Use of the Wilshire Equations to Correlate and Extrapolate Creep Data of Inconel 617 and Nimonic 105

Advanced power plant alloys must endure high temperatures and pressures for durations at which creep data are often not available, necessitating the extrapolation of creep life. A recently developed creep life extrapolation method is the Wilshire equations, with which multiple approaches can be used to increase the goodness of fit of available experimental data and improve the confidence level of calculating long-term creep strength at times well beyond the available experimental data. In this article, the Wilshire equation is used to extrapolate the creep life of Inconel 617 and Nimonic 105 to 100,000 h. The use of (a) different methods to determine creep activation energy, (b) region splitting, (c) heat- and processing-specific tensile strength data, and (d) short-duration test data were investigated to determine their effects on correlation and extrapolation. For Inconel 617, using the activation energy of lattice self-diffusion as QC* resulted in a poor fit with the experimental data. Additionally, the error of calculated rupture times worsened when splitting regions. For Nimonic 105, the error was reduced when heat- and processing-specific tensile strengths were used. Extrapolating Inconel 617 creep strength to 100,000 h life gave conservative results when compared to values calculated by the European Creep Collaborative Committee.


Introduction
Innovations in power generation require materials that are capable of withstanding high temperatures and stresses for at least 100,000 h of operation time. The high temperatures and pressures found in advanced power plants can induce creep failure in alloys. Consequently, alloys must be tested so that creep failures are avoided during service. However, data regarding creep of new advanced power plant alloys are often not available at times relevant to the required design life. In particular, nickel-based superalloys-promising alloys for ultra-supercritical power plant applications-have no creep rupture data at 100,000 h in the literature. The longest creep rupture test of a nickel-based superalloy known to the authors-an Inconel 617 specimen last reported at 90,936 h-is ongoing and is the result of a joint effort led by the U.S. Department of Energy and the Ohio Coal Development Office [1,2]. The same effort produced creep rupture data for Inconel 617 to 43,706 h (completed), and Inconel 740 to 30,957 h (completed) and 56,550 h (ongoing). In a similar effort, creep rupture data of Nimonic 105 was generated to 34,955 h [3,4]. Hence, extrapolating the creep life of these alloys is necessary to determine if they are suitable for use.
where t r is the time to rupture, C M is a constant, and α is the slope of log t r vs. log . ε m . This equation can be coupled with Equation (1), with α set equal to 1, to produce the following time-to-rupture-based version of the classic power law equation However, this equation is unreliable for predicting creep life at temperatures, stresses, and durations at which there are no experimental data available. This is due to the changing and difficult-to-predict stress exponent [10,11], which is a function of stress and temperature. Additionally, creep activation energy is a function of applied stress. Therefore, a creep activation energy that has been calculated in one stress region cannot be extrapolated to another. Many techniques to extrapolate creep life reliably from limited data have been proposed [12,13], including the relatively new Wilshire equation and the well-established Larson-Miller parameter equation.

Wilshire Equation
In 2007, Wilshire and Battenbough [5] developed a physically-based yet fairly simple method to represent creep life as a function of applied stress and temperature in uniaxial tests. The proposed equations are: where σ/σ TS is the ratio of applied stress to ultimate tensile strength, t r is time to rupture, . ε m is minimum creep rate, t ε is time to strain, Q * C is creep activation energy determined at constant σ/σ TS , R is the universal gas constant, T is absolute temperature, and k 1 , u, k 2 , v, k 3 , and w are fitting constants. Heat-and processing-specific tensile strength values may be used if data were collected from specimens of multiple heats. Applied stress can be normalized by yield strength (σ YS ), but normalization by ultimate tensile strength causes the stress ratio to always lie between zero and one. The boundaries of the Wilshire equations are

Larson-Miller Parameter Equation
A common method used to extrapolate creep life is the Larson-Miller parameter equation [14], which predates the Wilshire equation by half a century. The LMP equation is: where T is absolute temperature, C is a material constant, and t r is time to rupture. The LMP, a function of stress, is often described by the following fitting function.

