ELSEVIER

ht. J. Pres. Ves. & Piping 72 (1997) 177-191 0 1997 Elsevier Science Limited

Printed in Northern Ireland. All rights reserved

PII:SO308-0161(97)00044-6 0308-0161/97/$17.00

Some interpolative properties of the Monkman-Gralnt empirical relation in

2X3-1Mo steel tubes

M. Evans Department of Materials Engineering, University of Wales Swansea, Singleton Park, Swansea SA2 8PP, UK

(Received 16 April 1997; accepted 6 May 1997)

This paper assesses the suitability of using the Monkman-Grant empirical relation to predict the operating life for 2$Cr-1Mo steel tubes. This is done through an in-depth study of the interpolative properties of this empirical relation over a range of accelerated test conditions. The interpolative tests used in this paper include those for outliers, the distributional form for failure times, the constancy of variability in failure times: functional form mis- specification and those for the stability and predictive capability of the estimated relation. Whilst some batch to batch variation in results was observed, the Monkman-Grant relation was shown to be stable and showed no signs of consistently untder- or over-estimating the time to failure at the lower range of test conditi’ons. However, the nature of the Monkman-Grant relation was shown to be dependent upon the test temperature so that any use of this relation for the purpose of long term extrapolation must correctly identify and make use of this additional information.

0 1997 Elsevier Science Ltd.

1 INTRODUCTION

At present there exists a vast array of methods which can and have been used to predict the long term high temperature behaviour of materials. To extrapolate to typical design conditions many of these techniques have made use of just the failure times obtained under accelerated test conditions. This is certainly true of the many parametriclm7 techniques that have been extensively used in the past. Unfortunately, an inability to identify a suitably linear relationship between transformed rupture lives and transformed stresses and temperatures has resulted in predictions in which engineers have very little confidence.

More recently attempts have been made to improve predictions through the incorporation of aldditional information that can be gathered during any high temperature creep test. Thus the theta’ prediction approach uses the whole creep curve rather than just its end point. These models do in general yield life time predictions that are more realistic than those obtained from the parametric techniques but do require the continuous measurement of strain in a

creep test up to the point of rupture. Many of the extensive testing programs conducted over the last 30 years have not accumulated this information. How- ever, some of these studies have recorded the minimum creep rate associated with a particular rupture time.

Monkman and Grant’ have suggested that a linear relationship exists between the log time to rupture and the log minimum creep rate. If this is true then a very simple and powerful extrapolation procedure already exists because minimum creep rates are obtainable from long term non-accelerated creep tests well before any rupture actually takes place. Accelerated test data can be used to quantify the Monkman-Grant empirical relation for a specific material. Then the minimum creep rate from tests still running at design conditions can be used in combination with the identified relation to predict the operating life of that material.

This paper takes a first step to assessing the suitability of this approach to extrapolation by studying the interpolative properties of the Monkman-Grant relation at accelerated test condi-

178 M. Evans

tions for 2$Cr-1Mo steel tubes. A generalised linear model is applied to the data which is first tested for the existence of outlier or rogue failure times and then a number of diagnostic tests are carried out to assess the adequacy of the model over the full range of accelerated test conditions. In particular the data is tested to ensure that failure times are:

i. Log normally distributed. ii. Have a variance that is independent of the

minimum creep rate. iii. Are log linearly related to the minimum creep. iv. Vary with the minimum creep rate in a way that

is independent of the temperature conditions.

If necessary the model is amended in the light of these tests and then used to obtain in-sample life time predictions. The adequacy of these predictions is then assessed.

This paper deals with these aspects of interpolation along the following lines: Section 2 provides a brief review of existing parametric techniques, Section 3 outlines the generalised linear model of the Monkman-Grant relation together with the interpola- tion tests to be used, Section 4 briefly describes the two batches of data used to estimate any batch to batch variation in results and these results are then presented in Section 5, conclusions are presented in Section 6.

2 EXISTING PARAMETRIC TECHNIQUES

Parametric methods are also referred to as tempera- ture compensated time techniques in that they attempt to remove the influence of temperature on the time to rupture. For most techniques this is achieved through the use of the Arrhenius and Monkman-Grant’ equations. The Arrhenius equation often used to model creep as a rate process takes the form:

& = A exp(-QJRT) (14

where P, is the minimum creep rate, Qc is the activation energy, T is absolute temperature, R is the universal gas constant and A is a further parameter. Monkman and Grant have further suggested that time to failure (tF) increases as the minimum creep rate decreases in the following way:

B 'F-G so that tF=CAexp(~Q,,RT)Jm (lb)

where m and B are constants that purportedly differ only for different materials (m typically in the range O.l-10%).

