Enhancement of master curve method for ... - Springer Link

Journal of Mechanical Science and Technology 26 (9) (2012) 2727~2734 www.springerlink.com/content/1738-494x

DOI 10.1007/s12206-012-0739-2

Enhancement of master curve method for inhomogeneous material† Shin-Beom Choi1, Sung Choi2, Jae-Boong Choi3, Yoon-Suk Chang4,*, Min-Chul Kim1 and Bong-Sang Lee1 1 Korea Atomic Energy Research Institute, Daejon, 305-330, Korea Nano Science and Technology, Sungkyunkwan University, Suwon, 440-746, Korea 3 School of Mechanical Engineering, Sungkyunkwan University, Suwon, 440-746, Korea 4 Department of Nuclear Engineering, Kyung Hee University, Yongin, 446-701, Korea 2

(Manuscript Received January 31, 2012; Revised April 16, 2012; Accepted May 2, 2012) ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------

Abstract Although the well-known master curve method has been widely used to estimate the ductile to brittle transition temperature and to prevent subsequent brittle fracture of reactor pressure vessels, it has a limitation to determine reference temperatures on heat affected zone and weld metals. The present paper is to propose an enhanced master curve method. Prior to this, several schemes to provide the master curve of inhomogeneous materials such as bimodal master curve estimation, randomly inhomogeneous master curve estimation and single point estimation were reviewed to confirm their applicability. As a result, the single point estimation scheme was chosen as the basic algorithm to calculate the reference temperature and modified by adopting three T0sp parameters concept to enhance the accuracy. The proposed method can be used to inhomogeneous materials for more accurate reference temperature calculation of reactor pressure vessels even with dispersed fracture toughness test data. Keywords: Inhomogeneous material; Master curve; Reactor pressure vessel; Reference temperature; Single point estimation ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------

1. Introduction While the median line of the master curve provided in ASTM E1921 can be used to predict the brittle fracture behavior of ferritic steel in the DBTT (Ductile to Brittle Transition Temperature) region [1, 2], the well-known master curve shows different characteristics between homogeneous materials and inhomogeneous materials. The master curve has limited applicability, especially to inhomogeneous materials, such as HAZ (Heat Affected Zone) and weld metals, which have largely scattered fracture toughness test data. To overcome this limitation, K. Wallin et al. [3, 4] suggested several alternative master curve methods for inhomogeneous materials such as BMC (Bimodal Master Curve), RIMC (Randomly Inhomogeneous Master Curve) and SPE (Single Point Estimation). BMC and RIMC employ the MLM (Maximum Likelihood Method) to cover the inherent broad range between upper bound and lower bound. However, they required an additional complicated numerical procedure for obtaining fitting parameters to represent the master curve. Also, SPE is a very simple method that can be used to calculate the reference temperature *

Corresponding author. Tel.: +82 31 201 3323, Fax.: +82 31 202 8106 E-mail address: [email protected] Recommended by Associate Editor Jin Weon Kim © KSME & Springer 2012 †

of inhomogeneous material but it gives poor results when invalid data are included. On the other hand, since the master curve method includes complicated processes as shown in Fig. 1, a proven tool is required to calculate the reference temperature for DBTT estimation to prevent the brittle fracture. The Belgium Nuclear Research Centre of SCK·KEN made a webbased master curve evaluation program [5] which provides three useful algorithms to determine the master curve in accordance with ASTM E1921, BMC and RIMC. However, it does not have SPE method and the built-in procedures for BMC and RIMC require a fair knowledge to determine relevant optimized fitting parameters during the evaluation. In addition, this program does not provide master curve algorithms for homogeneous material with constraint loss affecting practical load-carrying capacity [6]. The aim of the present paper is to enhance the current master curve method and to automate reference temperature evaluation processes as part of it. In this context, the scheme of SPE is modified to calculate a more accurate reference temperature for inhomogeneous material and its applicability is confirmed by comparison with BMC scheme. Subsequently, to calculate the reference temperature for both homogeneous materials and inhomogeneous materials, a web-based program called as RT-CAL (Reference Temperature CALculation program) is developed. Particularly, ASTM E1921 and modified

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where, K01 and K02 are the characteristic toughness values for two constituents and pa is the probability of toughness belonging to the distribution 1. In contrast to the current MC (Master Curve), where only one parameter such as K0 is needed to be determined, BMC adopts three parameters such as K01, K02 and pa. In order to handle randomly censored multi-temperature data sets, the estimation should comply with the maximum likelihood procedure. The likelihood is expressed as n

L = ∏ f cδi i ⋅ Sc1i−δ i

(2)

i =1

where, fc is the probability density function and Sc is the survival function and δ is the censoring parameters. The probability density function and survival function have the following forms, respectively. f c = 4 ⋅ pa ⋅

  K − K ( K Jc − K min )3 Jc min exp −  ( K Jc − K min ) 4   K 01 − K min

  

4

  

(3)

Fig. 1. Procedure of master curve determination.

