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Materials Science and Engineering A 528 (2010) 500–512

Contents lists available at ScienceDirect

Materials Science and Engineering A

journa l homepage: www.e lsev ier .com/ locate /msea

multi-level model for the analysis and prediction of creep and creepracture of 2.25Cr–1Mo steel tubes

. Evans ∗

aterials Research Centre, School of Engineering, Swansea University, Swansea SA2 8PP, UK

r t i c l e i n f o

rticle history:eceived 1 July 2010eceived in revised form 9 September 2010

a b s t r a c t

The creep and creep rupture properties of 2.25Cr–1Mo steel tubes have been analysed using the Wilshireequations. The observed behaviour patterns are then discussed in terms of the dislocation processesgoverning creep strain accumulation. A suitable statistical framework for analysing both the single and

ccepted 9 September 2010

eywords:reepracture

multi batch data available on this material is then specified. It is shown that ignoring the hierarchicalnature present in many creep data bases, which has been the approach until now when using the Wilshireequations, leads to a serious and significant underestimate of the predicted safe life for this material. Themodel allows accurate predictions of long-term properties by extrapolation of short-term results for thissteel.

ainitic steelsulti level models

. Introduction

In general, when selecting alloy steels for large-scale compo-ents used in power and petrochemical plants, decisions are basedn the ‘allowable creep strengths’, normally calculated from theensile stresses causing failure in 100,000 h at the relevant serviceemperatures [1]. However, creep life measurements for structuralteels show considerable batch to batch variability so, in Europe,ests up to 30,000 h have often been completed for five melts ofach steel grade [2]. Yet, even when using these extensive results,he 100,000 h strength estimates depend on the methods adoptedo make the calculations [3,4], despite the international activitiesevoted to the assessment of different data analysis procedures5]. Moreover, for a series of 9–12% chromium steels developedor ultra super-critical power plant the allowable strengths haveeen reduced progressively as the test durations have increasedell beyond 30,000 h towards 100,000 and more [3,4].

In contrast to the problems encountered using traditionalarametric procedures for extrapolation of short-term creep liferoperties [5], a new methodology appears to allow accuraterediction of 100,000 h strengths by considering results with max-

mum lives up to 10,000 h or less [6–9]. This new approach, based

n normalising the applied stress (�) through the ultimate tensiletress (�TS) at the creep temperatures for each batch of materialnvestigated, has proved successful for a series of ferritic [9], bainitic8] and martensitic steels [7]. In addition, confidence in this method

∗ Tel.: +44 (0) 1792 295748; fax: +44 (0) 1792 295676.E-mail address: M.evans@swansea.ac.uk.

921-5093/$ – see front matter © 2010 Elsevier B.V. All rights reserved.oi:10.1016/j.msea.2010.09.032

© 2010 Elsevier B.V. All rights reserved.

is improved by interpretation of the property sets in terms of theprocesses governing creep and creep fracture [6]. Hence, in thepresent study, this new methodology is adopted for 2.25Cr–1Mosteel tubes using the creep data sheets produced at 773–923 K bythe National Institute for Materials Science (NIMS), Japan [10].

A single and multi batch analysis is carried out because in all theapplications of the Wilshire equations referenced above, the analy-sis has been carried out on multi batches of the same material usingtechniques (namely the use of linear least squares) that are reallyonly appropriate for single batches of material. This is because inmulti batch data sets the material properties (such as minimumcreep rates and times to failure) will not be independent of eachother, making least squares parameter estimates less efficient thanthey could be. The parameter estimates, on average, will thereforetend to differ more from the true values than is appropriate andconfidence intervals for safe life will be under estimated. Unfortu-nately, the size of this underestimation has never been quantified.

The aim of this paper is therefore not just to help validate thisnew methodology through an application to a material not yetsubjected to the Wilshire equations, but to also provide a moresuitable statistical framework that should be used when applyingthe Wilshire equations to multi batch data sets. To achieve thisaim, the paper is therefore structured as follows. Section 2 brieflydescribes the material used for this study and the nature of theNIMS data base available on this material. Section 3 provides a sin-

gle batch analysis of this material (using the MAF batch) that issimilar in structure to that in all the above referenced Wilshire etal. papers. It is however different in two important respects. Theframework adopted allows for the derivation of confidence limitsand it can cope with the presence of runout data that often exist in

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M. Evans / Materials Science an

ong term high temperature data. Section 4 applies a new type oftatistical model to the multi batch data, with the statistical modeleing described in detail within an Appendix A. The paper finishesith some conclusions and suggestions for future work.

. The data

The National Institute for Materials Science (NIMS) in Japan havearried out an extensive high temperature testing program involv-ng many different steel alloys. Creep Data Sheet No. 3B containsnformation on 2.25Cr–1Mo steel tubes [10]. Creep tests were con-ucted on solid cylindrical specimens with a gauge length of 30 mmt constant load with a loading accuracy of ±0.5%. This database isery hierarchical in nature, with test specimens being cut from dif-erent batches of the same material. These batches differ both in theype of heat treatment used and in their chemical compositions. Inotal, this data sheet contains twelve different batches of material,eferenced by NIMS as MAA through to MAN. Three of these batchesere obtained from hot extruded and cold drawn ingots that wereeat treated for 3600 s at 1193 K followed by 5400 s at 1013 K andir quenched. Three of these batches were obtained from rotaryierced and cold drawn ingots that were heat treated for 1200 s at200 K followed by 7800 s at 993 K and air quenched. Three of theseatches were obtained from rotary pierced and cold drawn ingotshat were heat treated for 4200 s at 1183 K followed by 4200 s at83 K and air quenched. The final three batches were obtained fromully annealed hot extruded and cold drawn ingots. The chemicalompositions of these different batches can be found in [10].

Fig. 1a shows the structure of the NIMS data base for this mate-ial. In this figure only the first and last few results are shown forach batch and the data base is sorted by batch then by temperaturend finally by stress to give an impression of the hierarchical naturef this data base. In total some 332 specimens were tested over 5 dif-erent temperatures and over a wide range of different stresses withhe resulting failure times varying from just a few hours througho over 100,000 h. At the time of publishing creep data sheet No.B (and indeed in subsequent online revisions), seven specimensemain on test. All that is known about these so-called runouts,s that failure will occur beyond the published censored time. Allhese runouts were within the MAF batch of material and mini-

um creep rates were only recorded for this batch of material. Forhe rest of the paper, the phrase sub sample will refer to all thexperimental results corresponding to failure times of less than0,000 h.

