Appl. Sci. 2022 , 12 , 1684
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Figure5. Examples for surface forms defined by the proposed model (continuous surface), compared to measured data of corrugated board (black circles). Histograms of the distance between the model surface and the measured data are also provided. MD refers to the machine direction, CD the cross direction of corrugated board production. 5. Discussion The model proposed in this article does not contain any assumptions about the inner structure of the board. Therefore, it is applicable to any corrugated board regardless of the number of layers or the form of the corrugation. The only relevant model assump- tions are the validity of the homogenization approach and the assumption of a position- independent interpolation function. The homogenization has already been proven as a good approximation in numerous previous publications. For the second assumption, the position-independent interpolation, arguments were provided in this work. This generality comes at the cost that no calculation method for the linear factors κ k can be given at this level of abstraction. The main result is the existence of a linear relationship between the displacement fields in the paper sheets and the curvature caused by the internal stress. While this result may seem weak, a complete functional relationship would probably offer only little additional benefit for the actual industrial practice. The necessary material data are not routinely measured in corrugated board production. Therefore, it appears to be a better approach to fit the linear factors directly to the measured curvatures as outlined in Section 3, especially since there are only three effective parameters in (16). The fact that the curvature depends on a linear combination of the displacements in the individual paper sheets implies that there must be more than one solution for eliminating warp in corrugated board. Only a weighted sum of all paper sheets occurs in the final equations, which means that in theory any effect can be accomplished by influencing only one layer. This seems plausible, as it would otherwise be hard to imagine how warp could be controlled in practice. Many plants rely completely on heuristics and experience of their employees to correctly balance the production parameters (various
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