Data
The Wilshire and Larson-Miller parameter equations both require the following data from creep rupture tests: temperature, applied stress, and time to rupture. Additionally, the Wilshire equation requires the material's ultimate tensile strength at each test temperature. Yield strength values are not included in the Wilshire equations, but according to Wilshire and Battenbough [5] can be used to group high-and low-stress data for analysis. Data were obtained from tables or extracted from plots in various publications using Dagra digitization software (version 2.0.12) [15].

Inconel 617
Inconel 617 tensile and yield strength data were obtained and extracted from a conference presentation [16]. The data is composed of multiple datasets and there is noticeable scatter in the plotted data. A trendline of the average tensile strength of the combined datasets was used for the extraction of data. Creep test data were extracted from five sources [1,[17][18][19][20], the longest of which extends to 43,706 h. A total of 420 creep rupture data points were extracted; 386 have a rupture time less than 10,000 h and 354 have an applied stress less than yield strength.

Nimonic 105
For Nimonic 105, tensile and yield strength values and creep rupture data were obtained from two technical reports [3,4] and email correspondence with one of their authors [21]. Specimens with the following processing conditions were tested at temperatures from 760 to 816 • C for up to 34,955 h: as-processed (AP), peak-aged (PA), and over-aged (OA). Heat-and processing-specific tensile and yield strength values are available at 760 and 816 • C. Linear interpolation was used to calculate tensile and yield strength values at intermediate temperatures. Of the 33 specimens, eight were aged at 774 • C for one or two years prior to testing. It is not known which of the eight specimens were aged for each timeframe, so tensile and yield strength values for both timeframes were averaged. Tensile and yield strength values at 760 and 816 • C for each heat and processing condition are shown in Table A1 of Appendix A. All data points have an applied stress less than yield strength and 16 data points have a rupture time less than 10,000 h. The overall scatter of the data is low compared to that of Inconel 617.

Wilshire Equation
Although three versions of the Wilshire equation exist, in this study Equation (4)-the Wilshire equation for time to rupture-is used. Several methods have been used in the literature to determine Q * C . In this work, Q * C was determined using multiple methods. First, Q * C values were determined using Arrhenius plots. This method is the most well-known, and it is assumed [22] that Wilshire used this method in his papers [4,5,[23][24][25][26][27][28][29][30][31]. For this approach, existing creep data is regressed and rupture times at constant stress ratios are calculated. This work uses least squares regression and stress ratios at every tenth value (i.e., σ/σ TS = 0.1, 0.2, 0.3, etc.) with suitable data at each test temperature. Next, an Arrhenius plot of the natural log of time to rupture vs. the inverse absolute temperature is generated. For each stress ratio, a Q * C value is defined as the slope of a line of best fit multiplied by the universal gas constant. An average is then taken of all Q * C values to determine the final Q * C value. Second, Q * C values were determined by optimizing the correlation of data on a Wilshire plot (ln[t r exp(−Q * C /RT)] vs. ln[−ln(σ/σ TS )]), as performed by Whittaker [32][33][34] and Jeffs [35]. When calculating Q * C using an Arrhenius plot, the delta between the stress ratios influences the final average Q * C value (e.g., σ/σ TS = 0.1, 0.2 and 0.3 compared to σ/σ TS = 0.1, 0.15, 0.2, 0.25, and 0.3). This presents a concern when calculating an average Q * C , since the stress ratios are not mathematically derived, but chosen based on the judgement of the user. Determining Q * C by optimizing the correlation of data on a Wilshire plot eliminates concern over the variance of Q * C at arbitrarily-chosen stress ratios. The technique used to optimize the correlation of data is generally not defined in the literature and may vary between authors. In this work, Q * C values ranging from 1 to 500 kJ/mol were iterated with a step size of 1 kJ/mol to find the best correlation, which was quantified by the coefficient of determination (R 2 ).