. Creep of bainitic 2.25Cr–1Mo steel tubes: a single batchnalysis

.1. Data description and mathematical representation

The MAF batch was chosen for individual analysis as this is thenly batch for which both minimum creep rates and times to failureere recorded. For over half a century, the dependencies of theinimum creep rate (ε̇m) and the creep fracture life (tf) on stress�) and temperature (T) in such data sets have been described usingower law equations of the form

M

tf= ε̇m = A�n exp

−QcRT

(1)

here R = 8.314 J mol−1 K−1 and M and A are model constants that

re material specific. Many variants of this description have alsoeen proposed in the literature including the Larson and Miller [11]nd Minimum Commitment [12] methods. From the ln(ε̇m)/ln(�)lots, the stress exponent (n) is calculated to decrease from aroundat high stresses at 773 K to almost 3 under low stresses at 923 K,

neering A 528 (2010) 500–512 501

with the activation energy for creep (Qc) ranging from about 280to 450 kJ mol−1 (see Fig. 2a) Similar trends were observed for theln(tf)/ln(�) relationships in Fig. 2b. Clearly, these unpredictablevariations in n and Qc make it impossible to estimate long-termproperties by extrapolation of the short-term data. The inappropri-ateness of equations like Eq. (1) is further revealed by the tendencyfor isothermal experimental data points to trace out curves ratherthan straight lines over wide stress ranges.

To consider the effects of normalising the applied stress, use wasmade of the NIMS �TS (tensile strength) values – some values forwhich are shown in Fig. 1. Using these results, the power law plotsin Fig. 2 were rationalised by normalising stress through the use of�TS so that Eq. (1) can be rewritten as

M

tf= ε̇m = A∗

(�

�TS

)nexp

(−Q

∗c

RT

)(2)

where A* /= A and �TS is the tensile strength at the creep test tem-perature. Adopting Eq. (2), the data sets at different temperaturesare superimposed onto a single curve through an activation energy(Q ∗c ) of 250 kJ mol−1 (see Fig. 3), where Q ∗

c is determined from thetemperature dependencies of ε̇m and tf at constant (�/�TS), ratherthan at constant � as in the determination of Qc with Eq. (1). How-ever, this procedure does not eliminate the decrease from n ∼= 9 to3, so the unknown curvatures of the ln(ε̇m) versus ln(�/�TS) andln tf versus log(�/�TS) plots in Fig. 3 still does not allow predictionof long-term properties by extrapolation of short-term results forthis steel.

In contrast, extended extrapolation to provide 100,000 h datafrom tests lasting up to 10,000 h or less does appear to be possibleusing the Wilshire equations [6–9] given by(�

�TS

)= exp

{−k2

[ε̇m exp

(Q ∗c

RT

)]u2}(3a)

(�

�TS

)= exp

{−k1

[tf exp

(−Q ∗c

RT

)]u1}(3b)

where the coefficients, k1 and k2 as well as u1 and u2, areeasily determined from plots of and ln[tf exp(−Q ∗

c /RT)] againstln[−ln(�/�TS)]. These plots are shown as Fig. 4.

Wilshire has never suggested any particular physical mecha-nism that could lead to Eq. (3a), except to emphasis the requirementthat the functional form chosen must have the property that theminimum creep rate tends to infinity as �/�TS tends to 1 and thatthe minimum creep rate tends to zero as �/�TS tends to 0. Eq. (3a)has exactly this property, although there are many other functionalforms that also have this property. Indeed, Evans [13] has experi-mented with different sigmoidal functions. To go from Eqs. (3a) to(3b) then requires acceptance of the well known Monkman–Grantrelation [14], shown for the MAF batch of this material in Fig. 1b.Haney et al. [15] has shown that for various martensitic and bainiticsteels, failure is often due to necking and Hoff [16] identifiedthe relationship between the Monkman–Grant plot and necking.Strictly speaking, Eq. (3b) follows from Eq. (3a) only when the expo-nent on ε̇m shown in Fig. 1b equals unity. More precisely Q ∗

c in Eq.(3b) should be replaced by Q ∗

c raised to the power of this exponentvalue. This refinement is ignored in this paper as the true valuefor this exponent over all the batches of steel contained in [10]is unknown and may well be close to unity given the substantialscatter present in Fig. 1a.

Essentially, the results shown in Fig. 4 are best described as trac-ing out three intersecting straight lines, so that it then becomes

necessary to explain the behavioural differences associated witheach straight line. On decreasing the applied stress and increas-ing the creep temperature, the first change, when looking at thefailure times if Fig. 4b, occurs at ln[−ln(�/�TS)] = −0.2 signifyingthat the creep lives are longer at lower stress than would be pre-

502 M. Evans / Materials Science and Engineering A 528 (2010) 500–512

F e tos

dAmai

ilitt1(t

ig. 1. (a) Structure of NIMS creep data sheet No. 3B. (b) The dependence of the timteel tubes. Best fit line obtained using least squares.

icted by linear extrapolation of results when ln[−ln(�/�TS)] < −0.2.further change then occurs when ln[−ln(�/�TS)] is around 0.40eaning that, in long-term tests at 873 K and above, the creep lives

re much shorter than would be expected on extrapolation of thentermediate stress/temperature data.

This rather neat description of the data can also be given a mean-ngful interpretation based on deformation mechanisms. Withn[−ln(�/�TS)] = −0.2 roughly corresponding to 0.85�PS (where �PSs the 0.2% proof stress), changes in the coefficients of Eqs. (3) seem

o occur when � < 0.85�PS, such that the creep lives are longer andhe creep rates are slower at lower stresses. As previously found forCr–0.5Mo ferritic steels [8], this appears to take place when � < �Ywhere �Y is the yield stress), as it seems reasonable to assumehat �Y ∼= 0.85�PS for this type of material. Thus, when � > �Y, dislo-

failure on the minimum creep rate at 773–923 K for the MAF batch of 2.25Cr–1Mo

cations multiply rapidly during the initial strain on loading at thecreep temperature, giving high creep rates and shorter creep lives.In contrast, when � < �Y, creep must occur not by the generationof new dislocations but by the movement of the dislocations pre-existing in the as-received bainitic microstructure (or within thegrain boundaries), giving slower creep rates and correspondinglylonger creep lives.

