端场Williams形式的展开式,采用半权函数法计算有限大压电体平面问题应力和电位移强度因子。对无穷大板含中心裂纹的情况下本文结果和已有结果进行了比较,表明本文方法得到的结果精度可靠。本文方法的最大优点是可以求解有限压电体的应力强度因子,并且需要的单元少,精度高,实用性好。

Crack tip field in planar piezoelectric problem is proposed with Stroh formulas. The matrices A and B are obtained via comparison between Lekhnitskii and Stroh relationships. Semi-weight function method is developed to determine stress intensity factors and electric displacement intensity factor. The stress intensity factors defined in terms of the integral form are derived from the Betti's reciprocal work theorem and expressed with stresses, displacement, electric displacement and electric potential on a path and the semi-weight function. The numerical results for a center-cracked plate under pure electric loading demonstrate the accuracy and practicality of the method, in which tight grids and high resolution near the crack tip are unnecessary.