J-STAGE Advance Publication date: 8 November, 2017Paper No.15-00694

© 2017 The Japan Society of Mechanical Engineers[DOI: 10.1299/mej.15-00694]

Vol.4, No.6, 2017Bulletin of the JSME

Mechanical Engineering Journal

Development of failure evaluation method for BWR Lower head in severe accident; high temperature creep test and creep

damage model Yoshihito YAMAGUCHI*, Jinya KATSUYAMA*, Yoshiyuki NEMOTO**, Yoshiyuki KAJI**,

Hiroyuki YOSHIDA** and Yinsheng LI* *Nuclear Safety Research Center, Japan Atomic Energy Agency (JAEA)

Shirakata 2-4, Tokai, Ibaraki, 319-1195 Japan E-mail: yamaguchi.yoshihito@jaea.go.jp

**Nuclear Science and Engineering Center, JAEA Shirakata 2-4, Tokai, Ibaraki, 319-1195 Japan

Abstract After the Fukushima Daiichi nuclear power plant accident due to the pacific coast of Tohoku earthquake and Tsunami, we have been developing an analysis method considering creep damage mechanisms based on three–dimensional analysis in order to estimate condition of the fuel debris and failure behavior of the reactor due to severe accidents for the early completion of the decommissioning of nuclear power plants. We have also been obtaining material properties that are not provided in existing databases or literature for the analysis. In this study, we measure the tensile and creep properties of low alloy steel, Ni–based alloy, and stainless steel at high temperatures near the melting points. Using experimental data, some parameters for the creep constitutive law and creep failure evaluation method are determined. It is confirmed that the estimated rupture time by the failure evaluation method agree well with the experimental one.

Keywords : Fukushima Daiichi, Severe accident, Uni–axial tensile and creep test, Creep constitutive law

1. Introduction

The Fukushima Daiichi nuclear power plant (NPP) Units 1–4 suffered serious damages because of station blackout

due to the 2011 off the pacific coast of Tohoku Earthquake and Tsunami. After the Fukushima Daiichi accident, for the early completion of the decommissioning of NPPs, the Japan Atomic Energy Agency is performing various research and development activities such as the fundamental data acquisition, development of analytical method for estimating inside reactor, etc.

For failure evaluation of the components of NPPs at high temperature caused by severe accidents, the influence of creep damage on the rupture of components should be taken into account (Rempe, 1993), (Willschutz, 2002). Therefore, we have been developing a failure evaluation method that can estimate rupture time and locations on the basis of finite element analysis (FEA) considering the creep damage behavior. Material properties at high temperature are necessary for the analysis and failure evaluation. Creep data at high temperatures were obtained by Rempe (1993), Willschutz (2002) and Kato (2007). However, maximum test temperatures were about 300 degree lower than melting points for each material. In addition, there are little data at high temperature about creep deformation that are necessary to develop the creep constitutive law.

In this study, a uniaxial tensile/creep testing machine was developed to measure the tensile and creep properties at high temperature for structural analysis and creep failure evaluation. The stress–strain curves and creep deformation/rupture properties were obtained for low alloy steel, austenitic stainless steel, and nickel based alloy that are used in the lower head of reactor pressure vessels (RPVs) including control rods, stub tubes, etc. Then, a useful creep constitutive law to estimate the high temperature creep behavior was determined and parameters for the creep constitutive law were obtained from experimental results. Meanwhile, the failure evaluation method can evaluate the creep damage was investigated on the basis of the creep constitutive law. FEA considering the failure evaluation

1

Received: 4 December 2015; Revised: 12 December 2016; Accepted: 31 October 2017

2© 2017 The Japan Society of Mechanical Engineers

Yamaguchi, Katsuyama, Nemoto, Kaji, Yoshida and Li, Mechanical Engineering Journal, Vol.4, No.6 (2017)

[DOI: 10.1299/mej.15-00694]

method was performed to estimate the rupture time for each experiment condition, and estimated rupture time was compared with experimental one.

2. High temperature tensile and creep tests 2.1 Experimental procedure

Low alloy steel ASTM SA533B, austenitic stainless steel ASTM A276 316 (Type 316SS), and nickel based alloy

ASTM B166 N06600 (Alloy 600) are materials constituting the lower head of RPV including control rods, stub tubes, welds, etc. These materials were used for tensile and creep tests in this study. The chemical compositions of these materials are listed in Table 1. Uniaxial tensile and creep tests were conducted using round bar specimens with a gauge length and gauge diameter of 30 mm and 6 mm, respectively, as shown in Fig. 1. These tests were performed at temperatures ranging from 800°C to 1300°C in argon gas atmosphere using a high temperature tensile and creep testing machine (Fig. 2). The specimen is heated by a high temperature resistance heating furnace that can heat the specimen to 1600°C. Each testing specimen is first held at the specified temperature for 30 min prior to applying the load. Elongation of gauge length is continuously measured using an optical contactless displacement sensor. The tensile tests were performed under a monotonic displacement controlled condition with a stroke rate of 0.9 mm/min. Yield stress and ultimate tensile strength were obtained from the measured stress–strain curve. Creep tests were performed under a load controlled condition with a constant load. The creep strain–time curves were obtained from the creep tests.