The time-temperature parameter (tF, T) used by Dorn’ and Larson and Miller’ can be derived from eqn (lb). As this parameter is purely a function of

stress, two-dimensional representations can be made. To illustrate let u(p) be a function describing the dependency of the minimum creep rate (&) on stress. Dorn assumed that the parameter A in eqn (la) can be made a function of stress so that:

B tF = {u(p). exp(-Q,/RT)}” (24

Taking natural logs through eqn (2a) then gives:

mQc h(tF) = In(B) - m ln[u(p)] + RT -PI

A plot of ln(t,) against l/T for a given stress should therefore give a line with slope (mQc)/R and an intercept that varies in sdme way with the stress level. Rearranging for ln[u(p)] gives:

In[u(p)] = {F + g-F} = (tF, T)” (2~)

where (tF, T)” is the time-temperature parameter suggested by Dorn. A plot of In(p) against (tF, T)” should give an indication of the functional form for ln[u(p)]. Such a plot requires information on the values for m, Q, and B.

Implicit within the work of Dorn is the assumption that the minimum creep rate is a multiplicative function of the form:

& = u(p). v(T) (24

where v(T) is a function describing the dependency of the minimum creep rate (is) on absolute temperature. Such a representation implies that the effects of stress and temperature on times to failure are separable, i.e. no interaction between the two is present.

Larson and Miller,’ instead of combining stress and temperature multiplicatively as in eqn (2d), assumed the activation energy to be a function of stress. This allows stress and temperature to interact in the determination of rupture time. This is supported by a tendency for lines of constant stress plotted in ln(tF) - l/T space to diverge. Under such an assumption eqn (lb) becomes:

B

fF={A. exp(-u(p)/RT)}”

Taking logs of eqn (3a) then gives:

(34

ln(tF) = {In(B) - m In(A)} + m. 414 RT WI

A plot of ln(t,) against l/T for a given stress should therefore give a line with slope (mQ,)/R and intercept In(B) - m In(A). This time however the slopes (not intercepts) will differ for different stress levels. Rearranging for u(p) gives:

m .u(P) R

= T[hl(t,) - c] = (tF, T)LM (3c)

Monkman-Grant empirical relation 179

where C = (In(B) -m h(A)) and (tF, T)LM is the time-temperature parameter suggested by Larson and Miller. A plot of u against (tF, T)LM should give an indication of the functional form for u(p). Such1 a plot requires information on the values for m, A and B. Larson and Miller assumed C to be a true cons’tant so that irrespective of the stress level In(&) converged to a value of C as l/T converged to zero. This has seemed a little too restrictive so that Murry3 a!ssumed that both Qc and A in eqn (la) are functions of stress so allowing C to vary with the stress level. This implies that stress alters both the slope and intercept (of lines plotted in l/T - ln(t,) space.

C can of course be a function of stress and temperature for totally different reasons. As some of the results of this paper show, B appears to depend on the absolute temperature so that C should allso vary with temperature.

A rather curious parametric method is that suggested by Manson and Haferd.4 They appear to reject the Arrhenius equation altogether by suggesting that for a constant stress level, ln(tF) varies linearly with T rather than l/T. They assumed that the slope of these lines is a function of stress but all such lines converged to a specific failure time ta at a temperature T,. That is:

ln(r,) = In(&) + &CT - T,)

Solving for u(p) gives:

U(P) = T - T,

In(b) - ln(t,> = (tF, T)MH (4b)

A generalisation of this has been provided by Manson and Brown’ who allow for a possibly non-linear relationship between ln(t,) and T for a given stress via a further parameter w as follows:

‘(‘) = 144 - WJ = ctF, T)MB

(T - T,)W (5)

Much of this independent work was brought together by Manson who suggested the use of the following generalised time-temperature parameter:

P’ Wd - Wt,) fdP> =- (T _ T,)w = (tF, QM (6)

If u = 0, w # 1 and T, # 0 then (tF, T)M = (t,?, T)MB. The additional restriction that w = 1 results in (tF, T)M = l/(t,, T)MH. On the other hand, if u = 0, w = -1 and T, = 0 then (tF, T)M = (tF, T)LM.

However, this is by no means the only way to generalise the Larson-Miller time-temperature para- meter. For example, Eyring7 has suggested that the Arrhenius equation lacks a theoretical derivation when factors other than temperature are important in determining a rate process. The Eyring model based

on chemical reaction rate theory and quantum mechanics suggests that:

es = AT* exp(-QJRT) (74

Whilst theory suggests that p should enter eqn (7a) in a specific way, (although no indication of the units of measurement for p is provided by the theory), invoking the Larson-Miller assumption that Q, is a function of stress instead results in:

m. U(P) R

= T[ln(t,) - C f (Y In(T)] = (tF, T)MLM (7b)

which collapses to the Larson-Miller model only when (Y = 0.

3 INTERPOLATIVE TESTS FOR THE MONKMAN-GRANT RELATION

3.1 A generalised linear model

Letting i represent the ith pairing of t, and ti,, a linear and stochastic version of the Monkman-Grant relation (lb) can be written as:

In( = B’ - m ln(Q + gei @a>

where B’ = In(B) and F is a scale parameter so that the eis are standardised but identical and indepen- dently distributed errors with a zero mean and constant variance, denoted E(e) = 0 and Var(e), respectively. Least squares estimates for B’, m and (T are given by:

N ;ij, (In( ’ ln(e,),} - $jl ln(t,), . jl ln(i& fi=

N 2 WtF)J2 . ii!1 ln(k)I 1’ @b)

i=l

5 In( P=j=’

$, WJ, -viz

N N @cl

;C [In( - (8’ - +Z ~d~s>i>l’ ,+* _ 1

Var(e) ,el N-2 (84

where N is the sample size. Unfortunately, Cox and Hinkley” have shown that the least squares estimates of B’, m and especially (T are much less efficient than the maximum likelihood estimates when k is small (less than 1). Of course for large k these least squares estimates are also maximum likelihood estimates and so are unbiased in large samples.

