Henryk G Pisarski a,1, Kim Wallin b
a Structural Integrity Technology Group, TWI, Granta Park, Gt Abington, Cambridge, CB1 6AL, UK
b VTT Manufacturing Technology, Technical Research Centre Finland, Espoo, 02044 VTT, Finland
1 Corresponding author. Tel. +44 (0) 1223 899000; Fax: +44 (0) 1223 893303
Published in Engineering Fracture Mechanics, vol.67, no.6. (2000) pp. 613-624.
The maximum likelihood (mml) procedure for the analysis of fracture toughness data generated at a single temperature that was developed for the SINTAP flaw assessment procedure is described. The procedure involves a series of steps to ensure that a Weibull toughness distribution fitted to experimental data is conservative. Validation is provided by experimental data from weld metal, HAZ and parent plate. Results are compared with those from assessments carried out to existing methods described in BSI PD6493:1991 and BS 7910:1999. It is concluded that for small data sets, the mml procedure provides a greater level of consistency and reduces selection of potentially non-conservative fracture toughness values.
For reasons of cost and/or restrictions on the amount of material available, fracture toughness is often determined using a limited number of specimens. The choice of an appropriate statistical distribution from limited data can be arbitrary and unreliable. Consequently, the selection of a characteristic value from such a distribution for use in a flaw assessment procedure may be unconservative, or at least give inconsistent assessments. However, some of the inconsistency can be removed by assuming that fracture is governed by a weak link process which follows a three parameter Weibull distribution. For non-austenitic steels this is given by:
P(K) is the cumulative probability of fracture toughness, K (MPam 0.5)
K o is the scale parameter (the 63 rd percentile of the distribution)
20 is the shift parameter in the Weibull distribution (MPam 0.5)
4 is the value of the shape parameter in the Weibull distribution for small scale yielding.
Equation  may be re-written as follows to provide an estimate of K with a given probability level once K o is known:
The distribution fitting procedure involves finding the optimum value of K o for a particular set of data. However, data censoring may be necessary if some of the test results do not result in fracture or the value of fracture toughness exceeds the theoretical capacity of the specimen, i.e. the result is no longer controlled by small-scale yielding. Unfortunately, the test results may be biased when specimens are extracted from inhomogeneous materials such as weld metals and heat affected zones (HAZs). Since every specimen in a series of tests is unlikely to sample local brittle zones (LBZs) that may be present, the fracture toughness distribution will be biased to the higher toughness regions that are present. Thus the fitted distribution could result in non-conservative assessments being made. To avoid this potential problem Wallin [1,2] proposed a maximum likelihood estimation procedure (mml procedure) based on the three parameter Weibull distribution in the SINTAP flaw assessment method  . The procedure is intended to provide conservative, but realistic, estimates of fracture toughness for use in defect assessment of safety critical components. This paper outlines the SINTAP mml procedure and illustrates it using fracture toughness values determined at single temperatures in weld metal, HAZ and parent plate. Furthermore, the results are compared with those obtained from procedures described in BSI PD6493:1991  and its replacement BS 7910:1999  . It is shown that the SINTAP mml procedure provides estimates of fracture that are conservative but still consistent with BS procedures. However, as the SINTAP procedure is based on a weak link model of fracture, it offers the possibility of establishing characteristic fracture toughness values for any thickness of interest from sub-sized specimens (e.g. specimen thickness less than original component or weld thickness).
2. SINTAP mml estimation procedure
The mml estimation procedure is used to obtain a lower bound scale parameter, K o
, which describes the Weibull fracture toughness distribution (see Eq.) for inhomogeneous materials. It proceeds via a series of stages that are described below. In this paper, the procedure is used to evaluate fracture toughness obtained at a single test temperature. However, the procedure can be applied to transition curve data, and further comment about this is made at the end of this section.
The first stage in the procedure is to check that all the data meet the acceptance criteria of the relevant testing standard. The fracture toughness values are converted into stress intensity factors with MPam 0.5 units. Where CTOD data are available from standard, deeply notched bend specimens, these may be converted into equivalent K values using:
|= yield strength (MPa)
= crack tip opening displacement (m)
= Young's modulus (MPa)
= Poisson's ratio
It should be noted that the constant 1.5 is an empirically derived approximation, but is considered appropriate for deeply notched bend specimens made from low work hardening ferritic steels. The constant depends on specimen geometry, crack size and work hardening.
Next, the specimen capacity limit (K cen) is determined from:
where b o (units in metres, m) is the initial ligament below the notch (W-a o) in the test specimen.
