Extrapolation

Abstract

The extrapolation of creep data to longer times is technically very important. The traditional way of extrapolating creep rupture data is to use time temperaturer parameters (TTPs). In this way data from several test temperatures are combined to a single master curve that can be used to assess rupture strengths at long times. Recently, there is much focus on machine learning techniques (neural networks, NNs). Both types of procedures can generate accurate results, but a detailed analysis is required. A good way to assess the quality of the results is to use the post assessment tests (PATs) developed by ECCC. Without such tests arbitrary results can be obtained. They are important for both TTPs and NNs. It has been shown that by putting requirements on the derivatives of the creep rupture curves, the PATs can more or less automatically be satisfied. In addition, the error in the extrapolated values should be estimated. Using the basic creep models presented in this book, prediction of rupture strength and ductility can be made in a safer way. It is demonstrated for Cu that accurate extrapolation of many order of magnitude in the creep rate can be made, which is never possible with empirical models.

14.1 Introduction

Many types of high temperature plants have a long design life. Modern fossil fired power plants are designed for 20–40 years of operation and nuclear power plants for 60–80 years. To ensure safe operation accurate creep data and other time dependent material property values must be available that cover such long design lives. Direct measurement of creep data for these extended times is not practical for several reasons. It is expensive and technically complicated to perform long term creep tests. Disturbances can take place that destroy the test results. In addition the material being tested can be outdated before the test is finished. Instead numerical time extrapolation of the experimental results must be carried out. Extrapolation of creep rupture data have been performed for many years. Long term values have been predicted from shorter time experimental data. It should be emphasized that systematic procedures must be applied in order to obtain accurate results.

The most common approach is to use time-temperature parameters (TTPs). With the help of a TTP, creep rupture data at several temperatures are combined to a single master curve, where the creep stresses are shown as a function of the TTP. From the master curve, time extrapolated values at lower temperatures can be derived from master curve values at higher temperatures. In this way, extrapolated values at most temperatures can be found by interpolation from the master curve. By being able to interpolate reduces the error in the analysis significantly. Many different TTPs exist, see [1]. It started with the Larson-Miller method in the 1950ties. Some TTPs will be listed in Sect. 14.2. Although many adjustable parameters are typically involved in the fit of the creep rupture data to the master curve, the TTPs are in general simple to use since the adjustable parameters can be obtained with linear algebra. The application of TTPs will be demonstrated in Sect. 14.2.

There are other types of methods than TTPs for extrapolation of creep rupture data. Two methods can be mentioned where also the form of the TTP is adjustable. The minimum commitment method [2] and the free temperature model [3]. In general it is difficult to reach more than a factor of three in time for accurate extrapolation with statistical methods like TTPs. With the free temperature model a factor of ten can be reached in many cases. This has been demonstrated for austenitic stainless steels. Other approaches for extended extrapolation also exist [4, 5]. Two other groups of methods should also be mentioned. With algebraic methods a creep stress versus rupture time equation is the starting point. This type of approach was popular in the former Soviet Union [6]. The German graphical techniques have successfully been used to perform accurate extrapolations [7]. Another graphical method is a former ISO-standard [8]. A more recent method that is popular is due to Wilshire [9]. In this approach the creep data is normalized with the tensile strength. This is valuable if the amount of scatter in the creep data can be reduced in this way.

It was recognized a long time ago that the extrapolation results significantly depended on the chosen TTP and on the degree of the polynomial used in the fit of the master curve [10]. Since different analysts often have different opinions about the choice of method, it means that the results are operator dependent and this was not considered to be an acceptable situation. For this reason, the European Collaborative Creep Committee (ECCC) was initiated. Within ECCC, a framework of systematic procedures has been developed that generate more consistent results. The ECCC program started in 1992 and is still ongoing in 2022. It has been known that it is not possible to pin point a specific TTP to be more advantageous than others. Instead, post assessment tests (PATs) have been formulated that are used to check that the applied method does not show unphysical behavior and that the extrapolated creep rupture strength values are not sensitive to limited changes in the input data. Due to the flexibility of the polynomial fit to the master curve, unphysical behavior often appears. The PATs represented a major step forward when they were introduced [11, 12]. Creep rupture data for many steels and alloys proposed by ECCC are now in European standards. However, there are still short comings of the proposed procedures and PATs. None of them are based on derivations from physical principles and in the past there has been no way of estimating the error in the results.

To improve the situation for these remaining limitations, two things have recently been proposed [13]. The first one is to put requirements on the first and second derivatives of the predicted creep rupture curve. The derivatives should be negative. In this way a correct physical behavior is often automatically achieved and the flexing of creep rupture curves is avoided. The second item is that principles for estimating the extrapolation error have been proposed. The use of these findings will be discussed in Sect. 14.3.

The use of a Neural Network (NN) is an alternative to TTP. NN is also called Artificial Neuron Network and is a part of deep learning. NNs are extensively applied in the development of Artificial Intelligence (AI). NNs are used in a large number of industrial applications such as autonomous driving, signal processing, risk assessment, pattern recognition of images, missile control, autopilots, etc. [14]. In contrast to TTP no basic parameter model is chosen. NNs consist of a number of functions, neurons, with adjustable parameters. NNs are sufficiently flexible that they can adapt to many types of functional dependence. Fitting of data with empirical models has found new interest with the wide spread use of NNs. NNs are in principle limited to interpolation of data. However, in practice they are used for extrapolation as well.

Complicated creep rupture behavior can be simulated with NNs even when special degradation mechanisms are present [15, 16]. The fit of the functional dependence implies that the number of adjustable parameters is larger than when using TTPs, often much larger. Liang et al. [17] have analyzed the creep rupture life of 9–12%Cr steels with NN; Ghatak et al. have modeled the creep rupture curve of HP40Nb steels with NN successfully [15]. Adductive NN has also been suggested for creep rupture prediction of 9–12%Cr steels by Wang et al. [18].

In fact, the debate for the extrapolation capability of NN has lasted for a long time. Including physical principles in the common NN is a good way to improve the extrapolation. This is called physics-constrained [19] or physics-informed NN [20] (PCNN or PINN). The extrapolation can be safely conducted by adding constraints or prior knowledge to the common NN.

Extrapolation from NN results must be common in industry. In spite of this it is difficult to find procedures for error analysis in the literature. The common method is to make a regression analysis between the predicted values and the source data and to determine the standard deviation between predictions and the observations. As will be shown in Sect. 14.3, this is not adequate at all for creep rupture data and probably not for many other types of applications either. For this reason new types of error analysis have been formulated recently, Sect. 14.3.

As was mentioned above, the introduction of requirements on the first and second derivative of the creep rupture curves can significantly simplify and improve the results of the extrapolation analysis with TTPs. For example, many of the ECCC PATs were found to be automatically satisfied if these requirements were introduced. It will be shown in Sect. 14.2 that these requirements are equally essential when NN-based modeling is carried out. This will be demonstrated in Sect. 14.3.

The methods discussed above are all empirical models. None of TTPs commonly used have been derived from basic physical principles. Both TTPs and NNs are just flexible expressions that can easily be fitted to the observations. Empirical models have the drawback that a large amount of data must be available to make safe predictions. Unless the amount of data is sufficient, the models cannot be used to identify the operating mechanisms [21,22,23]. To fully understand the underlying mechanisms, fundamental models should be applied where the contributions from different mechanisms are derived from basic physical principles. Such models have been derived for dislocation strengthening as well as precipitation and solid solution hardening. They have been used to predict the creep rate of austenitic stainless steels [21, 22, 24, 25]. Fundamental models have also been formulated for cavity nucleation and growth. By applying these models, brittle rupture can be modeled. Both creep rupture strength and ductility have successfully been computed for copper and austenitic stainless steels [22, 23, 26,27,28]. Some of these models will be discussed in Sect. 14.4. Fundamental models are of importance for example to meet the full design life of modern nuclear reactors with a planned design life of 60–80 years.

14.2 Empirical Extrapolation Analysis

14.2.1 Basic TTP Analysis

Extrapolation from a single curve gives quite an uncertain result in particularly if there are no requirements on the derivatives. It was recognized long ago that by combining creep data from several temperatures, the accuracy could be much improved. The size of the improvement will be analyzed in Sect. 14.3. The classical approach is to use a time temperature parameter (TTP). It is a function of absolute temperature and time. A polynomial in the logarithm of the creep stress is fitted to the TTP in such a way that the creep rupture data fall on a single curve, the master curve. A logarithm with the base 10 is most frequently used in studies on extrapolation and that practice will be followed in this chapter. The coefficients in the polynomial and a few constants in the TTP represent the adjustable parameters involved. For many TTPs the adjustable parameters can be determined with the help of linear algebra and are consequently easy to find. Many TTPs are available. Sources were given in Sect. 14.1. Examples of TTPs are listed in Table 14.1.

Table 14.1 Examples of time-temperature parameters (TTPs)

For the listed TTPs, T is the temperature, t the time, and C_NN and log(t_a) are adjustable parameters. The parameters T_a and r are given predefined values in general.

