Determination and Sensitivity Analysis of Target Flaw Sizes for Performance Demonstration of Cast Austenitic Stainless Steel Pipe by Probabilistic Fracture Mechanics

It is well known that cast austenitic stainless steel (CASS) used for primary coolant pipes of pressurized water reactor (PWR) plants is affected by thermal aging [1–3]. That is, thermal aging causes increases in tensile strength and decreases in fracture toughness during plant operation, which affects failure modes and allowable flaw depths of the CASS pipes. The ASME Section XI Code Case N-838 (CC N-838) [4] provides the target flaw sizes for Nondestructive Inspection in performance demonstration (PD) for ultrasonic examination system, which is based on fracture evaluations of flaws taking into consideration the effect of thermal aging.

A flaw is detected by volumetric examination during plant operation such as ultrasonic testing (UT). However, the coarse crystalline structure of CASS and its anisotropy cause low permeability of ultrasonic waves and low detectability. In recent years, the introduction of PD for ultrasonic examination systems in Japan was being examined by the Japanese Society for Nondestructive Inspection following the United States. For the introduction of the PD examination system, it is necessary to determine the target flaw sizes for the nondestructive testing. As a method for determining the target flaw sizes corresponding to the maximum flaw size without fracture, deterministic fracture mechanics is one candidate for application. However, there are several overconservatisms in the procedure, for example, use of fatigue flaw growth rate with 95% upper bound or a J-R curve from a high constraint laboratory specimen. As a result, the obtained target flaw sizes are small and difficult to detect by UT. Therefore, CC N-838 applied the probability fracture mechanics (PFM) [5] procedure to determine the target flaw sizes instead of deterministic fracture mechanics.

In the previous studies [6–8], a PFM analysis code, “PREFACE” was applied, and determination of target flaw sizes was attempted. Hayashi et al. [7] performed sensitivity analyses on the ferrite contents only in the fully thermal aged condition using PREFACE to obtain the target flaw sizes in the fully thermal aged condition. As a result, the target flaw sizes became the smallest at the minimum ferrite content, which is a condition of relatively larger fracture toughness and lower tensile strength. This was because the lower tensile strength caused plastic collapse. This report suggested that sensitivity analyses in arbitrary thermal aging conditions were needed to determine the actual minimum target flaw sizes.

Nishi et al. [8] performed sensitivity analyses using PREFACE with parameters of the ferrite contents and the thermal aging time. As a result, it was confirmed that the target flaw sizes became a minimum at lower ferrite content and relatively short thermal aging time, which is the condition with higher toughness and lower tensile strength than the initial values.

In the previous study, there were several points to be improved in the analysis conditions of PREFACE. First, the probabilistic distributions of the mechanical properties used in the analysis did not correspond to those of the CASS pipe of operating plants. The probabilistic distributions of each coefficient in prediction models for tensile properties and fracture toughness were applied. As a result, the mechanical properties had larger distribution than the actual distribution. Second, the correlation between tensile strength and fracture toughness was ignored and the target flaw sizes were calculated under the condition of low tensile strength and also low fracture toughness. In those cases, PFM analysis conditions were considered to be more conservative than the actual plant conditions. In order to set more realistic target flaw sizes, it is necessary to use probabilistic distributions of the mechanical properties considering correlations between tensile strength and fracture toughness of aged CASS.

On the other hand, there may be a less conservative condition in the previous studies using the PREFACE code. CASS straight parts of primary coolant pipe for PWR were manufactured by centrifugal casting and the other parts, such as elbows, were manufactured by static casting. Chopra [9] indicated that the thermal aging degradation behavior of CASS depends on the casting process. However, the existing thermal aging prediction models, the H3T, and true stress–true strain curve prediction method (TSS) models proposed by Kawaguchi et al. [10] are based on the database of the material properties of CASS manufactured by both casting processes. Similarly, Griesbach et al. [5] did not consider the effect of the differences of casting processes when they set the probabilistic distributions of the mechanical properties as random variables.

