# Study on the Prediction Method of the Ultra-Low-Cycle Fatigue Damage of Steel
## 2 Theoretical Model for ULCF of Structural Steel
### CVGM for ULCF  
On the basis of the void growth model, [PERSON] et al. [31] proposed a CVGM for predicting the ductile cracking of ULCF. For cyclic strain loading, the sign of stress triaxiality (\(T\)) changes, that is, it is positive under tension and negative under compression.  
Therefore, the cavity volume growth rate under cyclic strain loading can be expressed as Equation (4).  
\[\mathrm{d}r/r=\mathrm{sign}(T)C\exp{(\mathrm{l}1.5T]}\mathrm{d}\varepsilon_{p} \tag{4}\]  
\[\mathrm{sign}(T)=\left\{\begin{array}{ll}0&T<0\\ 1&T\geq 0\end{array}\right. \tag{5}\]  
where \(r\) is the instantaneous void radius, \(C\) represents the material constant, \(\mathrm{sign}(T)\) describes the sign of stress triaxiality shown as Equation (5). When \(T\) is positive, tensile strain exists in the material. Otherwise, compressive strain exists in the material.  
Void radius during cyclic strain loading can be obtained by integrating Equation (6).  
\[\ln{(r/r_{0})}_{cyclic}=\sum_{\textit{tensile}}C_{1}\int_{\varepsilon_{1}}^{ \varepsilon_{2}}\exp{(\mathrm{l}1.5T]}d\varepsilon_{p}-\sum_{\textit{ compressive}}C_{2}\int_{\varepsilon_{1}}^{\varepsilon_{2}}\exp{(\mathrm{l}1.5T]})d \varepsilon_{p} \tag{6}\]  
where \(\varepsilon_{1}\) and \(\varepsilon_{2}\) are the equivalent plastic strains at the beginning and end of the tension or compression cycle, respectively. \(C_{1}\) and \(C_{2}\) represent the expansion and contraction rates of micropores, respectively.  
Owing to the lack of sufficient experimental data, [PERSON] et al. [31] assumed that the expansion and contraction rates of micropores are the same, which are consistent with the expansion rates of micropores in the case of monotonic tension (\(C_{1}=C_{2}=C\)). Equation (13) can be changed to Equations (7) and (8).  
\[\ln{(r/r_{0})}_{cyclic}=C\cdot\left[\sum_{\textit{tensile}}\int_{\varepsilon_ {1}}^{\varepsilon_{2}}\exp{(\mathrm{l}1.5T]})d\varepsilon_{p}-\sum_{\textit{ compressive}}\int_{\varepsilon_{1}}^{\varepsilon_{2}}\exp{(\mathrm{l}1.5T]})d \varepsilon_{p}\right] \tag{7}\]  
\[VGI_{cyclic}=\sum_{\textit{tensile}}\int_{\varepsilon_{1}}^{\varepsilon_{2}} \exp{(\mathrm{l}1.5T]})d\varepsilon_{p}-\sum_{\textit{ compressive}}\int_{\varepsilon_{1}}^{\varepsilon_{2}}\exp{(\mathrm{l}1.5T]})d \varepsilon_{p} \tag{8}\]  
where \(VGI_{cyclic}\) is the hole expansion index [31].  
Under cyclic loading, critical hole expansion index \(VGI_{cyclic}^{\textit{critical}}\) is defined as follows [31]:  
\[VGI_{cyclic}^{\textit{critical}}=\eta\exp{(-\lambda_{CVGM}\varepsilon_{p}^{ \textit{accumulated}})} \tag{9}\]  
where \(\lambda_{CVGM}\) is the damage degradation parameter of materials under cyclic loading, \(\eta\) represents the monotonic void growth 'capacity', that is, toughness parameters, and \(\varepsilon_{p}^{accumulated}\) is the equivalent plastic strain at the beginning of each tension cycle.  
When hole expansion index \(VGI_{cyclic}\) exceeds critical hole expansion index \(VGI_{cyclic}^{\textit{critical}}\), the ductile cracking of ULCF occurs.  
\[VGI_{cyclic}\geq VGI_{cyclic}^{\textit{critical}} \tag{10}\]  
The fracture index \(FI_{CVGM}\) of CVGM is defined as Equation (11).  
\[FI_{CVGM}=VGI_{cyclic}/VGI_{cyclic}^{\textit{critical}} \tag{11}\]  
When fracture index \(FI_{CVGM}\geq 1\), the material has ULCF ductile cracks.