Table 1 Chemical compositions of test materials

Fig. 1 Geometry of round bar specimen

R1.5

140110

60

30

R3 R2 R1.5

Φ14

Φ10

Φ6

C Si Mn P S Cu Ni Cr Mo Fe SA533B 0.17 0.23 1.47 0.014 0.017 0.15 0.57 0.1 0.51 96.77

Type316SS 0.02 0.38 1.68 0.033 0.001 - 10.67 16.37 2.21 68.64 Alloy 600 0.05 0.09 0.29 0.006 0.002 0.01 74.47 15.63 - 8.75

2

2© 2017 The Japan Society of Mechanical Engineers

Yamaguchi, Katsuyama, Nemoto, Kaji, Yoshida and Li, Mechanical Engineering Journal, Vol.4, No.6 (2017)

[DOI: 10.1299/mej.15-00694]

Fig. 2 High temperature tensile and creep testing machine

2.2 Results of tensile tests

Data of yield stress and ultimate tensile strength obtained from the tensile tests for SA533B, Type 316SS, and

Alloy 600 at high temperature are shown in Table 2. The yield stress was determined using 0.2% strain offset from the linear elastic portion of the stress–strain curve. Yield stresses of each material decrease with increasing temperature at the range of 800°C to 1300°C. Temperature dependencies of yield stress for each material evaluated in this study are compared with data published in previous papers by Rempe (1993) and JNES (2012) and are shown in Figs. 3–5. The temperature dependency of yield stress obtained in this study is in accord with tendency of the previous data. The high temperature softening effect dominated the mechanical behavior of SA533B at 200°C–800°C, and the yield stress decreased significantly with increasing temperature. Similarly, for Type 316SS and Alloy 600, the high temperature softening effect dominated the mechanical behavior at 800°C–1000°C and 600°C–1000°C, respectively. For temperatures beyond the temperature regions which were applied to previous studies of each material, the yield stress decreased in a mild manner with increasing temperature. The same trend can also be recognized for the ultimate tensile strength.

Table 2 Experimental results of tensile tests.

Temperature control device

Load cellWater cooling rod

Heating furnace

Actuator

Heater

Specimen

Thermocouples

Optical contactless displacement sensorBackside

Temperature [°C] Yield stress [MPa] Ultimate tensile strength [MPa]

SA533B

800 35.8 48.1 1100 8.87 13.8 1200 4.70 7.80 1300 2.40 4.14

Type316SS

800 112 115 1000 24.3 30.8 1100 14.0 17.1 1200 7.55 9.78

Alloy 600

800 82.0 87.0 1000 30.0 38.4 1100 14.6 18.4 1200 7.70 11.9 1300 4.31 7.64

3

2© 2017 The Japan Society of Mechanical Engineers

Yamaguchi, Katsuyama, Nemoto, Kaji, Yoshida and Li, Mechanical Engineering Journal, Vol.4, No.6 (2017)

[DOI: 10.1299/mej.15-00694]

Fig. 3 Yield stress of SA533B at various temperatures

Fig. 4 Yield stress of Type 316SS at various temperatures

Fig. 5 Yield stress of Alloy 600 at various temperatures

2.3 Results of creep tests

Figures 6–8 show typical creep strain curves for SA533B, Type 316SS, and Alloy 600 at 1300°C, respectively. The

creep strain curve under constant stress condition displays three different stages as illustrated in Fig. 9. As shown in the figures, larger applied stress caused a shorter rupture time and higher creep strain rate at a given temperature. For the creep behavior at high temperature in the range of test temperature, the rupture time is short, and the tertiary creep stage is significant. The increase in the creep rate during the tertiary creep stage is due to the growth and coalescence of voids and necking generated by plastic deformation. The results of the creep tests are also summarized in Table 3.

