A comparison of failure assessment diagram options for Inconel 690 and Incoloy 800 nuclear steam generators tubes

Annals of Nuclear Energy 140 (2020) 107310

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A comparison of failure assessment diagram options for Inconel 690 and Incoloy 800 nuclear steam generators tubes Marcos A. Bergant a,⇑, Alejandro A. Yawny b, Juan E. Perez Ipiña c a

División Física de Metales, Instituto Balseiro, Centro Atómico Bariloche (CNEA), Av. Bustillo 9500, San Carlos de Bariloche 8400, Argentina División Física de Metales, Instituto Balseiro, Centro Atómico Bariloche (CNEA)/CONICET, Av. Bustillo 9500, San Carlos de Bariloche 8400, Argentina c Grupo Mecánica de Fractura, Universidad Nacional del Comahue/CONICET, Buenos Aires 1400, Neuquén 8300, Argentina b

a r t i c l e

i n f o

Article history: Received 3 May 2019 Received in revised form 28 November 2019 Accepted 6 January 2020

Keywords: Steam generator tubes Failure assessment diagram Structural integrity assessment FAD options Inconel 690 Incoloy 800

a b s t r a c t In the context of structural integrity assessment of cracked Steam Generator Tubes (SGTs), the Failure Assessment Diagram (FAD) has been proposed as a modern tool able to reduce the conservativeness of traditional plastic limit load analyses. As different FAD failure curves are available in the literature, this paper presents a comparison between the so-called Options 1, 2 and 3 FAD curves derived for Inconel 690 and Incoloy 800 nuclear SGTs. Detailed Option 3 curves are obtained from elastoplastic finite element analyses and experimental data from fracture tests. Material-specific Option 2 curves are drawn from stress vs. strain data of uniaxial tests of SGTs. It is shown that the simplest generic Option 1 curve is conservative but appropriate for structural integrity assessments of cracked nuclear SGTs in operation. Alternatively, Option 3 FAD curves must be used when an excessive conservatism shall be avoided. Ó 2020 Elsevier Ltd. All rights reserved.

1. Introduction Steam generators are heat exchangers consisting of a bundle of several thousands of thin-walled tubes arranged inside a pressure vessel. The steam generator tubes (SGTs) separate the primary and the secondary coolant circuits of a nuclear reactor and may represent up to 60% of the primary water pressure boundary. In case of a tube rupture, a potential leak of radioactive elements from the primary to the secondary circuit may take place. The experience in the nuclear industry has shown that crack-like defects may develop during service due to diverse mechanisms (EPRI, 2006). Therefore, the structural integrity assessment of these cracked SGTs has received great attention in the last decades. Diverse regulatory requirements were developed to ensure a low probability of spontaneous tube rupture under normal and accident conditions. Due to the high ductility of SGTs materials, former traditional tube repair or plugging criteria based on minimum wall thickness requirements were derived from plastic limit load analyses. This lead to the development of different recommendations like the so-called ‘‘40% criterion”, among others (IAEA, 2011). These criteria have been applied to different types of defects, but are known to be overly conservative for crack-like ⇑ Corresponding author. E-mail addresses: [email protected] (M.A. Bergant), yawny@cab. cnea.gov.ar (A.A. Yawny), [email protected] (J.E. Perez Ipiña). https://doi.org/10.1016/j.anucene.2020.107310 0306-4549/Ó 2020 Elsevier Ltd. All rights reserved.

defects. Therefore, newly revised fitness for service criteria were proposed in recent years to reduce the number of tubes unnecessarily removed from service. In this context, a research effort has been conducted in order to apply fracture mechanics methodologies to the structural integrity assessment of flawed tubes (Bergant et al., 2015a,2015b) and to estimate fracture toughness properties of actual SGTs (Bergant, 2016; Bergant et al., 2012, 2015c, 2016, 2017). Regarding the assessment procedures applied to cracked SGTs, the Failure Assessment Diagram (FAD) has been proposed in previous researches (Bergant et al., 2015a,2015b; Chang et al., 2006; Lee et al., 2001; Tonkovic et al., 2008; Wang and Reinhardt, 2003). As it is a comprehensive option encompassing both plastic collapse and fracture mechanics analyses, the FAD method is well suited for the evaluation of cracked austenitic SGTs due to their inherent high fracture toughness. Furthermore, its simple use, flexibility and robustness turned FAD the most widely adopted assessment method, being incorporated in most prominent construction codes and service guides such as API 579/ASME FFS-1 (API 579-1/ASME FFS-1, 2007), BS 7910 (BS 7910, 2013), R6 (R6, 2001) and SINTAP/FITNET (SINTAP/FITNET, 2004) procedures. The main characteristics of the three FAD options that have been developed are described in the next section. They are usually referred to as Option 1, 2 and 3 in the literature (Milne et al., 1988a). The three FAD curve options for Inconel 690 and Incoloy 800 SGTs materials are compared, and their appropriateness and

