Fitting small data sets in the lower ductile-to-brittle transition region and lower shelf of ferritic steels

Engineering Fracture Mechanics 98 (2013) 350–364

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Fitting small data sets in the lower ductile-to-brittle transition region and lower shelf of ferritic steels M.R. Wenman ⇑ Department of Materials, Imperial College London, London SW7 2AZ, United Kingdom

a r t i c l e

i n f o

Article history: Received 6 April 2011 Received in revised form 28 September 2012 Accepted 21 November 2012

Keywords: Weibull Master Curve Minimum toughness Steel Finite element

a b s t r a c t The use of a 3-parameter Master Curve approach where the Weibull modulus is 4 and a minimum toughness value of 20 MPa m1/2 is now well accepted and standardised. The standard allows the use of a minimum of six specimens, tested at a single temperature, to produce the Master Curve. However, the examination of a small data set can lead to difficulties when trying to fit to this standard formula especially if the toughness is larger than 20 MPa m1/2. A small data set obtained from lower shelf testing of a ferritic pressure vessel steel is used to explore some of the issues related to the Weibull modulus and minimum toughness and how a user might try to fit such data. Finite element models are used to show that enhanced minimum toughness values larger than 20 MPa m1/2 are unlikely to have arisen from warm pre-stressing. The implications for the Master Curve of the different fitting options are also briefly explored. Ó 2012 Elsevier Ltd. All rights reserved.

1. Introduction The introduction of the Master Curve concept for transition region and lower shelf brittle fracture of ferritic steels is now well accepted, and used to fit toughness data to steels, where only a small data set is available and this has been standardised in ASTM 1921 E [1]. Six specimens, tested at a single temperature, is the accepted minimum number needed to produce the Master Curve. Whilst it is accepted that a correction may need to be made for lower shelf toughness [2,3], the Master Curve approach is expected to fit data sets for all ferritic steels well. However, for small data sets, the well-known cumulative probability density function (CDF) for a 3-parameter Weibull probability of failure relationship, given in Eq. (1), does on occasion appear to present problems to the user. In this paper the values of the minimum toughness of 20 MPa m1/2 and the Weibull slope are explored for one small data set of a nuclear reactor pressure vessel forging steel (SA 508 class 3) tested at 155 °C to ensure brittle failure.

  4 K I  K min Pf ¼ 1  exp  K 0  K min

ð1Þ

where Pf is the probability of failure, KI is the mode I specimen fracture toughness, K0 is a scaling parameter, which is equal to a 0.632 probability of failure for a 2-parameter Weibull. In Weibull’s original 3-parameter distribution, K0 + Kmin would be equal to a 0.632 probability of failure where Kmin is the minimum toughness value. It should be noted that Wallin’s 3-parameter Weibull distribution differs from the standard form of a 3-parameter Weibull distribution often used in ceramics communities due to the additional Kmin term in the denominator to return K0 to the K value at 0.632 probability of failure, as it is in the 2-parameter form. ⇑ Tel.: +44 (0) 207 594 6763. E-mail address: [email protected] 0013-7944/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.engfracmech.2012.11.016

M.R. Wenman / Engineering Fracture Mechanics 98 (2013) 350–364

351

Nomenclature a0 B CDF E i K0 K0(1T) KI Kmin KJc KJc(med) KJ(i) m PDF Pf N S T T0 W

m

initial crack length specimen thickness cumulative (probability) distribution function Young’s modulus ith rank scaling parameter equal to the toughness at a 0.632 probability of failure for a 2-parameter Weibull distribution K0 scaled to a specimen thickness of a one inch thick compact tension specimen mode I specimen fracture toughness or mode I stress intensity factor minimum fracture toughness value measured critical fracture toughness value at the onset of cleavage calculated from the J integral value median value of the measured critical toughness value ith toughness value Weibull modulus (a measure of the distribution width) probability density function probability of failure number of data points in sample span between outer rollers for a 3 point bend specimen test test temperature the reference temperature used to locate the Master Curve on the temperature axis specimen width Poisson’s ratio

In turn this means the probability density function (PDF) for the 3-parameter Weibull used by Wallin is also modified from the standard form and is given in the following equation where m = 4.

 m1   m  m K I  K min K I  K min  exp  ðK 0  K min Þ K 0  K min K 0  K min

ð2Þ

Wallin’s paper in 1984 [4] set out the use of this minimum toughness value and the theoretical value of the Weibull modulus of 4, for a pre-fatigue cracked steel specimen. The minimum toughness value was introduced as it is not unreasonable to assume there must be a minimum toughness, below which failure will not occur, and it is proposed this could have a range of values from 10 to 35 MPa m1/2 for steels. The lack of a minimum toughness was a flaw with the 2-parameter Weibull for fitting to toughness data. However, as pointed out by Knott [5,6] the introduction of a minimum value directly into the Weibull distribution changes many of the important parameters. E.g. the Weibull modulus fitted to a 2 and 3 parameter Weibull distribution for the same data set have different Weibull moduli. Other authors have also studied whether Weibull is the correct distribution for modelling the failure of brittle materials containing flaws [7–9]. Todinov [7] concludes that Weibull does not have a physical basis for describing brittle fracture and gives a theoretical proof of this and then further suggests a new method of assessing the failure of materials [8]. As for the theoretical Weibull modulus of 4 predicted by Wallin the theory used was that according to McMeeking’s analysis [10], of a blunting sharp crack, that the effective microstructural volume taking part in a fracture, Veff, was related to the specimen thickness, B, angular stress field dependence and the distance ahead of the crack tip, Xeff through Eq. (3).