Unfortunately, the value for Var(e) depends on the shape of the distribution for e. For example, Evans” assumed, when studying replicated failure times, that e followed the transformed log gamma distribution suggested by Prentice.” Under this assumption:

Var(e) = k&(k) @e)

180 M. Evans

where $‘(k) is the trigamma function and k is some constant whose value determines the type of distribution for e. Because varying k alters the shape of the distribution, over restrictive assumptions about the nature of e can be avoided. For example, when k = 1, e has a standard extreme value distribution whose mean is zero and whose variance equals 1~~16, On the other hand e’ = u. e has an extreme value distribution with mean zero and variance (02r2)/6. Then as k+a, e tends to a standard normal distribution. More precisely, as k -+ M, qY(k)f 0 so that Var(e) = 1, i.e. e follows a standard normal distribution with zero mean and unit variance. e’ = (T. e therefore follows a normal distribution with mean zero and variance I?. Whatever the value for k, e is always the standardised value for e’.

Two strategies therefore suggest themselves. First, eqns (8) could be estimated using various values of k. There will then be a goodness of fit statistic for eqn @a) associated with each k value so that the estimates for k, B’, m and g are those associated with the best fit. This approach has been used by Evans.’ ’ The alternative approach selected for this paper is to estimate B’, m and (T from eqns (8) assuming e’ to be normally distributed (k = m) and then test the validity of this assumption against a very general alternative distribution for e’. Note that if e’ is normally distributed then tF is log normally distributed.

where

RSS = 2 ei2 = 2 (ln(tF)f - 8 ’ + A ln(&)J” (SC) i=l i=l

LDI therefore measure the extent to which the deletion of the ith pairing for tF and tis from the sample changes the fit of the Monkman-Grant relation to the data. If this change is statistically significant, then that ith observation is considered to be an outlier. Because LD, is distributed as a chi square distribution with v = 3 degrees of freedom when N is large, there is a 95% chance that the ith observation is an outlier if its LDi value exceeds 7.81.

3.2 Testing for outliers

A more general test for influence has quite recently been proposed by Cook.‘4 Instead of comparing the fit of the model when the ith pairing is included to when it is excluded, Cook compared the change in fit resulting from a change in the weight attached to the ith observation. Such case weighting corresponds to giving each value for In(+) a weight value (wi) somewhere between 0 and 1. Clearly wi = 0 implies that the ith failure time is deleted from the sample, whilst wi = O-5 corresponds to a downgrading of the importance of the ith failure time. wi could also increase or decrease with increasing failure times (heteroscedasticity) so that this test, unlike the LDi test, is less likely to diagnose a large failure time as an outlier if in fact heteroscedasticity is present in the sample. In this paper the implementation of Cook’s test is discussed in relation to eqn (8a) whilst fuller details of the test are available in Cook.i4 The test involves the construction of the following N X N vector:

One possible definition of an outlier is a data point that has an undue influence on the predictions obtained from a model or the parameters estimated for that model. It is of course possible for an influential data point not to be an outlier. This aside, one general measure of influence is the likelihood distance defined by Cook and Weisberg.13 They use the estimated log likelihood as a measure of how good a fit a model is to the data. The likelihood distance is defined by them to be:

2# = 2[A(i;))‘A’] VW

from the log likelihood of the perturbed model,

L(8 1 w) = - & $ wj[ln(tF), - B’ + m ln(Qi]* (lob) I 1

w is a N X 1 vector of weights associated with the N failure times. Now A is a N x 3 matrix of cross partial derivatives of L(f3 1 w) with respect to w evaluated at the least squares estimates for B’, m and u and for when wi = 1 for all i. Thus the first column of A is:

LD, = 2. ABS([L@) - L(&)]) *, 11 = a*w I w>

aB’ dw = 5 [el] (lOc>

where L(8) is the log likelihood of eqn (8a) obtained for k = 00 using the whole sample of data on tF and 2,. L(eJ is the log likelihood of eqn (8a) obtained for k = cc) when the ith values for tF and 2, are removed from the sample. In turn when k = m the log likelihood of eqn (8a) is related to the residual sum of squares (RSS) through:

where eI comes from eqn (SC). The second and third columns are therefore given by:

A- = d2L(m 1 w> = 1n($>. wi 12

dm aw ,.* Eerl u (104

A. = J2Ua I w> _ wj[e.2* 13

au aw -.3

u we>

~(8) = -N In(a) - N ln(&J - $ $ e12 (9b) z is a 3 X 3 matrix of second and cross partial 1 I derivatives of L(B) [in eqn (9b)] with respect to B’, m

Monkman-Grant empirical relation 181

and (T again evaluated using the least squares estimate of CT:

Let Cm,, represent the largest absolute eigenvalue associated with 2@ and I,,, the N X 1 eigenvector associated with C,,,. Cook suggests that an outlier is likely to be present if C,,, is vastly in excess of 2. If the ith element of I,,, is also found to be large this indicates that changes in the value of y associated with the ith observation for In(&) is relatively influential in determining the fit of the Monkman- Grant relation to the experimental data. As I,,, must be in the range -1 to +l the term relatively large is fairly easy to assess. On the other hand if the N values for Lax increase/decrease with increasing/decreasing values for In(&) this is evidence that the vaniance in ln(tF) is not constant and is a function of In(&). That is, heteroscedasticity is present.