Third, in some articles [5,[24][25][26][27][28][29]32,36] the reasonableness of the calculated value of Q * C is assessed by comparing it to an experimentally-measured or theoretically-calculated activation energy of lattice self-diffusion. If a value for the activation energy of lattice self-diffusion is known, it may be expedient to use this value as Q * C rather than calculate a value from the experimental data. After Q * C has been determined, the u and k 1 fitting constants are respectively defined as the slope and exponential of the y-intercept of a best fit line on a Wilshire plot. Multiple linear regions may be visible on this plot, which would suggest that the data be separated into regions to improve the goodness of fit of the Wilshire equation. Gray and Whittaker [37] point out that Wilshire split regions using two different methods. Regions were consistently split where σ was equal to σ YS , but in one case, Q * C and new fitting constants were recalculated for each region [33], while in another case, the original Q * C value was used and only the fitting constants were recalculated for each region [4]. The latter case is known to under-predict creep life [38], and the former case, which more accurately describes the underlying physical processes [37,39,40], was used by Whittaker and Wilshire to extrapolate the creep life of Grade 22, 23, and 24 steels [39]. In this study, both region-splitting techniques are used. Evans proposed a method to handle data from multiple batches [41], but this method requires a more complicated analysis that is outside the scope of this study, so it was not utilized.

Larson-Miller Parameter Equation
In the Larson-Miller parameter equation, the material constant C is often set to 20 [12]. However, C can be calculated if desired. The method described by Zhu et al. [42] was used to calculate C in this work. The accuracy of Equation (8), the LMP fitting function, increases with the number of terms that are used. For this study, four terms were deemed to be sufficient so the parameters B 0 , B 1 , B 2 and B 3 were obtained. With this information, t r can be estimated for any given combination of temperature and stress.
Following the method detailed by Zhu et al. [42], the matrix laboratory software (MATLAB, version 2018b) surface fitting tool was used to determine the LMP equation constants. The Larson-Miller parameter equation was arranged as: where z = log(t r ), x = T, y = log(σ). Creep rupture data for Inconel 617 and Nimonic 105 were split into two data sets for each alloy; one consisted of all data, while the other was limited to data with rupture times less than 10,000 h. The purpose of the limited data set is to show the efficacy of extrapolating short-term test data to longer times, as it has been claimed that the Wilshire equation is well-suited to do so [4,5,25,27,28,33]. For each data set, values of Q * C were determined using Arrhenius plots (shown in Tables 1 and 2), by optimizing the correlation of data on Wilshire plots (shown in Tables 3 and 4), and using the activation energy of self-diffusion of nickel in a nickel lattice, 292 kJ/mol [43]. Since tensile strength values for each heat and processing condition are available for Nimonic 105, the effect of their use compared to the use of average values was investigated. Average tensile strength values were determined at each temperature using tensile strength values of all heats and processing conditions. Wilshire plots for each case were generated to show the goodness of fit of the data. To improve the goodness of fit, Wilshire plots for Inconel 617 were split into two regions: σ < σ YS and σ ≥ σ YS . Wilshire plots for Nimonic 105 were not split into regions because no data points have an applied stress higher than yield strength and no visible break appears in the data. For Inconel 617, negative Q * C values were calculated using Arrhenius plots at two stress ratios, 0.4 and 0.5, for data with an applied stress less than yield strength. These negative Q * C values were omitted from average Q * C calculations. Representative Wilshire plots are shown in Figure 1. Plots of all cases are provided as Figures A1-A5 in Appendix A. For both alloys, Q * C values much lower than the activation energy of self-diffusion of nickel in a nickel lattice were occasionally obtained. Similarly, low Q * C values have been obtained by others [33,35,44].       For all cases, time to rupture was calculated at the stress and temperature of each experimental data point. The error of the calculated rupture times in hours was measured using mean squared error (MSE), defined as

Results and Discussion