V eff  B  X eff  X eff sin h and that because of the linear dependence between Xeff and

V eff ¼ a 

K 4I

ð3Þ K 2I

this leads to the volume dependence of

ð4Þ

where a is a constant. i.e. that theoretically the Weibull modulus for the sharp cracked case should be 4. Wallin went onto show two other ways in which the Weibull modulus was equal to 4. The first was using the local approach of Pineau [11] and the second was via the WST model by Wallin et al. [12]. Anderson et al. [13] have also published an account of where the m equal to 4 comes from. They show that m equal to 4 arises from weakest link theory for a 2-parameter Weibull distribution under J controlled fracture. However, they note that this would not hold for a 3-parameter Weibull distribution, which should have no theoretical basis for m to be fixed at 4. A Weibull modulus, m, equal to 4 was tested by Wallin by doing numerous theoretical tests by using an inverse transformation method in the interval 0–1 to generate values from a Weibull distribution according to Eq. (6), calculating the theoretical scatter for different numbers of tests and plotting these against real experimental results. This showed the tendency for the value of Weibull modulus to be scattered about 4, tending to 4, for large numbers of tests, and often substantially higher than 4 for small data sets but largely within the predicted theoretical scatter limits. Akbarzadeh et al. [14] have shown

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that Weibull modulus is not a material parameter but dependent on geometry and temperature. Millella and Bonora [15] showed that on tests of A508 class 3 steel m varied between 6 and 40, depending on notch depth and root radius. Although they did also show results of m = 4 for sharp cracked specimens matching the theoretical treatment of Wallin. Gerguri et al. [16] showed that for notched graphite bars the Weibull modulus had a value of 29 and that for unnotched bars it was 10 and indeed it is widely excepted that Weibull modulus is only a material parameter in the sense that it describes the material and process route to produce it. The value of Kmin for a 3 parameter Weibull is somewhat arbitrary as it cannot truly, of course, be measured by any reasonable means and Wallin and ASTM 1921 E suggest a value around 20 MPa m1/2. Anderson et al. [13] state that the reason for a value of 20 MPa m1/2 was due to the warm pre-stress effect introducing a compressive residual stress to the crack tip during fatigue pre-crack growth. Hence the minimum toughness would need to be greater than the final Kmax applied during pre-crack growth to overcome it. Scibetta et al. [17] reviewed the effect of pre-cracking on the fracture toughness of a well characterised forging steel. They showed that the pre-cracking Kmax could affect KIC and this was more likely for the lower tail of the fracture toughness curve. They also noted that the pre-cracking procedure must not alter the fracture process zone microstructure. Finally, they suggest, as described by Anderson et al. [13], that there is a limiting distance ahead of the crack tip of 7.5 lm where a microcrack cannot propagate and that Kmax should be limited according to this distance. Using McMeeking’s [10] analysis of the stress field ahead of the crack this distance of 7.5 lm equates to a final pre-crack load of about 20 MPa m1/2. Other authors [18–20] have studied the warm pre-stress effect on toughness through finite element (FE) modelling. Crack tip blunting was noted as a key feature of increased toughness with residual stress deemed a secondary effect. The Chell et al. [21] and Curry [22] models for the warm pre-stress effect are determined by the crack tip plastic zones sizes. Smith et al. [18] noting that if the plastic zone at the low temperature fracture is larger than the plastic zones created by preloading and unloading the models predict no benefit from warm pre-stressing. This argument is used later to show that the lower shelf toughness measured was not a result of warm pre-stress enhancement. Zhang and Knott [23] have argued that Kmin is likely to be variable between steels with some possibly showing much greater values of Kmin than 20 MPa m1/2 and some lower. Zhang and Knott also successfully showed that extrapolation of a badly fitted distribution down to probabilities of failure of 0.01% could give physically unrealistic results including negative values of toughness as seen in the fitting of the ‘‘Euro’’ toughness data set [3]. They further showed that a mixed microstructure (bainitic/martensitic in their example) was the most likely root cause of this. They pragmatically proposed extrapolation of minimum toughness to low probabilities is best made from the single phase toughness measurements of the lowest toughness phase. The author’s interest in the subject stems from a study of specimens containing residual stress where the measurement of a shift in the apparent toughness was the goal but only six reference specimens, without residual stress, were tested at a single test temperature. This was done as the Master Curve ASTM standard [1] allowed this option. This caused the problem that using the standard Kmin = 20 MPa m1/2 and m = 4 did not appear to fit these data and leaves the user wondering what to do next? Mirzaee-Sisan et al. [24] were similarly faced with this problem and the approach they took was to fit a new value of Kmin of 31 MPa m1/2 and stick to a Weibull slope of 4 but not much was made of their decision process to arrive at this. The aim of this paper is to briefly explore the different possibilities for fitting of the same small set of data and to show that it could actually have come from any number of different distributions. FE models are used to test the influence of a fatigue pre-cracking cycle has on the final toughness. The implications of this for minimum toughness are discussed in relation to the different Weibull fits made.