Further, with this material the creep process actually degradesthe initial bainitic microstructures, such that the original lath-like

structure entirely disappears in the long-term tests at the highestcreep temperatures. This can be clearly seen by comparing the as-received and ruptured micrographs contained in NIMS data sheetNo. 3B. Thus, the creep rate increases sharply, leading to a cor-responding fall in creep life, as the bainitic microstructure of this

M. Evans / Materials Science and Engineering A 528 (2010) 500–512 503

F e MAp re at 7E

mp

3

utw

l

l

wˇpft

ig. 2. (a) The stress dependence of the minimum creep rate at 773–923 K for tharameter values as given in Table 1a. (b) The stress dependence of the time to failuq. (6d) with the parameter values as given in Table 1a.

aterial degrades rapidly with increasing test duration and tem-erature. This explains the second break in the data shown in Fig. 4.

.2. Parameter estimation

All applications of the Wilshire equations referenced above havesed the technique of linear least squares to identify the values forhe parameters in Eqs. (3). The easiest way to visualise how thisorks is to rewrite Eqs. (3) as

n[ε̇m] + Q ∗c

[1RT

]= ˛0 + ˛1 ln

[−ln

(�

�TS

)]+ e∗ (4a)

n[tf ] − Q ∗c

[1RT

]= ˇ0 + ˇ1 ln

[−ln

(�

�TS

)]+ v∗ (4b)

ith ˛0 = −ln(k2)/u2, ˛1 = 1/u2, e* =�ee and with ˇ0 = −ln(k1)/u1,1 = 1/u1, v∗ = �vv. The random variables e* and v∗ are added toick up the stochastic nature of creep properties, i.e. the tendencyor failure times or creep rates to differ under identical settings ofhe experimental variables like stress and temperature (as seen in

F batch of 2.25Cr–1Mo steel tubes. The curves are drawn using Eq. (6c) with the73–923 K for the MAF batch of 2.25Cr–1Mo steel tubes. The curves are drawn using

Fig. 2). e and v are standardised values for these random variables.One way to allow the values for k1, k2, u1 and u2 to be different indifferent stress ranges is through the use of spline functions. Theseare continuous functions that allow for such differing parametervalues

ln[ε̇m]+Q ∗c

[1RT

]=a0 + a1�

∗ + a2[�∗ − �∗1]D1 + a3[�∗ − �∗

2]D2 + �ee(5a)

ln[tf ] − Q ∗c

[1RT

]= b0 + b1�

∗ + b2[�∗ − �∗1]D1 + b3[�∗ − �∗

2]D2 + �vv

(5b)

where �* = ln(−ln(�/�TS)), D1 = 0 when (�∗ − �∗1) ≤ 0 and D1 = 1

otherwise. D2 = 0 when (�∗ − �∗2) ≤ 0 and D2 = 1 otherwise. Con-

sequently, in the stress regime corresponding to �∗ ≤ �∗1 ˛0 = a0,

˛1 = a1, ˇ0 = b0, ˇ1 = b1. But in the stress regime correspondingto , ˛0 = a0 − a2�∗

1, ˛1 = a1 + a2, , ˇ1 = b1 + b2. Finally, in the stress

504 M. Evans / Materials Science and Engineering A 528 (2010) 500–512

F r the Mp )) on ld

rˇ

twspasa

aadwsei

ig. 3. (a) The dependence of ln(ε̇m exp(250,000/RT)) on ln(�/�TS) at 773–923 K foarameter values as given in Table 1a. (b) The dependence of ln(tf exp(−250,000/RTrawn using Eq. (6d) with the parameter values as given in Table 1a.

egime corresponding to , ˛0 = a0 − a2�∗1 − a3�∗

2, ˛1 = a1 + a2 + a3,0 = b0 − b2�∗

1 − b3�∗2, ˇ1 = b1 + b2 + b3.

This can be generalized to any number of breaks and not just thewo breaks observed in Fig. 4 for this material. Within this frame-ork, �∗

1 and �∗2 should be seen as a further parameter whose value

hould be determined from the data. In least squares analysis, thearameters of Eqs. (5) are chosen so as to minimize the sum (overll observations) of the squares of e* and v∗. In this procedure nopecification needs to be given as to how e* or v∗ (and thus ε̇m or tf)re actually distributed.

However this imposes some severe limitations on what can bechieved using Eqs. (3). First, because least squares does not requireny specification to made about how failure times or creep rates are

istributed, predictions from the Wilshire equations typically comeithout any confidence limits placed around then. Further, least

quares cannot deal with the presence of runout data. They mustither be ignored or the runout time treated as a failure time – andn either approach this can lead to misleading parameter estimates.

AF batch of 2.25Cr–1Mo steel tubes. The curves are drawn using Eq. (6c) with then(�/�TS) at 773–923 K for the MAF batch of 2.25Cr–1Mo steel tubes. The curves are

Using a maximum likelihood technique is a neat alternative toleast squares because not only are runouts a natural part of theestimation procedure, the need to specify a failure time distributionresults in predictions being made with levels of confidence. Theonly concern is that the nature of the creep rate and the failuretime distributions are unknown, so that the best approach is touse a very general specification for these distributions. Only thenis it possible to see which distribution, contained as a special casewithin this general specification, is actually supported by the data.One such general distribution, suggested by Bartlett and Kendall’s[17], is the log gamma distribution. More recently, this distributionhas been modified by Prentice [18] because in its original form thedistribution had no limits. In this modification, the random variable

e or v is taken to have the probability density function (PDFs) givenby

f (e) = ��−0.5

� (�)exp

(√�(e

�e

)− � exp

[e

�e√�

])(6a)

M. Evans / Materials Science and Engineering A 528 (2010) 500–512 505

Fig. 4. (a) The dependence of ln(ε̇m exp(250,000/RT)) on ln(−ln(�/�TS)) at 773–923 K for the MAF batch of 2.25Cr–1Mo steel tubes. The curves are drawn using Eq. (6c) withthe parameter values as given in Table 1a. (b) The dependence of ln(tf exp(−250,000/RT)) on ln(−ln(�/�TS)) at 773–923 K for the MAF batch of 2.25Cr–1Mo steel tubes. Thecurves are drawn using Eq. (6d) with the parameter values as given in Table 1a.

Table 1aThe values of the coefficients, ˛0 and ˛1 in Eq. (4a) and ˇ0 and ˇ1 in Eq. (4b) determined by maximum likelihood estimation using all the MAF batch data at 773–923 K for2.25Cr–1Mo steel tubes.