Momotani et al. [11] applied the H3T model and the TSS model to aged CASS manufactured by centrifugal casting and static casting, separately, and pointed out that the TSS model sometimes predicted much higher values for tensile properties than the actual aged material of the static CASS. In this case, the fracture evaluations of static CASS using the tensile properties predicted by the TSS model will be unconservative. Then, they proposed a modified TSS model, which is applicable to static CASS. By applying the different models to predict the tensile properties for centrifugal CASS and static CASS, it is expected that reasonable target flaw sizes will be obtained for each casting process.

In this study, PFM analyses for centrifugal and static CASS were carried out in order to obtain more reasonable target flaw sizes for the introduction of PD system in Japan. In the PFM analysis, more reasonable probabilistic distributions of mechanical properties were set by introducing the correlation between them and the lower limits of the tensile properties.

In order to determine the target flaw sizes for the PD examination system, PFM analysis was performed. The target flaw sizes were defined as the maximum flaw sizes, which had sufficiently low probability of fracture. In the PFM analysis, material properties were set as random variables. For more reasonable target flaw sizes, the following analysis conditions were adopted. First, to take into consideration the effect of the casting process on the material properties, thermal aging prediction methods were used for each casting process, centrifugal and static. Second, the probabilistic distributions of material properties were set reasonably based on residual analysis of the thermal aging CASS database (Refer to “Prediction Methods for the Mechanical Properties”). In addition, the correlations between the mechanical properties and the lower limits of the tensile properties were considered (Refer to “Advanced Sampling Methods for Mechanical Properties”). Finally, analysis conditions for Japanese CASS pipe and their chemical conditions in the PFM analysis were selected conservatively from among actual Japanese PWR plants (Refer to “Selection of Analysis Conditions for Japanese CASS Pipe”).

The fracture analysis was carried out by assuming a semi-elliptical surface flaw in the circumferential or axial direction on the inner surface of the pipe as shown in Fig. 1. In the analysis, the PFM analysis code, PREFACE [6–8], was used. The analysis flow in PREFACE is shown in Fig. 2. In PREFACE, the mechanical properties, i.e., tensile properties and fracture toughness, are set as random variables, and the failure probability is calculated by the Monte Carlo method. Flaw sizes, loads, chemical compositions, and thermal aging condition are set as fixed parameters. Each analysis case calculates failure probabilities by changing the flaw depth. The target size is defined as the flaw depth at which the failure probability is equal to a specified value (10⁻⁶ or 10⁻⁴).

In the fracture evaluation, the tensile properties (σ_y, σ_f, and Stress–Strain curve) and the fracture toughness (J_Ic, J₆, and J-R curve) were used as random variables. The probability distribution of each random variable was determined based on the prediction error for test data. The details are as follows.

The H3T model and TSS model were functions of the ferrite content and the chemical compositions. The ferrite content corresponding to the chemical composition was calculated based on ASTM A800 [12]. The ferrite content and the chemical composition were set as fixed parameters in PREFACE.

Note that N (μ, S²) is a normal distribution with the mean of μ and the standard deviation of S. Table 1 shows the standard deviations of the normalized prediction errors of yield stress, S_σy, and flow stress, S_σf.

The methods of fracture analysis for assumed flaws on the inner surface of pipe were based on the JSME rules on FFS. Two methods were applied in the fracture analysis: Two-parameters method and Limit-load method. Ductile fracture and plastic collapse due to assumed flaws were evaluated by using each method. The details of these two methods are described in  Appendix. The safety factors, SF, were not considered in the PFM analysis (SF = 1). The J-integral for the two-parameters method was calculated by the reference stress method [14]. It's important to note that flow stress affects both methods of fracture analysis. In the Two-parameter method, fracture become less likely with higher flow stress, because it causes higher stress of S-S curves, σ_SS, and lower J-integral. Also, in the Limit-load method, plastic collapse stress become higher with higher flow stress. On the other hand, yield stress affects only Two-parameter method. What these tensile properties have in common is that lower value causes higher fracture probability.

In order to determine more reasonable probability distributions of the mechanical properties, new sampling methods were introduced as described below.

