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The lower bound toughness procedure applied to the Euro fracture toughness dataset
Engineering Fracture Mechanics 69 (2002) 483–495 www.elsevier.com/locate/engfracmech
The lower bound toughness procedure applied to the Euro fracture toughness dataset J. Heerens *, M. Pfuff, D. Hellmann, U. Zerbst Institute of Materials Research, GKSS Research Centre Geesthacht, Max-Planck-Street, D-21502 Geesthacht, Germany Received 7 February 2000; received in revised form 21 July 2000; accepted 23 July 2000
Abstract The validity of a statistical method for estimating an engineering lower bound fracture toughness in the ductileto-brittle transition region is investigated using the Euro fracture toughness dataset generated in the European SM&T Project ‘‘Fracture Toughness of Steel in the Ductile-to-Brittle Transition Regime’’. The lower bound method is based on the empirical evidence that, in the low probability regime, the cumulative failure probability function tends to be a straight line rather than a curve as is the case for Weibull distributions. The investigation demonstrates that the engineering lower bound toughness values as predicted by the method are related to a cumulative cleavage failure probability lower than 2.5%. Such bound predictions can be achieved on the basis of a small number of cleavage toughness values measured at the temperature of interest. The results confirm the validity of the method. 2002 Elsevier Science Ltd. All rights reserved. Keywords: Ductile-to-brittle fracture transition; Lower bound fracture toughness; Failure probability; Pressure vessel steel; Test procedure
1. Introduction In the ductile-to-brittle transition regime, the fracture toughness of ferritic steels exhibits large scatter requiring statistical treatment of the data. In order to avoid the initiation of cleavage fracture in engineering structures, the lower bound regime of the cleavage toughness scatter is of particular practical interest. Substantial efforts have been undertaken in recent years to develop appropriate procedures for modelling the toughness scatter including its lower bound regime. The most advanced procedure has been recently presented by ASTM [1] which is based on the master curve concept [1–3]. In that procedure, the cleavage failure probability of laboratory specimens is modelled using a three parameter Weibull p distribution. This distribution has a ‘‘tail’’ in the low probability regime with a fixed origin of 20 MPa m. This origin is seen as a mathematical fitting parameter but is not necessarily a real physical toughness lower bound [4].
*
Corresponding author. Tel.: +49-4152-87-2610; fax: +49-4152-87-2534. E-mail address: [email protected] (J. Heerens).
0013-7944/02/$ - see front matter 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 0 1 3 - 7 9 4 4 ( 0 1 ) 0 0 0 6 9 - 8
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J. Heerens et al. / Engineering Fracture Mechanics 69 (2002) 483–495
Nomenclature b Jc Jc JLB Jmax Jmin J0 , K0 m p Pf b a0 B W T rY N n
ligament length J -integral related to unstable fracture mean value of Jc lower bound fracture toughness predicted by the procedure maximum Jc value due to the size criterion lowest Jc value of a dataset scale parameter of the Weibull distribution shape parameter of the Weibull distribution fraction of data which is rejected by the specimen size criterion cumulative failure probability correction factor for rejected toughness values initial crack length specimen thickness specimen width test temperature yield strength total number of data points of a dataset number of valid data points of a dataset
In Ref. [5] a statistical method for deriving an engineering lower bound toughness value was proposed. This lower bound is assessed using a modified Weibull distribution. The modified distribution has no tail in the low failure probability regime as it is the case for the conventional Weibull distribution. Based on empirical evidence, a straight line fit is used for modelling the low failure probability regime and the lower bound is formally obtained by extrapolating the straight line down to a failure probability of ‘‘zero’’. The method has the advantage of an easy application and can be used to estimate a toughness lower bound value of single temperature cleavage fracture toughness datasets. In Ref. [5] the method was applied to several types of ferritic steels. The results were promising. Nevertheless, the practical experience with this method is still limited and further validation of the method is necessary. For statistical reasons the experimental validation of a lower bound toughness assessment method requires a large experimental data base. Recently the Euro fracture toughness dataset was developed in a European Project [6]. It consists of about 800 fracture toughness tests performed on standardised compact specimens at different temperatures in the transition regime. Due to the large number of fracture toughness tests the dataset is seen to be ideal for validating statistical methods. In the present paper, the recent development on the lower bound method is presented and the validity of the method is illustrated on the basis of the Euro fracture toughness dataset.
2. The lower bound toughness method The procedure for assessing engineering lower bound toughness values is outlined in detail in Ref. [5]. In this section, the method will be briefly explained and some background information on its development will be given.
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Fig. 1. Principle of the lower bound procedure.