Ignore runouts Runouts included

Low stress Medium stress High stress Low stress Medium stress High stress

˛0 18.151 20.544 14.116 – – –[0.134] [0.333] [1.150]

˛1 −6.418 −18.385 −5.529 – – –[0.425] [1.582] [2.711]

�e 0.378 [0.047] –ˇ0 −18.336 −17.113 −14.549 −18.339 −17.111 −14.550

[0.173] [0.211] [0.396] [0.169] [0.203] [0.357]ˇ1 3.499 9.616 3.511 3.496 9.635 3.538

[0.369] [0.706] [1.067] [0.361] [0.670] [0.968]�v 0.373 [0.041] 0.369 [0.037]

Standard errors shown in parenthesis. �∗1 = 0.2 and �∗

2 = 0.50 in Eq. (5a). �∗1 = −0.2 and �∗

2 = 0.42 in Eq. (5b). �= ∞ for minimum creep rate and failure time data. (–) indicatesthat all the recorded minimum creep rates are uncensored.

5 d Engineering A 528 (2010) 500–512

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l

w

ω

wc

atig

l

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Table 1bThe values of the coefficients, ˛0 and ˛1 in Eq. (4a) and ˇ0 and ˇ1 in Eq. (4b)determined by maximum likelihood estimation using the MAF batch data whentf ≤ 10,000 h at 773–923 K for 2.25Cr–1Mo steel tubes.

Low stress Medium stress High stress

˛0 18.109 20.510 14.250[0.248] [0.742] [6.128]

˛1 −6.576 −18.579 −6.060[0.764] [3.577] [12.681]

�e 0.515 [0.170]ˇ0 −18.345 −17.138 −14.559

[0.233] [0.289] [0.746]ˇ1 3.489 9.522 3.380

[0.483] [0.984] [1.910]

06 M. Evans / Materials Science an

(v) = ��−0.5

� (�)exp

(√�( v�v

)− � exp

[v

�v√�

])(6b)

here � (�) is the gamma function and �e and �v are the param-ters that standardises the random variables e* and v∗. Thisistribution has been successfully applied by Evans [19] to data onCr–1Mo steel. Prentice has shown that when the parameter�= 1, vnd e have standard extreme value distributions (and so, for exam-le, failure times are then Weibull distributed). But when �= ∞, end v have standard normal distributions (and so failure times arehen log normally distributed). In this special case, �e and �v arehe standard deviations for the log of the minimum creep rate andhe log of the times to failure respectively. (further, when �= 1 andv = 1, v has a standard exponential distribution). The gamma dis-

ribution is also a special case. Any quantile (q) of this distributionused to form confidence limits) is then given by

n[ε̇m]p + Q ∗c

[1RT

]= a0 + a1�

∗ + a2[�∗ − �∗1]D1 + a3[�∗ − �∗

2]D2

+ ω�,q�e (6c)

n[tf ]p − Q ∗c

[1RT

]= b0 + b1�

∗ + b2[�∗ − �∗1]D1 + b3[�∗ − �∗

2]D2

+ω�,q�v (6d)

here

�,q =√� ln

(1

2��2

2�,q

)(6e)

ith �2 having a value that corresponds to the qth quantile of thehi square distribution with 2� degrees of freedom.

In maximum likelihood estimation, the parameters of Eqs. (5)re chosen so as to maximise the joint probability of observing allhe observed failure times and all the observed runout times. Thiss typically done by maximising a log likelihood function, whichiven Eq. (6b) has the form

n(Li) =N1∑i=1

{(�− 0.5)ln(�) − ln� (�) − ln(�v) +

√�vi − �evi/

√�}

+N2∑i=1

lnQ (�,�evi/√�) (7)

here Q() is the incomplete gamma integral and gives the proba-ility of specimens surviving a given length of time (natural log ofime in fact) and N1 are the number of failed specimens in the batchf data and N2 the number of runouts or unfailed specimens. As allhe recorded minimum creep rates are uncensored, the parametersf Eq. (5a) are obtained by maximising Eq. (7) without the last sum-ation term included (and with e replacing v and with �e replacing

v).

.3. Results

Table 1a shows the maximum likelihood estimates of Eqs. (4)hat were obtained using all the specimens tested within the MAFatch – including the runouts. The first point of interest about theseesults is that ignoring the runout results has very little impact onhe estimates made for ˇ0 and ˇ1. However, it does have a biggermpact on the estimate made for their standard errors. In particu-

ar, the use of runout times has the effect of lowering the estimate

ade for such standard errors which in turn will impact upon thestimates made for any confidence limits associated with the timeo failure. Secondly, the estimates made for ˇ0 differ significantlyrom each other over the differing stress regimes. The same is true

�v 0.466 [0.065]

Standard errors shown in parenthesis. �∗1 = 0.2 and �∗

2 = 0.50 in Eq. (5a). �∗1 = −0.2

and �∗2 = 0.42 in Eq. (5b). �= ∞ for minimum creep rate and failure time data.

ˇ1. Thirdly, within the high stress regimes the standard errors for˛1 and ˇ1 are relatively large reflecting the small number of datapoints in this regime and the larger scatter present in the data pointswithin this regime. Fourthly, the results in Table 1a are obtained for�= ∞ in Eqs. (6a) and (6b). This is the value for � that maximisedthe log likelihood given by Eq. (7) and so is the value most sup-ported by the data. This value implies that creep failure times arelog normally distributed under a given set of test conditions.

These values, in conjunction with Eqs. (5) and (6), are thenused to obtained the curves shown in Figs. 2–4. When consider-ing minimum creep rates, the fit of the curves to the data shownin Figs. 3a and 4a are very good with only one data point fallingoutside the 0.01–0.99 quantiles. These curves are again shown inFig. 2a but in the more familiar stress – minimum creep rate space.Again the fit obtained is good with all data points being within thesame qauntile range at 873 K. Similar conclusions can be drawn byinspecting the failure times shown in Figs. 2b, 3b and 4b.

