三维动态裂纹问题的超奇异积分方程法

Analysis of hypersingular integral equation method to 3D dynamic crack

基于弹性材料的动态基本方程,结合广义Betti-Rayleigh互易等式与时域下的边界积分方程,推导得到时域下的超奇异积分方程组。引入Laplace域下的动态基本解,将经过主部分析的积分核函数分解为静态和动态部分,其中动态积分核不具有奇异性。在裂纹前沿附近单元,采用与理论分析一致的平方根位移模型。结合Lubich时间卷积实现拉氏变换,采用配置点法计算超奇异积分,获得问题的数值解。并针对椭圆裂纹算例编写Fortran程序,得到冲击荷载作用下张开型裂纹的动态应力强度因子变化规律,数值结果稳定且收敛速度快。

Combining Betti-Rayleigh equation with dynamic boundary integral equation,the dynamic basic equation of elasticity is substituted into a couple of hypersingular integral equations.A basic solution in Laplace domain is introduced to derive the integral kernel function,which is divided into a static part and a dynamic part after the dominant analysis,and the dynamic part is nonsingular.According to the analytic theory of hypersingular integral equations,the square root models of displacement discontinuities in the element near the crack front are applied.Finally,with the Lubich convolution quadrature to implement dispose the Laplace transform and the collocation method to compute the hypersingular integral,the numerical solution is obtained.FORTRAN codes are programmed for examples of an elliptic crack,therefore the time variation of dynamic stress intensity factor of mode I crack under impact load is acquired.It is shown that the numerical results are stable and fast convergent.