# Numerical Homogenization of Single-Walled Corrugated Board with Imperfections
## 2 Materials and Methods
### Homogenization Technique  
The numerical homogenization method proposed by [PERSON] [17], and then extended by [PERSON] and [PERSON] [18] was utilized in the present study. This method is based on the provision of strain energy equivalence between the full representative volume element (RVE) of the corrugated board and the simplified shell model. The RVE is a small, periodically repeating segment of the full corrugated board structure model that is then simplified to a single layer of flat shell elements in such a way as to ensure the same behavior of the full and simplified model. Only the most important assumptions of the described method are presented below. The entire theoretical derivation of the constitutive model can be found in [18]. However, it is important to note that for the evaluation of homogenized properties for macroscopic analysis, the concept of statistical RVE (SRVE) can be utilized [32; 33; 34]. The nodal displacements can be determined while involving the finite element method (FEM). The FEM equation for linear analysis is shown below:  
\[\mathbf{K}_{e}\;\mathbf{u}_{e}=\mathbf{F}_{e}, \tag{3}\] \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Layers**} & \(E_{1}\) & \(E_{2}\) & \(\
u_{12}\) & \(G_{12}\) & \(G_{13}\) & \(G_{23}\) \\ \cline{2-7}  & **(MPa)** & **(MPa)** & **(-)** & **(MPa)** & **(MPa)** & **(MPa)** \\ \hline liners & 3326 & 1694 & 0.34 & 859 & 429.5 & 429.5 \\ fluting & 2614 & 1532 & 0.32 & 724 & 362 & 362 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Material data used in orthotropic constitutive model. Figure 2: Material orientation. where \(\mathbf{K}_{e}\) is a global stiffness matrix statically condensed to the external nodes of the RVE, \(\mathbf{u}_{e}\) is a displacement vector of the external nodes and \(\mathbf{F}_{e}\) is the vector of nodal forces applied to the external nodes. In Figure 3, the finite element mesh and nodes of the RVE are shown. The global stiffness matrix was obtained by applying the static condensation process. It consists in removing selected unknown degrees of freedom (DOF) and determining the stiffness matrix for a smaller number of DOFs (called primary unknowns). In the discussed model, the internal nodes were removed, and for the external nodes, the stiffness matrix was determined from the following formula:  
\[\mathbf{K}_{e}=\mathbf{K}_{ee}-\mathbf{K}_{ei}\;\mathbf{K}_{ii}^{-1}\;\mathbf{ K}_{ie}, \tag{4}\]  
where all the subarrays \(\mathbf{K}_{**}\) contain the stiffnesses for external (subscript \(e\)) and internal (subscript \(i\)) nodes, which in matrix form relate nodal displacements and nodal forces:  
\[ \begin{bmatrix}\mathbf{K}_{ee}&\mathbf{K}_{ei}\\ \mathbf{K}_{ie}&\mathbf{K}_{ii}\end{bmatrix}\begin{bmatrix}\mathbf{u}_{e}\\ \mathbf{u}_{i}\end{bmatrix} = \begin{bmatrix}\mathbf{F}_{e}\\ 0\end{bmatrix} . \tag{5}\]  
As a result of static condensation, the total elastic strain energy was reduced to the work of external forces on the corresponding displacements:  
\[E=\frac{1}{2}\;\mathbf{u}_{e}^{T}\;\mathbf{F}_{e}. \tag{6}\]  
The balance of the total strain energy between the full 3D RVE model and the simplified shell model was ensured by appropriate determination of nodal displacements and by enabling the membrane and bending behavior. The details can be found in [18]. The relationship between generalized displacements and generalized strains can be presented by the formula:  
\[\mathbf{u}_{i}=\mathbf{H}_{i}\;\varepsilon_{i}, \tag{7}\]  
where the \(\mathbf{H}_{i}\) matrix can be determined for each node (\(x_{i}=x\), \(y_{i}=y\), \(z_{i}=z\)):  
\[ \begin{bmatrix}u_{x}\\ u_{y}\\ u_{z}\\ \theta_{x}\\ \theta_{y}\end{bmatrix} _{i}= \begin{bmatrix}x&0&y/2&xz&0&yz/2&z/2&0\\ 0&y&x/2&0&yz&xz/2&0&z/2\\ 0&0&0&-x^{2}/2&-y^{2}/2&-xy/2&x/2&y/2\\ 0&0&0&0&-y&-x/2&0&0\\ 0&0&0&x&0&y/2&0&0\end{bmatrix} _{i} \begin{bmatrix}\varepsilon_{x}\\ \varepsilon_{y}\\ \tau_{xy}\\ \kappa_{x}\\ \kappa_{y}\\ \tau_{xx}\\ \gamma_{yx}\end{bmatrix} _{i}.