# 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
Previous LCF tests have established the relationship between fatigue life and strain using the Basquin-Manson-Coffin equation. While this equation offers simplicity and versatility, it calculates fatigue life solely based on strain as the life parameter, neglecting the influence of stress responses at elevated temperatures. To address this limitation, researchers have proposed alternative approaches that correlate the external energy put into a material with low-cycle fatigue life [39]. These energy-based methods are generally thought to better reveal the fundamental nature of low-cycle fatigue damage, offering a clearer picture of its physical significance and improved accuracy compared to the Manson-Coffin equation. This study employed the net tensile hysteresis energy method (the Ostergren method [40]), which was developed based on the cumulative plastic strain energy approach [41]. In the Ostergren method, fatigue failure is primarily caused by tensile stress. The mathematical formulation of the Ostergren method is expressed as follows [40]:  
\[t_{R}(\sigma,T)=k_{TF}\cdot\sigma^{-\alpha_{TF}} \tag{5}\]3 \[t_{R}(\sigma,T)=k_{TF}\cdot\sigma^{-\alpha_{TF}} \tag{5}\]4  
In this equation, \(c\) and \(d\) represent constant coefficients; \(\Delta W_{t}\) is the net tensile hysteresis energy; and the energy representations during low-cycle fatigue (LCF) and creep-fatigue hysteresis loop processes are illustrated by the shaded areas in Figure 9a,b, respectively. \[t_{R}(\sigma,T)=k_{TF}\cdot\sigma^{-\alpha_{TF}} \tag{5}\]5 Life prediction results using the TF method and the T-SEDE method: (**a**) the TF model and (**b**) the T-SEDE model. Specifically, Figure 9b demonstrates that the net tensile hysteresis energy in creep-fatigue processes is calculated based on the corresponding LCF condition. \(\sigma_{max}\) is the cyclic stress peak, and \(\Delta\varepsilon_{p}\) represents the plastic strain range. For medium-silicon molybdenum ductile iron, the fitting results of parameters c and d using the Ostergren method are presented in \[t_{R}(\sigma,T)=k_{TF}\cdot\sigma^{-\alpha_{TF}} \tag{5}\]6  
An analysis was conducted from an energy perspective. Ratio R1 was obtained by dividing the net tensile hysteresis energy by the creep inelastic strain energy in the creep-fatigue hysteresis loops at various strain amplitudes. The relationship between creep-fatigue life and R1 at different temperatures is presented in \[t_{R}(\sigma,T)=k_{TF}\cdot\sigma^{-\alpha_{TF}} \tag{5}\]7 The results show that at all temperatures, R1 gradually increases with an increasing strain amplitude, while the creep-fatigue life decreases with an increasing R1. This indicates that the fatigue damage caused by an increasing strain amplitude plays an important role in creep-fatigue damage. However, as the strain amplitude increases, the curves show different trends at different temperatures. For example, at 400 \({}^{\circ}\)C, R1 increases from 7.84 to 9.70 and the life decreases by 1364 cycles. In contrast, at 700 \({}^{\circ}\)C, R1 increases from 10.38 to 17.32, but the life only decreases by 865 cycles. This demonstrates that as temperature increases, the influence of net tensile hysteresis energy damage on life caused by strain amplitude changes decreases. Correspondingly, it also indicates that the influence of creep inelastic strain energy on life increases. These findings are consistent with the actual situation, where creep effects become stronger at higher temperatures. \[t_{R}(\sigma,T)=k_{TF}\cdot\sigma^{-\alpha_{TF}} \tag{5}\]8 Relationship between net tensile hysteresis energy and fatigue life for medium-silicon molybdenum ductile iron at various temperatures. \[t_{R}(\sigma,T)=k_{TF}\cdot\sigma^{-\alpha_{TF}} \tag{5}\]9 The influence of the ratio (R1) of net tensile hysteresis energy to creep inelastic strain energy on creep–fatigue life.