# Numerical Homogenization of Single-Walled Corrugated Board with Imperfections
###### Abstract
[PERSON] and [PERSON] used FEM to compare the strength of the packaging with a formula based on, e.g., the Poisson's ratio [11]. [PERSON] and [PERSON] performed FEM simulations to compare the obtained effective cardboard transverse shear moduli with analytical predictions and experimental results [12; 13]. [PERSON] and [PERSON] presented observations of buckling phenomena of corrugated cardboard boxes modeled with FEM [14]. In 2011, an overview of the analytical and numerical load capacity estimation of corrugated board packaging was presented by [PERSON] and [PERSON] [15]. The layered structure of the cardboard and the anisotropy of the paper make numerical modeling difficult due to the necessity of knowing the material parameters of all corrugated board layers. The solution to this problem is a process called homogenization. Homogenization allows to reduce the complicated cross-section of the corrugated board to one layer, which significantly simplifies the numerical model and shortens the computation time. Using this method, the equivalent stiffnesses and the effective thickness of the model should be determined to ensure the same behavior before and after homogenization. The analytical homogenization of corrugated board is based on the equations of the classical theory of material strength and the classical theory of laminates [16]. In numerical homogenization, the basic equations of the finite element method are also used. The authors in this work utilize the method proposed by [PERSON] [17] and then extended by [PERSON] and [PERSON] [18], which is based on an energy balance between a representative volume element and an equivalent plate. [PERSON] presented a representative element of heterogeneous and homogenized elements, which is based on the strain energy [19]. By means of periodic homogenization, [PERSON] et al. obtained equivalent membrane and bending characteristics of the plates [20]. In 2010, [PERSON] and [PERSON] determined the torsional stiffness of an orthotropic sandwich panel by splitting the panel into two beams [21]. In 2018, [PERSON] et al. presented a multiscale asymptotic homogenization for a composite material with a layered hierarchical structure [22; 23]. [PERSON] et al. proposed an approximation of the load capacity and deformation of a package while applying empirical observations [24]. The use of homogenization to analyze a thin-walled seating made of triple-wall corrugated cardboard is presented by [PERSON] et al. [25]. [PERSON] et al. investigated the effect of creasing or perforation on the stiffness of corrugated board [26]. [PERSON] et al. performed a sensitivity analysis in optimal design of single- and double-walled corrugated board packaging [1; 2]. [PERSON] et al. verified the accuracy and reliability of the homogenization methods in the static analysis of trapezoidally and sinusoidally corrugated panels [27]. The load capacity of the corrugated board depends on several factors. One of them is the imperfections of the cardboard layers resulting from the production process. In 1995, [PERSON] carried out parametric tests to investigate the sensitivity of the load capacity with the size of initial imperfections [28]. The influence of geometrical imperfections on the mechanical properties was presented by [PERSON] et al. [29]. [PERSON] modified the nonlinear buckling analysis of [PERSON] and [PERSON] orthotropic plates to account for initial cardboard imperfections [30]. [PERSON] and [PERSON] presented an analytical method of determining the bending stiffness of double-walled corrugated cardboard with imperfections [31]. In the present article, an approach to account for the geometric imperfections of corrugated cardboard in numerical homogenization of single-walled corrugated board is presented. The approach presented in this paper extends and generalizes the homogenization methods already presented in our previous works. This generalization concerns geometric imperfections built into the material.