The TTPs in Table 14.1 have been used for a number of decades. References to the original sources of the TTPs can be found in [1, 29, 30]. The extrapolation results depend on the chosen TTP and the degree of the polynomial that is used to fit the master curve. Consequently, it is critical how these quantities are selected. There is general experience that it is not possible to find the optimal TTP for a given analysis in advance [10]. Instead, a number of TTPs and polynomial degrees have to be tested to find a satisfactory solution.

With an example it will be shown how extrapolation with a TTP can be performed and how the result is checked and analyzed. A creep resistant 17Cr12NiTi austenitic stainless steel with the common designation 321H will be studied. The experimental data are taken from NIMS’ large collection. The studied creep data are shown in Fig. 14.1. Larson-Miller TTP is applied and the polynomial degree is selected to 6. The fitted master curve is shown in Fig. 14.2.

Fig. 14.1

Comparison of experimental and modeling (interpolated and extrapolated) values for creep rupture of the 17Cr12NiTi austenitic stainless steel (321H); Larson-Miller TTP model is used; ‘pred’ represents the interpolated values, ‘epol’ the extrapolated values, and ‘exp’ the experimental data at five test temperatures from 550 to 750 °C. Experimental data from NIMS [31]

Fig. 14.2

Master curve for the creep rupture data in Fig. 14.1. A polynomial of degree 6 in log(σ_creep) versus TTP (dashed curve --) is fitted to the experimental data (pluses +) at the five test temperatures 550–750 °C

The model values in Fig. 14.1 are taken directly from the master curve in Fig. 14.2. The part of the master in the range of the data is used for all temperatures except the highest one where it is necessary to take into account the extrapolated part of the master curve for large TTP values. The advantage of using a TTP is already obvious from this description. The extrapolated values at all temperatures except the highest one can be determined by interpolation along the master curve. This gives a more accurate result than when being forced to extrapolate from single curves. As a consequence the extrapolated values at the highest temperature involve a potentially higher error than the predicted values at other temperatures. The first requirement of a successful analysis is that good fit to the data is obtained, which should be directly evident from the comparison with source data (Fig. 14.1) and for the master curve (Fig. 14.2).

In many cases the analysis is performed without additional requirements. This often means that a number of attempts have to be made before smooth curves and a good fit can be obtained. However, the results in Figs. 14.1 and 14.2 have been determined with additional requirements. All creep strength versus rupture time representations with double logarithmic scale (which are referred to as creep rupture curves) must have a non-positive derivative, in practice a negative derivative. Otherwise the rupture time would not increase with decreasing creep strength which is unphysical. In addition, the absolute slope is not arbitrarily large.

The second derivative of creep rupture curves is in most cases negative, i.e. the absolute value of their slope increases with rupture time. The reason is that there is microstructural degradation for example through coarsening of precipitates. The exception is so-called sigmoidal behavior with a slightly S-shaped curve. Such rupture curves are typically the result of complex precipitation during the creep tests and there is a time period when the strength is not degrading but increasing. It is quite unusual that the sigmoidal appearance has a temperature dependence that is consistent with the creep rupture and can be described with a TTP so this case is not considered here. In fact, sigmoidal forms are suitable to handle with the type of basic models that are presented in Sect. 14.4. As a consequence it can be assumed that also the second derivative of the rupture curve is negative.

When computing the results in Fig. 14.1 and in Fig. 14.2, constraints on the creep rupture curves have been taken into account. The conditions are formulated as

$$ - m = \frac{{d\log t_{{\text{R}}} }}{d\log \sigma } \le - 1.5;\;\frac{{d^{2} \log t_{{\text{R}}} }}{{d\log \sigma^{2} }} \le 0 $$

(14.1)

$$ \frac{{d\log t_{{\text{R}}} }}{d\log T} \le 0 $$

(14.2)

As pointed out above it is a physical requirement that the first derivative of creep rupture curves is negative

$$ \frac{d\log \sigma }{{d\log t_{{\text{R}}} }} < 0 $$

(14.3)

$\frac{{d\log t_{{\text{R}}} }}{d\log \sigma }$ is the inverse of $\frac{d\log \sigma }{{d\log t_{{\text{R}}} }}$, and consequently the former derivative is also negative.

In Eq. (14.1) the second derivative can be expressed as the second derivative of the rupture curve

$$ \frac{{d^{2} \log t_{{\text{R}}} }}{{d\log \sigma^{2} }} = - \frac{{d^{2} \log \sigma }}{{d\log t_{{\text{R}}}^{2} }}/\left\{ {\frac{d\log \sigma }{{d\log t_{{\text{R}}} }}} \right\}^{3} $$

(14.4)

Equation (14.4) can be obtained with elementary calculus. Since the first derivative is assumed to be negative it follows that the two second derivatives have the same sign. Thus, the second criterion in Eq. (14.1) is verified. This ensures that the creep rupture curves have a negative second derivative. The second derivative of the rupture time is somewhat easier to compute than the second derivative of the creep stress.

With a Norton type of approach both the creep rate $\dot{\varepsilon }_{\sec }$ and the rupture time t_R can be represented with stress exponents (power-law creep)

$$ \dot{\varepsilon }_{\sec } = A_{{\text{N}}} \sigma^{{n_{{\text{N}}} }} ;\;t_{{\text{R}}} = B_{{\text{R}}} \sigma^{ - m} $$

(14.5)

The constant m is the inverse slope of the flow curve with a minus sign, see Eq. (14.1). The Modified Monkman-Grant equation relates the rupture strain ε_R to the strain rate and the rupture time [32]

$$ \dot{\varepsilon }_{\sec } t_{{\text{R}}} = C_{{{\text{MMG}}}} \varepsilon_{{\text{R}}} $$

(14.6)

Sundararajan lists values of C_MMG for a number of materials [33]. The values lie in the interval 0.1–0.64. If only secondary creep contributed to the rupture strain the constant C_MMG would be equal to unity. Another constant, the rupture ductility factor λ_R, is often used

$$ \lambda_{{\text{R}}} = \frac{{\varepsilon_{{\text{R}}} }}{{\dot{\varepsilon }_{\sec } t_{{\text{R}}} }};\;\lambda_{{\text{R}}} = 1/C_{{{\text{MMG}}}} $$

(14.7)

By comparing the definition of λ_R with the modified Monkman-Grant equation, it is evident that λ_R is just the inverse of the modified Monkman-Grant constant C_MMG. The value of the rupture ductility factor has been analyzed in more detail for modern 9Cr1Mo steels. The result is that λ_R ≈ 5 and increases with rupture time [34].

By combining Eqs. (14.5)–(14.7) and taking the logarithmic derivative, one finds that

$$ m - n_{{\text{N}}} = \frac{{d\log \lambda_{{\text{R}}} }}{d\log \sigma } - \frac{{d\log \varepsilon_{{\text{R}}} }}{d\log \sigma } $$

(14.8)

Since the rupture strain often increases with stress and λ_R decreases with stress for some alloys, the m value is smaller than the n_N value at least at lower stresses. ECCC has suggested a lower limit of m of 1.5, cf. Eq. (14.1). This is a characteristic feature of creep resistant steels. If the absolute value of the slope would be still higher and the m value lower, the steel would not be safe to use.

If the Omega model is satisfied for tertiary creep, the following relation is available from Eq. (12.37) if the reasonable assumption that ε_diff is large is made

$$ t_{{\text{R}}} \dot{\varepsilon }_{\min } = \frac{{e^{{ - n_{\Omega } \varepsilon_{0} }} f_{{{\text{red}}}}^{{n_{{\text{N}}} }} }}{{n_{\Omega } }}\,\, $$

(14.9)

If this equation is combined with the expression for the rupture strain, Eq. (12.38), one finds that

$$ t_{{\text{R}}} \dot{\varepsilon }_{\min } = \varepsilon_{{\text{R}}} f_{{{\text{red}}}}^{{n_{{\text{N}}} }} (1 - f_{{{\text{red}}}}^{{n_{{\text{N}}} }} )\,\, $$

(14.10)

This recovers obviously the modified Monkman-Gran relationship (14.6) with a constant that depends on the size of the imperfection of the specimen. Since ε₀ is about 0.01 and $f_\text{red}$ ≈ 0.99, the right hand side of Eq. (14.9) is often close to 1/n_Ω. From Eqs. (14.6), (14.7) and (14.9) one then finds that

$$ \frac{1}{{n_{\Omega } }}\, \approx C_{{{\text{MMG}}}} \varepsilon_{{\text{R}}} \, = \frac{{\varepsilon_{{\text{R}}} }}{{\lambda_{{\text{R}}} }} $$

(14.11)

This provides another way of estimating the slope n_Ω of log(strain rate) versus strain curves in the tertiary stage. This result is consistent with the finding that n_Ω increases with increasing rupture time for modified 9Cr1Mo steels (end of Sect. 12.4.2) in the same way as ε_R/λ_R does [34].

The first and second derivatives for the predicted rupture curves in Fig. 14.1 are illustrated in Fig. 14.3. The first derivative is given as the m value, Eq. (14.1).

