## 3 Results and Discussion  
### Influence of the Amount of Load F  
In the developed XFEM model, firstly the static loading with a value of \(F_{\max}\) was applied to initiate a crack in the spot-welded structure. As seen in Figure 2a, a crack with a length of 0.0005 mm appeared on the thinner adherend of the structure next to the spot-welding point, in compliance with the observations in the literature [8,11]. When cycling loading is applied this crack expands. Figure 2b,c demonstrates how the crack expands along the width of the spot-welded structure. The validation in the numerical model was performed using the experimental results from [19]. Figure 3a compares the obtained values for Case 1, where the cyclic loading with minimum and maximum values of 2000 N and 4000 N were applied to the spot-welded structure. It was observed that \(N_{f}\) equaled to 179,251 and 212,620 cycles from the experiment and XFEM simulations. We believe that they match each other reasonably well. In the experimental study in [19], the authors did not mention how \(N_{f}\) was defined. In the simulations, the \(N_{f}\) was taken to be the number of cycles when the crack length reached to the size of the nugget diameter, 0.45 mm, as can be seen in Figure 3a. This matches well with the experimentally observed crack pattern in [42]. To see the effect of \(F\) on the fatigue performance of spot-welded structure, two extra loading cases, Cases 2 and 3 (see Table 1), were also considered, where the \(F_{\max}\) was changed into 3600 N and 3200 N, respectively, by keeping the load ratio \(R\) constant. From the simulations, \(N_{f}\) was obtained to be 287,120 and 582,180 cycles, respectively, where they matched well with those obtained experimentally with 290,223 and 485,628 cycles, respectively (see Figure 4). The obtained crack patterns are presented in Figure 3b,c, where they are similar to each other and also not very different from the one with \(F_{\max}\) = 4000 N. When the \(F_{\max}\) was reduced by 10% and 20% (from 4000 N into 3600 N and 3200 N), the \(N_{f}\) increased by 73% and 244%. Overall, it was observed that the applied cycling load influenced the fatigue life significantly. \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & **Nugget Diameter, \(d\) (mm)** & \(F_{\min}\) (kN) & \(F_{\max}\) (kN) & \(R\) \\ \hline Case 6 & 4.5 & 3000 & 4000 & 0.75 \\ \hline Case 7 & 3.5 & 2000 & 4000 & 0.5 \\ \hline Case 8 & 4.0 & 2000 & 4000 & 0.5 \\ \hline Case 9 & 5.0 & 2000 & 4000 & 0.5 \\ \hline Case 10 & 5.5 & 2000 & 4000 & 0.5 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: _Cont._ \begin{table}
\begin{tabular}{c c c} \hline \hline
**Source** & **Property** & **Austenitic Steel** \\ \hline \multirow{4}{*}{[11]} & \(E\) (GPa) & 193 \\ \cline{2-3}  & \(v\) & 0.3 \\ \cline{2-3}  & \(\sigma_{max}\) (MPa) & 608 \\ \cline{2-3}  & \(S_{y}\) (MPa) & 400 \\ \hline \multirow{4}{*}{[41]} & \(G_{\rm 1c}\) (N/m) & 6500 \\ \cline{2-3}  & \(G_{\rm 2c}\) (N/m) & 6500 \\ \cline{2-3}  & \(G_{\rm 3c}\) (N/m) & 6500 \\ \hline \multirow{2}{*}{[This paper]} & \(c_{3}\) & \(7.5\times 10^{-8}\) \\ \cline{2-3}  & \(c_{4}\) & 1.75 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The material parameters for austenitic steel used in the simulations. Figure 3: The cracks formed around the spot weld (when \(N_{f}\) was reached) with the sizes equal to the nugget diameter for Cases 1–3 (**a–c**). Figure 2: The nucleation of the crack around the spot weld on the thinner adherend of the structure at the end of the static loading (**a**) and its expansion in the course of cyclic loading (**b**,**c**). The crack growth rate is of paramount importance in assessing the structural integrity and durability of materials and components. It provides critical information about the rate at which cracks propagate under cyclic loading conditions, which directly influences the remaining fatigue life and the potential for catastrophic failure.