Testing Model Structure Through a Unification of Some Modern Parametric Models of Creep: An Application to 316H Stainless Steel
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Abstract
It is important to be able to predict the creep life of materials used in power plants and in aeroengines. This paper develops a new parametric creep model that extends those put forward by Wilshire and Yang et al. by having them as restricted or special cases of a new generalized model. When this generalized model was applied to failure time data on 316H stainless steel it was found that neither of these established parametric models explained the greatest variation in the experimentally obtained times to failure. Instead, a version of this generalized model was most compatible with the experimental data. It was further found that the activation energy for this material changed at a normalized stress of 0.41 due to a change from the domination of dislocation movement within grains to movement within grain boundaries. Finally, when the generalized model was used to predict failure times beyond 5000 hours (using only the shorter test times), the new generalized model had better predictive capability at most temperatures.
1 Introduction
The selection of materials for high temperature applications in power plants is typically based on the requirement that creep failure should not occur under the prevailing operating conditions during plant lives of approximately 30 years. Although complex stresses and temperatures are often encountered by materials used in power generation, design decisions are generally made on the basis of allowable tensile creep strength. This strength is commonly taken to be 67 pct of the average stress (up to 1088 K).[1] At present, protracted and expensive test programs lasting 12-15 years are necessary to determine the required long-term strengths and lives. A reduction in this 12-15 years of ‘materials development cycle’ was therefore defined as the No. 1 priority in the 2007 UK Energy Materials—Strategic Research.[2]
With the aim of reducing this development cycle, a new group of parametric creep models has been developed in recent years that are characterized through their use of a normalized stress (defined as the ratio of stress to tensile strength, σ/σ_{TS}) for the determination of safe life. The rationale behind this new group of creep models is that by definition, failure will be instantaneous when stress is equal to a material’s tensile strength. Then when the material is subjected to a stress that is an infinitesimally small fraction of its tensile strength, the material should remain intact for a very long period of time. That is, t_{f} must vary from 0 towards ∞ as σ/σ_{TS} varies from 1 towards 0. Unfortunately, the rate at which this happens is not fully understood, and so this group of models assumes that the relationship between σ/σ_{TS} and t_{f} (at a fixed temperature) is given by an inverted S-shaped curve.[3,4] The models within the group are then differentiated by the mathematical function used to describe this inverted S-shaped curve, and so consequently these models can end up producing very different safe life predictions.
This paper aims to tackle this problem by specifying a generalized model that nests the creep model first put forward by Wilshire and Battenbough[3] and the model proposed by Yang et al.,[4] i.e., these two models are special cases within this more general model. Within such a framework, it is then possible to use some basic statistical tests to allow experimental creep data to determine the correct shape of the inverted S-shaped relationship between σ/σ_{TS} and t_{f}. Once identified, this shape can be used to obtain safe life predictions that are compatible with experimental data, rather simply using some ad hoc functional form for the S-shaped relationship. To achieve this aim, the paper is structured as follows. The next section gives a brief review of old and new parametric creep models and this is followed by a section deriving the generalized creep model together with a statistical test for the form of the inverted S-shaped curve. The next section then estimates the unknown parameters of these models together with a short discussion on likely failure mechanisms that could account for the values of these estimated parameters. There then follows a subsection comparing the predictive accuracy of the Wilshire, Yang et al. and generalized creep models when they are applied to experimentally derived failure times for 316H stainless steel. The paper ends with a conclusion section that outlines some areas for future work.