In Table 1b the same parameters were estimated but using onlythose specimens that failed at or before 10,000 h. In this reducedsample there are of course no runouts. This additional exercise wascarried out in order to assess the ability of the Wilshire equation topredict long-term creep properties by extrapolation of short-term(defined at 10,000 h or less) results for this steel. It can be seenby comparing Tables 1a and 1b that the estimates made for eachparameter are not severely affected by whether the full or partialsample is used. It is therefore not a great surprise to see that inFig. 5 the extrapolation to lower stresses and temperatures is verygood, with all results falling with the predicted 0.01–0.99 quantileat 873 K. Apart from at 923 K the median predictions also appear torun through the middle of the experimental data points.

4. Creep of bainitic 2.25Cr–1Mo steel tubes: a multi-batchanalysis

4.1. Data description and mathematical representation

All the above results were carried out on a single batch of mate-rial. In all the applications of the Wilshire equations referencedabove, the analysis has however been carried out on multi batchesof the same material using least squares techniques. Unfortunately,this is even more inappropriate than when analysing a single batchof material using least squares. The reason is that in such a situationthe observed minimum creep rates will not be independent of eachother (nor will the individual failure times). In creep data sheet No.

3B, there is a two level sampling structure involving the batchesselected for testing and the individual specimens tested for eachbatch – see Section 2. Now this batch to batch variability is typi-cally much greater that the within batch variability as can be seenthrough a comparison of Fig. 6, where the results from all twelve

M. Evans / Materials Science and Engineering A 528 (2010) 500–512 507

F f 2.25a

boalrmvefio

dbtdol

Fi

ig. 5. The stress dependence of the time to failure at 773–923 K for the MAF batch os given in Table 1b.

atches of 2.25Cr–1Mo steel are shown, with Fig. 2b that containsnly the MAF batch (no multi batch data on minimum creep ratesre available on creep data sheet No. 3B). Whilst estimation byeast squares remains consistent in the presence of dependent testesults, it will be less efficient than it could be. The parameter esti-ates, on average, will therefore tend to differ more from the true

alues than is appropriate and confidence intervals will be understimated. The size of this underestimation has never been quanti-ed because all of the published literature involving the applicationf the Wilshire equations has ignored this fundamental issue.

A potentially useful way to describe and analyse multi batchata is to allow the parameters of Eqs. (4) to differ from batch to

atch in a purely random fashion. This makes sense if it is believedhat the chemical compositions and heat treatments of each batchiffer from each other in a random fashion (see Section 2 for detailsf these differences). Within the lifetime statistical literature, multievel models (see Hougaard [20] for a detailed review) have been

ig. 6. The stress dependence of the time to failure at 773–923 K for all batches of 2.25Crn Table 2a (full sample estimates).

Cr–1Mo steel tubes. The curves are drawn using Eq. (6d) with the parameter values

developed to account for any dependence within clusters, such asdifferent batches of material. A multi-level version of Eqs. (4) hasthe form

ln[tf ]ij − Q ∗c

[1RT

]ij

= ˇ0j + ˇ1j ln[−ln

( �ij�TS

)]+ v∗

ij (8a)

where

ˇ0j = ˇ0 + ε0j, ˇ1j = ˇ1 + ε1j (8b)

and the new subscripts refer to the results from the ith specimencut from the jth batch of material and vij = �vv∗

ijis an indepen-

dent error term that has a distributional form given by Eq. (6b)

and a variance given by �2

v . To keep consistency, and to aid com-parisons with the above single batch results, the activation energyis not allowed to vary between batches and so again is taken to be250 kJ mol−1. However, it is a simple exercise to relax this constraintwithin the analytical framework outlined above. In data sheet No.

–1Mo steel. The curves are drawn using Eqs. (8) with the parameter values shown

5 d Engi

3otAbelvw

c

l

w

b

aa0so

D

4

ltagbthRtua

oIvi

b

we

4

Eoniac

08 M. Evans / Materials Science an

B there are m = 12 batches and within the jth batch the numberf specimens tested is variable and is given by nj. ˇk (k = 0–1) arehe coefficients of scientific interest and εkj are the random effects.s such, and as seen in Eq. (8b), the coefficients vary from batch toatch in a random fashion. In this multi level model, the randomffects, and therefore the coefficients themselves, are taken to fol-ow a joint Gaussian distribution with mean vector 0 and unknownariance–covariance matrix �, where � is a 2 × 2 symmetric matrixith variances on the diagonal and covariance’s off the diagonal

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

n[tf ]ij − Q ∗c

[1RT

]ij

= b0j + b1j�∗ij + b2j[�

∗ij − �∗

1]D1 + b3j[�∗ij − �∗

2]D2

+�vvij (8c)

here

0j = b0 + u0j, b1j = b1 + u1j, b2j = b2 + u2j, b3j = b3

+u3j (8d)

nd ukj (k = 0–3) are the random effects. These random effects aregain taken to follow a joint Gaussian distribution with mean vectorand unknown variance–covariance matrix D, where D is a 4 × 4

ymmetric matrix with variances on the diagonal and covariance’sff the diagonal

=

⎡⎢⎣�2u0�u01 �2

u1�u02 �u12 �2

u2�u04 �u13 �u23 �2

u3

⎤⎥⎦ (8e)

.2. Parameter estimation

Appendix A to this paper describes a simulated maximum like-ihood procedure that can be used to estimate values for D, �2

v andhe parameters b0–b3. The basic idea is to simulate values for ukjnd use these values to compute a number for the log likelihoodiven by Eq. (A7) at a given value for D, �2

v and the parameters0–b3. Then Newton–Raphson optimisation algorithms can be usedo find which values for these parameters maximise the log likeli-ood at the simulated values for ukj. This procedure is then repeatedtimes, where R is the number of iterations, to yield R values for

hese parameters, the averages of which are taken to be the sim-lated maximum likelihood estimates D̃, �̃v, b̃0 to b̃3 of D, �2

vnd the parameters b0–b3.

This procedure estimates the mean vector b (containing b0–b3)f the random process generating the vector bj (containing b0j–b3j).t is often useful to produce predictions of the batch parameterector as well, i.e. the bj. A posterior estimate that uses all thenformation available within the sample of data is given by

˜j =

∑Rr=1b̃jr jr∑Rr=1 jr

; b̃jr = b̃ + �̃ujr (9)

here jr is the exponential of Eq. (A5) that can be calculated forach batch during each of the R simulations.

