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Applied Mathematical Modelling 35 (2011) 2838–2854

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Applied Mathematical Modelling

journal homepage: www.elsevier .com/locate /apm

Obtaining confidence limits for safe creep life in the presence of multibatch hierarchical databases: An application to 18Cr–12Ni–Mo steel

M. Evans ⇑Materials Research Centre, School of Engineering, Swansea University, Swansea SA2 8PP, UK

a r t i c l e i n f o

Article history:Received 14 September 2010Received in revised form 23 November 2010Accepted 30 November 2010Available online 15 December 2010

Keywords:CreepFractureAustenitic steelsMultilevel models

0307-904X/$ - see front matter � 2010 Elsevier Incdoi:10.1016/j.apm.2010.11.072

⇑ Tel.: +44 0 1792 295748; fax: +44 0 1792 2956E-mail address: [email protected]

a b s t r a c t

The creep and creep rupture properties of 18Cr–12Ni–Mo steel tubes have been analysedusing the Wilshire equations. The observed behaviour patterns are then briefly discussedin terms of the dislocation processes governing creep strain accumulation. A suitable sta-tistical framework for analysing both the single and multi batch data available on thismaterial is then specified. It is shown that ignoring the hierarchical nature present in manycreep data bases, which has been the approach used until now when using the Wilshireequations, leads to a serious and significant underestimate of the predicted safe life for thismaterial. The model allows accurate predictions, with associated levels of confidence, oflong-term properties by extrapolation of short-term test results for this steel.

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1. Introduction

In general, when selecting alloy steels for large-scale components used in power and petrochemical plants, decisions arebased on the ‘allowable creep strengths’, normally calculated from the tensile stresses causing failure in 100,000 h at therelevant service temperatures [1]. However, creep life measurements for structural steels show considerable batch to batchvariability so, in Europe, tests up to 30,000 h have often been completed for five melts of each steel grade [2]. Yet, even whenusing these extensive results, the 100,000 h strength estimates depend on the methods adopted to make the calculations[3,4], despite the international activities devoted to the assessment of different data analysis procedures [5]. Moreover,for a series of 9–12% chromium steels developed for ultra super-critical power plant the allowable strengths have beenreduced progressively as the test durations have increased well beyond 30,000 h towards 100,000 and more [3,4].

In contrast to the problems encountered using traditional parametric procedures for extrapolation of short-term creep lifeproperties [5], a new methodology appears to allow accurate prediction of 100,000 h strengths by considering results withmaximum lives up to 5000 h or less [6–9]. Whilst this new approach based on normalising the applied stress (s) through theultimate tensile stress (sTS) at the creep temperatures for each batch of material investigated, has from a practical perspec-tive proved very successful at predicting creep lives for a series of ferritic [9], bainitic [8] and martensitic steels [7], it is notintended as a substitute for models that describe the physical processes involved in the generation and movement of thedislocations or the interaction of particles with sliding boundaries. In addition, confidence in this method is improved byinterpretation of the property sets in terms of the processes governing creep and creep fracture [6]. Hence, in the presentstudy, this new methodology is adopted for 18Cr–12Ni–Mo steel tubes using the creep data sheets produced at873–1023 K by the National Institute for Materials Science (NIMS), Japan [10]. To reduce fuel consumption and CO2 omis-sions from power plants, new high-temperature alloys are required to resist the increase in temperature and pressure

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76.

M. Evans / Applied Mathematical Modelling 35 (2011) 2838–2854 2839

needed to raise plant efficiencies and austenitic steels such as 18Cr–12Ni–Mo will play an important role provided that theycan be shown to have an acceptable creep strength at these raised temperatures.

A single and multi batch analysis is carried out because in all the applications of the Wilshire equations referenced above,the analysis has been carried out on multi batches of the same material using techniques (namely the use of linear leastsquares) that are really only appropriate for single batches of material This is because in multi batch data sets the materialproperties (such as minimum creep rates and times to failure) will not be independent of each other, making least squaresparameter estimates less efficient than they could be. The parameter estimates, on average, will therefore tend to differ morefrom the true values than is appropriate and confidence intervals for safe life will be under estimated. Unfortunately, the sizeof this underestimation has never been quantified.

The aim of this paper is therefore not just to identify whether the Wilshire equations can help reduce the developmentcycle for new austenitic steels (an objective recently identified as the No. 1 priority in the 2007 UK Strategic Research Agenda[11]) by seeing if these equations can successfully extrapolate existing short term data on the now well established austen-itic 18Cr–12Ni–Mo steel, but to also provide a more suitable statistical framework that should be used when applying theWilshire equations to multi batch data sets such as this one. To achieve these aims, the paper is therefore structured asfollows. Section 2 briefly describes the material used for this study and the nature of the NIMS data base available on thismaterial. Section 3 provides a single batch analysis of this material that is similar in structure to that in all the abovereferenced Wilshire et al. papers. It is however different in two important respects. The framework adopted allows forthe derivation of confidence limits and it can cope with the presence of runout data that often exit in long term hightemperature data. Section 4 applies a new type of statistical model to the multi batch data, with the statistical model beingdescribed in detail within an Appendix. The paper finishes with some conclusions and suggestions for future work.

2. The data

The National Institute for Materials Science (NIMS) in Japan have carried out an extensive high temperature testing pro-gram involving many different steel alloys. Creep data sheet No. 6B contains information on 18Cr–12Ni–Mo steel tubes [10].Tube specimens for the tensile and creep rupture tests were taken longitudinally from the as received boiler tubes at themiddle of the wall thickness and each test specimen had a diameter of 6mm with a gauge length of 30 mm. Further detailson this creep rupture testing programs can be found in Ref. [10]. This database is very hierarchical in nature, with test spec-imens being cut from different batches of the same material. These batches differ both in the type of heat treatment used andin their chemical compositions. In total, this data sheet contains nine different batches of material, referenced by NIMS asAAA through to AAN. Table 1 summarises the chemical compositions of these different batches.