0

50

100

150

200

250

300

350

400

450

0 500 1000 1500

Yiel

d st

ress

(MPa

)

Temperature (℃)

Present workRempe (1993)JNES (2012)

0

50

100

150

200

250

300

350

400

0 500 1000 1500

Yiel

d st

ress

(MPa

)

Temperature (℃)

Present workRempe (1993)JNES (2012)

0

50

100

150

200

250

0 500 1000 1500

Yiel

d st

ress

(MPa

)

Temperature (℃)

Present workRempe (1993)JNES (2012)

4

2© 2017 The Japan Society of Mechanical Engineers

Yamaguchi, Katsuyama, Nemoto, Kaji, Yoshida and Li, Mechanical Engineering Journal, Vol.4, No.6 (2017)

[DOI: 10.1299/mej.15-00694]

Larson–Miller parameter (LMP) (Larson, 1952) for each material was determined on the basis of rupture time. LMP was used for the creep failure evaluation. The value of LMP is usually expressed as LMP = T (C + log tr), where C is the material specific constant often approximated as 20, tr is the rupture time in hour, and T is the temperature in Kelvin. In this study, C was assumed to be 20. The LMP calculated using the above equation from the results of each test condition is related to the applied stress and is expressed by Eq. (1) for each material.

LMP = [log(σ) − 𝑐0] 𝑐1⁄ (1)

where c0 and c1 are the material constants as listed in Table 4 for each material. Figure 10 shows the relationship between LMP and stress of the creep rupture data obtained in this study and previous studies (Rempe, 1993), (Willschutz, 2002) for SA533B. A linear equation was determined on the basis of a best fitting of the creep data obtained in previous studies, and each curve was obtained for the standard deviations of 1σ and 2σ. Similarly, for Type 316SS and Alloy 600, creep data obtained in this study are compared with the previous data as shown in Figs. 11 and 12, respectively. Creep data obtained in this study have the same decreasing tendency as the previous data and agree with those by the previous workers within experimental uncertainty. c0 and c1 for each material correspond to the intercept and gradient on the best fit lines in Figs. 10–12, respectively. The best fit lines for each material are in good agreement with experimental data in the present work. Maximum errors of the intercept and gradient were less than 3% and 5%, respectively. Thus, scatters of c0 and c1 are less than 5%.

The minimum creep rate during the secondary creep stage is often related to the applied stress. This relationship is represented by the Norton creep law (Norton, 1929).

ε̇𝑚𝑚𝑚 = 𝐴σ𝑚 (2)

where σ is the applied stress. The material constants A and n in Eq. (2) at each temperature can be determined from the secondary creep behavior as follows:

𝐴 = ε̇𝑚𝑚𝑚1σ1𝑚 = ε̇𝑚𝑚𝑚2

σ2𝑚 (3)

𝑛 = log(ε̇𝑚𝑚𝑚1 ε̇𝑚𝑚𝑚2⁄ )log(σ1 σ2⁄ ) (4)

where ε̇𝑚𝑚𝑚1 and ε̇𝑚𝑚𝑚2 are the minimum creep rates for the constant stresses σ1 and σ2, respectively, at the same temperature. Typical results of fitting to the creep strain curve of each material using the Norton creep law are shown in Figs. 6–8. Although tertiary creep behavior is significant at the test temperature range in this study, the Norton creep law can simulate the creep behavior only during the secondary creep stage.

In order to simulate the creep deformation during the secondary and tertiary creep stages, we selected the Kachanov (1958) and Rabotnov (1958) (K–R) model, which was developed on the basis of a creep damage mechanism. The damage mechanism is a manifestation of microstructure degradation including transgranular damage and intergranular damage. Transgranular damage arises where slip bands of the plastic region under high stress and low temperature conditions. Intergranular damage is a microcracking process at the grain boundaries under low stress and high temperature conditions. The strain rate based on the K–R model is expressed as follows:

ε̇ = 𝐴 � σ1−ω

�𝑚

(5)

ω̇ = 𝐵 σ𝑘

(1−ω)𝑙 (6)

5

2© 2017 The Japan Society of Mechanical Engineers

Yamaguchi, Katsuyama, Nemoto, Kaji, Yoshida and Li, Mechanical Engineering Journal, Vol.4, No.6 (2017)

[DOI: 10.1299/mej.15-00694]

where A and n are the material constants for the secondary creep stage and are the same as the parameters from the Norton creep law. B, k, and l are tertiary creep damage constants and are derived from the creep tests. The damage parameter ω is calculated considering the creep damage. For a constant stress condition, the damage evolution equation in Eq. (6) can be integrated as follows:

∫ (1 − ω)𝑙𝑑ω = ∫ 𝐵σ𝑘𝑑𝑑𝑡𝑟0

ω𝑟0 (7)

where tr and ωr are the rupture time and damage parameter at the rupture condition, respectively. ω is a returned number between “0.0” and “1.0” during the tertiary creep stage. If the material is free from creep damage, ω equals “0.0.” In contrast, when the specimen reaches rupture, ω equals “1.0.” Setting ωr = 1 and performing the integration, one can obtain:

1𝑙+1

= 𝐵σ𝑘𝑑𝑟 (8)

Using Eq. (8) and the experimental data, the relations of B, k, and l can be obtained:

𝑘 = log(𝑡𝑟2 𝑡𝑟1⁄ )log(σ1 σ2⁄ ) (9)

𝐵(𝑙 + 1) = 1𝑡𝑟1σ1

𝑘 = 1𝑡𝑟2σ2

𝑘 (10)

where 𝑑𝑟1 and 𝑑𝑟2 are the rupture times corresponding to the applied stresses σ1 and σ2, respectively. Furthermore, let us assume k = l in Eq. (10), then the parameter B can be determined as follows:

𝐵 = 1𝑡𝑟1σ1

𝑘(𝑘+1)= 1

𝑡𝑟2σ2𝑘(𝑘+1) (11)

Figures 6–8 show typical results of fitting to the creep strain curve using the K–R model. It can be clearly seen from the figures that the K–R model can simulate the creep behavior during the secondary and tertiary creep stages. Parameters for each temperature and material were determined by a similar method, as shown in Table 5. The parameters obtained in this study were compared with previous data obtained at lower temperatures by Rempe (1993). Typical comparison results of parameters A, B, k, and n for the SA533B are shown in Figs. 13 and 14. Parameters A and B obtained in this study and provided in the previous research have the same tendency to increase and are continuous. Scatters of parameters A and B were evaluated by assuming that probabilistic distribution for A and B follow log-normal distributions. Errors of parameters determined for SA533B in the present work were less than 10%. For Type 316SS, maximum scatter of parameters was obtained at 1300°C and is about 20%. The parameters for Type 316SS at any temperature other than the 1300°C were in error by less than 10%. Errors of parameters for Alloy 600 were larger than errors obtained for other materials and determined parameters at any temperature other than 1200°C were in error by as much as about 20%. Parameters A and B have temperature dependency and increase with increase in temperature at the range of 600°C–1300°C. Parameters k and n do not have explicit temperature dependence. The same trends were recognized for the Type316SS and Alloy 600 as shown in Figs. 15–18.

As described above, we obtained the material and creep properties at high temperature necessary for the structural analysis and failure evaluation method based on LMP and K–R model. Further work is underway to increase the accuracy of these properties through obtaining further creep data.

6

2© 2017 The Japan Society of Mechanical Engineers

Yamaguchi, Katsuyama, Nemoto, Kaji, Yoshida and Li, Mechanical Engineering Journal, Vol.4, No.6 (2017)

[DOI: 10.1299/mej.15-00694]

Fig. 6 Comparison of experimental results and fitting results of SA533B at 1300°C

Fig. 7 Comparison of experimental results and fitting results of Type 316SS at 1300°C

Fig. 8 Comparison of experimental results and fitting results of Alloy 600 at 1300°C

Fig. 9 Basic creep curve

0

0.2

0.4

0.6

0.8

1

0 5 10 15 20 25St

rain

(-)

Time (h)

ExperimentNorton lawK-R model

3.15 MPa

2.27 MPa

0

0.1

0.2

0.3

0.4

0 0.5 1 1.5 2 2.5 3

Stra

in(-)

Time (h)

ExperimentNorton lawK-R model

4.7 MPa

3.95 MPa

0

0.2

0.4

0.6

0.8

0 2 4 6 8 10

Stra

in(-)

Time (h)

ExperimentNorton lawK-R model

4.96 MPa 4.36 MPa

×Failure

Primary secondary Tertiary

Time

Stra

in

7

2© 2017 The Japan Society of Mechanical Engineers

Yamaguchi, Katsuyama, Nemoto, Kaji, Yoshida and Li, Mechanical Engineering Journal, Vol.4, No.6 (2017)

[DOI: 10.1299/mej.15-00694]

Table 3 Experimental conditions and results of creep tests.

Table 4 Material constants used to calculate LMP.