.3. Results

The simulated maximum likelihood estimates of ˇ0 and ˇ1 inq. (8b) are shown in Table 2a. It can be seen that the treatment

f runouts has little influence on the resulting estimates. This isot surprising as all runouts are confined to the MAF batch so that

n this multi batch analysis the number of censored data points issmaller proportion of the data set than when analysis was just

onfined to the MAF batch. A comparison of Table 1a with Table 2a

neering A 528 (2010) 500–512

further reveals that taking into account the hierarchical nature ofthe data thorough use of a multi level model does alter the estimatesmade for ˇ0 and ˇ1. In particular, the estimates for these parame-ters are raised in the low and high stress regimes but lowered in themedium stress regime. However, the biggest effect is seen on theestimates made for the standard error of these parameters. Takinginto account the hierarchical nature of the data, results in a sub-stantial drop in the estimated standard errors as a large part of thescatter in the data is now picked up by the random effects them-selves. The estimate variance–covariance matrix D (when runoutsare included in the analysis) is estimated to be

D =

⎡⎢⎣

0.0180.037 0.150

−0.056 −0.158 0.2130.024 0.001 −0.089 0.247

⎤⎥⎦

This latter result means that estimated confidence bounds arealso very different once the dependency in the test results is cor-rectly modelled. This can be seen in Fig. 7a where the confidencelimits for failure times at 873 K derived from the multi level modelare compared to those shown in Fig. 2b derived using a single batchof material when all the results within each batch are analysed. Asexpected the median predictions from the single and multi batchanalysis are very similar over the full range of stresses. However,the confidence limits are very different – especially at the longestfailure times corresponding to the lowest stresses. This is quitesignificant because failure to take into account test dependencyservely under estimates the safe working life of this material. Forexample, if safe is defined as a 1% chance of failure, then at a stressof around 25 MPa, the single batch analysis suggest a life of around10 years, compared to around 5 years for the multi batch analysis.The curves in Fig. 6 are drawn using the multi level model of Eqs.(8) using full sample estimates and show the model gives a gooddescription of the data at the other test temperatures as well.

Another way to visualise this variability is using the batch bybatch estimates of the parameters of Eq. (8a) as derived using Eq.(9). These estimates are shown in Table 2b and they result in themedian prediction lines shown in Fig. 7b when the estimate for thesymmetric matrix D is given by

D =

⎡⎢⎣

0.0240.049 0.180

−0.076 −0.206 0.3040.035 0.038 −0.188 0.439

⎤⎥⎦

As can be seen from this figure, the variation in the batch tobatch predictions mimic well the actual batch to batch variation inthe data. A similar result was observed at the other temperaturesas well.

Fig. 8a is the multi batch equivalent of Fig. 4b, with the quantilepredictions being given now by Eqs. (2), (8b), and 8(c). Looking atthe experimental data it is clear that the scatter is different in thethree stress regimes, with the variability being markedly higher inthe lowest ln[−ln(�/�TS)] regime. This heterogeneity implies thatthe parameters of the Wilshire model vary from batch to batch andshould not be taken as constant. Further, nearly all of the experi-mental data falls within the 0.01–0.99 quantile predictions givenby the multi level model of Eq. (8b).

Finally, in Table 3 the same parameters of Eq. (8b) were estimatebut using only those specimens that failed at or before 10,000 h. Inthis reduced sample there are of course no runouts. This exercise

was carried out in order to assess the ability of the multi level ver-sion of the Wilshire equation to predict long-term creep propertiesby extrapolation of short-term (defined at 10,000 h or less) resultsfor this steel. It can be seen by comparing Table 3 with Table 2a thatthe estimates made for each parameter are not severely affected

M. Evans / Materials Science and Engineering A 528 (2010) 500–512 509

Table 2aThe values of the coefficients, ˇ0 and ˇ1 in Eq. (8b) determined by simulated maximum likelihood estimation using all the results within all the batches of data at 773–923 Kfor 2.25Cr–1Mo steel tubes.

Ignore runouts Runouts included

Low stress Medium stress High stress Low stress Medium stress High stress

ˇ0 −18.111 −17.113 −14.957 −18.127 −17.112 −14.986[0.033] [0.042] [0.053] [0.036] [0.046] [0.056]

ˇ1 3.859 8.8478 3.716 3.827 8.899 3.838

S

bsapdd

Fo

[0.113] [0.175] [0.191]�v 0.516 [0.005]

tandard errors shown in parenthesis. �∗1 = −0.2 and �∗

2 = 0.42 in Eq. (8c). �= ∞.

y whether the full or partial sample is used. It is therefore not aurprise to see that in Fig. 8b the extrapolation to lower stresses

nd temperatures is very good, with all results falling with theredicted 0.01–0.99 quantiles. Apart from at 923 K the median pre-ictions also appear to run through the middle of the experimentalata points. Also shown in Fig. 8b are the predictions obtained by

ig. 7. (a) A comparison of single and multi batch fits to the data at 873 K. Curves drawn frf the time to failure at 873 K for all batches of 2.25Cr–1Mo steel tubes. The curves are dr

[0.119] [0.187] [0.203]0.505 [0.004]

applying Eqs. (4b) and (5b) to those specimens from all batches thatfailed at or before 10,000 h. This classic application of the Wilshire

methodology was done for data at 873 K only, and it is clearly seenthat the predictions are inferior to those obtained from the multilevel version of this model – as given by Eqs. (8a) and (8c) – espe-cially at the very lowest stresses.

om results using all the data on 2.25Cr–1Mo steel tubes. (b) The stress dependenceawn using Eq. (9) with parameter values as given in Table 2b.

510 M. Evans / Materials Science and Engineering A 528 (2010) 500–512

Table 2bThe values of the individual batch coefficients,ˇ0j andˇ1j in Eq. (8a) determined by use of Eq. (9) using all the results within all the batches of data at 773–923 K for 2.25Cr–1Mosteel tubes.