When a material has a high σ_y, it is likely to have a higher σ_f. High strength materials generally have lower toughness, which is a negative correlation between the variation in tensile properties and fracture toughness. In previous analyses using the PREFACE code, those properties were independently generated as probabilistic variables despite those kinds of correlations.

To confirm these correlations, relations of each prediction error used for the calculation of the standard deviation of the mechanical properties were arranged. As a typical example, Figs. 6 and 7 show the relationship between the prediction errors in J_Ic-J₆ and σ_f-J_Ic of centrifugal CASS. The correlation coefficients in each combination are arranged in Table 3. The combinations of J_Ic-J₆, σ_y-σ_f, and σ_f-σ_SS gave positive correlation with the correlation coefficients, R, ranging from 0.61 to 0.78. On the other hand, the correlation between the tensile properties and the fracture toughness were relatively weaker, and the values of R ranged from −0.45 to 0.08.

Figure 8 shows the comparison of prediction error and sampling results considering the correlation by using Eqs. (9) and (10). The variation and the correlation in prediction errors data could be sampled equivalently by considering the correlation, and more reasonable samplings for mechanical properties were possible.

Thermal aging increases tensile strength. Caused by larger variations in tensile properties after thermal aging, in the sampling, tensile properties after thermal aging sometimes become smaller than the value assumed by variation before thermal aging, which differs from the actual situation because yield stress cannot decrease by thermal aging. In order to avoid these excessive conservatisms, it was considered that the probabilistic distributions of σ_y and σ_f after thermal aging have lower limits by the initial tensile properties before thermal aging, σ_y0 and σ_f0.

The standard deviations of the tensile properties before thermal aging are shown in Table 4. These were calculated by using the data of the mill sheets for Japanese PWR plants. Table 4 also shows the standard deviations after aging from Table 1. In static CASS, the standard deviations of σ_y and σ_f before thermal aging were smaller than after aging. The cumulative probability of the tensile properties before and after thermal aging under the representative material of static CASS and thermal aging conditions is as shown in Fig. 9. It indicates that if the probability distributions of the tensile properties only after thermal aging are used for sampling in PFM analysis, excessively low tensile properties may be sampled, which is lower than before thermal aging. In the PFM analysis, the cumulative probability density of the tensile properties of static CASS was defined by combinations of cumulative probability densities before and after thermal aging such as the red line in Fig. 9. This simplified combination model of cumulative probability densities does not correspond to actual variation in tensile properties, but it can avoid excessively lower sampling tensile properties and too conservative failure probabilities.

In such cases, C₁ value is almost same with J_Ic,s and C₂ value is almost zero in Eq. (4).

The previous studies [8] reported that failure probability is affected by the ferrite content calculated by chemical compositions and the thermal aging condition. This means that chemical compositions and the thermal aging time should be selected appropriately in order to conservatively determine the target flaw sizes.

Figure 10 shows the average values of σ_f and J_Ic predicted by the TSS model or the modified TSS model and the H3T model for static or centrifugal CASS pipe of Japanese PWR plants. In Fig. 10, the thermal aging times were changed in the range of 20,000–525,600 h (corresponding to 2–60 years) at the thermal aging temperature 325 °C. Because the CASS pipe of each plant has different chemical compositions, σ_f and J_Ic in Fig. 10 have large variations.

In this study, in order to obtain conservative target flaw sizes for the Japanese PWR plants except plants planed for decommissioning, several CASS pipes, which are assumed to have the highest failure probability, were selected for PFM analysis as described below.

Four CASS pipes from each casting process were selected for the PFM analysis, that is, material C-I to C-IV for centrifugal, and S-I to S-IV for static. Three conditions of the thermal aging time were set, short time (20,000 h), medium time (25,000 or 50,000 h), and long time (525,600 h) at the temperature of 325 °C. In the fracture analysis, ductile fracture occurs at lower fracture toughness (J_Ic, J₆) and lower tensile properties (σ_y, σ_f, σ_SS), and plastic collapse occurs at lower flow stress (σ_f). Therefore, the failure probability was assumed to become higher with a condition of lower tensile properties and lower fracture toughness. As shown in Fig. 10, σ_f and J_Ic values of selected CASS pipes designated by the colored symbols envelope the lower limits of those of for all pipes. It was expected that one of the selected pipes provided the highest failure probability and the smallest target flaw sizes among all CASS pipes. Table 5 summarizes the selected CASS pipes for PFM analysis.

