压电复合材料中正n边形孔边裂纹的反平面问题研究  

Anti-Plane Problem of Regular n-Polygon Hole with Radial Edge Crack in Piezoelectric Composites

作　　者：徐燕[1] 杨娟[2,3] XU Yan;YANG Juan(Xinhua College of Ningxia University,Yinchuan 750021,Ningxia,China;School of Preparatory Education for Nationalities,Ningxia University,Yinchuan 750002,Ningxia,China;School of Mathematics and Statistics,Ningxia University,Yinchuan 750021,Ningxia,China)

机构地区：[1]宁夏大学新华学院,宁夏银川750021 [2]宁夏大学民族预科教育学院,宁夏银川750002 [3]宁夏大学数学统计学院,宁夏银川750021

出　　处：《力学季刊》2021年第2期279-290,共12页Chinese Quarterly of Mechanics

基　　金：国家自然科学基金(12062022,42064004,11762016,11762017);宁夏高等学校科学研究项目(NGY2020013);宁夏自然科学基金(2020AAC03061,2020AAC03057);2019年宁夏一流基层教学组织(NXYLJCJXZ-21)。

摘　　要：本文基于Cauchy积分理论和Schwarz-Christoffel(SC)变换技术,针对压电复合材料中带一条裂纹的正n边形孔口缺陷的反平面断裂力学进行了探究.假设满足电不可通边界条件,利用Cauchy积分公式和留数定理,获得了任意正n边形裂尖处应力和电位移两个场强度因子以及全能量释放率的封闭形式的显式解.当正n边形边数取定时,所得解可退化为已有结果,以此验证方法的有效性.并通过数值算例,对比分析了n=3,n=4,n=5三种特殊情形对应的等效场强度因子和无量纲能量释放率与孔口边长、裂纹长度和受到的力、电载荷之间的曲线图.数值结果显示:正n边形孔洞的尺寸和裂纹长度均会促进裂纹扩展,且前者的影响更显著一些;正n边形边的数量增加会阻止裂纹的扩展;在电不可通边界条件下,机械载荷对裂纹的扩展始终贡献显著,电场对断裂行为的影响取决于机械载荷.本研究结果具有一般性,适用于任意正n边形孔边裂纹问题的求解,为压电复合材料元器件的优化设计和断裂特性分析提供了新思路.Based on the Cauchy integral theory and the Schwarz-Christoffel(SC)transformation,the anti-plane fracture mechanics for regular n-polygon hole with single radial edge crack in piezoelectric composites was investigated in this paper.Using Cauchy integral formula and the retention theorem,the problem of an arbitrary regular polygon hole with a radial edge crack was solved.Under the electrically impermeable boundary condition,we obtained the explicit solutions in closed form for the stress intensity factor,the electric displacement intensity factor and the total energy release rate near the crack tip of regular n-polygon orifice.When the number of regular n-polygon edges is given,the obtained results can be reduced to the well-known results,validating the proposed method.In addition,the numerical examples in special cases of n=3,n=4,n=5 were provided to illustrate the influences of the hole size,the crack geometric dimension,the force and the electric load on the equivalent field intensity factor and the dimensionless energy release rate.The numerical results are presented to show that,both the size of regular n-polygon hole and the crack length can promote the crack growth,with the former having a more significant effect on the crack growth.Moreover,the increase in the number of n-polygon edges will retard the crack propagation.Under the electrically impermeable boundary condition,the mechanical loads always contribute significantly to the crack propagation.The effect of electric fields on the fracture behavior is dependent on the mechanical load.The results are suitable for calculating the problem of an arbitrary regular polygon hole with a radial edge crack,thereby providing a new idea for the optimal design and fracture characteristic analysis of the piezoelectric components.

关 键 词：压电复合材料 保角映射 正n边形孔边裂纹 场强度因子 能量释放率 

分 类 号：O346[理学—固体力学]