# 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
This model, developed by [PERSON] [30,31] through extensive experimental data analysis, represents a modification of the SEDE model [36]. This model proposes that creep damage can be neglected during the initial phase of stress relaxation in the dwell period, which has a high inelastic strain rate. Consequently, the creep damage per cycle is calculated using Equation (8). In this equation, the inelastic strain energy density dissipation rate (\(\dot{w}_{in}\)) and the inelastic strain energy density (\(w_{f}(\dot{w}_{in},T)\)) are related as shown in Equation (9). \[d_{c}^{T-SEDE}=\int_{0}^{t_{e}}\left[\frac{\dot{w}_{in}}{\min\left[w_{f}(\dot{ w}_{in},T),w_{f,crit}(T)\right]}-\frac{\dot{w}_{in}}{w_{f,crit}(T)}\right] \mathrm{d}t \tag{8}\]  
\[w_{f}=B\exp\left(\frac{Q}{RT}\right)\dot{w}_{in}^{n} \tag{9}\]  
In this context, \(w_{f,crit}(T)\) is the critical inelastic strain energy density, which corresponds to the upper plateau of the inelastic strain energy density obtained from creep tests [37]. \(B\) and \(n\) are material-dependent parameters determined through fitting, \(R\) is the universal gas constant, and \(Q\) denotes the activation energy in the SEDE model [34]. At a constant temperature, the relationship between the inelastic strain energy density dissipation rate (\(\dot{w}_{in}\)) and the inelastic strain energy density (\(w_{f}(\dot{w}_{in},T)\)) in Equation (9) can be expressed as follows:  
\[w_{f}\left(\dot{w}_{in},T\right)=\varphi\dot{w}_{in}^{n} \tag{10}\]  
Here, \(\varphi\) and \(n\) represent material- and temperature-dependent model parameters, respectively, that are determined by fitting creep results at various temperatures. The inelastic strain energy density dissipation rate (\(\dot{w}_{in}\)) during the stress relaxation phase under strain control can be expressed using Equation (11), and the creep inelastic strain energy during a single cycle is represented by the shaded area in Figure 9c. \[\dot{w}_{in}=-\frac{\dot{\sigma}}{\mathbb{E}}\cdot\sigma \tag{11}\]By differentiating the Feltham stress relaxation formula (Equation (6)) with respect to the creep dwell time (\(t\)), the stress relaxation rate can be obtained as follows [38]:  
\[d_{c}=\int_{0}^{t_{c}}\frac{1}{t_{R}}\ \mathrm{d}t \tag{4}\]0  
Therefore, by combining Equations (4), (8), and (10), the creep-fatigue damage expression at a constant temperature can be derived as follows:  
\[d_{c}=\int_{0}^{t_{c}}\frac{1}{t_{R}}\ \mathrm{d}t \tag{4}\]1  
where  
\[d_{c}=\int_{0}^{t_{c}}\frac{1}{t_{R}}\ \mathrm{d}t \tag{4}\]2  
In this study, the creep damage parameters were determined by fitting \(\varphi\) and \(n\) in Equation (10) through a series of creep tests at various stress levels. The objective of this approach was to establish a relationship between the failure inelastic strain energy density and the instantaneous inelastic strain energy density dissipation rate. Specifically, for a given stress level (\(\sigma\)) in LCF tests, the values of \(\dot{w}_{in}\) and \(w_{f}\) can be obtained through the transformation from creep ductility (\(\varepsilon_{f}\)) and the creep rupture time (\(t_{R}\)), as expressed in the following equations [25]:  
\[d_{c}=\int_{0}^{t_{c}}\frac{1}{t_{R}}\ \mathrm{d}t \tag{4}\]3  
\[d_{c}=\int_{0}^{t_{c}}\frac{1}{t_{R}}\ \mathrm{d}t \tag{4}\]4  
The results of creep tests for medium-silicon molybdenum ductile iron at various temperatures and stress levels are presented in \[d_{c}=\int_{0}^{t_{c}}\frac{1}{t_{R}}\ \mathrm{d}t \tag{4}\]5  
\[d_{c}=\int_{0}^{t_{c}}\frac{1}{t_{R}}\ \mathrm{d}t \tag{4}\]6 Creep curves of medium-silicon molybdenum ductile iron at various temperatures: (a) 400 \({}^{\circ}\)C, (b) 600 \({}^{\circ}\)C, and (c) 700 \({}^{\circ}\)C.  
\[d_{c}=\int_{0}^{t_{c}}\frac{1}{t_{R}}\ \mathrm{d}t \tag{4}\]7 The shaded area within the hysteresis loop represents the energy contributing to fatigue/creep damage during a single cycle.