The Wilshire equation's goodness of fit (quantified by R 2 ) for each data set and error obtained by applying the calculated * values and fitting constants to all creep rupture data are presented in Tables For all cases, time to rupture was calculated at the stress and temperature of each experimental data point. The error of the calculated rupture times in hours was measured using mean squared error (MSE), defined as The Wilshire equation's goodness of fit (quantified by R 2 ) for each data set and error obtained by applying the calculated Q * C values and fitting constants to all creep rupture data are presented in Tables 5 and 6.   For Inconel 617, the activation energy of self-diffusion of nickel in a nickel lattice gave the worst goodness of fit and error in all cases. Creep activation energy values at stresses below yield strength are much lower than those above yield strength. Both methods calculated much lower Q * C values, ranging from 62 to 110 kJ/mol, than the published activation energy of self-diffusion of nickel in a nickel lattice, 292 kJ/mol. For data with an applied stress above yield strength, the calculated Q * C values are slightly lower, but still near the activation energy of self-diffusion of nickel in a nickel lattice. Contrary to expectations, region splitting worsened the error in all cases. Additionally, using data with rupture times less than 10,000 h to calculate Q * C and the fitting constants yielded errors similar to using all data. Regardless of the method used to calculate Q * C , relatively poor fits of the Wilshire equation to the Inconel 617 data were obtained. A similar issue of large data scatter of Inconel 617 has been reported by others [45,46].
For Nimonic 105, the goodness of fit and error improved dramatically from the use of heat-and processing-specific tensile strength values; for the data set with all data, the coefficient of determination increased from 0.88 to 0.95 and the mean squared error was reduced by about 70%. All three methods to determine Q * C gave a similar goodness of fit and error with all data. Compared to using all data, the calculated Q * C values are much lower and the goodness of fit is worse when using data with rupture times less than 10,000 h. Figure 2 shows the correlation of the Wilshire equation and error at each potential value of Q * C , which were obtained when using the correlation optimization method to determine Q * C .   Figure 6 is shown as stress normalized by tensile strength. Calculated times to rupture for each data point and method to determine * for Nimonic 105 are shown in Table A2 in Appendix A. When splitting the data into above-and below-yield stress regions, the time to rupture at the transition from one region to the other is not calculated to be the same value in each region. Due to this, the split-region calculations of the Wilshire equation can yield zero or two stress values at some rupture times. The Inconel 617 plots show the tendency of the singleregion rupture stress calculations to become more conservative than the split-region calculations as time increases.   Figure 6 is shown as stress normalized by tensile strength. Calculated times to rupture for each data point and method to determine Q * C for Nimonic 105 are shown in Table A2 in Appendix A. When splitting the data into above-and below-yield stress regions, the time to rupture at the transition from one region to the other is not calculated to be the same value in each region. Due to this, the split-region calculations of the Wilshire equation can yield zero or two stress values at some rupture times. The Inconel 617 plots show the tendency of the single-region rupture stress calculations to become more conservative than the split-region calculations as time increases.        From an engineering perspective, determining the average percentage difference between the calculated and experimentally-obtained rupture times is a reasonable way to show the tendency of the Wilshire equation to over-or under-predict creep life. For use by boiler pressure vessel design code organizations it is desirable that conservative estimations of creep life are produced. Average percentage difference is defined as For all cases, the average percentage difference was calculated at each temperature, and the results are shown in Figures 7-10.
From an engineering perspective, determining the average percentage difference between the calculated and experimentally-obtained rupture times is a reasonable way to show the tendency of the Wilshire equation to over-or under-predict creep life. For use by boiler pressure vessel design code organizations it is desirable that conservative estimations of creep life are produced. Average percentage difference is defined as For all cases, the average percentage difference was calculated at each temperature, and the results are shown in Figures 7-10.
