## 3 FE Modeling  
To construct the FE model, the model parameters described in Section 2 must be calibrated. The material used for modeling was SAC305, and the hardening parameters obtained from [37] are listed in Table 1. The fracture criterion parameters for SAC305 material were derived from the Johnson-Cook damage model [38] and are listed in Table 2. Figure 1,b show the hardening curve and fracture strain limit of the model, respectively.  
Figure 2 illustrates the 3D FE model for analyzing the deformation of the solder ball. The fine mesh was organized to ensure accuracy, especially in the corner zone of the solder. Three-dimensional linear elements with eight nodes (C3D8R) were used. In this model, the chip and PCB were assumed to be rigid bodies, while the solder ball was modeled as a deformable body. Tie constraints were applied at both interfaces where the solder ball contacts the PCB and chip. However, this approach is not entirely representative of the actual structure as it does not consider the IMC, which prevents the relative displacement difference at the solder-PCB and -chip contacts. Some studies [27; 28] have attempted to model IMC in failure simulations, but they only use elastic properties, which cannot yield sufficiently accurate results. To account for the IMC, a clear understanding of its inelastic properties is necessary, but this information is difficult to find in the literature and may require additional experimentation. This study focuses on the changes in inelastic deformation in the solder under varying external loading conditions while maintaining other conditions constant. Therefore, the model does not consider the IMC and leaves it for future work, as discussed in the Conclusions section. Figure 3 shows a schematic of the seven loading conditions, and Table 3 provides a detailed description of the loading cases. Each loading condition was applied as a single loading case involving shear, compression, and bending for cases 1 to 3, respectively. Dual loading cases were compared for cases 4 to 6. Lastly, case 7 involved complex loading, which combined all three loading conditions. The loading directions of shear, compression, and bending are shown in Figure 2 as red, blue, and green arrows, respectively.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline \(A\) & \(B\) & \(n\) \\ \hline
38 & 275 & 0.710 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Hardening parameters [34].  
Figure 1: Material behavior for SAC305 (**a**) hardening curve and (**b**) fracture strain with respect to triaxiality.  
Figure 3: Schematic diagram of single, dual, and complex loading conditions.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \multicolumn{1}{c}{**Case**} & \multicolumn{1}{c}{**Shear**} & \multicolumn{1}{c}{**Compression**} & \multicolumn{1}{c}{**Bending**} \\ \hline \multirow{3}{*}{Single loading} & Case 1 (S) & \(d=0.08\) mm & - & - \\ \cline{2-5}  & Case 2 (C) & - & \(d=0.08\) mm & - \\ \cline{2-5}  & Case 3 (B) & - & - & \(rev.=10^{\circ}\) \\ \hline \multirow{3}{*}{Dual loading} & Case 4 (SC) & \(d=0.04\sqrt{2}\) mm & \(d=0.04\sqrt{2}\) mm & - \\ \cline{2-5}  & Case 5 (SB) & \(d=0.08\) mm & - & \(rev.=10^{\circ}\) \\ \cline{2-5}  & Case 6 (CB) & - & \(d=0.08\) mm & \(rev.=10^{\circ}\) \\ \hline Complex loading & Case 7 (SCB) & \(d=0.04\sqrt{2}\)mm & \(d=0.04\sqrt{2}\)mm & \(rev.=10^{\circ}\) \\ \hline \hline \end{tabular}
\end{table}
Table 3: Mechanical loading cases.  
Figure 2: 3D FE model of solder joint: (**a**) parts dimensions and (**b**) section view of a solder ball with mesh distribution.