# Residual Life Prediction of Gas-Engine Turbine Blades Based on Damage Surrogate-Assisted Modeling
## 2 Methods
### Damage Prediction  
Turbine blades operate for extended periods under heavy loads in conditions of nonuniform heating and cyclic loading. Damage is a process that occurs in a material under stress and temperature, and eventually leads to failure [6]. Damage is assumed to be zero (\(D=0\)) when the material is new and equal to one (\(D=1\)) upon local stress rupture failure. The damage may be due to creep, low-cycle fatigue (LCF), or thermomechanical and high-cycle fatigue, which are some key damage modes. For high-temperature blades, other factors such as oxidation should also be considered [9; 10; 11].  
By comparing the value of the accumulated damage with its experimentally defined maximal allowable value, one can predict the residual life. Under conditions of the simultaneous action of several damage modes, the linear damage-accumulation (LDA) rule [12; 13]) can be used:  
\[D_{\tau}+D_{c}+D_{v}+\ldots+D_{j}=a \tag{1}\]  
where \(D_{\tau}\), \(D_{c}\), and \(D_{v}\) characterize the contribution of the effects of static, cyclic, and vibration loads to the total damage, and \(a\) is the material parameter. Other damage modes (\(D_{j}\)) can also be considered. Although the proposed method allows for us to consider various damage modes, because this study considered a GTE power plant, only static damage was considered:  
\[D_{\tau}=\frac{t_{e}}{\tau_{r}} \tag{2}\]  
where \(t_{e}\) is the actual time spent under conditions \(i\) (local values of temperature and von Mises stress), and \(\tau_{r}\) is the time to failure under condition \(i\); \(\tau_{r}\) can be determined using the Larson-Miller parameter (\(LMP\)) (Equation (3)).  
\[\tau_{r}=10^{\frac{LMP}{l}-C} \tag{3}\]  
Figure 2: Proposed gas-turbine-engine (GTE) model. LPC, low-pressure compressor; HPC, high-pressure compressor; LPT, low-pressure turbine; HPT, high-pressure turbine; Cp volume, chamber with constant volume.  
where \(T\) is absolute temperature (K) and \(C\) is the material constant. \(LMP\) is defined by the value of von Mises stress. Using the LDA rule is less accurate, as the history of the damage is not considered.  
During operation in a regime, stress redistribution and relaxation occur (Figure 3). If this process is not considered during strength calculations, the damage value may be several times higher.  
To calculate the kinetics of blade stress state, strain-hardening theory was used in the ANSYS(r) software.  
\[\dot{\epsilon}_{cr}=C_{1}\cdot\sigma^{C_{2}}\cdot\epsilon_{cr}^{C_{3}}\cdot e ^{-C_{4}/T} \tag{4}\]  
where \(\dot{\epsilon}_{cr}\) is creep-strain rate, \(\epsilon_{cr}\) is creep strain, and \(C_{1}...C_{4}\) are parameters dependent upon blade material.  
To calculate damage-taking stress relaxation into consideration, the following equation can be used:  
\[D_{\tau}=\int_{0}^{t_{\tau}}\frac{dt}{\tau_{r}(t)} \tag{5}\]  
where \(\tau_{r}(t)\) is the time to failure at the current temperature (\(T\)) and stress (\(\sigma\)) in the blade at moment of time \(t\). To calculate the damage, the stress-relaxation curve is divided into a set of segments, at the vertices of which time to failure is determined at the current stress level (\(\sigma_{ti}\)).