一维六方压电准晶中正n边形孔边裂纹的反平面问题

The Anti-Plane Problem of Regular n-Polygon Holes With Radial Edge Cracks in 1D Hexagonal Piezoelectric Quasicrystals

利用复变函数方法和Schwarz-Christoffel(SC)变换,构造了保形映射函数,研究了一维六方压电准晶中正n边形孔边裂纹的反平面问题.首先由一维六方压电准晶反平面问题的本构方程、平衡方程和几何方程推导得出其控制方程.在电不可导通边界条件下,应用Cauchy积分公式,得出任意正n边形孔边裂纹尖端附近应力强度因子和电位移强度因子的解析解,并针对n=3,5,6时,给出数值算例,可以看出这些特殊情形可退化为已有的结果.研究结果表明:等效场强度因子K的值随着孔边长a和裂纹长度L/a的增加而增大;孔洞的尺寸对等效场强度因子K的影响特别显著,易导致破坏.该文所给结果对计算等效强度因子具有一般性,适用于任意正n边形孔边裂纹的求解问题,从而为工程力学、材料的制备和应用等提供了良好的理论依据.

With the complex variable function method and the Schwarz⁃Christoffel transforma⁃tion,a conformal mapping function was constructed to address the anti⁃plane problem of regu⁃lar n⁃polygon holes with radial edge cracks in 1D hexagonal piezoelectric quasicrystals.Firstly,the constitutive equations,equilibrium equations and geometric equations for 1D hexagonal pie⁃zoelectric quasicrystals were employed to derive the governing equations subjected to anti⁃plane loading.Under the electrically nonconductive boundary conditions,the Cauchy integral formula was applied to solve the problem of an arbitrary regular polygon hole with a radial edge crack.The analytical solutions of the stress intensity factor and the electric displacement intensity fac⁃tor were obtained,and the numerical examples in special cases of n=3,5,6 were demonstrated to be consistent with the existing results.The research shows that,the value of the effective in⁃tensity factor increases with side length a and crack length L/a.Parameter a has a significant effect on effective field intensity factor K,which is critical to damage.The results are general and suitable for calculating the effective intensity factor of an arbitrary regular polygon hole with a radial edge crack,thereby providing a good theoretical basis for engineering mechanics,material preparation and application.