3.3 Testing e’ for normality

The omnibus test suggested by Doornik and Hansen” has as its basis the observation that if e has a. standard normal distributed in eqn @a) [and therefore ln(t,) is log normally distributed] then the degree of skewness (asymmetry around the mean of zero) should be zero and the degree of kurtosis (peakedens around the mean) should be 3. Writing mj = (l/N) Z: (ei -Z)‘, the sample estimates for the degree of skewness and kurtosis can be given, respectively, by:

%=G and b2=m, m2 nz;

Giving b2 a gamma dcbution and approximating _

the distribution tor Vb, by the Johnson system, Shenton and Bowman16 showed that in large samples:

J WC)’ and N(b2 - 3)2

6 24 (lib)

followed normal distributions provided that e’ is normally distributed. Hence the statistic

Ea = N%)* + Nbz - 3)2 P 6 24 WC)

will in large samples be chi square distributed with 2 degrees of freedom provided e’ is normally distributed. Doornik and Hansen have shown that

N(b2 - 3)2 24

approaches normality very slowly with

increasing N so that the test may have low power in small samples against the range of alternative distributions covered by the Johnson system.

Doornik and Hansen have derived transformations of skewness and kurtosis that are much closer to standard normal than those shown in eqn (llb). The transformation Zi for skewness is as in D’Agostino17 whilst the transformation Zs for kurtosis is based on the standard transformation from a gamma to a chi square distribution which in turn is translated into a standard normal using the Wilson-Hilferty cubed root transformation. The test statistic for normality suggested by Doornik and Hansen is therefore:

(114

which under the null of normality for e’ follows an asymptotic chi square distribution with 2 degrees of freedom. Any value for E, in excess of the critical value of 5.99 at the 5% significance level is an indication of non-normality.

The Johnson system contains distributions with a vast range of values for V& and b,, i.e. distributions with a great variety of degrees of skewness and kurtosis. Contained within the system are the beta, gamma, log normal and extreme value distributions to name but a few. Using Monte Carlo simulations, Doornik and Hansen have shown that in samples as small as 50, the E, statistics has very high power against the alternative that e’ follows a distribution within the Johnson system that is not normal. That is, the E, statistic will frequently reject normality when the distribution comes from the non-normal segment of the Johnson system.

A useful visualisation of this test can be obtained by dividing the values for e’ into N intervals of length h to obtain a simple histogram and then superimposing upon this the estimated density given as a smoothed function of the histogram using a normal kernel K(.). h is chosen to be:

1.068 h=----” IP2 We>

where 8, is the estimated standard deviation for e (not e’). This smoothed function can then be compared visually to the normal curve with a variance given by A2

u.

3.4 Testing for heteroscedasticity

A test attributed to White” involves the following specification for heteroscedasticity:

Uf = 6, + 6, In(&), + 8,[ln(E,),]’ + Ui (124

where ui is assumed normally distributed with mean zero and constant variance. This suggests a particular type of heteroscedasticity where the variability in e’

182 M. Euans

and therefore In(t,) depends on the log of the minimum creep rate and possibly its square. Notice that for constant variance only a0 should be statistically significant. Hence a significant student t statistic for a1 or a2 is an indication of heteroscedas- ticity. Alternatively, under the assumption of constant variance, the coefficient of determination R2 in eqn (12a) should be zero. Now N. R2 has a chi square distribution with 2 degrees of freedom under the assumption that only SO is significant. Thus a value for H, = N. R2 in excess of 5.99 suggests that heteros- cedasticity is present with 95% certainty. White has shown that for practical purposes (T? can be replaced with (er)’ without invalidating the test. Hence the test requires a least squares regression of the following;

e12 = a0 + S1 In(&), + S2[ln(&)J2 + vi (12b)

3.5 Testing for functional form mis-specification

Two types of functional form mis-specification are considered in this paper. The first relates to the assumption that the log of rupture time and minimum creep rate are linearly related. This is tested against the alternative that the relationship is a polynomial of up to order three. The RESET test proposed by Ramsey I9 involves comparing the residual sum of squares obtained from the estimation of eqn (8a)-RSS,--, with the residual sums of squares obtained with terms [log(8,)J2 and [log(&),]3 added to eqn @a)-RSS,. Under the null hypothesis that no polynomial terms are significant these two residual sums of squares should be the same. Under this null and assuming e’ to be normally distributed the following statistic has an F distribution with u1 = 2 and u2 = N - 4 degrees of freedom:

F R

= (RSS, - RSS,)/2 RSSJN - 4 W)

Non-linearity is accepted only if this F value is large enough to exceed the critical value for F at the 5% significance level.