For Inconel 617, calculations from 800 to 900 °C are generally not conservative and those beyond 1000 °C tend to be conservative regardless of the method used. Use of the self-diffusion activation energy of nickel in a nickel lattice as * gave the largest overpredictions of creep life for single-and split-region analyses. For Nimonic 105, the use of heat-and processing-specific tensile strength data usually-but not always-improved the average percentage difference.  Split regions -QC* = 292 Average percentage difference at each temperature for Inconel 617 using all data to calculate Q * C (kJ/mol).
For all cases, the average percentage difference was calculated at each temperature, and the results are shown in Figures 7-10.
For Inconel 617, calculations from 800 to 900 °C are generally not conservative and those beyond 1000 °C tend to be conservative regardless of the method used. Use of the self-diffusion activation energy of nickel in a nickel lattice as * gave the largest overpredictions of creep life for single-and split-region analyses. For Nimonic 105, the use of heat-and processing-specific tensile strength data usually-but not always-improved the average percentage difference.
Average percentage difference at each temperature for Inconel 617 using data with times to rupture less than 10,000 h to calculate Q * C (kJ/mol).
For Inconel 617, calculations from 800 to 900 • C are generally not conservative and those beyond 1000 • C tend to be conservative regardless of the method used. Use of the self-diffusion activation energy of nickel in a nickel lattice as Q * C gave the largest overpredictions of creep life for single-and split-region analyses. For Nimonic 105, the use of heat-and processing-specific tensile strength data usually-but not always-improved the average percentage difference.  The European Creep Collaborative Committee (ECCC) has extrapolated the creep life of various alloys to 100,000 h, including Inconel 617 [47]. As mentioned by Bullough, et al. [48], an interim Inconel 617B ECCC data sheet exists and a revision is in progress. A comparison of the creep strengths for rupture at 100,000 h specified by the ECCC and those obtained using the Wilshire equation are shown in Table 7 at temperatures that are common to both the ECCC's data sheet and the data used in this paper. The calculations of the Wilshire equation are closest to the values in the Inconel 617 ECCC data sheet when all data are treated as a single region and * is determined by optimizing the correlation of data on a Wilshire plot. Calculated creep strength values for rupture at 100,000 h for Nimonic 105 are presented in Table 8. For both alloys, the use of data with rupture times less than 10,000 h to calculate * generally resulted in lower calculated creep strengths for rupture at 100,000 h.  The European Creep Collaborative Committee (ECCC) has extrapolated the creep life of various alloys to 100,000 h, including Inconel 617 [47]. As mentioned by Bullough, et al. [48], an interim Inconel 617B ECCC data sheet exists and a revision is in progress. A comparison of the creep strengths for rupture at 100,000 h specified by the ECCC and those obtained using the Wilshire equation are shown in Table 7 at temperatures that are common to both the ECCC's data sheet and the data used in this paper. The calculations of the Wilshire equation are closest to the values in the Inconel 617 ECCC data sheet when all data are treated as a single region and * is determined by optimizing the correlation of data on a Wilshire plot. Calculated creep strength values for rupture at 100,000 h for Nimonic 105 are presented in Table 8. For both alloys, the use of data with rupture times less than 10,000 h to calculate * generally resulted in lower calculated creep strengths for rupture at 100,000 h. Average percentage difference at each temperature for Nimonic 105 using data with times to rupture less than 10,000 h to calculate Q * C (kJ/mol).
The European Creep Collaborative Committee (ECCC) has extrapolated the creep life of various alloys to 100,000 h, including Inconel 617 [47]. As mentioned by Bullough, et al. [48], an interim Inconel 617B ECCC data sheet exists and a revision is in progress. A comparison of the creep strengths for rupture at 100,000 h specified by the ECCC and those obtained using the Wilshire equation are shown in Table 7 at temperatures that are common to both the ECCC's data sheet and the data used in this paper. The calculations of the Wilshire equation are closest to the values in the Inconel 617 ECCC data sheet when all data are treated as a single region and Q * C is determined by optimizing the correlation of data on a Wilshire plot. Calculated creep strength values for rupture at 100,000 h for Nimonic 105 are presented in Table 8. For both alloys, the use of data with rupture times less than 10,000 h to calculate Q * C generally resulted in lower calculated creep strengths for rupture at 100,000 h.