Fig. 14.3

The first a and second b derivatives of the rupture time as a function of creep stress for the creep rupture curves at the five test temperatures 550–750 °C in Fig. 14.1. The m-value is minus the inverse of the first derivative (PAT 1.3)

The conditions in Eq. (14.1) require that the m value is larger than 1.5 and that the second derivative is not positive. These conditions are obviously fulfilled. The criterion in Eq. (14.2) ensures that the creep rupture curves at different temperatures do not cross. This criterion is usually not difficult to meet.

14.2.2 The ECCC Post-assessment Tests

As mentioned above the results of the extrapolation analysis depend on the chosen TTP and degree of the polynomial that is used to fit the master curve. The European Creep Collaborative Committee (ECCC) recognized that additional tools are needed to improve the possibility to select amongst all the alternatives. They proposed a number of Post Assessment Tests (PATs) that should be applied when the predicted rupture strengths have been generated. There are three sets of PATs [11, 12, 35]. The PATs are listed in Table 14.2. PATs 1.1–1.3 check the physical realism of the predicted creep rupture curves. A good fit to the data is required, and the derivative of the rupture curve should follow the measured values. PATs 2.1 and 2.2 assess that the result is uniform and unbiased and that data at specific stresses, temperatures, or casts do not behave in a different way from the rest of the prediction. The analysis is repeated in PATs 3.1 and 3.2 with some of the long term data removed to verify the stability of the prediction.

Table 14.2 ECCC post-assessment tests (PATs) for creep rupture data extrapolation (reproduced from [36] with permission from Elsevier)

The match of the predicted to the experimental data is illustrated in Figs. 14.1 and 14.2. The fit is fine which means that PAT 1.1 is satisfied. In Fig. 14.3, m > 1.5 and consequently PAT 1.3 is fulfilled.

In Fig. 14.4 the results of PAT 1.2 and PAT 3 are shown. In Fig. 14.4a there is no bending back or crossing of curves in spite of the wide range of stresses and rupture times and the fine temperature spacing. PAT 1.2 is verified. It should be demonstrated that the results are not sensitive to the data for the longest test times. This is studied with the help PAT 3.1 and PAT 3.2. In PAT 3.1 the same analysis is performed again but with 50% of the data points with rupture time larger than a tenth of the maximum observed rupture time randomly removed (culled). In PAT 3.2, the analysis is repeated again but this time with 10% of the data points with the lowest stresses culled. As can be seen in Fig. 14.4b only at the highest temperature there is a significant difference between the culled and the unculled curves. The difference should be less than 10% according to the ECCC recommendation and this is satisfied.

Fig. 14.4

a Representation of predicted creep rupture curves over an extended range of rupture times for a fine distribution of temperatures (PAT 1.2); b influence of culling of data on the predicted rupture curves; According to PAT 3.1 long term data are removed in the analysis, and in PAT 3.2 the data points with the lowest stresses at each temperature are not included

In Fig. 14.5 regression plots between the experimental and the predicted rupture times are shown. The purpose of this type of diagram is to demonstrate that the predicted values are close to the observed ones at both low and high stresses. Otherwise the regression line would deviate from the 1:1 line. Two sets of border lines are marked in the Figure. There are lines a factor of 2 above and below the 1:1 line. If there is limited scatter in the data set, the result would fall inside these lines, but that is typically not the case. The second set is located 2.5 σ_std above and below the 1:1 line. If the data show a normal distribution, no more than 2% of the data should fall outside these lines. This is obviously fulfilled in Fig. 14.5. The regression coefficient should be higher than 0.78 (0.92 in Fig. 14.5a). The regression plots are used to estimate the regression error. This will be further discussed in Sect. 14.3.

Fig. 14.5

Regression plots between observed and predicted rupture times. The regression line is marked. The black line represents a factor of 2 above and below the mean line. The red line describes a factor of 2.5 times the standard deviation above and below the mean line; a all temperatures (PAT 2.1); b Regression plots for individual temperatures

14.2.3 Use of Neural Network (NN)

Creep rupture data can be analyzed with a neural network [37]. It is sufficient to use a simple NN. The NN that has been applied is illustrated in Fig. 14.6. There are a hidden layer and an output layer. In the hidden layer there are 3–10 neurons (3 in the Fig. 3) and in the output layer one.

Fig. 14.6

Schematic structure of the neural network used to fit the creep rupture data (with Matlab)

Each neuron represents transfer functions, one for each input and one for each output. The type of transfer function is sigmoidal in the input layer and linear in the output layer. There are weight and base parameters in the transfer functions. They are used as adjustable parameters in the fitting process. Well established procedures for finding the values of the adjustable parameters are available [14]. The fitting process for NNs is usually called training. Of the experimental data 70% were set for training, 15% for testing, and 15% for validation. The Levenberg-Marquardt back propagation method was applied in the training of the network to minimize the Mean Squared Error (MSE). A random number stream was used in the NN fitting process to fix the output.

With n_input inputs, n_out outputs, and n_neur neurons in the hidden layers, the number of adjustable parameters (weight and base parameters) n_adj is

$$ n_{{{\text{adj}}}} = (n_{{{\text{input}}}} + 1 + n_{{{\text{out}}}} )n_{{{\text{neur}}}} + 1 $$

(14.12)

For creep rupture, the test temperatures and the stresses are the two inputs and the rupture times the output. The flexible NN model should give a good representation of the rupture times. According to Eq. (14.12), there are 13 adjustable parameters with 3 neurons in the hidden layer; and there are 41 with 10 neurons. In the framework of creep rupture, 13 adjustable parameters is already a large number [12]. With more neurons the fitting of the data is improved but overfitting may quickly be the result.

For precisely the same reasons as in the TTP analysis, the requirements on the derivatives in (14.1) and (14.2) should be fulfilled. This ensures that flexing and other unphysical behavior of the predicted creep rupture curves are avoided. Unfortunately expressions for the derivatives are not readily available in NN software. For this reason expressions for derivatives have been derived. Since the expressions are complex, the derivation has been placed in an Appendix (Sect. 14.6). An alternative is to repeat the computations many times until the constraints and other requirements are fulfilled. This is referred to as soft constrained machine learning [38].

The application of a NN model is illustrated in Fig. 14.7 for the austenitic stainless 18Cr10NiCu steel Super304H. It has also the common designation 304HCu. The predicted rupture strengths are compared with the observations.

Fig. 14.7

Comparison between observed and predicted rupture times for the creep-resistant austenitic stainless steel Super304H at four test temperatures from 600 to 750 °C. The prediction of the rupture times was made with a constrained NN model. “pred” is the modeling results within the experimental range; “epol” is the extrapolated results; “exp” is the experimental data of Super304H taken from [39]. Reproduced from [37] with the permission of Taylor & Francis

A good fit was possible to obtain in Fig. 14.7. It should be emphasized that many runs with different stream numbers were needed before a satisfactory result was obtained. The stream number fixes the random number generator so the same run can be repeated. In this way the adjustable parameters in the NN model are initiated with the same values. The initial values have obviously a significant effect on the result. The requirements on the derivatives simplify the search for a good fit that behaves in a physical correct way.

The derivatives of the rupture curves are presented in Fig. 14.8. In this case the derivatives are given as a function of rupture time instead of stress as in Fig. 14.3. But the same message is provided. The derivatives are negative and the m value is larger than 1.5 so the conditions in Eqs. (14.1) and (14.2) are satisfied.

Fig. 14.8

The first and second derivatives of rupture time with respect to the creep stress for the rupture curves in Fig. 14.7; a the m value and b the second derivative both as a function of rupture time. Reproduced from [37] with the permission of Taylor & Francis

The regression plot in Fig. 14.9 shows a narrower scatter band than in Fig. 14.5 for 321H. Most of the data fall inside the band for a factor of 2. The reason is most likely that the data for Super304H come from just one cast whereas the 321H data are from 9 casts. Only single points are outside the 2.5 σ_std limit in Fig. 14.9. PAT 2.1 is fulfilled since the regression line is close to the 1:1 line. An even distribution is shown for the extended curves in Fig. 14.9b. No crossing of curves and no bending back verify that PAT 1.2 is satisfied.

Fig. 14.9

Regression plot of predicted rupture time versus observed rupture time for Super304H with a regression line (dashed line) close to the 1:1 line. Two bands are given: ±2.5 times the standard deviation and a factor of ±2; b predicted rupture times over a wide range of stresses and rupture times to demonstrate that the curves are not bending back in an unphysical way. Reproduced from [37] with the permission of Taylor & Francis

Studies on several materials show that the PATs play an equally important role for NN models as in TTP analysis. That the results have a physical correct behavior cannot be ascertained without the application of the PATs. The use of constrained optimization with conditions on the derivatives of the creep rupture curves makes it much more straightforward to fulfill the requirements of the PATs.

14.3 Error Analysis in Extrapolation

14.3.1 Model for Error Analysis

Fitting a model to the observed creep rupture data is the start of all empirical models. The model must give a good fit to the data. This means that the model must be able to interpolate accurately between the data points. If TTPs are used the fit is to a single curve, the master curve. The deviation between the interpolated values and experimental data gives the first contribution to the extrapolation error.

A schematic creep rupture curve is shown in Fig. 14.10. The creep stress is plotted versus rupture time. The data points are scattered around a source curve with a random scatter of 50%. The curve will be used to estimate the interpolation error.