Fig. 1 shows the structure of the NIMS data base for this material. In this figure only the first and last few results areshown for each batch and the data base is sorted by batch then by temperature and finally by stress to give an impressionof the hierarchical nature of this data base. In total some 323 specimens were tested over four different temperatures andover a wide range of different stresses with the resulting failure times varying from just a few hours through to over 100,000hours. At the time of publishing creep data sheet No. 6B (and indeed in all subsequent online revisions), three specimensremained on test. All that is known about these so called runouts, is that failure will occur beyond the published censoredtime. All these runouts were within the AAL batch of material and minimum creep rates were only recorded for this batch ofmaterial.

3. Creep of austenitic 18Cr–12Ni–Mo steel tubes: a single batch analysis

3.1. Data description and mathematical representation

The AAL batch was chosen for individual analysis as this is the only batch for which both minimum creep rates and timesto failure were recorded, although not all the AAL specimens had a recorded minimum creep rate associated with them. For

Table 1Composition of 18Cr–12Ni–1Mo steel tubes.

Batch Chemical composition (mass percent)

C Si Mn P S Ni Cr Mo Cu Ti Al B N Nb + Ta60.08 61 62 60.045 60.03 10–14 16–18 2–3 – – – – – –

AAN 0.06 0.52 1.58 0.025 0.007 13.6 16.6 2.31 0.26 0.029 0.021 0.0007 0.024 0.01AAM 0.06 0.52 1.6 0.025 0.007 13.3 16.7 2.25 0.24 0.06 0.02 0.0008 0.034 0.01AAL 0.07 0.61 1.65 0.025 0.007 13.6 16.6 2.33 0.26 0.043 0.017 0.0011 0.025 0.01AAF 0.066 0.63 1.73 0.029 0.024 13.16 17.07 2.34 0.13 0.055 0.095 0.002 0.028 0.01AAE 0.078 0.62 1.6 0.021 0.008 13.51 16.9 2.08 0.16 0.037 0.031 0.001 0.028 0.01AAD 0.058 0.58 1.56 0.027 0.008 13.65 16.86 2.14 0.17 0.028 0.041 0.0009 0.025 0.01AAC 0.05 0.71 1.52 0.022 0.013 13.5 17.5 2.28 0.17 0.055 0.027 0.0013 0.035 0.02AAB 0.05 0.52 1.51 0.021 0.01 13.21 16.42 2.34 0.14 0.011 0.018 0.0005 0.034 0.01AAA 0.06 0.59 1.69 0.024 0.017 13.32 16.73 2.38 0.07 0.011 0.015 0.001 0.03 0.02

Fig. 1. Structure of NIMS creep data sheet No. 6B.

2840 M. Evans / Applied Mathematical Modelling 35 (2011) 2838–2854

over half a century, the dependencies of the minimum creep rate ( _em) and the creep fracture life (tf) on stress (s) and tem-perature (T) in such data sets have been described using a combination of power law equations and the Monkman–Grantrelation

Mtf

� �1=q

¼ _em ¼ Asn exp�Q c

RT; ð1aÞ

where R = 8.314 J mol�1 K�1 and q takes on the value 0.65 for this batch of material. From the lnð _emÞ= lnðsÞ plots, the stressexponent (n) is calculated to decrease from n ffi 10 to 6 with increasing test duration and temperature, with the activationenergy for creep (Qc) ranging from about 380 to 450 kJ mol�1 (see Fig. 2(a)). Similar trends were observed for the ln(tf)/ln(s)relationships in Fig. 2(b). Clearly, these unpredictable variations in n and Qc make it impossible to estimate long-term prop-erties by extrapolation of the short-term data. The inappropriateness of equations like Eq. (1a) is further revealed by the ten-dency for isothermal experimental data points to trace out curves rather than lines over a wide range of stresses.

Many variants of this description have also been proposed in the literature including the Larson–Miller [12] andMinimum Commitment [13] methods. Unfortunately, whilst these parametric procedures are fairly simplistic, Evans [14]has recently demonstrated, using 1Cr–1M0–0.25V steels as a test bed material, that they have very undesirable propertieswhen used to life materials using short term data sets. For example, the Larson–Miller model has the mathematical form

Fig. 2. (a) The stress dependence of the minimum creep rate at 873–1023 K for the AAL batch of 18Cr–12Ni–Mo steel tubes. The curves are drawn usingEq.(6c) with the parameter values as given in Table 2a. (b) The stress dependence of the time to failure at 873–1023 K for the AAL batch of 18Cr–12Ni–Mosteel tubes.The curves are drawn using Eq. (6d) with the parameter values as given in Table 2a.

M. Evans / Applied Mathematical Modelling 35 (2011) 2838–2854 2841

ln½tF � ¼ fln½M� � ln½A�g þ f ðsÞRT

; ð1bÞ

where f(s) is some function of stress. The form of this function is currently uncertain and in applications of the Larson–Millermodel this has been approximated by polynomials in stress or by log transformations (the latter leading to Minimum Com-mitment type models). As a result of this, when Eq. (1b) is used to obtain isothermal life time predictions, these curves have atendency to bend back on each other at high and more seriously low stress, making them very unsuitable for reliable remain-ing life estimation. In contrast, the structure of the equations in the new methodology is such that this phenomenon cannotoccur (see Eq. (3) below).

To consider the effects of normalising the applied stress, use was made of the NIMS sTS (tensile strength) values – somevalues for which are shown in Fig. 1. Using these results, the power law plots in Fig. 2 were rationalised by normalising stressthrough the use of sTS so that Eq. [1a] can be rewritten as

Mtf

� �1=q

¼ _em ¼ A�ðs=sTSÞn expð�Q �c=RTÞ; ð2Þ

2842 M. Evans / Applied Mathematical Modelling 35 (2011) 2838–2854

where A⁄– A. Adopting Eq. (2), the data sets at different temperatures are superimposed onto a single curve through an acti-vation energy Q �c

� �of 360 kJ mol�1 (see Fig. 3), where Q �c

� �is determined from the temperature dependencies of _em and tf at

constant (s/sTS), rather than at constant s as in the determination of Qc with Eq. (1a). However, this procedure does not elim-inate the decrease from n ffi 10 to 6, so the unknown curvatures of the ln( _em)/ln(s/sTS) and ln(tf)/log(s/sTS) plots in Fig. 3 stilldoes not allow prediction of long-term properties by extrapolation of short-term results for this steel.

