Finite element error estimation for crack tip singular elements

Abstract

A stress-based finite element error estimator is proposed for fracture mechanics analysis involving singular isoparametric crack tip finite elements. In the crack tip region the discretization error is estimated by the L 2 integral norm of the difference between the finite element effective stress function and the known asymptotic effective stress function derived from the Westergaard solution. In the remaining portion of the domain, the L 2 element error norm is based on the difference between the discontinuous finite element effective stress function and the C 0 continuous “smoothed” effective stress function. Numerical convergence studies are presented for the classical edge crack Mode I problem with six different singular and nonsingular element formulations at the crack tip region. The analysis results showed steady monotonic convergence of the L 2 norm error estimator for the critical six-noded triangular and eight-noded quadrilateral singular crack tip elements, whereas poor convergence behaviour is seen in the Zienkiewicz and Zhu stress-based energy norm error estimator applied to the same critical crack tip element. However, monotonic convergence was seen in all global L 2 and energy error norms for all mesh cases.