## 2 Methodologies  
In this study, three methods, including strain gauges, full-field shadow moire, and FEM analysis were employed for measuring and calculating the curvature and deformations (or warpages) of a PCB with surface-mount dual in-line memory module (DIMM) sockets under thermal loads. The test sample, provided by Inventec Corporation, is the PCB mounted with six surface-mount DIMM sockets (called the PCB assembly here) shown in Figure 1a, in which the bare PCB before the mounting sockets has the size of \(220\times 72\times 2.5\) mm, as shown in Figure 1b. During the reflow heating process, the DIMM sockets and PCB expand freely and differently due to the liquid-state (or paste-state) solder material without any constraint, as shown in Figure 2a. However, during the reflow cooling process, from 250 to 30 \({}^{\circ}\)C (equivalent with a thermal load of \(-220\)\({}^{\circ}\)C), the solder material becomes solidified and provides a constrained force between the DIMM sockets and the PCB. Therefore, the PCB and DIMM sockets would deform and warp simultaneously resulting from the CTE mismatch under a thermal load, as illustrated in Figure 2b. Figure 1: _Cont._  
Before the reflow process, back-to-back strain gauges were adhered to a specific position on both the top and bottom surfaces of the PCB, as shown in Figure 1a, for measuring the strains on the PCB during the reflow process. The strain data with the variation of temperature were further converted into bending strains (\(\varepsilon_{b}\)) and bending curvatures (\(k\)) of the PCB by Equations (1) and (2), respectively. Then, the out-of-plane deformation of the PCB under different temperatures could be further calculated by Equation (3) for a given curvature:  
\[\varepsilon_{b}=\frac{\left(\varepsilon_{top}-\varepsilon_{bottom}\right)}{2}; \tag{1}\]  
\[k=\frac{2\varepsilon_{b}}{h}; \tag{2}\]  
\[W=\frac{kx^{2}}{2}. \tag{3}\]  
where \(\varepsilon_{top}\) is the strain on the PCB surface with DIMM sockets, \(\varepsilon_{bottom}\) is the strain on the PCB back surface, \(h\) is the PCB thickness, and \(W\) is the maximum out-of-plane displacement (deformation) at the given distance \(x\) with constant curvature \(k\). Figure 1: Configurations of (**a**) the printed circuit board (PCB) with dual in-line memory module (DIMM) sockets (called a PCB assembly), in which (**b**) the bare PCB is with the size of \(220\times 72\times 2.5\) mm. Figure 2: (**a**) No constraint between DIMM socket and PCB during the reflow heating process, and (**b**) thermally-induced warpage of the PCB assembly due to a coefficient of thermal expansion (CTE) mismatch during the reflow cooling process. For measuring the out-of-plane deformation of the PCB either before or after the reflow process, an optical full-field shadow moire system was employed in this study. It is expected that a significant warpage change occurs on the PCB after the cooling reflow process (from 250 to 30 \({}^{\circ}\)C) due to the CTE mismatch. The measured deformations of the PCB assembly will further be compared with those from the strain gauge measurement and the FEM simulation. In the FEM simulation, an isothermal linear analysis was performed to calculate the CTE mismatch-induced PCB deformation or warpage. The material properties for the PCB, DIMM socket, and solder material are listed in Table 1. Among those material properties, the effective CTEs (\(\alpha\)) of the PCB and socket have been carefully measured using strain gauges in this study, but not for the solder material since it is relatively thin, while non-sensitive data such as the elastic modulus (\(E\)) and Poisson ratio (\(\
u\)) have been taken from the literature. The schematic drawing and the FEM model are also shown in Figure 3 with the related boundary conditions and meshes. An applied thermal load of \(\Delta T=-220\)\({}^{\circ}\)C (from 250 \({}^{\circ}\)C cooling down to 30 \({}^{\circ}\)C) is chosen as to mimic the thermal loading in the reflow cooling process.