Equation  ensures that fracture occurs under small-scale yielding conditions. Results from specimens that exceed this limit are censored; the censoring parameter, δ , is set at 0 and the fracture toughness values are set at K cen. Specimens that do not fracture are also censored and δ set at 0. Results from other specimens are not censored and δ is set at 1 for each specimen.
Next, the fracture toughness values are corrected to a reference thickness of 25mm using:
where B is the thickness (units in mm) of the specimen in which K was determined.
The values of K 25 and their associated censoring parameters are then used to make the first K o estimate. This is STEP 1 in the mml procedure (referred to as 'normal mml estimation') and K o is obtained from:
|= number of results in data set
= i th result
Note that the maximum likelihood procedure uses all the test results and that the K o estimate is biased towards the uncensored data.
If the material tested were homogeneous, the estimate of the K o from Eq. could be inserted into Eq. to provide a value of K, for a specified probability level. For the value to be used in a flaw assessment procedure, it is necessary to correct this fracture toughness for the appropriate component thickness (see Eq.) or crack front length, further discussion of this correction is made later. However, when the data are from inhomogeneous materials, further censoring is required; this is provided by the next two steps in the procedure.
STEP 2 in the mml procedure, (referred to as 'lower tail mml estimation') involves censoring all data above the 50 th percentile or median of the distribution (
), i.e. setting δ = 0 for all results above the 50 th
percentile. This ensures that the estimate of K o
is biased towards the lower tail of the toughness distribution so as to include results from specimens containing LBZs. Results from specimens which do not contain LBZs and that are likely to give high fracture toughness values are excluded by STEP 2. The censored data are assigned the median value of toughness given by:
After censoring, K o is re-estimated using Eq.. However, since Eq. and  both contain K o, STEP 2 is iterative with K o and K m being continually recalculated until a consistent minimum K o is obtained: convergence is usually achieved in less than 10 iterations.
The final step, STEP 3 (referred to as 'minimum value estimation'), requires an estimate of the K o using the minimum fracture toughness value in the data set. This is obtained from:
The values of K o from each of the three steps are now compared. The K o considered to characterise the fracture toughness distribution is the lowest determined from the three steps, except when K o from STEP 3 is not less than 90% of the lower of K o from STEPS 1 or 2. In this case, K o is the lowest of STEPS 1 or 2. Where steps, o from STEP 3 is less than 90% of the lower of STEPS 1 and 2, it would be conservative to use K o from STEP 3. However, since the procedure is highlighting an outlier, a judgement has to be made as to its significance. If the data set is large and the fit in STEP 1 or 2 is good, then the result from STEP 3 can be treated as representing an anomaly and can be ignored (the reason for the anomaly may require investigation). It is then reasonable to assume that the lower of STEPS 1 or 2 is characterising the fracture toughness distribution. The SINTAP procedure recognises this, and recommends that for more than 9 results, STEP 3 is unnecessary. However, if the data set is small, it would be unreasonable and potentially unconservative to ignore STEP 3. Nevertheless, if the result from STEP 3 is considered to be unsatisfactory then further testing should be conducted to better define the lower tail of the fracture toughness distribution. Finally, a further correction is made to adjust the estimate of K o for the sample size. This is given by:
where r = the number of specimens that fractured.
The procedure described above can, in principle, be applied to data in the fracture toughness transition regime. In this case the temperature dependence of fracture toughness for ferritic steels is described by the Master Curve (6) which is given by:
where T o is the transition temperature for a median fracture toughness (50 th percentile) of 100MPam 0.5 normalised to a 25mm thick specimen. The steps in the procedure are similar to those described above but involve estimating T o; details of the procedure are given elsewhere (3). Having established T o, the fracture toughness at a given temperature, specimen thickness and probability level can be obtained from:
However, in the examples given in the next section, the procedure is illustrated for tests conducted at a single test temperature.
Once the appropriate K o has been established, it is necessary to estimate a value of toughness that is going to be used in fracture mechanics analyses, K mat, from either Eq. or  for a specific probability. The choice of probability depends on the reliability required from the structural component. Typical probability values suggested by SINTAP (3) are the 5 th percentile for critical applications, where a consequence of failure could be loss of life, or, the 20 th percentile for less critical applications. Where a specific target probability of failure needs to be achieved, probabilistic fracture mechanics analyses can be conducted using the Weibull distributions established, or simplified procedures employing partial safety factors. The SINTAP procedure provides recommendations on the partial safety factors to use. With respect to fracture toughness, these are applied to the mean minus one standard deviation value; this is approximately the same as the 20 th percentile.