Temperature [°C] Stress [MPa] Time to failure [h] Minimum creep rate [%/h]

SA533B

1000 12.6 18.4 2.20 1000 17.0 6.91 5.55 1000 19.0 3.98 14.1 1200 4.50 10.40 1.81 1200 4.99 8.52 2.00 1200 5.97 3.06 7.72 1300 2.27 17.7 0.98 1300 2.61 11.3 2.66 1300 3.15 4.68 4.02

Type 316SS

800 75.0 20.4 1.05 800 87.0 5.36 4.82 927 33.0 6.40 3.49 927 36.3 4.24 5.87 928 40.5 1.97 10.9

1077 13.4 4.80 4.39 1077 14.5 4.45 4.75 1077 16.9 1.97 8.24 1200 7.00 2.10 5.55 1200 8.80 0.94 14.2 1300 3.95 2.44 6.86 1300 4.50 1.45 9.17 1300 4.70 0.68 15.4 1300 5.50 1.22 12.1 1350 4.00 0.19 23.1

Alloy 600

982 29.5 6.02 3.24 982 35.0 2.71 7.80 982 40.6 1.30 18.4

1093 11.3 18.3 0.79 1093 18.0 2.97 7.68 1093 22.2 0.76 21.5 1200 9.47 2.49 6.96 1200 11.0 1.57 11.2 1200 14.0 0.60 28.1 1300 4.36 7.02 1.89 1300 4.96 2.74 5.71

c0 c1 SA533B 4.502 -1.348×10–4

Type 316SS 5.307 -1.501×10–4 Alloy 600 4.636 -1.218×10–4

8

2© 2017 The Japan Society of Mechanical Engineers

Yamaguchi, Katsuyama, Nemoto, Kaji, Yoshida and Li, Mechanical Engineering Journal, Vol.4, No.6 (2017)

[DOI: 10.1299/mej.15-00694]

Fig. 10 Larson–Miller plot of creep rupture data for SA533B

Fig. 11 Larson–Miller plot of creep rupture data for Type 316SS

Fig. 12 Larson–Miller plot of creep rupture data for Alloy 600

E+04 2.0E+04 2.5E+04 3.0E+04 3.5E+04 4.0E+04

LMP (-)

Willschutz (2002)Rampe (1993)Present work

1.5 x 104 2.0 x 104 2.5 x 104 3.0 x 104 3.5 x 104 4.0 x 104

LMP (-)

103

102

101

100

Stre

ss (M

Pa)

Best fit line

-1σ

-2σ

+2σ Rempe (1993)

E+04 2.0E+04 2.5E+04 3.0E+04 3.5E+04 4.0E+04

LMP (-)

JNES (2012)Rampe (1993)Present work

1.5 x 104 2.0 x 104 2.5 x 104 3.0 x 104 3.5 x 104 4.0 x 104

LMP (-)

103

102

101

100

Stre

ss (M

Pa)

Best fit line

-1σ-2σ

+2σ Rempe (1993)

E+04 2.0E+04 2.5E+04 3.0E+04 3.5E+04 4.0E+04

LMP (-)

Rampe (1993)

Present work

1.5 x 104 2.0 x 104 2.5 x 104 3.0 x 104 3.5 x 104 4.0 x 104

LMP (-)

103

102

101

100

Stre

ss (M

Pa)

Best fit line

-1σ-2σ

+2σ Rempe (1993)

9

2© 2017 The Japan Society of Mechanical Engineers

Yamaguchi, Katsuyama, Nemoto, Kaji, Yoshida and Li, Mechanical Engineering Journal, Vol.4, No.6 (2017)

[DOI: 10.1299/mej.15-00694]

Table 5 Parameters of the K–R model.

Fig. 13 Parameters (A and B) of the K–R model for SA533B at various temperatures

Fig. 14 Parameters (k and n) of the K–R model for SA533B at various temperatures

0 500 1000 1500

Temperature (℃)

ParameterA B

Present workRempe (1993)

10-2

10-5

10-8

10-11

10-14

10-17

10-20

Para

met

er v

alue

(-)

0

2

4

6

8

10

12

0 500 1000 1500

Para

met

er v

alue

(-)

Temperature (℃)

Parameterk

Present workRempe (1993)

n

Temperature [°C]

Parameter of K–R model A B k n

SA533B 1000 1.00×10–5 3.59×10–6 3.23 3.04 1200 9.13×10–6 3.06×10–5 4.27 5.07 1300 2.87×10–4 4.04×10–4 4.05 4.31

Type 316SS

800 5.19×10–22 6.60×10–20 9.00 10.3 927 8.43×10–11 6.98×10–14 7.44 5.67

1077 3.14×10–6 3.01×10–8 5.24 3.6 1200 1.93×10–5 1.13×10–4 7.77 4.09 1300 1.17×10–4 2.13×10–6 7.31 4.64

Alloy 600

982 3.30×10–10 2.53×10–9 4.80 5.44 1093 5.70×10–8 9.40×10–9 5.45 4.89 1200 1.15×10–5 8.04×10–6 4.03 3.83 1300 6.04×10–8 3.73×10–7 7.29 8.59