If the fracture resistance is expressed in values of Jc , then the probability of failure may be described by a two parameter Weibull distribution: m Jc Pf ¼ 1 exp ð1Þ J0 It is common practice to fix the shape parameter m to have a value of 2 [7–10]. The only unknown quantity is the scale parameter, J0 , which has to be derived from a toughness dataset. As mentioned before and illustrated in Fig. 1 there is empirical evidence that the lower tail of the failure probability curve tends to follow a straight line. Therefore, a linear fit is used to model the scatter in the low probability regime. An engineering lower bound, JLB , is obtained by formally extrapolating the straight line down to a failure probability of ‘‘zero’’. The distribution function referring to this assumption is then given by 8 pffiffiffiffiffi > Pf 6 0:5 < Jln0 2 ðJc JLB Þ; 2 Pf ¼ ð2Þ J > ; Pf P 0:5 : 1 exp J0c p with the lower bound toughness, JLB . By noting that Jc ¼ J0 ln 2 for Pf ¼ 0:5 one may easily verify that Pf and its derivative are continuous with respect to Jc if JLB is related to J0 by JLB ¼
ln 2 0:5 pffiffiffiffiffiffiffiffi J0 ¼ 0:23 J0 ln 2
ð3Þ
Using Eqs. (2) and (3) the scale parameter J0 may be related to the mean value of Jc , Jc , leading to Jc ¼ 0:89 J0
ð4Þ
As a result, the lower bound toughness is obtained from the mean value by JLB ¼ 0:26 Jc
ð5Þ
It is known that increasing plastic deformation in the specimen leads to loss of constraint. In order to quantify this, the specimen size criterion
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Jmax ¼
1 brY 30
ð6Þ
is introduced. J -values greater than Jmax are considered as ‘‘invalid data’’ or ‘‘censored data’’. This is done because loss of constraint generally increases the fracture toughness. The distribution function Eq. (2) does not account for this effect. By this reason invalid data have to be rejected and must not to be used to determine Jc . Therefore, Jc is the arithmetical mean value of the valid toughness values, Jc , in a dataset. Note, a rejection of data points from a given dataset is shifting the lower bound prediction to the conservative side (low toughness side). In order to compensate for this shifting effect a correction term b is introduced JLB ¼ 0:26 b Jc
ð7Þ
with b ¼ 1 þ 1:286 p where p is the fraction of data which is rejected by the size criterion, Eq. (6). In Ref. [5] it is stated that the lower bound procedure shall not be applied to datasets with p larger than 70%. For the application of the method, it is of particular interest how the number of toughness data points influences the lower bound toughness prediction. This subject was investigated using computer simulations performed as follows: In Fig. 2 the lower bound prediction is shown on the basis of a cleavage toughness dataset of a pressure vessel steel. The entire dataset contains 16 data points and the predicted lower bound is 29.6 N/mm. In order to investigate the influence of the number of data points, samples of 3 and 6 data points were randomly selected from the entire dataset and the lower bound was determined for each sample. This was done 10,000 times for each sample size. In Fig. 3 the histograms of the predicted lower bounds are shown. It can be seen that the distribution is nearly normal. Similar simulations were also performed by selecting randomly data points from the modified Weibull distribution as defined by Eq. (2). The computer simulation indicated that the predicted lower bounds are approximately normally distributed and their standard deviation can be assessed as follows: p r 0:13 Jc = n ð8Þ
Fig. 2. Lower bound procedure applied to a toughness dataset.
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487
Fig. 3. Histogram of lower bound predictions using different sample sizes taken from the dataset in Fig. 2.
Fig. 4. Influence of sample size on the accuracy of the lower bound prediction.
where n is the number of valid (with respect to Eq. (6)) data points. Eq. (8) reveals that relatively good lower bound predictions can be achieved on the basis of small datasets as is shown graphically in Fig. 4.
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J. Heerens et al. / Engineering Fracture Mechanics 69 (2002) 483–495
3. Results and discussion The Euro fracture toughness dataset provides the fracture toughness of the quenched and tempered pressure vessel steel DIN 22NiMoCr37 in the ductile-to-brittle transition regime. Overall, the Euro dataset provides 27 individual toughness datasets related to 1=2T, 1T, 2T and 4T C(T) specimen which were tested at eight different temperatures [6]. The test temperatures cover the entire range from the lower shelf to the upper shelf. The individual toughness datasets contain about 30 data points. According to the data validity criterion, (Eq. (6)), it was found that four of the 27 toughness datasets do not contain any valid data points due to the fact that cleavage fracture of all specimens occurred at J > Jmax . These four datasets are from 1=2T