2 Parametric Creep Models
2.1 A Brief History
Summary of Some Parametric Creep Models for Predicting Time to Failure
Model Name | Single Comprehensive Equation |
---|---|
Orr–Sherby–Dorn[9] (OSD) | \( \ln \left( {t_{\text{f}} } \right) = a_{0} + a_{1} \ln (\sigma ) + a_{3} \frac{1}{T} \) |
Larson–Miller[8] (LM) | \( \ln \left( {t_{\text{f}} } \right) = a_{0} + a_{3} \frac{1}{T} + a_{4} \frac{\ln (\sigma )}{T} \) |
Minimum Commitment[10] (MC) | \( \ln \left( {t_{\text{f}} } \right) = a_{0} + a_{1} \ln (\sigma ) + a_{2} T + a_{3} \frac{1}{T} + b_{1} \sigma + b_{1} \sigma^{2} \) |
Manson and Haferd[11] (MH)* | \( \ln \left( {t_{\text{f}} } \right) = a_{0} + a_{1} f(\sigma ) + a_{4} \frac{f(\sigma )}{T}\left[ {T - a_{5} } \right] \) |
Manson and Brown[12] (MB)* | \( \ln \left( {t_{\text{f}} } \right) = a_{0} + a_{1} f(\sigma ) + a_{4} \frac{f(\sigma )}{T}\left[ {T - a_{5} } \right]^{{a_{6} }} \) |
Soviet Model 1[7] (SM1) | \( \ln \left( {t_{\text{f}} } \right) = a_{0} + a_{2} \ln (\sigma ) + a_{3} \frac{1}{T} + a_{4} \frac{\sigma }{T} \) |
Wilshire[3] (W) | \( \ln \left( {t_{\text{f}} } \right) = a_{0} + a_{1} \ln \left[ { - \ln \left( {\sigma /\sigma_{\text{TS}} } \right)} \right], + a_{3} \frac{1}{T} \) |
Yang et al.[4] (Y) | \( \ln \left( {t_{\text{f}} } \right) = a_{0} + a_{1} \ln \left[ {\sigma /\left( {\sigma_{\text{TS}} - \sigma } \right)} \right] + a_{3} \frac{1}{T} \) |
2.2 Normalized Stress Creep Models
2.2.1 Isothermal
Now, when k = 1, r^{*} = \( \ln \left[ {\frac{1}{r} - 1} \right], \) and so Eqs. [4a] and [4b], simplify into Eqs. [2a] and [2b] with α = α_{L} and β = β_{L}. Thus k = 1 results in the model proposed by Yang et al. Further, as k → ∞, r^{*} → \( \ln \left[ { - \ln \left( r \right)} \right] \) and so then Eqs. [4a] and [4b], simplify into Eqs. [3a] and [3b] with α = α_{W} and β = β_{W}. Thus, k tending to ∞ results in the model by Wilshire. Other inverted S-shaped curves exist for k values between these two limits. The different inverted S-shaped curves associated with the different values for k are shown in Figure 1(c), where k is rescaled to fall within the limits 0 to 1 through the transformation k^{*} = k^{−0.5} (so that when k = 1, k^{*} = 1 and as k → ∞, k^{*} → 0). In this rescaling, Yang et al.’s model corresponds to k^{*} = 1 and Wilshire’s model emerges when k^{*} = 0. Figure 1(c) shows the different shapes produced by different values for k^{*} in Eq. [4c] when β = 3 and α = 12.
2.2.2 Temperature compensated
Again, Yang et al.’s model corresponds to k^{*} = 1 and Wilshire’s model emerges when k^{*} = 0.
\( \sigma^{\text{c}}_{\text{j}} \) are critical values for the normalized stresses and so fall between 0 and 1. In partitioning, there are p creep regimes that occur in distinct ranges for the normalized stress and the p versions of Eq. [5c] then apply to each regime. Typically, p varies between 1 and 4 depending on the material being studied and the test conditions present in the creep data base on that material. Some of the first studies to appear in the literature include applications to Copper[3] and 1Cr-1Mo-0.25 steel[13] where p was found equal to 2 and where the activation energy was the same either side of \( \sigma_{1}^{\text{c}} \) (so \( Q_{\text{cj}} = Q_{c} \)). Later studies by Whittaker and Wilshire using 2.25Cr-1Mo steel[14] found p equal to 3 and again the activation energy was the same either side of \( \sigma_{1}^{\text{c}} \)and \( \sigma_{2}^{\text{c}} \) and Evans[15] found p = 2 but with a varying activation energy either side of \( \sigma_{1}^{\text{c}} \) when studying a 12Cr steel. Finally, Whittaker et al.[16] found p = 2 with varying activation energies around \( \sigma_{1}^{\text{c}} \) when studying a particular grade of Waspaloy.
