# Study on the Prediction Method of the Ultra-Low-Cycle Fatigue Damage of Steel
## 2 Theoretical Model for ULCF of Structural Steel
### Calibration of Material Parameters for CVGM
#### 2.5.2 Calibration of Material Parameters for CVGM by Finite Element Analysis  
The two-dimensional axisymmetric finite element models of a notched round bar specimen for base metal, HAZ and weld metal are established, as shown in Figure 3, to calibrate toughness parameter \(\eta\). The control deformation in the monotonic loading test is applied to the finite element analysis. Toughness parameter \(\eta\) can be calibrated by Equation (20).  
\[\eta=\int\limits_{0}^{\epsilon_{p}^{critical}}\exp{(1.5T)}\cdot d\varepsilon_{p} \tag{20}\]  
where \(\epsilon_{p}^{critical}\) is the critical equivalent plastic strain.  
Calibration results of the material parameters for \(\eta\) are shown in Table 2. The \(\eta\) value of the base metal is slightly larger than that of HAZ and significantly larger than that of weld metal. This finding indicates that the fracture toughness of the base metal is the strongest, followed by that of HAZ and the weld metal is the weakest. The large dispersion coefficient of weld metal indicates that the welding quality is unstable.  
In addition, the damage degradation parameter \(\lambda_{CVGM}\) can be calibrated by the cyclic loading test and finite element analysis. The loading form in the cyclic loading test is applied to the finite element model in the analysis, and hole expansion index \(VGI_{cyclic}\) is recorded when the cyclic number reaches the fatigue life \(N_{f}\) of test. Assuming that \(VGI_{cyclic}\) equals critical hole expansion index \(VGI_{cyclic}^{critical}\), \(\lambda_{CVGM}\) can be calibrated by Equation (16).  
Figure 2: Distribution of equivalent plastic strain and stress triaxiality at the root of the notch section. (**a**) Distribution of equivalent plastic strain. (**b**) Distribution of stress triaxiality.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Material** & **R/mm** & **Number** & \(\delta\)**/mm** & \(\eta\) \\ \hline \multirow{8}{*}{Base metal} & 4.25 & BLM-1 & 2.240 & 2.981 \\  & & BLM-2 & 2.262 & 3.000 \\ \cline{2-5}  & \multirow{2}{*}{3.0} & BMM-1 & 1.876 & 2.902 \\  & & BMM-2 & 1.917 & 2.965 \\ \cline{2-5}  & \multirow{2}{*}{1.5} & BSM-1 & 3.543 & 3.280 \\  & & BSM-2 & 3.477 & 3.163 \\ \cline{2-5}  & \multirow{2}{*}{4.25} & Average & 3.048 \\  & & Dispersion coefficient & 4.7\% \\ \hline \multirow{8}{*}{Heat-affected zone} & 4.25 & HLM-1 & 2.098 & 2.759 \\  & & HLM-2 & 1.934 & 2.576 \\ \cline{2-5}  & \multirow{2}{*}{3.0} & HMM-1 & 1.895 & 2.795 \\  & & HMM-2 & 1.922 & 2.947 \\ \cline{2-5}  & \multirow{2}{*}{1.5} & HSM-1 & 3.808 & 3.242 \\  & & HSM-2 & 4.009 & 3.536 \\ \cline{2-5}  & \multirow{2}{*}{4.25} & Average & 2.976 \\  & & Dispersion coefficient & 11.9\% \\ \hline \multirow{8}{*}{Weld metal} & 4.25 & WLM-1 & 1.821 & 2.711 \\  & & WLM-2 & 1.672 & 2.521 \\ \cline{1-1} \cline{2-5}  & \multirow{2}{*}{3.0} & WMM-1 & 1.371 & 2.591 \\ \cline{1-1}  & & WMM-2 & 1.306 & 2.454 \\ \cline{1-1} \cline{2-5}  & \multirow{2}{*}{1.5} & WSM-1 & 2.593 & 1.508 \\ \cline{1-1}  & & WSM-2 & 2.264 & 1.103 \\ \cline{1-1} \cline{2-5}  & \multirow{2}{*}{Average Dispersion coefficient} & 2.148 \\ \cline{1-1}  & & Dispersion coefficient & 31.2\% \\ \hline \hline \end{tabular}
\end{table}
Table 2: Calibration results of material parameters for \(\eta\).  
Figure 3: Two-dimensional axisymmetric finite element model of a notched round bar specimen.