压电压磁材料中正n边形孔边裂纹分析

基于电磁复合材料力学,运用Stroh型公式和复变函数方法,针对压电压磁材料中含正n边形孔边裂纹反平面问题进行了研究。利用Schwarz-Christoffel变换技术,结合Cauchy积分公式和留数定理,导出了磁电全非渗透型边界条件下任意正n边形裂纹尖端场强度因子和能量释放率的解析解。当缺失磁场时,所得解退化为已有结果,以此验证方法的有效性。通过数值算例,对比分析了n=3,n=4和n=5三种特殊情形对应的孔口边长、裂纹长度和受到的力、电和磁载荷对等效场强度因子和无量纲能量释放率的影响规律。研究结果发现,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;在磁电全非渗透型边界条件下,机械载荷始终促进裂纹的扩展,电位移载荷可以促进或抑制裂纹的扩展,磁载荷对裂纹的扩展贡献较少。本研究结果适用于任意正n边形孔边裂纹求解问题,为压电压磁材料元器件的优化设计和断裂特性分析提供了新思路。

Analysis of regular n-polygon hole with radial edge cracks in magneto-electro-elastic composites

Based on the mechanics of magneto-electro-elastic composite materials,using Stroh-type formula and complex variable function method,the anti-plane problem of regular n-polygon holes with radial edge cracks in piezoelectric-piezomagnetic materials was investigated in this paper.By means of the Schwarz-Christoffel transformation,combined with Cauchy integral formula and retention theorem,the problem of an arbitrary regular polygon hole with a radial edge crack was solved.The analytical solutions of the field intensity factor and energy release rate at the crack tip were obtained under the magnetoelectrically impermeable boundary condition.In the absence of magnetic field,the obtained results can be reduced to the well-known results in order to verify the validity of the method.In addition,the numerical examples in special cases of n= 3,n= 4,n= 5 were provided to show the influences of hole size,crack length and applied mechanical-electric-magnetic loads on the equivalent field intensity factor and the dimensionless energy release rate.The research shows that,both the size of the regular n-polygon holes and the crack length can promote the crack growth,and the former has a more significant effect on the crack growth.Moreover,the increase in the number of n-polygon edges will retard crack propagation.Under the magnetoelectrically impermeable boundary condition,the mechanical loads always accelerate the crack propagation,the electric displacements may promote or retard the crack propagation,and the magnetic loads have little contribution to crack growth.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 analysis of piezoelectric-piezomagnetic materials.