###### Abstract
[9] conducted the machine learning analysis to investigate the effect of the design pattern of the package substrate on the thermal properties that can effect deformation. [PERSON] et al. [10] investigated the geometrical effects of individual solder joint models under the shear loading. Parametric modeling and lab-scale experiments on rectangular single solders were conducted by [PERSON] et al. [11] and [PERSON] et al. [12], respectively. These studies mainly focused on the failure of solder joints based on the in-plane shear stress resulting from the coefficient of thermal expansion (CTE) mismatch between the layers. Failure can occur in various deformation modes, including shear, owing to the interaction of internal microvoids caused by externally applied loads. Several theories have been proposed to explain the void nucleation, growth, and coalescence processes leading to crack initiation and propagation. Furthermore, in the case of inelastic deformation or failure of materials, it is crucial to clearly define the history of the inelastic deformation path. However, previous studies have extensively examined the solder ball analysis, and most of them have focused on analyzing the effects of the entire PCB. Consequently, analyzing the precise evolution of the inelastic deformation path in a single solder ball will offer insights into the quantitative impact of external forces. Mechanical theories of plasticity and ductile fracture criteria can provide valuable insights into structure deformation and failure predictions. Solder joint failures occur through various deformation modes, such as shear, owing to the interaction between internal microvoids and external loads. Damage models involving the nucleation, growth, and coalescence of voids affected by pressure and Lode parameters have been introduced to simulate the ductile fracture [13; 14; 15; 16; 17]. Coalescence, driven by the maximum shear stress, further contributes to fracture. Recent studies by [PERSON] and [PERSON] introduced a fracture strain definition based on stress triaxiality and validated it experimentally [18]. Subsequently, [PERSON] et al. calibrated their results using multiple fracture models [19]. [PERSON] proposed a modified maximum force criterion (MMFC) to account for rupture and localized necking [20], whereas [PERSON] et al. and [PERSON] et al. presented a ductile fracture criterion considering microvoid interactions during plastic deformation [21; 22]. [PERSON] and [PERSON] presented an effective stress-based fracture limit model [23]. [PERSON] et al. presented a fracture modeling considering thermal softening [24], and [PERSON] proposed a fracture technique for the heterogeneous distribution of subscale features based on the strain energy density [25]. [PERSON] et al. recently addressed the calibration of the fracture parameters with respect to the initial crack size and external loads under fatigue fracture situations [26]. These fracture mechanics models can provide a comprehensive understanding of the solder deformation and therefore fracture under mechanical loadings. This work aims to investigate the evolution of the inelastic deformation path of solder joints under external force conditions through numerical simulations. The motivation for this research stems from the fact that many studies have discussed the occurrence of failure in solder joints when the maximum stress exceeds a critical value [27; 28]. However, there has been little discussion on how the stress varies significantly during the occurrence of inelastic deformation due to the change in the deformation path. Additionally, several studies have relied solely on the elastic properties during the stress calculation process. In addition, this study demonstrates that the inelastic deformation path varies depending on the combination of external loading conditions and has a significant influence on the occurrence of fracture. The external force conditions in this study are based on the fundamental modes of shear, compression, and bending, as well as their combinations.