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JMEPEG (2007) 16:220–223 DOI: 10.1007/s11665-007-9035-1

ÓASM International 1059-9495/$19.00

Three-term Asymptotic Stress Field Expansion for Analysis of Surface Cracked Elbows in Nuclear Pressure Vessels
Fernando Labbe (Submitted June 11, 2006) Elbows with a shallow surface cracks in nuclear pressure pipes have been recognized as a major origin of potential catastrophic failures. Crackassessment is normally performed by using the J-integral approach. Although this one-parameter-based approach is useful to predict the ductile crack onset, it depends strongly on specimen geometry or constraint level. When a shallow crack exists (depth crack-to-thickness wall ratio less than 0.2) and/or a fully plastic condition develops around the crack, the J-integral alone does not describecompletely the crack-tip stress field. In this paper, we report on the use of a three-term asymptotic expansion, referred to as the J–A2 methodology, for modeling the elastic-plastic stress field around a threedimensional shallow surface crack in an elbow subjected to internal pressure and out-of-plane bending. The material, an A 516 Gr. 70 steel, used in the nuclear industry, was modeled with aRamberg–Osgood power law and flow theory of plasticity. A finite deformation theory was included to account for the highly nonlinear behavior around the crack tip. Numerical finite element results were used to calculate a second fracture parameter A2 for the J–A2 methodology. We found that the used three-term asymptotic expansion accurately describes the stress field around the considered three-dimensionalshallow surface crack.


asymptotic stress fields, constraint, nonlinear fracture mechanics, pressure vessels

1. Introduction
Flaws in power plant pipes under normal operating conditions typically initiate on the inner surface of the elbow walls. They grow by fatigue and/or stress corrosion forming a surface crack. As a consequence, shallow surface cracks (depth crackto-thickness wallratio less than 0.2) are often encountered and hence have been recognized as a major origin of potential catastrophic failure. In order to prevent these failures, fracture mechanics methodologies are normally applied to evaluate the structural integrity of pressure pipes. The application of conventional fracture mechanics techniques relies on the notion that a single parameter, such as Jintegral,characterizes the resistance of the material to fracture (Ref 1, 2). The J-integral is used to measure the intensity of the stress and deformation fields on the singular one-term crack-tip solution for a nonlinear material. It is referred to as the HRR singularity field since Hutchinson and Rice (Ref 3) and Rosengren (Ref 4). However, the applicability of this oneparameter-based approach is limitedto high constraint crack geometry such as deep crack in bending loading. When shallow
Fernando Labbe, Mechanical Engineering Department, Universidad Tecnica Federico Santa Maria, Casilla 110-V, Valparaiso, Chile. Contact e-mail:

surface crack exists and/or extended plastic conditions around the crack are meet, J-integral parameter alone does not describe completely thefield around the crack tip. The limited ability of a single parameter J to fully characterize crack-tip conditions for certain geometry and load conditions is still an open problem. To solve it, an approach recently proposed is establishing a multi-term asymptotic crack-tip stress field expansion (Ref 5). A two-term asymptotic stress field expansion in fracture mechanics has been carried out by O’Dowd etal. (Ref 6-8) as well as Betegon and Hancock (Ref 9). In the two-parameter approach, denoted as J–Q methodology, a second parameter Q measures the degree of triaxiality and constraint of the stress field. Applications of J–Q methodology to cracked cylindrical specimen has been performed by Donoso (Ref 10) and, in the case of flawed pressure vessels, by Labbe (Ref 11). Both works have shown...
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