and 1T C(T) specimens tested at 0C and 1T and 2T C(T) specimens tested at 20C. All remaining datasets exhibit valid toughness data and were used to validate the engineering lower bound method. The main results of the analysis are listed in Table 1, showing the following quantities for all individual datasets: the predicted engineering lower bound toughness, JLB , the standard deviation, the ‘‘lowest measured toughness value’’, Jmin , the total number of data points, N , and the number of valid data points. In Fig. 5 the JLB -values and the corresponding lowest measured toughness values, Jmin , are compared graphically. The error bars in the diagram indicate the standard deviation of the predicted lower bounds by 2 r. It is seen that for 12 out of 23 cases the lower bound predictions are very close to the lowest measured toughness values. This is demonstrated by the fact that the error bars overlap the 1 to 1-line. In all remaining cases the lower bound prediction was found to be on the conservative side which means that the predicted lower bounds are lower than the lowest measured toughness values. In general cleavage fracture
Table 1 Results of the lower bound analysis T (C)
CT-specimen size (mm) (and its abbreviation)
154 154 154 110 91 91 91 91 60 60 60 60 40 40 40 20 20 20 20 20 0 0 þ20
W W W W W W W W W W W W W W W W W W W W W W W
a
¼ 25, B ¼ 12:5; (1=2T) ¼ 50, B ¼ 25; (1T) ¼ 100, B ¼ 50; (2T) ¼ 25, B ¼ 12:5 ¼ 25, B ¼ 12:5 ¼ 50, B ¼ 25 ¼ 100, B ¼ 50 ¼ 200, B ¼ 100; (4T) ¼ 25, B ¼ 12:5 ¼ 25, B ¼ 12:5a ¼ 50, B ¼ 25 ¼ 100, B ¼ 50 ¼ 25, B ¼ 12:5 ¼ 50, B ¼ 25 ¼ 100, B ¼ 50 ¼ 25, B ¼ 12:5 ¼ 50, B ¼ 25 ¼ 50, B ¼ 25b ¼ 100, B ¼ 50 ¼ 200, B ¼ 100 ¼ 100, B ¼ 50 ¼ 200, B ¼ 100 ¼ 200, B ¼ 100
Jmin (N/mm) (Experimental)
JLB 2r (N/mm) (Predicted)
N
Valid data points (Jc 6 Jmax )
2.8 3.4 3.8 8.1 19.9 14.9 19.6 13.8 34.0 38.4 46.7 30.4 61.6 49.1 40.0 78.9 139.0 102.0 58.6 112.0 174.3 126.5 405.0
1.87 0.34 1.77 0.30 1.59 0.30 7.90 1.06 15.27 2.74 13.22 2.74 10.08 1.84 8.29 2.14 27.13 4.50 41.1 6.31 26.02 4.46 36.92 6.74 67.83 13.65 54.47 9.20 33.80 6.17 52.01 16.69 113.06 18.79 89.88 17.94 63.84 11.37 45.76 11.82 150.18 23.46 107.95 26.99 128.23 77.40
31 34 30 55 31 34 30 15 31 31 34 30 30 32 30 31 30 20 30 15 30 16 15
31 34 30 55 31 34 30 15 29 20 34 30 6c 30 30 2c 11 8 29 15 16 16 2c
Specimens machined from 2TCT-specimens. Side grooved. c Dataset with p > 70%, all specimens: a0 =W 0:55. b
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Fig. 5. Comparison between predicted lower bound values and the lowest measured toughness values.
Fig. 6. Prediction of lower bound toughness in the lower shelf regime at: (a) 154C, specimen size W ¼ 100 mm, (b) 154C, specimen size W ¼ 50 mm, (c) 154C, specimen size W ¼ 25 mm.
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Fig. 7. Prediction of the lower bound toughness at 110C, specimen size W ¼ 25 mm.
Fig. 8. Prediction of the lower bound toughness at: (a) 91C, specimen size W ¼ 200 mm, (b) 91C, specimen size W ¼ 100 mm, (c) 91C, specimen size W ¼ 50 mm, (d) 91C, specimen size W ¼ 25 mm.
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491
Fig. 9. Prediction of the lower bound toughness at: (a) 60C, specimen size W ¼ 100 mm, (b) 60C, specimen size W ¼ 50 mm, (c) 60C, specimen size W ¼ 25 mm (machined form W ¼ 100 mm), (d) 60C, specimen size W ¼ 25 mm.
toughness datasets are specimen size dependent. This is in particular the case in the lower transition regime and in the upper transition regime. An increase of the specimen thickness typically reduces the mean J -value of a toughness dataset. In such cases, also the predicted lower bounds will show a similar size effect trend. This can be seen by comparing the JLB -values of different specimen sizes obtained at the each individual test temperatures, see Table 1. In Fig. 5 the lower bound prediction is compared with one single data point representing the lowest measured toughness value of the dataset. Let us now consider how the modified Weibull distribution fits with the entire toughness set. For this purpose all toughness datasets were converted into cumulative probability plots. This is done by ranking the toughness values and by calculating a failure probability for each individual data point as follows: Pfi ¼ ði 0:3Þ=ðN þ 0:4Þ;
16i6N
ð9Þ
where N is the total number of data points of the toughness dataset. Note, for statistical reasons, the invalid toughness values (Jc P Jmax ) have to be included when calculating N . Figs. 6–13 compare the experimentally derived cumulative probability functions (shown by the points) with the modified Weibull distribution, Eq. (2) (shown by the solid curve). The open symbols indicate the invalid data points according to Eq. (6). In Ref. [5] it is stated that the procedure shall not be applied to
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Fig. 10. Prediction of the lower bound toughness at: (a) 40C, specimen size W ¼ 100 mm, (b) 40C, specimen size W ¼ 50 mm, (c) 40C, specimen size W ¼ 25 mm.