Since the mml procedure derives from a weak link model of fracture, K mat must be adjusted for crack front length (K mat(1)). The SINTAP procedure recommends that this adjustment is applied to the Kmat value established for the reference thickness of 25mm (K mat(25)) using the following equation (this is similar to Eq.):
where 1 is the crack front length (in mm), when l < 2t, or 2t when l > 2t. t is the thickness of the component being assessed. This approach is different from that recommended in other flaw assessment procedures, where it is usual to employ fracture toughness results measured in specimens of the same thickness as the component being assessed [4,5] . Where this is not possible, because sub-size specimens are used, the implications are that the fracture toughness needs to be adjusted for the component thickness. Consequently, l in Eq. would be replaced by t.
3. Application of the mml procedure to experimental data
In order to illustrate the mml procedure, examples are given for experimental fracture toughness data obtained in a weld metal, weld HAZ and parent plate. Each data set is described and then results of the mml procedure are presented.
The weld metal data set was derived from a European round-robin weld metal fracture toughness testing programme 
. Tests were conducted on multipass submerged arc butt welds made from one side at a heat input of 4.5kJ/mm in a 50mm thick normalised C-Mn steel plate with a nominal yield strength of 340MPa. Various specimen geometries were tested at different temperatures. The data set selected here represents results obtained on full-thickness, rectangular section, deeply notched bend specimens (SE(B), Bx2B, a o
/W=0.5, where B=50mm). The specimens were through-thickness notched along the weld centre line and the tests conducted in accordance with BS 7448:Part 1:1991. As the specimens were in the 'as-welded' condition, local compression was carried out to relieve residual stresses and promote growth of straight fatigue cracks. Only those specimens that met the fatigue crack front straightness and minimum fatigue crack length requirements of BS 7448:Part 1:1991 are included in the data set. (It may be noted that local compression procedures are described in BS 7448:Part 2:1997. Also, the fatigue crack front straightness requirements are more relaxed in this standard compared with BS 7448:Part 1:1991). Other details of the test data are listed below:
Number of results, n = 27; number of specimens failing by cleavage: 21
B = 50mm
b o = 50mm
Test temperature: -20°C
Weld metal yield strength = 529MPa at -20°C.
Heat affected zone, HAZ
HAZ data sets were derived from the European ACCRIS project which examined the effect of yield strength mismatch on toughness  . Fracture toughness tests were conducted on X butt welds in a 48mm thick TMCP steel with a yield strength of 457MPa. Two series of welds were made by submerged arc welding, using different consumables, at a heat input of 3kJ/mm. In the ACCRIS programme, the data set from the first series is referred to as an 'even match' (EM) condition, although the weld metal yield strength overmatched that of the parent plate by about 24%. The data set from the second series of welds is referred to as an 'over match' (OM) condition and the weld metal yield strength overmatched the parent plate by 41%. Full thickness, square section specimens (SE(B), BxB, a o/W=0.3) were employed which were notched from the original plate surface into HAZ close to the weld fusion boundary. Post-test metallography was conducted to establish the actual microstructures at the fatigue crack tip and at fracture initiation. The data sets selected here include results from all the tests, irrespective of the results from post test metallography, and those from specimens in which post test metallography confirmed that the fatigue crack tip was located in, or fracture initiated from, the grain coarsened HAZ. In this material, this is the lowest toughness region of the HAZ. (Results from specimens with the fatigue crack tip in weld metal were included, provided that fracture initiation took place in the grain coarsened HAZ no further than 0.5mm from the crack tip). The tests were conducted in accordance with BS 7448:Part 1:1991 and CTOD values were reported. For the purpose of the present study, these have been converted into equivalent K values using Eq. and assuming parent plate yield strength. Since the crack depths corresponded to a o/W < 0.45, the equations for calculation of CTOD and K o are not strictly valid. However, for the purposes of the present analyses, the errors that use of this equation may cause does not affect the illustration of the assessment procedure used, nor the conclusions drawn. Other details about the data set are given below:
All data (OM and EM)
n = 45 for each set; all failed by cleavage
Achieving post test metallography criteria:
n = 39; EM
n = 29, OM
B = 48mm
b o = 33.6mm
Test temperature = -10°C
Weld metal yield strength = 569MPa EM condition, 643MPa OM condition
Parent plate yield strength = 457MPa
The parent plate data set represents results from a European round-robin fracture toughness programme carried out on a normalised C-Mn steel 
. The particular data set selected represents the so-called 'control' tests (replicate tests conducted under carefully controlled conditions). The tests were conducted on full-thickness, deeply notched bend specimens (SE(B) Bx2B, a o
/W = 0.5). The specimens were notched in the through-thickness (LT) direction. Further details are given below:
n = 108; specimens that fractured: 106
B = 50mm
b o = 50mm
Test temperature = -65°C
Yield strength at -65°C = 409MPa
4. Results of mml procedure assessment
The results from the mml procedure are summarised in Table 1
. The fracture toughness distributions predicted from each of the three steps are compared with the experimental data points (all for a reference thickness of 25mm) in Fig.1 to 4
. Each step involves a rotation of the fitted curve in an anti-clockwise direction when K o
predicted from STEP 1>STEP 2>STEP 3; as illustrated by the results from the parent plate, see Fig.1
. However, this sequence does not occur in every case and for the submerged arc weld and HAZ, STEP 2 provides the lowest K o
. Indeed, it turns out that for all the materials examined here the mml procedure shows that the K o
obtained at STEP 2 characterises each fracture toughness distribution. This is probably because the data sets are relatively large and the distribution is fairly even in the lower tail. With small data sets where scatter is high, the conclusion could be different and STEP 3 is likely to characterise the distribution. STEP 1, the normal mml procedure, does not always provide a good description of the data set as a whole and, in comparison with the other steps, provides an overestimate of K o
. This indicates that the Weibull distribution shape parameter of 4 is not optimum. Nevertheless, STEP 2 or STEP 3 does appear to provide a conservative description of the lower tail to the fracture toughness distribution.
