动载下裂纹应力强度因子计算的改进型扩展有限元法

相较于常规扩展有限元法(extended finite element method, XFEM), 改进型扩展有限元法(improved XFEM) 解决了现有方法线性相关与总体刚度矩阵高度病态问题, 在数量级上提升了总体方程的求解效率, 克服了现有方法在动力学问题中的能量正确传递、动态应力强度因子数值震荡、精度低下问题. 本文基于改进型XFEM, 采用Newmark 隐式时间积分算法, 重点研究了动载荷作用下扩展裂纹尖端应力强度因子的求解方法, 与静力学方法相比, 增加了裂纹扩展速度项与惯性项的贡献. 通过数值算例研究了网格单元尺寸、质量矩阵、时间步长、裂尖加强区域、惯性项、扩展速度项及相互作用积分区域J-domain的网格与单元尺寸对动态应力强度因子求解精度的影响, 验证了改进型XFEM计算动态裂纹应力强度因子方法的有效性. 针对文献中具有挑战性的 "I 型半无限长裂纹先稳定后扩展"问题, 改进型XFEM给出目前为止精度最好的动态应力强度因子数值解. 

ACCURATE COMPUTATION ON DYNAMIC SIFS USING IMPROVED XFEM

Compared to the standard extended finite element method (XFEM), the improved XFEM overcomes, in theory, the linear dependence and the ill-conditioning issues, and improves in magnitude of orders the efficiency of linear system solve. In particular, in dynamic crack propagation problems, thanks to the exclusion of the extra dofs on crack tip enriched nodes, the new method eliminates the issue of energy inconsistency or the inconservitive energy transfer caused by dof dynamics, and provides more accurate dynamic stress intensity factors (DSIFs) with much less numerical oscillation. To the best of our knowledge, numerical solution on DSIFs for crack propagation under dynamic loading remains engineeringly unsatisfied. In this paper, the extra-dof free XFEM is extended to implicitly dynamic crack propagation problems-still a remaining difficulty for the current XFEMs. The implicit Newmark algorithm is used for time discretization. A dynamic interaction integral method is employed for DSIFs for both stationary and moving cracks under dynamic loading. Compared with the interaction integral method for static cracks, the dynamic method considers the effects of crack growth speed and inertia terms. The paper investigates in detail the influences of element size, mass matrix formulation, time step size, crack tip enriched zone size, inertia term, crack growth speed, and J-domain mesh/cell sizes of the interaction integral. Numerical tests show satisified accuracy and efficiency of the new method for dynamic crack problems. In particular on the challenging benchmark problem "Mode I semi-infinite stationary and then moving crack", the new improved XFEM provides the best DSIFs up to the time of publication.