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Procedia Materials Science 3 (2014) 1687 - 1693

20th European Conference on Fracture (ECF20)

Constraint effects for a reactor pressure vessel subjected to pressurized thermal shock

Guian Qian*, V.F. Gonzalez-Albuixech, Markus Niffenegger

Paul Scherrer Institute, Nuclear Energy and Safety Department, Laboratory for Nuclear Materials, 5232 Villigen PSI, Switzerland

Abstract

Transferability of fracture toughness data obtained on small scale specimens to a full-scale cracked structure involves both inplane and out-of-plane constraint effects. Both in-plane and out-of-plane constraint effects of a crack in a reference reactor pressure vessel (RPV) subjected to pressurized thermal shock (PTS) are analyzed by two-parameter and three-parameter methods. T11 (the second term of William's extension acting parallel to the crack plane) generally displays a reversed relation to the stress intensity factor (SIF) with the transient time, which indicates that the loading (SIF) plays an important role on the inplane constraint effect. T33 (the second term of William's extension acting along the thickness) displays a different relation to T11 during the transient. The results demonstrate that both in-plane and out-of-plane constraint effect should be analyzed separately in order to describe precisely the stress distribution ahead of the crack tip. The local approach to fracture, i.e. a*-A* model is used to predict the in-plane and out-of-plane constraint effect by considering the micro mechanism of cleavage fracture.

Keywords: pressurized thermal shock, constraint loss, reactor pressure vessel, out-of-plane constraint, local approach to fracture;

1. Introduction

Transferability of fracture toughness data obtained on small scale specimens to a full-scale cracked structure is one of the key issues in integrity assessment of engineering structures. The reason that geometry and size of the tested specimens affects the fracture toughness is attributed to different stress and strain fields ahead of the crack tip. In order to consider the stress triaxiality of the crack tip, accurate two-parameter approaches, such as K-T (Williams, 1957) and J-Q (O'Dowd and Shih, 1991), have been developed. These approaches have been applied successfully in engineering designs though they are limited to describe the effect of the in-plane constraint on the crack-tip field and fracture toughness. In order to study the out-of-plane constraint effect on the structures, T33 was proposed by

* Corresponding author. Tel.: +41-56-3102865; fax: +41-56-3102199. E-mail address: guian.qian@psi.ch

2211-8128 © 2014 Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.Org/licenses/by-nc-nd/3.0/).

Selection and peer-review under responsibility of the Norwegian University of Science and Technology (NTNU), Department of Structural Engineering doi:10.1016/j.mspro.2014.06.272

Gao (Gao, 1992) and used by Wang (Wang, 2003) and Meshii and Tanaka (Meshii and Tanaka, 2010) to characterize the out-of-plane constraint for different structures. T33 is the second term of Williams extension along the thickness direction of the crack tip. It has a clear physical meaning and is consistent with T11 for constraint analysis.

For reactor pressure vessels (RPVs) in a pressurized water reactor, one potential challenging loading to the integrity is a pressurized thermal shock (PTS). PTS transients result in complicated 3D stresses along the RPV wall. If the stress intensity factor (SIF) of a postulated crack is too large this may lead to crack initiation and in the worst case even to the failure of the RPV. The integrity of RPV has been studied by using one or two-parameter methods (Shum et al., 1994; Kim et al., 2002; Qian and Niffenegger, 2013a, 2013b). In order to get a more precise result, the fracture toughness from the test standards should be adjusted to different points of the crack front by considering both the in-plane and out-of-plane constraint.

Therefore, in this paper, a 3D model of a RPV is used to study the stress distributions ahead of the crack tip. Both in-plane and out-of-plane constraint effects of the crack tip are quantified by Tn and T33. The constraint effects are also predicted by a local approach to fracture.