Table 1 - K o estimates (reference 25mm thick) from mml procedure
|Data set||n||STEP 1||K o, MPam 0.5||STEP 3|
|Note: In all cases, K o from STEP 2 characterises the distribution
Fig.1. MML procedure applied to parent plate results at -65°C
Fig.2. MML procedure applied to weld metal results at -20°C
Fig.3. MML procedure applied to all HAZ results from overmatched weld
Fig.4. MML procedure applied to censored (by post test metallography) HAZ results from overmatched weld
In Table 1, the effect of censoring the HAZ data by post test metallography to confirm that the HAZ was correctly tested can be noted. When the data set includes all the results from specimens nominally notched into the HAZ (i.e. results from post test metallography are ignored), higher values of K o are estimated for all steps compared with data sets which have been censored by post test metallography. It is clear that for the HAZ, from both the OM and EM welds, censoring provided by the mml procedure alone cannot be relied upon to provide a conservative, lower bound estimate of K o. This is because the data sets are of different sizes and, because high or low fracture results cannot be assumed to actually relate to the HAZ (the results could come from another region, such as parent plate or weld metal). Consequently, it is recommended that the mml procedure should be only applied to HAZ data which have been confirmed by post test metallography to test the HAZ correctly. (Procedures for conducting post test metallography on fracture toughness specimens are described in BS 7448:Part 2:1997).
In the subsequent analyses described here, only the HAZ data sets which have been censored by post test metallography are considered. In these analyses, K o obtained from Table 1 (from STEP 2) is used to derive fracture toughnesses from Eq. so that these can be compared with those predicted from the procedures described in BSI PD6493:1991 and BS 7910:1999.
5. Comparison of mml procedure with BS procedures
The procedures for estimating a characteristic fracture toughness for use in flaw assessment are given in Appendix A of BSI PD6493:1991 and Annex K of BS 7910:1999. Both are similar and two methods are given. The first method is intended for small data sets, typically involving less than 15 specimens, and is based on the minimum of three equivalent (MOTE) concept. The MOTE procedure requires that when there are between three and five results, the minimum value is used; between six and 10 results, the second lowest; and between 11 and 15 results, the third lowest. These characteristic values represent the 20 th percentile of the distribution with 50% confidence (approximately). In order to guard against excessive scatter, the BS procedures require the minimum to be not less than 70% of the average toughness (in terms of K) or the maximum to be no more than 1.4 times the average. If scatter is excessive, further testing is recommended. However, in many cases this is not practical and the user must base his analyses on the data available. The second procedure is recommended when there are more than 15 results, and involves fitting a statistical distribution (e.g. 1n-normal or Weibull, whichever fits best) to the data and selecting the mean minus one standard deviation for the flaw assessment.
In this section, predictions made from the mml and BS procedure are compared. In order to provide a consistent comparison criterion with the MOTE procedure, the K mat obtained from the mml procedure was the 20 th percentile of the Weibull distribution described by the appropriate K o. The procedure was as follows. For each set of test data, sub-sets of three, six, nine and 12 results were selected at random, and MOTE and mml procedures applied to each sub-set. This process was repeated 100 times for each sub-set. With the mml procedure, the small sample correction was applied to K o (see Eq.) before calculating the 20 th percentile, designated K mat.