2.2.3 Estimation and a statistical test for the stress function (i.e., for the value for k)
3 The Data
Variation of Tensile/0.2 Pct Proof Stress (MPa) with Temperature (K) and Batch
Temperature | Batch AaA | Batch AaB |
---|---|---|
873 | 413/120 | 405/114 |
923 | 362/123 | 357/128 |
973 | 294/109 | 292/116 |
1023 | 242/110 | 239/108 |
1073 | 190/92 | 188/101 |
1123 | 151/85 | 151/92 |
4 Results
4.1 The Wilshire Model (k^{*} = 0)
Whittaker et al.[19] carried out a detailed study of this material using the Wilshire model and found there to be two distinct creep regimes (p = 2). This finding was obtained using steel tubes, plate, and piping material rather than just the tube data used in this paper. Sticking with this finding, and using the estimation techniques described above when k^{*} = 0 (so leading to the Wilshire model), resulted in the largest R^{2} value being associated with \( \sigma_{1}^{\text{c}} \) = 0.41 and this is similar in value to that found by Whittaker et al. The resulting least squares estimates of the models parameters were then found to be
4.2 The Model by Yang et al. (k^{*} = k = 1)
Again working with p = 2, and using the estimation techniques described above when k^{*} = 1 (so leading to the Yang et al. model), resulted in the largest R^{2} value being associated with \( \sigma_{1}^{\text{c}} \) = 0.38—only marginally lower than when using the Wilshire model. The resulting least squares estimates of the models parameters were then found to be
The parameter estimates shown in Eq. [11] imply that when σ/σ_{TS} > 0.38, α_{w} =exp(−25.145) = 1.199E-11, β_{w} = 1/5.528 = 0.181 with an activation energy of 305 kJmol^{−1}. But at a normalized stress below 0.38 the activation energy drops to 181 kJmol^{−1} with α_{w} =exp(−9.763) = 5.754E-05 and β_{w} = 1/3.737 = 0.2676. So these activation energies are not dissimilar to that obtained when using the Wilshire model. The estimated parameters shown in Eq. [11] are based on all the experimental failure times. All the parameters are statistically significant at the 5 pct significance level and the student t values for δ_{0} through to δ_{2} also revealed that all these parameters are significantly different from zero at the 5 pct significance level. Consequently, at a normalized stress of 0.38 there is a real change in the value for α and the activation energy, but also and unlike in the Wilshire model, a real change in the slope parameter β. This structure is shown visually in Figure 3(b), where the fitted line has a steeper slope after the break point. The overall R^{2} value of 97.08 pct is a little higher than that associated with the Wilshire model.
4.3 The General Model
The parameter estimates shown in Eq. [12] imply that when σ/σ_{TS} > 0.41, α =exp(-23.971) = 3.886E-11, β = 1/6.413 = 0.1559 with an activation energy of 306 kJmol^{−1}. But at a normalized stress below 0.41 the activation energy drops to 182 kJmol^{−1} with α =exp(−9.223) = 9.874E-05 and β = 1/5.164 = 0.1936. So these activation energies are not dissimilar to that obtained when using the Wilshire model and the model by Yang et al. The estimated parameters shown in Eq. [12] are based on all the experimental failure times. All the parameters are statistically significant at the 5 pct significance level and the student t values for δ_{0} through to δ_{2} also revealed that all these parameters are significantly different from zero at the 5 pct significance level. Consequently, at a normalized stress of 0.41 there is a real change in the value for α and the activation energy, but also and unlike in the Wilshire model, a real change in the slope parameter β. This structure is shown visually in Figure 3(c), where the fitted line has a steeper slope after the break point. The overall R^{2} value of 97.16 pct is a little higher than that associated with the Wilshire model.
