1 Introduction

New grades of creep-resistant steels capable of operating at higher temperatures are required to further increase thermal efficiencies of steam turbine power plants. Development and application of such grades of steels inevitably require the accurate assessment of their maximum tensile stress under which creep rupture would not occur or creep deformation would not exceed 1 pct in at least 100,000 hours at a particular application temperature. It has been reported that, without results from prolonged tests, it would be difficult to accurately predict such long-term creep strengths with the conventional time–temperature parameter methods (for instance, the Larson–Miller parameter method).[1–4] As such, it has been the subject of an extensive research effort, aimed at defining a method of reliable data extrapolation that would allow long-term creep strengths to be predicted from short-term test data and hence substantially reduce the development cycle of creep-resistant steels for power plant applications.[1–8] The difficulties are caused by the changes of slopes in the plots of stress vs rupture time or in the plots of minimum creep rate vs stress.[9,10] And these problems are made more difficult by the fact that the stresses at which such slope changes occur, i.e., the transition stress, are not predictable at different temperatures, often leading to serious overestimations of long-term creep strengths.[9,10] However, it has been reported that, in the case of P92 and T92 steel grades, the transition stresses all occur at a common stress ratio at different temperatures,[1] i.e., at a testing stress to 0.2 pct offset yield stress ratio of approximately 0.5. But this phenomenon has not yet been understood and hence has not been interpreted so far.[1] Nevertheless, it is suggested that the long-term creep strengths of creep-resistant steels should only be estimated from creep test data measured at stresses lower than 50 pct of the 0.2 pct offset yield stress, i.e., at stresses below the proportional limit. The implication is that only prolonged tests lasting well over 10000 hours are useful for accurately estimating the 100,000-hour creep strengths. However, in the present study, it will be shown that this problem can be resolved and the phenomenon observed previously[1] can be explained if the new power law creep equation to be introduced in the following section is used to rationalize the creep test data. But, before presenting the full analysis, it is useful to briefly examine the basis for predicting the long-term creep strength and the difficulties encountered in using the conventional power law creep equation to rationalize creep test data. By doing so, the basis of the new power law creep equation to be introduced in the present study is explained.

2 Basis of Long-Term Creep Strength Prediction

In metal or metal alloy creep, it has been established that Monkman–Grant relationship can be used to extrapolate the long-term minimum creep rate data from short-term creep tests:[11]

$$ \acute{\varepsilon}_{ \hbox{min} } = Mt_{\text{r}}^{ - m}, $$
(1)

where έ min is the minimum creep rate under a tensile stress σ, t r the creep fracture time, and M and m are constants. Equation [1] is applicable in a wide range of stresses and temperatures.[1,12,13] Experimentally, the parameters of M and m can be determined from the plot of ln έ min vs ln t r measured from accelerated tests at high stresses and temperatures (short-term tests). They can then be used to make extrapolations to low stresses and temperatures (long-term tests). For instance, a number of studies have shown that when ln έ min is plotted against ln t r, all the data fall on a single straight line no matter whether they are measured from short- or long-term tests.[1,12–14] Thus, when the required minimum rupture time t r is specified, the minimum creep rate required of the alloy can be determined using Eq. [1] even if its parameters are determined only from short-term tests.

However, once the required minimum creep rate έ min is known, the corresponding creep strength has to be determined. The relationship between the minimum creep rate and tensile testing stress σ and temperature T is given by the power law creep equation, which is conventionally written as[15]

$$ \acute{\varepsilon}_{ \hbox{min} } = A\left( {\sigma /\sigma_{0} } \right)^{n} { \exp }[ - Q_{\text{c}} /(RT)], $$
(2)

where A is a constant, n the stress exponent, Q the activation energy of creep, R the gas constant, T the absolute temperature, and σ 0 is the reference stress, typically taken as the yield stress. The term (σ/σ 0) in the above equation may also be written as (σ/μ),[15] where μ is the shear modulus at testing temperature T. The values of A and n are usually estimated from the plot of ln έ min vs ln σ at constant temperatures and the value of Q c from the plot of ln έ min vs 1/T at constant stresses. However, it has been found that the values of A, n, and Q c estimated using these methods depend strongly on stress and temperature[1,16] and this dependence is unpredictable with Eq. [2]. Thus, Eq. [2] cannot be used to determine the long-term rupture strength if the values of its parameters are determined only from short-term tests.