toughness datasets which have more than 70% invalid data points. In Figs. 6–13 this limit is given by the dotted line at Pf ¼ 0:3. Considering all 23 datasets it is seen that the modified Weibull distribution fits quite well to the valid experimental datasets with six exceptions: Poor fits are observed for all three datasets measured at the lower shelf temperature T ¼ 154C, see Fig. 6a–c. At 154C the slope of the experimental curve is much steeper than the theoretical function (Eq. (2)). In addition, it is found that for all three datasets the predicted lower bound is significantly smaller than the lowest measured toughness value. This indicates that the modified Weibull distribution is not ideal for modelling lower shelf toughness scatter. There are three more cases where the slope of the theoretical function seems to be too small, see Figs. 8b, 10c and 12b. Note, in all six cases where the fit (Eq. (2)) was found to be poor, the predicted lower bounds remain conservative. This is important for practical applications and seems to indicate an inherent safety of the procedure. In the procedure, the engineering lower bound value is obtained by linear extrapolation of the modified Weibull distribution (Eq. (2)) down to a failure probability of ‘‘zero’’. If this approach is fully valid, no specimen should fail below such a lower bound value. Clearly, a full justification of this approach is impossible because it would require testing an infinite number of specimens to be tested. Nevertheless, let us consider to what extent the lower bound assumption is validated by the results of the present paper. The majority of the datasets investigated here consist of approximately 30 toughness values. The largest dataset (at 110C) has 55 data points. From Eq. (9) it can be easily assessed that the lowest toughness value of
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Fig. 11. Prediction of the lower bound toughness at: (a) 20C, specimen size W ¼ 200 mm, (b) 20C, specimen size W ¼ 100 mm, (c) 20C, specimen size W ¼ 50 mm, (d) 20C, specimen size W ¼ 50 mm (side grooved), (e) 20C, specimen size W ¼ 25 mm.
such a dataset is related to a failure probability of the order of 2.3% (N ¼ 30) and 1.3% (N ¼ 55). The results of the analysis show that for all 23 toughness datasets the lowest measured toughness value was found to be similar to, or even larger than, the predicted engineering lower bound. This finding seems to
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Fig. 12. Prediction of the lower bound toughness at: (a) 0C, specimen size W ¼ 200 mm, (b) 0C, specimen size W ¼ 100 mm.
Fig. 13. Prediction of the lower bound toughness at 20C, specimen size W ¼ 200 mm.
indicate that the engineering lower bounds as they are predicted by the method are related to a failure probability level which is similar to or smaller than 2.5%. This must be kept in mind when engineering lower bounds as predicted by this method are used in practical applications.
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4. Conclusions An analysis of the Euro fracture toughness dataset confirms the validity and applicability of the engineering lower bound toughness prediction procedure. In particular, the investigations reveal that the procedure has the following characteristics which are important for its practical application and possible standardisation: • The procedure predicts an engineering lower bound fracture toughness and its standard deviation from a cleavage toughness scatter band measured at a single test temperature using identical specimens. • The engineering lower bound is related to a failure probability lower than 2.5%. • A lower bound fracture toughness can be predicted from a small number of fracture toughness values. • Beyond possible application limits as found in the lower shelf regime, the procedure seems to be reliable in terms of predicting conservative lower bound toughness values. • The procedure can easily be implemented in current test procedures. Acknowledgements K. Erdmann, Dipl-Ing. J. Knaack and O. Kreienbring have conducted a large part of the testing work. K.-H. Balzereit prepared the equipment for measuring the test temperatures. The authors wish to thank the colleagues for their assistance. References [1] ASTM: Draft, test practice (method) for fracture toughness in the transition range, American Society for Testing and Materials, 1996. [2] Wallin K. The scatter in KIc results. Engng Fract Mech 1984;19:1085–93. [3] Wallin K. Statistical aspects of constraint with emphasis to testing and analysis of laboratory specimens in the transition regime. ASTM-Symposium on Constraint Effects in Fracture, ASTM STP, vol. 1171; 1991. p. 264–88. [4] McCabe DE, Merkle JG, Nanstad RK. A perspective on transition temperature and KJc data characterization. ASTM STP, vol. 1207; 1994. p. 215–32. [5] Zerbst U, Heerens J, Pfuff M, Wittkowsky BU, Schwalbe K-H. Engineering estimation of the lower bound toughness in the ductile to brittle transition regime. Fatigue Fract Engng Mater Struct 1998;21:1273–8. [6] Heerens J, Hellmann D. Final Report of the EU-Project MAT-CT-940080, Fracture toughness of steel in the ductile to brittle transition regime, GKSS-Research Center, Germany, 1999. [7] Slatcher S. A probabilistic model for lower-shelf fracture toughness – theory and application. Fatigue Fract Engng Mater Struct 1986;9:275–89. [8] Miyata T, Otsuka A, Katayama T. Probabilistic analysis of cleavage fracture and fracture toughness of steels. J Soc Mater Sci Jpn 1988;37:1191–6. [9] Anderson TL, Stienstra D. A model to predict the sources and magnitude of scatter in toughness data in the transition region. J Test Eval 1989;17:46–53. [10] Heerens J, Zerbst U, Schwalbe K-H. Strategy for characterizing fracture toughness in the ductile to brittle transition regime. Fatigue Fract Engng Mater Struct 1993;16:1213–30.
