## 3 Results and Discussion
### Stress-Strain Behavior Using Equivalent Materials Properties of Corrugated Paperboard  
In order to perform FEA on the flat crush resistance, the calculated equivalent mechanical properties of the corrugated paperboard samples were applied to the simplified FE models (Figure 17). The simplified model was discretized into 156,825 nodes and 102,400 mesh elements. The S-S behavior (flat crush resistance) of the simplified FE model was compared with experimental data of corrugated paperboard samples obtained by following ISO 3035 [27] (Figure 18). In case of the flat crush test of corrugated paperboard samples, the S-S curve was measured when a circular metal plate that has 90.6 mm in diameter and was attached the upper cross-head of tester exerted compressive force on corrugated paperboard sample (length \(\times\) width = \(30\times 30\) cm). The loading rate was \(12.5\pm 2.5\) mm/min.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Classifications** & **AB/F-DW** & **BB/F-DW** & **A/F-SW** \\ \hline \(F_{xy}\) & 1634.86 & 1726.43 & 1580.28 \\ \(F_{yz}\) & 83.64 & 156.53 & 138.97 \\ \(F_{xx}\) & 8.70 & 11.11 & 2.47 \\ \hline \hline \end{tabular}
\end{table}
Table 7: Reactional forces (N) for 3 sub-problems.  
Figure 16: Directional reaction force for the unit cell of A/F-SW corrugated paperboard, they should be list as: (**a**) _xy_-direction; (**b**) _yz_-direction; (**c**) _zx_-direction.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Material Properties** & **AB/F-DW** & **BB/F-DW** & **A/F-SW** \\ \hline Elastic modulus & \(E_{x}\) & 260.32 & 345.50 & 298.37 \\ (MPa) & \(E_{y}\) & 116.60 & 151.35 & 106.24 \\ \(E_{z}\) & 1.58 & 7.87 & 6.57 \\ \hline Shear modulus & \(G_{xy}\) & 58.02 & 81.25 & 58.83 \\ (MPa) & \(G_{yz}\) & 7.41 & 10.50 & 5.26 \\ \(G_{xx}\) & 3.95 & 6.70 & 2.65 \\ \hline Poisson’s ratio & \(v_{xy}\) & 0.173 & 0.177 & 0.189 \\ (-) & \(v_{yz}\) & 0.067 & 0.045 & 0.046 \\ \(v_{xx}\) & 0.006 & 0.026 & 0.027 \\ \hline \hline \end{tabular}
\end{table}
Table 9: Equivalent material properties for the corrugated paperboard samples.  
Sandpaper was attached to the upper and lower compression plates of the tester in contact with the test piece to reduce the experimental error by the swelling of the flute in the test piece of the corrugated paperboard. However, the FEA for the simplified model of corrugated paperboard samples was reproduced by simulating the same conditions as the experiment.  
When the upper surface of the model with a diameter of 90.6 mm was forcibly displaced at the same loading rate as the experiment in the z-direction, the reaction force transmitted to the lower surface of the model and S-S curves were measured and recorded. In FEA, the lower surface of the model was restricted to the translational and rotational motion in the \(x\), \(y\), and \(z\)-directions; however, the upper surface was restricted only to translational motion in \(x\) and \(y\)-directions.  
The flat crush behavior of the corrugated paperboard samples showed elastic-plastic behavior when it exceeded a certain strain; however, the calculated equivalent value was the result of the theoretical approach assuming the elastic region.  
As expected, the difference between the simulated and experimental values was observed in the region beyond the elastic region. In the actual behavior of the corrugated paperboard, the liner of the corrugated paperboard was deformed in the shear direction because of the change of the flute shape. Therefore, the experiment showed more significant deformation at smaller loads than ideal conditions such as FEA (Figure 18). As a result, when the equivalent mechanical properties calculated by FEA were used to simulate S-S behavior of corrugated paperboard, it was necessary consider the characteristics of the region where the strain increased without an increase in load due to the flute of the corrugated paperboard.