−4 ⋅ (1 − pa ) ⋅

ASTM E1921 with constraint loss are adopted in the module for homogeneous materials, and the SPE and modified SPE are used in the module for inhomogeneous materials. The remainder of this paper is organized as follows: In section 2, several schemes to assess the master curve of inhomogeneous materials are reviewed and, consequently, the advantages and disadvantages of each scheme are discussed. In section 3, details of the modified SPE and its applicability are described, and concluding remarks are derived in section 4. Finally, specific features and structure of the web-based program to calculate the reference temperature are introduced in the appendix.

  K − K ( K Jc − K min )3 min exp −  Jc 4 − ( K Jc − K min ) K K min   02

  K − K min Sc = pa ⋅ exp −  Jc K − K 01 min  

  

4

  

  K − K min + (1 − pa ) ⋅ exp −  Jc K − K 02 min  

  

4

  

  

4

  

(4)

The parameters are solved so as to maximize the likelihood given by Eq. (2). The numerical iterative process is simplified by taking the logarithm of the likelihood so that a summation equation is obtained as Eq. (5).

2. Brief review of master curve method n

2.1 Specific features of master curve for inhomogeneous material By perceiving that the present master curve does not cover largely scattered fracture toughness data of inhomogeneous material, K. Wallin et al. proposed several master curve methods for inhomogeneous materials such as BMC, RIMC and SPE to widen the range between upper bound and lower bound. In the case of BMC, when the data population consists of two combined master curve distributions, the total cumulative probability distribution can be expressed as a bimodal distribution form.   K − K min p f = 1 − pa ⋅ exp  −  Jc   K 01 − K min

  

4

  

(5)

i =1

RIMC was proposed by M. Scribetta and also based on the maximum likelihood. The random variable T0 is assumed to follow a Gaussian distribution characterized by mean T0MML and standard deviation σT0MML. The probability density function for T0 in this case is fT =

) 2   (T − T ⋅ exp − 0 0 MML . 2 σ T0 MML ⋅ 2π  2 ⋅ σ T0 MML  1

(6)

The conditional survival probability at T0 is the current MC expression.

  

  K − K min − (1 − pa ) ⋅ exp −  Jc   K 02 − K min

ln L = ∑ δ i ⋅ ln( f ci ) + (1 − δ i ) ⋅ ln( Sci ) 

4

  

(1)

  K − K min ST0 = exp −  Jc K − K 0 min  

  

4

  

(7)

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S.-B. Choi et al. / Journal of Mechanical Science and Technology 26 (9) (2012) 2727~2734 800

The total survival probability S is obtained by solving the integral and the corresponding total distribution function f is derived as follows:

∞

f = ∫ fT ⋅ fT0 ⋅ dT0 . −∞

(8) (9)

400 KJc(med)

200 5% Lower Bound

The parameters T0MML and σT0MML are then solved by maximizing Eq. (5) using Eqs. (8) and (9) as input parameters. The SPE method itself is very simple. After size adjustment, all non-censored values are used to determine individual T0 estimates in Eq. (10).

0 -150

-50

0

800 Test Data(-100°C, 10mm) Test Data(-80°C, 10mm) Test Data(-60°C, 10mm) Test Data(0°C, 10mm)

95% Upper Bound

400

1T

where KJc,25mm,i represents the fracture toughness data after the size-adjustment. The single point T0 value (T0SP) is then estimated as follows:

KJc(med)

200

0 -150

5% Lower Bound

-100

∑T i =1

r

0i

-50

0

Test temperature(℃)

r

T0 SP =

− 4°C

(b) RIMC

(11)

Fig. 2. Applicability of BMC and RIMC.