The lower bound toughness procedure applied to the Euro fracture toughness dataset J. Heerens *, M. Pfuff, D. Hellmann, U. Zerbst Institute of Materials Research, GKSS Research Centre Geesthacht, Max-Planck-Street, D-21502 Geesthacht, Germany Received 7 February 2000; received in revised form 21 July 2000; accepted 23 July 2000
Abstract The validity of a statistical method for estimating an engineering lower bound fracture toughness in the ductileto-brittle transition region is investigated using the Euro fracture toughness dataset generated in the European SM&T Project ‘‘Fracture Toughness of Steel in the Ductile-to-Brittle Transition Regime’’. The lower bound method is based on the empirical evidence that, in the low probability regime, the cumulative failure probability function tends to be a straight line rather than a curve as is the case for Weibull distributions. The investigation demonstrates that the engineering lower bound toughness values as predicted by the method are related to a cumulative cleavage failure probability lower than 2.5%. Such bound predictions can be achieved on the basis of a small number of cleavage toughness values measured at the temperature of interest. The results confirm the validity of the method. 2002 Elsevier Science Ltd. All rights reserved. Keywords: Ductile-to-brittle fracture transition; Lower bound fracture toughness; Failure probability; Pressure vessel steel; Test procedure
1. Introduction In the ductile-to-brittle transition regime, the fracture toughness of ferritic steels exhibits large scatter requiring statistical treatment of the data. In order to avoid the initiation of cleavage fracture in engineering structures, the lower bound regime of the cleavage toughness scatter is of particular practical interest. Substantial efforts have been undertaken in recent years to develop appropriate procedures for modelling the toughness scatter including its lower bound regime. The most advanced procedure has been recently presented by ASTM [1] which is based on the master curve concept [1–3]. In that procedure, the cleavage failure probability of laboratory specimens is modelled using a three parameter Weibull p distribution. This distribution has a ‘‘tail’’ in the low probability regime with a fixed origin of 20 MPa m. This origin is seen as a mathematical fitting parameter but is not necessarily a real physical toughness lower bound [4].
*
Corresponding author. Tel.: +49-4152-87-2610; fax: +49-4152-87-2534. E-mail address: [email protected] (J. Heerens).
0013-7944/02/$ - see front matter 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 0 1 3 - 7 9 4 4 ( 0 1 ) 0 0 0 6 9 - 8
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J. Heerens et al. / Engineering Fracture Mechanics 69 (2002) 483–495
Nomenclature b Jc Jc JLB Jmax Jmin J0 , K0 m p Pf b a0 B W T rY N n
ligament length J -integral related to unstable fracture mean value of Jc lower bound fracture toughness predicted by the procedure maximum Jc value due to the size criterion lowest Jc value of a dataset scale parameter of the Weibull distribution shape parameter of the Weibull distribution fraction of data which is rejected by the specimen size criterion cumulative failure probability correction factor for rejected toughness values initial crack length specimen thickness specimen width test temperature yield strength total number of data points of a dataset number of valid data points of a dataset
In Ref. [5] a statistical method for deriving an engineering lower bound toughness value was proposed. This lower bound is assessed using a modified Weibull distribution. The modified distribution has no tail in the low failure probability regime as it is the case for the conventional Weibull distribution. Based on empirical evidence, a straight line fit is used for modelling the low failure probability regime and the lower bound is formally obtained by extrapolating the straight line down to a failure probability of ‘‘zero’’. The method has the advantage of an easy application and can be used to estimate a toughness lower bound value of single temperature cleavage fracture toughness datasets. In Ref. [5] the method was applied to several types of ferritic steels. The results were promising. Nevertheless, the practical experience with this method is still limited and further validation of the method is necessary. For statistical reasons the experimental validation of a lower bound toughness assessment method requires a large experimental data base. Recently the Euro fracture toughness dataset was developed in a European Project [6]. It consists of about 800 fracture toughness tests performed on standardised compact specimens at different temperatures in the transition regime. Due to the large number of fracture toughness tests the dataset is seen to be ideal for validating statistical methods. In the present paper, the recent development on the lower bound method is presented and the validity of the method is illustrated on the basis of the Euro fracture toughness dataset.
2. The lower bound toughness method The procedure for assessing engineering lower bound toughness values is outlined in detail in Ref. [5]. In this section, the method will be briefly explained and some background information on its development will be given.
J. Heerens et al. / Engineering Fracture Mechanics 69 (2002) 483–495
485
Fig. 1. Principle of the lower bound procedure.