In contrast, extended extrapolation to provide 100,000 h data from tests lasting up to 5000 h or less does appear to bepossible using the Wilshire equations [6–9] given by

Fig. 3.Eq. (6c)

18Cr–1

ssTS

� �¼ exp �k2 _em exp

Q �cRT

� �� �u2�

; ð3aÞ

ssTS

� �¼ exp �k1 t1=q

f exp�Q �cRT

� �� �u1�

; ð3bÞ

(a) The dependence of ln( _em exp(360,000/RT)) on ln(s/sTS) at 873–1023K for the AAL batch of 18Cr–12Ni–Mo steel tubes.The curves are drawn usingwith the parameter values as given in Table 2a. (b) The dependence of ln t1:5

f expð�360;000=RTÞ �

on ln(s/sTS) at 873–1023 K for the AAL batch of

2Ni–Mo steel tubes. The curves are drawn using Eq. (6d) with the parameter values as given in Table 2a.

M. Evans / Applied Mathematical Modelling 35 (2011) 2838–2854 2843

where the coefficients, k1 and k2 as well as u1 and u2, are easily determined from plots of ln _em exp Q �c=RT� ��

and

ln t1:5f exp �Q �c=RT

� �h iagainst ln[�ln(s/sTS)]. These plots are shown as Fig. 4.

Essentially, the results shown in Fig. 4 are best described as tracing out two intersecting straight lines, so that it then be-comes necessary to explain the behavioural differences associated with each straight line. On decreasing the applied stressand increasing the creep temperature a change occurs at ln[�ln(s/sTS)] = 0.3 signifying that the creep lives are longer andminimum creep rates slower at lower stresses than would be predicted by linear extrapolation of results when ln[�ln(s/sTS)] < 0.3.

This rather neat description of the data can also be given a meaningful interpretation based on deformation mechanisms.With ln[�ln(s/sTS)] = 0.3 roughly corresponding to 0.85sPS (where sPS is the 0.2% proof stress), changes in the coefficients ofEq. (3) seem to occur when s < 0.85sPS, such that the creep lives are longer and the creep rates are slower at lower stresses. Aspreviously found for 1Cr–0.5Mo ferritic steels [8], this appears to take place when s < sY (where sY is the yield stress), as itseems reasonable to assume that sY ffi 0.85sPS for this type of material. Thus, when s > sY, dislocations multiply rapidly

Fig. 4. (a) The dependence of ln _em exp 360;000=RTð Þð Þ on ln(�ln(s/sTS)) at 873–1023 K for the AAL batch of 18Cr–12Ni–Mo steel tubes. The curves aredrawn using Eq. (6c) with the parameter values as given in Table 2a. (b) The dependence of ln t1:5

f expð�360;000=RTÞ �

on ln(�ln(s/sTS)) at 873–1023 K forthe AAL batch of 18Cr–12Ni–Mo steel tubes. The curves are drawn using Eq. (6d) with the parameter values as given in Table 2a.

2844 M. Evans / Applied Mathematical Modelling 35 (2011) 2838–2854

during the initial strain on loading at the creep temperature, giving high creep rates and shorter creep lives. In contrast, whens < sY, creep must occur not by the generation of new dislocations but by the movement of the dislocations pre-existing inthe as-received austenitic microstructure (or within the grain boundaries), giving slower creep rates and correspondinglylonger creep lives.

3.2. Parameter estimation

All applications of the Wilshire equations referenced above have used the technique of linear least squares to identify thevalues for the parameters in Eq. (3). The easiest way to visualise how this works is to rewrite Eq. (3) as

ln ½ _em� þ Q �c1

RT

� �¼ a0 þ a1 ln½� lnðs=sTSÞ� þ e�; ð4aÞ

ln½tf �q� Q �c

1RT

� �¼ b0 þ b1 ln½� lnðs=sTSÞ� þ v� ð4bÞ

with a0 = �ln(k2)/u2, a1 = 1/u2, e⁄ = ree and with b0 = �ln(k1)/u1, b1 = 1/u1, v⁄ = rvv and q = 0.65. The random variables e⁄ andv⁄ are added to pick up the stochastic nature of creep properties, i.e. the tendency for failure times or creep rates to differunder identical settings of the experimental variables like stress and temperature (as seen in Fig. 2). e and v are standardisedvalues for these random variables. One way to allow the values for k1, k2, u1 and u2 to be different in different stress ranges isthrough the use of spline functions. These are continuous functions that allows for such differing parameter values

ln½ _em� þ Q�c1

RT

� �¼ a0 þ a1s� þ a2½s� � s�1�D1 þ ree; ð5aÞ

ln½tf �q� Q �c

1RT

� �¼ b0 þ b1s� þ b2½s� � s�1�D1 þ rvv ð5bÞ

where s⁄ = ln(�ln(s/sTS)), D1 = 0 when s� � s�1� �

6 0 and D1 = 1 otherwise. Consequently, in the stress regime correspondingto s� 6 s�1 a0 = a0, a1 = a1, b0 = b0, b1 = b1. But in the stress regime corresponding to s� > s�1, a0 ¼ a0 � a2s�1, a1 = a1 + a2,

b0 = b0 � b2 s�1 and b1 = b1 + b2.This can be generalized to any number of breaks and not just the single break observed in Fig. 4 for this material. Within

this framework, s�1 should be seen as a further parameter whose value should be determined from the data. In least squaresanalysis, the parameters of Eq. (5) are chosen so as to minimize the sum (over all observations) of the squares of e⁄ and v⁄. Inthis procedure no specification needs to be given as to how e⁄ or v⁄ (and thus _em or tf) are actually distributed.

