Appl. Sci. 2022 , 12 , 1684
13of 15
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. Therefore, a practical approach can model the displacement with a quadratic or cubic polynomial in machine direction and a polynomial of degree three-to-five in cross-direction. This reduction to a limited number of degrees of freedom can also serve an important role for measuring warp. Not every deviation of the real surface of corrugated board is caused by warp. As mentioned above, warp is large-scale, systematic curvature. In reality, however, warp cannot be measured directly. The only quantity which can be measured directly is the shape of the surface. Here, other effects, such as washboarding (where the surface partially follows the form of the fluting), or simple surface defects such as dents, superimpose the warp. Therefore, when combined with a suitable approximation for the technically plausible displacements in the paper sheets, this model can serve as a definition to separate the warp from other effects. Everything that can be explained by the model with displacement functions taken from a sensible function space is considered to be warp and effects of the self-weight (which can be separated by the model, if required). Therefore, to analyze warp, the measured surface data would be fitted to the model, yielding a smooth idealized surface, which in turn can be used to improve process control. 6. Conclusions In this article, a novel method for modeling warp in corrugated cardboard has been developed. This allows more insight into both measurement and control of warp during the production of corrugated board. It gives a more detailed mathematical description of warp than the metrics currently used in the industry. The latter generally reduce the entire effect of warp to a single number, whereas the proposed model allows reconstruction of a 2D surface even from a limited number of sample points. At the same time, the results of applying the model are different from a simple surface scan of the corrugated board (using, e.g., a coordinate-measuring machine), since the latter would also include other surface defects (such as washboarding) which have different reasons and therefore need to be treated differently. The model, on the other hand, can act as a filter extracting the warp selectively, by implicitly defining a set of physically plausible surface forms for warped corrugated board. The model does not require knowledge of the material parameters or the details of the corrugated structure, which makes it easily applicable in industrial production and processing of cardboard, as the mechanical properties are usually measured using destructive tests and, due to the nature of the raw materials, subject to variations even
Made with FlippingBook - Online magazine maker