## 3 Results and Discussion
### Calculation of Equivalent Material Properties of Corrugated Paperboards  
In order to obtain the equivalent mechanical properties of the corrugated paperboard depending on flute type, the physical properties (Table 3) of linerboard and corrugating medium were applied to the FE model for the unit cells (Figure 5) of corrugated paperboard samples. Since the contact surface conditions of FEA were the contact areas between the linerboard and the flute of corrugated paperboard, the number of contact surfaces in DW and SW corrugated paperboards was set to 4 and 2, respectively. As aforementioned, the interface between the flute and liner of the corrugated paperboard was considered to be joined for the convenience of FEA. First, to obtain the equivalent elastic modulus, the reaction force was measured through FEA with each constraint given to the three sub-problems (Figure 3). The reaction forces (\(F_{xi}\), \(F_{yi}\), and \(F_{zi}\)) were measured in the state in which forced displacement of 1 mm was applied in the \(x\), \(y\), and \(z\)-directions, respectively. In this case, all degrees of freedom were constrained in all directions except for the direction in which the forced displacement was applied: the \(y\) and \(z\)-directions when measuring \(F_{xi}\), the \(x\), and \(z\)-directions when measuring \(F_{yi}\), and the \(x\) and \(y\)-directions when measuring \(F_{zi}\). The measured directional reaction force for unit cells of corrugated paperboard samples through FEA are shown in Figures 11-13. In addition, the measurement results of the reaction force for the three sub-problems are provided in Table 4. The measured reaction forces through FEA were not significantly different from the measured reaction forces in tension and compression test. The corrugated paper board is mainly subjected to compressive force when it is applied to the product packaging and cushioning. Therefore, the reaction force measured under compressive load was applied to the analysis of the equivalent elastic modulus for the corrugated paperboard samples. The equivalent elastic modulus for the corrugated paperboard samples could be calculated by applying the reaction forces listed in Table 4 to Equations (2) and (5)-(7). The calculated results are summarized in Table 5. For example, the equivalent elastic modulus for AB/F-DW corrugated paperboard was calculated as follows. The coefficient (\([\alpha]_{ii}\), relative ratio) was calculated by substituting the reaction forces obtained from FEA for each sub-problem into Equation (2). The coefficient for each sub-problem was calculated by applying the cross-sectional area and the strain of AB/F-DW corrugated paperboard (Table 5) and arbitrary load (\([P]_{ii}=100\) N). The calculated coefficients are listed in Table 6. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Classifications** & **AB/F-DW** & **BB/F-DW** & **A/F-SW** \\ \hline \multirow{2}{*}{\(x\)-axis} & \(F_{x1}\) & \(-\)583.58 & \(-\)593.22 & \(-\)284.74 \\  & \(F_{y1}\) & \(-\)163.41 & \(-\)169.07 & \(-\)105.49 \\  & \(F_{z1}\) & \(-\)5.46 & \(-\)29.17 & \(-\)11.32 \\ \hline \multirow{2}{*}{\(y\)-axis} & \(F_{x2}\) & \(-\)163.26 & \(-\)167.66 & \(-\)105.55 \\  & \(F_{y2}\) & \(-\)3374.73 & \(-\)3306.89 & \(-\)2931.99 \\  & \(F_{z2}\) & \(-\)3.43 & \(-\)14.59 & \(-\)12.45 \\ \hline \multirow{2}{*}{\(z\)-axis} & \(F_{x3}\) & \(-\)7.88 & \(-\)47.09 & \(-\)41.81 \\  & \(F_{y3}\) & \(-\)5.40 & \(-\)29.04 & \(-\)44.49 \\ \cline{1-1}  & \(F_{z3}\) & \(-\)17.97 & \(-\)120.86 & \(-\)177.82 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Reaction forces (N) for 3 sub-problems. Figure 12: Directional reaction force for the unit cell of BB/F-DW corrugated paperboard, they should be list as: (**a**) \(x\)-direction; (**b**) \(y\)-direction; (**c**) \(z\)-direction. Figure 13: Directional reaction force for the unit cell of A/F-SW corrugated paperboard, they should be list as: (**a**) \(x\)-direction; (**b**) \(y\)-direction; (**c**) \(z\)-direction.