where r is the number of valid data. 2.2 Limitation of master curve method for inhomogeneous material To check validity of alternative master curve methods, fracture toughness data at -100, -80, -60 and 0°C were used [5]. As depicted in Fig. 2, BMC and RIMC show good applicability for covering large scatters because they adopt several fitting parameters. Especially, the BMC method adopts three parameters, K01, K02 and pa. This means that the fitting algorithm of BMC is more complicated than that of the present MC. In comparison with the current method adopting Weibull distribution, BMC and RIMC employ complex alternatives, such as the bimodal distribution and MLM, to widen their ranges between the lower bound and upper bound. However, the built-in procedures for BMC and RIMC require an optimized algorithm to determine relevant fitting parameters. It means that the resulting parameters may lead to lack of consistency as well as depend on evaluation experiences and skills. Meanwhile, SPE is a very simple scheme but its applicability decreases when invalid fracture toughness data are used to calculate the reference temperature. When all fracture toughness data satisfy the criteria in ASTM E1921, this method has excellent applicability as shown in Fig. 3(a). Contrary to this promising case, Fig. 3(b) shows the poor applicability of SPE when valid data and inva-

50

(a) BMC

600

(10)

-100


KJc (MPa√m)

− 30  K ln  Jc ,25 mm,i  70  T0i = Ti −  0.019

95% Upper Bound

1T

−∞

KJc (MPa√m)

∞

S = ∫ fT ⋅ ST0 ⋅ dT0

600

Test Data(-100°C, 10mm) Test Data(-80°C, 10mm) Test Data(-60°C, 10mm) Test Data(0°C, 10mm)


(a) All valid data


(b) Valid data and invalid data Fig. 3. Applicability of SPE.

50

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S.-B. Choi et al. / Journal of Mechanical Science and Technology 26 (9) (2012) 2727~2734 1000 800

1T

KJc (MPa√m)

lid data coexist. Several largely scattered fracture toughness data were not covered because of the narrow range between upper bound and lower bound. Therefore, SPE should be improved to calculate the reference temperature for inhomogeneous materials and the details of this modification are explained in the next section.

3. Enhancement master curve method for inhomogeneous material

Test Data(-95°C, 10mm) Test Data(-130°C, 10mm) Test Data(-120°C, 10mm)

BMC(95% UB)

600

Interim SPE(95% UB)

400

Interim SPE(med)

200

Interim SPE(5% LB)

BMC(med)

BMC(5% LB)

0 -190

3.1 Proposal of modified single point estimation

-150

-110

-70

-30

10


In order to expand the master curve method for inhomogeneous materials, the original SPE represented in Eq. (11) was modified to cover largely scattered fracture toughness data as follows:

(a) T0SP(lower) 1000 800

Test Data(-95°C, 10mm) Test Data(-130°C, 10mm) Test Data(-120°C, 10mm)

Interim SPE(95% UB)

∑T i =1

0i

N r

N

∑T T0 SP ( mean ) = [( i =1 N

(12)

− 4°C 0i

1T

T0 SP ( lower ) =

∑T − 4) + ( i =1 r

0i

KJc (MPa√m)

r

− 4)]/2 °C

(13)

BMC(95% UB)

600

Interim SPE(med)

400

Interim SPE(5% LB) BMC(med)

200

N

BMC(5% LB)

∑T T0 SP ( upper ) =

i =1

r

0i

− 4°C

(14)

where N is the number of all data, T0SP(lower) is the reference temperature for lower bound, T0SP(mean) is the reference temperature for median line and T0SP(upper) is the reference temperature for upper bound of the modified SPE, respectively. The proposed method not only preserves the T0SP parameter in the original SPE scheme but also adopts the lower bound, upper bound and median line like those in the BMC scheme. Thereby, three candidates of bounding curve sets can be defined according to the new reference temperatures through Eqs. (12)-(14). Fig. 4 represents two resultant candidates of bounding curve sets along the corresponding test data to explain how the modified SPE scheme is established to deals with the large scatters. Fig. 4(a) compares the lower bounds, upper bounds and median lines derived from the interim SPE with T0SP(lower) and BMC. On the whole, the range between the interim SPE (5% LB) and SPE (95% UB) curves was relatively narrow comparing with that of the BMC (5% LB) and BMC (95% UB) curves. Particularly, the interim SPE (95% UB) curve was too low to encompass the scattered data while the interim SPE (5% LB) curve was comparable to the BMC (5% LB) curve. It means that T0SP(lower) is suitable only to define the lower bound of modified SPE. Fig. 4(b) compares three curves derived from the interim SPE with T0SP(upper) and BMC. Dissimilar to the previous case, the interim SPE (5% LB) curve could not encompass the scattered data while the interim SPE (95% UB) curve was comparable to the BMC (95% UB) curve. Also, T0SP(upper) is suitable only to define the upper bound of modified SPE.