2. Methodology 2.1. Fracture toughness testing and analysis Six 10 by 10 mm standard pre-cracked SE(B) specimens with a 2 mm deep starter slit notch cut by electro-discharge machining and two standard 1T compact tension C(T) specimens were made from a ring forging of SA508 class 3 steel. Microstructural orientation effects had previously been investigated by performing optical microscopy of the 3 orientations and by hardness measurements. None of the above indicated any observable difference due to orientation and a fine grained tempered bainitic microstructure was observed in all orientations providing a suitably quasi-homogeneous material. Pre-fatigue cracking was carried out at room temperature by applying an R-ratio of 0.1 with a reducing load in four steps. The final load was such that a Kf value of 20 MPa m1/2 was obtained for an a/W ratio of 0.5. This value is somewhat higher than the final Kf value allowed in the most recent Master Curve standard of 15 MPa m1/2. For this reason there was some investigation of effect of crack length and warm pre-stress on the toughness value measured to check if there was any correlation suggesting an effect of compressive crack tip residual stress or crack tip blunting on the final toughness value measured. Pre-fatigue crack lengths were monitored by both the potential difference method and occasional optical crack length measurements of both sides of the specimens by surface replicas. Final crack lengths were all in the range of 0.48–0.55 a0/W with a mean of 0.503. The specimens were tested using a servo-hydraulic machine with a 100 kN load cell and a cold bath cooled with liquid nitrogen to 155 ± 3 °C. For the SE(B) specimens a span of 40 mm was use in 3 point bending. More information on the material and its tensile properties can be found in [25].

M.R. Wenman / Engineering Fracture Mechanics 98 (2013) 350–364

353

All values of Jc were converted to KJc using the standard equation of

K Jc ¼

rffiffiffiffiffiffiffiffiffiffiffiffiffiffi Jc E 1  m2

ð5Þ

2.2. Data fitting A data set of 100 points was artificially generated by the inverse transformation method converting a uniformly distributed number in the interval [0,1] into a random toughness value, from a 3-parameter Weibull, as used by Wallin via Eq. (6).

  1=m 1 K I ¼ K min þ ðK 0  K min Þ ln 1  Pf

ð6Þ

where KI is the stress intensity factor, K0 is the stress intensity factor at 63.2% probability of failure, Pf is the probability of failure and m is the Weibull modulus (equal to 4 according to Wallin’s work). These Pf values can then be ranked using various methods. Here the median rank method was used according to Eq. (7) where i is the ith rank and N is the number of data points in the sample.

i  0:3 N þ 0:4

ð7Þ

The simulated 100 data point set and the experimental data were tested for various fits of 2-parameter and 3-parameter Weibull distributions using a non-linear Levenburg–Marquardt method. The data was also tested for normality by plotting onto normal probability paper and a least squares best fit was made to both sets of data. Data was then used to produce the Master Curve using the formulae provided in the ASTM standard [1] but for generalised values of m and Kmin. K0 is calculated from the following equation:

" K0 ¼

 m #1=m n X K JðiÞ  K min þ K min N i¼1

ð8Þ

where 20 and 4 have been replaced by the generalised form of Kmin and m respectively. The K0 values are then used to obtain the estimate of KJc(med) from

K JcðmedÞ ¼ K min þ ðK 0ð1TÞ  K min Þðln 2Þ1=m

ð9Þ

The general equation for the Master Curve and its tolerance bounds is then given by

  1=m 1 K Jcð0xxÞ ¼ K min þ ln f11 þ 77 exp ½0:019ðT  T 0 Þg 1  0  xx

ð10Þ

where (0xx) is the probability of interest i.e. 0.50 for the Master Curve itself and 0.05 and 0.95 for the 5% and 95% tolerance bounds respectively. This gives the recognised form of the Master Curve of

K JcðmedÞ ¼ 30 þ 70 exp ½0:019ðT  T 0 Þ

ð11Þ

where 30 and 70 are constants calculated from Eq. (10) and referred to in more general terms as A and B from now on. For the Master Curve A + B is equal to 100 and for K0 A + B = 108. The curve shape was derived in [26] and verified for irradiated material in [27]. T0 is the reference temperature used to locate the Master Curve on the temperature axis i.e. the test temperature at which the median KJc value from 1T specimens is 100 MPa m1/2. It is calculated, for single temperature tests from

T0 ¼ T 



   K JcðmedÞ  A 1 ln 0:019 B

ð12Þ

where T is the single test temperature of the experimental data. 2.3. Finite element modelling Abaqus was used to create a series of plane strain 2D FE models of the SE(B) specimen geometry, of a single pre-cracking load-unload cycle (assuming a final crack depth of 5 mm), followed by cooling to 155 °C and then reloading to a typical fracture load. The models contained plastic data at room temperature given in [25] combined with low temperature data from a similar steel from [28]. The yield stress and UTS values with temperature are given in Table 1. This was provided to Abaqus for linear kinematic hardening plasticity. (Note it is not possible to use combined hardening with temperature–displacement elements in Abaqus Standard). The Young’s modulus and Poisson’s ratio were modelled as 200 GPa and 0.3 respectively. The mesh was made of quadrilateral 2D plane strain coupled temperature–displacement elements (CPE4T) and was concentrated to a crack tip element size of 0.3 lm. A blunted crack tip radius of 10 lm was used. The model

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Table 1 Experimental tensile data used for the linear kinematic hardening model for FE modelling. Yield stress (MPa)

UTS (MPa)

Plastic strain at UTS

Temperature (°C)

875 700 625 600 500 485

1142 1040 965 940 840 825

0.2 0.3 0.3 0.325 0.34 0.345

196 180 160 120 80 20

used symmetry on the x-plane, along the crack ligament, so that only half the 3 point bend specimen was modelled for computational efficiency. Fig. 1 shows the mesh and overall model setup. Specimens were pre-loaded to maximum load of 940 N at 20 °C, which is half the calculated load to achieve an applied KI value of 20 MPa m1/2 using Eq. (13).