To overcome the difficulties described above, various models have been suggested.[8,17–22] The most notable ones are those suggested recently by Wilshire and co-workers, i.e., the Wilshire equations.[14,20–22] One of these Wilshire equations describes the stress and temperature dependence of the minimum creep rate, which is written as

$$ \sigma /\sigma_{\text{TS}} = { \exp }\{ - k_{2} (\acute{\varepsilon}_{ \hbox{min} } { \exp }\left[ {Q_{\text{c}} /\left( {RT} \right)} \right])^{v} \}, $$
(3)

where k 2 and v are constants and σ TS is the tensile strength at creep temperature. The equation above is introduced on the consideration that it can meet two boundary conditions: έ min → ∞ when σ → σ TS and έ min → 0 when σ → 0. It is useful to note that, for Eq. [3] to meet these two boundary conditions, the value of v must be negative.

However, the conventional power law creep equation, i.e., Eq. [2], can also be modified to satisfy the two boundary conditions stated above. This can be done by substituting the reference stress σ 0 in Eq. [2] by the term (σ TS − σ), i.e., if the new power law creep equation is written as

$$ \acute{\varepsilon}_{ \hbox{min} } = A[\sigma /(\sigma_{\text{TS}} - \sigma )\left] {^{n} { \exp }} \right[ - Q_{\text{c}} /\left( {RT} \right)]. $$
(4)

It can be verified that the equation above satisfies the conditions of έ min → 0 when σ → 0 and έ min → ∞ when σ → σ TS. Nevertheless, the physical significance of the tensile strength σ TS in Eq. [4] requires clarification because it must be a uniquely defined and measurable physical quantity. It is known that the values of tensile strength at high temperatures depend on strain rate. However, there must be a minimum tensile stress under which creep fracture would occur instantly, i.e., t r = 0 or έ min = ∞ when this tensile stress is applied. In practical reality, this tensile stress may be taken as the tensile strength measured using a sufficiently fast strain rate. Thus, the tensile strengths σ TS in Eq. [4] at different temperatures can be regarded as the ultimate tensile strengths measured under sufficiently high strain rate condition, i.e., at a minimum strain rate beyond which the tensile strength measured no longer depends on the strain rate.

Mathematically, Eq. [4] can be approximated by Eq. [1] when σ → 0, and by Eq. [3], i.e., the Wilshire equation, when σ → σ TS,[23] i.e., the conventional power law creep equation [Eq. 1] and the Wilshire equation [Eq. 3] may be used to approximately describe the respective two extreme cases of the new power law creep equation [Eq. 4]. In a previous study, Eq. [4] was applied to predict the long-term creep strength of a grade of 11Cr steel.[23] In the present study, it will be used to rationalize creep test data of 9Cr creep-resistant steels and predict their long-term creep strengths from short-term test data. The steel grades studied here (9Cr) differ from the one investigated previously (11Cr) mainly in steel composition, but all of them have a similar initial microstructure with the same tempered martensite matrix.[24] It will be shown that the phenomenon that was observed but could not be explained previously for this grade of steels can be clarified with the new power law creep equation.

3 Creep Data Analyzed and the Data Used in the Analysis

The comprehensive creep test data measured for P92 and T92 steel grades by the National Institute for Materials Science (NIMS), Japan,[24] are used in this study. These data are compiled in the creep data sheet no. 48A of the NIMS. To start with, all the creep data in the data sheet are used in the analysis. As this data sheet contains long-term creep test data with testing durations lasting over 100,000 hours in some cases, it is ideally suited for demonstrating the applicability of new power law creep equation in rationalizing creep test data. And, then, only those data with creep rupture times shorter than 5000 hours are used to predict the 100,000-hour creep rupture strengths at different temperatures and hence to demonstrate the predictive capability of new power law creep equation.