However, this imposes some severe limitations on what can be achieved using Eq. (3). First, because least squares doesnot require any specification to be made about how failure times or creep rates are distributed, predictions from the Wilshireequations typically come without any confidence limits placed around then. Further, least squares cannot deal with the pres-ence of runout data. They must either be ignored or the runout times treated as a failure times – and in either approach thiscan lead to misleading parameter estimates.

Using a maximum likelihood technique is a neat alternative to least squares because not only are runouts a natural part ofthe estimation procedure, the need to specify a failure time distribution results in predictions being made with levels of con-fidence. The only concern is that the nature of the creep rate and the failure time distributions are unknown, so that the bestapproach is to use a very general specification for these distributions. Only then is it possible to see which distribution, con-tained as a special case within this general specification, is actually most supported by the data. One such general distribu-tion, suggested by Bartlett and Kendall’s [15], is the log gamma distribution. More recently, this distribution has beenmodified by Prentice [16] because in its original form the distribution had no limits. In this modification, the random variablee or v is taken to have the probability density function (PDF’s) given by

f ðeÞ ¼ kk�0:5

CðkÞ expffiffiffikpðe=reÞ � k exp e= re

ffiffiffikp �h i

; ð6aÞ

f ðvÞ ¼ kk�0:5

CðkÞ expffiffiffikpðv=rvÞ � k exp v rv

ffiffiffikp �.h i

; ð6bÞ

where C(k) is the gamma function and re and rv are the parameters that standardises the random variables e⁄ and v⁄. Thisdistribution has been successfully applied by Evans [17] to data on 1Cr–1Mo steel. Prentice has shown that when the param-eter k = 1, v and e have standard extreme value distributions (and so, for example, failure times are then weibull distributed).But when k =1, e and v have standard normal distributions (and so failure times are then log normally distributed). In thisspecial case, re and rv are the standard deviations for the log of the minimum creep rate and the log of the times to failurerespectively. (Further, when k = 1 and rv = 1, v has a standard exponential distribution). The gamma distribution is also aspecial case. Any quantile (q) of this distribution (used to form confidence limits) is then given by

M. Evans / Applied Mathematical Modelling 35 (2011) 2838–2854 2845

ln½ _em�q þ Q �c1

RT

� �¼ a0 þ a1s� þ a2½s� � s�1�D1 þxk;qre; ð6cÞ

ln½tf �qq� Q �c

1RT

� �¼ b0 þ b1s� þ b2½s� � s�1�D1 þxk;qrv ; ð6dÞ

where

xk;q ¼ffiffiffikp

ln12k

v22k;q

� �ð6eÞ

with v2 having a value that corresponds to the qth quantile of the chi square distribution with 2k degrees of freedom.In maximum likelihood estimation, the parameters of Eq. (5) are chosen so as to maximise the joint probability of

observing all the observed failure times and all the observed runout times. This is typically done by maximising a loglikelihood function, which given Eq. (6b) has the form

lnðLiÞ ¼XN1

i¼1

ðk� 0:5Þ lnðkÞ � ln CðkÞ � lnðrvÞ þffiffiffikp

v i � kev i=ffiffikpn oþXN2

i¼1

ln Q k; kev i=ffiffikp �

; ð7Þ

where Q() is the incomplete gamma integral and gives the probability of specimens surviving a given length of time (naturallog of time in fact) and N1 are the number of failed specimens in the batch of data and N2 the number of runouts or unfailedspecimens. As all the recorded minimum creep rates are uncensored, the parameters of Eq. (5a) are obtained by maximisingEq. (7) without the last summation term included (and with e replacing v and with re replacing rv).

3.3. Results

Table 2a shows the maximum likelihood estimates of Eq. (4) that were obtained using all the specimens tested within theAAL batch – including the runouts. The first point of interest about these results is that ignoring the runout results has anoticeable impact on the estimates made for b0 and b1 in the low stress regime where the runout specimens occur, eventhough there are only three such specimens. Further, the use of runout times has the effect of lowering slightly the estimatemade for such standard errors which in turn will impact upon the estimates made for any confidence limits associated withthe time to failure. Secondly, the estimates made for b0 differ significantly from each other over the two stress regimes. Thesame is true for b1. Thirdly, the minimum creep rates appear to have a weibull distribution, with k = 1 being most supportedby this data set. On the other hand, failure times appear to have a generalised gamma distribution with k = 50 being mostsupported by this data set (although in practice there is only a small difference between the shapes of the distributions asso-ciated with these two values for k). This difference could be attributable to the much smaller number of observations on _em

compared to tf. Clearly however, the normal distribution is not suitable for modelling either of these material properties.These values, in conjunction with Eqs. (5) and (6), are then used to obtained the curves shown in Figs. 2–4. When con-

sidering minimum creep rates, the fit of the curves to the data shown in Fig. 3a and Fig. 4 a are very good with only one datapoints falling outside the 0.01–0.99 quantiles. These curves are again shown in Fig. 2a but in the more familiar stress –minimum creep rate space. Again the fit obtained is good with all but one data point being within the same qauntile rangeat 973 K. Similar conclusions can be drawn by inspecting the failure times shown in Figs. 2b, 3b and 4b.

In Table 2b the same parameters were estimate but using only those specimens that failed at or before 5000 h. (Theparameters of Eq. (5a) were not estimated from this reduced sample because too few results existed to reliably identify

Table 2aThe values of the coefficients, a0 and a1 in Eq. (4a) and b0 and b1 in Eq. (4b) determined by maximum likelihoodestimation using all the AAL batch data at 873–1023 K for 18Cr–12Ni–Mo steel tubes.

Ignore runouts Runouts included

High stress Low stress High stress Low stress

a0 29.151 31.558 – –[0.126] [0.472]

a1 �8.516 �17.430 – –[0.750] [1.844]

re 0.423 [0.089] –b0 �14.391 �15.745 �14.392 �15.861

[0.042] [0.181] [0.039] [0.175]b1 6.521 10.888 6.510 11.249

[0.154] [0.587] [0.138] [0.566]rv 0.270 [0.026] 0.244 [0.024]

Standard errors shown in parenthesis. s�1 ¼ 0:30 in Eqs. (5a) and (5b). k = 1 for minimum creep rate and k = 50 forfailure time data. – Indicates that all the recorded minimum creep rates are uncensored.

