## 3 Size Effects on Mechanical Properties of Solder Joints
### Fracture and Creep Deformation of Solder Joints  
[PERSON] et al. [55] investigated size and constraint effects on the fracture behavior of Sn3.0 Ag0.5 Cu and Sn37 Pb solder joints by simulations with elastic-plastic and dynamic fracture methods. The damage-equivalent stress area and energy release distribution, which strongly depend on the diameter-to-thickness ratio (\(d/t\)) of the solder joints, were used to characterize the fracture mechanism and position. They found that the solder joints with a small \(d/t\) were apt to fracture at the middle of the solder layer, while the joints with a large \(d/t\) tended to fracture at the solder/IMC interface. [PERSON] et al. [56] developed a theoretical model based on the local stress singularity at the bi-material corner to analyze the relationship between the microstructure size and macro-fracture toughness of lead-free solder joints. The IMC grain size effect on the fracture toughness of solder joints was stated as follows [34]:  
\[K_{Q}=K_{QQ}\bigg{(}1+\frac{d}{d_{0}}\bigg{)}^{\lambda_{IMC}-\lambda_{s}} \tag{4}\]  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Solder and Substrate Type** & **V/A Ratio (\(\mu\)m)** & **Shear Strength (MPa)** & **Reference** \\ \hline \multirow{4}{*}{Cu/Sn/Cu} & 49 & 23.7 & \multirow{4}{*}{[27]} \\  & 102 & 22.7 & \\  & 300 & 21.3 & \\  & 550 & 18.6 & \\ \hline \multirow{4}{*}{Cu/Sn/Ni} & 47 & 21.9 & \multirow{4}{*}{[27]} \\  & 100 & 22.2 & \\  & 300 & 17.7 & \\  & 550 & 17.1 & \\ \hline \multirow{4}{*}{Cu/Sn-3.5 Ag/Cu} & 156 & 110.6 & \multirow{4}{*}{[50]} \\  & 186 & 80.3 & \\ \cline{1-1}  & 243 & 62.1 & \\ \cline{1-1}  & 326 & 37.5 & \\ \hline \multirow{4}{*}{Cu/Sn3.0 Ag0.5 Cu} & 192 & 56.8 & \multirow{4}{*}{[52]} \\  & 288 & 52.9 & \\ \cline{1-1}  & 384 & 48.7 & \\ \cline{1-1}  & 480 & 48.1 & \\ \cline{1-1}  & 576 & 46.8 & \\ \hline \multirow{4}{*}{Fe-Ni/Sn/Fe-Ni} & 18 & 13.9 & \multirow{4}{*}{[53]} \\ \cline{1-1}  & 48 & 14.8 & \\ \cline{1-1}  & 300 & 10.9 & \\ \hline \hline \end{tabular}
\end{table}
Table 2: Shear strengths of lead-free solder joints with different V/A ratios.  
where \(K_{Q}\) is the predicted interfacial fracture toughness and \(d\) is the representative length of the interfacial IMC grain. The parameters \(K_{QQ}\) and \(d_{0}\), which are dependent on the specimen material and geometry, can be obtained by numerical fitting. The parameters \(\lambda_{IMC}\) and \(\lambda_{s}\), which can be theoretically and numerically determined, are the order of displacement singularity at the corner of the IMC grain and the solder joint, respectively. This size effect was suggested to be valid in solder joints with scallop-type IMCs [56; 57; 58; 59], as shown in Figure 9.  
Size effects on the creep deformation of line-type Cu/Sn3.0 Ag0.5 Cu/Cu solder joints were studied by [PERSON] et al. [60] using DMA. They found that the creep exponents ranged from 4.2 to 3.1 and the creep activation energy varied from 90 to 66 kJ/mol as the joint thickness increased from 50 to 200 \(\upmu\)m. Lattice diffusion was deemed to be the dominant creep mechanism, since the creep exponents were closed to the theoretical exponents of lattice diffusion. The findings by [PERSON] et al. [60] are in accordance with the bump height effect on solder hardening [44].  
Size effects on solder joints are usually complicated in terms of various compositions and microstructures. Therefore, a multi-scale microstructure-based modeling method, which can capture the hardening and aging effects of mixed intermetallic phases on the constitutive behavior, is becoming an important approach for exploring the relationship between the microstructure and mechanical behavior of solder joints [61]. However, so far the multi-scale modeling method is only appropriate in simulating the mechanical behavior of solder layers which incorporate fine intermetallic particles. When a relatively weak interface strength is referred to, this method may not be applicable.