If the fracture resistance is expressed in values of Jc , then the probability of failure may be described by a two parameter Weibull distribution: m Jc Pf ¼ 1 exp ð1Þ J0 It is common practice to fix the shape parameter m to have a value of 2 [7–10]. The only unknown quantity is the scale parameter, J0 , which has to be derived from a toughness dataset. As mentioned before and illustrated in Fig. 1 there is empirical evidence that the lower tail of the failure probability curve tends to follow a straight line. Therefore, a linear fit is used to model the scatter in the low probability regime. An engineering lower bound, JLB , is obtained by formally extrapolating the straight line down to a failure probability of ‘‘zero’’. The distribution function referring to this assumption is then given by 8 pffiffiffiffiffi > Pf 6 0:5 < Jln0 2 ðJc JLB Þ; 2 Pf ¼ ð2Þ J > ; Pf P 0:5 : 1 exp J0c p with the lower bound toughness, JLB . By noting that Jc ¼ J0 ln 2 for Pf ¼ 0:5 one may easily verify that Pf and its derivative are continuous with respect to Jc if JLB is related to J0 by JLB ¼
ln 2 0:5 pffiffiffiffiffiffiffiffi J0 ¼ 0:23 J0 ln 2
ð3Þ
Using Eqs. (2) and (3) the scale parameter J0 may be related to the mean value of Jc , Jc , leading to Jc ¼ 0:89 J0
ð4Þ
As a result, the lower bound toughness is obtained from the mean value by JLB ¼ 0:26 Jc
ð5Þ
It is known that increasing plastic deformation in the specimen leads to loss of constraint. In order to quantify this, the specimen size criterion
486
J. Heerens et al. / Engineering Fracture Mechanics 69 (2002) 483–495
Jmax ¼
1 brY 30
ð6Þ
is introduced. J -values greater than Jmax are considered as ‘‘invalid data’’ or ‘‘censored data’’. This is done because loss of constraint generally increases the fracture toughness. The distribution function Eq. (2) does not account for this effect. By this reason invalid data have to be rejected and must not to be used to determine Jc . Therefore, Jc is the arithmetical mean value of the valid toughness values, Jc , in a dataset. Note, a rejection of data points from a given dataset is shifting the lower bound prediction to the conservative side (low toughness side). In order to compensate for this shifting effect a correction term b is introduced JLB ¼ 0:26 b Jc
ð7Þ
with b ¼ 1 þ 1:286 p where p is the fraction of data which is rejected by the size criterion, Eq. (6). In Ref. [5] it is stated that the lower bound procedure shall not be applied to datasets with p larger than 70%. For the application of the method, it is of particular interest how the number of toughness data points influences the lower bound toughness prediction. This subject was investigated using computer simulations performed as follows: In Fig. 2 the lower bound prediction is shown on the basis of a cleavage toughness dataset of a pressure vessel steel. The entire dataset contains 16 data points and the predicted lower bound is 29.6 N/mm. In order to investigate the influence of the number of data points, samples of 3 and 6 data points were randomly selected from the entire dataset and the lower bound was determined for each sample. This was done 10,000 times for each sample size. In Fig. 3 the histograms of the predicted lower bounds are shown. It can be seen that the distribution is nearly normal. Similar simulations were also performed by selecting randomly data points from the modified Weibull distribution as defined by Eq. (2). The computer simulation indicated that the predicted lower bounds are approximately normally distributed and their standard deviation can be assessed as follows: p r 0:13 Jc = n ð8Þ
Fig. 2. Lower bound procedure applied to a toughness dataset.
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487
Fig. 3. Histogram of lower bound predictions using different sample sizes taken from the dataset in Fig. 2.
Fig. 4. Influence of sample size on the accuracy of the lower bound prediction.
where n is the number of valid (with respect to Eq. (6)) data points. Eq. (8) reveals that relatively good lower bound predictions can be achieved on the basis of small datasets as is shown graphically in Fig. 4.
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3. Results and discussion The Euro fracture toughness dataset provides the fracture toughness of the quenched and tempered pressure vessel steel DIN 22NiMoCr37 in the ductile-to-brittle transition regime. Overall, the Euro dataset provides 27 individual toughness datasets related to 1=2T, 1T, 2T and 4T C(T) specimen which were tested at eight different temperatures [6]. The test temperatures cover the entire range from the lower shelf to the upper shelf. The individual toughness datasets contain about 30 data points. According to the data validity criterion, (Eq. (6)), it was found that four of the 27 toughness datasets do not contain any valid data points due to the fact that cleavage fracture of all specimens occurred at J > Jmax . These four datasets are from 1=2T and 1T C(T) specimens