- Email: [email protected]
Reliable analysis and extrapolation of creep rupture data
Accepted Manuscript Reliable analysis and extrapolation of creep rupture data J. Bolton PII:
S0308-0161(17)30015-7
DOI:
10.1016/j.ijpvp.2017.08.001
Reference:
IPVP 3641
To appear in:
International Journal of Pressure Vessels and Piping
Received Date: 6 January 2017 Revised Date:
2 July 2017
Accepted Date: 19 August 2017
Please cite this article as: Bolton J, Reliable analysis and extrapolation of creep rupture data, International Journal of Pressure Vessels and Piping (2017), doi: 10.1016/j.ijpvp.2017.08.001. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
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ACCEPTED MANUSCRIPT Reliable analysis and extrapolation of creep rupture data J. Bolton, 01-07-‘17 [email protected]
Abstract:
Keywords: Creep rupture, rupture model, extrapolation, P-NID method
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1. Introduction:
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The P-NID (Parametric, Numerical Isothermal Datum) method of extrapolating creep rupture data has been applied to the four large datasets recently analysed by the European Creep Collaborative Committee in order to re-evaluate its own recommended procedure. It is demonstrated that the P-NID method provides a very reliable basis for extrapolation.
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The progressive development of high-temperature plant since the 1950s has depended upon the development of creep-resistant materials; and the provision of design data for their long-term rupture properties has relied upon extrapolation from shorter-term rupture tests. Numerous techniques of modelling and extrapolation have been developed over that period for that purpose. The models employed, whether long-established or of recent origin, and whether or not they purport to represent creep mechanisms, are generally in the form of a mathematical equation for rupture life as an explicit function of temperature and stress. Each contains a number of numerical coefficients that are determined by fitting to the data. However, such models have achieved only intermittent success, being found to fit data satisfactorily for one material but less well for another.
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It is not unusual for researchers within different organisations to favour different types of model, or different procedures for model optimisation, with the result that wide divergences may be found between predictions of long-term life. In consequence, it is likely that some extrapolations that form the basis of current design data are less reliable than design engineers suppose. It has long been the aim of analysts concerned by such divergences to establish a greater consistency of practice, both in the comparative review of candidate models and in the application of validity tests.
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Much work has been done to improve matters by the European Creep Collaborative Committee (ECCC), which has recommended a set of validity tests applicable to any mathematical model, but this approach has met with only partial success. As a result of known problems within this procedure, the relevant ECCC Working Group (WG1) performed a recent re-evaluation of their own recommendations using large datasets for 2.25Cr1Mo steel, 11%CrMoVNb steel, Type 304 stainless steel and Incoloy 800 as test cases. Their results were published in [1]. The present article reports a similar exercise in which a fundamentally different approach, the P-NID method [2, 3] was applied to the same datasets, which were made available for that purpose by ECCC. The method is described in the context of its application to each material in the following sections, and a general synopsis of the method is given in Appendix A. Comparisons are made between P-NID models for these datasets and models proposed by ECCC [1]. Reliability in extrapolation is tested by constructing models from short-term data and comparing their predictions to long-term data omitted from the modelling process.
2. Analysis of 2.25Cr1Mo steel data: 2.1 Determination of median rupture lives The ECCC dataset for 2.25Cr1Mo includes the results of 1016 tests to rupture at temperatures o o between 450 C and 650 C, after the elimination of anomalous and duplicated results. It incorporates
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ACCEPTED MANUSCRIPT results from a number of different sources on a large number of sample heats, tested to widely varying extents. The dataset is far from ideal because rupture lives are not known for a consistent set of samples tested over the same range of conditions, and the diversity in sampling and test conditions introduces potential problems in modelling.
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The potential problems are illustrated by Fig.1 which shows rupture curves for all 33 samples o (designated D1, D2, etc) tested at 550 C and at a minimum of two stress levels. The logarithmic stress scale of the figure is expanded for clarity and shows horizontal gridlines only at an illustrative series of stresses. Table 1 shows the median lives and geometric mean lives (mean of logarithmic lives) determined from tests at those specific stresses. Data at nominally 176.5 MPa include all data at 176 and 177 MPa, and data at nominally 78.5 MPa include all data at 78 and 79 MPa. o
Table 1: Summary of rupture data at specific stresses, 2.25Cr1Mo at 550 C 206 3 141 130
176.5 22 314 321
137 16 1590 1570
108 21 10,300 9390
78.5 9 37,200 41,540
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Test stress, MPa Number of samples tested Median rupture life, h Mean rupture life, h
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The tabulated median and mean rupture lives agree to within about 10%, which is a small fraction of the observed scatter. But it can be seen in Fig.1 that these lives are not representative of a consistent group of samples at each stress; for example, (i) A relatively large number of stronger samples were not tested at 206 MPa but were tested only at higher and lower stresses; and a relatively large number of weaker samples were tested only at lower stresses. (ii) A relatively large number of stronger samples were not tested at 78-79 MPa but were tested only at higher stresses.
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Such inconsistencies of sampling are a common problem in the analysis of materials data. However, the typical trend may be more confidently established by incorporating the additional information obtainable by interpolation between tests at higher and lower stresses, or by extrapolation from tests at higher stresses only or at lower stresses only.
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Where data were available only at higher and lower stresses, the curves for individual samples were interpolated. The curves in Fig.1 are automatic EXCEL fits to the available data, and some of the curve shapes suggest that some of the data may be unreliable. But no reason was found to discount particular data points, so all points were treated as valid and all interpolations were treated as valid. It was not necessary for the interpolations to be precise, only to determine whether each curve passed above or below a trial value of median life.
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Where data were available only at higher stresses, or only at lower stresses, the curves for individual samples were projected along upper and lower lines consistent with the general trend of other samples. It was not necessary for those projections to be precise, only to determine whether they could pass only below or only above a trial value of median life. For most samples either the upper projection passed below but not credibly above a trial median, or vice versa. A minority of cases were indeterminate and those were discounted. Thus the population of data available for the determination of median values of rupture life was greatly expanded relative to Table 1, as shown in Table 2. The median lives in this table were determined simply by counting the total number of samples to either side of a trial value, and adjusting the trial value until a balance was achieved. o
Table 2: Number of samples below/above median rupture life, 2.25Cr1Mo at 550 C Stress, MPa Median life, h Stress specific (below/above)
206 125 1/2
176.5 295 10/12
137 1650 8/7
108 10,000 9/12
78.5 63,000 7/2
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ACCEPTED MANUSCRIPT Interpolated (below/above) Extrapolated (below/above) Total results (below/above)
0/6 12/5 13/13
0/2 4/1 14/15
4/6 1/0 13/13
5/2 1/1 15/15
3/0 2/11 12/13
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The median lives determined at stresses of 176.5, 137 and 108 MPa differ very little from those in Table 1, because the numbers of curves interpolated or extrapolated were relatively few and largely counterbalanced. The median life at 206 MPa is changed little because the relatively large numbers of interpolated and extrapolated curves counterbalanced each other. But the median life at 78.5 MPa is much increased owing to the large number of curves for stronger samples that could confidently be projected to rupture lives greater than 63,000 h. The consistency of those projections with the trends exhibited by other samples can be seen in Fig.1.
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The effect of including extrapolated results at 78.5 MPa, which increases median life from 37,200 h to 63,000 h, demonstrates the importance of the additional information in establishing median lives for a broadly consistent group of samples. Such information is effectively discarded in methods based upon statistical criteria, such as the ECCC approach, which treat every data point as if it were for a unique sample. o
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Median lives at 550 C were similarly determined at stresses of 156, 98, and 88 MPa; and median o rupture lives were determined in the same way at temperatures of 475, 500, 600 and 650 C. The dataset was thus condensed to median rupture lives at 33 conditions of stress and temperature. Median values were used in this study as the basis of model formulation because they are readily determined by the simple counting process described above. It would be laborious to model and extrapolate curves for individual samples to the specific lives necessary for the calculation of mean values, to no advantage. The differences between median and geometric mean values are expected to be small, as for example in Table 1, and to be insignificant in the context of establishing lowerbound values, e.g. at 95% Confidence as discussed later in Section 2.4. 2.2 Selection of parametric equations
=
(1a)
=
−
1⁄ − 1⁄
(1b)
is a numerical coefficient, with units of absolute temperature.
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Where
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Having established a set of median data, the next step in the P-NID procedure is to find a pair of parametric equations that resolve the data at all test temperatures into a single, isothermal rupture curve at a convenient datum temperature. A series of candidate pairs of equations, corresponding to a series of established rupture models is given in Appendix B. The parametric equations selected for the 2.25Cr1Mo data were those of the Orr-Sherby-Dorn (OSD) model, for which stress and rupture time at test temperature (absolute) are transformed to parametric stress and parametric rupture time at datum temperature by the relationships,
The OSD model was chosen because it resulted in satisfactory convergence of all median data into a narrow band as shown in Fig.2, the value of being selected for optimum visual convergence. The o figure is a parametric plot of the median data transformed to a datum temperature of 550 C, for = 42,500 K. The residual disparities between data obtained at different temperatures are in the region of +/- 20% of rupture time, which is no greater than disparities commonly found between duplicate tests on specimens taken from the same sample. A number of alternative pairs of parametric equations were considered but were rejected because they failed to produce convergence within a sufficiently narrow band. 2.3 Definition of datum rupture curve The final stage in constructing a P-NID model is to select a series of stress and rupture time coordinates defining the datum curve of Fig.2. The usual procedure is to select 11 such points, from which the full median rupture curve is generated by a piecewise cubic spline as described in Appendix C. The model so defined (OSD-NID) is shown in Fig.2, together with a tolerance envelope. The tolerance envelope has an upper limit at stress x1.025 and rupture time x1.10 and a lower limit at
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ACCEPTED MANUSCRIPT stress /1.025 and rupture time /1.10, as recommended in [3]. A full numerical definition of the model curve may be found in Appendix D.
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A more conventional comparison of the OSD-NID model to the median data is presented in Fig.3, showing the model and data separately at each test temperature. The correspondence between model and data is very satisfactory across the range of test results. Hence interpolation and extrapolation from this model may be considered valid anywhere within the ranges of parametric stress, parametric time and temperature in Fig.2. But it ceases to be valid beyond the lowest-stress and longest-term data point, which is the parametric boundary of existing data. Thus there is no basis o for extrapolation at 650 C, but at lower temperatures the model curve is expected to remain reliably o accurate up to the same parametric limit, e.g. to about 130,000 h at 600 C (Fig.3) or to about o o 6 o 7 o 200,000 h at 590 C. The equivalent limits at 550 C (2.5.10 h), 500 C (7.0.10 h) and 475 C 8 (4.4.10 h) are beyond the time-scale of practical interest. 2.4 Comparison of P-NID model to distribution of data for individual samples o
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A comparison of the OSD-NID model to individual sample data at 550 C is shown in Table 3, in the same fashion as Table 2. Rupture lives defined by the model are within about 10% of the median values in Table 2 and the differences between sample numbers in the different categories in Tables 2 & 3 are small. o
Stress, MPa Model rupture life, h Stress specific (below/above) Interpolated (below/above) Extrapolated (below/above) Total (below/above)
206 141 1/1 0/6 12/5 13/12
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Table 3: Number of samples below/above OSD-NID model rupture life, 2.25Cr1Mo at 550 C 176.5 323 11/11 0/2 5/1 16/14
137 1600 8/8 4/7 1/0 13/15
108 10,550 10/10 5/2 0/1 15/13
78.5 62,000 7/2 3/0 2/11 12/13
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The combined sum of samples within any sub-population, below and above the median, is not always the same in Table 3 as in Table 2. That is because some stress-specific and interpolated results coincided with the median or with the model and fell into neither category, and some extrapolated results fell into neither category with any certainty.
