# Incorporating Transverse Normal Strain in the Homogenization of Corrugated Cardboards
## 4 Validation
### Homogenized Model for Corrugated Cardboard  
The detailed finite element model of a single-wall corrugated cardboard is established in Abaqus using S4R elements and compared with the homogenized model using the proposed element. The flutes and liners of the detailed model are bonded by sharing the same nodes. The material and geometry parameters are shown in Table 3. The size of the model is 80 mm \(\times\) 80 mm \(\times\) 4 mm, and it contains 10 complete corrugated periods in MD. Different load cases including stretching, bending, twisting, and shearing on MD and CD are applied to the detailed model and the homogenized model. Kinematic coupling constraints are used on the boundary of both models to apply consistent boundary conditions and loads. The displacement results on the control point are shown in Table 4. It can be seen that the overall error between the detailed model and the homogenized model is within 8%, indicating that the extended homogenization method is accurate. Figure 10 compares contour plots of both models under CD bending. The homogenized model demonstrates excellent global consistency with the detailed model; a refined mesh is unnecessary. The user element can be visualized like native elements thanks to dummy elements. Otherwise, only thin cross lines would be seen in Figure 10.  
Table 5 compares the computational scale and runtime of both models in a CD-bending simulation. The size of equations systems to be solved directly relates to the number of total DOFs, which drops massively when using a homogenized model. Therefore, the runtime can be reduced by over 50%, significantly enhancing the FEA efficiency of corrugated cardboard. The load case selected for comparison here is basic and the model scale remains relatively small, with most of the computational time spent on preprocessing. \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Layer**} & \multirow{2}{*}{**Thickness (mm)**} & \multicolumn{6}{c}{**Elastic Constant (MPa)**} & \multirow{2}{*}{**Period (mm)**} & \multirow{2}{*}{**Height (mm)**} \\ \cline{3-3} \cline{5-10}  & & \(E_{1}\) & \(E_{2}\) & \(\
u_{12}\) & \(G_{12}\) & \(G_{13}\) & \(G_{23}\) & \\ \hline Liner & 0.29 & 3326 & 1694 & 0.34 & 860 & 60 & 48 & – & – \\ Flute & 0.30 & 2614 & 1532 & 0.33 & 792 & 47 & 43 & 8 & 4 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Elastic property of raw paper. Figure 9: The distribution of transverse normal stress. Therefore, this efficiency gain can be viewed as a conservative baseline; it will become more pronounced for larger-scale models. To evaluate through-thickness deformation, displacements along the line \(y\) = 40 mm in the CD-bending simulation are extracted and plotted in Figure 11. Thickness reduction in the detailed model is calculated as the deflection difference between corresponding nodes on the upper and lower liners. The curve in Figure 11 exhibits significant fluctuations due to the local warping of the liners, which is particularly pronounced near boundaries caused by insufficient constraints; nevertheless, the thickness is reduced overall as expected. Based on the displacement field assumption, the thickness reduction in the proposed element is defined as \(h\psi\). Figure 11 demonstrates that the proposed element accurately captures the average thickness reduction of the detailed model. Additionally, to directly evaluate the element's capability in characterizing through-thickness deformation, an out-of-plane compression test was conducted. The loading and boundary conditions were applied as follows: for the detailed model, the bottom liner was fully constrained while the top liner was coupled to a control point under concentrated loading. For the homogenized model, all nodal DOFs except \(\psi\) and \(\zeta\) were constrained, with a uniformly distributed load applied. Although the displacement DOFs are constrained, thickness-direction deformation can be still captured through DOF \(\psi\) and \(\zeta\).