K¼

PS BW

3=2

f

a

0

ð13Þ

W

where P is the load applied, S is the span between the outer rollers, B is the specimen thickness and W is the specimen width. The compliance function f(a0/W) is defined in the ASTM standard [1]. The load was applied to the reference point of the central roller, whilst the top roller was fixed, and ramped according to a sinusoidal curve to the maximum force and then the load was fully removed. The specimen was then cooled to 155 °C before reloading to a final load of 2500 N (equivalent to a KI value of around 54 MPa m1/2 and the maximum fracture load of any of the eight specimens tested). The key objective of this was to show whether there was likely to be an enhanced toughness value on the lower shelf as a result of warm pre-stressing as the final Kmax used in pre-cracking was 20 MPa m1/2, which could have influenced the final measured toughness. This was assessed by measuring the plastic zone sizes after pre-loading to 20 MPa m1/2 at room temperature and the plastic zone size at the fracture load and temperature. Several other crack tip radii of 1, 2 and 5 lm were also assessed and 10 lm was selected. The maximum crack tip plastic strain after warm pre-load, for the 10 lm crack tip radius, was 0.28. All smaller crack tip radii tested showed much larger plastic strains suggesting further blunting would have occurred. 3. Results and discussion 3.1. Finite element modelling Fig. 2 shows that there is no correlation between the fatigue pre-crack length and the final toughness value measured for the SE(B) specimens, which gives some confidence that pre-cracking load has not unduly influenced the final toughness

fixed

x-symm y x

load

Fig. 1. Geometry, boundary conditions and mesh detail of the 2D plane strain FE model of an SE(B) specimen.

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M.R. Wenman / Engineering Fracture Mechanics 98 (2013) 350–364

54

KJc (MPa m1/2)

53 52 51 50 49 48 47 4.8

4.9

5.0

5.1

5.2

5.3

5.4

5.5

5.6

Pre-crack length (mm) Fig. 2. Pre-crack length versus measured KJc for the SE(B) specimens showing no noticeable correlation between crack length and fracture toughness obtained.

value measured. Fig. 3 shows the result from the plane strain FE model giving the plastic zone sizes on loading a crack, with an initial blunted root radius of 10 lm, to a KI value of 20 MPa m1/2 (based on the half-load applied of 940 N) and then on reloading at 155 °C to a typical fracture load. The plastic zone size was measured as 179 lm in its longest dimension and 110 lm across, perpendicular to that direction, after pre-loading at room temperature (see, Fig. 3a). On cooling the plastic zone remains the same size and it begins to increase in size on reloading at a load of 1273 N (K equivalent to 27.1 MPa m1/2). It does not start to increase in size at a KI value of 20 MPa m1/2 due to the increase in yield stress at 155 °C compared to room temperature. However, at the final load of approximately 2500 N (KI equivalent of 54 MPa m1/2) the plastic zone dimensions are much larger having measurements of 1116 lm in the longest direction and 560 lm perpendicular to it. It is clear that the plastic zone size increases by many times on reloading at low temperature and hence there is unlikely to be a warm pre-stress effect on the lower shelf toughness despite the fatigue pre-cracking load being larger than allowed in the standard. This is a significant result when considering the genuine value of Kmin for this steel later. 3.2. Fracture toughness testing and analysis The results of the toughness data are given in Table 2. and shown in Fig. 4 fitted with a simple 2-parameter Weibull distribution giving m = 22.0 and K0 = 51.2 MPa m1/2. Fig. 5 shows the same data fitted by linear least squares regression, which gives similar values of m = 23.3 and K0 = 51.3 MPa m1/2. Fig. 6 shows the data fitted to 3 different distributions. First it is clear that for this sparse data set it does not appear to fit a value of m = 4 and Kmin = 20 MPa m1/2 (black solid line). The red dashed curve shows the result of fitting a distribution where Kmin is fixed at a value of 20 MPa m1/2 and m is fitted by a non-linear least squares algorithm giving an R2 value of 0.975. This predicts an m value of 13.3. The blue dot-dashed curve shows the result of holding the Weibull modulus, m, fixed at a value of 4 and fitting, by the same method, the Kmin value of 41.3 MPa m1/ 2 is obtained and an R2 value of 0.982. Both these curves clearly give a good fit to the data. Fig. 7 shows the two distributions, predicted from their probability density functions, plotted together with the 2-parameter Weibull and the distribution of Wallin (Kmin = 20 MPa m1/2, m = 4). Wallin’s predicted distribution is clearly much broader than the other 3 fits to the data,

Crack tip Crack tip y x

x = 179 µm

(a)

y = 110 µm

(b)

Fig. 3. Plots of equivalent plastic strain from the FE model showing the size of the crack tip plastic zone after a single cycle (a) pre-loading to 940 N (20 MPa m1/2) at room temperature and (b) after unloading and cooling to 155 °C and then reloading to 2500 N (54 MPa m1/2).

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Table 2 Experimental fracture toughness test data. Charpy-sized SE(B) specimens with toughness of less than 50 i.e. specimen 6 was not size corrected to 1T all others were. Specimen no.

Specimen type

Test temperature (°C)

KJc (MPa m1/2)

K(1T) (MPa m1/2)

Ductile crack growth (lm)

1 2 3 4 5 6 7 8

SE(B) SE(B) SE(B) SE(B) SE(B) SE(B) CT(1T) CT(1T)

155 155 155 155 155 155 155 155

57.5 62.4 59.1 57.2 61.4 48.1 46.5 50.6

49.7 53.6 51.0 49.4 52.8 48.1 46.5 50.6

0 0 0 0 0 0 0 0

Test Data 2 Parameter Weibull m = 22.0, K 0 = 51.2 1.0

0.8

Pf

0.6

0.4

0.2

0.0 46

48

50

52

54

KJc (MPa m1/2) Fig. 4. Non-linear regression fit to experimental fracture toughness data of a 2-parameter Weibull distribution.

