## 2 Modeling  
### General Model Assumptions  
Corrugated board is produced by gluing together several--usually at least 3--paper sheets (see Figure 2a for an example). Every second sheet is pressed into a corrugated form before gluing it to the flat sheets to form a fluting. Warpage is caused by internal stresses in this composite. Alternatively, these stresses can also be regarded as the result of deformation of individual sheets at the time of gluing in relation to the later state of equilibrium (Figure 3). The reason for this distortion--temperature, humidity, and mechanical prestressing of the sheets would be obvious parameters in the actual production process--is irrelevant for modeling; it is sufficient to note that such a distortion state exists. Figure 1: Typical production defects of corrugated board. The overall board is warped. The two magnified details show a damaged top liner (**I**) and washboarding (**II**). The aim of the presented model is to separate large-scale warping from those (and similar) small-scale effects. The corrugated board is modeled as an orthotropic plate residing in the _xy_-plane with the corrugation direction aligned to one of the coordinate axes (Figure 2b). The warp is, as mentioned above, the systematic large-scale deviation from the ideal planar form, not including any local defects. It is seen as the result of the internal strain based on linear Kirchhoff theory. This means that the displacement _w_(_x,y_) is considered to be small enough to be neglected and all calculations can be performed as if the plate was flat. It should be noted that the approach of this model is to take the well-established homogenization approach for corrugated structures and apply it backwards in a certain sense: as shown in the literature [16; 17; 18], it is possible to approximate the overall behavior of a corrugated board by a homogenous, orthotropic plate. Therefore, the proposed model assumes that internal forces, strains, and moments can be approximated as in a homogenous plate as well. Besides the internal stress, the only forces considered are the weight of the board and the reaction forces of the underground. Since the model is intended to be used to characterize warp, it seems reasonable to assume the board to be under measurement conditions with no external loads. The self-weight, however, needs to be included in calculations: while corrugated structures are generally known for providing high stiffness at low specific weight, the size of corrugated boards creates lever arms that can cause visible deformation under the weight of the board. By only considering situations without external loads, the model can be simplified: the board is only under bending load (compression in z-direction can be neglected compared to the warping), while in-plane deformation is not considered. Therefore, extension-bending coupling--which would otherwise be relevant if the board is not symmetric--does not have any effect relevant to the considerations of the article and can thus be neglected. Since there are no external in-plane forces, classical buckling is not included in the model. The compressive forces for buckling must stay in plane while the plate is deformed  
Figure 3: Model assumption for the deformation of the individual paper sheets during production (dashed line) compared to the later equilibrium state (solid line). This causes a position-dependent displacement (\(u\),\(v\))\({}^{\mathrm{T}}\). Figure 2: Possible example structure of corrugated board with two (different) flutings (**a**) and the equivalent orthotropic plate model (**b**). so that a lever arm exists within the deformed plate. The internal forces discussed here, however, will always be aligned with the plate. The in-plane deformation of the board is neglected since it is, contrary to warpage, not relevant in industrial practice. For the rest of this section, the relevant equations for modeling orthotropic Kirchhoff plates under orthogonal surface load are summarized based on standard textbook knowledge (e.g., [22] (pp. 193-208)).