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Nomenclature a c E EPFM FAD FEA J Jel Jmat Jq KI Kmat Kr LEFM Lr Lr max

length or depth of surface crack-like flaw half-length of through-wall crack-like flaw Young’s modulus elastic plastic fracture mechanics failure assessment diagram finite element analysis J-integral elastic component of J-integral material fracture toughness in terms of J-integral J-integral value at the beginning of stable crack growth mode I stress intensity factor material fracture toughness in terms of K parameter toughness ratio linear elastic fracture mechanics load ratio cut-off value of the load ratio

limitations are discussed. In previous works of the present authors (Bergant et al., 2015a,2015b), the simplest and more generic Option 1 curve was adopted for a first evaluation. These results will be now reanalyzed in this work considering the more detailed Option 2 and 3 FAD curves. Finally, the implications of using different FAD curve options for structural integrity assessments of flawed SGTs are shown by considering an actual example case. 2. The FAD approach The application of the FAD methodology requires the estimation of two parameters, i.e., the toughness ratio Kr and the load ratio Lr. Given a component with a crack of size a, the Kr and Lr parameters for a load P can be calculated as (Anderson, 2005):

Kr ¼

K I ðP; aÞ K mat

ð1Þ

Lr ¼

rref P ¼ PL rys

ð2Þ

where KI is the mode I stress intensity factor, Kmat is the material fracture toughness, PL is the limit load (also referred to as yield load), rys is the yield strength and rref is a reference stress that usually is interpreted as the effective primary stress acting on the unflawed net section area of the cracked component. Therefore, rref can be estimated as the product of rys with the ratio P/PL, where the limit load PL is defined as the maximum load that a cracked structure made of perfectly plastic material with a limiting stress equal to the yield strength could sustain (Anderson, 2005). As will be shown in the following, this approach for the definition of PL and rref introduces a geometry dependence into the FAD curve (Anderson, 2005). Although other different forms for PL and rref have been proposed in order to reduce the geometry dependence (Anderson, 2005; API 579-1/ASME FFS-1, 2007; Zerbst et al., 2007,2009,2012), current standardized FAD methods still use the traditional definition based on limit or yield load solutions (API 579-1/ASME FFS-1, 2007; BS 7910, 2013; R6, 2001; SINTAP/ FITNET, 2004). The present work aims to analyze the impact of this definition in the geometry dependence of the FAD curves for SGTs. The FAD has a cut-off value Lr max related to yielding failure conditions that depend on the material behaviour. For materials with continuous yielding such as those used in SGTs, the cut-off value Lr max usually represents the plastic collapse condition and is defined as (Anderson, 2005):

MSLB NO P PTWC Ri SGT t TWC

main steam line break normal operation applied load part-through-wall crack inside tube radius steam generator tube tube thickness through-wall crack true strain at the reference stress rref Poisson’s ratio flow stress reference stress ultimate tensile strength yield strength

eref m rf rref ru rys

Lr max ¼

rf rys

ð3Þ

where rf is the flow stress. The Kr and Lr parameters define the coordinates of an assessment point in the FAD. The relative position between this point and the failure line Kr = f(Lr) determines the safety margins regarding fracture, plastic collapse and critical crack length. The Kr = f(Lr) function represents the failure condition and all points located below the line are considered safe (Anderson, 2005). Fig. 1 shows a FAD defining the safe and potential unsafe regions delimited by the failure line. It can be seen that for low fracture toughness materials and low applied loads, i.e., low Lr and high Kr, the linear-elastic fracture mechanics (LEFM) is applicable. Here, the failure corresponds to brittle fracture. On the other hand, for high toughness materials and high applied loads, i.e., high Lr and low Kr, failure occurs by plastic collapse. For the intermediate situations between these two extremes, the fracture will be preceded by plastic deformation thus making necessary the consideration of elastic-plastic fracture mechanics (EPFM). The FAD curve may be directly related to the driving force J of the EPFM and it can be shown that under certain circumstances the FAD and J might be considered equivalent approaches (Ainsworth, 1984; Bloom, 1983). Some authors introduced certain approximations to obtain FAD curves that are independent of geometry and the degree of strain hardening of the material (Milne et al., 1988a). These approaches tend to make more conservative the FAD curve, while still considerably favouring the robustness and easiness of the method. In terms of the J-integral parameter, fracture avoidance can be expressed as:

