This paper evaluates eight different analytical procedures used in determining elastic stress intensity factors for gradient or nonlinear stress fields. From a fracture viewpoint, the main interest in this problem comes from the nuclear industry where the safety of the nuclear system is of concern. A fracture mechanics analysis is then required to demonstrate the vessel integrity under these postulated accident conditions. The geometry chosen for his study is that of a 10-in. thick flawed plate with nonuniform stress distribution through the thickness. Two loading conditions are evaluated, both nonlinear and both defined by polynomials. The assumed cracks are infinitely long surface defects. Eight methods are used to find the stress intensity factor: 1–maximum stress, 2–linear envelope, 3–linearization over the crack length from ASME Code, Section XI, 4–equivalent linear moment from ASME Code, Section III, Appendix G for thermal loadings, 5–integration method from WRC 175, Appendix 4 for thermal loadings, 6–8-node singularity (quarter-point) isoparametric element in conjunction with the displacement method, 7–polynomial method, and 8–semi-infinite edge crack linear distribution over crack. Comparisons are made between all eight procedures with the finding that the methods can be ranked in order of decreasing conservatism and ease of application as follows: 1–maximum stress, 2–linear envelope, 3–linearization over the crack length, 4–polynomial method, and 5–singularity element method. Good agreement is found between the last three of these methods. The remaining three methods produce nonconservative results.
Addition of sets via symmetric polynomials — A polynomial method