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A comprehensive check on the OSD-NID model is provided by Fig.4. This is a parametric plot of every individual test result, and is useful in two respects. Firstly, data neglected in the construction of the model, at temperatures for which there were few test results or for samples for which there were too o few test results, may be compared to the model. For example, data at 450 – 454 C may be compared o to the model established from data at higher temperatures. The relatively sparse data at 450 – 454 C are visibly consistent with the model. Initially neglected data at a number of intermediate temperatures may also be seen to be consistent with the model. Secondly, it is important in the context of engineering design to appreciate the extent to which a weaker sample may fall short of the median rupture life. A median line is of little practical value without some quantitative allowance for the shortfall, and it is useful to represent the scatter in data as a margin relative to the median. A boundary representing, for example, 95% confidence may be drawn for stress and time margins that exclude the lower 5% of the total data. The boundary shown in Fig.4 was constructed at median stress / 1 + and median rupture time / 1 + 4 , for = 0.13. The form of these margins is consistent with the tolerance envelope discussed above. The boundary can be seen to exclude a fairly consistent proportion of data points across the range of parametric time. It is possible that a model based on geometric mean lives would produce a line somewhat above or below a model based on median lives. But the margin required for 95% Confidence would necessarily increase or decrease accordingly, without significant alteration to the 95% Confidence line. 2.5 Comparison of ECCC model to data for individual samples and to P-NID model
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ACCEPTED MANUSCRIPT A model judged in the ECCC re-evaluation exercise [1] to be accurate and reliable in extrapolation (ECCC MCm) is also shown in Fig.3. This model can be seen to diverge substantially from the median rupture data in many areas. o
A comparison between the ECCC model and data at 550 C is presented in Table 4 in the same fashion as in Table 3. o
Table 4: Number of samples below/above ECCC MCm model rupture life, 2.25Cr1Mo at 550 C 206 282 3/0 4/2 15/1 22/3
176.5 670 18/3 1/1 7/0 26/4
137 2580 13/3 9/2 1/0 23/5
108 8400 8/13 4/3 0/1 12/17
78.5 37,500 5/4 2/1 2/11 8/16
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Stress, MPa Model rupture life, h Stress specific (below/above) Interpolated (below/above) Extrapolated (below/above) Total (below/above)
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Large imbalances between the total number of samples below and above the model rupture lives confirm that this model is an inaccurate representation of the material data. Fig.1 confirms graphically that the model lives at 206, 176.5, 137 and 78.5 MPa are at variance with the data. The ECCC MCm model cannot therefore be a reliable basis for extrapolation.
3. Analysis of 11%CrMoVNb steel data:
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It is common, for the purposes of engineering design, for models of the kind discussed to be o extrapolated to 200,000 hours. Extrapolation of the OSD-NID model at 550 C gives a rupture stress at 200,000 h of 61 MPa; but at that stress the ECCC MCm model predicts a rupture life of 118,000 hours. The evidence presented here indicates that the OSD-NID model is the more reliable of the two, and hence implies that the ECCC MCm model underestimates rupture life at that stress by 40%.
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The ECCC dataset for 11%CrMoVNb steel includes the results of 305 tests to rupture at temperatures o o between 425 C and 600 C. Results were available for a large number of sample casts, tested to widely varying extents. The dataset contains much of the 12%CrMoVNb dataset modelled and discussed elsewhere [4, 5] but omits some of that data and includes other data, so the two datasets are significantly different and should not be confused. A P-NID model of the 11%CrMoVNb data was created following the same procedure as described above for the 2.25Cr1Mo steel. o
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Fig.5 shows the data for 33 samples tested at 550 C and illustrates the determination of median data at stresses of 392, 309, 247, 196 and 140 MPa. Table 5 presents the distribution of sample data relative to estimated median lives at the selected stresses. o
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Table 5: Number of samples below/above median rupture life, 11%CrMoVNb at 550 C Stress, MPa Median life, h Stress specific (below/above) Interpolated (below/above) Extrapolated (below/above) Total (below/above)
392 300 3/2 1/4 4/2 8/8
309 4120 6/7 7/7 1/0 14/14
247 24,000 4/4 7/7 0/0 11/11
196 50,500 1/4 8/4 0/1 9/9
140 90,000 3/0 0/0 3/7 6/7
Many of these samples were not tested at the lower stresses, with the consequence that median rupture lives at those stresses were determined by a relatively small proportion of the total number of samples. Nevertheless, as far as can be judged from Fig.5, the samples tested were representative of the general population. At each stress the balance between samples below and above the median life is satisfactory.
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ACCEPTED MANUSCRIPT o
Median lives at 550 C were similarly determined at stresses of 355, 170, and 154 MPa; and median o rupture lives were determined in the same way at 475, 500 and 600 C. The dataset was thus condensed to median rupture lives at 23 conditions of stress and temperature. A pair of parametric equations was found that produced satisfactory convergence of the median data o at a datum temperature, as shown in Fig.6 at 550 C. The parametric relationships chosen were those for a Bolton A (BTA) model, as below. =
−
−
⁄
=
(2a)
−
−
⁄
(2b)
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Where and are numerical coefficients, with units of absolute temperature, determined so as to obtain optimum convergence. Other symbols have the same meaning as for eqns. 1a and 1b. The parametric rupture plot of Fig.6 demonstrates that, for = 220 K and = 32 K, the median data at all temperatures coalesce satisfactorily into a single curve. Residual disparities are in the region of +/- 20% of rupture time, which is no greater than the disparities found between duplicate tests.
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As in the case of 2.25Cr1Mo, a numerical model of the parametric rupture curve of Fig.6 was created from 11 selected co-ordinates of stress and rupture time, as tabulated in Appendix D, and the piecewise cubic spline described in Appendix C. The model so defined (BTA-NID) is shown in Fig.6 together with the same tolerance envelope, between stress x1.025 and rupture time x1.10 and stress /1.025 and rupture time /1.10. A conventional comparison of the BTA-NID model to the median data is presented in Fig.7, showing the model and data separately at each test temperature. The correspondence between model and data is very satisfactory across the range of test temperatures. At each temperature, model extrapolation beyond the available data, but within the parametric boundary, is justified by the accuracy with which the chosen parametric relationships (eqn.s 2a & 2b and the values of and ) represent the data. o
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A comparison of the BTA-NID model to the distribution of individual sample data at 550 C is shown in Table 6. Rupture lives defined by the model are mostly within 10% of the median values in Table 5 and the differences between sample categorisation in Tables 5 & 6 are small. o
Table 6: Number of samples below/above BTA-NID model life, 11%CrMoVNb at 550 C 392 243 3/2 1/4 3/2 7/8
309 4370 6/7 8/6 1/0 15/13
247 24,100 4/3 7/7 0/0 11/10
196 50,100 0/4 8/4 0/1 8/9
140 87,700 3/0 0/0 3/7 6/7
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Stress, MPa Model rupture life, h Stress specific (below/above) Interpolated (below/above) Extrapolated (below/above) Total (below/above)
Fig.8 is a parametric plot of all the data, in which the 95% confidence boundary is drawn in the same way as in Fig.4, but for = 0.11. Fig.7 also shows a model (ECCC OSD3) judged in the ECCC re-evaluation exercise to be accurate and reliable in extrapolation. The OSD3 model curves can be seen to depart substantially from the median rupture data at the higher temperatures. A comparison between the OSD3 model and sample o data at 550 C is presented in Table 7. o
Table 7: Number of samples below/above ECCC OSD3 model life, 11%CrMoVNb data at 550 C Stress, MPa Model rupture life, h Stress specific (below/above) Interpolated (below/above) Extrapolated (below/above) Total (below/above)
392 650 5/0 2/3 5/0 12/3
309 4350 6/7 8/6 1/0 15/13
247 16,500 2/6 6/8 0/0 8/14
196 45,300 0/5 6/6 0/1 6/12
140 127,000 3/0 0/0 7/1 10/1
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ACCEPTED MANUSCRIPT Large imbalances between the total number of samples below and above the model rupture lives confirm that this model is an inaccurate representation of the material data. It is evident from visual inspection of Fig.5 that the median life at 140 MPa must lie somewhere in the region of 90,000 h, and that a model life at that stress of 127,000 h is unrepresentative of the available data. It is likewise evident that a model life of 650 h at 392 MPa is unrepresentative of the data. o
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In terms of the stress to cause rupture in 200,000 hours, extrapolation of the OSD3 model at 550 C gives a stress of 116 MPa, but at that stress the BTA model gives a rupture life of 125,000 hours. The evidence presented here indicates that the BTA model is the more reliable of the two, and hence indicates that the OSD3 model overestimates rupture life at that stress by 60%.
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4. Analysis of Type 304H stainless steel data:
The ECCC dataset for this steel includes the results of 782 tests to rupture at temperatures between o o 482 C and 899 C. Specimens tested were from a large number of samples.
o
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Analysis of the data followed the same procedure as for the two steels discussed above and is summarised below in terms of a corresponding series of plots, Fig.s 9, 10 & 11, and a brief accompanying text. Fig.9 is a parametric plot at 650 C of the median data at the major test temperatures. Satisfactory convergence was obtained with the parametric relationships of a Minimum Commitment (MC) model, as below. =
=
(3a)
−
−
⁄
−
1⁄ − 1⁄
(3b)
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Where (150 K) and (36,000 K) are numerical coefficients and other symbols have the same meaning as for eqns. 1a and 1b. o
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The tolerance envelope is dimensioned as before. Median data at 732 C fall somewhat outside this tolerance, but that is attributable to the samples tested at that temperature being different from those o tested at other temperatures. The gradient of median data at 732 C is nonetheless consistent with gradients at adjacent temperatures and provides confirmation of the continuity of the parametric rupture curve.
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Fig.10 is a comparison of the MC-NID model to median data at each temperature and also shows a model (ECCC SM1) endorsed by ECCC [1] as accurate and reliable in extrapolation. The SM1 model deviates significantly from the median data at high stresses and at low stresses. In some areas, extrapolation of the SM1 model predicts only half the rupture life predicted by the MC-NID model. Fig.11 is a parametric plot of data scatter relative to the median model, showing that relatively sparse o o data for test temperatures below 550 C and above 800 C are entirely consistent with the MC-NID model defined within that range. The 95% confidence boundary is drawn for = 0.10.
5. Analysis of Incoloy 800 data: The ECCC dataset for Incoloy 800 includes the results of 495 tests to rupture at temperatures o o between 500 C and 1050 C. Specimens from a large number of samples were tested. Analysis of the data followed the same procedure as for the steels discussed above and is summarised below in terms of a series of plots, Fig.s 12, 13 & 14, and a brief accompanying text.
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Fig.12 is a parametric plot at 700 C of the median data at the major test temperatures. Satisfactory convergence was obtained with the parametric relationships of a Manson-Haferd (MH) model, as below. =
=
(4a)
−
⁄
−
−
−
/
−
(4b)
16
Where (280 K) and (2.10 h) are numerical coefficients and other symbols have the same meaning as for eqns. 1a and 1b.
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The tolerance envelope is dimensioned as before and bounds all the median data. Fig.13 is a comparison of the MH-NID model to median data at each major test temperature and also shows a model (ECCC MH3, [1]) presented as accurate and reliable in extrapolation. The differences between the two models are fairly small within the range of the existing median data, but the MH-NID model conforms more closely to that data and thus appears the more reliable basis for extrapolation. In some areas, extrapolation of the ECCC MH3 model predicts only half the rupture life predicted by the MH-NID model. o
6. Extrapolation from restricted data:
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Fig.14 shows that data neglected in the construction of the MH-NID model, at 1050 C and at some intermediate temperatures, are consistent with the model. The 95% confidence boundary is drawn for = 0.12.
The ability of a procedure to produce a model that is reliable in extrapolation can be tested by modelling a restricted dataset that omits all longer-term data, and then comparing the resultant model to the omitted, longer-term data.