JðP; aÞ < J mat

ð4Þ

Here, Jmat represents the material fracture toughness in terms of the J-integral parameter given by:

J mat ¼

K 2mat E0

ð5Þ

where Kmat is the material fracture toughness in terms of the K parameter, E is the Young’s modulus, m is the Poisson’s ratio and E0 = E/(1  m2) or E0 = E for plane strain or plane stress states, respectively. For the extreme condition of small scale yielding, where the LEFM is valid, failure is avoided if Kr < 1, while for plastic collapse failure is avoided if Lr < Lr max. For the more common intermediate

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Fig. 1. Schematic FAD illustrating the safe and unsafe regions.

situations, the criterion to prevent failure can be expressed combining Eqs. (1), (4) and (5), as suggested in (Ainsworth, 1996), i.e.:

K r < f 3 ðLr Þ ¼


1=2 J el J

ð6Þ

where Jel is the elastic component of the J-integral parameter. The subscript 3 is used in Eq. (6) as this expression for f(Lr) is known as Option 3 in the structural integrity assessment procedures based on the FAD methodology. The evaluation of Eq. (6) requires the calculation or measurement of J and Jel for the cracked component at different loading levels thus taking into account the effects of material behaviour, component geometry and type of loading. Due to that is considered to be the most rigorous option for the evaluation of cracked structures (Anderson, 2005). However, this analysis may become complex and costly given the need for detailed elastoplastic finite element analyses (FEAs). This has led to the development of some approximate but more practical procedures for the determination of FAD curves. For instance, Option 2 FAD curve is defined by (Ainsworth, 1996):

f 2 ðLr Þ ¼

E eref L3 rys þ r Lr rys 2 E eref

!1=2 ð7Þ

where eref is the true strain at the reference stress rref that can be determined from stress vs. strain curves of the material. It can be seen that the Option 2 curve, while still depending on the material behaviour, becomes independent of the component geometry and loading type (Anderson, 2005; Milne et al., 1988a). A further step in direction of simplification is given by the simplest Option 1 FAD curve given by (API 579-1/ASME FFS-1, 2007; BS 7910, 2013; R6, 2001):

   f 1 ðLr Þ ¼ 1  0:14 L2r 0:3 þ 0:7 expð0:65L6r Þ

ð8Þ

This curve is independent of both material behaviour and component geometry and was determined as a lower bound empirical fit of Option 2 curves for several materials (Ainsworth, 1996; Milne et al., 1988b). Option 1 expression allows performing integrity assessments of cracked components knowing only the yield strength rys and ultimate tensile strength ru of the material that define the limit Lr max. The following sections present a comparison between Options 1, 2 and 3 FAD curves derived for Inconel 690 and Incoloy 800 nuclear SGTs. Detailed Option 3 curves were obtained from elastoplastic FEA results and experimental data from fracture tests, while material-specific Option 2 curves are drawn from stress vs. strain data of uniaxial tests of SGTs performed at room temperature and at 300 °C (Bergant, 2016). The specific chemical composition (wt.%) and dimensions (external diameter, wall thickness, in mm) of the SGTs materials considered in the present work are given as Inconel 690: 61Ni29Cr-9Fe, 15.88, 0.97 and Incoloy 800: 33Ni-22Cr-42Fe, 15.88, 1.13.

3. Generic Option 1 and material-specific Option 2 FAD curves The mathematical expression for the Option 1 curve used in this work is given in Eq. (8), although other slightly different functions have been also proposed (Anderson, 2005). In the following, the Option 1 curve is included in the figures to compare it with the other FAD curve options. Option 2 curves were derived using Eq. (7) and the axial stress vs. strain curves obtained for Inconel 690 and Incoloy 800 SGTs at 24 and 300 °C presented in Fig. 2. Fig. 3 shows a comparison between the generic Option 1 and the material-specific Option 2 curves. It can be seen that the Option 1

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Fig. 2. (a) True stress–strain curves for Inconel 690 and Incoloy 800 at 24 and 300 °C, (b) detail of the curves in the elastic to elastic–plastic transition range.