There is growing suspicion that not all test results of ln(tF) and In(&) fall nicely around a single straight line. Instead it appears that there may be a separate linear line for each test temperature (but not for each stress level). This can be explicitly tested by introducing dummy variables into eqn @a) and using the student t statistic associated with these dummy variables. Suppose there are 4 different test temperatures used in a particular batch of creep test results and that temperature shifts the intercept B’ in eqn (8a) rather then the slope m. Then eqn (Sa) can be modified as follows:

ln(tF)i = B:Bi 2 Dii - wz ln(Qj + uei W) j=2

where 0, takes on a value of unity at temperature level j but zero otherwise. Consequently, B’ represents the intercept of the Monkman-Grant relationship at the lowest of all the temperature test levels, B’ + B: the intercept at the next lowest temperature test level and so on up to B’ + Bh for the intercept at the highest temperature test level. Based on the work of Dobes and Malicka2’ these B,! values can be equated to the average of the. total creep strains associated with all test specimens failing at temperature j.

Provided the residuals ei are standard normal, each B,f also follows a normal distribution, and the means of these distributions can be said to be zero at the 5% significance level provided

for each B,f is less than the critical value for t with N - 2 degrees of freedom. 8,; is the standard error for BI. Hence the Monkman-Grant relation will be temperature independent if each t value is below its critical value.

3.6 Adequacy of in-sample life time predictions

A final diagnostic test is to see whether the model identified from the tests above is capable of prediction. To do this the data is ordered from the lowest to highest failure time. Equation (13b) (which may or may not include dummy variables depending on the t test statistics) is then estimated using the first two-thirds of the sorted sample. Having done this two prediction tests can be carried out.

The first involves using the estimated equation from the first two-thirds of sorted data to obtain predictions for the remaining one-third of the failure times. That is:

ln(?,)i = fi ’ + B,! 2 Dji - rfz ln(iJi /=2

i = :N + 1,. . . , N (14a)

where B ‘, Bj and r+? are estimates made from the sorted sample using only the first 2/3N sorted data points. The prediction error is the mistake made for each test result:

ei = f?ej = ln(t,) - loll = ln(t,)

- {b’+Bi,$2Djj-&ln(e,)l) i=gN+l,...,N

(14b)

Provides that no heteroscedasticity is present, the standard error of the prediction error is given by:

SE(el) = d&2xL!(X’X)-‘xi + c2 (14c)

Monkman-Gmnt empirical relation 183

where 4 THE DATA

1 In(&), Dr, . . . D,,

1 In(Q D,, . . . Dq2 1

and q can of course be zero. Each prediction can therefore be given a student t value (provided that the errors are normally distributed):

which should always be less than the critical t value if the prediction is not significantly different from the actual failure time.

The second test is graphical in that it plots the evolution of the estimated parameters in eqn (13b) as one additional observation is added to the initial (but sorted) two-thirds sample until the end of the sample is reached. If these estimates change as additional observations are added this is an indication that the Monkman-Grant relation does not hold universally over all failure times-especially higher failure times associated with smaller stresses and temperatures. This procedure is called recursive least squares estimation.

The main material analysed in this paper is 2.25Cr-1Mo fabricated header pipe supplied by Babcock.2 The piping, which had an outside diameter of 500mm and a wall thickness of 60 mm, was normalised at 960°C and tempered 695°C. In total, 30 specimens were tested at the accelerated temperature of 783K (205-280 MPa), 813K (HO-250MPa), S38K (145-220 MPa), 853K(135-190 MPa) and 873K(120- 175 MPa). The composition of the steel (% weight) was O.l4C, 0.24Si, 0*47Mn, 0*013P, O.O32S, 1.02M0, 0.34Ni and less than OeOlAl with balanced Fe.

To obtain a feel for batch to batch variability this data was compared with 2.25Cr-1Mo virgin super- heated header pipe with an outside diameter of 114mm and a wall thickness of 19mm. This data, accumulated by Willis22 as part of a Ph.D. program involved 53 specimens tested at 78313 (220-350 MPa), 813K (190-300 MPa), 838K (160-275 MPa), 853K (160-245 MPa) and 873K (160-235 MPa). There was no heat treatment prior to testing. These two sets of test results will from now on be referred to as batch A and batch B, respectively.

5 RESULTS

5.1 Batch A

Data from batch A is shown in Fig. 1. The 30 pairings for times to failure and minimum creep rates can be

NATURAL LOG OF FAILURE TIME In($), s

L

-

1 k

t +

8 813K A 838K - 873K

I

I I -Best Fit Line j

I

c

, 1 17

,

Fig. 1. The Monkman-Grant relation for 2.2.5Cr-1Mo steel tubes (Batch A).