Steel composition, processing history, and creep property measurement methods are described in detail in the data sheet. Briefly, both steel grades are of 9Cr-1.8W-0.5Mo-V-Nb-B type with similar chemical composition but slightly different heat treatment history; P92 is a pipe grade and T92 is a tube grade. Minimum creep rates and creep rupture times were measured from tensile creep tests on specimens with a gage length of 50 mm and a diameter of 10 mm at the stresses of 20 to 320 MPa and the temperatures from 823 K to 1023 K (550 °C to 750 °C). Tensile strengths were measured at temperatures of 823 K, 873 K, 923 K, 973 K, and 1023 K (550 °C, 600 °C, 650 °C, 700 °C, and 750 °C), but not at the temperatures of 848 K, 898 K, and 948 K (575 °C, 625 °C, and 675 °C). These data were interpolated by the present authors by assuming that the temperature dependence of tensile strength is linear in the range of two vicinity temperature points at which the tensile strengths were measured experimentally. Table I summarizes the tensile strength data used in this study. All the tensile strength data in the data sheet were measured using a strain rate of 1.25 × 10−3/s; no data are available at this stage that are measured at higher strain rates. Thus, these data are used in the present study by assuming that the strain rate used to measure them is sufficiently fast as required in Eq. [4]. Nevertheless, it would be useful to carry out further tests at higher strain rates to confirm whether this is the case or not.

Table I Tensile Strengths of P92 and T92 Steels at Different Temperatures Used in the Analysis
Full size table

4 Results and Discussion

4.1 Rationalization of Minimum Creep Rate Data Using the New Power Law Creep Equation

The new power law equation [Eq. 4] can also be written in the following form:

$$ { \ln }\{ \acute{\varepsilon}_{ \hbox{min} } { \exp }\left[ {Q_{\text{c}} /\left( {RT} \right)} \right]\} \, = { \ln }A + n{ \ln }[\sigma /(\sigma_{\text{TS}} {-}\sigma )] $$
(5)

which predicts a straight line on the plot of ln{έ minexp[Q c/(RT)]} vs ln[σ/(σ TS − σ)] if the activation energy Q c of creep is known. This value can be determined by measuring έ min at different temperatures under constant σ/(σ TS − σ). But the creep data in the NIMS creep data sheet are not measured under this condition. Nevertheless, after a careful examination of the data, it was found that there were creep datasets in the data sheet that have an approximately constant value of ln[σ/(σ TS − σ)] and these creep datasets were selected for estimating the activation energy Q c. The plots of lnέ min vs 1/T were then made using these datasets and the results are shown in Figure 1, where the numerical figures shown next to the data points are the values of ln[σ/(σ TS − σ)]. It shows a set of near-parallel lines for both steel grades. The average activation energy calculated from the slope of these straight lines is approximately 367 kJ mol−1 for both P92 and T92 steel grades.

Fig. 1
figure 1

Plot of ln έ min vs 1/T at three different ln σ/(σ TS − σ) levels with each level having an approximately same ln σ/(σ TS − σ) value: (a) for P92 and (b) for T92

Full size image

In fact, using the creep data available in the data sheet, the activation energy of creep Q c can also be estimated using a curve fitting procedure. In this procedure, a preliminary value of Q c is selected to make the plot of ln{έ minexp[Q c/(RT)]} vs ln[σ/(σ TS − σ)] and an optimizing routine is carried out to obtain the highest value of coefficient of determination (r 2) by the least squares method. The Q c values obtained from this procedure were 378 and 364 kJ mol−1 for P92 and T92, respectively. These values are only slightly different from those estimated from the plots in Figure 1, demonstrating that this curve fitting procedure can also be used to obtain the value of Q c. But the average value of Q c estimated from Figure 1 for both 92 and T92 steel grades, i.e., 367 kJ mol−1, is used in the following analysis.

Using the average value of Q c estimated from Figure 1, the plots of ln{έ minexp[Q c/(RT)]} vs ln[σ/(σ TS − σ)] are made (Figure 2). All the data fall on two straight lines with the slope changing at ln[σ/(σ TS − σ)] = −0.667 and −0.458, which correspond to σ/σ TS = 0.339 and 0.387 for P92 and T92, respectively. The fact that all the data fall on two straight lines when a single activation energy value is used to make the plots is significant; it suggests that the activation energy does not change either with testing stress or with temperature in both cases. This is in contrast to the results obtained when the conventional power law creep equation is used to rationalize the creep data, which gave two different values of Q c: one for the low stress and the other for the high stress range.[15]