CC N-838 provides the target flaw sizes for different flaw lengths and load levels for each flaw direction (circumferential or axial) and service condition (A & B or C & D), in which load levels and safety factors are different. In this study, the analysis conditions were set so as to make tables similar to those of the JSME rules on FFS. The flaw length and the load were set as parameters shown in Table 6. The load parameters were defined as (P_m + P_b + P_e)/S_m with changing P_b for circumferential flaws and P_h/S_m with changing P_h for axial flaws. Here, S_m is the design stress, which is defined as one-third of ultimate stress or two-thirds of yield stress, for example,. The PFM analysis were carried out for each combination of load parameters and flaw length parameters.

As shown in Table 7, the allowable failure probabilities in each service condition of A & B or C & D were assumed with reference CC N-838 [4,5]. The sampling numbers in the PFM analysis were set to be N = 5 × 10⁷ or 5 × 10⁵ according to the allowable failure probabilities in the service condition of A & B or C & D.

Common analysis conditions such as the pipe geometry, the thermal aging temperature, and the material constants were set as follows:

• Outer pipe diameter, D_o: 882 mm, Pipe thickness, t: 72.7 mm

• Thermal aging temperature, T_age: 325 °C

• Young's modules, E: 174,000 MPa, Poisson ratio, ν: 0.3

The PFM analysis were carried out by the procedure shown in Fig. 2, and the thickness ratio of the maximum flaw depth satisfying the allowable failure probability, which is called the target flaw size, a/t, were calculated for each load and flaw length. Since the target flaw sizes of centrifugal CASS exceeded those of the static CASS under almost all conditions, the details of the latter results are described below.

As a typical example of the results of PFM for circumferential flaws, Fig. 11 shows the relationship between target flaw sizes and stress ratios (P_m + P_b + P_e)/S_m of S-IV for 30 deg. flaw angle in the service condition C & D. The thermal aging time at which the target flaw sizes are the minimum values was 525,600 h for S-IV. The same analyses were performed for the other materials, and the thermal aging times which had the minimum target flaw sizes were determined as below:

• S-I and S-II: at 20,000 h (S-I-Short and S-II-Short)

• S-III: 20,000 h for intermediate stress ratio, or 525,600 h for high and low stress ratios (S-III-Short or S-III-Long)

• S-IV: at 525,600 h (S-IV-Long)

The comparison of the target flaw sizes of each material in these thermal aging times (S- III-Short are shown as a representative) is shown in Fig. 12. The materials which had the minimum target flaw sizes were S-II-Short for medium stress ratios ((P_m + P_b + P_e)/S_m = 1.4–2.0) or S-IV-Long for lower and higher stress ratios ((P_m + P_b + P_e)/S_m = 1.0–1.2 and 2.2–2.6). In almost other flaw angles, the same materials had the minimum target flaw sizes. S-II-Short was the material which had the second lowest tensile property, and S-IV-Long was the material, which had the lowest fracture toughness. The dominant failure modes were ductile fracture in all conditions, and failure probabilities caused by plastic collapse were relatively lower. In the service condition A & B, the dominant thermal aging time and failure modes for minimizing the target flaw sizes were similar to those in the service condition C & D.

As a typical example of the results of PFM for axial flaws, Fig. 13 shows the result of the target flaw sizes in S-IV with normalized flaw length ℓ/√ (R_mt) = 1.0 in service condition C & D. The trend of the target flaw sizes is similar to Fig. 11, and S-IV-Long had the minimum target flaw sizes. The thermal aging time at which the target flaw sizes were the minimum are shown as below:

• S-I and S-II: at 20,000 h (S-I-Short and S-II-Short)

• S-III and S-IV: at 525,600 h (S-III-Long and S-IV-Long)

A comparison of the target flaw sizes of each material at these thermal aging times is shown in Fig. 14. The materials having the minimum target flaw sizes were S-IV-Long for lower stress ratios (P_h/S_m = 0.6–1.8) or S-I-Short for higher stress ratios (P_h/S_m = 2.0–2.4). S-I-Short had the lowest tensile properties, and S-IV-Long had the lowest fracture toughness. These tendencies were the same in the other flaw lengths. Plastic collapse was the dominant failure mode only for the higher stress ratio, and ductile fracture was dominant for the other stress ratios.