184 M. Evans

given an index number based on the value for ln(tF). against this index number based on the ordered failure Thus the In(&) - In(&) pairing corresponding to the times. All the distance estimates are well below the lowest failure time is given an index number of 1 critical value at the 5% significance level of 7.8. The whilst the In(t,) - In(&) pairing corresponding to the largest of these estimates corresponds to the sixth highest failure time is given an index number of 30. smallest failure time obtained at a temperature of

The 30 likelihood distance estimates from the 813K. This data point is also highlighted in Fig. 1 Monkman-Grant relation are plotted in Fig. 2(a) (note it is the sixth failure time when reading from left

8

5% Critcal value for LD,

5 I In(tF)= 12.637, K = 813 I

15

Index of ordered failure times

Fig. 2(a). Likelihood distance estimates from the Monkman-Grant relation (Batch A).

’ 1------ --I--.-.- ---‘----.-----.. - --. - .~._ . .._._ ~.. . . . ._

0.9 -

0.7 -

ln(tF)= 12.637, K = 813 0.6 -

Index of order failure times

Fig. Z(b). Plot of l,,, against index of ordered failure times (Batch A).

Monkman-Grant empirical relation 185

to right). There therefore seems little evidence to suggest that an outlier data point is present. in this batch of tests. This is confirmed by Cook’s test. The largest absolute eigenvalue of C,,, = 3.84 d.oes not indicate extreme local sensitivity to changes in the weights associated with each failure time. Neither are any of the I,,, values in Fig. 2(b) excessively large-none exceed 0,5 in value and assuming all the data points are equally influential these I,,, values would take on a value of 0.18.

Having concluded that no outliers are present the least squares estimates, assuming normality., for B’ and m of the Monkman-Grant relation are s#hown in Table 1 together with their student f statistics. This best fitting line is shown in Fig. 1 and explains nearly 97% of the variation in ln(tr). Although ln(t,:) = 12.63 is the data point furthest from this best fitting line, it does not stand out as an outlier in relation to this line.

The omnibus tests for normality, E: = 1.13 and E, = 1.01 both suggest that the values for e’ from the estimated Monkman-Grant relation can be treated as

coming from a normal distribution so that tF can be treated as log normally distributed. Figure 3 gives a visual impression of these errors. The histogram constructed using eqn (lle) is roughly symmetric and the smoothed density approximates that for a normally distributed error that has a mean of zero and a standard deviation of CT = 0.222. Surprisingly this approximation is better at the tail ends of the distribution.

Table 2 contains the results obtained from the application of White’s tests for heteroscedasticity- eqn (12b). The student t values associated with 6, and 8, suggest that the variability in e’ varies neither with In(&) nor with its square. This is further confirmed by the statistic H, = 0.9 which is well below the critical value of 5.99 (at the 5% significance level).

Next the terms In(&)’ and ln(Q3 were added to eqn (8a) to test for the existence of a non-linear relationship between ln(tr) and In(&). The resulting statistic FR = 1.36. The critical value for FR with u, = 2 and u2 = 26 degrees of freedom is 3.39 at the 5%

Table 1. Least squares estimates for the Momkman- Grant relation using Batch A test data (a;ssuming

k=m) Table 2. White’s test for heteroscedasticity in the

Monkman-Grant relation (Batch A)

B’ B in CT

Estimated parameter -1.237 0.29 -0.915 0.222 t statistic -2.3” _ -28.43s 5.74

R2 = 96.65%

Estimated parameter t statistic

R’= 3.0%

0.315 0.026 0.001 0.21 0.14 o-11

* Statistically significant at the 5% significance level. R2 is the coefficient of determination.

* Statistically significant at the 5% significance level. R2 is the coefficient of determination.

Normal density with mean zero and o = 0.222 \

\

Smoothed density /’ / \

-. 7 -.6 -.5 -.4 -.3 -.2 -.l 0 .l .2 .3 .4 .5 .lj e

Fig. 3. Histogram and smoothed. density function for e’ from the MO&man-Grant relation

186 M. Evans

Table 3. Assessing the temperature dependency of the Monkman-Grant relation (Batch A)

B’ m u B; B; B; B;

Estimated parameter -1.464 -0.925 0.215 -0.097 0.187 0.092 0.078 t statistic -2.58” -2w3* - -0.77 1.4718 0.73 0.61

R2 = 97.3%

* Statistically significant at the 5% significance level, i statistically significant at the 10% significance level, $ statistically significant at the 20% significance level. R* is the coefficient of determination. B’ is the intercept associated with a temperature of 783K. B’ + B; is the intercept associated with a temperature of 813K and so on up to B’ + BA which is the intercept associated with a temperature of 873K.

significance level. FR is well below this critical value so there is no evidence to suggest the existence of a polynomial relation.

The results of estimating eqn (13b) using the least squares method is shown in Table 3. None of the dummy variables associated with the largest four test temperatures were significant at the 5% or 10% significance levels. That is, increases in temperature did not shift the intercept value of B’ = -1-464 which is associated with the lowest test temperature of 783K. Only at the 20% significance level does the intercept appear to shift marginally to -1.28 at a temperature of 838K. It therefore appears to be the case that the results shown in Table 1 adequately capture the relationship existing between ln(tr) and In(&).