Fig. 2
figure 2

Plot of ln έ minexp[Q c/(RT)] vs ln [σ/(σ TS − σ)]: (a) for P92 and (b) for T92

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The solid straight lines in Figure 2 represent the least squares fits to the data points in respective σ/(σ TS − σ) ranges. In the case of P92 steel grade, the results are

$$ \acute{\varepsilon}_{ \hbox{min} } = \, 2.90 \times 10^{19} [\sigma /(\sigma_{\text{TS}} - \sigma )]^{7.4} { \exp }[ - 44142.4/T],\;{\text{for}}\;\sigma /\sigma_{\text{TS}} > \, 0.339 $$
(6)
$$ \acute{\varepsilon}_{ \hbox{min} } = \, 2.92 \times 10^{18} [\sigma /(\sigma_{\text{TS}} - \sigma )]^{3.9} { \exp }[ - 44142.4/T],\;{\text{ for}}\;\sigma /\sigma_{\text{TS}} < \, 0.339, $$
(7)

and in the case of T92 steel grade, they are

$$ \acute{\varepsilon}_{ \hbox{min} } = \, 1.32 \times 10^{19} [\sigma /(\sigma_{\text{TS}} - \sigma )\left] {^{7.7} { \exp }} \right[ - 44142.4/T], \, \;{\text{for}}\,\sigma /\sigma_{\text{TS}} > \, 0.387 $$
(8)
$$ \acute{\varepsilon}_{ \hbox{min} } = \, 2.19 \times 10^{18} [\sigma /(\sigma_{\text{TS}} - \sigma )\left] {^{3.8} { \exp }} \right[ - 44142.4/T], \, \;{\text{for}}\;\sigma /\sigma_{\text{TS}} < \, 0.387. $$
(9)

In all the numerical equations above, έ min and stresses are in units of pct h−1 and MPa, respectively. It is clear that the stress exponents in respective stress ranges are similar for both steel grades, but the values of their constant A differ slightly. In both cases, the stress exponent n decreases from about 7 to 8 in the high stress range to approximately 4 in the low stress range, but these values do not depend on temperature. Again, these results are completely different from those obtained using the conventional power law creep equation.[1,16] For instance, when the conventional power law creep equation was used to rationalize the data,[16] the stress exponents of P92 and T92 steel grades depended on both stress and temperature with a value in the range from 11 to 22 and 4 to 8 in the high and low stress ranges, respectively.

Using the numerical Eqs. [6] through [9], the dependence of minimum creep rate on stress can now be constructed and compared with the measurement data. This comparison is made in Figure 3 where the points are experimental data and solid straight lines are the predictions of Eqs. [6] through [9]. It shows a series of parallel straight lines matching reasonably well with experimental data with slopes or stress exponents changing at a common σ/σ TS stress ratio at different temperatures. Thus, in the case of P92 steel, it is now possible to predict that the stress exponent n would change to a lower value if creep tests were conducted at stresses below 137 MPa at 823 K (550 °C) and 127 MPa at 848 K (575 °C) although there were no data measured in these stress ranges at these temperatures. Similar predictions can be made in the case of T92 steel. As the tensile strength is proportional to 0.2 pct offset yield strength, it follows that the change in the slopes of straight lines in Figure 3 occurs also at a constant ratio of test stress to 0.2 pct offset yield strength. This is a feature that was observed, but could not be interpreted, previously.[1] It is now clear that it is a phenomenon expected by the new power law creep equation because, according to Eq. [4], if the stress exponent n changes, it must change at a common σ/σ TS stress ratio at different temperatures. These results demonstrate convincingly that the new power law equation can be used to rationalize the minimum creep rate data over a wide range of stresses and temperatures.

Fig. 3
figure 3

Comparisons of the predictions with experimental data: (a) for P92 and (b) for T92

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4.2 Estimation of Long-Term Creep Strength Using the New Power Law Creep Equation

Having determined the parameters in the new power law equation in high and low stress ranges, the long-term creep strengths at different temperatures can be readily estimated once the dependence of creep rupture time t r on the minimum creep rate έ min is quantified. This can be easily done by plotting lnέ min vs lnt r. Such plots are made using all the creep test data in the data sheet and the results are shown in Figure 4. It is clear that the Monkman–Grant relationship is followed by all the data measured at different stresses and temperatures. The straight lines are the least squares fit to the experimental data points. The results are as follows:

Fig. 4
figure 4

Dependence of creep rupture time (t r) on the minimum creep rate έ min: (a) for P92 and (b) for T92

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$$ {\text{for P92}},{ \ln }\acute{\varepsilon}_{ \hbox{min} } = - 1.1016{ \ln }t_{\text{r}} + \, 2.0066 $$
(10)
$$ {\text{for T92}},{ \ln }\acute{\varepsilon}_{ \hbox{min} } = - 1.1414{ \ln }t_{\text{r}} + \, 2.0969, $$
(11)

where έ min and t r are in units of pct h−1 and h, respectively.