Table 2bThe values of the coefficients, b0 and b1 in Eq. (4b) determined by maximum likelihoodestimation using the AAL batch data when tf 6 5000 h at 873–1023 K for 18Cr–12Ni–Mosteel tubes.

High stress Low stress

b0 �14.393 �15.923[0.047] [0.252]

b1 6.509 11.446[0.160] [0.814]

rv 0.266 [0.029]

Standard errors shown in parenthesis. s�1 ¼ 0:30 in Eq. (5b). k = 50 for failure time data.

Fig. 5. The stress dependence of the time to failure at 873–1023 K for the AAL batch of 18Cr–12Ni–Mo steel tubes. The curves are drawn using Eq. (6d) withthe parameter values as given in Table 2b (sub sample).

2846 M. Evans / Applied Mathematical Modelling 35 (2011) 2838–2854

the value of the parameters in both stress regimes). In this reduced sample there are of course no runouts. This additionalexercise was carried out in order to assess the ability of the Wilshire equation to predict long-term creep properties byextrapolation of short-term (defined at 5000 h or less) results for this steel. It can be seen by comparing Table 2a and b thatthe estimates made for each parameter are not severely affected by whether the full or partial sample is used. It is thereforenot a great surprise to see that in Fig. 5 the extrapolation to lower stresses and temperatures is very good, with all but oneof the results falling with the predicted 0.01–0.99 quantiles at 973 K. At all the other temperatures the median predictionsappear to run through the middle of the experimental data points (apart from at the highest stresses and temperature).

4. Creep of austenitic 18Cr–12Ni–Mo steel tubes: a multi batch analysis

4.1. Data description and mathematical representation

All the above results were carried out on a single batch of material. In all the applications of the Wilshire equations ref-erenced above (including that recently carried out by Evans [18] on this material), the analysis has however been carried outon multi batches of the same material using least squares techniques. Unfortunately, this is even more inappropriate thanwhen analysing a single batch of material using least squares. The reason is that in such a situation the observed minimumcreep rates will not be independent of each other (nor will the individual failure times). In creep data sheet No. 6B, there is atwo level sampling structure involving the batches selected for testing and the individual specimens tested for each batch –see Section 2 above. Now this batch to batch variability is typically much greater that the within batch variability as can beseen through a comparison of Fig. 6, where the results from all nine batches of 18Cr–12Ni–Mo steel are shown, with Fig. 2bthat contains only the AAL batch. (No multi batch data on minimum creep rates are available on creep data sheet No. 6B.)Whilst estimation by least squares remains consistent in the presence of dependent test results, it will be less efficient than it

Fig. 6. The stress dependence of the time to failure at 873–1023 K for all batches of 18Cr–12Ni–Mo steel. The curves are drawn using Eq. (8) with theparameter values shown in Table 3a (full sample estimates).

M. Evans / Applied Mathematical Modelling 35 (2011) 2838–2854 2847

could be. The parameter estimates, on average, will therefore tend to differ more from the true values than is appropriate andconfidence intervals will be under estimated. The size of this underestimation has never been quantified because all of thepublished literature involving the application of the Wilshire equations has ignored this fundamental issue.

It is also worth pointing out that the scatter present in this austenitic steel is much larger than that present in the NIMSdata bases for its ferritic, bainitic and martensitic steels. A potentially useful way to describe and analyse multi batch data isto allow the parameters of Eq. (4) to differ from batch to batch in a purely random fashion. This makes sense if it is believedthat the chemical compositions of each batch differ from each other in a random fashion. Within the lifetime statisticalliterature, multilevel models (see Hougaard [19] for a detailed review) have been developed to account for any dependencewithin clusters, such as within different batches of material. A multilevel version of Eq. (4) has the form

ln½tF �ijq¼ b0j þ b1jln½�lnðsij=sTSÞ� þ b2j

1RT

� �ijþ v�ij ð8aÞ

where

b0j ¼ b0 þ e0j; b1j ¼ b1 þ e1j; b2j ¼ b2 þ e2j ð8bÞ

and the new subscripts refer to the results from the ith specimen cut from the jth batch of material and v ij ¼ rvv�ij is an inde-pendent error term that has a distributional form given by Eq. (6b) and a variance given by r2

V . As there are now nine dif-ferent batches that may have randomly different activation energies associated with them, Eq. (8) allow for such variation bymaking b2 a parameter to be estimated. Theoretically, b2 should be similar in value to that given for Q �c , with there beingrandom variation in this value from batch to batch – as picked up by the b2j values. In data sheet No. 6B there are m = 9batches and within the jth batch the number of specimens tested is variable and is given by nj. bk (k = 0–2) are the coefficientsof scientific interest and ekj are the random effects. In this multilevel model, the random effects, and therefore the coefficientsthemselves, are taken to follow a joint Gaussian distribution with mean vector 0 and unknown variance–covariance matrixX, where X is a 3 � 3 symmetric matrix with variances on the diagonal and covariance’s off the diagonal. As such, and asseen in Eq. (8b), the coefficients vary from batch to batch in a random fashion.

