# Incorporating Transverse Normal Strain in the Homogenization of Corrugated Cardboards
###### Abstract  
Homogenization researches for corrugated cardboard are predominantly based on plate theories assuming constant thickness, such as the Reissner-Mindlin plate. However, corrugated cardboard is prone to significant deformation in the thickness direction. To address this limitation, the present work proposes an improved plate element designed by expanding the deflection function to the quadratic term of the thickness coordinate, enabling a linearly varied transverse normal strain. Furthermore, an extension of the established homogenization method is developed to derive the constitutive matrix. The element is implemented via the Abaqus user subroutine UEL. Validation demonstrates that the proposed element effectively characterizes a linearly varied transverse normal strain and stress. Simulation results from the homogenized model applying the proposed element and extended homogenization method are compared with those from detailed models. The comparisons confirm the efficiency and accuracy of the proposed approach. corrugated cardboard; homogenization; finite element analysis; Abaqus user subroutines; UEL 2122122312  
Paper sheet and corrugated cardboard are both orthotropic material with three principal directions: machine direction (MD), cross direction (CD), and thickness direction (ZD). For paper sheet, MD is the direction that paper comes out of the papermaking machine, CD lies perpendicular to MD within the sheet plane, and ZD is normal to the sheet surface. For corrugated cardboard, MD is perpendicular to the corrugation profile, while CD is parallel to it, as shown in Figure 1. MD exhibits higher fiber alignment, resulting in better mechanical performance than CD for both paper sheet and corrugated cardboard. When modeling corrugated cardboard in FEA, a trade-off exists between geometry complexity and computational efficiency. It is difficult and tedious to model every flute and liner. Although it might help to capture local deformations, it incurs prohibitive computational costs, particularly for large-scale problems. Therefore, characterizing mechanical properties at the macroscale and homogenizing the cardboard into an equivalent orthotropic plate is preferable. Analytical methods for corrugated cardboard homogenization are broadly classified into two categories. The first employs structural mechanics frameworks to derive equivalent properties. Early work by [PERSON] [1] provided analytical expressions using beam theory under Kirchhoff plate theory. [PERSON] [2] extended this methodology to sinusoidal corrugated cardboard. [PERSON] [3] derived the transverse shear stiffness for diverse corrugation shapes via curved beam theory. [PERSON] [4] applied an energetic approach to compute equivalent properties under Reissner-Mindlin plate theory. [PERSON] [5] decomposed the torsional problem of orthotropic cardboard into the torsion of two orthogonal beams to obtain the equivalent torsional stiffness. The second category utilizes laminated plate theory with coordinate transformation, which was originally developed for fiber orientation in composites, adapted to corrugated structure with modified rotation axes. [PERSON] [6] established fundamental procedures for this method under the framework of classical laminated plate theory (CLPT), later refined by [PERSON] [7] to first-order shear deformation theory (FSDT) with corrections. [PERSON] [8] further extended it to double-wall corrugated cardboard. This approach is accepted by many researchers [9; 10] due to its simplicity. In addition to analytical methods, there are numerical [11] and experimental [12] methods of corrugated cardboard homogenization. \begin{table}
\begin{tabular}{c c c c} \hline
**Flute Type** & **Height (mm)** & **Period (mm)** & **Corrugation Factor** \\ \hline A & 4.8 & 8.0–9.5 & \(\approx\)1.50 \\ B & 3.2 & 5.5–6.5 & \(\approx\)1.40 \\ C & 4.0 & 6.8–7.9 & \(\approx\)1.45 \\ E & 1.6 & 3.0–3.5 & \(\approx\)1.25 \\ F & 0.8 & 1.9–2.6 & \(\approx\)1.25 \\ \hline \end{tabular}
\end{table} 
Table 1: Parameters of common flute types.