# Creep-Fatigue Behavior and Life Prediction of Medium-Si-Mo Ductile Iron
## 3 Results and Analysis
### Creep-Fatigue Interaction Behavior
#### 3.2.2 Investigation of Creep-Fatigue Life Prediction Models  
Internationally, the linear damage summation (LDS) method is widely used to predict creep-fatigue life [28]. Based on this approach, fatigue and creep damage are calculated independently for each cycle, with material failure occurring once the cumulative damage reaches one. Typically, the half-life cycle is considered the representative steady-state cycle, and all stress and strain data are derived from this period. The creep-fatigue life can be predicted using Equation (3) in these conditions. Considering that damage estimation for low-cycle fatigue is relatively mature, this study focused on accurate assessment of creep damage to improve the predictive capability of the LDS theory under creep-fatigue conditions. Specifically, the fatigue damage per cycle was calculated as the reciprocal of the low-cycle fatigue test life (\(N_{f0}\)) for an identical strain range. Subsequently, the creep damage per cycle was evaluated using the time fraction (TF) method [29] and the strain energy density dissipation method [30; 31]. \[N_{c-f}=\frac{1}{d_{f}+d_{c}} \tag{3}\]  
In Equation (3), \(N_{c-f}\) is the creep-fatigue interaction life, \(d_{f}\) is the fatigue damage per cycle, and \(d_{c}\) indicates the creep damage per cycle. In the time fraction (TF) method, the creep damage per cycle is evaluated using Equation (4) [32; 33]. \[d_{c}=\int_{0}^{t_{c}}\frac{1}{t_{R}}\ \mathrm{d}t \tag{4}\]  
In this equation, \(t_{R}\) indicates the creep rupture times corresponding to different stresses and \(t_{c}\) is the creep time of a single cycle. Under constant temperature conditions, the relationship between the creep rupture time (\(t_{R}\)) and the applied stress (\(\sigma\)) is expressed by Equation (5) [34]. \[t_{R}(\sigma,T)=k_{TF}\cdot\sigma^{-\alpha_{TF}} \tag{5}\]  
In this equation, \(k_{TF}\) and \(\alpha_{TF}\) are temperature- and material-dependent parameters that are obtained by fitting the creep stress values with corresponding rupture times. However,  
Figure 8: Stabilized hysteresis loops under various creep–fatigue conditions. (**a**) Stabilized hysteresis loops at 400 \({}^{\circ}\)C under various strain amplitudes, (**b**) stabilized hysteresis loops at 600 \({}^{\circ}\)C under various strain amplitudes, (**c**) stabilized hysteresis loops at 700 \({}^{\circ}\)C under various strain amplitudes, (**d**) stabilized hysteresis loops at 600 \({}^{\circ}\)C with various hold times, and (**e**) stabilized hysteresis loops at 700 \({}^{\circ}\)C with various hold times. for strain-controlled fatigue tests with dwell periods, the stress during creep does not remain constant but undergoes relaxation while maintaining constant strain. This phenomenon, essentially caused by creep, represents an opposite effect compared to constant-stress creep. Previous researchers have conducted extensive studies on stress relaxation, leading to the Feltham equation, which is widely used in engineering applications [35]. \[\sigma=\sigma_{0}\cdot\left[1-B^{\prime}\cdot\ln(bt+1)\right] \tag{6}\]  
In this equation, \(B^{\prime}\) and \(b\) are model parameters in the empirical stress relaxation formula. \(\sigma_{0}\) represents the initial stress of relaxation, and \(t\) denotes the relaxation time from the beginning of the dwell period. During the stress relaxation phase, Equation (6) describes the stress reduction during the dwell time, which can be combined with Equation (5) to obtain the creep rupture times corresponding to different stress levels. By integrating Equations (4)-(6), a creep-fatigue life prediction (\(N_{c-f}^{TF}\)) can be obtained using the TF method [34]. \[N_{c-f}^{TF}=\left\{\frac{1}{N_{f0}}+\int_{0}^{t_{c}}\frac{1}{k_{TF}}\cdot \left[\sigma_{0}-B^{\prime}\sigma_{0}\cdot\ln(bt+1)\right]^{\alpha_{TF}}\mathrm{ d}t\right\}^{-1} \tag{7}\]  
Building on the LDS theory, this study also adopted an alternative strain energy density dissipation model.