The MOTE results (K mat values) were corrected for a common thickness of 25mm using Eq., in order to facilitate comparison with the mml results (also for 25mm).
The results from these calculations are presented in Fig.5 to 7 for the parent plate, weld metal and even match HAZ. It should be noted that because the weld metal and even match HAZ data sets are relatively small, the results in Fig.5 and 6 contain a number of repeat K mat values. Also included in the figures are the 20 th percentile K mat predictions from both BS and mml procedures made from the complete data set. For the BS procedure, various statistical distributions were fitted to the censored experimental data using a maximum likelihood method. Results from specimens that did not fracture were censored. However, censoring did not mean the result was ignored. In all cases the 1n-normal distribution was found to provide the best fit, and the fitted parameters together with the 20 th percentile values are given in Table 2.
Fig.5. MML and MOTE procedures for weld metal at -20°C compared for 100 simulations for each sub-set up to 12 results; above 12, 20 th percentile values are shown for both SINTAP and BSI procedures
Fig.6. MML and MOTE procedures for HAZ even match at -10°C compared for 100 simulations for each sub-set up to 12 results; above 12, 20 th percentile values are shown for both SINTAP and BSI procedures
Fig.7. MML and MOTE procedures for parent plate at -65°C compared for 100 simulations for each sub-set up to 12 results; above 12, 20 th percentile values are shown for both SINTAP and BSI procedures
Table 2 - Ln-normal distribution parameters for experimental data
|Data set||n||Mean*||Std Dev*||K mat, BS|
20 th percentile
|K mat, mml|
20 th percentile
* For 1n-normal distribution, actual thickness, as per BS 7910 procedure
** Values for a reference thickness of 25mm
Figures 5-7 clearly show that for the simulations, a higher mean fracture toughness is obtained with the MOTE procedure compared with the mml procedure. Scatter is also higher for the MOTE procedure than the mml procedure, especially when the number in the sub-set is 6 or less, but decreases for larger sub-sets. This implies that the mml procedure reduces the risk of overestimating fracture toughness compared with the MOTE procedure, especially for small data sets, e.g. 12 or less. When the data set comprises only three results, both procedures show that a very wide range of fracture toughness estimates is possible. Actual toughness can be significantly overestimated or underestimated, although large overestimates are less likely with the mml procedure. Indeed, these analyses illustrate the problems of attempting to characterise fracture toughness with only three test results.
When the results from the whole data set are compared (for the present purposes, this represents the whole population), the 20 th percentile predictions from both procedures are similar, despite the fact that the mml procedure is based on a 3 parameter Weibull distribution with a shape parameter of 4, and the BS procedure is based on a 'best-fit' ln-normal distribution.
Nevertheless, Figs.5-7 illustrate the particular strength of the mml procedure when dealing with small data sets, in that, in comparison with the MOTE procedure, it reduces the likelihood of overestimating fracture toughness but does not eliminate it. However, the results do illustrate the potential problems of relying on the results from data sets as small as three. The fact that engineering critical assessments (ECAs) employing fracture toughness data obtained from three tests have not proved to be unsafe, as far as is known, must imply that either the assessment procedure contains inherent conservatisms, or that other input data are conservative. It is suspected that in practice, both factors play a rôle. Indeed, this is in keeping with the philosophy adopted in ECAs, which is fracture avoidance rather than prediction of the critical conditions for fracture. Nevertheless, for safety critical components reliance on three test results would appear to be unacceptable, unless high safety factors are included in other input parameters to the flaw assessment.
The SINTAP mml procedure for inhomogeneous materials tested at a single temperature has been described and the results assessed using experimental data from weld metal, HAZ and parent plate. The results have been compared against results obtained from procedures described in BSI PD6493:1991 and BS 7910:1999. The following conclusions are drawn:
- The mml procedure appears to work well provided that the data sets are not too inhomogeneous, e.g. parent plate and weld metal.
- When the mml procedure is used on highly inhomogeneous materials such as HAZs, the data should be censored by post test metallographic procedures to ensure results are representative of the HAZ and results from other regions are excluded, otherwise erroneous fracture toughness estimates will be made.
- For large data sets (>15), the procedures given in the BSI documents, which are based on characteristic value taken from a fitted statistical distribution, give similar results to the mml procedure when the comparison is based on the 20 th percentile of the K mat distributions.
- For small data sets (12 results or less), the mml procedure provides more consistent estimates of K mat compared with the MOTE procedure described in the BSI documents. In particular, in comparison with the MOTE procedure, the mml procedure reduces the risk of overestimating the 'true' fracture toughness of the material.
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Presented with permission from Elsevier