Batch Low stress Medium stress High stress

ˇ0j ˇ1j ˇ0j ˇ1j ˇ0j ˇ1j

MAA −18.093 4.253 −17.142 9.008 −14.936 3.757MAB −18.185 3.766 −17.147 8.957 −15.013 3.877MAC −18.034 4.476 −17.135 8.975 −14.956 3.789MAD −18.164 3.796 −17.135 8.941 −15.002 3.863MAE −18.166 3.814 −17.138 8.952 −14.995 3.848MAF −18.141 3.969 −17.138 8.983 −14.957 3.790MAG −18.179 3.766 −17.141 8.958 −14.996 3.852MAH −18.182 3.783 −17.145 8.968 −15.001 3.864MAJ −18.183 3.799 −17.148 8.974 −14.996 3.85MAL −18.176 3.765 −17.141 8.941 −15.010 3.868MAM −18.101 4.172 −17.143 8.961 −14.989 3.831MAN −18.079 4.239 −17.132 8.971 −14.98 3.846

Fig. 8. (a) The dependence of ln(tf exp(−250,000/RT)) on ln(−ln(�/�TS)) at 773–923 K for all batches of 2.25Cr–1Mo steel tubes. The curves are drawn using Eq. (8b) with theparameter values as given in Table 2a. (b) The stress dependence of the time to failure at 773–923 K for all batches of 2.25Cr–1Mo steel. The solid and dotted curves are drawnusing Eqs. (8) with the parameter values in Table 3 (sub sample estimates). The dashed curves are drawn by applying Eq. (5b) (estimates using least squares) to all batchesof the sub sample data.

M. Evans / Materials Science and Engi

Table 3The values of the coefficients, ˇ0 and ˇ1 in Eq. (8b) determined by maximum likeli-hood estimation using all the results within all the batches of data when tf ≤ 10,000 hat 773–923 K for 2.25Cr–1Mo steel tube.

Low stress Medium stress High stress

ˇ0 −18.015 −17.132 −14.979[0.038] [0.042] [0.053]

ˇ1 3.831 8.971 3.846

S

5

nmtfpe2cdad

tfbtecucm(t

A

A

s

l

w

w

a

x

tt

V

ai

[0.119] [0.175] [0.191]�v 0.584 [0.006]

tandard errors shown in parenthesis. �∗1 = −0.2 and �∗

2 = 0.42 in Eq. (8c). �= ∞.

. Conclusions

The creep and creep fracture behaviour of 2.25Cr–1Mo steels,ormally characterized by power law descriptions of the mini-um creep rates and rupture lives, are rationalised by normalizing

he applied stress through the ultimate tensile stress determinedrom high-strain-rate tensile tests carried out at the creep tem-eratures. Using these new relationships, termed the Wilshirequations, these rationalization procedures allow the data set for.25Cr–1Mo steel to be discussed in terms of the dislocation pro-esses controlling creep strain accumulation and the differentamage phenomena causing tertiary creep and eventual failures-received bainitic microstructures degrade with increasing testuration and temperature.

Moreover, these new relationships also enable long-term datao be predicted by extrapolation of short term results. A statisticalramework was developed for the analysis of both single and multiatch data sets, which enable for the first time confidence limitso be placed around the predictions obtained using the Wilshirequations. It was shown that a failure to allow for the hierarchi-al nature of creep data bases, results in a serious and significantnderestimate of the predicted safe life corresponding to operatingonditions. Areas for future work include the extension of the aboveodel to three or more levels so that results for different products

such as bar, plate and tubes) and from different laboratories forhe same material can be analysed using a single model.

ppendix A.

.1. Test result dependency and heteroscedasticity

Estimation of the muti level model given by Eqs. (8) is nottraightforward. To see this note that Eq. (8a) can be written as

n[tf ]ij − Q ∗c

[1RT

]ij

= b0 + bk3∑k=1

xkij +wij (A1)

here

ij = �vvij + u0j +3∑k=1

ukjxkij (A2)

nd where

1ij = �∗ij; x2ij = [�∗

ij − �∗1]D1; x3ij = [�∗

ij − �∗2]D2 (A3)

The random error term (wij) in Eq. (A1) is clearly a function ofhe test conditions and its variance, Var[wij], is given by (assuminghat the ukj and vij are independent of each other)

3

ar[wij] = �2v Var[vij] + Var[u0j] +

∑k=1

Var[ukj]x2kij (A4)

nd so also varies with the test conditions. This heteroscedastic-ty has important implications for parameter estimation. It means

neering A 528 (2010) 500–512 511

that if the linear least squares formula is used to estimate the bkparameters in Eq. (A1), by minimising

∑w2ij, the resulting esti-

mates will be unbiased but inefficient (have higher variances thanif Var[wij] were constant). Further, the estimates made of the stan-dard errors of these parameters will be biased, so invalidatingany standard tests of statistical significance or any constructedconfidence limits. Hence some other estimation procedure isrequired.

A.2. Simulated maximum likelihood (SML)

It follows from Eqs. (6b), (8a), and (8b) that the log likelihood forthe nj observations in batch j (of which there are N1j failure timesand N2j runout times so that nj = N1j + N2j) is the sum of the logs ofthe densities (see Greene [21] for a similar derivation)

ln(Lj¦xj; uj) =N1j∑i=1

{(�− 0.5)ln(�) − ln� (�) − ln(�v)

+√�vij − �evij/

√�}

+N2j∑i=1

lnQ (�,�evij/√�) (A5)

where

vij =

[ln[tf ]ij − Q ∗

c

[1/RT

]ij

−{b0 + bk

∑3

k=1xkij

}−{u0j +

∑3

k=1ukjxkij

}�v

](A6)

and uj is a vector containing the k values for u0j to u3j and xj is amatrix containing the nj observations on the variables x1ij to x3ij.The log likelihood for the full sample, when there are m batches inall, is then

ln(L) =m∑j=1

ln(Lj¦xj; uj) (A7)

In principle this log likelihood could be maximised with respectto b0, all the bk and �v. However, there are two problems with thisapproach. First, the variance–covariance matrix for uj (i.e. D) is notpresent and so cannot be estimated. This is easily solved throughthe following standardisation process

�∗kzkj = ukj if k = 0

�∗kzkj + �∗

(k−r)k

k∑r=1

z(k−r)j = ukj otherwise(A8)

and �∗(k−r)k and �∗

kare further parameters whose values are

unknown. The relationship between the random vector uj and itsstandardised equivalent, zj, is therefore given by uj = �zj, where �is a lower triangular matrix made up of the above �∗

kand �∗

(k−r)kparameters

� =

⎡⎢⎣�∗

u0�∗

u01�∗

u1�∗

u02�∗

u12�∗

u2�∗

u04�∗

u13�∗

u23�∗

u3

⎤⎥⎦ (A9)

The full variance–covariance matrix for uj is then found byletting ��′ = D, where D is given by Eq. (8e). Eq. (A8) can be substi-tuted into Eq. (A6) and then Eq. (A7) can be maximised with respectto b0, all the bkj, � and �v. D can then be found from ��′. The com-plication in doing this is of course the fact that the zj are unobserved

and so the log likelihood cannot be computed directly. The solutionrequires zj to be integrated out of the above conditional likeli-hood, but such integrals typically do not have an analytical solution.However, integrals of this nature can be adequately evaluated bysimulation methods.