Fig. 14.10

Schematic creep rupture curve with the creep stress as a function of the rupture time in a double logarithmic diagram

The data is fitted with a polynomial of degree 5. To simulate the situation for a creep rupture curve, the condition for the first derivative is taken into account according to the first criterion in Eq. (14.1). The polynomial lies well inside the scatter band but it is moving from one side of the source curve to the other side. This means that the second derivative changes sign. In the curve designated fit with constraints the second derivative is also assumed to be negative according to the second criteria in Eq. (14.1). Since this condition is fulfilled for most rupture curves, see Sect. 14.2, the focus will be on this case. It will be referred to as the constrained one (although the pure polynomial fit is also constrained to some degree).

The average error σ_devcon in the fit with the constraints can be estimated from basic principles in statistics [40]

$$ \log \sigma_{{{\text{dev}}\,{\text{con}}}} = \log \sigma_{{{\text{rnd}}}} /\sqrt {n_{{\text{d}}} } $$

(14.13)

n_d is the number of data points. σ_rnd the amount of scatter in the data points measured as the deviation of the difference between the data points and the source curve.

$$ \log \sigma_{{{\text{rnd}}}} = \sqrt {\sum\limits_{i} {(\log \sigma_{i} - \log \sigma_{{{\text{source}}}} )^{2} /(n_{{\text{d}}} - 1)} } $$

(14.14)

A polynomial fit is flexing around the source up to n_p times, where n_p is the degree of the polynomial. Within each range of flexing the principles of Eq. (14.13) can be applied. The average error σ_devuncon becomes

$$ \log \sigma_{{{\text{devuncon}}}} = \log \sigma_{{{\text{rnd}}}} /\sqrt {n_{{\text{d}}} /n_{{\text{p}}} } $$

(14.15)

Consequently, the error is raised by a factor of $\sqrt {n_{{\text{p}}} }$ in relation to Eq. (14.13). The derived errors in Eqs. (14.13) and (14.15) have been verified with thousands of test runs. Logarithms are used in Eqs. (14.13) and (14.15), which shows that relative errors are derived.

In Fig. 14.11a it is illustrated what happens if one tries to extrapolate the curves in Fig. 14.10. For the polynomial fit a partially unphysical result is obtained and the curve is almost bending upwards. This is avoided in the constrained fit. In Fig. 14.11b another case is illustrated for the same source curve but with a new set of randomly generated data points. It is evident that extrapolation from a single curve can give significant variations.

Fig. 14.11

Schematic creep rupture curve with the creep stress as a function of the rupture time in a double logarithmic diagram; a extrapolated from Fig. 14.10; b parallel example from the same source curve

When a significant extrapolation is made, the result is controlled by the highest order term in the polynomial. In the error analysis, this part is taken into account. It can be expressed as

$$ \log e_{{{\text{ext}}}} = A(\log (t_{{{\text{ext}}}} /t_{{{\text{ref}}}} ))^{{n_{{\text{p}}} }} $$

(14.16)

where t_ext is the extrapolated rupture time. The reference time t_ref is chosen as the minimum rupture time t_min included in the analysis. A is a constant that is a function of the degree n_p of the polynomial, the number of data points n_d, and the half-width of scatter band σ_rnd. The value of A has been determined with the help of a large number of test runs. The following expressions have been found for the constrained and unconstrained cases

$$ A_{{{\text{con}}}} = \frac{{a_{0} \log \sigma_{{{\text{rnd}}}} }}{{\sqrt {n_{{\text{d}}} } (\log (t_{{{\text{max}}}} /t_{\min } ))^{{n_{{\text{p}}} }} }};\;A_{{{\text{uncon}}}} = \frac{{a_{0} \log \sigma_{rnd} }}{{\sqrt {n_{d} /n_{p} } (\log (t_{\max } /t_{\min } ))^{{n_{p} }} }} $$

(14.17)

t_min and t_max are the minimum and maximum rupture times in the analysis. a₀ is an empirical factor that has been found to be about unity. A safety factor of 2.5 in the width of the scatter band is included in the value of a₀.

The constant A and consequently the error from Eq. (14.17) are directly proportional to the logarithm of the half-width of the scatter band and inversely proportional to the square root of the number of data points. For the unconstrained case the error is higher by a factor that is equal to the square root of the degree of the polynomial. These basic factors are the same as in Eqs. (14.13) and (14.15).

In Fig. 14.12 an example is given for how the error increases with rupture time. There is a rapid increase. If extrapolation by a factor of 3 in time is assumed which is a common requirement, it can be seen from the Figure that this corresponds to an error of 20% in the constrained case and 50% in the unconstrained case. It is obvious that extrapolation from a single curve can generate large errors.

Fig. 14.12

Error in stress after extrapolation. Results for extrapolation with constrained derivatives are compared with unconstrained ones. t_min = 0.01 kh and t_max = 20 kh. Redrawn from [36] with permission of Taylor & Francis

14.3.2 Error Analysis with PATs

The ECCC PATs are valuable tools to assess if the analysis has worked in a satisfactory way and the predicted values show a correct physical behavior. However, they do not give a direct measure of the accuracy even if all PATs are satisfied. However, with the help of the analysis above this can be achieved. The interpolation error e_interp from the master curve can be expressed as

$$ \log e_{{{\text{interp}}}} = \frac{{\log \sigma_{{{\text{rnd}}}} }}{m(\sigma )}\sqrt {\frac{{n_{{{\text{temp}}}} }}{{n_{{\text{d}}} }}} $$

(14.18)

Logarithms are used in Eq. (14.18), since the modeling of the creep rupture curves is usually analyzed in log scales. The error in stress is requested, whereas σ_rnd gives the scatter in the time direction. To take this into account, the m(σ) value is introduced in Eq. (14.18). As marked the m value is stress dependent, see Fig. 14.3. The number of temperatures in the analysis n_temp is included for the same reasons as the degree of the polynomial in Eq. (14.15).

The regression lines in Fig. 14.5 do not follow the 1:1 line precisely. This deviation results in an error that can be expressed as

$$ \log e_{{{\text{regr}}}} = b_{0} + (b_{1} - 1)\log t_{{\text{R}}} $$

(14.19)

where b₀ and b₁ are adjustable parameters describing the regression line. In Fig. 14.5b, the regression lines are evaluated at each temperature. This means that b₀ and b₁ are temperature dependent. As can be seen the regression line at individual temperatures can deviate from the 1:1 line much more than the mean line.

The expressions for the errors e_interp and e_regr in Eqs. (14.18) and (14.19) are often close to unity. The corresponding differences from unity E_interp and E_regr and their sum are therefore introduced

$$ E_{{{\text{interp}}}} = e_{{{\text{interp}}}} - 1;\,\,\,\,E_{{{\text{regr}}}} = e_{{{\text{regr}}}} - 1$$

(14.20)

$$ E_{{{\text{tot}}}} = e_{{{\text{interp}}}} e_{{{\text{regr}}}} - 1 $$

(14.21)

Table 14.3 Error estimates for values interpolated from the master curve in Fig. 14.2

For the case in Fig. 14.1 the three types of errors are given for 321H in Table 14.3.

The interpolation error in Table 14.3 is between 4 and 6%. The error due to bias in the prediction is referred to as the regression error which takes values from 7 to 10%. The total error is between 12 and 18%. The strength values are multiplied by the total relative errors to find the uncertainties in the stress values that are given as plus-minus additions.

To find the extrapolated values at the highest temperature, it is necessary to extrapolate from the master curve. The error in this case is given by Eqs. (14.13) and (14.14)

$$ \log e_{{{\text{extrap}}}} = \frac{{a_{0} \log \sigma_{{{\text{rnd}}}} }}{{\sqrt {n_{{\text{d}}} } }}\left[ {\frac{{\log (t_{{{\text{ext}}}} /t_{{{\text{ref}}}} )}}{{\log (t_{{{\text{max}}}} /t_{\min } )}}} \right]^{{n_{{\text{p}}} }} $$

(14.22)

t_ext is the extrapolated rupture time, and t_min and t_max the range of experimental rupture times. a₀ = 1 and t_ref = t_min are chosen as explained above. The relative error E_extrap is obtained as in Eq. (14.20) from

$$ E_{{{\text{extrap}}}} = e_{{{\text{extrap}}}} - 1 $$

(14.23)

The regression error is determined in the same way as for the other temperatures and the total error from Eq. (14.21). The results are shown in Table 14.4.

Table 14.4 Error estimates for values extrapolated from the master curve (321H)

The total errors are larger in Table 14.4 than in Table 14.3. This is precisely as expected since at the highest temperature there is extrapolation from the master curve. The degree of the polynomial plays an important role. The extrapolation error is significantly increased when the degree of the polynomial is raised from 4 to 6. The corresponding effect is small in Table 14.3. The absolute error for the strength is obtained in the same way as in Table 14.3, i.e. by multiplying the relative total error by the stress value and giving the result as a plus-minus addition.

The errors in Table 14.4 are so large that the extrapolated values are of little technical value. The reason for the large values is that the data range at the highest temperature is quite limited. That this gives a large error is directly evident from Eq. (14.22). In addition the longest test duration at the highest temperature is fairly short, see Fig. 14.1. The case demonstrates the value of the error estimates, although already from Fig. 14.1 it is evident that the accuracy would be lower at the highest temperature. However, not until the error has been computed, one can draw the conclusion that the error is so large that the extrapolated results at the highest temperature are not very useful.