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Figure 15 is a parametric plot of median rupture data for 2.25Cr1Mo restricted to 20,000 hours. It is identical to Fig.2 except for the exclusion of longer-term data. Sufficient data remain to show convincing convergence and overlap of data obtained at all temperatures. The OSD-NID(R) model shown in Fig.15 is based on this restricted dataset, but is nevertheless identical to the OSD-NID model of Fig.3 based on the full dataset - because the fit to the restricted dataset could not be visibly improved by selecting an alternative pair of parametric equations, by adjusting the coefficient value of the parametric equations, or by re-selecting the points that define the datum curve. The parametric o boundary of the model also remains unchanged as it is fixed by data at 650 C at less than 20,000 hours.
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A marginally different coefficient value might have been chosen for model OSD-NID(R), and different points might have been selected to define the datum curve – but not such that any resultant model could have been noticeably different from the OSD-NID model of the full dataset. Hence, as illustrated in Fig.16, the omitted longer-term data provide direct evidence of the model accuracy over a two-fold o o extrapolation at 600 C, a three-fold extrapolation at 550 C and nearly four-fold extrapolations at 500 o o C and 475 C. Fig.s 17 & 18 provide a similar demonstration for 11%CrMoVNb, with data restricted to 24,000 h. A new model of the restricted data, BTA-NID(R), removed the slight inflection of the original BTA-NID o model at about 75,000 h (at 550 C) but was otherwise barely distinguishable from the original. The o only significant change was that the parametric limit was reduced to about 115,000 h (at 550 C) in o consequence of the exclusion of the longest-term data at 600 C. The excluded longer-term data at o temperatures below 600 C provide direct evidence of model accuracy over a nearly four-fold o o extrapolation at 550 C and a nearly five-fold extrapolation at 500 C. A similar exercise for Type 304 SS showed that a model of data restricted to 12,000 h was accurate o over a nearly five-fold extrapolation at 600 C.
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ACCEPTED MANUSCRIPT No similar exercise could be performed for Incoloy 800 because any significant restriction of the data led to insufficient overlap of the parametric data at different temperatures, and thus there remained insufficient evidence of model continuity.
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Table 8 summarises extrapolations from P-NID models of restricted data, comparing them to the longest-term test data available. The first six rows show comparisons for median data from the ECCC datasets, as discussed above. Also shown, in italics, are the lives according to ECC models based on unrestricted data. The following eight rows show comparisons for median data for other datasets. The final four rows are for individual samples of 2.25Cr1Mo that have been tested over exceptionally long times. Results for datasets other than the ECCC datasets may be found in [2, 3, 5, 6 and 7]. Some are unpublished except in Researchgate website postings or are unreported in any form. Table 8: Comparison of P-NID models of restricted data to omitted, longer-term data
11%CrMoVNb Median data
304H SS Median data Ni15%CrTiAl Median data
Data restriction <20,000 h None <20,000 h None <20,000 h None
550 C, 140 MPa o 500 C, 310 MPa o 600 C 98 MPa
o
BTA OSD3 BTA OSD3 MC SM1
<24,000 h None <24,000 h None <12,000 h None
x 3.8 None x 4.8 None x 4.6 None
87,700 h 127,000 h 115,000 h 81,200 h 55,700 h 40,500 h
o
MH
<8000 h
x 2.4
18,800 h
16,800 h
1.12
MH
<8000 h
x 2.1
17,000 h
15,000 h
1.13
800 C 50 MPa o 700 C 200 MPa o
12%CrMoVNb Median data
500 C 309 MPa
9%CrMo Median data
600 C 80 MPa
1%CrMoV Median data 2.25Cr plate Median data
500 C 118 MPa
Model life 62,000 h 37,500 h 76,500 h 64,300 h 75,000 h 57,300 h
Test life 63,000 h
85,000 h 95,000 h 90,000 h
130,000 h 60,000 h
Model life/ Test life 0.98 0.60 0.90 0.76 0.79 0.60 0.97 1.41 0.88 0.62 0.93 0.68
BTA
<12,000 h
x 4.2
50,000 h
60,000 h
0.83
BTA
<15,000 h
x 5.7
85,000 h
80,000 h
1.06
550 C 137 MPa
BTB
<10,000 h
x 9.7
97,000 h
92,000 h
1.05
o
MH
<15,000 h
x 13
200,000 h
200,000 h
1.00
650 C 78 MPa o 600 C 118 MPa
BTB
<5000 h
x 15
73,000 h
66,000 h
1.11
BTB
<5000 h
x 23
115,000 h
105,000 h
1.10
o
OSD
<11,000 h
x 8.8
97,000 h
100,000 h
0.97
o
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o
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18%Cr Median data
Extrp’n multiple x 3.1 None x 3.8 None x 3.8 None
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Model type OSD MCm OSD MCm OSD MCm
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2.25Cr1Mo Median data
Test Condition o 550 C, 78.5 MPa o 500 C, 127 MPa o 475 C 170 MPa
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Material
o
2.25Cr1Mo Sample D2
550 C 62 MPa
2.25Cr1Mo Sample D1
500 C 123 MPa
o
OSD
<7000 h
x 19
130,000 h
104,000 h
1.25
2.25Cr1Mo Sample J12
550 C 78 MPa o 500 C 118 MPa
o
OSD
<4200 h
x 25
103,000 h
131,000 h
0.79
OSD
<4200 h
x 29
120,000 h
142,000 h
0.85
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ACCEPTED MANUSCRIPT The tabulated results show satisfactory agreement between longer-term tests and P-NID models of restricted data over extrapolations of up to almost thirty-fold. The extrapolations shown in the first six rows are substantially closer to the longer-term test data than ECCC models based on unrestricted data. o
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A parametric plot for the greatest of these extrapolations, for 2.25Cr1Mo sample J12 at 500 C, is shown in Fig.19. The longest-term data for this sample were taken from Ref.8, which records rupture lives for a number of low-stress tests recorded as unbroken in the ECCC dataset. These additional results were not included in the determination of 2.25Cr1Mo median data.
7. Discussion: 7.1 Accuracy of P-NID models
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The principle of extrapolation from a P-NID model is that, if it accurately represents the systematic dependence of rupture time on temperature and stress under all conditions of testing, it will do so under any other condition within the parametric limits of the test data. Hence the reliability of such a model in extrapolation depends fundamentally upon the accuracy with which the model succeeds in representing the available data. The scope for error at each stage of model construction is discussed below.
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The determination of median rupture time at each temperature and stress is a simple process of counting test results falling above and below an estimated median that, in itself, leaves no scope for error. There is scope for some subjectivity in the projection of individual sample data to a stress at which that sample has not been tested: but it is not necessary to estimate what the projected life might be, only to decide whether that life can confidently be expected to fall below or above an estimated median. The scope for error is therefore small, as may be verified by inspection of Fig.s 1 & 5 and Tables 2 & 5. There is also some scope for inconsistencies between median values where there are disparities between the numbers or identities of samples tested under different conditions of temperature and stress, which is a problem common to any modelling procedure. This is compensated to some extent in the P-NID procedure by parametric comparison of median lives determined at different temperatures, for different groups of samples, as in Fig.s 2, 6, 9 & 12. As noted with reference to data for Type 304 SS in Fig.9, the gradient displayed by a disparate group of samples may provide a useful guide at parametric times where there is a lack of data for consistent samples.
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The selection of a pair of parametric equations introduces some possibility of error if the analyst neglects to apply strict limits to the divergence between parametric data obtained at different temperatures. But the recommended tolerance envelope, the origin of which may be found in [3], allows little scope for any error of that kind. It is possible that more than one pair of parametric equations might achieve satisfactory parametric convergence, leaving some doubt over which is the more reliable basis for extrapolation. But, as discussed in [3], the divergence in extrapolation between such models is likely to be small. Once a pair of parametric equations has been selected and the optimum values of their coefficients have been determined, there is no further scope for error. A numerical datum relationship between stress and rupture time allows precise fitting to the data, regardless of how complex the shape of the observed relationship may be. In summary, there is some scope for subjectivity in the creation of a PNID model, but the constraints within the procedure ensure that any divergence between models produced by independent analysis must necessarily be small. As discussed in the preceding section, models based on restricted median data have shown satisfactory accuracy over up to 23-fold extrapolations in rupture time, and data for individual samples have shown satisfactory accuracy over up to 29-fold extrapolation. It has been proposed here that 95% confidence lines may be constructed relative to median rupture curves at median stress /(1 + ) and median rupture life /(1 + 4 ). Appropriate values of for the
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It is a key feature of the P-NID approach that the temperature terms in the parametric equations are validated by the convergence of data when plotted parametrically, as in Fig.s 2, 6, 9 and 12. By contrast, it is common in procedures for generating a single master equation that no specific test of validity is applied to those terms. The consequences are illustrated in Fig.20, which is a parametric plot for the ECCC MCm model of 2.25Cr1Mo material. The data are plotted in accordance with the 15 MCm parametric equations (Appendix B) and the coefficient value ( = 2.76.10 h) chosen by the investigators. Most of the data fall outside the tolerance envelope and the interweaving of data trends at different temperatures reveals that the temperature terms of this model are unsatisfactory. They define a systematic dependence on temperature that does not accord with the material behaviour. The same observations apply to the ECCC OSD3 model of 11%CrMoVNb material ( = 37,465 K), as shown in Fig.21.
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Similar observations apply, to a lesser degree, to the ECCC SM1 model of 304H SS and, to a lesser degree again, to the ECCC MH3 model of Incoloy 800. The corresponding parametric plots are not included here, but systematic discrepancies between model and data can be seen in Fig.s 10 and 13 respectively. A similar case is discussed in [7], which examined the selection by investigators at Swansea University of a Wilshire model [8] to represent data for a single heat of 2.25Cr1Mo steel. The parametric plot of Fig.22 reveals that the temperature terms of the Wilshire model, partitioned by the investigators into three regions and here designated WS-WSp, fail to represent the material behaviour with sufficient accuracy. This case illustrates the inconsistencies that may be overlooked if the investigators have a prior preference for a particular master equation.
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The second key feature of the P-NID approach is that stress dependence is represented in a piecewise manner, permitting precise replication of the observed data trends regardless of how complex their shapes may be. The advantages are evident in Fig.s 3 and 7, where the data delineate curves that defy accurate replication within the constraints of a conventional, single master equation. The discrepancies between the ECCC model curves and the data illustrate the point. The P-NID approach generates the equivalent of a master equation, but with the flexibility permitted by the equivalent of 12 or 13 coefficients as in Appendix D. In principle a rupture curve may be created that extends back to UTS values at very short times, but that has not formed part of this study.
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The third key feature of the approach is that it inherently defines the limit of valid extrapolation. The parametric boundary, which is the locus of the parametrically longest rupture time (and lowest stress) and equivalent points at other temperatures, is a strict and logical limit to valid extrapolation. All P-NID model curves plotted in this paper terminate at this boundary, unless curtailed by the limited extent of the time axis. Conventional methods, of the master equation type, have no in-built means of defining a validity limit and rely largely upon common practice.
8. Conclusion: A study of four large datasets, for 2.25Cr1Mo, 11%CrMoVNb, 304H SS and Incoloy 800, has shown that the P-NID method of rupture data modelling and extrapolation is both accurate and reliable. The study identified no detectable susceptibility of this method to significant error. The results of this and similar studies have demonstrated the reliability of this method over extrapolations of up to nearly thirty-fold from restricted data. Comparisons made to a widely-used procedure for model selection and extrapolation, the ECCC recommended method, indicated that the P-NID method is significantly more reliable.
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References:
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[1] M. Spindler, P. Lombardi, G. Merckling, A. Riva, M. Ortolani, M. Norman, M. Talik, S. Holmstrom and M. Schwienheer, A re-evaluation of the ECCC guidance for the assessment of full size creep rupture datasets, 3rd International ECCC Conference on Creep & Fracture in High Temperature Components Design & Life Assessment, Rome 5-7 May 2014. 798-808; INAIL, ISBN 978-88-7484510-1, 2017 (Incorporated as Appendix 1 in ECCC Recommendations – Vol 5 Part 1a, Generic recommendations and guidance for the assessment of full-size creep rupture data sets, Issue 6, 0705-2014, on ECCC website). [2] J. Bolton, A visual perspective of creep rupture extrapolation, Materials at High Temperatures, 30(2), 87-98, 2012.
