# Study on the Prediction Method of the Ultra-Low-Cycle Fatigue Damage of Steel
## 2 Theoretical Model for ULCF of Structural Steel
### Material Property  
The constitutive model of the base metal and weld specimens under cyclic loading is mixed hardening model [32]. The model includes two parts, namely, nonlinear equivalent hardening and kinematic hardening, as shown in Figure 1. The model can simulate the strength evolution of materials under cyclic loading. Chaboche mixed hardening model can reflect the strength evolution of materials under cyclic loading.  
The part of nonlinear equivalent hardening is represented by the change in yield surface. \(\sigma^{0}\) is the nonlinear function of equivalent plastic strain \(\varepsilon^{p}\).  
\[\sigma^{0}=\sigma|_{0}+Q_{\infty}[1-\exp{(-be^{p})}] \tag{12}\]  
where \(\sigma^{0}\) represents the size of the initial yielding surface, \(\sigma|_{0}\) is the yield stress when the equivalent plastic strain is 0, \(Q_{\infty}\) is the maximum change value in yield surface and \(b\) is the rate of change of the yield surface with the increase in equivalent plastic strain. The yield surface size \(o_{i}^{0}\) at circle \(i\) (\(i=1\), 2, 3,...) is obtained by Equation (13).  
\[\sigma_{i}^{0}=\frac{\sigma_{i}^{t}-\sigma_{i}^{c}}{2} \tag{13}\]  
where \(\sigma_{i}^{t}\) and \(\sigma_{i}^{c}\) are the maximum tensile and compressive stresses at circle \(i\), respectively.  
\(\varepsilon_{i}^{p}\) is the equivalent plastic strain corresponding to \(\sigma_{i}^{0}\).  
\[\varepsilon_{i}^{p}=\frac{1}{2}(4i-3)\Delta\varepsilon^{p} \tag{14}\]  
where \(\Delta\varepsilon^{p}\) is the plastic strain range, which can be calculated by Equation (15).  
\[\Delta\varepsilon^{p}\approx\Delta\varepsilon-2\sigma_{1}^{t}/E \tag{15}\]  
where \(\Delta\varepsilon\) is the total strain.  
The variation law of back stress \(\alpha\) with equivalent plastic strain \(\varepsilon^{p}\) is given in Equations (16) and (17).  
\[\alpha_{k}=\frac{C_{k}}{\gamma_{k}}[1-\exp(-\gamma_{k}\varepsilon^{p})]+ \alpha_{k,1}\exp(-\gamma_{k}\varepsilon^{p}) \tag{16}\]  
\[\alpha=\sum_{k=1}^{N}\alpha_{k} \tag{17}\]  
Figure 1: Chaboche mixed hardening model. (**a**) Nonlinear equivalent hardening. (**b**) Kinematic hardening.  
where \(\alpha_{k}\) is the '\(k\)'\(th\) back stress, \(\alpha_{k,1}\) is the '\(k\)'\(th\) back stress at the first data point and \(C_{k}\)/\(\gamma_{k}\) is the maximum change value in the back stress. \(\gamma_{k}\) determines the rate at which the back stress increases with the rise in plastic strain. Given that the shape of the curve varies greatly with different strain ranges, a more accurate curve can be obtained through multi-back stress superposition.  
In data pairs (\(\sigma_{i}\), \(\varepsilon_{i}^{p}\)), back stress value \(\alpha_{i}\) can be determined from Figure 1b.  
\[\alpha_{i}=\sigma_{i}-\sigma^{s} \tag{18}\]  
where \(\sigma^{s}\) is the average value of first data point \(\sigma_{1}\) and last data point \(\sigma_{n}\).  
The equivalent plastic strain \(\varepsilon_{i}^{p}\) is obtained by the following equation:  
\[\varepsilon_{i}^{p}=\varepsilon_{i}-\frac{\sigma_{i}}{E}-\varepsilon_{p}^{0} \tag{19}\]  
where \(\varepsilon_{1}^{p}\) equals 0, and \(\varepsilon_{p}^{0}\) is the equivalent plastic strain value when the curve passes through the strain axis. Data pairs (\(\alpha_{i}\), \(\varepsilon_{1}^{p}\)) and (\(\alpha_{1}\), 0) should satisfy Equations (16) and (17). Material parameters \(C_{k}\) and \(\gamma_{k}\) can be obtained by fitting the test data.  
According to the above-mentioned method, the Chaboche mixed hardening model parameters of Q345 qC base metal, heat-affected zone (HAZ) and weld metal are calibrated, and the calibration results are shown in Table 1.