0 -190

-150

-110

-70

-30

10


(b) T0SP(upper) Fig. 4. Determination of lower bound and upper bound for modified SPE.

Advantage of the proposed method is to obtain reasonable results comparable to the BMC in spite of its inherent simplicity. Therefore, in the modified SPE, the lower bound curve is determined by using Eq. (12) and the upper bound curve is determined by using Eq. (14). Further, the median line can be determined by using Eq. (13) as the average value of T0SP(lower) and T0SP(upper). 3.2 The applicability of modified single point estimation To confirm the applicability of the modified SPE, two sets of fracture toughness data of inhomogeneous materials were used [5]; Table 1 is for verification with BMC and Table 2 is for validation with SPE. At first, Fig. 5 represents the comparison of results between the modified SPE and BMC. As shown in the figure, the modified SPE was appropriate for covering largely scattered fracture toughness data of inhomogeneous materials because this method can widen the range between upper bound and lower bound. Particularly, the reference temperature predicted by the current MC, BMC and the modified SPE were -102.7°C, -117.9°C and -114°C. This means that the proposed method properly evaluate the reference temperature than the current MC and give a similar result to the BMC in spite of its simple scheme.

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Table 1. Fracture toughness data set A.

Table 2. Fracture toughness data set B.

No.

Test temp. (°C)

Thickness (mm)

KJc measured (MPa√m)

1

-95

10

143.5

2

-95

10

174.5

3

-95

10

113.0

4

-95

10

151.1

5

-130

10

43.6

6

-130

10

49.6

7

-130

10

48.7

8

-130

10

94.4

9

-120

10

57.6

10

-120

10

54.1

Fig. 5. Comparison of results between BMC and modified SPE.

Fig. 6 shows the adjustment of the modified SPE by comparing it with the original SPE. As depicted in Fig. 6(a), the original SPE could not encompass several fracture toughness data obtained from different test temperatures varying from 20°C to -140°C and specimen sizes varying from 0.18T-CT (compact tension) to 1T-CT. On the other hand, as shown in Fig. 6(b), the modified SPE adopting three simple equations covered well the significantly scattered data. In a word, the modified SPE showed a good applicability even when valid and invalid fracture toughness data coexisted. These results can be used to determine more accurate reference temperature and to predict realistic fracture behavior of inhomogeneous materials.

4. Conclusion In this research, the current master curve method was enhanced to cover the largely scattered facture toughness data of inhomogeneous materials. The scheme of SPE was chosen and modified to properly widen its range between upper bound and lower bound. The applicability of the modified SPE was verified by comparing its results with BMC and

Size 1T 1T 1T 1T 1T 1T 1T 1T 1T 1T 1T 1T 1T 1T 1T 1T 1T 1T 1T 1T 1T 1T 1T 1T 1T 1T 1T 0.4T 0.4T 0.4T 0.4T 0.4T 0.4T 0.4T 0.4T 0.4T 0.4T 0.4T 0.4T 0.4T 0.4T 0.4T 0.4T 0.4T 0.4T 0.4T 0.18T 0.18T 0.18T 0.18T 0.18T 0.18T 0.18T

Test temp. σys (MPa) (°C) -20 537 -20 537 -20 537 -20 537 -50 550 -50 550 -50 550 -50 550 -50 550 -70 560 -70 560 -70 560 -70 560 -70 560 -70 560 -70 560 -100 575 -100 575 -100 575 -100 575 -100 575 -40 546 -40 546 -60 555 -60 555 -80 564 -80 564 -50 550 -50 550 -20 537 -20 537 -100 575 -100 575 -100 575 -100 575 -100 575 -100 575 -50 550 -50 550 -50 550 -50 550 -50 550 -50 550 -165 612 -165 612 -165 612 -140 597 -140 597 -140 597 -140 597 -140 597 -140 597 -140 597

KJc (MPa√m) 88.6 416.7 395.6 524.2 128.4 114.6 146.7 94.6 412.4 136.9 117.1 122.1 137.4 127.3 129 113.8 111.7 96.9 173.4 96.4 131.5 87.3 80.2 86.2 87.2 99.7 81.1 124.5 322.9 347.7 414.6 81 161.3 92.9 135.1 116.8 225.5 306 359.1 393 340.6 394.4 335.9 56.1 40.9 37.9 107.6 111.1 91 98.1 84.7 48.7 101.2