Fig. 3. Option 1 and experimental Option 2 curves for SGTs at 24 and 300 °C.

and Option 2 curves for both materials and testing temperatures result almost identical. 4. Material and geometry-specific Option 3 FAD curves Option 3 curves, Eq. (6), can be derived from elastoplastic FEA results or experimental measurements of the J-integral parameter. Using both alternatives, this section presents Option 3 curves for the SGTs studied in this work. 4.1. Option 3 FAD curves obtained from elastoplastic FEA results Elastoplastic FEAs were performed with Abaqus 6.12-1 and using the Incoloy 800 SGTs studied in previous works (Bergant, 2016; Bergant et al., 2012, 2015a, 2015b, 2015c, 2016,2017). The tubes were modelled with 15.88 mm outside diameter and 1.13 mm wall thickness. The stress vs. strain curve at room temperature was adopted to model the mechanical behaviour. Circumferential and longitudinal part-through-wall cracks (PTWCs) and through-wall cracks (TWCs) were considered, as all of them were

found in service (EPRI, 2006). On the other hand, the experience in the nuclear industry has shown that the most relevant loading conditions that can lead to structural failures of flawed SGTs are combinations of tension and bending for circumferential cracks and internal pressure for longitudinal cracks (EPRI, 2006). These service conditions were adopted for numerical simulations. Numerical 3D models for different crack geometries are presented in Fig. 4, i.e., circumferential and longitudinal semi-elliptical PTWCs in Fig. 4 (a) and (b), and circumferential and longitudinal TWCs in Fig. 4 (c) and (d), respectively. Only stationary cracks were analyzed, so stable crack growth was not allowed during FEA simulations. Using symmetry considerations, one-quarter of the tubes were modelled applying proper boundary conditions. Tubes with circumferential cracks were loaded in tension and bending, while internal pressure was applied for longitudinal flawed tubes. In all cases, the tubes were loaded until the plastic collapse. Focused meshes were designed to provide detailed resolution of the near-tip stress-strain fields, using 3D 20-node quadratic brick elements with reduced integration and finite strain analysis. Specific details of the meshes are presented in Fig. 4. From the FEA results, the J-integral values for rising loads were obtained. For PTWCs, the maximum value of J-integral in the crack front was used, while for TWCs an averaged value in the crack front was adopted. FEA models were also used to calculate the limit loads PL required for the estimation of Lr according to Eq. (2). For each of the cracked geometries, the limit load PL was estimated for the collapse condition, that corresponded to the load reached in the last load increment step exhibiting convergence. Fig. 5 shows Option 3 curves from numerical results, again compared with the Option 1 definition. A general good agreement between the curves can be seen, although Option 3 curves tend to be lower than Option 1 curve. This result indicates that the Option 1 curve is not conservative, at least for the analyzed situations. Fig. 5 also presents loading paths for semi-elliptical circumferential and longitudinal cracks taken from a previous work (Bergant et al., 2015b) where expressions from (API 579-1/ASME FFS-1, 2007) were used (for the sake of brevity, they are not repeated here). Mechanical and fracture properties measured for the SGTs used in this work and reported in (Bergant et al., 2012, 2016,2017) were used to define Kr, Lr and Lr max parameters. It can be seen that the failure mode for semi-elliptical PTWCs is biased to plastic collapse so that the critical condition is given by Lr max. This is a consequence of the typical high fracture toughness

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Fig. 4. FEA meshes for cracked SGTs: (a) circumferential PTWC with a/t = 0.75 and c/a = 3, (b) longitudinal PTWC with a/t = 0.50 and c/a = 4, (c) circumferential TWC with 2c = 12 mm and (d) longitudinal TWC with 2c = 20 mm.

Fig. 5. Option 1 and FEA Option 3 curves for semi-elliptical PTWCs. Loading lines are taken from (Bergant et al., 2015b).

and relatively low strength of SGTs. The previous research where experimental data were analyzed using the FAD methodology confirmed that the Lr max parameter is well suited for predicting the plastic collapse of flawed SGTs with PTWCs (Bergant et al., 2015b). In that work also a similar behaviour was observed for

constant depth and infinite length cracks. These types of cracks are of particular interest since they were found in nuclear SGTs susceptible to stress corrosion cracking phenomena (IAEA, 2011), being the reason why most of the assessment criteria were developed for constant depth cracks.