Using the numerical Eqs. [6] through [11], the creep strength for any specified creep lifetime in the temperature range from 823 K to 923 K (550 °C and 650 °C) can now be estimated for P92 and T92 steel grades. It is necessary to point out that which equation should be used for making such estimates will depend not only on specified t r but also on temperature. The estimations for the 100,000-hour creep rupture strengths at different temperatures are made and the results are listed in the second and third columns in Table II. In these cases, the equations applicable in the low stress range, i.e., Eqs. [7] and [9], are used to make the estimations at the temperatures of 873 K and 923 K (600 °C and 650 °C), but at 823 K (550 °C), those applicable in the high stress range are used, i.e., Eqs. [6] and [8]. Otherwise, much higher estimations would be made, which are unrealistic. This is a clear example illustrating that even in the case of estimating the long-term creep strength, the selection of equations for making the estimates depends on both the required t r and temperature. Again, this finding is in contrast to the conclusion made previously with the conventional data analysis approaches, which suggested that the data measured in the high stress range should not be considered when making long-term estimates.[1,9] At the outset of estimations, it may not be always clear whether the equation for high or low stress range should be used to make the calculations. In such a situation, the calculations can be made using equations for both high and low stress ranges and the lower value is taken as the required long-term creep strength.

Table II Predictions of 105-Hour Creep Rupture Strength for P92 and T92 Steel Grades
Full size table

The accuracy of the estimations listed in Table II may be checked by comparing them with those reported in the literature.[1,9,13] For instance, based on the minimum creep rate measurements at different temperatures and the quantified Monkman−Grant relationship, Ennis et al. estimated that the creep strength of P92 steel grade is 55 MPa at 923 K (650 °C) and 115 MPa at 873 K (600 °C).[13] The estimates listed in Table I are in close agreement with these results. This agreement is not surprising as both sets of estimations are based on parameters determined from long-term test data, some of which are from tests lasting well over 25000 hours. However, it demonstrates again that the new power law creep equation can be used to rationalize the long-term creep test data.

4.3 Predictive Capability of the New Power Law Creep Equation

To evaluate the predictive capability of the new power law equation, the creep data in the same data sheet are used, but the analysis is carried out on the basis of two assumptions. One is that only those data with creep rupture times shorter than 5000 hours are available and the other is that the values of activation energy of creep (Q c) are unknown for the two steel grades. In effect, a blind test is used to evaluate the capability of the new power law creep equation in predicting the long-term creep properties.

With the above-stated constraints imposed on the use of data in the data sheet, no sufficient data are available to allow the activation energy of creep (Q c) to be estimated from the plot of lnέ min vs 1/T at constant σ/(σ TS − σ). This value is then estimated using a curve fitting procedure to obtain an optimum fit to the plot of ln{έ minexp[Q c/(RT)]} vs ln[σ/(σ TS − σ)] as described previously. The Q c values obtained from this procedure are 345 and 324 kJ mol−1 for P92 and T92, respectively. It is noted that these values are all lower than the ones estimated previously using the same procedure for these two steel grades. As the data measured at rupture times >5000 hours are removed from the analysis, the accuracy of the estimates may understandably be affected. But these values are still well within the expected range for this type of materials. Thus, these two values are used in the subsequent analysis in this blind test as described above.

Using the respective Q c values estimated above for the two steel grades, the plots of ln{έ minexp[Q c/(RT)]} vs ln[σ/(σ TS − σ)] are shown in Figure 5. Again, all the data fall on two straight lines with slopes changing at ln[σ/(σ TS − σ)] = −0.556 and −0.340, which correspond to σ/σ TS = 0.364 and 0.415 for P92 and T92, respectively. The straight lines in Figure 5 are the least squares fit to the experimental data points. For P92 steel grade, they are

$$ \acute{\varepsilon}_{ \hbox{min} } = \, 1.70 \times 10^{18} [\sigma /(\sigma_{\text{TS}} - \sigma )\left] {^{6.9} { \exp }} \right[ - 42578.8/T],{\text{ for}}\;\sigma /\sigma_{\text{TS}} > \, 0.364 $$
(12)
$$ \acute{\varepsilon}_{ \hbox{min} } = \, 4.43 \times 10^{17} [\sigma /(\sigma_{\text{TS}} - \sigma )\left] {^{4.5} { \exp }} \right[ - 42578.8/T], \, \;{\text{for}}\,\sigma /\sigma_{\text{TS}} < \, 0.364. $$
(13)