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[3] J. Bolton, The potential for major extrapolation of creep rupture and creep strain data, Materials at High Temperatures, 31(2), 109-120, 2014.
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[4] British Standards Institute, PD6605-1:1998, Guidance on methodology for assessment of stressrupture data – part 1: Procedure for derivation of strength values, 1998 (withdrawn in 2011). [5] J. Bolton, The extrapolation of creep rupture data by PD6605 – An independent case study, International Journal of Pressure Vessels & Piping, 88, 158-165, 2011. [6] Unpublished results (Researchgate website posting: J. Bolton, Accuracy of major extrapolations in creep rupture life for an Austenitic steel, Aug 2015) [7] Unpublished results (Researchgate website posting: J. Bolton, Critical examination of a Wilshire model for creep rupture, Jul 2015)
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[8] Z. Abdallah, V. Gray, M. Whittaker and K. Perkins, A critical analysis of the conventionally employed creep lifing methods, Materials, 5, 3371-3398, 2014. [9] F.R. Larson and J. Miller, A time-temperature relationship for rupture and creep stresses, Trans. ASME, 74, 765-775, 1952.
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[10] S.S. Manson and A.M. Haferd, A linear time-temperature relation for extrapolation of creep and stress rupture data, NACA TN 2890, 1953. [11] A. Mendelson, E. Roberts and S.S. Manson, Optimisation of time-temperature parameters for creep and stress rupture, with application to data from German co-operative long-term creep program, NASA TN D-2975, 1965.
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[12] S.S. Manson and W.F. Brown, Time-temperature stress relations for rupture and creep stresses, Proc ASTM, 53, 683-719, 1953. [13] L.R. Orr, O.D. Sherby and J.E. Dorn, Correlation of rupture data for metals at elevated temperatures, Trans ASME, 46, 113, 1954. [14] S.S. Manson and U. Muraldihan, Analysis of creep rupture data for five multi-heat alloys by the Minimum Commitment method, Research project 638-1, EPRI CS-3171, 1983. th
[15] I.I. Trunin, N.G. Golobova and E.A. Loginov, Proc. 4 Int. Symp. On heat-resistant metallic materials, Mala Fatra, CSSR, 168, 1971. [16] B. Wilshire and M.B. Bache, Proc. ECCC Creep Conf., Creep and fracture in high temperature components, ISBN 978-1-60595-005-1, 2009.
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ACCEPTED MANUSCRIPT Appendix A: Summary of the P-NID method of extrapolation The P-NID method for modelling and extrapolating median rupture data proceeds by the following steps: 1. Scrutinise the available test data and delete any result that is clearly anomalous in relation to other data. Replace duplicate test results by a logarithmic mean rupture time.
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2. Plot log stress against log rupture time at each test temperature, distinguishing each different sample, as for example in Fig.s 1 & 5. As far as practically possible, data at each temperature should represent a similar group of samples. Data at temperatures for which there are only relatively few test results should initially be neglected, and data for samples tested at only one stress at any particular temperature should also initially be neglected.
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3. For each test temperature determine a median rupture life at a series of chosen stresses, as for example in Tables 2 & 5. Wherever possible, utilise data for any sample not tested at a chosen stress to infer its approximate rupture life at that stress. It is only necessary to infer whether a sample falls above or below an estimated median, not to infer a rupture life with any precision. Nor, as in Tables 2 & 5, is it necessary to determine median rupture lives with any great precision.
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4. Select a pair of parametric equations, see Appendix B, and transform the median data at all temperatures to a convenient datum temperature. Determine the numerical coefficients of the parametric equations by trial to produce satisfactory convergence, as in Fig.s 2, 6, 9 & 12. Convergence is satisfactory only if the median data fall within tolerance limits of a) stress x1.025 and time x1.10, and b) stress x1/1.025 and time x1/1.10 from a mean line passing through them. 5. If satisfactory convergence cannot be achieved by trial variation of the numerical coefficients then reject the initially selected parametric equations and try a different pair. Experience suggests that one or other of the pairs listed in Appendix B will prove satisfactory but, exceptionally, it may be necessary to partition the model into temperature ranges with different coefficient values, as in [3].
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6. Select a series of stress and rupture time co-ordinates to delineate a datum model of the converged parametric median data. Enter the selected values into an interpolation routine to define the intervals between points, as described in Appendix C, and compare the resultant curve to the median data, as in Fig.s 2, 6, 9 & 12. Adjust the selected datum values of stress and rupture time as necessary to improve correspondence with the data and generate a smooth curve.
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7. Transform the numerical datum model to each test temperature, via the parametric equations, and compare to the data, as in Fig.s 3, 7, 10 & 13. Make any necessary further adjustments to the selected datum values of stress and rupture time.
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8. Extrapolated rupture properties may then be determined within the validity limits of the parametric boundary, i.e. within the locus of the lowest parametric stress and longest parametric rupture time of the isothermal datum model and the equivalent points at different temperatures. 9. Plot all available data parametrically, as in Fig.s 4, 8, 11 & 14, to check that any data initially excluded are consistent with the model. A 95% confidence line may be defined relative to the median line, as stress / 1 + and median rupture time / 1 + 4 , by selecting a value of to exclude the lower 5% of test results. The above summary describes the P-NID approach starting from an initial determination of median rupture lives. The approach is equally applicable to modelling and extrapolation of data for any individual sample, as described in [6, 7].
Appendix 2: Parametric equations The P-NID procedure for constructing a model of rupture data requires the selection of a pair of parametric equations, relating stress and rupture time at absolute temperature to datum stress and datum time at datum absolute temperature , that satisfactorily represent the observed
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=
⁄
+
−
⁄
⁄
1−
empirical coefficient:
=
=
−
+
−
/
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b) Manson-Haferd [10], MH −
coeff.s:
c) Mendelson-Roberts-Manson [11] =
=
⁄
−
−
+
⁄
−
−
coeff.s:
=
+
−
−
/
−
/
coeff.s
=
1⁄ − 1⁄
+ =
−
⁄
−
−
g) Modified Minimum Commitment, MCm =
=
/
+
( /
=
)
i) Wilshire [16], WS =
⁄
+
/
⁄
+ .
coeff.s:
+
1⁄
1⁄ − 1⁄
+
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−
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,
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,
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−
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coeff.s:
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,
coeff.:
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h) Trunin-Golobova-Loginov A [15], SM1 =
,
coeff.:
f) Minimum Commitment [14], MC =
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,
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d) Manson-Brown [12] =
,
is its value at the datum temperature. ,
coeff.s:
k) Bolton B [2], BTB =
+
1⁄ − 1⁄
=
+
1⁄ − 1⁄
coeff.s:
,
Appendix C: Spline function for numerical isothermal datum rupture curves A numerical isothermal datum curve, e.g. as shown in Fig.2 for the OSD-NID model, is a piecewisecubic spline passing through a series of stress, rupture time co-ordinates , selected to delineate the parametric rupture curve at the datum temperature. First a logarithmic gradient ! is calculated at the intermediate point of each three successive points, designated ", #, $, as below. ! %
&' ( ) &' ( )
= * +
&' (, -&' ( ) &' ( -&' ( ,
*
-.
&' ( -&' ( ) &' ( -&' ( )
+
&' ( -&' ( )
&' ( -&' ( ,
*
-.
&' (, -&' ( ) &' ( -&' ( , )
/
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2
42' = > 32' =
2 2
*2' =
+
−
' '
= *2' + 32'
9
0
− 32'
: − 29
1
− 362' @9
− 242' 0
−
0
+ 42'
0
0
5
: −9
− 362' 9
0
0
0
− 42' 9
1
5
:
3HG =
− E
G
*HG =
− 24HG E
− F
G
I
E
− 24OP M
N
−
O9
− 3OP
M
− 4OP 9
F
,
− E
E
JA=I F
J
M
M
−
1
,
1
,
2,
'
and the cubic
−
1
1
:<
:
7
7
:
F
,
G
and the quadratic equation for
5
J F
5
5
M
,
= *OP + 3OP
M
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O
−
,
− E
− 4HG I
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3OP =
M
E
+ 4HG
For a final interval KL the known quantities are terms of is,
Where, 4OP = −8
,
= *HG + 3HG
F
− 3HG
E
0
−
0
, 7
0
: AB=829
5
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5
0
+ 62'
:<=9
1
,
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−
: − 62' 9
For an initial interval CD the known quantities are
in terms of
5
0
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the known quantities are
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Thus for each intermediate interval equation for in terms of is,
N
:<=9
M
,
N
,
N
,
O
+ 4OP M
−
and the quadratic equation for
in
5 N
5
:
5
:
This spline fitting procedure was automated in an EXCEL spreadsheet. The procedure performs the same function as standard spreadsheet graphics but has the advantage that it permits explicit interpolation within any interval, whereas the equations behind standard EXCEL graphics are not known to the user. Experience suggests that it generates a smoother and more easily controlled curve than is obtained from standard EXCEL graphics. The datum curve is transformed to any other temperature, e.g. as in Fig. 3, by inversion of the parametric relationships of the model, e.g. eqn.s 1a & 1b.
Appendix D: Numerical values defining P-NID models for each of the four ECCC datasets Numerical values in the table below define P-NID models for each of the four ECCC datasets over the parametric range of the available rupture data. Each model is defined at a convenient datum temperature by 11 values of median rupture time, corresponding to 11 convenient stresses. The
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2.25Cr1Mo OSD-NID (Fig.2) o = 550 C (823 K)
11%CrMoVNb BTA-NID (Fig.6) o = 550 C (823 K)
Type 304H Stainless MC-NID (Fig.9) o = 650 C (923 K)
Incoloy 800 MH-NID (Fig.12) o = 700 C (973 K)
Parametric coeff’t.s: P = 220 K = 32 K
Parametric coeff’ts: = 150 K = 36,000 K
Parametric coeff’ts: = 280 K 16 = 2.00.10 h
Stress MPa 315 227 170 136 120 103 82 65 51 38 24.5
Stress MPa 410 370 300 275 245 205 170 150 135 110 83
Stress MPa 260 220 195 169 141 120 99 80 65.5 53.5 39
Stress MPa 360 250 180 66 50 39 29 19 14 10 6.5
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Time h 2 10 30 100 300 1000 4000 15,000 50,000 150,000 600,000
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Time h 90 600 5500 10,000 25,000 45,000 66,000 79,000 93,000 140,000 270,000
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Parametric coeff’t: = 42500 K
Time h 0.005 1 20 10,000 40,000 150,000 600,000 2,200,000 7,000,000 7 2.0.10 7 5.0.10
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Data 550C
MC-NID 550C
ECCC SM1 550C
Data 600C
MC-NID 600C
ECCC SM1 600C
Data 650C
MC-NID 650C
ECCC SM1 650C
Data 700C
MC-NID 700C
ECCC SM1 700C
Data 732C
MC-NID 732C
ECCC SM1 732C
MC-NID 800C
ECCC SM1 800C
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Stress, MPa
1000
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Fig.10: Median rupture data for 304 SS and models
10 1.E+01
1.E+02
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1.E+03
Time, h
1.E+04
1.E+05
1.E+06
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Fig.16: Model of 2.25Cr1Mo data at <20,000 h compared to omitted longer-term data
475C Restricted OSD-NID(R) 475C 475C Omitted 500C Restricted
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OSD-NID(R) 500C
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550C Restricted
600C Omitted 650C Restricted
10 1.E+01
1.E+02
AC C
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OSD-NID(R) 650C
1.E+03
Time, h
1.E+04
1.E+05
1.E+06
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Comprehensive description of P-NID approach to rupture extrapolation Demonstration of modelling accuracy for four large rupture datasets by P-NID method Demonstration of extrapolation accuracy from models of time-restricted datasets Comparison of P-NID models and ECCC models for the same data
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• • • •
S0308-0161(17)30015-7
DOI:
10.1016/j.ijpvp.2017.08.001
Reference:
IPVP 3641
To appear in:
International Journal of Pressure Vessels and Piping
Received Date: 6 January 2017 Revised Date:
2 July 2017
Accepted Date: 19 August 2017
Please cite this article as: Bolton J, Reliable analysis and extrapolation of creep rupture data, International Journal of Pressure Vessels and Piping (2017), doi: 10.1016/j.ijpvp.2017.08.001. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
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ACCEPTED MANUSCRIPT Reliable analysis and extrapolation of creep rupture data J. Bolton, 01-07-‘17 [email protected]
Abstract:
Keywords: Creep rupture, rupture model, extrapolation, P-NID method
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1. Introduction:
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The P-NID (Parametric, Numerical Isothermal Datum) method of extrapolating creep rupture data has been applied to the four large datasets recently analysed by the European Creep Collaborative Committee in order to re-evaluate its own recommended procedure. It is demonstrated that the P-NID method provides a very reliable basis for extrapolation.
