# Analysis of Fatigue Performance of Spot-Welded Steel T-Profiles Under Cyclic Torsional Loading
## 2 XFEM Modelling
Figure 2: The details about the XFEM modelling of the T-profile assembly subjected to cyclic torsional loading. The simulations employed the direct cyclic approach in Abaqus 2021/Standard, combining the XFEM and linear elastic fracture mechanics (LEFM). The crack's nucleation and evolution were modeled using the Paris law, which relates the fracture energy release rate to crack growth. \(G_{pl}=0.85\)\(G_{eq,\ c}\) and \(G_{thresh}=0.01\)\(G_{eq,\ c}\) were the definitions of the maximum energy release rate and the lower limit, respectively. \(G_{eq,\ c}\) stands for the critical equivalent strain energy release rate, where the following mode-mix criterion (using the linear power law model) was used to calculate it. \[\frac{G_{eq}}{G_{eq,c}}=\left(\frac{G_{1}}{G_{1c}}\right)^{\alpha_{1}}+\left( \frac{G_{2}}{G_{2c}}\right)^{\alpha_{2}}+\left(\frac{G_{3}}{G_{3c}}\right)^{ \alpha_{3}} \tag{2}\]  
The opening (Mode I), first shear (Mode II), and second shear (Mode III) critical energy release rates in this model are represented by the symbols \(G_{1c}\), \(G_{2c}\), and \(G_{3c}\), respectively. In the analysis, it is assumed that the coefficients \(\alpha_{1}\), \(\alpha_{2}\) and \(\alpha_{3}\) have a value of 1.0. The fatigue fracture in the structure is measured during the low-cycle fatigue study by \(\Delta G\), which is the difference in energy release rates between the minimum and maximum loads. The Paris Law, which is written as \(da/dN=c_{3}\Delta G^{c_{4}}\), was used to calculate the progress rate of the crack per cycle. The crack length is denoted by the letter \(a\), the cycle number is \(N\), and the material constants are \(c_{3}\) and \(c_{4}\)[12]. In order to appropriately depict the beginning and progression of the crack, it is crucial to satisfy the condition \(G_{pl}>G_{max}>G_{thresh}\) in the computations. Following the start of a crack, the following steps are taken: one element was released at the interface at the end of each cycle \(N\), allowing the software to advance the crack length (\(a_{N}\)) progressively from the current cycle to \(a_{N+\Delta N}\). \(\Delta N_{j}\), where j is the node that corresponds to the crack tip and \(N\) is the number of cycles required to produce failure in each interface element at the crack tip. The determined node spacing at the interface elements close to the crack tips (\(\Delta a_{Nj}=a_{N+\Delta N}-a_{N}\)) and the material constants \(c_{3}\) and \(c_{4}\) were used in this calculation. The goal of the analysis was to release one or more interface elements at the end of each stable loading cycle. Thus, \(\Delta N_{min}=\min\) (\(\Delta N_{j}\)), which indicates the number of cycles necessary for the crack to propagate across its element length, \(\Delta a_{Nmin}=\min\) (\(\Delta a_{Nj}\)), was found to be the element that required the fewest cycles for  
Figure 3: XFEM flowchart [12]. propagation. As a result, zero stiffness and constraint were achieved when the element requiring the fewest cycles was found to be suitable for release. According to [12], with the interface element released during the next cycle, a new relative fracture energy release rate was computed for the interface elements at the crack tip. The material constants used in the simulations are listed in Table 1. They are mainly the elastic modulus, Poisson's ratio, tensile strength, the critical energy release rates in different modes (\(G_{1c}\), \(G_{2c}\) and \(G_{3c}\)), and the material constants used in the Paris Law (\(c_{3}\) and \(c_{4}\)). When \(T\) and \(-T\) were applied during cyclic torsional loading, the spot-welded assembly's deformed shape was recorded, as shown in Figure 4.