Again, spline functions can be used to allow for differing coefficient values between different stress regimes

ln½tf �ij ¼ b0j þ b1js�ij þ b2j½s�ij � s�1�D1 þ b3j1

RT

� �ijþ rvv ij; ð8cÞ

where

b0j ¼ b0 þ u0j; b1j ¼ b1 þ u1j; b2j ¼ b2 þ u2j; b3j ¼ b3 þ u3j ð8dÞ

and ukj (k = 0–3) are the random effects. These random effects are again taken to follow a joint Gaussian distribution withmean vector 0 and unknown variance–covariance matrix D, where D is a 4 � 4 symmetric matrix with variances on the diag-onal and covariance’s off the diagonal

Table 3The valat 873–

b0

b1

b2

rv

Standar

2848 M. Evans / Applied Mathematical Modelling 35 (2011) 2838–2854

D ¼

r2u0

ru01 r2u1

ru02 ru12 r2u2

ru04 ru13 ru23 r2u3

266664

377775 ð8eÞ

4.2. Parameter estimation

The Appendix to this paper describes a simulated maximum likelihood procedure that can be used to estimate values forD, r2

V and the parameters b0–b3. Whilst the least squares procedure is more well known when studying fracture, Pascual andMeeker [20] have used maximum likelihood techniques to estimate the fatigue – limit parameter in nickel-base super alloy,Evans [21] has used these techniques for estimating fracture probability relations in ceramics at high temperatures andEvans [22] used maximum likelihood to estimate creep failure distributions in the presence of censored data. The basic ideais to simulate values for ukj and use these values to compute a number for the log likelihood given by Eq. (A.7) at a givenvalue for D, r2

V and the parameters b0–b3. Then Newton–Raphson optimisation algorithms can be used to find which valuesfor these parameters maximise the log likelihood at the simulated values for ukj. This procedure is then repeated R times,where R is the number of iterations, to yield R values for these parameters, the averages of which are taken to be the sim-ulated maximum likelihood estimates ~D; ~rv ;

~b0; to; ~b3 of D, r2V and the parameters b0–b3.

This procedure estimates the mean vector b (containing b0–b3) of the random process generating the vector bj (containingb0j–b3j). It is often useful to produce predictions of the batch parameter vector as well, i.e. the bj. A posterior estimate thatuses all the information available within the sample of data is given by

~bj ¼PR

r¼1~bjrwjrPR

r¼1wjr

; ~bjr ¼ ~bþ ~Kujr; ð9Þ

where wjr is the exponential of Eq. (A.5) that can be calculated for each batch during each of the R simulations.

4.3. Results

The simulated maximum likelihood estimates of b0 and b1 in Eq. (8b) are shown in Table 3a It can be seen that the treat-ment of runouts has little influence on the resulting estimates. This is not surprising as all runouts are confined to the AALbatch so that in this multi batch analysis the number of censored data points is a smaller proportion of the data set thanwhen analysis was just confined to the AAL batch. A comparison of Table 2a with Table 3a further reveals that taking intoaccount the hierarchical nature of the data through use of a multilevel model does alter the estimates made for b0 and b1.Irrespective of the stress regime the estimate made for b0 is much lower. The estimate made for b2 is a little lower than360 kJ mol�1 suggesting that batch AAL has an above average activation energy for this material. In the high stress regimethe estimate made for b1 is larger, but the opposite is true in the low stress regime. However, the biggest effect is seen on theestimates made for the standard error for b1. Taking into account the hierarchical nature of the data, results in a substantialdrop in the estimated standard errors as a large part of the scatter in the data is now picked up by the random effects them-selves. The estimate variance-covariance matrix D (when runouts are included in the analysis) is estimated to be

D ¼

0:767�0:037 0:059�0:138 �0:167 1:470�5:796 0:303 0:768 44:245

26664

37775

This latter result means that estimated confidence bounds are also very different once the dependency in the test resultsis correctly modelled. This can be seen in Fig. 7 a where the confidence limits for failure times at 973 K derived from the

aues of the coefficients, b0 and b1 in Eq. (8b) determined by simulated maximum likelihood estimation using all the results within all the batches of bata1023K for 18Cr–12Ni–Mo steel tubes.

Ignore runouts Runouts included

High stress Low stress High stress Low stress

�20.415[0.151]

�20.089[0.164]

�20.693[0.158]

�20.369[0.168]

9.678[0.035]

8.6266[0.210]

9.709[0.035]

8.666[0.192]

351.394 [1.143] 353.575 [1.192]0.794 [0.010] 0.7907 [0.009]

d errors shown in parenthesis. s�1 ¼ 0:30 in Eq. (8c). k = 50.

Fig. 7. (a) A comparison of single and multi batch fits to the data at 973 K. Curves drawn from results using all the data on 18Cr–12Ni–Mo steel tubes. (b)The stress dependence of the time to failure at 973 K for all batches of 18Cr–12Ni–Mo steel tubes. The curves are drawn using Eq. (9) with parameter valuesas given in Table 3b.

Table 3bThe values of the individual batch coefficients, b0j and b1j in Eq. (8a) determined by use of Eq. (9) using all the results within all the batches of data at 773–923Kfor 873–1023 K for 18Cr–12Ni–Mo steel tubes.

Batch High stress Low stress

b0j b1j b2j b0j b1j b2j

AAA �21.003 10.117 355.686 �20.336 8.064 355.686AAB �21.671 10.165 360.480 �21.309 8.374 360.480AAC �20.724 9.712 353.831 �20.328 8.632 353.831AAD �21.358 10.055 358.104 �20.821 8.723 358.104AAE �22.329 9.776 364.986 �22.105 9.492 364.986AAF �22.355 9.888 365.328 �22.159 9.188 365.328AAL �22.139 9.893 363.791 �21.685 9.000 363.791AAM �20.975 10.184 355.369 �20.593 8.377 355.369AAN �20.321 10.242 350.132 �19.784 7.786 350.132

M. Evans / Applied Mathematical Modelling 35 (2011) 2838–2854 2849

2850 M. Evans / Applied Mathematical Modelling 35 (2011) 2838–2854

multilevel model are compared to those shown in Fig. 2b derived using a single batch of material when all the results withineach batch are analysed. The median predictions from the single and multi batch analysis are very similar for stresses in ex-cess of 70 MPa. Below this stress the AAL batch of material appears to behave differently to some of the other batches so thatthe median predictions obtained using all the batches of data diverge from those obtained using the AAL batch.

The confidence limits are also very different – especially at the longest failure times corresponding to the lowest stresses.This is quite significant because failure to take into account test dependency severely under estimates the safe working life ofthis material. For example, if safe is defined as a 1% chance of failure, then at a stress of around 40 MPa, the single batchanalysis suggest a life of around 8000 days, compared to around 500 days for the multi batch analysis. The curves inFig. 6 are drawn using the multilevel model of Eq. (8) using full sample estimates and show the model gives a good descrip-tion of the data at the other test temperatures as well.