tested at 0C and 1T and 2T C(T) specimens tested at 20C. All remaining datasets exhibit valid toughness data and were used to validate the engineering lower bound method. The main results of the analysis are listed in Table 1, showing the following quantities for all individual datasets: the predicted engineering lower bound toughness, JLB , the standard deviation, the ‘‘lowest measured toughness value’’, Jmin , the total number of data points, N , and the number of valid data points. In Fig. 5 the JLB -values and the corresponding lowest measured toughness values, Jmin , are compared graphically. The error bars in the diagram indicate the standard deviation of the predicted lower bounds by 2 r. It is seen that for 12 out of 23 cases the lower bound predictions are very close to the lowest measured toughness values. This is demonstrated by the fact that the error bars overlap the 1 to 1-line. In all remaining cases the lower bound prediction was found to be on the conservative side which means that the predicted lower bounds are lower than the lowest measured toughness values. In general cleavage fracture
Table 1 Results of the lower bound analysis T (C)
CT-specimen size (mm) (and its abbreviation)
154 154 154 110 91 91 91 91 60 60 60 60 40 40 40 20 20 20 20 20 0 0 þ20
W W W W W W W W W W W W W W W W W W W W W W W
a
¼ 25, B ¼ 12:5; (1=2T) ¼ 50, B ¼ 25; (1T) ¼ 100, B ¼ 50; (2T) ¼ 25, B ¼ 12:5 ¼ 25, B ¼ 12:5 ¼ 50, B ¼ 25 ¼ 100, B ¼ 50 ¼ 200, B ¼ 100; (4T) ¼ 25, B ¼ 12:5 ¼ 25, B ¼ 12:5a ¼ 50, B ¼ 25 ¼ 100, B ¼ 50 ¼ 25, B ¼ 12:5 ¼ 50, B ¼ 25 ¼ 100, B ¼ 50 ¼ 25, B ¼ 12:5 ¼ 50, B ¼ 25 ¼ 50, B ¼ 25b ¼ 100, B ¼ 50 ¼ 200, B ¼ 100 ¼ 100, B ¼ 50 ¼ 200, B ¼ 100 ¼ 200, B ¼ 100
Jmin (N/mm) (Experimental)
JLB 2r (N/mm) (Predicted)
N
Valid data points (Jc 6 Jmax )
2.8 3.4 3.8 8.1 19.9 14.9 19.6 13.8 34.0 38.4 46.7 30.4 61.6 49.1 40.0 78.9 139.0 102.0 58.6 112.0 174.3 126.5 405.0
1.87 0.34 1.77 0.30 1.59 0.30 7.90 1.06 15.27 2.74 13.22 2.74 10.08 1.84 8.29 2.14 27.13 4.50 41.1 6.31 26.02 4.46 36.92 6.74 67.83 13.65 54.47 9.20 33.80 6.17 52.01 16.69 113.06 18.79 89.88 17.94 63.84 11.37 45.76 11.82 150.18 23.46 107.95 26.99 128.23 77.40
31 34 30 55 31 34 30 15 31 31 34 30 30 32 30 31 30 20 30 15 30 16 15
31 34 30 55 31 34 30 15 29 20 34 30 6c 30 30 2c 11 8 29 15 16 16 2c
Specimens machined from 2TCT-specimens. Side grooved. c Dataset with p > 70%, all specimens: a0 =W 0:55. b
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Fig. 5. Comparison between predicted lower bound values and the lowest measured toughness values.
Fig. 6. Prediction of lower bound toughness in the lower shelf regime at: (a) 154C, specimen size W ¼ 100 mm, (b) 154C, specimen size W ¼ 50 mm, (c) 154C, specimen size W ¼ 25 mm.
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Fig. 7. Prediction of the lower bound toughness at 110C, specimen size W ¼ 25 mm.
Fig. 8. Prediction of the lower bound toughness at: (a) 91C, specimen size W ¼ 200 mm, (b) 91C, specimen size W ¼ 100 mm, (c) 91C, specimen size W ¼ 50 mm, (d) 91C, specimen size W ¼ 25 mm.
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Fig. 9. Prediction of the lower bound toughness at: (a) 60C, specimen size W ¼ 100 mm, (b) 60C, specimen size W ¼ 50 mm, (c) 60C, specimen size W ¼ 25 mm (machined form W ¼ 100 mm), (d) 60C, specimen size W ¼ 25 mm.
toughness datasets are specimen size dependent. This is in particular the case in the lower transition regime and in the upper transition regime. An increase of the specimen thickness typically reduces the mean J -value of a toughness dataset. In such cases, also the predicted lower bounds will show a similar size effect trend. This can be seen by comparing the JLB -values of different specimen sizes obtained at the each individual test temperatures, see Table 1. In Fig. 5 the lower bound prediction is compared with one single data point representing the lowest measured toughness value of the dataset. Let us now consider how the modified Weibull distribution fits with the entire toughness set. For this purpose all toughness datasets were converted into cumulative probability plots. This is done by ranking the toughness values and by calculating a failure probability for each individual data point as follows: Pfi ¼ ði 0:3Þ=ðN þ 0:4Þ;
16i6N
ð9Þ
where N is the total number of data points of the toughness dataset. Note, for statistical reasons, the invalid toughness values (Jc P Jmax ) have to be included when calculating N . Figs. 6–13 compare the experimentally derived cumulative probability functions (shown by the points) with the modified Weibull distribution, Eq. (2) (shown by the solid curve). The open symbols indicate the invalid data points according to Eq. (6). In Ref. [5] it is stated that the procedure shall not be applied to
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Fig. 10. Prediction of the lower bound toughness at: (a) 40C, specimen size W ¼ 100 mm, (b) 40C, specimen size W ¼ 50 mm, (c) 40C, specimen size W ¼ 25 mm.