In service condition A & B, S-IV-Long had the minimum target flaw sizes in any flaw lengths and loads, and the dominant fracture mode was ductile unstable fracture.

Based on the results of the PFM analysis in all conditions, target flaw size tables were arranged. The target flaw sizes were determined as the values corresponding to the combination of the minimum target flaw sizes for each material, for example, S-II-Short and S-IV-Long shown in Fig. 15. Tables 8 and 9 show target flaw size tables of circumferential flaws for centrifugal and static CASS. Tables 10 and 11 show those of axial flaws for centrifugal and static CASS. The centrifugal CASS had larger target flaw sizes than the static CASS in almost conditions. This is because of the higher averages and smaller standard deviations of the tensile properties in centrifugal CASS.

As a result of the PFM analysis, the static CASS pipes, which had the minimum target flaw sizes, are summarized in Table 12. Among them, S-I-Short or S-II-Short were materials having the lower tensile properties, and S-IV-Short or S-IV-Long were materials having the lower fracture toughness. These results indicate that the dominant mechanical properties for failure were different in each condition.

Then, simple PFM analyses were performed and the dominant mechanical properties for failure were confirmed. In order to identify the combination of mechanical properties failed in PFM samplings, failure or not were analyzed in the typical conditions of stress levels and flaw depths for materials which had the minimum target flaw sizes. Conditions of stress levels and flaw depths in these analyses are shown in Table 13. In these analyses, the sampling number is set to 50,000 times for the convenience of the analysis.

Figure 16 shows the combination of the sampled mechanical properties, σ_f and J_Ic, and those of failure. In the case of High stress/Small flaw as shown in Fig. 16(a), ductile fracture occurred when both σ_f and J_Ic were located near lower limits. On the other hand, in the case of Intermediate stress/Medium flaw as shown in Fig. 16(b), ductile fractures occurred when σ_f were located at the lower limit. These results indicated that dominant mechanical properties are different in each stress or flaw size. That is, it is supposed that only tensile properties are dominant at intermediate stress and medium flaw size, but both tensile properties and fracture toughness affect the ductile fracture at high stress and small flaw size.

Figure 17 shows the combination of the sampled mechanical properties and those of failure. In the case of High stress/Small flaw as shown In Fig. 17(a), sampling results failed only when σ_f were located at the lower limit, and in this case, the dominant failure mode was plastic collapse. On the other hand, in cases of Intermediate stress/Medium flaw as shown in Fig. 17(b), the fracture toughnesses are located at the lower limit of failure sampling results. These results indicate that failure modes were different at higher stress or intermediate stress. That is, the fracture toughness was dominant in intermediate stress because dominant failure mode was ductile fracture, while only σ_f affects failure at high stress because plastic collapse could easily occur.

In CC N-838 [4,5], the target flaw sizes were calculated by PFM analysis for primary coolant pipe made of CASS in the United States, and the target flaw size tables were arranged. These were compared with the tables analyzed in this study.

The comparison of the PFM analysis conditions between this study for static CASS and CC N -838 is shown in Table 14. There are many differences in the analysis conditions, for example, material conditions (chemical composition, casting process, thermal aging time, and temperature), fracture evaluation methods, and variations in mechanical properties and size of CASS pipes. The comparison of the PFM analysis results between this study for static CASS and CC N-838 is shown in Fig. 18 for circumferential flaws and Fig. 19 for axial flaws. Compared with CC N-838, the target flaw sizes in this study had the tendencies described below.