Another way to visualise this variability is using the batch by batch estimates of the parameters of Eq. (8a) as derivedusing Eq. (9). These estimates are shown in Table 3b and they result in the median prediction lines shown in Fig. 7b whenthe estimate for the symmetric matrix D is as given above. As can be seen from this figure, the variation in the batch to batch

Fig. 8. (a) The dependence of ln t1:5f exp �354;000=RTð Þ

�on ln(�ln(s/sTS)) at 873-1023K for all batches of 18Cr–12Ni–Mo steel tubes. The curves are drawn

using Eq. (8) with the parameter values as given in Table 3a. (b) The stress dependence of the time to failure at 873–1023 K for all batches of 18Cr–12Ni–Mosteel. The curves are drawn using Eq. (8) with the parameter values in Table 4 (sub sample estimates).

Table 4The values of the coefficients, b0 and b1 in Eq. (8b) determined by maximum likelihoodestimation using all the results within all the batches of data when tf 6 5000 h 873–1023 Kfor 18Cr–12Ni–Mo steel tubes.

High stress Low stress

b0 �21.393 [0.128] �21.137 [0.137]b1 9.648 [0.032] 8.820 [0.153]b2 358.077 [0.997]rv 0.862 [0.004]

Standard errors shown in parenthesis. s�1 ¼ 0:30 in Eq. (8c). k = 50.

M. Evans / Applied Mathematical Modelling 35 (2011) 2838–2854 2851

predictions mimic well the actual batch to batch variation in the data. A similar result was observed at the other tempera-tures as well.

Fig. 8(a) plots ln[t1:5f exp(�354,000/RT)] against ln[�ln(s/sTS)], where 354,000 is the estimate made for the average acti-

vation energy over all batch shown in Table 3a. The large degree of scatter present in the failure times for this material isclearly evident, but despite this only a handful of data points fall outside the predicted 0.01–0.99 quantiles.

Finally, in Table 4 the same parameters of Eq. (8b) were estimate but using only those specimens that failed at or before5000 h. In this reduced sample there are of course no runouts. The estimate made for D was

D ¼

0:485�0:025 0:040�0:051 �0:075 0:966�3:828 0:180 0:229 30:669

26664

37775

This exercise was carried out in order to assess the ability of the Wilshire equation to predict long-term creep properties byextrapolation of short-term (defined at 5000 h or less) results for this steel. It can be seen by comparing Table 4 with Table 3athat the estimates made for each parameter are not severely affected by whether the full or partial sample is used. It is there-fore not a surprise to see that in Fig. 8 the extrapolation to lower stresses and temperatures is very good, with all but theexpected 3 or 4 results falling with the predicted 0.01–0.99 quantiles. Apart from at the lowest stresses at 973 K and1023 K the median predictions also appear to run through the middle of the experimental data points.

5. Conclusions

The creep and creep fracture behaviour of 18Cr–12Ni–Mo steels, normally characterized by power law descriptions of theminimum creep rates and rupture lives, are rationalized by normalising the applied stress through the ultimate tensile stressdetermined from high-strain-rate tensile tests carried out at the creep temperatures. Using these new relationships, termedthe Wilshire equations, these rationalization procedures allow the data set for 18Cr–12Ni–Mo steel to be discussed in termsof the dislocation processes controlling creep strain accumulation and the different damage phenomena causing tertiarycreep and eventual failure.

Moreover, these new relationships also enable long-term data to be predicted by extrapolation of short term results. Astatistical framework was developed for the analysis of both single and multi batch data sets, which enable for the first timeconfidence limits to be placed around the predictions obtained using the Wilshire equations. It was shown that a failure toallow for the hierarchical nature of creep data bases, results in a serious and significant underestimate of the predicted safelife corresponding to operating conditions. Areas for future work include the extension of the above model to three or morelevels so that results for different products (such as bars and plates as well as tubes) and from different laboratories for thesame material can be analysed using a single model.

Appendix A

A.1. Test result dependency and heteroscedasticity

Estimation of the multilevel model given by Eq. (8) is not straightforward. To see why note that Eq. (8a) can be written as

ln tf�

ij

q¼ b0 þ bk

X3

k¼1

xkij þwij ðA:1Þ

where

wij ¼ rvv ij þ u0j þX3

k¼1

ukjxkij ðA:2Þ

2852 M. Evans / Applied Mathematical Modelling 35 (2011) 2838–2854

and where

x1ij ¼ s�ij; x2ij ¼ ½s�ij � s�1�D1; x3ij ¼1000RTij

ðA:3Þ

The random error term (wij) in Eq. (A.1) is clearly a function of the test conditions and its variance, Var[wij], is given by(assuming that the ukj and vij are independent of each other)

Var½wij� ¼ r2vVar½v ij� þ Var½u0j� þ

X3

k¼1

Var½ukj�x2kij ðA:4Þ

and so also varies with the test conditions. This heteroscedasticity has important implications for parameter estimation. Itmeans that if the linear least squares formula is used to estimate the bk parameters in Eq. (A.1), by minimising Rw2

ij , theresulting estimates will be unbiased but inefficient (have higher variances than if Var[wij] were constant). Further, the esti-mates made of the standard errors of these parameters will be biased, so invalidating any standard tests of statistical signif-icance or any constructed confidence limits. Hence some other estimation procedure is required.

