# Numerical Homogenization of Single-Walled Corrugated Board with Imperfections
###### Abstract  
Numerical homogenization is an excellent tool for the quick simplification of complex structures with a model that is much simpler and, at the same time, accurately reflects the mechanical behavior of the original model. Corrugated cardboard modeling, with all geometrical nuances preserved, is a complicated and time-consuming process. The transfer of a full 3D model of corrugated board composed of two flat layers and a corrugated middle layer to one layer only, with substitute elastic parameters, greatly simplifies this process. Because the individual layers of corrugated cardboard are made of paper with a grammage in the range of 80-200 g/m\({}^{2}\), i.e., very thin plates, they are slightly buckled even in the initial configuration. These imperfections affect the equivalent parameters that are obtained in the homogenization process. This paper presents an approach of taking into account these imperfections when creating a simplified model. The numerical homogenization method based on the equivalence of elastic energy between a representative volumetric element (i.e., a part of a full 3D model) and an equivalent plate were applied. Different shapes of imperfections were analyzed in order to account for the buckling modes, notably for a specific unit deformation and curvature. Finally, one form of initial imperfections was proposed, which most accurately reflects the decrease in all plate stiffnesses. corrugated cardboard; imperfections; numerical homogenization; strain energy equivalence; orthotropic plate; plate stiffness properties +
Footnote †: journal: Applied Sciences  
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Footnote 1: [[https://www.mdpi.com/journal/applsci](https://www.mdpi.com/journal/applsci)]([https://www.mdpi.com/journal/applsci](https://www.mdpi.com/journal/applsci))  
22 September 202  
to assess the load capacity of boxes, including compressive, tensile and bending strength. The most common tests are the box compression test (BCT) and the edge crush test (ECT). The load capacity of the packaging has been the subject of research conducted by many scientific groups for many years. As early as 1952, a formula for the strength of the packaging which takes into account the box perimeter, the overall ring crush strength and the box and corrugated layer constants was proposed [3]. [PERSON] presented the relationship between the critical force in BCT and the dimensions of the packaging based on the Concora liner test and empirical constants [4]. In 1963, [PERSON] et al. proposed an approach that became one of the most popular methods of estimating the load capacity of packaging with a rectangular base [5]. The formula is based on the ECT value, box perimeter, bending stiffness of the walls and correction factors. The use of empirically determined correction factors reduces the versatility of the method because it is necessary to compute new values of these factors for each board quality and each packaging model. In 1968, a formula for estimating the strength of the regular package was presented. As the [PERSON] formula, it was based on the package perimeter, the ECT and plate bending stiffness, and, additionally, also took into account the buckling factor, stacking time and load ratio [6]. In 1985, [PERSON] et al. proposed other constants and exponents of the [PERSON] formula [7], and in 1987, [PERSON] et al. modified the [PERSON] formula for more cardboard cutting methods and equipment [8]. [PERSON] presented a formula for the load capacity of the packaging, which depends on several important factors, such as the grammage of the cardboard layers, cardboard thickness, take-up factor, average corrugation count, perimeter and type of the box, moisture of the walls and printed ratio [9]. In 1993, [PERSON] and [PERSON] proposed an extension of the [PERSON] formula to several other types of boxes, taking into account the width and depth of the package [10]. The finite element method (FEM) is often used for numerical analysis of packaging strength.