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The progressive development of high-temperature plant since the 1950s has depended upon the development of creep-resistant materials; and the provision of design data for their long-term rupture properties has relied upon extrapolation from shorter-term rupture tests. Numerous techniques of modelling and extrapolation have been developed over that period for that purpose. The models employed, whether long-established or of recent origin, and whether or not they purport to represent creep mechanisms, are generally in the form of a mathematical equation for rupture life as an explicit function of temperature and stress. Each contains a number of numerical coefficients that are determined by fitting to the data. However, such models have achieved only intermittent success, being found to fit data satisfactorily for one material but less well for another.
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It is not unusual for researchers within different organisations to favour different types of model, or different procedures for model optimisation, with the result that wide divergences may be found between predictions of long-term life. In consequence, it is likely that some extrapolations that form the basis of current design data are less reliable than design engineers suppose. It has long been the aim of analysts concerned by such divergences to establish a greater consistency of practice, both in the comparative review of candidate models and in the application of validity tests.
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Much work has been done to improve matters by the European Creep Collaborative Committee (ECCC), which has recommended a set of validity tests applicable to any mathematical model, but this approach has met with only partial success. As a result of known problems within this procedure, the relevant ECCC Working Group (WG1) performed a recent re-evaluation of their own recommendations using large datasets for 2.25Cr1Mo steel, 11%CrMoVNb steel, Type 304 stainless steel and Incoloy 800 as test cases. Their results were published in [1]. The present article reports a similar exercise in which a fundamentally different approach, the P-NID method [2, 3] was applied to the same datasets, which were made available for that purpose by ECCC. The method is described in the context of its application to each material in the following sections, and a general synopsis of the method is given in Appendix A. Comparisons are made between P-NID models for these datasets and models proposed by ECCC [1]. Reliability in extrapolation is tested by constructing models from short-term data and comparing their predictions to long-term data omitted from the modelling process.
2. Analysis of 2.25Cr1Mo steel data: 2.1 Determination of median rupture lives The ECCC dataset for 2.25Cr1Mo includes the results of 1016 tests to rupture at temperatures o o between 450 C and 650 C, after the elimination of anomalous and duplicated results. It incorporates
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ACCEPTED MANUSCRIPT results from a number of different sources on a large number of sample heats, tested to widely varying extents. The dataset is far from ideal because rupture lives are not known for a consistent set of samples tested over the same range of conditions, and the diversity in sampling and test conditions introduces potential problems in modelling.
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The potential problems are illustrated by Fig.1 which shows rupture curves for all 33 samples o (designated D1, D2, etc) tested at 550 C and at a minimum of two stress levels. The logarithmic stress scale of the figure is expanded for clarity and shows horizontal gridlines only at an illustrative series of stresses. Table 1 shows the median lives and geometric mean lives (mean of logarithmic lives) determined from tests at those specific stresses. Data at nominally 176.5 MPa include all data at 176 and 177 MPa, and data at nominally 78.5 MPa include all data at 78 and 79 MPa. o
Table 1: Summary of rupture data at specific stresses, 2.25Cr1Mo at 550 C 206 3 141 130
176.5 22 314 321
137 16 1590 1570
108 21 10,300 9390
78.5 9 37,200 41,540
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Test stress, MPa Number of samples tested Median rupture life, h Mean rupture life, h
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The tabulated median and mean rupture lives agree to within about 10%, which is a small fraction of the observed scatter. But it can be seen in Fig.1 that these lives are not representative of a consistent group of samples at each stress; for example, (i) A relatively large number of stronger samples were not tested at 206 MPa but were tested only at higher and lower stresses; and a relatively large number of weaker samples were tested only at lower stresses. (ii) A relatively large number of stronger samples were not tested at 78-79 MPa but were tested only at higher stresses.
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Such inconsistencies of sampling are a common problem in the analysis of materials data. However, the typical trend may be more confidently established by incorporating the additional information obtainable by interpolation between tests at higher and lower stresses, or by extrapolation from tests at higher stresses only or at lower stresses only.
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Where data were available only at higher and lower stresses, the curves for individual samples were interpolated. The curves in Fig.1 are automatic EXCEL fits to the available data, and some of the curve shapes suggest that some of the data may be unreliable. But no reason was found to discount particular data points, so all points were treated as valid and all interpolations were treated as valid. It was not necessary for the interpolations to be precise, only to determine whether each curve passed above or below a trial value of median life.
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Where data were available only at higher stresses, or only at lower stresses, the curves for individual samples were projected along upper and lower lines consistent with the general trend of other samples. It was not necessary for those projections to be precise, only to determine whether they could pass only below or only above a trial value of median life. For most samples either the upper projection passed below but not credibly above a trial median, or vice versa. A minority of cases were indeterminate and those were discounted. Thus the population of data available for the determination of median values of rupture life was greatly expanded relative to Table 1, as shown in Table 2. The median lives in this table were determined simply by counting the total number of samples to either side of a trial value, and adjusting the trial value until a balance was achieved. o
Table 2: Number of samples below/above median rupture life, 2.25Cr1Mo at 550 C Stress, MPa Median life, h Stress specific (below/above)
206 125 1/2
176.5 295 10/12
137 1650 8/7
108 10,000 9/12
78.5 63,000 7/2
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ACCEPTED MANUSCRIPT Interpolated (below/above) Extrapolated (below/above) Total results (below/above)
0/6 12/5 13/13
0/2 4/1 14/15
4/6 1/0 13/13
5/2 1/1 15/15
3/0 2/11 12/13
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The median lives determined at stresses of 176.5, 137 and 108 MPa differ very little from those in Table 1, because the numbers of curves interpolated or extrapolated were relatively few and largely counterbalanced. The median life at 206 MPa is changed little because the relatively large numbers of interpolated and extrapolated curves counterbalanced each other. But the median life at 78.5 MPa is much increased owing to the large number of curves for stronger samples that could confidently be projected to rupture lives greater than 63,000 h. The consistency of those projections with the trends exhibited by other samples can be seen in Fig.1.
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The effect of including extrapolated results at 78.5 MPa, which increases median life from 37,200 h to 63,000 h, demonstrates the importance of the additional information in establishing median lives for a broadly consistent group of samples. Such information is effectively discarded in methods based upon statistical criteria, such as the ECCC approach, which treat every data point as if it were for a unique sample. o
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Median lives at 550 C were similarly determined at stresses of 156, 98, and 88 MPa; and median o rupture lives were determined in the same way at temperatures of 475, 500, 600 and 650 C. The dataset was thus condensed to median rupture lives at 33 conditions of stress and temperature. Median values were used in this study as the basis of model formulation because they are readily determined by the simple counting process described above. It would be laborious to model and extrapolate curves for individual samples to the specific lives necessary for the calculation of mean values, to no advantage. The differences between median and geometric mean values are expected to be small, as for example in Table 1, and to be insignificant in the context of establishing lowerbound values, e.g. at 95% Confidence as discussed later in Section 2.4. 2.2 Selection of parametric equations
=
(1a)
=
−
1⁄ − 1⁄
(1b)
is a numerical coefficient, with units of absolute temperature.
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Where
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Having established a set of median data, the next step in the P-NID procedure is to find a pair of parametric equations that resolve the data at all test temperatures into a single, isothermal rupture curve at a convenient datum temperature. A series of candidate pairs of equations, corresponding to a series of established rupture models is given in Appendix B. The parametric equations selected for the 2.25Cr1Mo data were those of the Orr-Sherby-Dorn (OSD) model, for which stress and rupture time at test temperature (absolute) are transformed to parametric stress and parametric rupture time at datum temperature by the relationships,
The OSD model was chosen because it resulted in satisfactory convergence of all median data into a narrow band as shown in Fig.2, the value of being selected for optimum visual convergence. The o figure is a parametric plot of the median data transformed to a datum temperature of 550 C, for = 42,500 K. The residual disparities between data obtained at different temperatures are in the region of +/- 20% of rupture time, which is no greater than disparities commonly found between duplicate tests on specimens taken from the same sample. A number of alternative pairs of parametric equations were considered but were rejected because they failed to produce convergence within a sufficiently narrow band. 2.3 Definition of datum rupture curve The final stage in constructing a P-NID model is to select a series of stress and rupture time coordinates defining the datum curve of Fig.2. The usual procedure is to select 11 such points, from which the full median rupture curve is generated by a piecewise cubic spline as described in Appendix C. The model so defined (OSD-NID) is shown in Fig.2, together with a tolerance envelope. The tolerance envelope has an upper limit at stress x1.025 and rupture time x1.10 and a lower limit at
4
ACCEPTED MANUSCRIPT stress /1.025 and rupture time /1.10, as recommended in [3]. A full numerical definition of the model curve may be found in Appendix D.
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A more conventional comparison of the OSD-NID model to the median data is presented in Fig.3, showing the model and data separately at each test temperature. The correspondence between model and data is very satisfactory across the range of test results. Hence interpolation and extrapolation from this model may be considered valid anywhere within the ranges of parametric stress, parametric time and temperature in Fig.2. But it ceases to be valid beyond the lowest-stress and longest-term data point, which is the parametric boundary of existing data. Thus there is no basis o for extrapolation at 650 C, but at lower temperatures the model curve is expected to remain reliably o accurate up to the same parametric limit, e.g. to about 130,000 h at 600 C (Fig.3) or to about o o 6 o 7 o 200,000 h at 590 C. The equivalent limits at 550 C (2.5.10 h), 500 C (7.0.10 h) and 475 C 8 (4.4.10 h) are beyond the time-scale of practical interest. 2.4 Comparison of P-NID model to distribution of data for individual samples o
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A comparison of the OSD-NID model to individual sample data at 550 C is shown in Table 3, in the same fashion as Table 2. Rupture lives defined by the model are within about 10% of the median values in Table 2 and the differences between sample numbers in the different categories in Tables 2 & 3 are small. o
Stress, MPa Model rupture life, h Stress specific (below/above) Interpolated (below/above) Extrapolated (below/above) Total (below/above)
206 141 1/1 0/6 12/5 13/12
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Table 3: Number of samples below/above OSD-NID model rupture life, 2.25Cr1Mo at 550 C 176.5 323 11/11 0/2 5/1 16/14
137 1600 8/8 4/7 1/0 13/15
108 10,550 10/10 5/2 0/1 15/13
78.5 62,000 7/2 3/0 2/11 12/13
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The combined sum of samples within any sub-population, below and above the median, is not always the same in Table 3 as in Table 2. That is because some stress-specific and interpolated results coincided with the median or with the model and fell into neither category, and some extrapolated results fell into neither category with any certainty.