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It should be added that fracture toughness values Kmat used in (Bergant et al., 2015b) were obtained by testing specimens with relatively long TWCs (Bergant et al., 2012, 2016,2017), while it is expected that crack resistance of smaller cracks under low constraint conditions is higher, as found in (Bergant et al., 2016) for short TWCs. In this case, the slope of the loading lines in Fig. 5 will be even lower, thus favouring the failure by plastic collapse. Therefore, for either short or even long PTWCs, structural integrity assessments based only on plastic collapse analyses should be sufficient. The construction of detailed Option 3 FAD curves is not justified for the assessment of these types of cracks. Fig. 6 shows Option 3 curves from FEA results for longitudinal TWCs in tubes with internal pressure. On the other hand, Fig. 7 presents Option 3 curves from numerical results for circumferential TWCs in tubes subjected to tensile and bending loads. 4.2. Option 3 FAD curves obtained from experimental data Fig. 8 displays experimental Option 3 curves for circumferential TWCs in tubes subjected to tensile and bending loads. Test records for load, crack length and fracture toughness obtained from experimental researches were used to estimate the curves (Bergant, 2016; Bergant et al., 2012, 2016,2017). It is worth noting that the comparison of experimental curves for typical SGTs made of Inconel 690 and Incoloy 800 alloys indicates that there is a strong effect related to the material behaviour. Considering that FEAs were performed for Incoloy 800 SGTs, the experimental and numerical J-based Option 3 FAD curves can be compared for this material. It can be seen that there is a very good agreement between the experimental based and the numerically derived curves for tensile loading and crack lengths 2c in the range between 10 and 24 mm. For the bending configuration and 2c = 24 mm, the numerically obtained FAD curve is lower and thus more conservative than the experimental one, although both still in reasonable agreement. This difference might be attributed to the straight crack front assumption considered for FEAs of tubes with TWCs in contrast with the crack tunneling effect observed during fracture tests (Bergant et al., 2016). Despite this, the general trends related to the crack length and loading condition are coincident for both numerical and experimental Option 3 FAD curves.

The Option 3 curves in Figs. 6–8 show a greater dependence on geometry for long TWCs than that observed for PTWCs and short cracks in Fig. 5. These results show that for TWC lengths longer than 10 mm approximately, Option 3 curves tend to be higher than Option 1 for Lr > 1, increasing the safe region in the FAD. The longer the crack length, the higher the Option 3 curves. Therefore, Option 1 curve becomes conservative as the crack increases its length. 4.3. Option 3 FAD curves with the normalized definition for Lr As it was shown in the previous sections, the definition of Lr in terms of limit load or yield load solutions leads to a geometry dependence in the Option 3 FAD curves. Considering Eqs. (6)–(8), this means that the fracture driving forces are not well estimated through the simpler Option 1 or Option 2 FAD curves. The consequence of using these options is a reduction in the accuracy of fracture analyses, which is usually compensated with an increase in the degree of conservatism of the assessments. On the other hand, the use of more detailed FAD curves may become prohibitive in parametric studies due to the cost associated with FEA simulations. Given this context, alternative approaches for normalizing the loading parameter Lr have been proposed to reduce the geometry dependence of FAD curves (Anderson, 2005). In this way, the practical advantages of simpler options can be exploited giving acceptable accurate analyses. The most accepted proposals were developed to be applied in FAD based procedures (Anderson, 2005) and crack driving force scheme analyses such that in SINTAP/FITNET procedure (Zerbst et al., 2007,2009,2012). It should be noted that the method proposed in (Anderson, 2005) has already been included in API 579/ ASME FFS-1 guide (API 579-1/ASME FFS-1, 2007). These proposals rely on the same basic idea, i.e., the geometry dependence of FAD (or crack driving force) curves can be reduced by forcing them to pass through the same point at a specific loading level given by Lr. If the curves have a similar shape, they will also be in close agreement at other Lr values. This is performed by modifying the load parameter Lr through the introduction of geometry factors affecting the reference stress rref (Anderson, 2005) or by the definition of a reference load instead of the limit load PL in Eq. (2) (Zerbst et al., 2007,2009,2012). It is worth emphasizing that in this approach there is no necessity of a precise limit load expression

Fig. 6. Option 1 and FEA Option 3 curves for longitudinal TWCs. Loading lines are taken from (Bergant et al., 2015b).

