一维六方准晶中纳米尺度正三角形孔边裂纹的反平面问题

Anti-plane Problem of Nano-cracks Emanating from a Regular Triangular Nano-hole in One Dimensional Hexagonal Quasicrystals

基于Gurtin-Murdoch界面/表面理论和复变函数方法,结合保角映射技术研究了一维六方准晶含纳米尺度的正三角形孔边三裂纹反平面剪切问题,分别得到了声子场和相位子场的应力、应力强度因子封闭表达式,同时还获得了能量释放率的解析解。数值算例分析了无量纲化应力强度因子、无量纲化能量释放率在无穷远处声子场和相位子场应力作用下的变换规律。数值结果表明:无量纲化声子场应力强度因子随着无穷远处声子场应力的增大而逐渐增大并缓慢趋于稳定值,随无穷远处相位子场应力的增大而减小;无量纲化相位子场应力强度因子随着无穷远处声子场应力的增大而缓慢增加,随无穷远处相位子场应力的增大而缓慢减小且最终趋于稳定;当裂纹长度和三角形孔边长之比给定时,随着无穷远处应力的增大,无量纲化能量释放率也随之增大;表面效应作用下,声子场和相位子场之间存在耦合效应;无量纲化的应力强度因子和能量释放率均存在着显著的尺寸依赖性;缺陷尺寸的不断增大会减小表面效应对周围应力场的影响。

With the aid of the Gurtin-Murdoch interface/surface model and the complex function method,the anti-plane shear problem of three equal cracks emanating from a regular triangular hole in nano-scale for one dimensional hexagonal quasicrystals was discussed by a conformal mapping.The closed-expressions of stress and stress intensity factors of the phonon and phason field were obtained respectively.Meanwhile,the analytical solution of the energy release rate was given.By using numerical examples,the impact of the infinite phonon/phason field stress on the dimensionless phonon/phason field stress intensity factors and dimensionless energy release rate were discussed at nano-scale.The numerical results show that the dimensionless phonon field stress intensity factor increases with the increasing of the finite phonon field stress,while decreases with the increasing of the phason field stress at infinity,which eventually tends to stable.The dimensionless phonon field stress intensity factor increases slowly with the increasing of the infinite phonon field stress firstly,then decreases gradually with the increasing of the infinite phason field stress,which is to stabilize finally.When the ratio of crack length to triangular hole side length is given,the dimensionless energy release rate increases with the increase of stress at infinity.As considering the surface effect,the coupling effect is produced between phonon field and phason field,and the dimensionless stress intensity factors and the dimensionless energy release rate are affected by the stress at infinity.With the increase of defect size,the influence of the surface effect is decreasing.