2

Ln[ln(1/(1-Pf))]

1

m = 23.3 r 2 = 0.975 K0 = 51.3

0

-1

-2

-3 3.82 3.84 3.86 3.88 3.90 3.92 3.94 3.96 3.98 4.00

ln (K) Fig. 5. Linear least squares regression fit to the experimental fracture toughness data of a 2-parameter Weibull distribution.

which despite a variation in Weibull modulus of between 4 and 22 are very similar in width. In fact the 2-parameter Weibull fit and the 3-parameter fit where m is 13.3 appear almost identical. There is a small but noted difference in the lower tail of the 3-parameter fit where Kmin = 41.3 MPa m1/2. This last fit predicts a toughness of 42.3 at a failure probability of 104. Fig. 8 shows the result of generating, as Wallin did, a distribution which fits m = 4 and Kmin = 20 MPa m1/2 for 100 data points. This was then non-linear regression fitted to a 3-parameter Weibull distribution to give m = 4, K0 = 47.9 MPa m1/2 and Kmin = 20.8 MPa m1/2 to check this has successfully produced the correct distribution. On the same plot are the eight experimental data points and one of the 3-parameter fits for comparison. Table 3 gives the eight data points obtained from

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M.R. Wenman / Engineering Fracture Mechanics 98 (2013) 350–364

Data K min = 20, m = 4, K0 = 52.8 K min = 20, m = 13.3, K 0 = 51.2 K min = 41.3, m = 4, K 0 = 51.1 1.0

0.8

Pf

0.6

0.4

0.2

0.0 44

46

48

50

52

54

56

KJc (MPa m1/2) Fig. 6. Non-linear regression fits to the experimental fracture toughness data of 3-parameter Weibull distributions, including where m and Kmin are fixed according to the Wallin model and where either Kmin or m are allowed to vary.

2 P-Weibull, m =22, K0 =51.3 Kmin = 20, m = 4, K 0 = 52.8 Kmin = 20, m = 13.3, K0 = 51.2 Kmin = 41.3, m = 4, K 0 = 51.1 0.18 0.025

0.16

0.020

Pf

0.12

0.015

Pf

0.14

0.10 0.08

0.010 0.005 0.000 36

38

40

42

44

K (MPa m1/2 )

0.06 0.04 0.02 0.00 30

35

40

45

50

55

60

KJc (MPa m1/2) Fig. 7. Plots of the probability density functions shown in Figs. 6 and 4. It is clear that all show a near normal (bell-shaped) form and that the 2-parameter and the two fitted 3-parameter distributions are very similar whilst the m = 4 and Kmin = 20 PDF is a much broader distribution.

testing the A508 forging material together with the median rank failure probability. The question is could these data have come from a distribution of m = 4, Kmin = 20 MPa m1/2? From the 100 generated points that make up the distribution, which conforms to Wallin’s distribution, this is true if eight equivalent data points can be extracted/sampled, which closely match the values of the eight real test data points. Extracted values are shown in Table 4 alongside both the new and original median rank and rank numbers they possessed in the whole population of 100 points in brackets. Clearly the median ranks are now very different. This is illustrated in Fig. 9 where the eight extracted data points are plotted both with original rank and fitted to a cumulative probability density function with m = 4 and Kmin = 20.8 MPa m1/2 and with the new rank, as if they were the only 8 points available, as in my case for the A508 forging material. From this it appears as though my sample could have indeed come from a distribution conforming to Wallin’s theory where m is always 4 and Kmin is in the region of 20 MPa m1/2. Comparing the fits to the eight sampled data points to a fit of the experimental data (Figs. 8 and 9) shows that for both if m = 4 and K0 and Kmin are fitted the values are near identical. However, what of course is not clear is whether the experimental data did come from this theoretical distribution of Wallin’s.

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100 simulated data points K min = 20.8, m = 4, K 0 = 47.9 Experimental Data Fit K min = 41.3, m = 4, K 0 = 51.1 1.0

0.8

Pf

0.6

0.4

0.2

0.0 30

40

50

KJc (MPa

60

m1/2)

Fig. 8. 100 Data points generated from inverse transformation method to fit a theoretical distribution of m = 4 and Kmin = 20 MPa m1/2. The 3-parameter non-linear fit to the data shows it is close to the theoretical values. Also plotted are the eight experimental fracture toughness data points fitted to a 3parameter Weibull where Kmin is allowed to be fitted and m is fixed.

Table 3 Experimental test data in both as measured and 1T size corrected forms with median rank probability of failure. Specimen Rank No.

Ranked KJc (MPa m1/2)

Ranked K(1T) (MPa m1/2)

Median Rank (Pf)

1 2 3 4 5 6 7 8

46.5 48.1 50.6 57.2 57.5 59.1 61.4 62.4

46.5 48.1 49.4 49.7 50.6 51.0 52.8 53.6

0.083 0.202 0.321 0.440 0.560 0.679 0.798 0.917

Mean St. Dev.