For circumferential flaws: Fig. 18 

• In the case of 10⁻⁶ in failure probability, corresponding to service condition A & B in this study, this study had nearly same sizes in the case of 2θ = 30–40 deg. and smaller sizes in the case of 2θ = 60 deg.

• In the case of 10⁻⁴ in failure probability, corresponding to service condition C & D in this study, this study had smaller sizes.

For axial flaws: Fig. 19 

• This study had larger sizes in the case of ℓ/√(R_mt) = 0.2–1.0 and nearly the same sizes in the case of ℓ/√(R_mt) = 2.0.

The target flaw sizes in this study tended to be smaller for circumferential flaws and larger for axial flaws. The difference in fracture evaluation methods is considered to be the main reason of this tendency. The methods of fracture evaluation and calculating J-integral applied in this study were relatively severe for circumferential flaw, compared with the J-T method in CC N-838. Therefore, especially in the case of 10⁻⁴ in failure probability, target flaw sizes for circumferential flaws were smaller in this study.

On the other hand, in the case of 10⁻⁶ in failure probability, the difference in target flaw sizes for circumferential flaws between this study and CC N-838 becomes smaller. This result is considered to be mainly caused by the variations in mechanical properties. Especially, the lower limit of the tensile properties was set in this study as described in the section “Advanced Sampling Methods for the Mechanical Properties,” so very small tensile property was not sampled in the PFM analysis, and then failure became less likely to happen and target flaw sizes became larger. The effect became more significant when increasing sampling number to calculate smaller failure probability.

Consequently, in the case of 10⁻⁶ in failure probability, the target flaw sizes for circumferential flaws in this study became relatively larger.

The size of the pipe in the PFM analysis also affects the target flaw size. The pipe thickness in this study was about 1.2 times thicker than CC N-838 and flaw depth, a, was 1.2 times larger for the same normalized target flaw sizes, a/t. Deep flaws have higher J-integral, which causes higher failure probability and smaller target flaw sizes.

Although the magnitude of target flaw sizes was different between this study and CC N-838, the trends in the relationship between the target flaw sizes and the stress ratios are similar despite many differences in the analysis conditions. It indicates that these PFM analysis methods, conditions and results are appropriate.

In order to obtain reasonable target flaw size tables for the PD examination system in Japan, fracture evaluation of flaws in centrifugal or static CASS pipes by PFM analysis code, PREFACE, was carried out. The main results are as shown below.

1. For PFM analysis, the variations in mechanical properties for the analysis (such as standard deviations, correlations, lower limits) were arranged based on prediction errors for actual test data. In addition, the TSS model and the modified TSS model were applied as the thermal aging prediction model in order to consider the differences in tensile properties for each casting process, centrifugal or static.

2. In order to estimate the conservative target flaw sizes for all PWR plants planned to be restarted in Japan, PFM analyses were performed for typical CASS pipes, and the target flaw sizes were calculated respectively for centrifugal CASS and static CASS. Based on the results of PFM analyses, the target flaw size tables for PD examination system were arranged by using the combination of the minimum target flaw sizes of each pipe. In almost all conditions, the target flaw sizes are smaller in static CASS than centrifugal CASS.

3. Depending on the flaw direction and stress ratio, the CASS pipe material, which is characterized by chemical composition, thermal aging time and casting process, having the minimum target flaw sizes were different. It was because the failure mode and dominant mechanical properties could be switched depending on the flaw direction and stress ratio. It indicates that PFM evaluations under the conditions of different thermal aging time are necessary to provide conservative failure probabilities and target flaw sizes.

4. This study and the ASME Code Case N-838 have similar trends in the relationship between the target flaw sizes and the stress ratios. The magnitude of the relation of the target flaw sizes between this study and CC N-838 are different in each flaw direction, flaw length, or allowable failure probability. It is affected by differences in the conditions of material (chemical composition, casting process, thermal aging time and temperature), fracture evaluation methods, the variations in mechanical properties, and size of CASS pipe.

