磁电弹性材料中含有带四条裂纹的正方形孔的反平面断裂问题  

The Anti-plane Fracture Problem of Four Edge Cracks Emanating from a Square Hole in Magnetoelectroelastic Materials

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

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

出　　处：《工程数学学报》2024年第1期175-185,共11页Chinese Journal of Engineering Mathematics

基　　金：国家自然基金科学基金(12062022);宁夏高等学校科学研究项目(NGY2022112).

摘　　要：针对反平面剪切力作用下,磁电弹性材料中含有带四条裂纹的正方形孔口的断裂问题进行了深入地探索。基于线弹性断裂理论和Riemann-Schwarz解析延拓定理,利用复变函数方法和留数定理,通过构造合适的数值保角映射函数,将解析函数边值问题转化为柯西积分方程组进行解答,获得了磁电非渗透边界条件下裂纹尖端断裂力学参数的显式表达式。通过与已有结果的对比,验证了方法的有效性。利用数值算例描述了孔洞尺寸、四条裂纹长度和机–电–磁载荷等因素对裂纹扩展参数的影响规律。结果表明:水平右裂纹对孔口裂纹扩展影响更显著;垂直裂纹影响水平裂纹的扩展趋势;在磁电非渗透边界条件下,随着机械载荷的增大,裂纹尖端的应力强度因子变大,而电载荷和磁载荷对裂纹的扩展与机械载荷的大小和方向密切相关。该分析方法为求解复杂多连通域的智能复合材料问题提供了一条有效途径,研究成果为含裂纹的复合材料或结构的优化设计提供了科学依据。The fracture problem of the magnetic electro-elastic material with four cracks in a square hole is investigated under the action of anti-plane shear force.Based on the linear elastic fracture theory and the Riemann-Schwarz analytical continuation theorem,by construct-ing a suitable numerical conformal mapping function and the residue theorem,the boundary value problem of the analytical function is transformed into Cauchy integral equations,then the explicit expression of fracture mechanics parameters at the crack tip under magnetoelectric impermeable boundary conditions is obtained.The effectiveness of the method is verified by comparing with the existing results.The effects of hole size,four crack lengths and mechano-electro-magnetic load on crack propagation parameters are described by numerical examples.The results show that the effect of the horizontal right crack on the crack growth is more signif-icant.Vertical crack affects the propagation trend of horizontal crack.Under magnetoelectric impermeable boundary conditions,the stress intensity factor of crack tip increases with the in-crease of mechanical load.However,the propagation of crack under electric and magnetic load is closely related to the magnitude and direction of mechanical load.The analytical method developed provides an effective way to solve the problem of intelligent composite materials in complex multi-connected domains,and the research results provide a scientific basis for the optimal design of composite materials or structures with cracks.

关 键 词：复变函数方法 磁电弹性材料 正方形孔边裂纹 场强度因子 能量释放率 

分 类 号：O346.1[理学—固体力学]