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Fig. 7. Option 1 and FEA Option 3 curves for circumferential TWCs. Loading lines are taken from (Bergant et al., 2015b).

at large Lr values, the numerical estimation of the J-integral becomes more difficult (Zerbst et al., 2009). On the other hand, the material-specific curve f2 is usually adopted as the simplest option to introduce the effect of the material behaviour in the crack driving force estimations. The previous approach has been successfully implemented in some FAD based analyses (Anderson, 2005) and crack driving force estimations within the SINTAP/FITNET procedure (Zerbst et al., 2009,2012). In order to study its applicability in the context of the present work, the case of tubes with longitudinal TWCs subjected to internal pressure in Fig. 6 is analyzed in detail as this is the most relevant condition from a practical point of view. Fig. 9 presents again an Option 2 and original Option 3 FAD curves from Figs. 3 and 6, respectively. Load parameters Lr of Option 3 curves were normalized according to Eq. (9), and the resulting curves were also included in Fig. 9 for comparison. It can be seen that the normalization method forced Option 3 curves to coincide with Option 2 curve at Lr = 1. For the shorter

Fig. 8. Option 1 and experimental Option 3 curves for circumferential TWCs for Inconel 690 and Incoloy 800 SGTs.

but a reference load which is consistent with the models used to predict the crack driving forces acting in a given geometry (Zerbst et al., 2009). Geometry factors or reference loads are usually obtained by the comparison between J-based curves f3 in Eq. (6) and materialspecific curves f2 in Eq. (7), and forcing them to coincide at Lr = 1, i.e.:

 J  ¼ f 2 ðLr ¼ 1Þ2 J el Lr ¼1

ð9Þ

Eq. (9) is satisfied modifying the parameter Lr by a proportional factor which is dependent on the geometry and loading type, but also on the material’s strain hardening exponent (Anderson, 2005). Although the load ratio value defining the coincidence point of curves is arbitrary, Lr = 1 is usually chosen in different proposals (Anderson, 2005; Zerbst et al., 2009,2012). This is because Eq. (7) is insensitive to Lr at low values, while in gross plasticity conditions

Fig. 9. Original (dotted lines) and normalized (solid lines) Option 3 FAD curves for TWCs and experimental Option 2 FAD curve for Incoloy 800.

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TWCs, i.e., 2c = 10 and 20 mm, the original Option 3 curves closely agree with Option 2 at Lr = 1 and, therefore, the normalization produces slight changes in the curves. On the other hand, the curve for 2c = 30 mm is strongly modified by the normalization procedure. In general terms, it can be concluded that the proposed approach leads to a partial reduction of the geometry effect in the curves, even though it remains important in the normalized Option 3 curves. A possible explanation of this is that the shapes of FAD curves corresponding to Options 2 and 3 are not similar. Thus, the coincidence condition at Lr = 1 given in Eq. (9) is not sufficient to guaranty good agreement between curves at other Lr values. Taking into account the case of PTWCs, it could be foreseen that the normalization procedure applied to the curves in Fig. 5 would lead to a good agreement between FAD Options since the shape of them are in closer agreement. However, this analysis is not relevant as the failure mode for PTWCs is biased to plastic collapse as mentioned earlier. It is clear that an important disadvantage of the method outlined here is the need for additional FEA results for the specific geometry and loading conditions. This drawback may be overcome in some applications where a combination of primary and secondary loads or variations of material behaviour are considered, provided that a good agreement between the driving forces curves is achieved (Zerbst et al., 2009). This is not the case shown in Fig. 9. Therefore, in the context of the structural integrity assessment of cracked SGTs through the FAD approach, using the normalization method presented in this section seems worthless. Detailed FEA simulations are recommended instead to obtain J-based FAD curves if the excessive conservatism of the simpler FAD options shall be avoided.

5. Discussion on the application of FAD options for structural integrity assessments of cracked SGTs This section discusses some implications of the previous results in structural integrity assessments of flawed SGTs. Fig. 5 presented results of Option 3 FAD curves for PTWCs, showing that Option 1 curve may not be conservative. However, as discussed earlier, failure conditions for PTWCs and relatively short cracks are controlled by plastic collapse, and therefore the nonconservativeness of Option 1 does not influence the analysis. It should also be noted that PTWCs may easily develop into TWCs in the case of stable and/or subcritical crack growth. This idea can be asserted taking into account the thin walls of typical SGTs and the high fracture toughness and ductility of SGTs materials. Therefore, the analyses of PTWCs are of particular interest to avoid leakage as they become TWCs, but from a stability point of view, only the latter are usually of concern (EPRI, 2006). Furthermore, it is a common practice to model PTWCs as TWCs for stability evaluations based on fracture mechanics methodologies (IAEA, 2011). Regarding TWCs, Figs. 6–8 show Option 3 curves for different geometry and loading conditions obtained from FEA results and fracture toughness experimental data. It was concluded that Options 1 and 3 agree in case of crack lengths 2c = 10 mm approximately, while when the crack length increases, Option 3 curves tend to be higher. In order to provide some insight of the consequences of the previous results, numerical examples of integrity assessments of SGTs using the FAD approach are provided below. At this point, it is worth noting that all the previous analyses were performed considering a single crack. However, flaws caused by stress corrosion cracking in steam generator tubes often form networks of multiple cracks oriented normal to the maximum principal stresses (IAEA, 2011). If these individual cracks were close to one another, interac-