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A comprehensive check on the OSD-NID model is provided by Fig.4. This is a parametric plot of every individual test result, and is useful in two respects. Firstly, data neglected in the construction of the model, at temperatures for which there were few test results or for samples for which there were too o few test results, may be compared to the model. For example, data at 450 – 454 C may be compared o to the model established from data at higher temperatures. The relatively sparse data at 450 – 454 C are visibly consistent with the model. Initially neglected data at a number of intermediate temperatures may also be seen to be consistent with the model. Secondly, it is important in the context of engineering design to appreciate the extent to which a weaker sample may fall short of the median rupture life. A median line is of little practical value without some quantitative allowance for the shortfall, and it is useful to represent the scatter in data as a margin relative to the median. A boundary representing, for example, 95% confidence may be drawn for stress and time margins that exclude the lower 5% of the total data. The boundary shown in Fig.4 was constructed at median stress / 1 + and median rupture time / 1 + 4 , for = 0.13. The form of these margins is consistent with the tolerance envelope discussed above. The boundary can be seen to exclude a fairly consistent proportion of data points across the range of parametric time. It is possible that a model based on geometric mean lives would produce a line somewhat above or below a model based on median lives. But the margin required for 95% Confidence would necessarily increase or decrease accordingly, without significant alteration to the 95% Confidence line. 2.5 Comparison of ECCC model to data for individual samples and to P-NID model
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ACCEPTED MANUSCRIPT A model judged in the ECCC re-evaluation exercise [1] to be accurate and reliable in extrapolation (ECCC MCm) is also shown in Fig.3. This model can be seen to diverge substantially from the median rupture data in many areas. o
A comparison between the ECCC model and data at 550 C is presented in Table 4 in the same fashion as in Table 3. o
Table 4: Number of samples below/above ECCC MCm model rupture life, 2.25Cr1Mo at 550 C 206 282 3/0 4/2 15/1 22/3
176.5 670 18/3 1/1 7/0 26/4
137 2580 13/3 9/2 1/0 23/5
108 8400 8/13 4/3 0/1 12/17
78.5 37,500 5/4 2/1 2/11 8/16
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Stress, MPa Model rupture life, h Stress specific (below/above) Interpolated (below/above) Extrapolated (below/above) Total (below/above)
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Large imbalances between the total number of samples below and above the model rupture lives confirm that this model is an inaccurate representation of the material data. Fig.1 confirms graphically that the model lives at 206, 176.5, 137 and 78.5 MPa are at variance with the data. The ECCC MCm model cannot therefore be a reliable basis for extrapolation.
3. Analysis of 11%CrMoVNb steel data:
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It is common, for the purposes of engineering design, for models of the kind discussed to be o extrapolated to 200,000 hours. Extrapolation of the OSD-NID model at 550 C gives a rupture stress at 200,000 h of 61 MPa; but at that stress the ECCC MCm model predicts a rupture life of 118,000 hours. The evidence presented here indicates that the OSD-NID model is the more reliable of the two, and hence implies that the ECCC MCm model underestimates rupture life at that stress by 40%.
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The ECCC dataset for 11%CrMoVNb steel includes the results of 305 tests to rupture at temperatures o o between 425 C and 600 C. Results were available for a large number of sample casts, tested to widely varying extents. The dataset contains much of the 12%CrMoVNb dataset modelled and discussed elsewhere [4, 5] but omits some of that data and includes other data, so the two datasets are significantly different and should not be confused. A P-NID model of the 11%CrMoVNb data was created following the same procedure as described above for the 2.25Cr1Mo steel. o
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Fig.5 shows the data for 33 samples tested at 550 C and illustrates the determination of median data at stresses of 392, 309, 247, 196 and 140 MPa. Table 5 presents the distribution of sample data relative to estimated median lives at the selected stresses. o
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Table 5: Number of samples below/above median rupture life, 11%CrMoVNb at 550 C Stress, MPa Median life, h Stress specific (below/above) Interpolated (below/above) Extrapolated (below/above) Total (below/above)
392 300 3/2 1/4 4/2 8/8
309 4120 6/7 7/7 1/0 14/14
247 24,000 4/4 7/7 0/0 11/11
196 50,500 1/4 8/4 0/1 9/9
140 90,000 3/0 0/0 3/7 6/7
Many of these samples were not tested at the lower stresses, with the consequence that median rupture lives at those stresses were determined by a relatively small proportion of the total number of samples. Nevertheless, as far as can be judged from Fig.5, the samples tested were representative of the general population. At each stress the balance between samples below and above the median life is satisfactory.
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Median lives at 550 C were similarly determined at stresses of 355, 170, and 154 MPa; and median o rupture lives were determined in the same way at 475, 500 and 600 C. The dataset was thus condensed to median rupture lives at 23 conditions of stress and temperature. A pair of parametric equations was found that produced satisfactory convergence of the median data o at a datum temperature, as shown in Fig.6 at 550 C. The parametric relationships chosen were those for a Bolton A (BTA) model, as below. =
−
−
⁄
=
(2a)
−
−
⁄
(2b)
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Where and are numerical coefficients, with units of absolute temperature, determined so as to obtain optimum convergence. Other symbols have the same meaning as for eqns. 1a and 1b. The parametric rupture plot of Fig.6 demonstrates that, for = 220 K and = 32 K, the median data at all temperatures coalesce satisfactorily into a single curve. Residual disparities are in the region of +/- 20% of rupture time, which is no greater than the disparities found between duplicate tests.
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As in the case of 2.25Cr1Mo, a numerical model of the parametric rupture curve of Fig.6 was created from 11 selected co-ordinates of stress and rupture time, as tabulated in Appendix D, and the piecewise cubic spline described in Appendix C. The model so defined (BTA-NID) is shown in Fig.6 together with the same tolerance envelope, between stress x1.025 and rupture time x1.10 and stress /1.025 and rupture time /1.10. A conventional comparison of the BTA-NID model to the median data is presented in Fig.7, showing the model and data separately at each test temperature. The correspondence between model and data is very satisfactory across the range of test temperatures. At each temperature, model extrapolation beyond the available data, but within the parametric boundary, is justified by the accuracy with which the chosen parametric relationships (eqn.s 2a & 2b and the values of and ) represent the data. o
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A comparison of the BTA-NID model to the distribution of individual sample data at 550 C is shown in Table 6. Rupture lives defined by the model are mostly within 10% of the median values in Table 5 and the differences between sample categorisation in Tables 5 & 6 are small. o
Table 6: Number of samples below/above BTA-NID model life, 11%CrMoVNb at 550 C 392 243 3/2 1/4 3/2 7/8
309 4370 6/7 8/6 1/0 15/13
247 24,100 4/3 7/7 0/0 11/10
196 50,100 0/4 8/4 0/1 8/9
140 87,700 3/0 0/0 3/7 6/7
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Stress, MPa Model rupture life, h Stress specific (below/above) Interpolated (below/above) Extrapolated (below/above) Total (below/above)
Fig.8 is a parametric plot of all the data, in which the 95% confidence boundary is drawn in the same way as in Fig.4, but for = 0.11. Fig.7 also shows a model (ECCC OSD3) judged in the ECCC re-evaluation exercise to be accurate and reliable in extrapolation. The OSD3 model curves can be seen to depart substantially from the median rupture data at the higher temperatures. A comparison between the OSD3 model and sample o data at 550 C is presented in Table 7. o
Table 7: Number of samples below/above ECCC OSD3 model life, 11%CrMoVNb data at 550 C Stress, MPa Model rupture life, h Stress specific (below/above) Interpolated (below/above) Extrapolated (below/above) Total (below/above)
392 650 5/0 2/3 5/0 12/3
309 4350 6/7 8/6 1/0 15/13
247 16,500 2/6 6/8 0/0 8/14
196 45,300 0/5 6/6 0/1 6/12
140 127,000 3/0 0/0 7/1 10/1
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ACCEPTED MANUSCRIPT Large imbalances between the total number of samples below and above the model rupture lives confirm that this model is an inaccurate representation of the material data. It is evident from visual inspection of Fig.5 that the median life at 140 MPa must lie somewhere in the region of 90,000 h, and that a model life at that stress of 127,000 h is unrepresentative of the available data. It is likewise evident that a model life of 650 h at 392 MPa is unrepresentative of the data. o
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In terms of the stress to cause rupture in 200,000 hours, extrapolation of the OSD3 model at 550 C gives a stress of 116 MPa, but at that stress the BTA model gives a rupture life of 125,000 hours. The evidence presented here indicates that the BTA model is the more reliable of the two, and hence indicates that the OSD3 model overestimates rupture life at that stress by 60%.
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4. Analysis of Type 304H stainless steel data:
The ECCC dataset for this steel includes the results of 782 tests to rupture at temperatures between o o 482 C and 899 C. Specimens tested were from a large number of samples.
o
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Analysis of the data followed the same procedure as for the two steels discussed above and is summarised below in terms of a corresponding series of plots, Fig.s 9, 10 & 11, and a brief accompanying text. Fig.9 is a parametric plot at 650 C of the median data at the major test temperatures. Satisfactory convergence was obtained with the parametric relationships of a Minimum Commitment (MC) model, as below. =
=
(3a)
−
−
⁄
−
1⁄ − 1⁄
(3b)
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Where (150 K) and (36,000 K) are numerical coefficients and other symbols have the same meaning as for eqns. 1a and 1b. o
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The tolerance envelope is dimensioned as before. Median data at 732 C fall somewhat outside this tolerance, but that is attributable to the samples tested at that temperature being different from those o tested at other temperatures. The gradient of median data at 732 C is nonetheless consistent with gradients at adjacent temperatures and provides confirmation of the continuity of the parametric rupture curve.
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Fig.10 is a comparison of the MC-NID model to median data at each temperature and also shows a model (ECCC SM1) endorsed by ECCC [1] as accurate and reliable in extrapolation. The SM1 model deviates significantly from the median data at high stresses and at low stresses. In some areas, extrapolation of the SM1 model predicts only half the rupture life predicted by the MC-NID model. Fig.11 is a parametric plot of data scatter relative to the median model, showing that relatively sparse o o data for test temperatures below 550 C and above 800 C are entirely consistent with the MC-NID model defined within that range. The 95% confidence boundary is drawn for = 0.10.
5. Analysis of Incoloy 800 data: The ECCC dataset for Incoloy 800 includes the results of 495 tests to rupture at temperatures o o between 500 C and 1050 C. Specimens from a large number of samples were tested. Analysis of the data followed the same procedure as for the steels discussed above and is summarised below in terms of a series of plots, Fig.s 12, 13 & 14, and a brief accompanying text.
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Fig.12 is a parametric plot at 700 C of the median data at the major test temperatures. Satisfactory convergence was obtained with the parametric relationships of a Manson-Haferd (MH) model, as below. =
=
(4a)
−
⁄
−
−
−
/
−
(4b)
16
Where (280 K) and (2.10 h) are numerical coefficients and other symbols have the same meaning as for eqns. 1a and 1b.
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The tolerance envelope is dimensioned as before and bounds all the median data. Fig.13 is a comparison of the MH-NID model to median data at each major test temperature and also shows a model (ECCC MH3, [1]) presented as accurate and reliable in extrapolation. The differences between the two models are fairly small within the range of the existing median data, but the MH-NID model conforms more closely to that data and thus appears the more reliable basis for extrapolation. In some areas, extrapolation of the ECCC MH3 model predicts only half the rupture life predicted by the MH-NID model. o
6. Extrapolation from restricted data:
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Fig.14 shows that data neglected in the construction of the MH-NID model, at 1050 C and at some intermediate temperatures, are consistent with the model. The 95% confidence boundary is drawn for = 0.12.
The ability of a procedure to produce a model that is reliable in extrapolation can be tested by modelling a restricted dataset that omits all longer-term data, and then comparing the resultant model to the omitted, longer-term data.
