###### Abstract  
Solder joints of ball grid arrays (BGA) have been widely used to connect electronic components to printed circuit boards (PCBs) and are often subjected to mechanical stress. Several studies have been conducted on the mechanical reliability of solder joints. While these studies have been useful in the industry, detailed studies on how the inelastic deformation path of the solder ball joints evolves under specific loading conditions have not been sufficiently reported. This study aims to understand how the inelastic deformation path evolves when a solder joint is subjected to a constant external force by utilizing the theory of mechanics. It has also been found that the mechanical failure is strongly influenced by the evolution history of the deformation modes in materials. For this study, an elastoplastic constitutive model and a ductile fracture criterion were implemented into the vectorized user-defined material (VUMAT) subroutine of the ABAQUS program for finite element (FE) analysis. With the model, the evolution of the inelastic deformation path of a single solder ball under different loading conditions was numerically analyzed. Three loadings (shear, compression, and bending) were chosen as the basic loading conditions. In addition, combinations of the basic loadings resulted in three dual loadings and one complex loading. The simulation results showed that the shear and bending caused the fracture for both single and dual loadings, but when combined with compression, the fracture was suppressed. The results indicate that fracture is not solely determined by the magnitude of equivalent plastic strain but also by the evolution of inelastic deformation mode. This research offers an improved understanding of the significance of the inelastic deformation path and fracture. inelastic deformation path; solder joint; BGA; ductile fracture criterion; finite element method +
Footnote †: journal: Applied  
0000-0001-8800-0001-8800-0001-8800-0001-8800-0001-8800-0001-8800-0001-8800-0001-8800-0001-8800-0001-88000-0001-88000-0001-88000-00  
is essential to gain a thorough understanding of the mechanical reliability of the BGA packages and the solder ball connections to ensure the long-term functionality and reliability of electronic systems. The mechanical failure of solder balls can occur during various processes and conditions, such as reflow processing and thermal cycling testing throughout the life cycle of an integrated circuit (IC) package module. Several studies have investigated the failure mechanisms of solder joints under various conditions. For instance, [PERSON] et al. [4] explored the failure position of solder in a BGA-type system in package (SiP) under thermal cycling, highlighting the dominance of creep strain over plastic strain and the shift in the failure position from the center to the corner of the BGA. [PERSON] et al. [5] investigated the failure mechanism of SnAgCu/SnPb mixed solders during the manufacturing process and identified two failure modes: intermetallic compound (IMC) cracks and shrinkage within the solder matrix and near the IMC layer. [PERSON] et al. [6] studied the interaction between thermal cycling and drop impact conditions, revealing that drop impact during thermal cycling accelerates solder joint failure owing to the increased accumulated creep strain. The strain-to-fracture ratio decreased with an increasing number of drop impacts. [PERSON] et al. [7] emphasized that thermal effects, from the solder manufacturing process to assembly tests such as thermal cycling, play a critical role in the microvoid behavior, leading to crack propagation. Reliability prediction methods for solder joints have been proposed based on creep, fatigue, and drop impact analyses at both the package and board levels. [PERSON] and [PERSON] [8] performed thermal cycling tests and predicted the life cycle of fine-pitch BGA (fBGA) packages using elastoplastic creep (EPC) models. [PERSON] et al.