## 3 Results and Discussion
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multicolumn{1}{c}{**Topography**} & \multicolumn{1}{c}{**Mean Width**} & \multicolumn{1}{c}{**Mean Thickness**} & \multicolumn{1}{c}{**Area (\(\mu\)m\({}^{2}\))**} & \multicolumn{1}{c}{**Gauge Factor**} \\ \hline Stacked & Five layers & 209.39 & 404.36 & 84.67 & 3.098 \\ layers & Three layers & 203.15 & 175.08 & 35.57 & 3.051 \\  & Single layer & 163.34 & 63.52 & 10.37 & 2.471 \\ \hline \multirow{4}{*}{Width} & Seven & & & & \\ droplets & 266.60 & 62.12 & 16.56 & 2.589 \\  & Four droplets & 175.29 & 68.23 & 11.96 & 2.471 \\  & Two droplets & 151.59 & 58.42 & 8.86 & 2.047 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Measured widths, thicknesses and corresponding gauge factors of printed strain-gauge patterns. Figure 8: Gauge factor as a function of the width of printed strain-gauge patterns: (a) 2.047 for two droplets, (b) 2.471 for four droplets and (c) 2.589 for seven droplets. Figure 7: Gauge factor as a function of the number of stacked layers of printed strain-gauge patterns: (a) 2.471 for a single layer, (b) 3.051 for three layers and (c) 3.098 for five layers. The cross-sectional areas of the printed strain gauges and corresponding sensitivity factors are summarized in Table 2. Figure 9 shows that the gauge factor was proportional to the cross-sectional area of the printed strain-gauge pattern. The correlation was mathematically modelled as \(y=0.4167\text{ln}(x)+1.3837\), where \(x\) is the area in microns squared and \(y\) is the gauge factor. The coefficient of determination (\(R^{2}\)) was 0.8383. The correlation between the geometrical variation in the conductive pattern and the gauge factor was mathematically derived based on some assumptions: (1) the geometrical deformation occurred in the linear elastic region and (2) the conductive pattern was isotropic. The resistance of the pattern with the initial geometry was calculated as  
\[R=\rho_{e}\big{(}\
icefrac{{L}}{{A}}\big{)}=\rho_{e}\big{(}\
icefrac{{L}}{{H \text{W}}}\big{)} \tag{3}\]  
The geometries were expressed using the following initial conditions and variations:  
\[H =H_{0}+\text{d}H \tag{4}\] \[W =W_{0}+\text{d}W\] (5) \[L =L_{0}+\text{d}L \tag{6}\]  
where \(\rho_{e}\) is the specific electrical resistance, \(L\), \(A\), \(H\) and \(W\) are the length, cross-sectional area, height and width, respectively, and \(\text{d}L\), \(\text{d}A\), \(\text{d}H\) and \(\text{d}W\) are their corresponding variations. Then, the variation of resistance can be calculated as  
\[\text{d}R=R-R_{0}=\rho_{e}\frac{(L_{0}+\text{d}L)}{(H_{0}+\text{d}H)(W_{0}+ \text{d}W)}-\rho_{e}\frac{L_{0}}{H_{0}W_{0}} \tag{7}\]  
The strains within the cross section are  
\[\varepsilon_{w}=\
icefrac{{dW}}{{W_{0}}},\ \varepsilon_{H}=\
icefrac{{dH}}{{H_{0}}} \tag{8}\]  
[PERSON]'s ratio yields  
Figure 8:0  
Figure 8:1 Gauge factor as a function of cross-sectional area of printed strain-gauge patterns. The black dots and green squares are for the width and the number of stacked layers, respectively. Since \((\varepsilon_{w}+\varepsilon_{H}+\varepsilon_{W}\varepsilon_{H})\ll 1\), Equation (10) can be rearranged as  
Figure 8:2  
Combining Equations (1), (9) and (11) gives  
Figure 8:3  
Equation (12) indicates that the gauge factor can be determined from the longitudinal strain and Poisson's ratio of the conductive pattern. To compare the gauge factors measured experimentally, we used a strain of 2.5% and Poisson's ratio for bulk Ag of 0.37. This yielded a gauge factor of 1.742, which was lower than the measured range from 2.047 to 3.089, as indicated in Figure 8:4 We could speculate that this discrepancy in gauge factor is attributed to the porosity of sintered printed Ag layer. In [22], the porosity of material influences tensile mechanical properties, e.g., elastic modulus, ultimate strength, and elongation. Then, the porosity could affect the gauge factor as well as Poisson's ratio of sintered Ag layer in Equations (9) and (12).