KJc1T (MPa√m) 88.6 416.7 395.6 524.2 128.4 114.6 146.7 94.6 412.4 136.9 117.1 122.1 137.4 127.3 129 113.8 111.7 96.9 173.4 96.4 131.5 87.3 80.2 86.2 87.2 99.7 81.1 103.2 261.1 280.8 333.9 68.6 132.5 78 111.5 96.9 183.4 247.5 289.6 316.9 275 318 271.6 48.7 36.7 34.3 77.2 79.5 66.3 71 62.2 38.8 73

KJc(limit) (MPa√m) 311 311 311 311 316 316 316 316 316 320 320 320 320 320 320 320 326 326 326 326 326 314 314 318 318 322 322 209 204 204 208 214 211 218 211 217 217 207 212 206 204 207 209 211 225 220 153 155 155 153 149 154 145

δi 1 0 0 0 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 1 1 1 1 1 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1

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(a) SPE


(b) Modified SPE Fig. 6. Comparison of results between original SPE and modified SPE.

original SPE. In addition, the relevant processes to determine the reference temperature were automated for both homogeneous and inhomogeneous materials. The key features are summarized in the following appendix.

Acknowledgments The authors are grateful for the support provided by a grant from Korea Atomic Energy Research Institute and Post Brain Korea 21 Center of Sungkyunkwan University.

[5] www.sckcen.be/REACTORSAFETY/MECHANICAL TESTING/ MASTERCURVE. [6] S. B. Choi, Y. S. Chang, Y. J. Kim, M. C. Kim and B. S. Lee, Correction of constraint loss in fracture toughness measurement of PCVN specimens based on fracture toughness diagram, Journal of Mechanical Science and Technology, 24 (2010) 687-692. [7] BAW-2308, Revision 1, "Initial RTNDT of Linde 80 weld materials," Framatome ANP (2003). [8] Docket No. 50-305, Safety evaluation by the office nuclear reactor regulation regarding amendment of the Kewaunee nuclear power plant license to include the use of a master curve-based methodology for reactor pressure vessel integrity assessment, US Nuclear Regulatory Commission (2004). [9] KAERI/CM-1249, Development of numerical analysis techniques based on damage mechanics and fracture mechanics, Sungkyunkwan University (2009). [10] K. C. Koppenhoefer and R. H. Dodds Jr., Loading rate effects on cleavage fracture of pre cracked CVN specimens: 3-D studies, Engineering Fracture Mechanics, 58 (1997) 249-270. [11] J. P. Petti and R. H. Dodds Jr., Coupling of the Weibull stress model and macroscale models to predict cleavage fracture, Engineering Fracture Mechanics, 71 (2004) 2079-2013. [12] M. Nevalainen and R. H. Dodds Jr., Numerical investigation of 3-D constraint effect on brittle fracture in SE(B) and C(T) specimens, International Journal of Fracture, 74 (2004) 2079-2103. [13] J. P. Petti and R. H. Dodds Jr., Constraint comparisons for common fracture specimens: C(T)s and SE(B)s, Engineering Fracture Mechanics, 71 (2004) 2677-2683.

Appendix: A useful tool to evaluate the master curves A web-based master curve evaluation program, RT-CAL, was developed by employing both the author’s previously and currently proposed master curve methods to obtain the reference temperature more easily. Specifically, ASTM E1921, modified ASTM E1921, SPE and modified SPE each contained in two modules for homogeneous materials and inhomogeneous materials.

References [1] E1921-05, Standard test method for determination of reference temperature, T0, for ferritic steel in the transition range, American Society of Testing Materials (2005). [2] K. Wallin, Master curve analysis of ductile to brittle transition region fracture toughness round robin data, VTT Publications, 327 (1998). [3] K. Wallin, P. Nevasmaa, A. Laukkanen and T. Planman, Master curve analysis of inhomogeneous ferritic steels, Engineering Fracture Mechanics, 77 (2004) 2329-2346. [4] M.A. Sokolov and H. Tanigawa, Application of the master curve to inhomogeneous ferritic/martensitic steel, Journal of Nuclear Materials, 367-370 (2007) 587-592.