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Figure 15 is a parametric plot of median rupture data for 2.25Cr1Mo restricted to 20,000 hours. It is identical to Fig.2 except for the exclusion of longer-term data. Sufficient data remain to show convincing convergence and overlap of data obtained at all temperatures. The OSD-NID(R) model shown in Fig.15 is based on this restricted dataset, but is nevertheless identical to the OSD-NID model of Fig.3 based on the full dataset - because the fit to the restricted dataset could not be visibly improved by selecting an alternative pair of parametric equations, by adjusting the coefficient value of the parametric equations, or by re-selecting the points that define the datum curve. The parametric o boundary of the model also remains unchanged as it is fixed by data at 650 C at less than 20,000 hours.
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A marginally different coefficient value might have been chosen for model OSD-NID(R), and different points might have been selected to define the datum curve – but not such that any resultant model could have been noticeably different from the OSD-NID model of the full dataset. Hence, as illustrated in Fig.16, the omitted longer-term data provide direct evidence of the model accuracy over a two-fold o o extrapolation at 600 C, a three-fold extrapolation at 550 C and nearly four-fold extrapolations at 500 o o C and 475 C. Fig.s 17 & 18 provide a similar demonstration for 11%CrMoVNb, with data restricted to 24,000 h. A new model of the restricted data, BTA-NID(R), removed the slight inflection of the original BTA-NID o model at about 75,000 h (at 550 C) but was otherwise barely distinguishable from the original. The o only significant change was that the parametric limit was reduced to about 115,000 h (at 550 C) in o consequence of the exclusion of the longest-term data at 600 C. The excluded longer-term data at o temperatures below 600 C provide direct evidence of model accuracy over a nearly four-fold o o extrapolation at 550 C and a nearly five-fold extrapolation at 500 C. A similar exercise for Type 304 SS showed that a model of data restricted to 12,000 h was accurate o over a nearly five-fold extrapolation at 600 C.
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ACCEPTED MANUSCRIPT No similar exercise could be performed for Incoloy 800 because any significant restriction of the data led to insufficient overlap of the parametric data at different temperatures, and thus there remained insufficient evidence of model continuity.
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Table 8 summarises extrapolations from P-NID models of restricted data, comparing them to the longest-term test data available. The first six rows show comparisons for median data from the ECCC datasets, as discussed above. Also shown, in italics, are the lives according to ECC models based on unrestricted data. The following eight rows show comparisons for median data for other datasets. The final four rows are for individual samples of 2.25Cr1Mo that have been tested over exceptionally long times. Results for datasets other than the ECCC datasets may be found in [2, 3, 5, 6 and 7]. Some are unpublished except in Researchgate website postings or are unreported in any form. Table 8: Comparison of P-NID models of restricted data to omitted, longer-term data
11%CrMoVNb Median data
304H SS Median data Ni15%CrTiAl Median data
Data restriction <20,000 h None <20,000 h None <20,000 h None
550 C, 140 MPa o 500 C, 310 MPa o 600 C 98 MPa
o
BTA OSD3 BTA OSD3 MC SM1
<24,000 h None <24,000 h None <12,000 h None
x 3.8 None x 4.8 None x 4.6 None
87,700 h 127,000 h 115,000 h 81,200 h 55,700 h 40,500 h
o
MH
<8000 h
x 2.4
18,800 h
16,800 h
1.12
MH
<8000 h
x 2.1
17,000 h
15,000 h
1.13
800 C 50 MPa o 700 C 200 MPa o
12%CrMoVNb Median data
500 C 309 MPa
9%CrMo Median data
600 C 80 MPa
1%CrMoV Median data 2.25Cr plate Median data
500 C 118 MPa
Model life 62,000 h 37,500 h 76,500 h 64,300 h 75,000 h 57,300 h
Test life 63,000 h
85,000 h 95,000 h 90,000 h
130,000 h 60,000 h
Model life/ Test life 0.98 0.60 0.90 0.76 0.79 0.60 0.97 1.41 0.88 0.62 0.93 0.68
BTA
<12,000 h
x 4.2
50,000 h
60,000 h
0.83
BTA
<15,000 h
x 5.7
85,000 h
80,000 h
1.06
550 C 137 MPa
BTB
<10,000 h
x 9.7
97,000 h
92,000 h
1.05
o
MH
<15,000 h
x 13
200,000 h
200,000 h
1.00
650 C 78 MPa o 600 C 118 MPa
BTB
<5000 h
x 15
73,000 h
66,000 h
1.11
BTB
<5000 h
x 23
115,000 h
105,000 h
1.10
o
OSD
<11,000 h
x 8.8
97,000 h
100,000 h
0.97
o
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o
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18%Cr Median data
Extrp’n multiple x 3.1 None x 3.8 None x 3.8 None
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Model type OSD MCm OSD MCm OSD MCm
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2.25Cr1Mo Median data
Test Condition o 550 C, 78.5 MPa o 500 C, 127 MPa o 475 C 170 MPa
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Material
o
2.25Cr1Mo Sample D2
550 C 62 MPa
2.25Cr1Mo Sample D1
500 C 123 MPa
o
OSD
<7000 h
x 19
130,000 h
104,000 h
1.25
2.25Cr1Mo Sample J12
550 C 78 MPa o 500 C 118 MPa
o
OSD
<4200 h
x 25
103,000 h
131,000 h
0.79
OSD
<4200 h
x 29
120,000 h
142,000 h
0.85
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ACCEPTED MANUSCRIPT The tabulated results show satisfactory agreement between longer-term tests and P-NID models of restricted data over extrapolations of up to almost thirty-fold. The extrapolations shown in the first six rows are substantially closer to the longer-term test data than ECCC models based on unrestricted data. o
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A parametric plot for the greatest of these extrapolations, for 2.25Cr1Mo sample J12 at 500 C, is shown in Fig.19. The longest-term data for this sample were taken from Ref.8, which records rupture lives for a number of low-stress tests recorded as unbroken in the ECCC dataset. These additional results were not included in the determination of 2.25Cr1Mo median data.
7. Discussion: 7.1 Accuracy of P-NID models
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The principle of extrapolation from a P-NID model is that, if it accurately represents the systematic dependence of rupture time on temperature and stress under all conditions of testing, it will do so under any other condition within the parametric limits of the test data. Hence the reliability of such a model in extrapolation depends fundamentally upon the accuracy with which the model succeeds in representing the available data. The scope for error at each stage of model construction is discussed below.
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The determination of median rupture time at each temperature and stress is a simple process of counting test results falling above and below an estimated median that, in itself, leaves no scope for error. There is scope for some subjectivity in the projection of individual sample data to a stress at which that sample has not been tested: but it is not necessary to estimate what the projected life might be, only to decide whether that life can confidently be expected to fall below or above an estimated median. The scope for error is therefore small, as may be verified by inspection of Fig.s 1 & 5 and Tables 2 & 5. There is also some scope for inconsistencies between median values where there are disparities between the numbers or identities of samples tested under different conditions of temperature and stress, which is a problem common to any modelling procedure. This is compensated to some extent in the P-NID procedure by parametric comparison of median lives determined at different temperatures, for different groups of samples, as in Fig.s 2, 6, 9 & 12. As noted with reference to data for Type 304 SS in Fig.9, the gradient displayed by a disparate group of samples may provide a useful guide at parametric times where there is a lack of data for consistent samples.
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The selection of a pair of parametric equations introduces some possibility of error if the analyst neglects to apply strict limits to the divergence between parametric data obtained at different temperatures. But the recommended tolerance envelope, the origin of which may be found in [3], allows little scope for any error of that kind. It is possible that more than one pair of parametric equations might achieve satisfactory parametric convergence, leaving some doubt over which is the more reliable basis for extrapolation. But, as discussed in [3], the divergence in extrapolation between such models is likely to be small. Once a pair of parametric equations has been selected and the optimum values of their coefficients have been determined, there is no further scope for error. A numerical datum relationship between stress and rupture time allows precise fitting to the data, regardless of how complex the shape of the observed relationship may be. In summary, there is some scope for subjectivity in the creation of a PNID model, but the constraints within the procedure ensure that any divergence between models produced by independent analysis must necessarily be small. As discussed in the preceding section, models based on restricted median data have shown satisfactory accuracy over up to 23-fold extrapolations in rupture time, and data for individual samples have shown satisfactory accuracy over up to 29-fold extrapolation. It has been proposed here that 95% confidence lines may be constructed relative to median rupture curves at median stress /(1 + ) and median rupture life /(1 + 4 ). Appropriate values of for the
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It is a key feature of the P-NID approach that the temperature terms in the parametric equations are validated by the convergence of data when plotted parametrically, as in Fig.s 2, 6, 9 and 12. By contrast, it is common in procedures for generating a single master equation that no specific test of validity is applied to those terms. The consequences are illustrated in Fig.20, which is a parametric plot for the ECCC MCm model of 2.25Cr1Mo material. The data are plotted in accordance with the 15 MCm parametric equations (Appendix B) and the coefficient value ( = 2.76.10 h) chosen by the investigators. Most of the data fall outside the tolerance envelope and the interweaving of data trends at different temperatures reveals that the temperature terms of this model are unsatisfactory. They define a systematic dependence on temperature that does not accord with the material behaviour. The same observations apply to the ECCC OSD3 model of 11%CrMoVNb material ( = 37,465 K), as shown in Fig.21.
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Similar observations apply, to a lesser degree, to the ECCC SM1 model of 304H SS and, to a lesser degree again, to the ECCC MH3 model of Incoloy 800. The corresponding parametric plots are not included here, but systematic discrepancies between model and data can be seen in Fig.s 10 and 13 respectively. A similar case is discussed in [7], which examined the selection by investigators at Swansea University of a Wilshire model [8] to represent data for a single heat of 2.25Cr1Mo steel. The parametric plot of Fig.22 reveals that the temperature terms of the Wilshire model, partitioned by the investigators into three regions and here designated WS-WSp, fail to represent the material behaviour with sufficient accuracy. This case illustrates the inconsistencies that may be overlooked if the investigators have a prior preference for a particular master equation.
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The second key feature of the P-NID approach is that stress dependence is represented in a piecewise manner, permitting precise replication of the observed data trends regardless of how complex their shapes may be. The advantages are evident in Fig.s 3 and 7, where the data delineate curves that defy accurate replication within the constraints of a conventional, single master equation. The discrepancies between the ECCC model curves and the data illustrate the point. The P-NID approach generates the equivalent of a master equation, but with the flexibility permitted by the equivalent of 12 or 13 coefficients as in Appendix D. In principle a rupture curve may be created that extends back to UTS values at very short times, but that has not formed part of this study.
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The third key feature of the approach is that it inherently defines the limit of valid extrapolation. The parametric boundary, which is the locus of the parametrically longest rupture time (and lowest stress) and equivalent points at other temperatures, is a strict and logical limit to valid extrapolation. All P-NID model curves plotted in this paper terminate at this boundary, unless curtailed by the limited extent of the time axis. Conventional methods, of the master equation type, have no in-built means of defining a validity limit and rely largely upon common practice.
8. Conclusion: A study of four large datasets, for 2.25Cr1Mo, 11%CrMoVNb, 304H SS and Incoloy 800, has shown that the P-NID method of rupture data modelling and extrapolation is both accurate and reliable. The study identified no detectable susceptibility of this method to significant error. The results of this and similar studies have demonstrated the reliability of this method over extrapolations of up to nearly thirty-fold from restricted data. Comparisons made to a widely-used procedure for model selection and extrapolation, the ECCC recommended method, indicated that the P-NID method is significantly more reliable.
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References:
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[1] M. Spindler, P. Lombardi, G. Merckling, A. Riva, M. Ortolani, M. Norman, M. Talik, S. Holmstrom and M. Schwienheer, A re-evaluation of the ECCC guidance for the assessment of full size creep rupture datasets, 3rd International ECCC Conference on Creep & Fracture in High Temperature Components Design & Life Assessment, Rome 5-7 May 2014. 798-808; INAIL, ISBN 978-88-7484510-1, 2017 (Incorporated as Appendix 1 in ECCC Recommendations – Vol 5 Part 1a, Generic recommendations and guidance for the assessment of full-size creep rupture data sets, Issue 6, 0705-2014, on ECCC website). [2] J. Bolton, A visual perspective of creep rupture extrapolation, Materials at High Temperatures, 30(2), 87-98, 2012.