4.4 Deformation Mechanisms Behind the General Model
Whittaker et al.[19] have found for this material that the processes responsible for creep change at stresses above and below the yield stress. They found that when the stress exceeds the yield stress, creep occurs by movement of new dislocations generated during the plastic component of the initial strain on loading (ε_{0}), whereas creep takes place by grain boundary zone deformation when fully elastic ε_{0} values only are recorded when the stress is below the yield stress. This is confirmed by the results of this paper, where below a normalized stress of 0.41, all the test stresses are below the 0.2 pct proof stresses (that are approximately equal to the yield stress) shown in Table I. This suggests that for these materials all creep models based on the normalized stress should be of the partitioned variety with one break around the yield stress.
Whittaker et al.[19] further found that under all stress/temperature conditions, failure occurred by cavitation. If cavity development is strain controlled, then the creep life is determined principally by boundary zone deformation. Assuming that a comparable level of grain boundary zone deformation is necessary to cause failure, the contribution of grain deformation to the overall creep rate decreases with decreasing stress. If this is the case then this should be accompanied by a decrease in the strain at rupture as well. This is what is actually observed in creep data sheet 14B[18] where, for example, at a temperature of 1023 K the percentage elongation at rupture drops from 88 pct as a normalized stress of 0.3 to just 45 pct at a normalized stress of 0.1. The reader is referred to Whittaker et al.[19] for further details on creep mechanisms and microstructure.
4.5 Predictive Performance
Predictive Performance of Various Models of Creep Life
Model | k* = 1 (Yang et al.) | k* = 0 (Wilshire) | k* = 0.70 (General) | |||
---|---|---|---|---|---|---|
Temperature (K) | RMPSE (Pct) | U | RMPSE (Pct) | U | RMPSE (Pct) | U |
873 | 62.85 | 0.31 | 76.58 | 0.36 | 60.05 | 0.30 |
923 | 58.90 | 0.29 | 62.27 | 0.31 | 58.39 | 0.29 |
973 | 30.72 | 0.15 | 32.81 | 0.15 | 29.39 | 0.13 |
1023 | 45.97 | 0.28 | 51.51 | 0.41 | 45.82 | 0.33 |
1073 | 45.01 | 0.23 | 56.01 | 0.35 | 46.69 | 0.28 |
All | 46.49 | 0.25 | 53.80 | 0.39 | 46.53 | 0.27 |
Table III also reveals some interesting differences in predictive performance at specific temperatures. Based on a comparison of Theil’s U, all models predict failure times best at a mid-temperature of 973 K, with the general model performing slightly better than the other two nested models. All models predict failure times worst at the temperatures of 873 K and 1023 K. At 873 K the general model is the better of the three models, while at 1023 K the model by Yang et al. is the better performing one, with the general model performing much better than the Wilshire model (U = 0.3 < 0.36) but about as well as the model by Yang et al. Actually, at the highest two temperature, the model by Yang et al. has better U statistic.
5 Conclusions
- 1.
There is a large and statistically significant change in the activation energy for 316H stainless steel at a normalized stress of 0.41 (from around 306 kJmol^{−1} to around 182 kJmol^{−1}).
- 2.
This change is attributed to stress test conditions being below the yield stress when the normalized stress is below 0.41, and so there is a change away from dislocation movement within grains to movement within grain boundaries—where the activation energy is considerable lower.
- 3.
The Wilshire model[3] was found to be incompatible with the experimental data at the 5 pct significance level, and while the model by Yang et al.[4] was compatible with the data, it was not a model that was most supported by the data. Such a model was the generalized equation developed in this paper with k ≈ 2.
- 4.
The Wilshire model[3] produced life time predictions at test conditions leading to lives in excess of 5000 hours that were less accurate than those obtained from the generalized model and the model put forward by Yang et al.[4] Over all test temperatures the generalized model produced the most accurate life time predictions.
Areas for future work include the application of this new generalized model to other materials within the NIMS creep data base to see if a k value of around 2 is appropriate for these materials as well. It may turn out to be the case that k (like the Monkman–Grant parameter M) is materials specific. When then comparing the predictive accuracy of this new model with existing models in the literature, it would be important to use the same number of implied creep regimes and use the same data sets/test conditions.
Notes
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