The datasets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request.

a =

flaw depth

C₁ =

constant of J-R curve after thermal aging

C₂ =

constant of J-R curve after thermal aging

D_o =

outer diameter

E =

Young's modulus

J_Ic =

fracture toughness which is J_R value at Δa_Ic on a J-R curve

J_Ic,p =

J_Ic after thermal aging predicted by H3T model

J_Ic,s =

sampled value of J_Ic by the Monte Carlo method

J_R =

fracture toughness on a J-R curve after thermal aging

J₆ =

J_R value at Δa = 6 mm

J_6,p =

J₆ after thermal aging predicted by H3T model

J_6,s =

sampled value of J₆ by the Monte Carlo method

l =

flaw length of an axial flaw

N =

total sampling number

N_f =

sampling number of failures

P_b =

primary bending stress

P_e =

secondary bending stress

P_h =

hoop stress

P_m =

membrane stress

p_f =

failure probability

R =

correlation coefficient

R_i =

inner radius (=Ro-t)

R_m =

medium radius (=(Ro + Ri)/2)

R_o =

outer radius (=Do/2)

S_JIc =

standard deviation of J_Ic

S_J6 =

standard deviation of J₆

S_m =

design stress

S_σf =

standard deviation of σ_f

S_σSS =

standard deviation of σ_SS

S_σy =

standard deviation of σ_y

SF =

safety factor

S_X, S_Y =

standard deviation of X or Y

t =

thickness

T_age =

thermal aging temperature

γ_f =

increasing ratio of σ_f by thermal aging (=σ_f/σ_f0)

γ_y =

increasing ratio of σ_y by thermal aging (=σ_y/σ_y0)

Δa =

flaw extension on a J-R curve

Δa_Ic =

flaw extension at J_R = J_Ic

ε_SS =

strain on a stress-strain curve after thermal aging

θ =

half flow angle of a circumferential flaw

μ_X, μ_Y =

average value of X or Y

ν =

Poisson's ratio

σ_f =

flow stress, which is average of tensile strength and yield stress

σ_f,p =

σ_f after thermal aging predicted by TSS model or modified TSS model

σ_f0 =

initial flow stress before thermal aging

σ_f0,p =

σ_f0 predicted by TSS model or modified TSS model

σ_f,s =

sampled value of σ_f by the Monte Carlo method

σ_y =

yield stress

σ_y,p =

σ_y after thermal aging predicted by TSS model or modified TSS model

σ_y,s =

sampled value of σ_y by the Monte Carlo method

σ_SS =

stress on a stress-strain curve after thermal aging

σ_SS,avg =

average value of σ_SS predicted by Table 2 

σ_SS,s =

sampled value of σ_SS by the Monte Carlo method

σ_y0 =

initial yield stress before thermal aging

σ_y0,p =

σ_y0 predicted by TSS model or modified TSS model

CASS =

cast austenitic stainless steel

FFS =

fitness-for-service

H3T model =

fracture toughness prediction method

JSME =

The Japan Society of Mechanical Engineers

PD =

performance demonstration

PFM =

probability fracture mechanics

PWR =

pressurized water reactor

TSS model =

true stress-true strain curve prediction method

UT =

ultrasonic testing

The details of methods of fracture analysis based on the JSME rules on FFS are shown below.

Ductile fracture is evaluated by Two-parameter method. This method uses two kinds of parameters K_r and S_r relating to fracture mechanics and mechanics of materials, and is fundamentally equivalent to the J-T criterion for evaluation of ductile instability. Fracture assessment curve and ductile crack growth line are calculated, and fracture will occur when these don't intersect. Fracture assessment curve is defined by S_r and K_r and is calculated by increasing stresses for the initial flaw size. Ductile crack growth line is defined by S_r′ and K_r′ and is calculated by increasing the flaw depth for constant stresses.

S_r and S_r′ are defined as following equations.

ε_SS,ref is a strain value on the S-S curve corresponding to σ_SS,ref. K is a stress intensity factor calculated by an equation for a semi-elliptical inner surface crack in a cylinder.

Plastic collapse is evaluated by Limit-load method. Fracture will occur when the applied stress, P_bapp for circumferential flaws or P_happ for axial flaws, exceeds plastic collapse stress, $Pb0′$ or $Ph0′$⁠.