55.3 6.1

50.21 2.32

Table 4 Data extracted from 100 point simulated population with both original rank and sample rank. Specimen rank no. (original rank in brackets)

Ranked K(1T) (MPa m1/2)

Original median rank (Pf) (N = 100)

New median rank (Pf) (N = 8)

1(52) 2(68) 3(73) 4(75) 5(77) 6(78) 7(85) 8(89)

46.6 48.0 49.4 49.8 50.6 50.8 52.7 53.3

0.515 0.674 0.724 0.744 0.764 0.774 0.844 0.884

0.083 0.202 0.321 0.440 0.560 0.679 0.798 0.917

Mean St Dev

50.15 2.23

To further examine the likely distribution from which the experimental data was obtained Fig. 10 shows both the 100 simulated data points and the eight experimental test data points plotted on normal probability paper. The fits are again very good with R2 values of >0.98 for both sets of data confirming that even when m = 4 a Weibull is essentially the same as a normal distribution. The quality of fit to a normal distribution also suggests that this is a single microstructure steel as mixed microstructures usually show poor fits to normal distributions [23,29]. The 100 simulated data points give a mean value of 45.4 MPa m1/2 and a standard deviation of 6.9 MPa m1/2. From the normal fits extrapolation to very low failure probabilities can easily be made. Using the normal fit the generated data has a K value of 19.3 MPa m1/2 and the experimental data a value

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100 simulated data points K min = 20.8, m = 4, K 0 = 47.9 Extracted data points (rank based on N = 8) Extracted data points (original rank N = 100) K min = 41.5, m = 4, K 0 = 51.0 1.0

0.8

Pf

0.6

0.4

0.2

0.0 30

40

50

60

KJc (MPa m1/2) Fig. 9. Same data as shown in Fig. 8 but with the white circles now highlighting extracted data points, equivalent to the experimental data, as if it was a sample of the 100 data point population (N = 100). For comparison the same data is then re-ranked with N = 8 and plotted again and fitted in a similar manner to the experimental data shown in Fig. 8 showing the similarity.

Fracture test data Fit to normal probabiity 100 Simulated data points Fit to simulated data 99 98 95

Probability of failure (%)

90 80 70 50 30 20 10 5

R2 = 0.997

2 1 0.5 0.2 0.1 0.05

R2 = 0.983

0.01 20

30

40

50

60

KJc (MPa m1/2) Fig. 10. The simulated 100 data point population and the eight experimental fracture toughness data points plotted on normal probability paper and fitted to normal distributions showing that both distributions conform to a normal distribution shape with R2 values greater than 0.98 in both cases.

of 40.6 MPa m1/2 at a failure probability of 0.01% (104). Again this points to the fact that based pragmatically on the evidence available the data has a lower shelf value of 40 MPa m1/2 and assuming a value of 20 MPa m1/2 is overly pessimistic. In fact whilst the normal distribution has no lower limit, which is itself a limitation, you would have to arrive at a failure probability of 1031 to achieve a toughness of 20 MPa m1/2 for the test data.

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It is shown from the data in Fig. 11 and the original plots of Wallin [4] that increasing the value of Kmin has the effect of reducing the Weibull modulus as noted by Zhang and Knott [23]. As seen from Fig. 11 it is clearly possible to fit distributions that have a very similar width but very different Weibull modulus just by changing the Kmin parameter. Fig. 11 is the reproduced figure used by Wallin to justify the m = 4 value when Kmin = 20 MPa m1/2. This also includes theoretical limits generated for the distribution. Added to the plot is the point my data would have generated fitted to a Kmin = 20 and m = 13.3 distribution. It is clearly outside the theoretical bounds of Wallin’s data suggesting it does not conform to this distribution. Overall, the combination of analysis by FE modelling, fitting in a variety of ways for Weibull and fitting normal distributions suggest that the minimum toughness of the steel here is nearer 40 than 20 MPa m1/2. This suggests that if you wish to pragmatically use a 3-parameter Weibull you should probably choose to fit the minimum toughness and keep m = 4 even though there is no justification for choosing m with this value. However, a normal distribution also seems to give a reasonable estimate of the minimum toughness at a reasonable probability of failure of 104. Care must be taken however, in ensuring that there is no warm-press effect or heterogeneous microstructures (i.e. this approach can only be used for single microstructure steels) before accepting an enhanced minimum toughness. To do so without these steps would be dangerous with regard to structural integrity analysis. 3.3. Master Curve analysis Next the distributions are used to generate 3 Master Curves and their associated tolerance bounds of 5% and 95% assuming that (1) Kmin = 20 and m = 4, (2) assuming it is better to fit the eight data points to the best curve possible where m = 4 and Kmin = 41.3 MPa m1/2 and (3) where Kmin is fixed at 20 MPa m1/2 and m was fitted to 13.3. The first thing to note is that the Master Curve coefficients A and B quoted as 30 and 70 in Eq. (11) are those that you obtain from m = 4 and Kmin = 20. i.e. if you generalise the formula then the constants are calculated from Eq. (10) but where you set the probability in the denominator to 0.5 for the Master Curve itself and repeat this for 0.95 and 0.05 probability tolerance bounds. This gives a set of constants shown in Table 5. Of course A + B were originally fitted to give a summed fracture toughness of 100 MPa m1/2 and this then defines the T0 calculation. The parameters given for A and B in Table 5 using Eq. (10) do not of course meet the criterion of A + B = 100 when Kmin and m are varied making them invalid. The results for the other parameters K0, KJc(med) and T0 are shown in Table 6. As a consequence of the KJc(med) = 50.08 for case (2) it is not possible to calculate a T0 for this fit as the formula in Eq. (12) now contains the log of a negative number. As the predicted T0 for both case (1) and (3) are only different by 2 degrees the average of the two has been used for case (2) giving T0 of 83.56. The validity of this approach is not explored. T0 can of course also be calculated for the three cases assuming A and B are 30 and 70 and these are given in last column of Table 6 and called original values and perhaps for case (1) this is a more sensible approach. However, it is clear that the original Master Curve parameters are problematic if you assume Kmin is not 20 MPa m1/2. This is somewhat contradictory to the proposition that the constants are essentially material independent [26]. Figs. 12a and 12b show the Master Curves for the 3 fits to the data using values from Tables 5 and 6. It is clear that the fit for case 2 does not represent the data at all because the constants are not fitted properly. Whereas both fits keeping

K min = 20, m= 13

Fig. 11. Fig. 6 after Wallin [4] showing experimental data plotted for a Kmin value of 20 shows theoretical bounds for the Weibull modulus, m, with increasing data set size. Added is the data point for the eight experimental fracture toughness tests made here for which a Weibull distribution of Kmin = 20 and m = 13.3 has been added to the plot and is clearly not within the theoretical bounds.

