# Torsional Fatigue Performance of a Spot-Welded Structure: An XFEM Analysis
## 3 Results and Discussion  
In this section, the underlying deformation mechanisms of the spot-welded box subjected to torsional loading were investigated. Figure 3 demonstrates the deformed shapes of the box with displacement profile when it was subjected to min (\(T_{\text{min}}=-T\)) and maximum (\(T_{\text{max}}=T\)) loadings for \(R=-1.0\). While the displacements are reaching their maximums at the side torsioned, those of another end are 0. In the simulations, it was observed that the spot-welding spots on the overlapping surfaces of the welded box served as the cracks' nucleation point, and as the structure underwent cyclic loads, they subsequently spread out along its width. Here, the cracks mainly emerged at the spot-welded twins' at the fifth and sixth rows from the fixed side. The cracks formed at the overlapping surfaces reached the free surfaces after a significant number of cycles. Figure 4 depicts the cracks nucleated at the overlapping and free surfaces of the upper and lower adherends when \(R=0\) at 500,000 cycles. Obviously, the cracks formed and expanded at the joined surfaces of the upper and lower adherends more significantly than those at the free surfaces.  
Figure 3: Deformed shape of the spot-welded box when \(T_{\text{min}}=-T\) (**a**) and \(T_{\text{max}}=T\) (**b**) was applied during fatigue torsional loading (deformation was scaled 100 times for better visualization).  
Figure 2: The resulting crack initiation sites when coarse (**a**) and medium (**b**) mesh sizes were used in the simulations.  
For the validation of the developed XFEM model of the spot-welded box torsioned, the experimental data presented in [17] were used. Apart from the box with the geometry presented in Section 2, a thicker adherent twin with 2.0 mm thickness, where they are joined with 6.5 mm diameter spot welds was also considered. Experimentally and numerically obtained failure cycles (\(N_{f}\)) are presented in Table 2. In the simulations, \(N_{f}\) was taken to be the number of cycles at which the crack length reached the nugget diameter. Overall, experimentally and numerically obtained failure cycles for two different configurations are matching each other. That validates the developed numerical model. That also shows the accuracy of the selected material constants \(c_{3}\) and \(c_{4}\). The work in [17] also indicated that the 5 th and 6 th spot weld twins were the two most critical locations subject to fatigue failure. Ref. [17] shows the experimentally observed crack evolved around the 5 th spot welding; this agrees pretty well with our conclusions, which are shown in Figure 4. It is worth mentioning that internal/residual stresses and variations in temperature and microstructural characteristics of the material introduced during the welding process were not taken into account while modeling the spot-welds.  
In this work, different model parameters were varied to explore the fatigue behavior of spot-welded box torsioned. They are mainly the total number of welds used in the joining, the nugget diameter (\(D\)), and the load ratio (\(R\)).  
\begin{table}
\begin{tabular}{c c c c} \hline \hline  & Experiments [17] & XFEM \\ \hline \(t\) & \(D\), \(T\) & \(N_{f}\) & \\ \hline
1.5 mm & 5.5 mm, \(\pm\)225 Nm & 45,834 & 42,375 \\ \hline
2.0 mm & 6.5 mm, \(\pm\)320 Nm & 28,000 & 26,950 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Experimentally and numerically obtained failure cycles for two different cases.  
Figure 4: The illustration of the cracks that spread around the spot weld at the top and lower adherends’ overlapping and opposing surfaces at \(N\) = 500,000 cycles (\(R\) = 0.0).