# Numerical Homogenization of Single-Walled Corrugated Board with Imperfections
## 3 Results  
The corrugated cardboard model without initial imperfections and six models with six buckling modes were analyzed. Table 3 shows an example of the \(\mathbf{H}_{k}\) matrix computed for the RVE without imperfections (see Equation (12)). Figure 4: Imperfection modes in the case of: (**a**) compression in the MD, (**b**) bending in the MD, (**c**) compression in the CD, (**d**) bending in the CD, (**e**) in-plane shear and (**f**) twisting. \begin{table}
\begin{tabular}{c c c c c c c c} \hline  & & \multicolumn{3}{c}{**A (N/mm)**} & \multicolumn{3}{c}{**B (N)**} & \multicolumn{3}{c}{**R**} \\  & & \multicolumn{3}{c}{**\& B (N)**} & \multicolumn{3}{c}{**\& D (N-mm)**} & \multicolumn{3}{c}{**(N/mm)**} \\  & & **1** & **2** & **3** & **1** & **2** & **3** & **4** & **5** \\ \hline \hline
**A** (N/mm) & **1** & **2234** & **401** & **0** & \(-\)**18** & \(-\)**5** & **0** & & \\
**\& B** (N) & **2** & **401** & **1684** & **0** & \(-\)**5** & \(-\)**2** & **0** & & \\
**\& B** (N) & **3** & **0** & **0** & **687** & **0** & **0** & & \\
**B** (N) \& **4** & **1** & \(-\)**18** & \(-\)**5** & **0** & **3291** & **574** & **0** & & \\
**D** & **2** & \(-\)**5** & \(-\)**2** & **0** & **574** & **2047** & **0** & & \\
**(N-mm)** & **3** & **0** & **0** & **0** & **0** & **865** & & & \\
**R** & **4** & & & & & & & 104 & 0 & 95 \\ (N/mm) & **5** & & & & & & & & \\ \end{tabular}
\end{table} 
Table 3: Constitutive stiffness matrix \(\mathbf{H}_{k}\) for the cardboard model without imperfections. It should be noted that in the **B** subarrays non-zero components are formed, which is caused by the asymmetry of the cross-section of the corrugated cardboard. Table 10 shows the stiffness values of the main diagonal matrices that are presented in Tables 3-9 for the model without imperfections (REF) and the six models with buckling modes (M1-M6). The imperfection modes were marked in the order presented in Figure 4, i.e., M1--compression in the MD, M2--bending in the MD, M3--compression in the CD, etc. Omitting the components \((*)_{12}\) does not introduce an error in the analysis of the results, because these components depend on the components \((*)_{11}\), \((*)_{22}\) and on the Poisson's ratio. The **B** subarray was also disregarded, but its influence was taken into account by determining the uncoupled **D** matrix from Equation (13). The components of the **R** matrix were also omitted because their values remained almost unchanged for each buckling mode. Figures 5-10 show the reduction of the individual stiffnesses for the analyzed six imperfection modes compared to the reference value that was computed from the model without imperfection. Table 11 presents the differences between the stiffnesses for models with imperfections and the stiffnesses of the reference model (without imperfections). \begin{table}
\begin{tabular}{l c c c c c c c} \hline  & **REF** & **M1** & **M2** & **M3** & **M4** & **M5** & **M6** \\ \hline \(A_{11}\) (N/mm) & 2234 & 2144 & 2156 & 2140 & 2161 & 2233 & 2204 \\ \(A_{22}\) (N/mm) & 1684 & 1683 & 1682 & 1653 & 1659 & 1684 & 1652 \\ \(A_{33}\) (N/mm) & 687 & 686 & 686 & 684 & 685 & 686 & 685 \\ \(D_{11}\) (N \(\cdot\) mm) & 3290 & 2966 & 2979 & 3013 & 3040 & 3289 & 3208 \\ \(D_{22}\) (N \(\cdot\) mm) & 2047 & 1948 & 1944 & 1931 & 1937 & 2049 & 1980 \\ \(D_{33}\) (N \(\cdot\) mm) & 865 & 817 & 816 & 826 & 827 & 865 & 854 \\ \hline \end{tabular}
\end{table} 
Table 10: Selected stiffnesses for models without imperfections and with six imperfection modes.