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中国工业与应用数学学会会刊
主管：中华人民共和国教育部
主办：西安交通大学
ISSN 1005-3085  CN 61-1269/O1
工程数学学报

工程数学学报 ›› 2016, Vol. 33 ›› Issue (2): 184-198.doi: 10.3969/j.issn.1005-3085.2016.02.008

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一维六方压电准晶中带三条不对称裂纹的圆形孔口的反平面问题(英)

杨  娟1,2,    李  星2,   丁生虎2   

  1. 1- 宁夏大学民族预科教育学院，银川 750002
    2- 宁夏大学数学计算机学院，银川 750021
  • 收稿日期:2014-05-29 接受日期:2015-05-07 出版日期:2016-04-15 发布日期:2016-06-15
  • 通讯作者: 李 星 E-mail: li_x@nxu.edu.cn
  • 基金资助:
    国家自然科学基金 (11362018; 11261045)；高等学校博士点专项科研基金 (20116401110002)；宁夏大学自然科学基金 (ZR15026).

Anti-plane Analysis of a Circular Hole with Three Unequal Cracks in One-dimensional Hexagonal Piezoelectric Quasicrystals

YANG Juan1,2,  LI Xing2,  DING Sheng-hu2   

  1. 1- Institute of Ethnic Preparatory Education, Ningxia University, Yinchuan 750002
    2- Institute of Mathematics and Computer Science, Ningxia University, Yinchuan 750021
  • Received:2014-05-29 Accepted:2015-05-07 Online:2016-04-15 Published:2016-06-15
  • Contact: X. Li. E-mail address: li_x@nxu.edu.cn
  • Supported by:
    The National Natural Science Foundation of China (11362018; 11261045); the Specialized Research Fund for the Doctoral Program of Higher Education of China (20116401110002); the Natural Science Foundation of Ningxia University (ZR15026).

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摘要： 本文利用复变函数方法和保角映射，研究一维六方压电准晶材料中带不对称三裂纹的圆形孔口的的断裂问题．根据准晶压电材料基本方程的基础上，利用点群的对称性和一维六方准晶的线性压电效应，导出了一维六方准晶压电材料反平面问题的控制方程，并结合Cauchy积分公式，得到电非渗透与电渗透边界条件下的裂纹尖端场强度因子的解析表达式．当改变裂纹长度和孔口半径时，所得结果可以模拟出一些新裂纹模型．在不考虑电载荷作用时，所得结果和原有结果是一致的．通过数值算例讨论了材料的几何参数对场强度因子的影响，得出水平裂纹长度和圆半径可以促进裂纹增长．本研究为工程中材料的制备与应用将提供可靠的理论价值．

关键词: 压电准晶, 不对称三裂纹, 圆孔, 保角映射, 场强度因子, 复变方法

Abstract:

This paper employes variable function method and the technique of conformal mapping to discuss the anti-plane problem of a circular hole with three unequal cracks in a one-dimensional (1D) hexagonal piezoelectric quasicrystal. Based on the piezoelectricity fundamental equations of quasicrystal materials and the symmetry of 1D hexagonal quasicrystal and its linear piezoelectricity effect, 1D hexagonal quasicrystal control equations of anti-plane problem are derived. Applying Cauchy integral formula, the analytical expressions for the crack tip filed intensity factors are presented with the assumption that the crack are electrical impermeable and electrical permeable. With the variation of the hole-size and the crack length, some of the new model of crack are obtained. In the absence of the electric load, the results match with the classical ones. The numerical results indicate the effects of geometric parameters on the field intensity factors. It is verified that the horizontal crack length and the circle radius can easily promote crack growth. Research on such issues will provide reliable theoretical value for the engineering materials preparation and application. 

Key words: piezoelectric quasicrystals, three unequal cracks, circular hole, conformal mapping, the field intensity factors, complex variable method

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