## 1 Superalloy
### Creep Life Prediction Model
#### 1.5.2 Modern Creep Model Approaches  
Traditional creep models, which are based on the power law equation, are focused on extrapolating predictions using existing data trends. When using the traditional creep model, if the dominant creep deformation mechanism varies with temperature and stress, the creep life at low temperatures cannot be accurately predicted with results of creep tests performed at higher temperatures.  
In recent years, many studies have been conducted to improve the predictability of creep life by solving the above problem. In particular, the method of normalizing stress to tensile strength is used often. Tensile strength is used rather than yield strength because it is the maximum strength that can be applied in the creep test and the value of tensile strength can be obtained more easily than that of yield strength. Normalized power law and Wilshire model are creep models normalizing with tensile strength [48; 49].  
* Normalized power law equation  
The improvement of the stress term using the normalization factor from the power law model is called the normalized power law model. Such normalization factors include yield strength, tensile strength, and elastic modulus, but tensile strength is mainly used. This allows all creep data to be represented in the stress range from \(\sigma/\sigma_{TS}=0\) to \(\sigma/\sigma_{TS}=1\), and the temperature dependence of creep activation energy (\(Q_{c}\)) is improved.  
Normalized power law gives as Equation (6)  
\[\dot{\varepsilon}_{ss}\frac{M}{t_{r}}=A\left(\frac{\sigma}{\sigma_{TS}}\right) ^{n}\exp\left(\frac{-Q_{c}{}^{*}}{RT}\right) \tag{6}\]  
where \(\varepsilon_{ss}\) is the steady-state creep rate at temperature (\(T\)), \(Q_{c}{}^{*}\) is creep activation energy, \(A\) is constant, \(n\) is creep exponent, and \(M\) is Monkman-Grant constant. \(A\) and \(n\) value can be derived by linear relationship fitting of the x-axis with \(\ln(\sigma/\sigma_{TS})\) and the y-axis with \(\ln(\varepsilon_{ss}\exp(Q_{c}{}^{*}/RT))\). When representing the linear relationship of the NPL model, it is important to find the optimal \(Q_{c}{}^{*}\) value. If an optimal NPL linear relationship is made with experimental short-term creep data, it is possible to predict long-term creep life by extrapolation. When the deformation mechanism changes for each stress in the creep data, there may be multiple slopes in the NPL linear relationship [47]. With this model, it can be seen that the predictability is superior to the LMP model when predicting long-term creep life [45; 47].  
* Wilshire approach The Wilshire model has been introduced to examine the long-term rupture strength by extrapolating the short-term creep measurements. The formula has been modified based on the normalized power law to make linear relationship between function of stress and rupture time.  
\[\frac{\sigma}{\sigma_{TS}}=\exp\left\{-k_{1}\left[t_{r}\exp\left(\frac{-Q_{c}{}^ {*}}{RT}\right)\right]^{u}\right\} \tag{7}\]  
where \(\sigma\) is the creep stress, \(\sigma_{TS}\) is the tensile strength, \(t_{r}\) is the creep rupture time, \(Q_{c}{}^{*}\) is the creep activation energy, \(R\) is the ideal gas constant, and \(k_{1}\) and \(u\) are constants. \(k_{1}\)and \(u\) can be derived by linear relationship fitting the x-axis with \(\ln(t_{r}\exp(-Q_{c}{}^{*}/RT))\) and the y-axis with \(\ln(-\ln(\sigma/\sigma_{TS}))\)[50; 51; 52; 53]. The Wilshire model is influenced by the \(Q_{c}{}^{*}\) value like the NPL model in the fitting of the linear relationship. Multiple slopes also exist according to change creep deformation mechanism [47]. However, in the case of the Wilshire model, it is possible to derive a creep life regardless of the Monkman-Grant relationship. In predictability regime, the Wilshire model shows better long-term creep life prediction accuracy compared to LMP, NPL model [47].