A comparison between different TTPs is made in Table 14.5 to further investigate extrapolation errors. The study is for the austenitic stainless steel Sanicro 25 and is taken from [36]. The results for five TTPs are shown at 700 °C for two rupture times 100000 and 200000 h. The abbreviations of the five methods can be found in Table 14.1.

Table 14.5 Error estimates for values interpolated from the master curve with different TTPs for the austenitic stainless steel Sanicro 25 (22Cr25Ni4W1.5Co3CuNbN). Reproduced from [36] with permission of Taylor & Francis

The predicted strengths vary from 92 to 97 MPa at 100000 h and from 78 to 84 MPa at 200000 h. The five methods are associated with fairly similar error estimates. Two of the methods (Larson-Miller and Manson-Succop) give slightly higher predicted stresses. From the generated PATs for the TTPs (not shown) one finds that

• For these two methods, the fit at low stresses to the master curve results in a slight over-prediction of the stresses.

• The m-value at low stresses is about 4 for the two methods and 3 for the others.

• At low stresses the second derivative of the rupture time is about −2 at low stresses for the two methods instead of −5 for the others.

These three sets of observations are not unrelated. They simply show that the rupture curves bend down slightly less for Larson-Miller and Manson-Succop methods at low stresses than the other methods. The error analysis indicates that all the five methods give acceptable results but one of the OSD, SA and GS methods should be chosen if conservative values are desirable.

It has been assumed indirectly in the error analysis that the TTPs are valid also for extrapolated values. Some support for this assumption is that PAT 1.3 is satisfied when some of the long term or low stress points from the data set are removed and predicted values can be repeated, see Fig. 14.4.

14.3.3 Error Analysis with NN

In Table 14.6 some of the creep data for the investigated cast of Super304H are given that are essential for the error analysis. The shortest and longest rupture time at each temperature as well as an approximate strength value at the longest rupture time are provided. In addition the relative regression error is shown.

Table 14.6 Some creep properties of Super304H (reproduced from [37] with permission of Taylor & Francis)

Contrary to the case for TTPs, the creep rupture curves at different temperatures must be considered as individuals. Thus, each temperature must be analyzed as a single curve when the extrapolation error should be estimated with Eq. (14.22). The extrapolation error is multiplied by the regression error in the same way as in the TTP analysis. The total errors in the strength are listed in Table 14.7 as plus-minus additions. When the errors are very large, no decimal is given in the error because it does not have any significance. In some cases the errors are close or even exceeding the values they are associated with. In that case no values are given. The error depends on the number of adjustable parameters n_adj. When using Eq. (14.22) the polynomial degree n_p is assumed to correspond to n_p = n_adj − 2 since that is the case in the TTP analysis. Naturally, the error increases with extrapolated rupture time. In many cases the estimated error also increases rapidly with the number of adjustable parameters. This is an indication that the number of neurons in the hidden layer should be kept as low as possible. It is not common to use networks with smaller n_adj values than 13. This is an important observation because the number of adjustable parameters can easily become quite large in NN analyses. The total error is often quite large and typically larger than in a TTP analysis. The reason is that the extrapolation occurs from single curves with the NN method whereas at most temperatures the extrapolated values can be found from interpolation along the master curve with a TTP method.

Table 14.7 Error analysis for creep rupture prediction with the constrained NN model for Super304H. Reproduced from [37] with permission of Taylor & Francis

14.4 Basic Modeling of Creep Rupture Curves

14.4.1 General

To predict and extrapolate creep rupture data, empirical models have been used for many years. Such models are described in Sect. 14.2. Many such models are well established. They are in principle easy to use. However, if precise results are needed they have to be combined with post assessment tests and error analysis. These additions require a significant computational effort. This has the consequence that it is tempting to ignore these additions but then the results would be quite uncertain.

Nowadays it is possible to model and describe the development of the microstructure during high temperature service. There are also basic models for mechanical properties available. By combining these two types of models the mechanical properties including creep can be predicted.

Empirical models require large data set in order to be used for predictions of properties. With just a limited data set the models act as more or less arbitrary mathematical expressions that are fitted to the data. In such cases it would be very risky to generalize and extrapolate the results. An empirical model with say three or more adjustable parameters can represent many sets of experimental observations. It is very unlikely that a good fit to the data ensures that the model describes the physics of the observations.

To avoid these problems several steps must be taken:

• The models must be derived from basic physical principles.

• All the parameters in the models should be well defined and it should be specified how they should be determined.

• No adjustable parameters should be involved that are fitted to the mechanical properties.

There are many models in the creep literature that are derived from basic principles but with some parameters that are fitted to the data. It is not quite as risky to apply such models as fully empirical ones, but numerous examples exist in the literature where such models have been applied and questionable or incorrect results have been obtained. Further analysis of this issue can be found in Chap. 1. A summary of basic models that can be used for creep rupture is given below. Most of these models are derived in other parts of the book.

14.4.2 Secondary Creep Rate

The main contribution to the creep strength comes from the dislocation density. An accurate description of the dislocation density is therefore essential. Equation (2.17) describes the development of the dislocation density ρ

$$ \frac{d\rho }{{d\varepsilon }} = \frac{{m_{{\text{T}}} }}{{bc_{L} }}\rho^{1/2} - \omega \rho - 2\tau_{{\text{L}}} M_{{{\text{climb}}}} \rho^{2} /\dot{\varepsilon } $$

(14.24)

ε is the strain, m_T the Taylor factor, b Burger's vector, c_L a constant and L_s the “spurt” distance which the dislocation moves in each elementary release during deformation for example from a Frank-Read source. ω is the dynamic recovery constant, τ_L the dislocation line tension, and $\dot{\varepsilon }$ the creep strain rate. M_climb the dislocation mobility is given by Eq. (2.34)

$$ M_{{{\text{climb}}}} (T,\sigma ) = \frac{{D_{{{\text{s}}0}} b}}{{k_{{\text{B}}} T}}e^{{\frac{{\sigma b^{3} }}{{k_{{\text{B}}} T}}}} e^{{ - \frac{{(Q_{{{\text{self}}}} + Q_{{{\text{sol}}}} )}}{{R_{G} T}}}} f_{{{\text{clglide}}}} (\sigma ) $$

(14.25)

where T is the absolute temperature, σ the applied stress, D_s0 the pre-exponential coefficient for self-diffusion, Q_self the activation energy for self-diffusion, k_B Boltzmann’s constant, and R_G the gas constant. Q_sol the contribution to the activation energy from solid solution hardening has been added to Eq. (14.25). Q_sol is equal to the maximum interaction energy between solute atoms and dislocations, Eq. (6.10)

$$ Q_{{{\text{sol}}}} = U_{i}^{\max } = \frac{1}{\pi }\frac{{(1 + \nu_{{\text{P}}} )}}{{(1 - \nu_{{\text{P}}} )}}G\Omega_{0} \delta_{i} $$

(14.26)

where $G$ is the shear modulus, ν_P the Poisson’s ratio of the material, Ω₀ the atomic volume of the parent metal, and δ_i the linear misfit of the element i. M_climb has a strong temperature dependence from the activation energies. M_climb can also have strong stress dependence in particular in the power-law break down regime. This was explained in Sect. 2.6.4. This stress dependence is described with the function $f_\text{clglide}$, Eq. (2.50)

$$ f_{{{\text{clglide}}}} (\sigma ) = \exp \left( {\frac{Q}{{R_{{\text{G}}} T}}\left( {\frac{\sigma }{{R_{\max } }}} \right)^{2} } \right) $$

(14.27)

The name of the factor $f_\text{clglide}$ is somewhat misleading. Earlier it was thought that the factor was due to the influence of glide. However, the factor is now derived only assuming climb, see Sect. 2.6.4.

The secondary creep rate can be obtained from Eq. (14.24) directly since the dislocation density is constant in that stage and consequently its strain derivative vanishes.

$$ \dot{\varepsilon }_{\sec } = 2\tau_{{\text{L}}} M_{{{\text{climb}}}} \rho^{3/2} /(\frac{{m_{{\text{T}}} }}{{bc_{{\text{L}}} }} - \omega \rho^{1/2} ) $$

(14.28)

Taylor’s Eq. (2.29) gives the contribution σ_disl from the dislocation to the strength

$$ \sigma_{{{\text{disl}}}} = \alpha m_{{\text{T}}} Gb\rho^{1/2} = \sigma - \sigma_{{\text{i}}} $$

(14.29)

where σ is the applied stress and α ≈ 0.19 a constant. σ_i represents other strength contributions than from the dislocations. They will be exemplified below. Using Taylor’s Eq. (2.29), Eq. (14.28) can be formulated in terms of the applied stress which is the common way of expressing the creep rate

$$ \dot{\varepsilon }_{\sec } = h_{\sec } (\sigma - \sigma_{{\text{i}}} );\,\,\,\,h_{\sec } (\sigma ) = 2\tau_{{\text{L}}} M_{{{\text{climb}}}} (T,\sigma )\frac{{\sigma^{3} }}{{(\alpha m_{{\text{T}}} Gb)^{3} }}/\left( {\frac{{m_{{\text{T}}} }}{{bc_{L} }} - \omega \frac{\sigma }{{\alpha m_{{\text{T}}} Gb}}} \right)$$

(14.30)

This can be considered as a Norton equation where the stress dependence of the creep rate is given. At low stresses the stress dependence is from the σ³ factor but at the high stresses the main contribution is from the f_clglide factor, Eq. (14.27), in the expression for M_climb.