Comparison of Calculations of the Wilshire and Larson-Miller Parameter Equations
The Larson-Miller parameter equation was used to provide calculations for comparison with the calculations of the Wilshire equation. Equation (9) and the MATLAB surface fitting tool were used to correlate the experimental data to the LMP equation and the resulting coefficients and goodness of fit are shown in Table 9.  0.949 5.58 × 10 6 1 Data treated as a single region with Q * C calculated using Arrhenius plots; 2 Heat-and processing-specific tensile strength data with the self-diffusion activation energy of nickel in a nickel lattice as Q * C .
A comparison of the tendency for each equation to over-or under-predict creep life-quantified as the average percentage difference between calculated and experimentally-obtained rupture times-is shown in Figures 11 and 12. For Inconel 617, the over-and under-predictions of the LMP equation are generally less severe than those of the Wilshire equation, except at the two lowest temperatures.
As shown in Table 11 the calculated rupture time for both equations was conservative for each alloy compared to the longest experimental test data points. The LMP equation yielded more conservative estimates than the Wilshire equation.
As shown in Table 11 the calculated rupture time for both equations was conservative for each alloy compared to the longest experimental test data points. The LMP equation yielded more conservative estimates than the Wilshire equation.  The percentage differences of the calculated rupture time for the longest test duration of each alloy is defined as Percentage Difference = t r,calculated − t r,experimental t r,experimental × 100.
As shown in Table 11 the calculated rupture time for both equations was conservative for each alloy compared to the longest experimental test data points. The LMP equation yielded more conservative estimates than the Wilshire equation. 1 Data treated as a single region with Q * C calculated using Arrhenius plots; 2 Heat-and processing-specific tensile strength data with the self-diffusion activation energy of nickel in a nickel lattice as Q * C .
Calculated creep strengths for rupture at 100,000 h using the LMP and Wilshire equations are presented in Tables 12 and 13. Figures 13 and 14 show experimental creep data, calculations of the LMP equation, and calculations of the Wilshire equation using the Q * C value that yielded the lowest error. For ease of displaying the calculations, the y axis of Figure 14 is stress normalized by tensile strength, and the stresses calculated using the LMP equation are normalized by average tensile strength values. Calculated times to rupture for each data point and method to determine Q * C for Nimonic 105 are shown in Table A2 in Appendix A. The form of the stress function used in the LMP equation can result in multiple values of stress to be calculated at a single rupture time; however, this issue was not observed for either data set. For Inconel 617, the calculations of the Wilshire equation are more conservative than those of the LMP equation at failure times approaching and beyond 100,000 h. Nimonic 105 exhibited the same behavior at 800 and 850 • C. For Inconel 617, both the Wilshire and LMP equations predicted 100,000 h creep strengths lower or only slightly higher than the values calculated by the ECCC, including the temperature range of 800 to 900 • C where both equations overpredicted creep life vs. the experimental data (see Figure 11).

Conclusions
This study investigated multiple methods to determine Q * C , the effect of region-splitting, the use of short-term creep rupture data to extrapolate creep life, and the use of heat-and processing-specific tensile strength values, all of which are techniques that have been used or proposed to increase the accuracy of the Wilshire equation.