A.1 Technical background of the program The master curve method requires complicated processes for calculating the reference temperature. As mentioned, the program made by SCK·KEN has two major limitations: one is that its master curve method does not deal with homogeneous materials with constraint loss. The other is that it requires SPE as well as additional optimized procedures in relation to BMC and RIMC for inhomogeneous materials. With regard to the former limitation, several miniature specimens have been used to perform relevant post-irradiation testing with consideration of the restricted space of the surveillance capsule located in an RPV (Reactor Pressure Vessel) as


(a) The main window Fig. A.1. Surveillance capsule in RPV.

shown in Fig. A.1. However, fracture toughness data and load-carrying capacity were influenced by the non-standard size of the miniature specimen, which is known as the constraint loss effects. So, the master curve of homogeneous materials obtained from the procedure in accordance with ASTM E1921 has a limitation even if it provides a size adjusting equation. Especially, PCVN (Pre-cracked Charpy V-Notched) bias [6] due to the constraint loss effects has to be considered when the reference temperature is determined under a pressurized thermal shock condition to ensure the safety of an operating RPV. According to BAW-2308 [7] and US NRC (Nuclear Regulatory Council) report [8], consideration of the PCVN bias is very helpful for reducing the conservatism. Also, in the author’s previous work [9], a modified master curve method for homogeneous materials by considering the PCVN bias was already suggested through a fracture toughness diagram concept. It is one of alternatives that considers the constraint loss effects, which was derived from systematic finite element analyses based on simple TSM (Toughness Scale Method) proposed by R.H. Dodds Jr. et al. [10-13]. Subsequently, the equation for size effect predictions embodied in ASTM E1921 with constraint loss was proposed as below K Jc ( x ) = K min

 α B0  + [ K Jc (0) − K min ]   Bx 

 B  α = 1.998 ×  x   1 mm 

−0.682

(b) The input window

(c) The calculation window

(A1)

where the scale factor α obtained from the fracture toughness diagram was utilized to quantify the constraint loss effects. So, ASTM E1921 and modified ASTM E1921 were adopted to develop the master curve module for homogenous material. With regard to the latter limitation, even if omitted the details to avoid duplicity with the previous section, the modified SPE was proposed and verified to calculate the reference temperature of inhomogeneous materials. So, those were used to develop the module for inhomogeneous materials.

(d) The output window Fig. A.2. Screen of RT-CAL program.

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A.2 Structures of the program The reference temperature calculation program is based on the procedure in Fig. 1 and consists of two modules. In case of the homogeneous material module, basically, MC based on ASTM E1921 with PCVN bias was adopted. Moreover, Eq. (A1) was used to consider the constraint loss for modified MC. Both of them have the same procedure except for KJc(med) calculation. In case of the inhomogeneous material module, the difference between the original and modified SPEs was the reference temperature calculation. So, Eq. (10) was used for SPE while Eqs. (12)-(14) were used for the modified SPE to calculate the reference temperature. Fig. A.2(a) shows the main window of RT-CAL. In the input window of RT-CAL, the thickness of specimen, initial remaining ligament and test temperature are inputted to determine the validity of test data. Relevant data summarized in Table 2 are necessary to determine the validity of test data. When KJc derived from the test data is less than KJc(limit) value, the test data are regarded as valid. Also, KJc datum is invalid if the test data exceeds KJc(limit). Fig. A.2(b) shows the input window of RT-CAL. In the calculation window of RT-CAL, valid datum is expressed by 1 and invalid datum is expressed by 0. Also, the invalid datum is replaced by KJc(limit) to calculate the reference temperature for both homogeneous and inhomogeneous materials. Fig. A.2(c) shows the calculation window of RT-CAL, in which the calculated data such as censoring results, number of valid data, K0, KJc(med) and reference tempera-

ture per each method are summarized. In the output window of RT-CAL, calculation results such as the reference temperature, 5% lower bound, 95% upper bound and median line are shown in a graph, as depicted in Fig. A.2(d). According to ASTM E1921, minimum six fracture toughness data are needed to derive the master curve. So, if the number of fracture toughness data is not sufficient, the warning message appears under the ‘Remark’ of the output window of RT-CAL.

Shin-Beom Choi received his Ph.D. from the School of Mechanical Engineering in 2011 at Sungkyunkwan University in Korea. Currently, he is a senior researcher at Korea Atomic Energy Research Institute. His research interest is computational fracture mechanics.

Yoon-Suk Chang received his Ph.D. from the School of Mechanical Engineering in 1996 at Sungkyunkwan University. Currently, he is a Professor in the Department of Nuclear Engineering at Kyung Hee University. His current research interests are computational mechanics, fracture mechanics and damage mechanics.