5 d Engi

bribaeouav

opMaft%

A

lfitm%ofnafbtiatdowd

utu

[[[[[

[

[[[[[

12 M. Evans / Materials Science an

More formally, estimates for b0 all the bk, �v and � are obtainedy maximising Eq. (A7) with z̈j replacing zj where z̈j contains 4andom draws from the standard normal distribution. This max-misation procedure is then repeated R times yielding R values for0 all the bk, �v and �. The average of these R values can be takens the simulated maximum likelihood estimates for these param-ters, and the standard deviation in these R values as an estimatef their standard errors. This estimation procedure can be repeatedsing different values for �, and the value for � that maximises theverage log likelihood over all R iterations is chosen as the correctalue for �.

This is quite a straightforward procedure in practice becausebtaining random draws from Eq. (6b) is easily implemented withinopular and commercially available software packages such asicrosoft Excel [22] or more specialised Econometric software such

s Regression Analysis of Time Series [23] or RATS for short. In theormer package this can be done using the Normsinv(rand()) func-ions and in the latter package this can be done directly using theran(1) function.

.3. Confidence intervals

A simulation approach can also be used to obtain confidenceimits for the multi level model of Eqs. (8). This approach requiresrst simulating values for vij from Eq. (6b) to obtain �̃vvij wherehe tilde represents the simulated maximum likelihood (SML) esti-

ate for �v. In the RATS package this can be done directly using therangamma(�) function. This will return a random draw from ane parameter gamma distribution, so that its natural log is a drawrom the one parameter log gamma distribution. Subtracting theatural log of �, and then multiplying by the square root of � givesrandom draw from the PDF given by Eq. (6b), i.e. a random value

or vij . This last transformation is required for the log gamma distri-ution to be non-degenerate as � tends to infinity and for its meano be zero. Even without specialised software like RATS, obtain-ng random values for vij is straightforward. All that is required isn ability to draw random numbers from a standard normal dis-ribution because the value of a variable drawn from a chi squareistribution with � degrees of freedom is equal to half the valuef a variable drawn from the one parameter gamma distributionith �/2 degrees of freedom (see Johnson and Kotz [24] for more

etails).

This simulation approach then requires simulating values for uj,sing uj = �̃zj where the tilde refers to the SML estimate of � andhe zj are simulated in the way described above. Then b is simulatedsing.

[

[[[

neering A 528 (2010) 500–512

b = b̃ + {�/m}zj(A10)

where b is a vector made up of b0 and all the bk. The tilde refersto the SML value for b. These simulated values for b, uj and vij arethen inserted into Eq. (A11) to simulate the value for ln[tf] that isassociated with a particular value for x.

ln[tf ]ij = Q ∗c

[1RT

]+ b0 + bk

3∑k=1

xkij + �vvij + u0j +3∑k=1

ukjxkij (A11)

Repeating this a large number of times yields an empirical distri-bution for ln[tf] and therefore tf at a particular value for x. From this,any quantile of the empirical distribution can be obtained to rep-resent confidence limits around the simulated median prediction.Repeating this for other values of the test variables completes thecalculations required to obtain predictions with confidence limitsfor tf.

References

[1] ASME Boiler & Pressure Vessel Code, Sec. II, Part D, Appendix I, ASME 2004.[2] J. Hald, Mater. High Temp. 21 (2004) 41–46.[3] W. Bendick, J. Gabrel, in: I.A. Shibli, S.R. Holdsworth, G. Merckling (Eds.), Proc.

ECCC Int. Conf. on Creep and Fracture of High Temperature Components –Design and Life Assessment Issues, DEStech Publ., London, 2005, pp. 406–418.

[4] L. Cipolla, J. Gabrel, Proc. First Int. Conf. on Super High Strength Steels, AIM,Rome (CD-Rom), 2005.

[5] S.R. Holdsworth, et al., in: I.A. Shibli, S.R. Holdsworth, G. Merckling (Eds.), Proc.ECCC Int. Conf. on Creep and Fracture of High Temperature Components –Design and Life Assessment Issues, DEStech Publ., London, 2005, pp. 380–393.

[6] B. Wilshire, A.J. Battenbough, Mater. Sci. Eng. A 443A (2007) 156–166.[7] B. Wilshire, P.J. Scharning, Int. Mater. Rev. 53 (2008) 91–104.[8] B. Wilshire, P.J. Scharning, Mater. Sci. Technol. 24 (2008) 1–9.[9] B. Wilshire, P.J. Scharning, Int. J. Press. Vessels Piping 85 (2008) 739–743.10] NIMS Creep Data Sheet No. 3B, 1986.11] F.R. Larson, J. Miller, Trans. ASME 5 (1952) 174.12] S.S. Manson, U. Muraldihan, Research Project 638-1 EPRI Cs-3171 July, 1983.13] M. Evans, J. Mater. Sci. 43 (18) (2008) 6070–6080.14] F.C. Monkman, N.J. Grant, in: N.J. Grant, A.W. Mullendore (Eds.), Deformation

and Fracture at Elevated Temperature, MIT Press, Boston, 1963.15] E.M. Haney, F. Dalle, M. Sauzay, L. Vincent, I. Tournié, L. Allais, B. Fournier, Mater.

Sci. Eng. A 510–511 (2009) 99–110.16] N. Hoff, J. Appl. Mech. 20 (1) (1953).17] M.S. Bartlett, D.G. Kendall, J. Roy. Stat. Soc. Suppl. 8 (1946) 128–138.18] R.L. Prentice, Biometrika 61 (1974) 539–544.19] M. Evans, Mater. Sci. Technol. 26 (3) (2010) 309–317.20] P. Hougaard, Analysis of Multivariate Survival Data, Springer-Verlag, New York,

2000.

21] W.H. Greene, Econometric Analysis, 6th ed., Prentice-Hall, New York, 2008

(Chap. 17).22] Microsoft Excel, Microsoft Office, 2007.23] RATS:Version 7, By Estima, Evanston, USA, 2010.24] N.L. Johnson, S. Kotz, Continuous Univariate Distributions, vols. 1 and 2,

Houghton Mifflin, Boston, MA, 1970.