The large temperature span of Inconel 617 data, over 500 • C, may be the cause for relatively poor fits of both the Wilshire and LMP models to the data. At higher temperatures, greater than 850 • C, microstructural changes that affect creep strength may be accelerated, or other strength degradation phenomena might become significant that do not occur at lower temperatures. For Inconel 617, the mean squared error of the calculated creep life using the self-diffusion activation energy of nickel in a nickel lattice as Q * C was about an order of magnitude greater than the other methods to determine Q * C . The large scatter and temperature range may have exacerbated potential error introduced by using the self-diffusion activation energy as Q * C , rather than using a value calculated from the data. With the well-behaved Nimonic 105 data, a very similar goodness of fit was obtained using any of the three methods for determining Q * C . It is possible that calculations of the Wilshire equation are not significantly affected by the Q * C value when a small data set with low scatter is used. If a high degree of fit to the data is required, the authors recommend that self-diffusion activation energy not be used as Q * C due to the potential for large error, as seen for Inconel 617. Using the Q * C value that provided the lowest error, the longest time to rupture for both alloys was underpredicted, which is much more desirable than overprediction. Contrary to expectations, treating the data as a single region-rather than splitting the data into above-and below-yield stress regions-provided the lowest mean squared errors for Inconel 617. The use of heat-and processing-specific tensile strength values greatly improved the goodness of fit of the Wilshire equation to the Nimonic 105 data and reduced the error in all cases. For Inconel 617, the use of data with rupture times less than 10,000 h to extrapolate creep life gave roughly similar goodness of fit and error compared to using all data. Surprisingly, for Nimonic 105 the use of data with rupture times less than 10,000 h to extrapolate creep life resulted in a significant reduction in goodness of fit and error compared to using all data, e.g., R 2 of 0.880 versus 0.950 when Q * C was determined using the correlation optimization method. Considering that the longest Nimonic 105 experimental data point was 34,995 h time to rupture, this result would suggest that further investigation should be made of the ability of the Wilshire equation to accurately predict long-term creep strength from short-term creep rupture strength data.
In its basic form, the Wilshire equation is a simple method to quickly estimate long-term creep life using only three fitting constants-yet if an extensive analysis with a high level of precision is required, its complexity can be increased to improve the statistical fit of the Wilshire equation to available data. Evans [41,[49][50][51][52][53][54] has proposed more sophisticated methods of fitting the Wilshire equation to complex data sets, including the handling of data collected from specimens of multiple batches [41], determining Q * C as a function of temperature [41], statistically determining the number of stress regions [49], and utilizing additional batch characteristics [50]. The Wilshire equation is modular in that many combinations of these methods can be used, which gives it flexibility for a wide variety of applications. If only a preliminary estimate of long-term creep strength (e.g., at 100,000 h or longer design life) is needed, such as in the early stages of new alloy development, use of the Wilshire equation in its original form with Q * C equal to the activation energy of self-diffusion would probably be sufficient. More complex analyses (which, in essence, increase the number of fitting constants) would be needed if the intent is to use the Wilshire equation for component design or for establishing long-term creep strength values for design codes, instead of the equations now used in various design codes, and which contain more than three fitting constants.           Figure A17. Split region calculations of the Wilshire equation for Nimonic 105 using heat-and processing-specific tensile strength values with Q * C determined using Arrhenius plots.   Figure A18. Calculations of the Wilshire equation for Nimonic 105 using heat-and processing-specific tensile strength values with Q * C determined by optimizing the correlation of data on a Wilshire plot.   Figure A19. Calculations of the Wilshire equation for Nimonic 105 using heat-and processing-specific tensile strength values with the self-diffusion activation energy of nickel in a nickel lattice as Q * C .  Data Set = Data with t r < 10,000 h, Average TS values, Q * C Determination Method = Self-diffusion Activation, Q * C = 292, u = 0.111, k 1 = 19.9. Case 10: Data Set = Data with t r < 10,000 h, Heat-and processing-specific TS values, Q * C Determination Method = Arrhenius Plot, Q * C = 207, u = 0.126, k 1 = 8.32. Case 11: Data Set = Data with t r < 10,000 h, Heat-and processing-specific TS values, Q * C Determination Method = Correlation Optimization, Q * C = 235, u = 0.122, k 1 = 11.6. Case 12: Data Set = Data with t r < 10,000 h, Heat-and processing-specific TS values, Q * C Determination Method = Self-diffusion Activation, Q * C = 292, u = 0.114, k 1 = 20.9.