10

2© 2017 The Japan Society of Mechanical Engineers

Yamaguchi, Katsuyama, Nemoto, Kaji, Yoshida and Li, Mechanical Engineering Journal, Vol.4, No.6 (2017)

[DOI: 10.1299/mej.15-00694]

Fig. 15 Parameters (A and B) of the K–R model for Type 316SS at various temperatures

Fig. 16 Parameters (k and n) of the K–R model for Type 316SS at various temperatures

Fig. 17 Parameters (A and B) of the K–R model for Alloy 600 at various temperatures

0 500 1000 1500

Temperature (℃)

ParameterA B

Present workRempe (1993)

10-2

10-5

10-8

10-11

10-14

10-17

10-20

Para

met

er v

alue

(-)

0

2

4

6

8

10

12

0 500 1000 1500

Para

met

er v

alue

(-)

Temperature (℃)

Parameterk

Present workRempe (1993)

n

0 500 1000 1500

Temperature (℃)

ParameterA B

Present workRempe (1993)

10-2

10-5

10-8

10-11

10-14

10-17

10-20

Para

met

er v

alue

(-)

11

2© 2017 The Japan Society of Mechanical Engineers

Yamaguchi, Katsuyama, Nemoto, Kaji, Yoshida and Li, Mechanical Engineering Journal, Vol.4, No.6 (2017)

[DOI: 10.1299/mej.15-00694]

Fig. 18 Parameters (k and n) of the K–R model for Alloy 600 at various temperatures

3. Creep analysis and creep damage evaluation 3.1 Details of the failure evaluation method

We have proposed a failure evaluation method based on the thermal–elastic–plastic–creep analysis and creep

failure evaluation for BWR lower head of RPV due to severe accidents. The creep analysis is performed using nonlinear elastic–plastic models by FINAS/STAR ver.2013r140611 (ITOCHU, 2013). The Norton creep law, which is expressed by Eq. (2), was used for the creep analysis. In the failure evaluation method, the creep damage variable D was calculated for each element using creep damage models based on “considère,” strain, LMP, or Kachanov damage criteria (NEA, 2003), (Koundy, 2005). Damage variable D corresponds to ω for the K–R model. These failure evaluation criteria are explained as follows:

(1) “Considère” criterion

The “considère” criterion is stress–based. If the equivalent stress exceeds the threshold values, the element is judged to reach the failure condition. In this study, damage variable D is defined as follows:

𝐷(𝑑) = σ𝑒𝑒(𝑑) σ𝑓𝑓𝑚𝑙⁄ (𝑇(𝑑)) (12)

were σeq is the equivalent stress and σfail is threshold values of failure stress having a temperature dependence. σfail for each material is the same as that is described in the previous paper (Nicolas, 2002).

(2) Strain criterion

The strain criterion is similar to “Considère” criterion, but the factor used to equivalent strain. If the equivalent strain exceeds the threshold values, the element is judged to reach the failure. The damage variable D based on the strain criterion is defined as following equation:

𝐷(𝑑) = ε𝑒𝑒(𝑑) ε𝑓𝑓𝑚𝑙⁄ (𝑇(𝑑)) (13)

where εeq is equivalent strain. εfail is threshold value and is set to be 120% in this study. εfail is the same as that is described in the previous paper (Nicolas, 2002).

(3) LMP criterion The creep damage model using the LMP criterion calculates the damage variable D by estimating the creep rupture

time. The damage variable D is expressed as the ratio of time to the rupture time. The rupture time is calculated by using LMP as following equation:

0

2

4

6

8

10

12

0 500 1000 1500Pa

ram

eter

val

ue (-

)

Temperature (℃)

Parameterk

Present workRempe (1993)

n

12

2© 2017 The Japan Society of Mechanical Engineers

Yamaguchi, Katsuyama, Nemoto, Kaji, Yoshida and Li, Mechanical Engineering Journal, Vol.4, No.6 (2017)

[DOI: 10.1299/mej.15-00694]

𝑑𝑟 = 10[LMP/𝑇−𝐶] (14)

where tr is the rupture time, and T is the temperature in Kelvin. In this study, C was assumed to be 20. Using the rupture time, variation of damage variable ∆Di for each increment, i, is calculated by Eq. (15).

∆𝐷 𝑚(𝑑) = ∆𝑑𝑚 𝑑𝑟⁄ (15)

where ∆ti is increment time. The damage variable D for each increment is obtained by Eq.(16) as summation of D(ti-1) and ∆Di.

