## 3 An Energy-Based Fatigue Life Prediction Method
### Energy-Based Fatigue Life Model  
In this paper, the intrinsic dissipation \(d_{1}\) is regarded as the fatigue indicator. Once the fatigue damage is produced by each cyclic load, the dissipated energy is accumulated. In order to simplify calculation of the accumulated intrinsic dissipation \(d_{1c}\), the time \(t\) could be converted into the term \(N/f\) (\(f=N/t\)). Then, the total accumulated intrinsic dissipation \(d_{1c}\) could be obtained from the definite integral of the intrinsic dissipation \(d_{1}\) during the whole life cycle:  
\[d_{1c}=\int_{0}^{N_{f}}d_{1}/fdN \tag{5}\]  
Based on the damage evolution law, the total accumulated intrinsic dissipation \(d_{1c}\) has clear physical meaning, which represents the energy tolerance to failure of a kind of material applied by the cyclic loading. In other words, it is equal to the energy tolerance to failure \(E_{c}\). In the Equation (4), it is noticed that the term \(\partial\theta/\partial t\) would be zero if the asymptotic portion of temperature increase in stage II \(\theta_{AS}\) was constant. A large number of studies [23; 24] support this conclusion for most pure materials. However, the welding process brings impurities and porosities, which results in non-linear fatigue damage evolution behavior and varying thermal dissipation during the whole fatigue process. Some researchers [22; 23; 24; 25; 26; 27] also found there is the linear temperature evolution after the material gets into a stabilized stage II, as shown in Figure 9. Then, the energy tolerance to failure \(E_{c}\) could be evaluated like this:  
\[E_{c}=d_{1c}=\int_{0}^{N_{f}}\frac{\rho C}{f}(\frac{\partial\theta}{\partial N }+\frac{\theta}{\tau})dN\]  
\[E_{c}=\int_{0}^{N_{f}}\frac{\rho C}{f}(\frac{\partial\theta}{\partial N}+\frac{ \theta_{AS}+\lambda N}{\tau})dN \tag{8}\]  
In fact, the number of cycles in both initial stage I and final stage III only accounts for a small part of whole life cycles, and could be ignored in the following definite integral of Equation (8). Then, its integrating result is indicated like this:  
\[E_{c}=\frac{\rho C\lambda}{2f\tau}{N_{f}}^{2}+\frac{\rho C\lambda}{f}N_{f}+ \frac{\rho C\theta_{AS}}{f\tau}N_{f} \tag{9}\]  
According to Equation (9), the fatigue life \(N_{f}\) related with dissipated energy could be calculated as the following equation:  
\[N_{f}=\frac{f\tau}{\rho C\lambda}(\sqrt{2\frac{\rho C\lambda}{f\tau}E_{c}+( \frac{\rho C}{f}\lambda+\frac{\rho C\theta_{AS}}{f\tau})^{2}}-\frac{\rho C}{f} \lambda-\frac{\rho C\theta_{AS}}{f\tau}) \tag{10}\]  
Based on Equation (10), the cyclic loading had to at least reach into the early stage II for calculating the asymptotic portion of temperature increase \(\theta_{AS}\) and the temperature increment rate \(\lambda\). Once these two parameters were defined, the remaining lifetime or fatigue damage could be derived according to Equation (10) regardless of whether the thermal measurement is suspended or not. In order to calculate intrinsic dissipation, the thermo-physical parameters and loading frequency needed to calculate the intrinsic dissipation are given in Table 3. It is worth mentioning that the time parameter \(\tau\) was ascertained by calculating the heat loss from the peak temperature to the room temperature, also listed in Table 3.  
Furthermore, from the above Equation (9), the energy tolerance to failure \(E_{c}\) could be estimated once the fatigue life and loading conditions are determined. Combined, the thermal evolution in the whole lifespan under constant stress amplitudes with thermo-physical parameters, and the energy tolerances to failure \(E_{c}\) for different stress amplitudes are listed in Table 4. It is very interesting to note that all energy tolerances to failure \(E_{c}\) are close to 1.0 \(\times\) 10\({}^{9}\) J/m\({}^{3}\). The energy tolerance to failure \(E_{c}\) represents the inherent nature of the material or structure in some way and is independent of loading