Appl. Sci. 2022 , 12 , 1684
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the width of the board, whereas [14] uses the quadratic width of the board along with a normalization factor to ensure a dimension-free result. While those metrics can serve as a quality feature to distinguish between acceptable and non-acceptable amounts of warp, reducing a surface to a single number neglects a lot of information. This information could otherwise be used to improve the control of the manufacturing process.
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. In order to simplify the rather complex structure of corrugated board, homogenization is often applied, among others in [4,5,9–11]. This technique has already been extensively studied and been proven to represent the behavior of various corrugated structures with high accuracy. Literature therefore offers a range of different approximations to calculate the stiffness parameters of an equivalent orthotropic plate from the properties of the base material of corrugated plates. In [15], different homogenization models are compared using finite-element methods. The method proposed in [16] is not only verified with regard to the displacement under load, but also with regard to the internal forces and moments of the board. The approach outlined in [17] is checked for validity in borderline cases, such as a corrugation height of zero, against the respective analytic solutions. In [18], a combination of classical lamination theory and an equivalent energy method is used with a special focus on corrugated laminates. Recently, [19] included the effects of creasing and perforation on stability in homogenization approaches, and [20] derives effective torsional and transversal stiffness parameters for homogenization of corrugated cardboard. The mentioned applications of the homogenization technique are intended to analyze the reaction of corrugated board to external loads. This article, in contrast, focuses on internal stress in order to model warp, which is not covered by the models in the existing literature. In [21], warp in fused deposition modeling (FDM) rapid prototyping is modeled. The basic situation is somewhat similar to corrugated board, as rapid prototyping also works with clearly defined layers. Since workpieces manufactured this way can have arbitrary shape, they cannot be modeled as plates, contrary to corrugated board. This prevents an easy inclusion of forces—namely, their own weight—in the model. Moreover, the model in [21] utilizes the fact that the primary cause of warp in FDM processes, thermal contraction during the cooling phase, can directly be expressed by simple physical relations. The situation in corrugated board is more complex, which forces this article to take a more general approach.
2. Modeling 2.1. 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.
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