Next the stability of the results shown in Table 1 are assessed. Figures 4(a) and 4(b) plot the evolution in the values of B’ and m as one additional pairing is continually added to an initial sample of 20 until all 30

readings are included. Note that the data is sorted from lowest to highest failure time so that in Fig. 1 the best fitting line is updated by adding data points further and further out along the horizontal axis. There is no strong tendency for either m or B’ to continually increase or decrease in value as larger failure times are added into the equation. m, for example, increases from a value of -0.968 when the lowest 20 failure times are included in the analysis to -0.915 when all 30 tests are included (a 5% change). A similar picture for (T emerged from this recursive estimation.

Finally, the forecasting ability of the Monkman- Grant relation is tested. In Fig. 5, eqn (8a) is estimated using the smallest 20 failure times only. This is then extrapolated (dotted line) to the remaining 10 highest failure times. The prediction errors (which are essentially the gaps between each data point and the extrapolated line) together with their

-2 standard deviations for B’

0.5 ’ I

20 21 22 23 24 25 26 27 28 29 30

Index of ordered failure times

Fig. 4(a). Recursive estimation of B’ in the Monkman-Grant relation (Batch A).

I -1.24

Monkman-Grant empirical relation 187

-1.1

-1.05

-1

-0.95 E

2 al $ -0.9 E .d

9 -0.85

-0.8

-0.75

-0.7

*. -_ --. --_. + 2 standard deviations for m

-. ______----.. --._ ______------ -.

.___.- _.-- -. -. *.

*. ---..__ .______.-~--~~~.~.._____ -----____

__-*... -----____ ___--- .-._ _.-- __.~-.-.~..._______...

--_ _ . _ - *-. -...

-. ._ -- __-- .__--- *_______.--.-* ._.--

- 2 standard deviations for m

-4

-Ii -0.915

_! I

20 21 22 23 24 25 26 27 28 29

Index of ordered failure times

Fig. 4(b). Recursive estimation for m in the Monkman-Grant relation (Batch A).

30

associated t values are shown in Table 4. None of the forecast errors are statistically significant at the 5% significance level and neither is there a tendency for such errors to become larger at ever increasing times to failure. This picture is consistent with the relative stability of B’ and m shown in Fig. 4.

5.2 Batch B

This set of test results was used to assess the size of any batch to batch variation for 2*25Cr-1Mo steel tubes. A comparison of Tables 1 and 5 reveals that the Monkman-Grant relation is quantitatively similar

-14 NATURAL LOG OF FAILURE TIME In(+), s

1

‘--l-i;:

I

. 813K

. 853K

1 Step ahead predictions

Fig. 5. Predictive capability of the Monkman-Grant relation (Batch A).

188 44. Evans

Table 4. Predictive capability of the Monkman- Grant relation (Batch A)

Index W4 14&d, e,! t:,

21 14.537 14.749 -0.212 0.874 22 14.662 15.039 -0.377 1.5123 23 14.759 14.55 0.209 O-877 24 14.884 14.767 0.117 0.48 25 14.967 14.803 0.164 0.672 26 15.549 16.035 -0.486 1.728t 27 15.73 15.872 -0.142 0.517 28 15.841 15.745 0.096 0.353 29 15.938 16.198 -0.26 0.906 30 16.16 16.415 -0.255 0.864

* Statistically significant at the 5% significance level, t statistically significant at the 10% significance level, $ statistically significant at the 20% significance level.

between batches. For example, the two values for m (-0.915 and -0.859) are not statistically different from one another. However, three major differences have been identified between these two batches of test results. First, the test data in batch B is much ‘dirtier’ in two respects.

(i) An outlier appears to be present. In Fig. 6(a) the LDi statistic associated with In(&) = 14.814 exceeds the critical test value (this test was carried out at a temperature of K = 783) and l,,, in Fig. 6(b) for this test result is relatively high-in excess of O-5

?-

I-

i-

j-

(assuming all the data points are equally influential these l,,, values would take on a value of 0.14). This data point was therefore removed and the resulting best fit line (obtained using least squares) is shown in Fig. 7 and Table 5. The outlier highlighted in Fig. 7 is clearly a rogue in that the other test results at this temperature with a similar .C5 have much lower failure times.

(ii) Even after the removal of this outlier the scatter of test results in batch B is much greater than that in batch A. This shows up in a comparison of Figs 1 and 7 as well as the higher RZ values (almost 4% points higher) shown in Table 5 compared to Table 1.

The second major difference is that whilst the errors from the Monkman-Grant relation shown in Table 5 are normally distributed and homoscedastic (Ei = 0.13, H, = O-24) and the relationship is linear (& = O-17), there appears to be a dependency of this relationship on temperature. In Table 6 the parameter in front of the dummy variable associated with a temperature of 813K (Bi) appears statistically significant at the 5% level. Thus the intercept of the relationship increases by 0.50 from the zero value at a temperature of 783K. In Fig. 8 two solid lines are shown-the higher one for the (lowest 36 failure times) data obtained at 813K and the lower one for the (lowest 36 failure time) data obtained at all other temperatures. The data points associated with a

5% Critical value for LDi

In(tF) = 14.814, K= 783

5 10 15 20 25 30 35 40 45 50 55

Index of ordered failure times

Fig. 6(a). Likelihood distance estimates from the Monkman-Grant relation (Batch B).

