A Precise Time-Domain Expanding Crack-Tip Element for 3D Planar and Axial Dynamic Cracks

Unlike two-dimensional (2D) crack problems of which the analytical symplectic eigensolutions are available, the analytical symplectic solution of three-dimensional (3D) planar and axial dynamic cracks has not been derived yet. We thus propose a trial displacement field for the 3D planar and axial crack problem using the existing solutions of the 2D crack problems and construct a new crack-tip element on that basis. The recently developed precise integration scheme for 3D planar and axial cracks under dynamic loading (Hu, X. F., Chen, W. H., Zhang, P. and Yao, W. A. [2021] "An explicit crack-tip element for stationary dynamic cracks," Theor. Appl. Fract. Mech. 112, 102886) is further enhanced. A precise time-domain expanding algorithm is employed for the temporal discretization. All the field variables are expanded using a series expansion scheme, and the dynamic problem is then transformed into a series of quasi-static elastic problems. All the sub-elastic problems are related in a recursive manner. Taking advantage of a smart automatic truncating approach, the number of expanding terms can be determined automatically subjected to a prescribed error tolerance. A few numerical examples are investigated to demonstrate the developed method, numerical solutions indicate that the time dependent stress intensity factors (SIFs) can be calculated in a simple way and the present method has higher efficiency than the recently developed time integration scheme while having comparable accuracy.