正交弹性材料中双周期裂纹反平面问题的封闭解

A closed-form solution for the anti-plane problem of anisotropic elastic materials with doubly periodic cracks

研究了无限大正交弹性材料中含双周期裂纹的反平面问题,其基本胞元含有三条裂纹,且三条裂纹的中心恰好位于一等腰三角形顶点。运用椭圆函数、保角变换理论、施瓦兹公式获得了该问题应力场的封闭解,并得到了裂纹尖端处的应力强度因子。该问题结果取特殊情形退化对应于单排共线周期裂纹的解答。通过数值算例,分析了双周期裂纹归一化的应力强度因子随双周期裂纹的横向间距和纵向间距之比b/a分别取10、5、2、1时的变化曲线。结果表明:对于一定的横向间距,应力奇异因子随纵向间距的增大而减小,但随着纵向间距的增大,纵向间距对应力奇异因子的影响变得不明显;对于一定的纵向间距,应力奇异因子随横向间距的减小而减小,但随着横向间距的减小,横向间距对应力奇异因子的影响变得不明显。

The anti-plane problem of infinite orthogonal anisotropic elastic materials with doubly periodic cracks is studied.The basic unit cell of doubly periodic problem contains three cracks which centers are on the top of an isosceles triangle.By using methods of the conformal mapping,the elliptical function and the Schwarz＇s formula,the solution of the stress field is obtained in closed-form and thereby,the stress intensity factors are derived.The solutions of collinear periodic cracks problem can be evolved from the general solution as a special case for this problem.Numerical results are presented to show the effects of the microstructure parameters of doubly periodic problem on the stress intensity factors under b/a equals to 10,5,2 and 1,respectively.From the numerical results,some conclusions can be given： the stress intensity factor tends to increase with the decreasing of the longitudinal spacing when the horizontal spacing is given.However,as the longitudinal spacing increases,the effect of the longitudinal spacing on the stress intensity factor tends to be less sensitive.On the other hand,the stress intensity factor tends to decrease with the decreasing of the horizontal spacing when the longitudinal spacing is given.However,as the horizontal spacing decreases,the effect of the horizontal spacing on the stress intensity factors tends to be less sensitive.