At present, the XFEM in ABAQUS generally uses the virtual crack closure technique (VCCT) to calculate the energy release rate, and its accuracy is higher than that of the stress intensity factor calculated by the node displacement extrapolation method. It also has a good ability to solve fatigue fracture problems involving linear, nonlinear, and even anisotropic materials [29]. In this paper, an FCP equation based on the energy release rate considering the influence of the stress ratio is first derived. Then, three material parameters in the equation are determined based on tests of bridge steel Q345 qD under different stress ratios. Finally, a user subroutine based on the XFEM that considers the influence of the stress ratio is defined. The effects of the stress ratio on FCP are discussed. The numerical simulation results considering and not considering the influence of the stress ratio are compared with the experimental results. 2 Deriving an FCP Equation Considering the Stress Ratio \(R\) Based on the Energy Release Rate \(G\)  
An FCP calculation based on the Paris equation without considering the stress ratio \(R\) is shown in Equation (1):  
\[\frac{da}{dN}=C_{0}\cdot\Delta K^{m_{0}} \tag{1}\]  
where \(C_{0}\), \(m_{0}\) are material parameters, \(\Delta K\) is the stress intensity factor range, \(a\) is the crack length, \(N\) is the number of cycles, and \(da/dN\) is the FCP rate. Based on the Paris equation, [PERSON] [9] proposed a fatigue crack growth equation that considered the stress ratio \(R\) based on \(\Delta K\), as shown in Equation (2):  
\[\frac{da}{dN}=C\left(\frac{\Delta K}{\left(1-R\right)^{1-\gamma}}\right)^{m} \tag{2}\]  
where \(C\), \(\gamma\), \(m\) are material parameters and \(R\) is the stress ratio. For opening mode cracks, the relationship between the energy release rate \(G\) and stress intensity factor \(K\) is shown in Equation (3) [31]:  
\[G=\frac{1}{E^{\prime}}K^{2} \tag{3}\]  
where \(E^{\prime}=E\) for plane stress, \(E^{\prime}=E/(1-\
u^{2})\) for plane strain, \(E\) is material elasticity modulus, and \(\
u\) is Poisson's ratio. Figure 1: Meshing at the crack tip in different numerical simulation methods. (**a**) ANSYS [25]; (**b**) ZENCRACK [30]; (**c**) XFEM [28]. The stress ratio \(R\) and intensity factor range can be calculated as Equation (4):  
\[R=K_{\text{min}}/K_{\text{max}},\ \Delta K=K_{\text{max}}-K_{\text{min}} \tag{4}\]  
Thus, the energy release rate range is expressed in Equation (5) based on the relationship between \(G\) and \(K\) in Equation (3). \[\Delta G=G_{\text{max}}-G_{\text{min}}=\frac{K_{\text{max}}^{2}-K_{\text{min}} ^{2}}{E^{\prime}} \tag{5}\]  
Substituting Equation (4) into Equation (5), Equation (6) is obtained. \[\Delta G=\frac{1}{E^{\prime}}\cdot\frac{1+R}{1-R}\Delta K^{2} \tag{6}\]  
\(\Delta K\) can be solved from Equation (6) and substituted into Equation (2). Thus, as shown in Equation (7), an FCP equation considering the stress ratio \(R\) based on the energy release rate G, which is named the da/dN-\(\Delta G\)-R equation, is obtained. \[\frac{da}{dN}=C\Bigg{(}E^{\prime}\cdot\Delta G\cdot\frac{(1-R)^{2\gamma-1}}{(1 +R)}\Bigg{)}^{\frac{\pi}{2}} \tag{7}\]  
It can be seen from the derivation process that the material parameters \(C\), \(\gamma\), \(m\) in Equation (7) are the same as those in Equation (2). Therefore, a standard compact tensile test using the specifications outlined in [32] can be used to obtain the material parameters in the da/dN-\(\Delta G\)-R equation.