## 2 XFEM Modelling  
A numerical analysis was conducted to evaluate the fatigue life of a spot-welded structure. The analysis employed the direct cyclic approach in Abaqus/Standard, incorporating the extended finite element method (XFEM) and linear elastic fracture mechanics (LEFM) to simulate crack propagation. The Paris Law, which relates the fracture energy release rate to crack growth, was utilized to model the initiation and progression of the crack. The maximum energy release rate (\(G_{pl}\)) and the lower limit (\(G_{thresh}\)) were defined as \(G_{pl}\) = 0.85 \(G_{eq,~{}c}\) and \(G_{thresh}\) = 0.01 \(G_{eq,~{}c}\)[38], respectively, where \(G_{eq,~{}c}\) represents the critical equivalent strain energy release rate. The calculation of \(G_{eq,~{}c}\) was based on the mode-mix criterion using the linear power law model. The calculation is performed according to the mode-mix criterion, utilizing the linear power law model [39]. \[\frac{G_{eq}}{G_{eq,~{}c}}=\left(\frac{G_{1}}{G_{1c}}\right)^{\alpha_{1}}+ \left(\frac{G_{2}}{G_{2c}}\right)^{\alpha_{2}}+\left(\frac{G_{3}}{G_{1c}} \right)^{\alpha_{3}} \tag{1}\]  
In this model, the critical energy release rates in the opening (Mode I), first shear (Mode II), and second shear (Mode III) modes are denoted as \(G_{1c}\), \(G_{2c}\), and \(G_{3c}\), respectively. The coefficients \(\alpha_{1}\), \(\alpha_{2}\), and \(\alpha_{3}\) are assumed to have a value of 1.0 in the analysis. During the low-cycle fatigue analysis, the difference in energy release rates between the minimum and maximum loads, referred to as \(\Delta G\), serves as a measure of the fatigue crack in the structure. The growth rate of the crack per cycle was determined using the Paris Law, expressed in Equation (2), where _'a'_ represents the crack length, \(N\) is the cycle number, and \(c_{3}\) and \(c_{4}\) are material constants [38]. \[\frac{da}{dN}=c_{3}(\Delta G)^{c_{4}} \tag{2}\]  
In Abaqus, the crack propagation rate was defined in terms of the energy release rate rather than the stress intensity factor, resulting in slightly different definitions for \(c_{3}\) and \(c_{4}\) compared to the classical Paris Law constants \(C\) and \(m\) (\(da/dN=C\Delta K^{m}\)). The relationship between these constants is given as the following:  
\[c_{3}=CE^{c_{4}}, \tag{3}\]  
\[c_{4}=\frac{m}{2}. \tag{4}\]  
It is important to satisfy the inequality given in Equation (5) in the computations to accurately represent the onset and growth of the crack. \[G_{pl}>G_{max}>G_{thresh} \tag{5}\]  
After crack initiation, the procedure proceeded as follows: The software incrementally advanced the crack length (\(a_{N}\)) from the current cycle to \(a_{N+\Delta N}\) by releasing one element at the interface after each cycle \(N\). The number of cycles needed to cause failure in each interface element at the crack tip was determined as \(\Delta N_{j}\), where \(j\) represents the node corresponding to the crack tip. This calculation relied on the material constants \(c_{3}\) and \(c_{4}\), along with the determined node spacing at the interface elements near the crack tips as given in Equation (6). \[\Delta a_{Nj}=a_{N+\Delta N}-a_{N} \tag{6}\]  
At the end of each stable loading cycle, the analysis aimed to release at least one interface element. In this regard, the element that required the least number of cycles for propagation was identified, denoted as \(\Delta N_{min}\) = min (\(\Delta N_{j}\)), indicating the number of cycles needed for the crack to propagate over its element length, \(\Delta a_{Nmin}\) = min (\(\Delta a_{Nj}\)). Consequently, the element with the fewest required cycles was determined to be ready for release, resulting in zero stiffness and constraint. During the subsequent cycle, a new relative fracture energy release rate was calculated for the interface elements at the crack tip due to the release of the interface element, as stated in [38].