A closed-form method for integrating weight functions for part-through cracks subject to Mode I loading

Numerical integration of weight functions tends to be computationally inefficient because of the singularity in a typical weight function expression. An alternative technique has been developed for surface and corner cracks, which greatly improves both efficiency and accuracy of K_I estimates. Exact analytical solutions for the weight function integral are obtained over discrete intervals, and then summed to obtain the stress intensity factor. The only numerical approximation in this approach is the way in which the variation in stress between discrete known values is treated. Closed-form weight function integration methods are presented for three approximations of the stress distribution: (1) constant stress over each integration interval, (2) a piecewise linear representation, and (3) a piecewise quadratic fit. A series of benchmark analyses were performed to validate the approach and to infer convergence rates. The quadratic method is the most computationally efficient, and converges with a small number of integration increments. The piecewise linear method gives good results with a modest number of stress data points on the crack plane. The constant-stress approximation is the least accurate of the three methods, but gives acceptable results if there are sufficient stress data points.