Reference stress based J estimation formula for cracked cylinders with a wide range of radius-to thickness ratios: Part II- circumferential surface cracks

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Highlights

  • Propose reference stress based J estimation equations for circumferential surface and through-wall cracked cylinders.

  • Can be applied to cylinders with the radius-to-thickness ratio ranging from three up to seventy.

  • Same correction factor can be applied to both axial tension and global bending loading conditions.

  • Be valid for constant-depth shape of both internal and external surface cracks with high strain hardening materials.

  • Check validity by comparing with systematic finite element results.

Abstract

In this paper, reference stress based J estimation equations are presented for circumferential constant-depth surface and through-wall cracked cylinders with the radius to thickness ratio ranging from three up to seventy. The equations are developed based on extensive finite element (FE) analysis results. Firstly it is shown that application of existing J estimation equations is limited to thick-walled cylinder problems and can give non-conservative J estimates for a cylinder with a large radius-thickness ratio. To accommodate the effects of the crack depth, length and the radius-to-thickness ratio, correction factors for the reference stress are proposed based on FE results. By comparing with FE results, it shows that the proposed solutions can be applied to both an external and internal crack; and to combined axial tension and global bending.

Introduction

A multi-purpose canister made of austenitic stainless steels for long-term dry storage of spent nuclear fuel [1] is known to be vulnerable to chloride-induced stress corrosion cracking [2]. To assess crack stability analysis based on elastic-plastic fracture mechanics analysis, one needs an engineering elastic-plastic J estimation method, for instance, based on the reference stress approach [3] currently embedded in many codes and standards [[4], [5], [6], [7]]. It should be noted that the radius-to-thickness ratio of typical multi-purpose canisters is up to around seventy [1], which is much larger than typical values (less than twenty) in nuclear and thermal power plant piping components. In this respect, it is important to investigate applicability of existing reference stress based J estimation equations to cracked cylinders with large radius-to-thickness ratios.

In Part I [8], applicability of the reference stress based J estimation equations to axial surface and through-wall cracked cylinders under internal pressure, with the radius-to-thickness ratio ranging from five up to seventy, was investigated via detailed elastic-plastic finite element (FE) analysis. It was found that the use of the Tresca limit pressure given in R6 [4] could provide satisfactory results for all cases considered. This conclusion was valid for both inner and external axial surface cracks; for both semi-elliptical and constant-depth axial surface cracks.

As a companion paper, this paper (Part II) considers circumferential surface and through-wall crack cases under axial tension or global bending moment. Most of existing works on the references stress based J estimation method are limited to through-wall cracks with the radius-to-thickness ratio less than twenty [[9], [10], [11], [12], [13]] where the authors proposed an improved reference stress based J estimation equations for bending and combined pressure and bending. In Refs. [14], the influence functions for the GE/EPRI J estimation equations were given for a cylinder with the radius-to-thickness ratio of only thirty and fifty, which has limited application range. Furthermore in practical applications, surface cracks are more relevant than through-wall cracks.

For a surface crack, existing works were only for the internal surface crack [13,[15], [16], [17]]. In Ref. [13,15], a reference stress based J estimation equation was given for the internal surface crack (both constant depth and semi-elliptical shapes) but were limited to the radius-to-thickness ratio less than twenty. In Refs. [16,17], the influence functions for the GE/EPRI J estimation equations and the reference stress based J estimation equation were given for the internal semi-elliptical surface crack with the radius-to-thickness from five to sixty. However, the proposed solutions are strictly valid for the internal semi-elliptical crack under only tension loading, as will be discussed later in Section 3.2. Furthermore, it is not clear how to define the semi-elliptical shape for a cylinder with a large radius-to-thickness ratio (this will be also discussed later in Section 3.2). When a crack occurs due to stress corrosion cracking, it tends to have irregular shape and thus it would be more convenient to impose constant-depth surface crack rather than semi-elliptical one. As calculated J tends to be higher for the constant depth crack than that for the semi-elliptical one, the assumption of constant depth crack is conservative (some results will be given in the main text later). Survey of existing works suggests that more systematic analysis is needed to confirm the applicability of the existing J estimation equations and, if they are not applicable to cracked cylinders with wide ranges of the radius-to-thickness ratio, a modification should be proposed.

This paper presents a reference stress based J estimation equation for circumferential cracked cylinders with a wide range of radius to thickness ratios. By comparing with detailed FE analysis results, it will be shown that existing reference stress based J estimation equations for thick-walled cylinders can give non-conservative estimation results. Accordingly a modification will be proposed to determine proper reference stress solutions. The proposed solutions can be applied to tension and bending; to external and internal surface cracks. They are developed for constant depth surface cracks and thus conservatively used to semi-elliptical cracks. Section 2 explains FE analysis performed in this work. Based on the discussion on applicability of the existing solutions in Section 3, modification to the reference stress is proposed for external circumferential surface and through-wall cracks in Section 4 by comparing with FE results for tension loading. In Section 4, it will be shown that the proposed modification also works for bending and for internal cracks. The presented work is concluded in Section 5.

Section snippets

Geometry and loading

A general purpose FE program, ABAQUS 2016 version [18], was used for detailed elastic-plastic FE analysis. A cylinder with either circumferential through-wall or circumferential constant-depth surface crack (Fig. 1) was considered. For a long external crack, it is difficult to define a semi-elliptical shape, due to the limit in its length, as will be discussed further in Section 3.2 and in Fig. 7 and thus only the constant-depth surface cracks were considered in the present work. Both the

Reference stress based J estimation equation

In the reference stress method [4], the elastic-plastic J can be estimated fromJJe=Eεrefσref+12(σrefσy)2σrefEεrefσref=NNLσy=Lrσywhere the reference strain εref denotes the strain at the reference stress σref in the true stress-strain curve of the material; the load ratio Lr is defined by the ratio of the applied axial tension N and a limit load NL; and Je denotes the elastically calculated value of J. In R6 [4], the following limit load solution NL is recommended for a circumferential

Modification of reference stress solution for 3≤ri/t ≤ 10

To propose a reference stress for a wide range of ri/t ratio, the existing solution for through-wall crack, Eq. (5), is firstly extended to surface cracks. Noting that the through-wall crack is the limiting case of the constant-depth surface crack for a/t→1, the following extension of the modification factor is proposed for the constant-depth surface crackNOR=γ(θπ,at)NL;γ(θπ,at)=0.82+0.75(θπat)+0.42(θπat)2

Estimated J values using Eq. (3) with Eq. (7) are compared with FE results for ri/t = 3

Conclusion

In this paper, reference stress based J estimation equations are proposed for circumferential constant-depth surface and through-wall cracked cylinders with the radius to thickness ratio ranging from three up to seventy. The solutions cover all ranges of the surface depth including through-wall crack and can be applied both to internal and to external surface cracks. They can be also applied to axial tension and global bending, and thus combined bending and tension.

This paper firstly compares

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgment

This research was supported by National Research Foundation of Korea (NRF) grant funded by the Korea government (Ministry of Science and ICT) (NRF-2016M2A8A1952771 and NRF-2019M2D2A2048296).

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