And for T92 steel grade, they are

$$ \acute{\varepsilon}_{ \hbox{min} } = \, 4.39 \times 10^{16} [\sigma /(\sigma_{\text{TS}} - \sigma )\left] {^{7.3} { \exp }} \right[ - 38970.4/T],{\text{ for}}\;\sigma /\sigma_{\text{TS}} > \, 0.415 $$
(14)
$$ \acute{\varepsilon}_{ \hbox{min} } = \, 1.33 \times 10^{16} [\sigma /(\sigma_{\text{TS}} - \sigma )\left] {^{3.8} { \exp }} \right[ - 38970.4/T], \, \quad {\text{for}}\,\sigma /\sigma_{\text{TS}} < \, 0.415. $$
(15)
Fig. 5
figure 5

Plot of έ minexp[Q c/(RT)] vs ln [σ/(σ TS − σ)] for the data with t r < 5000 h: (a) for P92 and (b) for T92

Full size image

Again, in Eqs. [12] through [15] above, έ min and stresses are in units of pct h−1 and MPa, respectively. For both steel grades, the stress exponent decreases from about 7 to approximately 4 as the creep stress decreases, which is consistent with the behavior observed previously when all the data in the data sheet are included in the analysis (see Eqs. [6] through [9]). But it can be noted that the values of constant A in the high and low stress ranges are all much lower than those obtained previously (see, again, Eqs. [6] through [9]).

Those creep data with t r < 5000 hours also follow the Monkman−Grant relationship but with slightly different parameter values as compared with those determined previously when all the data are included in the analysis. The numerical equations obtained for the two steel grades are as follows:

$$ {\text{for P92}},{ \ln }\acute{\varepsilon}_{ \hbox{min} } = - 1.0763{ \ln }t_{\text{r}} + \, 1.8843 $$
(16)
$$ {\text{for T92}},{ \ln }\acute{\varepsilon}_{ \hbox{min} } = - 1.1132{ \ln }t_{\text{r}} + \, 1.9583, $$
(17)

where, again, έ min and t r are in units of pct h−1 and h, respectively. It is useful to point out that the differences in numerical values between Eqs. [10] and [16] and between Eqs. [11] and [17] are caused by the experimental data scattering but not by the change in creep deformation or fracture mechanisms as it can be seen in Figure 4 that both short (t r < 5000 hours)- and long (t r > 5000 hours)-term data fall on a single straight line on the plot of ln έ min vs ln t r with no convincing evidence showing any change in the slope of the plot.

The predictions of the 100,000-hour creep strength made using the numerical Eqs. [12] through [17] are listed in the last two columns in Table II. Again, equations applicable in the low stress range are used for predictions at 873 K and 923 K (600 °C and 650 °C), but those valid in the high stress range are employed for predictions at 823 K (550 °C). In general, slightly more conservative predictions are made in comparison with those estimated earlier when all the data available in the data sheet are used in the analysis, even though the predictions made in both sets of analysis are same at 923 K (650 °C) in the case of P92. The fact that slightly more conservative predictions are made demonstrates that the new power law equation can be used to predict the long-term creep strength once its parameters are determined from short-term test data because the slightly more conservative predictions would not jeopardize the safety of designs. However, it is necessary to point out that using Eq. [4] to predict long-term creep strengths from short-term creep test can only be possible if these short-term tests are carried out over a sufficiently wide range of temperatures and stresses so that all the stress ranges in which the stress exponent changes its value can be revealed on the plot of ln{έ minexp[Q c/(RT)]} vs ln[σ/(σ TS − σ)].

Finally, it is necessary to point out again that all the analyses given above were made using the tensile strength data measured by NIMS using a rate of 1.25 × 10−3 pct/s.[24] Further studies are needed to confirm whether these data will be higher at higher strain rates and accordingly to investigate the sensitivity of Eq. [4] to the tensile strength data measured at higher strain rates.