tion effects could render them more deleterious than the individual cracks would be. For such cases, rather conservative rules are provided in the recommended procedures in order to render a multiple crack situation into a case of a single dominant crack. This is the criterion adopted in the present work. As an example case for analysis, it was assumed a SGT of a conventional pressurized water reactor steam generator made of Inconel 690, with 19.05 and 1.09 mm outside diameter and wall thickness, respectively. As the most important loading condition for structural integrity performance assessments is the pressure differential across the tube wall during normal operation and accident conditions (EPRI, 2006), tubes subjected to internal pressure with longitudinal TWCs are postulated. Two loading conditions are usually considered for structural integrity assessments of SGTs, i.e., normal operation (NO) and main steam line break (MSLB) in which the secondary pressure drops down to atmospheric pressure (EPRI, 2006). The approximate differential pressures across the tube wall are, respectively, DPNO = 9MPa and DPMSLB = 18 MPa (Majumdar, 1999). Safety margins for permissible defect sizes are 3 and 1.43 under normal operation and accident conditions (EPRI, 2006). Therefore, the flawed tubes must bear 3  DPNO = 27 MPa and 1.43  DPMSLB = 25 MPa for continued operation. It is also common to evaluate other pressure levels that are associated with specific plant designs or regulation requirements, which can be as lower as the normal operation condition (Erhard et al., 2012). Therefore, internal pressures of 9 and 27 MPa are considered as limiting cases. The assessment points coordinates (Lr, Kr) in the FAD diagram, Eqs. (1) and (2), were calculated using solutions for the mode I stress intensity factor KI and the reference stress rref given in (API 579-1/ASME FFS-1, 2007) for tubes with longitudinal TWCs subjected to internal pressure, see also (Bergant et al., 2015b). The mechanical properties for Inconel 690 at 300 °C extracted from (Bergant et al., 2017) were used. The fracture toughness for longitudinal TWCs was defined in terms of the J-resistance or Kresistance curves shown in Fig. 10 (Bergant et al., 2017). The equivalence between J-integral and the stress intensity factor K is given by Eq. (5), and E = 193 GPa. Jq = 202 kJ/m2 is the J-integral value at the beginning of stable crack growth and can be converted to Kmat = 197 MPa.m1/2. Yield stress and tensile strength at high temperature are, respectively, 218 and 554 MPa (Bergant et al., 2017). The failure condition using the FAD methodology was determined considering two events, i.e., the beginning of stable crack growth and the ductile crack instability.

Fig. 10. J and K resistance curves for longitudinal cracks in Inconel 690 SGTs at 300 °C, taken from (Bergant et al., 2017).

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Stable crack growth is predicted when the crack driving force reaches the particular value of fracture toughness that characterizes this process, Eq. (4). In terms of FAD, this condition corresponds to the case where the assessment point (Kr, Lr) lies in the failure curve. Crack instability condition is achieved when the increment of the driving force is higher than the rate of change of the material tearing resistance (Anderson, 2005). When toughness is characterized by resistance curves, assessment points in the FAD approach are estimated updating the crack driving force and the material toughness with crack growth (Anderson, 2005). As the crack grows, the assessment points form a curve that moves downward and to the right initially due to the rising of the resistance curve, that generally is steeper than the increment of the crack driving force at the beginning of stable crack growth. The curve may reach a minimum and then increase as the crack driving force starts to rise more than the resistance curve as the crack lengthens. The critical condition is predicted when the assessment curve is tangent to the FAD curve, defining the critical crack length. If the initial crack length is shorter, the assessment curve moves to the left in the FAD, and the crack may then grow in a stable manner until it is arrested at the point where the assessment curve intersects the FAD line. On the other hand, if the crack length is larger, the crack growth is unstable. Fig. 11 presents the FAD predictions of both failure conditions defined, i.e., beginning of stable crack growth and crack instability, using Option 1 and Option 3 curves (Option 2 curves were not considered in the analysis as they are, from a practical point of view, equivalent to the Option 1 curve). The critical crack lengths 2 cc for internal pressures of 9 and 27 MPa are summarized in Table 1. In the case of high pressure condition, i.e., 27 MPa, the failures can be predicted using just the Option 1. This is because the critical crack lengths are shorter than 10 mm and there are no significant differences between Option 1 and Option 3 curves. On the other hand, for internal pressure of 9 MPa, the critical crack lengths are considerably larger and, besides the reference Option 1 curve, the Option 3 curve obtained for a TWC of 30 mm shown in Fig. 6 was applied. It was selected as the critical crack lengths are very close to 30 mm. Using Option 3 curve, the critical crack length increased close to 40% regarding the Option 1 prediction.