Nomenclature

a crack depth

A* stressed area

B specimen thickness

2c crack length

E Young's modulus

K, K Mode I linear elastic stress intensity factor

Kj stress intensity factor derived from J-integral

R RPV radius

t vessel wall thickness

tc cladding thickness

tb base thickness

ti transient time

T-stress, Tn Second term of William's extension along x direction

T33 Second term of William's extension along z direction

To reference temperature in Master Curve method

V* stressed volume

W specimen width

V Poisson's ratio

a* critical stress

0 angle of elliptical crack

MLOCA medium loss-of-coolant accident

PTS pressurized thermal shock

RPV reactor pressure vessel

SIF stress intensity factor

2. Methods for in-plane and out-of-plane constraint analyses

The K-T method is generally used for in-plane constraint analyses. The K-T concept considers both the first (singular) and second (non-singular) term of the Williams extension of the crack front stress field (in terms of the polar coordinate r and 9). By considering the second term of Williams extension, T33 is used for out-of-plane constraint analysis, as (Nakamura and Parks, 1992)

0L . 0 . 30

cos — I 1 - sin — sin-

2 ^ 2 2

0Í, . 0 . 3G cos — I 1 + sin —sin— 2 ^ 2 2

2vcos — 2

.00 30

sin—cos — cos-

T33 = ES33 +vT11 ■

In this study, Tu and T33 are used to analyze the in-plane and out-of-plane constraint effects of a surface crack in a RPV subjected to PTS transients.

3. RPV integrity analysis

3.1. Physical model

The physical model of this study is shown in Fig. 1 (a). The RPV containing a crack is assumed to be subjected to PTS. A medium loss-of-coolant (MLOCA) is postulated in this study. The history of the water temperatures, pressures and heat transfer coefficients between water and inner wall of the RPV for the two transients are shown in Fig. 1 (b). The transient is obtained from thermal hydraulic calculations with the RELAP code.

3.2. Fracture mechanics analysis

The thermal and stress analyses are firstly performed. For thermal analysis, heat flow through the inner surface of

the vessel is determined from the water temperature and heat transfer coefficient and it is assumed to be zero (adiabatic boundary conditions) at the outer surface of the vessel. The vessel wall is assumed to be at a uniform initial temperature. These boundary and initial conditions for thermal analysis are also shown in Fig. 1 (a). By taking advantage of symmetry (boundary condition for structural mechanics analysis), one quarter of the RPV is modelled, as shown in Fig. 2 (a). The temperature distribution through the vessel wall is obtained in the thermal analysis and is used for fracture mechanics analysis.

The quadratic 20-node hexahedron (brick) element is used for the finite element simulation. In order to simulate the stress singularity for elastic materials, the brick element is converted to a wedge element (in Abaqus it is called C3D20 element). Only the beltline region of the vessel, which is exposed to higher neutron irradiation fluence, is considered in this analysis. The thermo-mechanical properties of the base material and cladding for thermal mechanical analysis at different temperatures are listed in (Qian et al., 2013).

Fig. 2. (a) 3-D model of the beltline region of the RPV for thermal analysis. Due to the symmetry conditions, only one quarter of the circumference is modelled. The R^plane, RZ planes indicated as arrows are symmetrical planes; (b) KI distribution along the crack front.

Figure 2 (b) shows the SIF distributions around the crack tip during the two transients. During the MLOCA transient, the SIF generally decreases with crack angle and then increases to its maximum value at the deepest point (0=rn/2).

4. In-plane and out-of-plane constraint analyses

The interaction integral is used to calculate T-stress (T11) and the results are shown in Fig. 3 (a). T11generally increases with the crack front angle and then decreases to the minimum value. Constraint is regarded as a structural obstacle against plastic deformation induced mainly by the geometrical and physical boundary conditions. It is obvious that the free surface has no constraining effect, while in the interior of the crack (0<i><n/2), the constraint is high corresponding to the governing plain strain condition. The in-plane constraint is controlled by the crack size, ligament and loadings. Thus, the application of material toughness based on plane strain specimens for the RPV yields a conservative result. The results presented in Fig. 3 (a) also indicate that the lowest level of in-plane constraint at the deepest point occurs for a time period of 1000 second for the MLOCA. Therefore, the material volume in the vicinity of the crack tip (in the so-called process zone) of the RPV is subjected to a less severe stress state. Under certain conditions, this factor may result in the apparent enhancement of the fracture toughness measured on the RPV. It is worth to note that T11 is positive (or zero) for plane strain and normally negative for plane stress and the condition between plane strain and plane stress. In order to quantify the elastic in-plane

constraint effect of the crack tip on fracture toughness of the material, Wallin (Wallin, 2001) developed a relation between T11 and the Master Curve transition temperature T0 based on a large amount of databases. A simple relation between T0deep obtained from deeply cracked (high constraint) bars and T0 linked to shallow crack specimens (low constraint conditions) was proposed in (Wallin, 2001).