𝐷(𝑑𝑚) = 𝐷(𝑑𝑚−1) + ∆𝐷 𝑚(𝑑) (16)

LMP for each conditions are calculated by Eq.(1) and parameters listed in Table 4.

(4) Kachanov damage criterion

The creep damage model based on the Kachanov damage criterion calculates the damage variable D as follows:

𝐷 = ∫ �̇�(𝑑)𝑑𝑑𝑡0 (17)

�̇�(𝑑) = �σM

𝐴𝑘(1−𝐷)�𝑅

(18)

where σM is the Mises equivalent stress and R has the same value as k in the K–R model. Ak is a temperature dependent material parameter, which is calculated using the parameters of the K–R model as follows:

𝐴𝑘 = 𝐵−1𝑘 (19)

If the creep damage variable D reaches “1.0,” the element is judged to reach the failure condition. Creep model parameters, which were derived from the experiments and obtained in previous papers, were used for this creep damage model. This model can be applied to the failure evaluation for SA533B in the temperature range of 627°C–1300°C. Similarly, for Type 316SS and Alloy 600, this model can be applied in temperature range of 800°C–1300°C and 732°C–1300°C, respectively.

3.2 Creep analysis and creep damage evaluation

Elastic–plastic–creep analyses corresponding to the uniaxial creep tests were conducted using an axisymmetric

model as shown in Fig. 19. The FEA model consists of the four–node quadrilateral axisymmetric QUAD4 elements for the test specimen. The total number of elements and nodes are 9633 and 9263, respectively. Material properties and creep model parameters, which were derived from the experiments, were used for this analysis.

The typical analytical results of the creep deformation and damage for SA533B at 1300°C are shown in Fig. 20. The horizontal axis represents the normalizing time in accordance with the experimental rupture time. The left vertical axis expresses the strain value obtained from the experiment and creep analysis. The right represents the estimated value of the creep damage. Solid symbols are the strain values of each elapsed time which is obtained from the experiment and creep analysis. It is clearly shown in Fig. 20 that the creep analysis using the Norton creep law can

13

2© 2017 The Japan Society of Mechanical Engineers

Yamaguchi, Katsuyama, Nemoto, Kaji, Yoshida and Li, Mechanical Engineering Journal, Vol.4, No.6 (2017)

[DOI: 10.1299/mej.15-00694]

describe the secondary creep behavior. Broken or dotted lines in Fig. 20 are the time history of the creep damages that are calculated on the basis of the four types of criteria for estimating the creep rupture time. When the damage variable of all elements on the cross section reaches “1.0,” the specimen is judged as ruptured. As shown in Fig. 20, creep rupture time can be estimated with the best accuracy using the Kachanov criterion. On the other hand, the estimated results of the rupture time using the “considère” and strain criteria are longer than the experimental results. Cause of overestimating of the rupture time is an underestimating of the stress and strain in the structural analysis which cannot express the elongation and necking of a specimen using the Norton creep law. The creep rupture time of SA533B under other conditions in Table 3 were estimated using each criterion and were compared with the experimental results in the same way. Figures 21–23 show the normalizing rupture time, which were estimated for all conditions for SA533B, Type 316SS and Alloy 600. As shown in Fig. 21, the creep rupture time can be estimated using the LMP and Kachanov criteria. However, estimating results using the LMP criterion have a larger variation than the one using the Kachanov criterion. Similar trends were also observed for the rupture estimation results of other materials as shown in Figs. 22 and 23. From these results, we concluded that the Kachanov criterion provides the most accurate results and LMP is also applicable for rupture time estimation.

To assess progress of severe accident and to predict rupture time and location of reactors due to severe accident, Katsuyama et al. (2016) developed finite element model of RPV lower head including, penetrations, and welds, and four types of creep damage failure evaluation methods of Kachanov and LMP criterions confirmed in this study. Through the analyses, it was shown that failure occurred only near the penetrations under simulated conditions.

Fig. 19 Analysis model of round bar specimen

Fig. 20 Comparison of experimental and rupture estimation results of SA533B at 1300°C

14

2© 2017 The Japan Society of Mechanical Engineers

Yamaguchi, Katsuyama, Nemoto, Kaji, Yoshida and Li, Mechanical Engineering Journal, Vol.4, No.6 (2017)

[DOI: 10.1299/mej.15-00694]

Fig. 21 Comparison of rupture times between each failure model and experiments for SA533B

Fig. 22 Comparison of rupture times between each failure model and experiments for Type 316SS

Fig. 23 Comparison of rupture times between each failure model and experiments for Alloy 600