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M.R. Wenman / Engineering Fracture Mechanics 98 (2013) 350–364 Table 5 Master Curve constants A and B predicted from Eq. (10). Data fit used

A

B

Curve

Kmin = 20, m = 4 Kmin = 20, m = 4 Kmin = 20, m = 4 Kmin = 41.3, m = 4 Kmin = 41.3, m = 4 Kmin = 41.3, m = 4 Kmin = 20, m = 13.3 Kmin = 20, m = 13.3 Kmin = 20, m = 13.3

25.2 30.0 34.5 46.5 51.3 55.8 28.8 30.7 32.0

36.6 70.3 101.3 36.6 70.3 101.3 61.6 74.9 83.6

5% Master Curve (50%) 95% 5% Master Curve (50%) 95% 5% Master Curve (50%) 95%

Table 6 Predicted Master Curve parameters. Data fit used

K0 (MPa m1/2)

KJc(med) (MPa m1/2)

T0 (°C)

T0 (°C) original

Kmin = 20, m = 4 Kmin = 41.3, m = 4 Kmin = 20, m = 13.3

50.44 50.92 51.08

47.78 50.08 50.24

82.86 – 84.26

82.86 89.3 89.7

Kmin = 20 MPa m1/2, m = 4 Kmin = 41.3 MPa m1/2, m = 4 Kmin = 20 MPa m1/2, m = 13.3 Experimental Data

KJc(med) (MPa m1/2)

250

200

150

100

50

0 -200 -180 -160 -140 -120 -100

-80

-60

-40

Temperature (ºC) Fig. 12a. Master Curves fitted to the parameters in Tables 5 and 6 together with the experimental data showing that setting Kmin greater than 20 in Eq. (10) produces a meaningless Master Curve.

Kmin = 20 MPa m1/2 do fit the data presumably because the empirical relationship used to derive it, Eq. (10), has constants fitted to this Kmin value. The difference between Wallin’s Master Curve and that of case (3) are the gradient of the latter is slightly greater as shown in Figs. 13a and 13b. The main difference between case (1) and (3) is of course the 5% and 95% confidence bounds, which are much tighter for m = 13.3 than for m = 4. For m = 13.3 the confidence bounds are tight to the data as might be expected as it was fitted to it. It is notable that for the case most likely to be correct Kmin = 41.3 MPa m1/2 and m = 4 it is not possible to fit a Master Curve using the standard approach and yet despite the small data set from which it was derived it is hard to believe it is a sample from a population where m = 4 with a Kmin of 20 MPa m1/2. It is clear that the Master Curve approach is not always satisfactory given either a small data set or when the minimum toughness is likely to be substantially greater than 20 MPa m1/2.

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K min = 20 MPa m1/2, m = 4 K min = 41.3 MPa m1/2, m = 4 K min = 20 MPa m1/2, m = 13.3 Experimental Data

KJc(med) (MPa m1/2)

100

80

60

40

-200

-180

-160

-140

-120

-100

Temperature (ºC) Fig. 12b. Magnified version of Fig. 12a in the region of interest.

Kmin = 20 MPa m1/2, m = 4 Kmin = 20 MPa m1/2, m = 13.3 Experimental Data Kmin = 20 MPa m1/2, m = 4 (0.05) Kmin = 20 MPa m1/2, m = 4 (0.95) Kmin = 20 MPa m1/2, m = 13.3 (0.05) Kmin = 20 MPa m1/2, m = 13.3 (0.95)

KJc(med) (MPa m1/2)

250

200

150

100

50

0 -200 -180 -160 -140 -120 -100

-80

-60

-40

Temperature (ºC) Fig. 13a. Master Curves including 5% and 95% tolerance bounds for Kmin = 20 MPa m1/2 and m = 4 and m = 13.3. This shows the Master Curves are similar but the tolerance bounds for m = 13.3 are much tighter.

4. Conclusions 2D FE plane strain analysis has been used to show that the toughness enhancement on the lower transition region (155 °C) of the steel tested is unlikely to be due to the warm pre-stress effect. Plastic zones sizes at the fracture load at this lower temperature were many times the size of those generated due to room temperature pre-cracking even at a final prefatigue cracking load of 20 MPa m1/2. Fitting of the small data set both by 3-parameter Weibull and normal distributions suggest that the actual minimum toughness of the steel used here is closer to 40 MPa m1/2 rather than 20 MPa m1/2. There is no clear method for fitting data that does not conform to that of Wallin’s method. The user is faced with a choice of many possible Weibull or normal fits. Here, whilst several Weibull distributions could be fitted, a normal distribution was

M.R. Wenman / Engineering Fracture Mechanics 98 (2013) 350–364

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KJc(med) (MPa m1/2)

70

60

50

40

30 -200

-180

-160

-140

-120

-100

Temperature (ºC) Fig. 13b. Magnified version of Fig. 13a in the region of interest.