Practically all high temperature alloys have contributions from solid solution hardening (SSH) and/or precipitation hardening (PH) to the creep strength. SSH enters the equation in two ways. It gives a contribution Q_sol to the activation energy for the creep rate, Eq. (14.26). The other part gives a drag stress that contributes to σ_i. This depends on if slowly or fast diffusion elements are involved. The case with slowly diffusing solute will be covered first. There are several expressions for the drag stress in Sect. 6.4. The one that is most commonly valid is Eq. (6.20)

$$ \sigma_{i}^{drag} = \frac{{v_{{{\text{climb}}}} c_{i0} \beta^{2} }}{{bD_{i} k_{B} T}}I(z_{0} ) $$

(14.31)

c_i0 is the average concentration of solute i, and D_i the diffusion constant for solute i. I(z₀) is an integral of z₀ = b/r₀k_BT where r₀ is the dislocation core radius. I(z₀) is given by Eq. (6.21). The climb velocity v_climb and the strength parameter σ_i are found from, Eqs. (6.14) and (6.15)

$$ v_{{{\text{climb}}}} = M_{{{\text{climb}}}} b\sigma;\,\,\,\,\beta_{i} = bU_{i}^{\max }$$

(14.32)

For fast diffusion elements, the solute must break away from the dislocations. The necessary stress is, Eq. (6.28)

$$ \sigma_{{{\text{break}}}} = \frac{{U_{i}^{\max } }}{{b^{3} }}\int {c_{i}^{{{\text{dyn}}}} dz} $$

(14.33)

$c_{i}^{{{\text{dyn}}}}$ describes the distribution of solutes around the dislocations, Eq. (6.13).

The validity of the expressions (14.31) and (14.33) for SSH have been demonstrated in Sects. 6.4 and 6.5. The influence of Mg on creep rate in Al–Mg alloys is illustrated in Fig. 6.4 and the effect of P on the creep rate of Cu in Fig. 6.6.

The starting point for the precipitation hardening is the Orowan strength, Eq. (7.3)

$$ \sigma_{{\text{O}}} = \frac{{m_{{\text{T}}} C_{{\text{O}}} Gb}}{{\lambda_{{\text{s}}} }} $$

(14.34)

where λ_s is the mean particle spacing, Eq. (7.4) and C_O = 0.8 a constant. Equation (14.33) gives the contribution from particles at ambient temperatures. At elevated temperatures small particles can be climbed across without contributing to the strength. Only particles larger than a critical radius r_crit give an addition to the strength

$$ \sigma_{{{\text{part}}}} = \frac{{C_{{\text{O}}} Gbm_{{\text{T}}} }}{{\lambda_{{{\text{crit}}}} }} = \sigma_{{\text{O}}} e^{{ - k(r_{{{\text{crit}}}} - r_{0} )/2}} $$

(14.35)

where λ_crit is the mean spacing between the particles larger than r_crit, k ≈ 1/$\overline{r}$ is the slope of the particle size distribution and $\overline{r}$ is the mean particle size. r_crit is given as, Eq. (7.9)

$$ r_{{{\text{crit}}}} = M_{{{\text{climb}}}} (T,\sigma )b^{2} \sigma \lambda_{{\text{F}}} \frac{\rho }{{\dot{\varepsilon }_{\sec } m_{{\text{T}}} }} $$

(14.36)

The Friedel particle spacing λ_F can be determined from Eqs. (7.6) and (7.14)

$$ \left( {\frac{{\lambda_{{\text{s}}} }}{{\lambda_{{\text{F}}} }}} \right)^{3} = \frac{{\alpha_{{{\text{cl}}}} }}{{\alpha_{{{\text{cl}}}} + 2C_{{\text{O}}} }};\,\,\,\,\alpha_{{{\text{cl}}}} = \frac{{2\overline{r}}}{{3\lambda_{{\text{s}}} }} = \sqrt {\frac{{2f_{V} }}{3\pi }}$$

(14.37)

f_V is the volume fraction of all particles.

The applicability of Eq. (14.35) has been demonstrated in Sect. 7.4.3 for Cu–Co alloys. This is shown in Figs. 7.5 and 7.8. Cu–Co is a suitable system to analyze the effect of particles on the creep strength, since well-defined particles can be formed and the influence of SSH is quite small.

14.4.3 Creep Strain Curves

In the literature much focus has been placed on the prediction and analysis of the creep rate in the secondary stage. However, both primary and tertiary creep is of importance in many applications. It has been demonstrated in Sect. 12.4 that the whole creep strain versus time curves (creep strain curves) can be derived from the creep rate in the secondary stage. It is possible to simplify this approach somewhat [36]. The creep rate for the whole creep curve can then be expressed by using Eq. (14.30)

$$ \dot{\varepsilon } = h_{\sec } (\sigma_{{{\text{creep}}}} );\;\;\sigma_{{{\text{creep}}}} = \sigma_{{{\text{true}}}} + \sigma_{{{\text{nom}}}} - \sigma_{{{\text{disl}}}} - \sigma_{{\text{i}}} $$

(14.38)

The only difference in comparison with Eq. (14.30) is that an effective creep stress, Eq. (12.19), is introduced. σ_true and σ_nom are the true and nominal applied stress. It is essential to take the true stress into account because otherwise there would be no tertiary creep. The nominal creep stress must also be included. Otherwise the creep curves close to ambient temperatures could not be explained. They have the same form as at elevated temperatures in spite of the fact that stress exponent can be as high as 50. This is further discussed in Sect. 8.4. During primary creep, the dislocation density increases and thereby the dislocation strength σ_disl. When the secondary stage is reached, the stresses balance and σ_creep is equal to the true stress and Eq. (14.30) is recovered.

In the book there are several models for primary and tertiary creep. For example, there are more advanced models taking the substructure into account. However, the principles are the same. Examples of computed creep strain curves can be found in Sect. 12.5.3.

14.4.4 Cavitation

Basic models for nucleation and growth of creep cavities have been presented in Chap. 10 and summarized in Sect. 13.3.1. The cavity nucleation rate can be expressed as, Eq. (13.5)

$$ \frac{{dn_{{{\text{cav}}}} }}{dt} = \frac{{0.9C_{{\text{s}}} }}{{d_{{{\text{sub}}}} }} \left(\frac{{g_{{{\text{sub}}}} }}{{d_{{{\text{sub}}}}^{2} }} + \frac{{g_{{{\text{part}}}} }}{{\lambda^{2} }} \right)\dot{\varepsilon } = B_{{\text{s}}} \dot{\varepsilon } $$

(14.39)

n_cav is the number of cavities, d_sub the subgrain size, $\dot{\varepsilon }$ the creep strain rate, λ the interparticle spacing in the grain boundaries and C_s a constant. The fraction of active nucleation sites is given by the factors g_sub and g_part. The main feature of Eq. (14.39) is that the nucleation rate is proportional to the creep strain rate, which has been observed for many materials.

Two types of growth of cavities are considered: diffusion controlled and strain controlled. It was early on recognized that for diffusion control, the growth rate cannot be faster than creep deformation of the matrix. This is referred to as constrained growth. The general equation for constrained growth is, Eq. (13.6)

$$ \frac{{dR_{{{\text{cav}}}} }}{dt} = 2D_{0} K_{{\text{f}}} (\sigma_{{{\text{red}}}} - \sigma_{0} )\frac{1}{{R_{{{\text{cav}}}}^{2} }} $$

(14.40)

R_cav is the cavity radius in the grain boundary plane and σ₀ the sintering stress. The grain boundary diffusion parameter D₀ is equal to δD_GBΩ/k_BT where δ is the grain boundary width, D_GB the grain boundary self-diffusion coefficient, and Ω the atomic volume. k_B is the Boltzmann’s constant and T the absolute temperature. The factor K_f ≈ 0.2 is given in Eq. (10.12). The reduced stress σ_red can be determined from Eq. (13.7)

$$ 2\pi D_{0} K_{{\text{f}}} (\sigma_{{{\text{red}}}} - \sigma_{0} )n_{{{\text{cav}}}} R_{{{\text{cav}}}} + \dot{\varepsilon }(\sigma_{{{\text{red}}}} ) = \dot{\varepsilon }(\sigma ) $$

(14.41)

$\dot{\varepsilon }(\sigma_{{{\text{red}}}} )$ and $\dot{\varepsilon }(\sigma )$ are the creep rates at the reduced and applied stress.

Equations (14.39) and (14.40) have been applied successfully to model cavity nucleation and growth in austenitic stainless steels. This is illustrated in Figs. 10.5 and 10.8.