A.2. Simulated maximum likelihood (SML)

It follows from Eqs. (6b), (8a) and (8b) that the log likelihood for the nj observations in batch j (of which there are N1j

failure times and N2j runout times so that nj = N1j + N2j) is the sum of the logs of the densities (see Greene [23] for a similarderivation)

lnðLjjxj; ujÞ ¼XNij

i¼1

ðk� 0:5Þ lnðkÞ � ln CðkÞ � lnðrvÞ þffiffiffikp

v ij � keVij=ffiffikpn oþXN2j

i¼1

lnQ k; keVij=ffiffikp �

ðA:5Þ

where

v ij ¼

ln½tf �ijq � b0 þ bk

P3k¼1

xkij

� � u0j þ

P3k¼1

ukjxkij

�

rv

26664

37775 ðA:6Þ

and uj is a vector containing the k values for u0j to u3j and xj is a matrix containing the nj observations on the variables x1ij tox3ij. The log likelihood for the full sample, when there are m batches in all, is then

lnðLÞ ¼Xm

j¼1

lnðLjjxj; ujÞ ðA:7Þ

In principle this log likelihood could be maximised with respect to b0, all the bk and rv. However, there are two problemswith this approach. First, the variance–covariance matrix for uj (i.e. D) is not present and so cannot be estimated. This is eas-ily solved through the following standardisation process

r�ukzkj ¼ ukj if k ¼ 0

r�ukzkj þ r�uðk�rÞkPkr¼1

zðk�rÞj ¼ ukj otherwiseðA:8Þ

and r�uðk�rÞk and r�uk are further parameters whose values are unknown. The relationship between the random vector uj andits standardised equivalent, zj, is therefore given byuj = Kzj, where K is a lower triangular matrix made up of the above ruk�r)k

and ruk parameters

K ¼

r�u0

r�u01r�u1

r�u02r�u12

r�u2

r�u04r�u13

r�u23r�u3

26664

37775 ðA:9Þ

The full variance–covariance matrix for uj is then found by letting KK0 = D, where D is given by Eq. (8e). Eq. (A.8) can besubstituted into Eq. (A.6) and then Eq. (A.7) can be maximised with respect to b0, all the bkj, K and rv. D can then be foundfrom KK0. The complication in doing this is of course the fact that the zj are unobserved and so the log likelihood cannot becomputed directly. The solution requires zj to be integrated out of the above conditional likelihood, but such integralstypically do not have an analytical solution. However, integrals of this nature can be adequately evaluated by simulationmethods.

M. Evans / Applied Mathematical Modelling 35 (2011) 2838–2854 2853

The basic steps involved are:

1. Obtain at random k = 1–4 draws from the standard normal distribution using the well known Box–Muller [24] method.These values then define zk1 in Eq. (A.8), i.e. for j = 1.

2. Randomly select values for r�uðk�rÞk and r�uk in Eq (A.8).3. Use Eq. (A.8) together with the random values obtained in steps 1 and 2 above to obtain values for uk1.4. Randomly select k = 1–4 values for bk in Eq. (A.6) together with a random value for b0 and rv in this equation.5. Insert the values obtained from steps 3 and 4 into Eq. (A.6) to calculate N11 values for vi1 and then insert the values for vi1

into Eq. (A.5) to obtain a numeric value for the log likelihood associated with batch j = 1.6. Repeat steps 1, 3 and 5 a total of m times (i.e use the same r�uðk�rÞk, r�uk, b0 , bk and rv values) to obtain numeric values for

zkj and ukj (j = 2–m) and from these calculate the log likelihoods associated with all the other batches (j = 2–m). Then addup these individual log likelihoods to obtain a numeric value for the log likelihood of the full sample, i.e. ln(L) in Eq. (A.7).

7. Then run a standard numerical optimisation procedure such as to find values for b0, all the bk, rv, r�uðk�rÞk and r�uk thatmaximise ln(L).

8. Finally, this maximisation procedure is then repeated R times yielding R values for b0 all the bk, rv, r�uðk�rÞk and r�uk. Theaverage of these R values can be taken as the simulated maximum likelihood estimates for these parameters, and thestandard deviation in these R values as an estimate of their standard errors.

This estimation procedure can be repeated using different values for k, and the value for k that maximises the average loglikelihood over all R iterations is chosen as the correct value for k. This is quite a straightforward procedure in practicebecause obtaining random draws from Eq. (6b) is easily implemented within popular and commercially available softwarepackages such as Microsoft Excel [25] or more specialised Econometric software such as Regression Analysis of Time Series[26] or RATS for short. In the former package this can be done using the Normsinv(rand()) functions and in the latter packagethis can be done directly using the %ran(1) function.

A.3. Confidence intervals

A simulation approach can also be used to obtain confidence limits for the multilevel model of Eq. (8). This approach re-quires first simulating values for vij from Eq. (6b) to obtain ~rvv ij where the tilde represents the simulated maximum likeli-hood (SML) estimate for rv. In the RATS package this can be done directly using the %rangamma(k) function. This will returna random draw from a one parameter gamma distribution, so that its natural log is a draw from the one parameter log gam-ma distribution. Subtracting the natural log of k, and then multiplying by the square root of k gives a random draw from thePDF given by Eq. (6b), i.e. a random value for vij. This last transformation is required for the log gamma distribution to be nondegenerate as k tends to infinity and for its mean to be zero. Even without specialised software like RATS, obtaining randomvalues for vij is straightforward. All that is required is an ability to draw random numbers from a standard normal distribu-tion because the value of a variable drawn from a chi square distribution with k degrees of freedom is equal to half the valueof a variable drawn from the one parameter gamma distribution with k/2 degrees of freedom (see [27] for more details).

This simulation approach then requires simulating values for uj, using uj ¼ ~Kzj, where the tilde refers to the SML estimateof K and the zj are simulated in the way described above. Then b is simulated using

b ¼ ~bþ f~K=mgzj ðA:10Þ

where b is a vector made up of b0 and all the bk. The tilde refers to the SML value for b. These simulated values for b, uj and vij

are then inserted into Eqs. (A.11) to simulate the value for ln[tf] that is associated with a particular value for x

ln½tf �ijq¼ b0 þ bk

X3

k¼1

xkij þ rvv ij þ u0j þX3

k¼1

ukjxkij: ðA:11Þ

Repeating this a large number of times yields an empirical distribution for ln[tf] and therefore tf at a particular value for x.From this, any quantile of the empirical distribution can be obtained to represent confidence limits around the simulatedmedian prediction. Repeating this for other values of the test variables completes the calculations required to obtain predic-tions with confidence limits for tf..

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