also fitted and used to extrapolate to low probabilities of failure which suggested that a Kmin of 40 MPa m1/2 was appropriate. For a failure probability of 104, a pragmatic failure probability, the steel tested had a predicted toughness of 40.6 MPa m1/2. If a Weibull fit is chosen it seems sensible to fit to a minimum toughness and fix m = 4 rather than let m vary and stick to a minimum toughness of 20 MPa m1/2. This however, means that the standard Master Curve approach may not be used to predict the transition toughness from these six tests alone as previously suggested and a new method is needed for tougher steels. Acknowledgements The author would like to thank Dr. Alexander Price for the experimental data used in this paper and the MoD for the funding for that original work. I would also like to acknowledge useful discussions with Dr. Luc Vandeperre on Weibull analysis of ceramics and Prof. John Knott for his input. I would finally like to acknowledge the financial support of EDF Energy through a Research Fellowship scheme. References [1] ASTM. Test method for determination of reference temperature, T0, for ferritic steels in the transition range. ASTM-E-1921-02. West Conshohocken: ASTM, PA; 2002. [2] Wallin K. Macroscopic nature of brittle fracture. J Phys 1993;3(IV):575–84. [3] Wallin K. Master Curve analysis of the ‘‘euro’’ fracture toughness data set. Engng Fract Mech 2002;69:451–81. [4] Wallin K. The scatter in KIC-results. Engng Fract Mech 1984;19:1085–93. [5] Knott JF. Deterministic and probabilistic modelling of brittle fracture mechanisms in ferritic steel. Fatigue Fract Engng Mater Struct 2006;29:714–24. [6] Knott JF. Local approach concepts and the microstructures of steels. Engng Fract Mech 2008;75:3560–9. [7] Todinov MT. Is Weibull distribution the correct model for predicting probability of failure initiated by non-interacting flaws? Int J Solids Struct 2009;46:887–901. [8] Todinov MT. The cumulative hazard stress density as an alternative to the Weibull model. Int J Solids Struct 2010;47:3286–96. [9] Danzer R, Supancic P, Pascual J, Lube T. Fracture statistics of ceramics – Weibull statistics and deviations from Weibull statistics. Engng Fract Mech 2007;74:2919–32. [10] McMeeking RM. Finite deformation analysis of crack-tip opening in elastic–plastic materials and implications for fracture. J Mech Phys Solids 1977;25:351–81. [11] Pineau A. Review of fracture micromechanisms and a local approach to predicting crack resistance in low strength steels. In: 5th International conference on fracture; 1981. p. 553–577. [12] Wallin K, Saario T, Törrönen K. Statistical model for carbide induced brittle fracture in steel. Met Sci 1984;18:13–6. [13] Anderson TL, Stienstra D, Dodds RH. A theoretical framework for addressing fracture in the ductile-to-brittle region. In: Landes JD, McCabe DE, Boulet JAM, editors. Fracture Mechanics. ASTM STP 1207, vol. 24. Philadelphia, USA: ASTM; 1994. p. 186–214. [14] Akbarzadeh P, Hadidi-Moud S, Gourdarzi AM. Global equations for Weibull parameters in a ductile-to-brittle transition region. Nucl Engng Des 2009;239:1186–92. [15] Millella PP, Bonora N. On the dependence of the Weibull exponent on geometry and loading conditions and its implications on the fracture toughness probability curve using a local approach criterion. Int J Fract 2000;104:71–87. [16] Gerguri S, Fellows LJ, Durodola FJ, Fellows NA, Hutchinson AR, Dickerson T. Prediction of brittle fracture of notched graphite and silicon nitride bars. Appl Mech Mater 2004;1–2:113–9. [17] Scibetta M, Lucon E, Walle EV, Valo M. Towards a uniform pre-cracking procedure for fracture toughness testing. Int J Fract 2002;117:287–96. [18] Smith DJ, Hadidimoud S, Fowler H. The effects of warm pre-stressing on cleavage fracture. Part 2: Finite element analysis. Engng Fract Mech 2004;71:2033–51. [19] Stöckl H, Böschen R, Schmitt W, Varfolomeyev I, Chen JH. Quantification of the warm pre-stressing effect in a shape welded 10 MnMoNi 5-5 material. Engng Fract Mech 2000;67:119–37. [20] Chen JH, Wang VB, Wang GZ, Chen X. Mechanism of effects of warm prestressing on apparent toughness of precracked specimens of HSLA steels. Engng Fract Mech 2001;68:1669–86. [21] Chell GG, Haigh JR, Vitek V. A theory of warm pre-stressing: experimental validation and the implications for elastic plastic failure criteria. Int J Fract 1981;17:61–81.

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[22] Curry DA. A micro-mechanistic approach to the warm pre-stressing of ferritic steels. Int J Fract 1981;17:335–43. [23] Zhang XZ, Knott JF. The statistical modelling of brittle fracture in homogeneous and heterogeneous steel microstructures. Acta Mater 2000;48:2135–46. [24] Mirzaee-Sisan A, Truman CE, Smith DJ, Smith MC. Interaction of residual stress with mechanical loading. Engng Fract Mech 2007;74:2864–80. [25] Wenman MR, Price AJ, Steuwer A, Chard-Tuckey PR, Crocombe A. Modelling and experimental characterisation of a residual stress field in a compact tension specimen. Int J Pres Ves Pip 2009;86:830–7. [26] Wallin K. Fracture toughness transition curve shape for ferritic structural steels. In: Teoh ST, Lee KH, editors. Proceedings of the Joint FEFG ICF International Conference on the Fracture of Engineering Materials and Structures. Singapore: Elsevier; 1991. p. 83–8. [27] Wallin K. Irradiation damage effects on the fracture toughness transition curve shape for reactor pressure vessel steels. Int J Pres Ves Pip 1993;55:61–79. [28] Wenman MR. PhD thesis. University of Birmingham; 2004. [29] Todinov MT, Novovic M, Bowen P, Knott JF. Modelling the impact energy in the ductile/brittle transition region of C–Mn multi-run welds. Mat Sci Engng A 200(A287):116–124.