Fig. 11. FAD analyses with Options 1 and 3 curves for SGTs under internal pressure.

Table 1 Summary of critical crack lengths 2 cc obtained from FAD analyses in Fig. 11.

DPNO = 9 MPa

3  DPNO = 27 MPa

Failure condition:

Option 1

Option 3

Options 1 and 3

Beginning of stable crack growth Ductile crack instability

21.8 mm

29.4 mm

6.4 mm

22.3 mm

30.9 mm

6.7 mm

For each condition analyzed, it can be seen that critical crack lengths for the beginning of stable crack growth are just shorter than those for crack instability. Although this result is obvious, it should be noted that the difference is negligible from a practical point of view. This is because, in near fully plastic regime, slight increases of crack length (or applied load) leads to large rises in the applied driving force (Anderson, 2005). Therefore, the instability condition is achieved after small amounts of stable crack growth. 6. Conclusions The FAD methodology has been proposed as a failure criterion for cracked SGTs. Compared with a traditional plastic limit load criterion given by a vertical line in Lr = 1 in the FAD diagram, it can be seen that the FAD approach reduces the conservatism as it takes advantage of the safe region for Lr > 1. The FAD methodology has been applied for typical austenitic SGTs using FEA techniques and experimental results of previous researches (Bergant, 2016; Bergant et al., 2012, 2016,2017), and the three options of FAD curves have been evaluated and compared. The following conclusions could be drawn from the study: - Option 2 curves were obtained from axial stress vs. strain tensile tests performed with Inconel 690 and Incoloy 800 SGTs at 24 and 300 °C. All the resulting FAD curves were almost identical to the Option 1 curve. - Option 3 curves showed some dependence regarding the crack size. It was found that for semi-elliptical PTWCs and TWCs shorter than 10 mm, the Option 3 curves and Option 1 curve are in reasonable agreement. Some Option 3 curves were slightly lower than Option 1, and therefore the latter can be considered as non-conservative. However, loading paths for these relatively short cracks showed that the failure is close to plastic collapse due to the high fracture toughness and low strength properties of typical SGT materials (Bergant et al., 2015a,2015b). As the plastic collapse is only controlled by the Lr max parameter, the non-conservativeness of Option 1 is not an issue. - Option 3 curves for TWCs exhibited more geometry dependence for crack lengths longer than 10 mm. The Option 3 curves tend to be higher than Option 1 curve for Lr > 1, and consequently, the latter is conservative. The curves obtained from FEA modelling and the derived from experimental results showed a good agreement. Furthermore, an important effect of material behaviour on FAD curves obtained from experimental data was found. - Following a proposal in API 579-1/ASME FFS-1 guide, a recognized normalization approach for Lr was applied to Option 3 curves for TWCs to reduce the geometry effect. However, the resulting curves still exhibited a significant geometry dependence, which allows concluding that the approach is not suitable for TWCs in SGTs. - The generic FAD Option 1 curve was found to be appropriate enough for structural integrity assessments of cracked nuclear SGTs under most relevant loading conditions. The use of the

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simplest Option 1 avoids the need of costly FEA techniques. This constitutes an important advantage when several assessments for loading conditions and crack sizes are required to determine safety margins. - The geometry and material-specific Option 3 curve must be used to reduce the excessive level of conservatism associated with Option 1 curve, especially under circumstances where critical crack lengths are relatively large, i.e., approximately 10 mm for the conditions studied in this work. The possibility of taking advantage of this additional margin should be demonstrated for each specific loading case and SGT assessed.

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