## 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 assumptions 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 \(\kappa_{k}\) 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  
Figure 5: 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. temperatures, moisture, prestressing, etc.). This would hardly be possible if there were only one solution which results in good (i.e., warp-free) corrugated board within the space of possible production parameters. Note that this does not contradict the conclusion drawn from Equation (26), which was that the equivalent strains must completely disappear for the board to be warp-free. The equivalent strains in (26) that must disappear are still linear combinations of the strains in the individual paper sheets, and the latter are directly and individually influenced by production parameters. Establishing an analytic relationship between the displacement in the paper sheets and the curvature also allows inclusion of knowledge about the production process. Assume, for example, that there are a certain number of nozzles installed in the machine to moisturize the paper for processing. Then, moisture in the paper and therefore the displacement caused during subsequent drying, can only have a certain number of degrees of freedom. A simple but general approach is to model the displacement functions as polynomials of a low degree, which was successfully done in the examples in Section 4. As mentioned above, the board is produced in a continuous process. In machine direction, the individual sheet can therefore be seen as a local approximation similar to a Taylor series approximation, since control of production parameters varies only slowly compared to the production speed. In cross-direction, there are a limited number of devices interacting with the paper sheets. As long as this number is sufficiently low to prevent numerical instabilities, a simple polynomial interpolation in-between will probably yield acceptable results.