# Torsional Fatigue Performance of a Spot-Welded Structure: An XFEM Analysis
## 2 XFEM Modelling  
In this study, XFEM, linear elastic fracture mechanics, and Paris Law were integrated together to simulate fatigue crack growth. In XFEM analysis, a displacement vector function \(u\) with the partition of unity enrichment is approximated as:  
\[u(x)=\sum_{I\in N}N_{I}(x)\left[u_{I}+H(x)a_{I}+\sum_{\alpha=1}^{4}F_{\alpha}(x )b_{I}^{\alpha}\right] \tag{1}\]  
where \(N_{I}(x)\) represents the typical nodal shape functions; \(u_{I}\), the typical nodal displacement vector associated with the continuous portion of the finite element solution, is the first term on the right-hand side of the above equation. The second term in the equation is the result of multiplying the nodal enriched degree of freedom vector (\(a_{I}\)) by the associated discontinuous jump function \(H(x)\) across the crack surfaces. The third term is the product of \(b_{I}^{\alpha}\), the nodal enriched degree of freedom vector, and the related elastic asymptotic crack-tip functions \(F_{\alpha}(x)\). Here, all of the nodes in the model are covered by the first term on the right; nodes whose shape function support is cut by the crack interior only fall under the second term; and nodes whose shape function support is cut by the crack tip only fall under the third term. Additional information on it can be obtained elsewhere [16]. Figure 1a shows a three-dimensional XFEM model of the spot-welded box profile under torsional loading. It was developed using Abaqus 6.14. The box was subjected to the torsional loading from one end with a length of 70 mm, while the other end with the same size was fixed in all directions. This load was applied to the reference point shown in Figure 1a, located at the geometrical center of the loaded part of the box, and this point was  
controlling the concerned region through the coupling constraint defined as in the software. XFEM was defined as every point of both adherends joined together. To simulate spot welding, the interfaces of the two adherends on the welding side were specified as master and slave surfaces using the tie constraint module included in Abaqus. The dimensions of the XFEM model are shown in Figure 1a. These dimensions were selected based on those of experiments performed in [11; 17]. The model was validated using these experimental results. A brief self-descriptive equation used in XFEM calculations is presented in Table 1. Its details can be found elsewhere [8; 16]. To replicate the fatigue loading, a direct cyclic analysis was performed using 25 Fourier terms (to directly acquire the stabilized cyclic response, the direct cyclic technique combines a modified Newton approach with a Fourier representation of the solution and the residual vector). In order to induce crack formation at the stress concentration areas, \(T_{\text{max}}\) (the maximum torque value applied in the fatigue analysis) was used in a static stage prior to the loading cycle. Figure 1: Details about the XFEM model of a spot-welded box exposed to torsional fatigue loading (**a**) and the FE mesh used in the analysis (**b**). Details of the mesh used in the model are presented in Figure 1b. A finer mesh was utilized in the welding region, while a coarser one was considered in the other portions of both adherends. To ensure accuracy of the model, a mesh convergence study was performed. Three simulations with three different mesh sizes, namely 0.000375 mm, 0.000750 mm, and 0.0015 mm, were performed when the box was subjected to static \(T_{\text{max}}\) loading. The maximum von Mises stress value obtained from each model, observed in the element next to the welding regions, was compared. The concerned change was less than 3.5% when the mesh size was changed from 0.00075 mm to 0.000375 mm, which was more than 10.0% when the mesh was changed from the coarsest one into the medium one. The model was therefore discretized with an element size of 0.00075 mm.