toughness datasets which have more than 70% invalid data points. In Figs. 6–13 this limit is given by the dotted line at Pf ¼ 0:3. Considering all 23 datasets it is seen that the modified Weibull distribution fits quite well to the valid experimental datasets with six exceptions: Poor fits are observed for all three datasets measured at the lower shelf temperature T ¼ 154C, see Fig. 6a–c. At 154C the slope of the experimental curve is much steeper than the theoretical function (Eq. (2)). In addition, it is found that for all three datasets the predicted lower bound is significantly smaller than the lowest measured toughness value. This indicates that the modified Weibull distribution is not ideal for modelling lower shelf toughness scatter. There are three more cases where the slope of the theoretical function seems to be too small, see Figs. 8b, 10c and 12b. Note, in all six cases where the fit (Eq. (2)) was found to be poor, the predicted lower bounds remain conservative. This is important for practical applications and seems to indicate an inherent safety of the procedure. In the procedure, the engineering lower bound value is obtained by linear extrapolation of the modified Weibull distribution (Eq. (2)) down to a failure probability of ‘‘zero’’. If this approach is fully valid, no specimen should fail below such a lower bound value. Clearly, a full justification of this approach is impossible because it would require testing an infinite number of specimens to be tested. Nevertheless, let us consider to what extent the lower bound assumption is validated by the results of the present paper. The majority of the datasets investigated here consist of approximately 30 toughness values. The largest dataset (at 110C) has 55 data points. From Eq. (9) it can be easily assessed that the lowest toughness value of
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Fig. 11. Prediction of the lower bound toughness at: (a) 20C, specimen size W ¼ 200 mm, (b) 20C, specimen size W ¼ 100 mm, (c) 20C, specimen size W ¼ 50 mm, (d) 20C, specimen size W ¼ 50 mm (side grooved), (e) 20C, specimen size W ¼ 25 mm.
such a dataset is related to a failure probability of the order of 2.3% (N ¼ 30) and 1.3% (N ¼ 55). The results of the analysis show that for all 23 toughness datasets the lowest measured toughness value was found to be similar to, or even larger than, the predicted engineering lower bound. This finding seems to
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Fig. 12. Prediction of the lower bound toughness at: (a) 0C, specimen size W ¼ 200 mm, (b) 0C, specimen size W ¼ 100 mm.
Fig. 13. Prediction of the lower bound toughness at 20C, specimen size W ¼ 200 mm.
indicate that the engineering lower bounds as they are predicted by the method are related to a failure probability level which is similar to or smaller than 2.5%. This must be kept in mind when engineering lower bounds as predicted by this method are used in practical applications.
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4. Conclusions An analysis of the Euro fracture toughness dataset confirms the validity and applicability of the engineering lower bound toughness prediction procedure. In particular, the investigations reveal that the procedure has the following characteristics which are important for its practical application and possible standardisation: • The procedure predicts an engineering lower bound fracture toughness and its standard deviation from a cleavage toughness scatter band measured at a single test temperature using identical specimens. • The engineering lower bound is related to a failure probability lower than 2.5%. • A lower bound fracture toughness can be predicted from a small number of fracture toughness values. • Beyond possible application limits as found in the lower shelf regime, the procedure seems to be reliable in terms of predicting conservative lower bound toughness values. • The procedure can easily be implemented in current test procedures. Acknowledgements K. Erdmann, Dipl-Ing. J. Knaack and O. Kreienbring have conducted a large part of the testing work. K.-H. Balzereit prepared the equipment for measuring the test temperatures. The authors wish to thank the colleagues for their assistance. References [1] ASTM: Draft, test practice (method) for fracture toughness in the transition range, American Society for Testing and Materials, 1996. [2] Wallin K. The scatter in KIc results. Engng Fract Mech 1984;19:1085–93. [3] Wallin K. Statistical aspects of constraint with emphasis to testing and analysis of laboratory specimens in the transition regime. ASTM-Symposium on Constraint Effects in Fracture, ASTM STP, vol. 1171; 1991. p. 264–88. [4] McCabe DE, Merkle JG, Nanstad RK. A perspective on transition temperature and KJc data characterization. ASTM STP, vol. 1207; 1994. p. 215–32. [5] Zerbst U, Heerens J, Pfuff M, Wittkowsky BU, Schwalbe K-H. Engineering estimation of the lower bound toughness in the ductile to brittle transition regime. Fatigue Fract Engng Mater Struct 1998;21:1273–8. [6] Heerens J, Hellmann D. Final Report of the EU-Project MAT-CT-940080, Fracture toughness of steel in the ductile to brittle transition regime, GKSS-Research Center, Germany, 1999. [7] Slatcher S. A probabilistic model for lower-shelf fracture toughness – theory and application. Fatigue Fract Engng Mater Struct 1986;9:275–89. [8] Miyata T, Otsuka A, Katayama T. Probabilistic analysis of cleavage fracture and fracture toughness of steels. J Soc Mater Sci Jpn 1988;37:1191–6. [9] Anderson TL, Stienstra D. A model to predict the sources and magnitude of scatter in toughness data in the transition region. J Test Eval 1989;17:46–53. [10] Heerens J, Zerbst U, Schwalbe K-H. Strategy for characterizing fracture toughness in the ductile to brittle transition regime. Fatigue Fract Engng Mater Struct 1993;16:1213–30.