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[3] J. Bolton, The potential for major extrapolation of creep rupture and creep strain data, Materials at High Temperatures, 31(2), 109-120, 2014.
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[4] British Standards Institute, PD6605-1:1998, Guidance on methodology for assessment of stressrupture data – part 1: Procedure for derivation of strength values, 1998 (withdrawn in 2011). [5] J. Bolton, The extrapolation of creep rupture data by PD6605 – An independent case study, International Journal of Pressure Vessels & Piping, 88, 158-165, 2011. [6] Unpublished results (Researchgate website posting: J. Bolton, Accuracy of major extrapolations in creep rupture life for an Austenitic steel, Aug 2015) [7] Unpublished results (Researchgate website posting: J. Bolton, Critical examination of a Wilshire model for creep rupture, Jul 2015)
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[8] Z. Abdallah, V. Gray, M. Whittaker and K. Perkins, A critical analysis of the conventionally employed creep lifing methods, Materials, 5, 3371-3398, 2014. [9] F.R. Larson and J. Miller, A time-temperature relationship for rupture and creep stresses, Trans. ASME, 74, 765-775, 1952.
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[10] S.S. Manson and A.M. Haferd, A linear time-temperature relation for extrapolation of creep and stress rupture data, NACA TN 2890, 1953. [11] A. Mendelson, E. Roberts and S.S. Manson, Optimisation of time-temperature parameters for creep and stress rupture, with application to data from German co-operative long-term creep program, NASA TN D-2975, 1965.
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[12] S.S. Manson and W.F. Brown, Time-temperature stress relations for rupture and creep stresses, Proc ASTM, 53, 683-719, 1953. [13] L.R. Orr, O.D. Sherby and J.E. Dorn, Correlation of rupture data for metals at elevated temperatures, Trans ASME, 46, 113, 1954. [14] S.S. Manson and U. Muraldihan, Analysis of creep rupture data for five multi-heat alloys by the Minimum Commitment method, Research project 638-1, EPRI CS-3171, 1983. th
[15] I.I. Trunin, N.G. Golobova and E.A. Loginov, Proc. 4 Int. Symp. On heat-resistant metallic materials, Mala Fatra, CSSR, 168, 1971. [16] B. Wilshire and M.B. Bache, Proc. ECCC Creep Conf., Creep and fracture in high temperature components, ISBN 978-1-60595-005-1, 2009.
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ACCEPTED MANUSCRIPT Appendix A: Summary of the P-NID method of extrapolation The P-NID method for modelling and extrapolating median rupture data proceeds by the following steps: 1. Scrutinise the available test data and delete any result that is clearly anomalous in relation to other data. Replace duplicate test results by a logarithmic mean rupture time.
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2. Plot log stress against log rupture time at each test temperature, distinguishing each different sample, as for example in Fig.s 1 & 5. As far as practically possible, data at each temperature should represent a similar group of samples. Data at temperatures for which there are only relatively few test results should initially be neglected, and data for samples tested at only one stress at any particular temperature should also initially be neglected.
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3. For each test temperature determine a median rupture life at a series of chosen stresses, as for example in Tables 2 & 5. Wherever possible, utilise data for any sample not tested at a chosen stress to infer its approximate rupture life at that stress. It is only necessary to infer whether a sample falls above or below an estimated median, not to infer a rupture life with any precision. Nor, as in Tables 2 & 5, is it necessary to determine median rupture lives with any great precision.
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4. Select a pair of parametric equations, see Appendix B, and transform the median data at all temperatures to a convenient datum temperature. Determine the numerical coefficients of the parametric equations by trial to produce satisfactory convergence, as in Fig.s 2, 6, 9 & 12. Convergence is satisfactory only if the median data fall within tolerance limits of a) stress x1.025 and time x1.10, and b) stress x1/1.025 and time x1/1.10 from a mean line passing through them. 5. If satisfactory convergence cannot be achieved by trial variation of the numerical coefficients then reject the initially selected parametric equations and try a different pair. Experience suggests that one or other of the pairs listed in Appendix B will prove satisfactory but, exceptionally, it may be necessary to partition the model into temperature ranges with different coefficient values, as in [3].
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6. Select a series of stress and rupture time co-ordinates to delineate a datum model of the converged parametric median data. Enter the selected values into an interpolation routine to define the intervals between points, as described in Appendix C, and compare the resultant curve to the median data, as in Fig.s 2, 6, 9 & 12. Adjust the selected datum values of stress and rupture time as necessary to improve correspondence with the data and generate a smooth curve.
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7. Transform the numerical datum model to each test temperature, via the parametric equations, and compare to the data, as in Fig.s 3, 7, 10 & 13. Make any necessary further adjustments to the selected datum values of stress and rupture time.
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8. Extrapolated rupture properties may then be determined within the validity limits of the parametric boundary, i.e. within the locus of the lowest parametric stress and longest parametric rupture time of the isothermal datum model and the equivalent points at different temperatures. 9. Plot all available data parametrically, as in Fig.s 4, 8, 11 & 14, to check that any data initially excluded are consistent with the model. A 95% confidence line may be defined relative to the median line, as stress / 1 + and median rupture time / 1 + 4 , by selecting a value of to exclude the lower 5% of test results. The above summary describes the P-NID approach starting from an initial determination of median rupture lives. The approach is equally applicable to modelling and extrapolation of data for any individual sample, as described in [6, 7].
Appendix 2: Parametric equations The P-NID procedure for constructing a model of rupture data requires the selection of a pair of parametric equations, relating stress and rupture time at absolute temperature to datum stress and datum time at datum absolute temperature , that satisfactorily represent the observed
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=
⁄
+
−
⁄
⁄
1−
empirical coefficient:
=
=
−
+
−
/
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b) Manson-Haferd [10], MH −
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c) Mendelson-Roberts-Manson [11] =
=
⁄
−
−
+
⁄
−
−
coeff.s:
=
+
−
−
/
−
/
coeff.s
=
1⁄ − 1⁄
+ =
−
⁄
−
−
g) Modified Minimum Commitment, MCm =
=
/
+
( /
=
)
i) Wilshire [16], WS =
⁄
+
/
⁄
+ .
coeff.s:
+
1⁄
1⁄ − 1⁄
+
=
−
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,
⁄
−
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coeff.s:
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,
coeff.:
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h) Trunin-Golobova-Loginov A [15], SM1 =
,
coeff.:
f) Minimum Commitment [14], MC =
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,
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d) Manson-Brown [12] =
,
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coeff.s:
k) Bolton B [2], BTB =
+
1⁄ − 1⁄
=
+
1⁄ − 1⁄
coeff.s:
,
Appendix C: Spline function for numerical isothermal datum rupture curves A numerical isothermal datum curve, e.g. as shown in Fig.2 for the OSD-NID model, is a piecewisecubic spline passing through a series of stress, rupture time co-ordinates , selected to delineate the parametric rupture curve at the datum temperature. First a logarithmic gradient ! is calculated at the intermediate point of each three successive points, designated ", #, $, as below. ! %
&' ( ) &' ( )
= * +
&' (, -&' ( ) &' ( -&' ( ,
*
-.
&' ( -&' ( ) &' ( -&' ( )
+
&' ( -&' ( )
&' ( -&' ( ,
*
-.
&' (, -&' ( ) &' ( -&' ( , )
/
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2
42' = > 32' =
2 2
*2' =
+
−
' '
= *2' + 32'
9
0
− 32'
: − 29
1
− 362' @9
− 242' 0
−
0
+ 42'
0
0
5
: −9
− 362' 9
0
0
0
− 42' 9
1
5
:
3HG =
− E
G
*HG =
− 24HG E
− F
G
I
E
− 24OP M
N
−
O9
− 3OP
M
− 4OP 9
F
,
− E
E
JA=I F
J
M
M
−
1
,
1
,
2,
'
and the cubic
−
1
1
:<
:
7
7
:
F
,
G
and the quadratic equation for
5
J F
5
5
M
,
= *OP + 3OP
M
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*OP =
O
−
,
− E
− 4HG I
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3OP =
M
E
+ 4HG
For a final interval KL the known quantities are terms of is,
Where, 4OP = −8
,
= *HG + 3HG
F
− 3HG
E
0
−
0
, 7
0
: AB=829
5
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5
0
+ 62'
:<=9
1
,
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−
: − 62' 9
For an initial interval CD the known quantities are
in terms of
5
0
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the known quantities are
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Thus for each intermediate interval equation for in terms of is,
N
:<=9
M
,
N
,
N
,
O
+ 4OP M
−
and the quadratic equation for
in
5 N
5
:
5
:
This spline fitting procedure was automated in an EXCEL spreadsheet. The procedure performs the same function as standard spreadsheet graphics but has the advantage that it permits explicit interpolation within any interval, whereas the equations behind standard EXCEL graphics are not known to the user. Experience suggests that it generates a smoother and more easily controlled curve than is obtained from standard EXCEL graphics. The datum curve is transformed to any other temperature, e.g. as in Fig. 3, by inversion of the parametric relationships of the model, e.g. eqn.s 1a & 1b.
Appendix D: Numerical values defining P-NID models for each of the four ECCC datasets Numerical values in the table below define P-NID models for each of the four ECCC datasets over the parametric range of the available rupture data. Each model is defined at a convenient datum temperature by 11 values of median rupture time, corresponding to 11 convenient stresses. The
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2.25Cr1Mo OSD-NID (Fig.2) o = 550 C (823 K)
11%CrMoVNb BTA-NID (Fig.6) o = 550 C (823 K)
Type 304H Stainless MC-NID (Fig.9) o = 650 C (923 K)
Incoloy 800 MH-NID (Fig.12) o = 700 C (973 K)
Parametric coeff’t.s: P = 220 K = 32 K
Parametric coeff’ts: = 150 K = 36,000 K
Parametric coeff’ts: = 280 K 16 = 2.00.10 h
Stress MPa 315 227 170 136 120 103 82 65 51 38 24.5
Stress MPa 410 370 300 275 245 205 170 150 135 110 83
Stress MPa 260 220 195 169 141 120 99 80 65.5 53.5 39
Stress MPa 360 250 180 66 50 39 29 19 14 10 6.5
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Time h 2 10 30 100 300 1000 4000 15,000 50,000 150,000 600,000
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Time h 90 600 5500 10,000 25,000 45,000 66,000 79,000 93,000 140,000 270,000
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Parametric coeff’t: = 42500 K
Time h 0.005 1 20 10,000 40,000 150,000 600,000 2,200,000 7,000,000 7 2.0.10 7 5.0.10
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Data 550C
MC-NID 550C
ECCC SM1 550C
Data 600C
MC-NID 600C
ECCC SM1 600C
Data 650C
MC-NID 650C
ECCC SM1 650C
Data 700C
MC-NID 700C
ECCC SM1 700C
Data 732C
MC-NID 732C
ECCC SM1 732C
MC-NID 800C
ECCC SM1 800C
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Stress, MPa
1000
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Fig.10: Median rupture data for 304 SS and models
10 1.E+01
1.E+02
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Time, h
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1.E+05
1.E+06
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Fig.16: Model of 2.25Cr1Mo data at <20,000 h compared to omitted longer-term data
475C Restricted OSD-NID(R) 475C 475C Omitted 500C Restricted
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OSD-NID(R) 500C
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550C Restricted
600C Omitted 650C Restricted
10 1.E+01
1.E+02
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OSD-NID(R) 650C
1.E+03
Time, h
1.E+04
1.E+05
1.E+06
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Comprehensive description of P-NID approach to rupture extrapolation Demonstration of modelling accuracy for four large rupture datasets by P-NID method Demonstration of extrapolation accuracy from models of time-restricted datasets Comparison of P-NID models and ECCC models for the same data
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• • • •