4.4 Dependence of Creep Rupture Time t r on Testing Stress and Temperature

Substituting Eq. [1] into Eqs. [4] for έ min, the new power law equation can be written in a different form, giving the relationship between creep rupture time t r and creep stress and temperature:

$$ t_{\text{r}}^{ - m} = \, \left( {A/M} \right) \, [\sigma /(\sigma_{\text{TS}} - \sigma )\left] {^{n} { \exp }} \right[ - Q_{\text{c}} /\left( {RT} \right)]. $$
(18)

Rearranging the above equation gives

$$ t_{r}^{ - 1} = A^{\prime} \, [\sigma /(\sigma_{TS} - \sigma )\left] {^{{n^{\prime}}} \exp } \right[ - Q_{r} /\left( {RT} \right)], $$
(19)

where A’ = (A/M)1/m, n’ = n/m, and Q r = Q c/m are all new constants. Thus, the stress exponent and activation energy in Eq. [19] differ from those in Eq. [4] by a factor of m as defined in the Monkman–Grant relationship. Because n and Q c in Eq. [9] are termed stress exponent of creep and activation energy of creep, n’ and Q r in Eq. [19] may be appropriately termed stress exponent of creep rupture and activation energy of creep rupture, respectively. It can be easily verified that Eq. [19] can meet the boundary conditions of t r → ∞ when σ → 0 and t r → 0 when σ → σ TS.

It is necessary to point out that only one of the two equations, i.e., Eqs. [4] and [19], is independent as the two are related by Eq. [1], i.e., the Monkman–Grant relationship. But they can be used independently to rationalize the dependence of minimum creep rate έ min and creep rupture time t r on testing stress σ, respectively. This will be useful because, compared with creep test, the creep rupture test is much cheaper and easier to run as in such tests the creep deformations with time are not measured. Thus, in many practical applications, it is a preferred test and in such a situation, Eq. [19] can be used to rationalize the test results.

In the present cases of P92 and T92 steel grades, substituting Eq. [10] into Eqs. [6] and [7] for έ min and, similarly, substituting Eq. [11] into Eqs. [8] and [9] for έ min, one can easily obtain the numerical equations that quantify the relationship between creep rupture time and testing stress and temperature, allowing the creep rupture strength to be calculated directly from any specified creep rupture time in the temperature range from 823 K to 1023 K (550 °C and 750 °C) for these two steel grades.

5 Conclusions

The extensive sets of creep data measured over a wide range of stresses and temperatures for P92 and T92 steel grades have been rationalized using the new power law creep equation. The main conclusions can be summarized as follows:

  1. 1.

    For each of these two steel grades, all the data fall on two straight lines on the plot of ln{έ minexp[Q c/(RT)]} vs ln[σ/(σ TS − σ)] with the slope changing at constant σ/(σ TS − σ) or at a fixed σ/σ TS ratio and hence dividing the stress dependence of the minimum creep rate into high and low stress ranges; in both of these stress ranges, a single value of activation energy of creep Q c applies.

  2. 2.

    The stress exponent in the high stress range is about 7 to 8 and it is approximately 4 in the low stress range, but it does no depend on temperature.

  3. 3.

    The activation energy of creep Q c can be determined either from the plot of ln έ min vs 1/T at constant σ/σ TS ratios if the creep tests are conducted under this stress condition, or using a curve fitting procedure to produce an optimum fit to the plot of ln{έ minexp[Q c/(RT)]} vs ln[σ/(σ TS − σ)].

  4. 4.

    The predictive capability of the new power law creep equation has been demonstrated using a blind test, which resulted in slightly more conservative predictions for the 100,000-hour creep rupture strengths at different temperatures, confirming that the new power law equation can be used to predict the long-term creep strength once its parameters are determined from short-term tests carried out in a sufficiently wide range of temperatures and stresses.

  5. 5.

    The new power law creep equation is expected to be equally applicable to all grades of creep-resistant ferritic steels, as these grades of steels all share the same microstructure as P92 and T92 steel grades of this study, i.e., they all have a tempered martensite matrix strengthened by fine particle precipitations. The wider applicability of the new power law creep equation to other types of metals and alloys requires further confirmation although there appears to be no reason to suggest that it may not be able to do so.