There are several derived expressions for strain controlled growth. The most well-known ones are due to Rice and Tracey and to Cocks and Ashby modified by Wen and Tu, see Sect. 13.4.2. Unfortunately, these expressions are difficult to use for predictions. The start value for the cavity size has a large effect on the results but there is no obvious way of selecting the value. Furthermore, the model by Rice and Tracey gives quite a limited growth rate in the uniaxial case and the Cocks and Ashby model does not fulfill the criterion for constraint growth. A model based on grain boundary sliding, Eq. (10.24) avoids these problems but it needs further experimental verification for general use.

14.4.5 Rupture Criteria

A distinction is made between brittle and ductile rupture. Brittle rupture is assumed to occur when the cavitated grain boundary area reaches a sufficient fraction. The cavitated grain boundary fraction A_cav can be computed from the expression for cavity nucleation and growth, Eqs. (14.39) and (14.40). The result is given in Eq. (13.8)

$$ A_{{{\text{cav}}}} = \int\limits_{0}^{t} {\frac{{dn_{{{\text{cav}}}} }}{dt^{\prime}}(t^{\prime})} \pi R_{{{\text{cav}}}}^{{2}} (t,t^{\prime})dt^{\prime} $$

(14.42)

Several studies indicate that the local critical value for brittle rupture is A_cav ≈ 0.25.

The results in Sect. 12.5 suggest that a plastic instability initiates ductile rupture, for tensile specimens necking. Only very close to the rupture a fully developed waist is formed. The prediction of necking requires creep strain data. If creep strain curves are not available, they could be predicted with the help of Eq. (14.38). Unfortunately, the applicability of this equation has been documented mainly for Cu. The alternative is to assume ductility exhaustion, and use a fixed creep rupture elongation value as failure criterion. As will be seen below, this seems to work well.

14.4.6 Extensive Extrapolation of the Creep Rate for Cu

In the present book it has been emphasized that basic models can improve the possibility to predict and extrapolate results. This was illustrated for Al in Sect. 5.7 and for Cu-OFP in Sect. 5.8.1. Another example will be given here for Cu-OFP where this capability of extrapolation in time is also demonstrated in a dramatic way. The case was originally presented in [41], but it is reanalyzed here with the primary creep model in Sect. 5.5 [42].

With conventional creep testing techniques creep rates down to about 1 × 10⁻¹² 1/s can be measured. This requires for example that the testing temperature in the laboratory is controlled within 2 °C. Ho carried out creep tests at much lower stationary creep rates for Cu-OFP [43]. Tests were performed at 20–100 MPa at 95 °C, at 20–60 MPa at 125 °C and at 20–40 MPa at 150 °C. The model in Sect. 5.5 can at least approximately describe all the experimental results. The parameter value σ_y/K is taken from the room temperature data in [41], and the ω value is computed from Eq. (5.35). It is checked that the criterion in Eq. (5.36) is fulfilled. Two examples are given in Fig. 14.13. Four tests were performed at 19.9 MPa and six tests at 60 MPa. In spite of the low stresses, creep is clearly present in all the tests.

Fig. 14.13

Creep strain versus testing time for Cu with 50 ppm P (Cu-OFP) at 95 °C. The creep strain according to Eq. (5.32) is compared to experimental data from Ho [43]; a 19.9 MPa; b 60 MPa. Redrawn from [42] with permission of Elsevier

From Fig. 14.13 one might think that creep is approaching the secondary stage. But that is not at all the case. The lowest creep rates in Fig. 14.13a are about 1 × 10⁻¹² 1/s and in Fig. 14.13b about 3 × 10⁻¹¹ 1/s. From Eq. (14.30), the stationary creep rate can be estimated to 1.3 × 10⁻²² and 1.4 × 10⁻¹⁹ 1/s in Fig. 14.13a, b, respectively. Such creep rates are far outside the interval where they can be measured. For a specimen with a gauge length of 50 mm, 1.3 × 10⁻²² 1/s represents a displacement of less than one lattice spacing in a million years. The basic stationary creep model can obviously handle creep rates at least from 180 MPa at 75 °C of 1.4 × 10⁻⁷ (Fig. 6.6) to 20 MPa at 95 °C of 1.3 × 10⁻²² 1/s. This represents a range of validity of 15 orders of magnitude. It clearly demonstrates the value of the basic creep model.

In Fig. 12.1 the stress exponent is about n_N = 60 deep in the power-law break down regime. In Fig. 14.13a, the stress exponent is n_N = 3.3 and in Fig. 14.13b n_N = 8.3. Thus, the model provides valuable information even when there is a transfer from one creep regime to another. These results have profound implications.

• Basic models have the potential to extrapolate results by orders of magnitude in time. This should be compared with empirical statistical methods where a factor of 3 typically can be reached and in special cases a factor of 10.

• Meaningful results can be obtained even when there is a transfer from one testing regime to another. With basic models it is much safer to generalize and extrapolate findings.

• The technical consequences are of immense significance. Cu-OFP will be used in canisters for disposal of spent nuclear fuel in Finland and Sweden. The canisters have a design life of 100000 years. Cu-OFP has been creep tested for up to 3 years. An extrapolation by almost 5 orders of magnitude is required. The results above demonstrate that the basic creep model is valid for such a time extrapolation factor with a good margin and that the model can safely be used to predict the creep properties of the canisters.

14.4.7 Creep Rupture Predictions for Austenitic Stainless Steels

With the equation in Sects. 14.4.2–14.4.5 the rupture time can be predicted. This has been applied to austenitic stainless steels in several papers [21, 25, 44]. Ductile rupture is handled with ductility exhaustion. A constant elongation at rupture of 0.2 is assumed. This value is lower than the observed values for ductile rupture. On the other hand the strain computation is based on the secondary creep rate, Eq. (14.30) that underestimates the total strain. At present there is not sufficient data available to take primary and tertiary creep into account as well.

Brittle rupture is based on the cavitated area fraction in the grain boundaries A_cav, Eq. (14.41). When A_cav reaches 0.25, brittle rupture is assumed to occur. The criterion for ductile or brittle rupture that is met first is considered to control the type of rupture that takes place.

Results for the 18Cr12NiNb steels (347H) are shown in Fig. 14.14. In Fig. 14.14a only ductile rupture is taken into account but in Fig. 14.14b both types of rupture are included.

Fig. 14.14

Creep rupture strength values as a function of rupture time for the austenitic stainless steel 18Cr12NiNb (347H); a ductile rupture based on ductility exhaustion, Eq. (14.30); b ductile rupture as in a and brittle rupture assuming a fixed cavitated area fraction at failure, Eq. (14.42). Experimental data from [45] at temperatures between 600 and 750 °C. Redrawn from [21] with permission of Elsevier

The difference between the two types of rupture is not very large. Only at low stresses and long times there is a significant difference. Cavitation reduces the rupture times in that situation. The general behavior of the rupture curves can be seen to be represented in a reasonable way.

In Fig. 14.15, the corresponding results for the 17Cr12Ni2Mo steel 316H are illustrated. The influence of ductile and brittle rupture is much the same as for 347H. The experimental data is represented quite well in particular at long times.

Fig. 14.15

Creep rupture strength values as a function of rupture time for the austenitic stainless steel 17Cr12Ni2Mo (316H); a ductile rupture based on ductility exhaustion, Eq. (14.30); b ductile rupture as in a and brittle rupture assuming a fixed cavitated area fraction at failure, Eq. (14.42) Experimental data from [45] at temperatures between 600 and 750 °C. Redrawn from [21] with permission of Elsevier

The variation of the slope in Figs. 14.14 and 14.15 follows the observations well. The absolute value of slope was designated m in Eq. (14.1). The m value is given in Fig. 14.16a for the curves in Fig. 14.14 for 347H.

Fig. 14.16

Creep rupture behavior of the austenitic stainless 18Cr12NiNb steel 347H; a stress exponent m for creep rupture versus rupture time at the temperature 600 °C (top) to 750 °C (bottom); b contribution to the rupture strength from dislocations, precipitates and elements in solid solution. b Redrawn from [21] with permission of Elsevier

If the modified Monkman-Grant relation was strictly followed, m would be equal to the stress exponent for the creep rate n_N. However, m is typically larger than n_N at least for modest rupture times. Figure 14.16 illustrates the contributions to the rupture strength. For 347H, the dislocations give the largest contribution which is often the case. 347H is precipitation hardened with Nb(C, N). It clearly gives a significant contribution to the strength. From solid solution hardening, there is only a small effect.

Successful predictions of the creep ductility taking both brittle and ductile rupture into account were presented in Sect. 13.3. For example, results corresponding to Fig. 14.15 were shown in Fig. 13.6. These results illustrate that the ductility is also possible to compute with basic models.

14.5 Summary

• The use of time-temperature parameters (TTPs) is the classical way to extrapolate creep rupture data to longer times. With the help of TTPs the rupture data are fitted to a single curve, the master curve. Extrapolation at most temperatures can be handled by interpolation along the master curve. In this way the extrapolation from single curves is avoided that gives a less accurate result. Only at the highest temperature this is necessary.

• An alternative to TTPs is to use a neural network (NN). It is necessary to choose a simple NN to minimize the number of adjustable parameters involved. The error can increase rapidly with the number of parameters.