The out-of-plane constraint is analyzed by T33 according to Eq. 2. Fig. 3 (b) shows the distribution of T33 at the crack front during the MLOCA (9=0, r=0). It is seen that T33 generally decreases with the crack angle, followed by a slight increase with the crack angle and then decreases to the deepest point of the crack front. T33 is negative at the deepest point of the crack front, which indicates a loss of constraint at the deepest point of the crack front. In addition, T33 is the highest for a time of 1000 second and the lowest for a time of 11990 second. This trend is reversed to that of T11. It demonstrates that during the transient out-of-plane constraint is different from the in-plane constraint. Therefore, they both should be considered separately.

1000 [s]

■ -♦- 6000 [s]

-□- 8500 [s]

-■- 11990 [si

1000 800 600 400 200 0 -200 -400 -600 -800 1000

MLOCA -V- 1000 [s] -♦- 6000 [s] -D- 8500 [s] ■ 11990 [s]

0.0 0.2 0.4 0.6 0.8 1.0

Crack front angle [2Q/n]

0.4 0.6 0.8 1.0

Crack front angle [2®/m]

Fig. 3. (a) Tn distribution along the crack front; (b) T33 distribution along the crack front.

5. Local approach to fracture

In this Section, the a*-A* model is used to understand the micro mechanisms of the constraint effect. The a*-A* model (and its extensions) is based on the following two hypotheses (Bonade et al., 2008; Mueller and Spatig, 2009): (1) brittle fracture is triggered when a critical area A* (or volume V*) of material encompasses a critical stress a*, and (2) the critical values a* and A* (or V*) are temperature independent material properties.

In order to investigate crack depth effect on the in-plane constraint, A vs. Kj for short (a/W=0.1) and deep (a/W=0.5) cracks are studied. The subsized (0.18T) CT specimens having a width W of 9 mm and the crack depth a to specimen width W ratio (a/W) of 0.5 are modelled (2D version of Fig. 4). It is seen in Fig. 5 (a) that for a same Kj, A* for the short crack is more than 2 orders of magnitude smaller than for the deep crack. The decreased area A* for the short crack indicates the in-plane constraint loss, which on the other hand leads to a higher fracture toughness.

In order to quantify the out-of-plane constraint effect, finite element analyses for 3D CT specimens of different thickness (B=4.5 mm and 2.25 mm) with a deep crack (a/W=0.5) are conducted, as shown in Fig. 4. The stressed volume V for both specimens is shown in Fig. 5 (b). It is seen that for a same KJ, V for the thin specimen is more than 1 order of magnitude smaller than that for the thick one. The smaller stressed volume V for the thin specimen indicates a less severe crack tip stress distribution. This is the out-of-plane constraint loss due to the lack of stress triaxiality, which in turn leads to a toughness increase. In all, micro mechanics analysis shows that the crack depth and loading plays an important role in the in-plane constraint effect. The thickness at the crack tip has an important impact on the out-of-plane constraint effect. These trends are in agreement with the macro mechanics analyses in Section 4.

Fig. 4. 3D finite element model with a deep crack and the detail of crack tip.

Kj [MPama5] KJ [MPama5]

(a) (b)

Fig. 5. (a) A* vs. Kj for CT specimens with deep and shallow cracks T=-40 °C; (b) V* vs. Kj for thick and thin specimens at T=-40 °C. 6. Conclusions

Both in-plane and out-of-plane constraint effect at the crack tip are analyzed by two-parameter and three-parameter methods for PTS loading of a RPV. For in-plane constraint effects, the crack ligament and loading plays an important role. The variation of T11 with transient time is generally reversed to that of stress (or SIF). For out-of-plane constraint effects, T33 shows a different relation to T11 during the transient. Constraint loss occurs during the transient at the deepest point of the crack front. Constraint at the deepest point of the crack front is lower than that at the surface point. Since constraint is pointwise, K-T11-T33 (K is for crack driving force analysis, T11 is for in-plane

constraint analysis and T33 is for out-of-plane constraint analysis) provides a consistent method for the constraint analysis. Both in-plane and out-of-plane constraint effects are interpreted by the o*-A* model. In all, the crack depth and loading plays an important role in the in-plane constraint effect. The crack tip thickness has an important impact on the out-of-plane constraint effect.

Acknowledgement

The authors are grateful for the financial support of the PISA Project provided by the Swiss Federal Nuclear Safety Inspectorate (ENSI) (DIS-Vertrag Nr. H-100668).

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