1

0.9

0.8

07

06

0.5 ::

2 0.4

0.:

0.:

0:

I

-0

Monkman-Grant empirical relation 189

_ _ _. _ _.. _ _ _ - _ _ ̂. _ _-.., . -_ ..-- .___ .__ .---- .----.----. .-. -- ---- --

In(tF) = 14.814, K= 783

Index of ordered failure times

Fig. 6(b). Plot of l,,, against the index of ordered failure times (Batch B).

Table 5. Least squares estimates for the Monkman-Grant influential data points in batch B) a number of relation using Batch B test data (assuming k = m) conclusions can be identified.

3’ B nz o-

Estimated parameter -0.706 0.494 -0.859 0.411 t statistic -1.32 - -26.09* 10.14"

R2 = 93.16%

* Statistically significant at the 5% significance level, t statistically significant at the 10% significance level. R2 is the coefficient of determination.

temperature of 813K (circles) are consistently above the lower solid line and fit better around the upper line.

The third major difference is that the predictive ability of the Monkman-Grant relation shown in Table 6 is slightly poorer for batch B. This shows up in Table 7 where one prediction is significantly in error at the 5% level and two at the 20% level. However, there is no tendency for the predictive errors to increase with higher failure times. This is seen clearly in Fig. 8.

i. Rupture times appear to be log normally distributed over the range of accelerated test conditions.

ii. Over the range of accelerated test conditions the Monkman-Grant relation appeared to be linear. Whilst there was no evidence that the variability in creep rupture life depended on the minimum creep rate, temperature conditions do appear to influence the intercept of the relation.

iii. Perhaps most importantly as far as extrapolation is concerned, the predictive ability of the Monkman- Grant relation appears to be good. There was no evidence to suggest that the estimated relation consistently over- or under-estimated the higher failure times obtained in each batch of tests.

This last conclusion suggests that the Monkman- Grant relation should be subjected to a more stringent set of extrapolation tests. Namely that the relations identified over the accelerated test conditions of this paper should be used to extrapolate out to 10” and 1O’h for rupture life. It is the intention of this author to investigate such issues in the near future.

6 CONCLUSIONS

ACKNOWLEDGEMENTS Whilst some batch to batch differences were identified in the Monkman-Grant relation and in the data sets themselves (most disturbing being some highly

The author would like to thank Professor R. W. Evans of the IRC-Swansea for access to, and permission to

90 M. Evans

ra

ln(tF) = 14.8

5 1 k ‘7

.I

Fig. 7. The Monkman-Grant relation for 2.25Cr-1Mo steel tubes (Batch B).

Table 6. Assessing the temperature dependency of the Monkman-Grant relation (Batch B)

B’ m CT B; BS B; &

Estimated parameter -0.588 -0.842 0.36 0.498 0.109 0.203 -0.221 t statistic -1.0 -25.95* 264x 0.57 1.13 -1.06

R= = 95.16%

* Statistically significant at the 5% significance level, t statistically significant at the 10% significance level. R ’ is the coefficient of determinant. B’ is the intercept associated with a temperature of 783K. B’ + B; is the intercept associated with a temperature of 873K.

-12

i

2

-17

8 -18 -1

$ -19

2 -20

-21

NATURAL LOG OF FAlLUFZ TIME In&), s ,

\ 9 11 1 1

‘14 1 1 19

. l ’

,e

f

n 783K - ._

l 8I3K l r.- 1 '\ l

A 838K '.. . . '\

4 /

- 873K

I step ahead pre.5ctmns (excluding 8 13K) I ‘...

-- lstepahead pddmm(at8l3Kody~ I

/

I / I

Fig. 8. Predictive capability of the Monkman-Grant relation (Batch B).

Monkman-Grant empirical relation 191

Table 7. Predictive capabilitjl of the Monkman- Grant relation (Batch B)

Index W,), MF), ’ e, CL

37 14.038 14.061 -0.023 38 14.074 13.625 0.449 39 14.149 13873 O-276 40 14.222 13.823 0.399 41 14.352 14.576 -0.224 42 14.426 14.151 0.275 43 14.518 14.517 0.001 44 14.648 14.967 0.151 45 14.703 14.607 0.096 46 14.74 14.359 0.381 47 14851 14.111 0.74 48 15.092 15.072 0.02 49 15.332 15.102 0.23 50 15.369 15.191 0.178 51 16.183 15,191 0.992 52 16.312 15.567 0.745 53 16.46 16.49 -0.03

0.‘057 1.1 0.608 0.967 0.526 0.602 0.003 0.354 0.208 0.904 1.623$ 0.045 0.482 0.402 2.239” 1,639$ O”O57

* Statistically significant at the 5% significance level, t statistically significant at the 10% significance level, !: statistically significant at the 20% significance level.

use, the extensive set of test results contained within their creep data bank.

REFERENCES

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4. Manson, S. S. and Haferd, A. M., A linear time-temperature relation for extrapolation of creep and stress-rupture data. NACA TN 2890. 1953.

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