# Incorporating Transverse Normal Strain in the Homogenization of Corrugated Cardboards
###### Abstract
Figure 1: Principal directions of corrugated cardboard. Although FSDT improves upon CPT by incorporating transverse shear effects, both theories still assume thickness incompressibility. Consequently, these homogenization schemes are suitable for scenarios with negligible through-thickness deformation (i.e., transverse normal strain), such as the box compression test, where the load is primarily borne by the side panels and peak load is governed by panel buckling. However, we cannot overlook the fact that corrugated cardboard is prone to significant through-thickness deformation, which enables cushioning usage. Extensive research exists on its out-of-plane properties: [PERSON] [13] conducted nonlinear analyses on the out-of-plane compression; [PERSON] [14] investigated vibration stiffness and damping; and [PERSON] [15] studied energy absorption under varying humidity. Nevertheless, most studies on transverse normal strain remain stand-alone efforts, being incompatible with the aforementioned homogenization approaches. There are scenarios wherein it is necessary to incorporate transverse normal strain into the homogenized plate model, yet related studies remain scarce. For instance, in a carton fully filled with products, vertical loading may induce significant through-thickness deformation in flaps, rendering plate elements insufficient. Similarly, in simulations of carton drop tests or tip-over tests, various criteria dictate that the box may impact the floor with its face, edge, or corner. In such cases, all deformation components must be considered to prevent biased results. In fact, the mechanism of drop test is so complex that it is still a tricky problem in industry. Beyond packaging, corrugated cardboard finds applications in numerous scenarios such as sustainable furniture. It is valuable to pursue a general solution that can be applied to diverse situations. Therefore, the established homogenization method using laminated plate theory is extended in the present work. It should be noted that a common misconception may arise: capturing through-thickness deformation cannot be achieved merely by switching from shell elements to solid or solid-shell elements. Indeed, elements such as SC8R and CSS8 in Abaqus (2022) [16], SOLSH190 in ANSYS (2024 R2) [17], are capable of characterizing the transverse normal strain. These elements have the same topology and degrees of freedom (DOFs) as solid elements and are developed by applying the displacement constraints of shell theory to enable a good performance on a thin-walled structure problem as well as transverse normal strain [18; 19]. However, the corrugated structure is not a homogeneous material but one with cavities, which is both materially and structurally orthotropic. The sectional moment of inertia on MD and CD are inherently unequal, while a topology of a solid element implies an equality. As [PERSON] [11] notes: \"due to the corrugated structure, the core cannot be reduced to an equivalent orthotropic material valid for both bending and stretching\". Consequently, corrugated cardboard should be homogenized with plate elements because it is a 2D plane abstracted from a 3D solid, and the discrepancy between in-plane and flexural stiffnesses can be solved by specifying the constitutive matrix directly. Two features provided by Abaqus can be used to specify the constitutive matrix: General Shell Stiffness section [20] and user subroutine UGENS [21]. However, these can only be applied on a conventional shell (planar element), which is rooted in FSDT [16]. To overcome this limitation, a custom element is required. From a theoretical perspective, incorporating transverse normal strain necessitates manipulating the displacement functions \(u\), \(v\) and \(w\) to include high order terms of the thickness coordinate, which can be broadly concluded as parts of high-order shear deformation theory (HSDT). [PERSON] [22] developed a unified framework enabling displacement expansions to any order.