# Study on the Prediction Method of the Ultra-Low-Cycle Fatigue Damage of Steel
## 3 ULCF Life Prediction for the Q345 qC Base Material and Weld Joints
### ULCF Predictions for Base Material and Weld Joints
#### 3.2.2 Fatigue Life Prediction Based on the CDM Model  
The developments of the damage variable \(D\) of the base metal and weld specimens at strain ranges of 7%, 8%, 9% and 10% are shown in Figures 7 and 8, respectively. The base metal and weld specimens are not damaged when cumulative equivalent plastic strain \(\varepsilon\) is less than uniaxial plastic strain threshold \(\varepsilon_{th}\). Therefore, damage variable \(D\) equals 0 at the beginning of the cycle. The base metal and weld specimens exhibit damage when \(\varepsilon\) is larger than \(\varepsilon_{th}\). According to Equation (8), only the damage caused by tensile strain is considered. Thus, \(D\) exponentially increases with a stepped pattern. Under the same strain range, the \(D\) value of the cross section centres of the base metal and weld specimens reach \(D_{cr}\) earlier than that of the corresponding cross section edges. This result indicates that the predicted cracks originate from the cross section centre. CDM can predict not only the fatigue life \(N_{f}\) but also the number of cycles to initial damage \(N_{0}\). In addition, the \(N_{f}\) and \(N_{0}\) values of the base metal and weld specimens decrease gradually with the increase in total strain range. Given that \(dD\) is proportional to \(d\varepsilon^{p+}\), as shown in Equation (8), a large total strain range results in large \(dD\) and \(d\varepsilon^{p+}\), which leads to low \(N_{0}\) and \(N_{f}\). Figure 6: 3D finite element model of the weld specimen. The comparisons of the predicted and test lives for the base metal and weld specimens are shown in Tables 4 and 5, respectively. \(N_{f,C}^{p}\) and \(N_{f,E}^{p}\) are the prediction fatigue lives at the cross section centre and edge, respectively. \(N_{f}\) and \(N_{r}\) are the test fatigue and fracture lives, respectively. Given that the value of \(N_{f,C}^{p}\) is less than the value of \(N_{f,E}^{p}\), \(N_{f,C}^{p}\) and \(N_{f,E}^{p}\) can be considered the predicted fatigue and fracture lives, respectively. Similarly, \(N_{f,\text{WM}}^{p}\) is less than \(N_{f,\text{HAZ}}^{p}\). Thus, \(N_{f,\text{WM}}^{p}\) and \(N_{f,\text{HAZ}}^{p}\) can be considered the predicted fatigue and fracture lives, respectively. The error of the predicted and test fatigue lives decreases with the increase in strain. The proposed CDM is suitable for predicting the ductile fracture characteristics of materials, and a large strain range leads to highly ductile fracture characteristics of steel. Thus, the predicted fatigue and fracture lives are highly accurate in the case of large strain range. In the case of base metal and weld specimens, the predicted fatigue life is less than that in the test fatigue life. According to the comparisons in Tables 4 and 5, the predicted and test lives of the weld specimen are all less than those of the base metal specimen. Figure 8: Development of the damage variable \(D\) of the weld specimen at strain ranges of (**a**) 7% and (**b**) 10%. Figure 7: Development of the damage variable \(D\) of the base metal specimen at strain ranges of (**a**) 7% and (**b**) 10%. The relationships between the fatigue and fracture lives of the base metal and weld specimens in the CDM model are shown in Tables 6 and 7. \(N^{p}\!/N^{p}\!_{r}\) is the ratio of predicted fatigue life to the predicted fracture life, whilst \(N\!/N\!_{r}\) is the ratio of the test fatigue life to the test fracture life. For the base metal specimen, the value of \(N^{p}\!/N^{p}\!_{r}\) is very near the value of \(N\!/N\!_{r}\), both of which are over 90%. The fatigue life of the base metal specimen takes up most of the fracture life. When crack initiates, the crack will expand rapidly and fracture after several cycles. For the weld specimen, the value of \(N^{p}\!/N^{p}\!_{r}\) is between 71.0% and 78.8%, whilst the test value is between 87.5% and 96.3%. Therefore, the crack initiation life of the weld specimen still occupies most of the fracture life.