4. Conclusions Tensile and creep tests at high temperature were performed to obtain the tensile and creep properties, which have

not been obtained at high temperature in previous studies. The existing material database was expanded. For expressing the creep deformation in the tertiary creep stage, we discussed an optimum creep law on the basis of experimental results. We confirmed that the K–R model can simulate the creep behavior during the secondary and tertiary creep stages. The necessary parameters of the K–R model for expressing the creep deformation at high temperature were

0

1

2

3

4

5

6

"Considère" Strain LMP Kachanov

Estim

ated

/Exp

erim

enta

l rup

ture

tim

e 1000℃, 17 MPa

1000℃, 12.6 MPa

1200℃, 4.5 MPa

1200℃, 5.97 MPa

1200℃, 4.99 MPa

1300℃, 2.27 MPa

1300℃, 3.15 MPaNot

rupt

ure

0

1

2

3

4

5

6

"Considère" Strain LMP Kachanov

Estim

ated

/Exp

erim

enta

l rup

ture

tim

e 800℃, 75 MPa800℃, 87 MPa927℃, 33 MPa927℃, 36.3 MPa927℃, 40.5 MPa1077℃, 13.4 MPa1077℃, 14.5 MPa1077℃, 16.9 MPa1200℃, 7.0 MPa1200℃, 8.8 MPa1300℃, 3.95 MPa1300℃, 4.5 MPa1300℃, 4.7 MPa1300℃, 5.5 MPa

Not

rupt

ure

0

1

2

3

4

5

6

"Considère" Strain LMP Kachanov

Estim

ated

/Exp

erim

enta

l rup

ture

tim

e 982℃, 29.5 MPa982℃, 35 MPa982℃, 40.6 MPa1093℃, 11.3 MPa1093℃, 18.0 MPa1093℃, 22.2 MPa1200℃, 9.47 MPa1200℃, 11.0 MPa1200℃, 14.0 MPa1300℃, 4.36 MPa1300℃, 4.96 MPa

Not

rupt

ure

15

2© 2017 The Japan Society of Mechanical Engineers

Yamaguchi, Katsuyama, Nemoto, Kaji, Yoshida and Li, Mechanical Engineering Journal, Vol.4, No.6 (2017)

[DOI: 10.1299/mej.15-00694]

obtained in this study. For obtaining the most effective creep damage model for the creep analysis, elastic–plastic–creep analyses

corresponding to the experiments and creep failure evaluation were conducted using the material and creep properties that were obtained in this study. The estimated results pertaining to the creep rupture time were compared with the experimental results. It is confirmed that the Kachanov criterion is appropriate for creep failure evaluation.

References ITOCHU Techno-Solutions Corporation, 2013, FINAS/STAR theoretical manual, Ver. 2013. Japan Nuclear Energy Safety Organization (JNES), 2012, “Project of Integrity Assessment of Flawed Components with

Structural Discontinuity (IAF) Material Properties Data Book at High Temperature for dissimilar metal welding in Reactor Pressure Vessel,” JNES-RE-2012-0024. (In Japanese).

Kachanov L. M., 1958, “Time to rupture process under creep conditions,” lzv Akad. Nank. SSSR, 8, pp. 26-31. Kato S., Hasebe S. and Yoshida E., 2007, “Evaluation of creep strength on SUS304 under off-normal over heating,”

JAEA-Research 2007-091. (In Japanese) Katsuyama J., Yamaguchi Y., Nemoto Y., Kaji Y. and Yoshida H., “Development of failure evaluation method for BWR

lower head in severe accident; Creep damage evaluation based on thermal-hydraulics and structural analyses”, Mechanical Engineering Journal, Vol. 3 (2016) No. 3 p. 15-00682, DOI:10.1299/mej.15-00682.

Koundy V., Durin M., Nicolas L., and Combescure A., 2005, “Simplified modeling of a PWR reactor pressure vessel lower head failure in the case of a severe accident,” Nuclear Engineering and Design, 235, pp 835-843.

Larson F. R., and Miller J., 1952, “A time-temperature relationship for rupture and creep stress,” Transactions ASME, Vol. 74, pp. 765-771.

Norton F.H., 1929, “The Creep of Steel at High Temperature,” McGraw-Hill, New York. Nuclear Energy Agency Committee on the Safety of Nuclear Installations, NEA/CSNI/R (2003)1. Rabotnov Y. N., 1969, “Creep problems in structural members," North Holland: North-Holland Publ. Co. Rempe J. L., et al., 1993, “Light Water Reactor Lower Head Failure Analysis,” NUREG/CR-5642. Willschutz H. G., Altstadt.E., 2002, “Generation of a High Temperature Material Data Base and its Application to

Creep Tests with French or German RPV-steel”, FZR-353.

16