点群6一维六方准晶椭圆孔口与半无限裂纹反平面弹性问题的解析解  

Analytic Solutions of Anti-plane Elasticity of Point Group 6 of One-dimensional Hexagonal Quasicrystals with Elliptical Hole and Semi-infinite Cracks

作　　者：施志昱[1] 刘官厅[1] 

机构地区：[1]内蒙古师范大学数学科学学院,内蒙古呼和浩特010022

出　　处：《数学的实践与认识》2012年第6期111-121,共11页Mathematics in Practice and Theory

基　　金：国家自然科学基金项目(10761005);内蒙古自然科学基金(2009MS0102)

摘　　要：导出了点群6-维六方准晶反平面弹性问题的控制方程.利用复变方法,给出了点群6-维六方准晶在周期平面内的反平面弹性问题的应力分量以及边界条件的复变表示,通过引入适当的保角变换,研究了点群6-维六方准晶中带有椭圆孔口与半无限裂纹的反平面弹性问题,得到了椭圆孔口问题应力场的解析解,给出了半无限裂纹问题在裂纹尖端处的应力强度因子的解析解.在极限情形下,椭圆孔口转化为Griffith裂纹,并得到该裂纹在裂尖处的应力强度因子的解析解.当点群6-维六方准晶体的对称性增加时,其椭圆孔口与半无限裂纹的反平面弹性问题的解退化为点群6mm-维六方准晶带有椭圆孔口与半无限裂纹的反平面弹性问题的解。The control equations of the anti-plan elasticity problem of point group 6 of onedimensional （1D） hexagonal quasicrystals（QCs） are obtained.By using the complex variable function method,complex representations of the components of stress and boundary conditions of the anti-plane elasticity problem of point group 6 of 1D hexagonal QCs in the cycle plane are given.By introducing the appropriate conformal mapping,the analytic solution of the anti-plane elasticity problem of point group 6 of 1D hexagonal QCs with an elliptical hole is obtained.The stress intensity factors（SIFs） at the crack tips of that problem with semi-infinite cracks are also derived.In the limiting case,the elliptical hole can be reduced to the Griffith crack and the SIFs at the crack tip can be easily obtained.Under conditions of more symmetry,the anti-plane elasticity problem of point group 6 of 1D hexagonal QCs with an elliptical hole and semi-infinite cracks can be degenerated to that of point group 6mm of 1D hexagonal QCs.

关 键 词：点群6 一维六方准晶 椭圆孔口 无限裂纹 应力强度因子 

分 类 号：O753.3[理学—晶体学]