Three-dimensional finite element analyses have been conducted to calculate the stress intensity factors for internal and external low aspect ratio longitudinal semi-elliptic cracks in pipes with a ratio of internal radius to thickness of 10. The stress intensity factors are presented for the deepest and surface points for semi-elliptic cracks with aspect ratios of 0·05 and 0·1, and a/t values of 0·2, 0·5, and 0·8. Uniform, linear, parabolic or cubic stress distributions were applied to the crack face. The results for uniform and linear stress distributions were combined with corresponding results for higher aspect ratio surface cracks and edge cracks to derive the weight function over the entire range 0 ≤ a/c ≤ 1 and 0 ≤ a/t ≤ 0·8. The weight function is then verified against finite element data for the parabolic or cubic stress distributions. Results were within 5% for both the surface point and the deepest point.

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Journal International Journal of Pressure Vessels and Piping
Wang, X, & Lambert, S.B. (S. B.). (1996). Stress intensity factors and weight functions for longitudinal semi-elliptical surface cracks in thin pipes. International Journal of Pressure Vessels and Piping, 65(1), 75–87. doi:10.1016/0308-0161(94)00160-K