• The ECCC post assessment tests (PATs) are quite valuable to show that an extrapolation analysis has worked and that the results show a correct physical behavior. This applies to both TTPs and NN.

• Creep rupture curves, i.e. creep stress versus rupture times show some characteristic features. Their first derivative is always negative. This applies also to the second derivative except for so-called sigmoidal behavior but that is not considered in this chapter. For empirical extrapolation, the creep data are fitted to a model with a number of adjustable parameters. To improve the fit and the physical realism of the predicted rupture time, constrained optimization with conditions on first and second derivatives is recommended. It has been shown that many of the PATs are automatically satisfied in this way.

• Formulae for error estimates are presented. Expressions for the relative errors of interpolation, extrapolation and regression are given. These error estimates make it much simpler to assess the quality of an evaluation. Both interpolation and extrapolation from a master curve are covered as well as NN.

• Basic models for creep rupture have been presented throughout the book. The main equations for brittle and ductile rupture are summarized in Sect. 14.4.

• In principle, basic modeling should make it possible to significantly improve the possibility to generalize and extrapolate results. For Cu-OFP it has been possible to demonstrate that this is in fact the case. It was verified that meaningful extrapolation of many orders of magnitude in time is possible. This makes it possible to safely compute the creep properties over such extended periods as 100000 years, which has been fully utilized in canisters for disposal of spent nuclear waste.

• Basic creep rupture predictions for austenitic stainless are summarized. It is demonstrated that experimental creep rupture can be well reproduced.

Appendix: Derivatives in Neural Network Models (Reproduced from [37] with Permission)

The rupture curves, i.e., the creep stress plotted versus the observed rupture time, must show the same behavior as described for the TTP analysis, the first and second derivative must fulfill the requirements in Eqs. (14.1) and (14.2). If the criteria in these equations are satisfied unphysical flexing of the predicted rupture curve is prevented. The criteria are of the same importance when NN models are used.

The derivatives of predicted curves are not available in general in NN programs, so expressions for the derivatives are derived below. The NN model is represented with the matrix formalism given by Hagan et al. [14]. The input p to output a¹ from the first layer in the network is given by

$$ {\mathbf{v}}_{k}^{1} = \sum\limits_{i} {{\mathbf{W}}_{ki}^{1} {\mathbf{p}}_{i} } + {\mathbf{b}}_{i}^{1} $$

(14.43)

$$ {\mathbf{a}}_{k}^{1} = \varphi^{1} ({\mathbf{v}}_{k}^{1} ) $$

(14.44)

p has components corresponding to the number of inputs n_input. There is a weight factor W¹ to each neuron for each input. Thus, the matrix W¹ has the dimension n_neuron × n_input where n_neuron is the number of neurons in the first layer. For each neuron there is also a component in the base vector b¹. The linear combination in Eq. (14.43), the transfer input v¹ is fed into the transfer function φ¹, which results in the output a¹ from the first layer. φ¹ is a scalar function. The output from the first layer is the input to the next layer. The qth layer in the network can be expressed as:

$$ {\mathbf{v}}_{k}^{q} = \sum\limits_{i} {{\mathbf{W}}_{ki}^{q} {\mathbf{a}}_{i}^{q - 1} } + {\mathbf{b}}_{k}^{q} $$

(14.45)

$$ {\mathbf{a}}_{k}^{q} = \varphi^{q} ({\mathbf{v}}_{k}^{q} ) $$

(14.46)

The superscript refers to the number of the layer in the network. Thus the output from layer q − 1 namely a^q−1 is the input to layer q. This generates a transfer input v^q, which is then used to form the output from layer q.

The formalism set up in Eqs. (14.43)–(14.46) will now be employed to derive the expression for derivatives with respect to the input variables in the vector p. An iterative procedure will be considered in the sense that if the parameters for one layer are known the parameters in the next layer can be derived. The derivative of the transfer input in the first layer can be obtained from Eq. (14.43)

$$ \frac{{d{\mathbf{v}}_{j}^{1} }}{{d{\mathbf{p}}_{m} }} = {\mathbf{W}}_{jm}^{1} $$

(14.47)

Equation (14.45) gives the transfer input from layer q − 1 to layer q

$$ \frac{{d{\mathbf{v}}_{k}^{q} }}{{d{\mathbf{p}}_{m} }} = \sum\limits_{i} {} {\mathbf{W}}_{ki}^{q} \frac{{d{\mathbf{a}}_{i}^{q - 1} }}{{d{\mathbf{p}}_{m} }} $$

(14.48)

The derivative of the output a^q from layer q takes the form

$$ \frac{{d{\mathbf{a}}_{k}^{q} }}{{d{\mathbf{p}}_{m} }} = \frac{{d\varphi^{q} }}{dv}({\mathbf{v}}_{k}^{q} )\frac{{d{\mathbf{v}}_{k}^{q} }}{{d{\mathbf{p}}_{m} }} $$

(14.49)

Notice that there is a summation over index i in Eq. (14.48) but no summation over the repeated index k in Eq. (14.49). The rule of automatic summation over repeated indices is not applied. When there is a summation, it is explicitly indicated. If Eqs. (14.48) and (14.49) are combined, an expression is obtained where the derivative of the transfer input in one layer is directly related to transfer input in the previous one.

$$ \frac{{d{\mathbf{v}}_{k}^{q} }}{{d{\mathbf{p}}_{m} }} = \sum\limits_{i} {{\mathbf{W}}_{ki}^{q} \frac{{d\varphi^{q - 1} }}{dv}({\mathbf{v}}_{i}^{q - 1} )\frac{{d{\mathbf{v}}_{i}^{q - 1} }}{{d{\mathbf{p}}_{m} }}} $$

(14.50)

With the Eqs. (14.47), (14.49) and (14.50) the derivatives of transfer input and output can be computed for a layer from the corresponding values in the previous layers.

The second derivatives can be derived in a similar way. From Eq. (14.43), it is evident that the second derivative of the transfer input vanishes in the first layer.

$$ \frac{{d^{2} {\mathbf{v}}_{j}^{1} }}{{d{\mathbf{p}}_{m} d{\mathbf{p}}_{n} }} = 0 $$

(14.51)

Derivating Eq. (14.50) gives the following result

$$ \frac{{d^{2} {\mathbf{v}}_{k}^{q} }}{{d{\mathbf{p}}_{m} d{\mathbf{p}}_{n} }} = \sum\limits_{i} {\left\{ {{\mathbf{W}}_{ki}^{q} \frac{{d^{2} \varphi^{q - 1} }}{{dv^{2} }}({\mathbf{v}}_{i}^{q - 1} )\frac{{d{\mathbf{v}}_{i}^{q - 1} }}{{d{\mathbf{p}}_{m} }}\frac{{d{\mathbf{v}}_{i}^{q - 1} }}{{d{\mathbf{p}}_{n} }} + {\mathbf{W}}_{ki}^{q} \frac{{d\varphi^{q - 1} }}{dv}({\mathbf{v}}_{i}^{q - 1} )\frac{{d^{2} {\mathbf{v}}_{i}^{q - 1} }}{{d{\mathbf{p}}_{m} d{\mathbf{p}}_{n} }}} \right\}} $$

(14.52)

The second derivative of the output from layer q can be obtained from Eq. (14.49)

$$ \frac{{d^{2} {\mathbf{a}}_{j}^{q} }}{{d{\mathbf{p}}_{m} d{\mathbf{p}}_{n} }} = \frac{{d^{2} \varphi^{q} }}{{dv^{2} }}({\mathbf{v}}_{j}^{q} )\frac{{d{\mathbf{v}}_{j}^{q} }}{{d{\mathbf{p}}_{m} }}\frac{{d{\mathbf{v}}_{j}^{q} }}{{d{\mathbf{p}}_{n} }} + \frac{{d\varphi^{q} }}{dv}({\mathbf{v}}_{j}^{q} )\frac{{d^{2} {\mathbf{v}}_{j}^{q} }}{{d{\mathbf{p}}_{m} d{\mathbf{p}}_{n} }} $$

(14.53)

Since NN software where conditions on the derivatives could be introduced was not found, a new NN program was written from scratch. The network consists of layers of neurons. The number of neurons in each layer could be chosen. The values of the weights and bases were determined with the help of an algorithm for back propagation. A contribution Δerr to the mean square error was added

$$ \Delta {\text{err}} = c_{1} \varphi_{{{\text{logsig}}}} \left( {\frac{{da^{Q} }}{{dp_{m} }}} \right) + c_{2} \varphi_{{{\text{logsig}}}} \left( {\frac{{d^{2} a^{Q} }}{{dp_{n}^{2} }}} \right) $$

(14.54)

where Q is the final layer in the network, and c₁ and c₂ are constants. φ_logsig is a logsig function. If the derivatives are positive, there is a contribution to the error function that the training of the network will try to remove.

In the NN models for the prediction of the creep rupture times, two layers were used. The first hidden layer contained 3–10 neurons with a logsig transfer function for φ¹. The second output layer with one output had one neuron with a linear transfer function φ². With a final linear transfer function, the first term in Eq. (14.53) vanishes in this case. With one output, a^Q becomes a scalar as indicated in Eq. (14.54).