applied sciences
Article Modeling Warp in Corrugated Cardboard Based on Homogenization Techniques for In-Process Measurement Applications
Markus Beck * and Gerhard Fischerauer
Chair of Measurement and Control Systems, Faculty of Engineering Science, University of Bayreuth, Universitätsstraße 30, 95447 Bayreuth, Germany; Gerhard.Fischerauer@uni-bayreuth.de * Correspondence: Markus.Beck@uni-bayreuth.de Featured Application: The model proposed in this article lays the foundation for cost-efficient in- process measurement of warp in corrugated cardboard manufacturing and, subsequently, improved process control to reduce waste. Abstract: A model for describing warp—characterized as a systematic, large-scale deviation from the intended flat shape—in corrugated board based on Kirchhoff plate theory is proposed. It is based on established homogenization techniques and only a minimum of model assumptions. This yields general results applicable to any kind of corrugated cardboard. Since the model is intended to be used with industrial data, basic material properties which are usually not measured in practice are summarized to a few parameters. Those parameters can easily be fitted to the measurement data, allowing the user to systematically identify ways to reduce warp in a given situation in practice. In particular, the model can be used both as a filter to separate the warp from other surface effects such as washboarding, and to interpolate between discrete sample points scattered across the surface of a corrugated board sheet. Applying the model only requires height measurements of the corrugated board at several known (not necessarily exactly predetermined) locations across the corrugated board and acts as an interpolation or regression method between those points. These data can be acquired during production in a cost-efficient way and do not require any destructive testing of the board. The principle of an algorithm for fitting measured data to the model is presented and illustrated with examples taken from ongoing measurements. Additionally, the case of warp-free board is analyzed in more detail to deduce additional theoretical conditions necessary to reach this state.
Citation: Beck, M.; Fischerauer, G. Modeling Warp in Corrugated Cardboard Based on Homogenization Techniques for In-Process Measurement Applications. Appl. Sci. 2022 , 12 , 1684. https://doi.org/ 10.3390/app12031684
Keywords: corrugated board; homogenization; orthotropic plate; warping
Academic Editor: Andrea Dorigato
Received: 29 November 2021 Accepted: 3 February 2022 Published: 6 February 2022
1. Introduction Despite ubiquity in daily life and high production volume (45 × 10 9 m 2 with 1.6% annual growth in Europe as of 2019 [1]), corrugated cardboard and especially its produc- tion receive only a little attention in scientific research. The existing research is mainly focused on mechanical properties such as buckling under load [2], bending, and torsional and transverse shear stiffness [3]. In [4,5], the effect of partially crushing the corrugated structure—a problem that routinely occurs while printing or otherwise processing corru- gated board—on mechanical properties is studied numerically, analytically and with lab tests, while differentiating the structure of the corrugation. This focus on mainly mechanical properties of corrugated board is also present in the standardized testing procedures applied in corrugated board industry. Ref. [6] gives an overview over usual testing procedures, including flat crush test, edge crush test, and tests for puncture and bursting resistance. It should be noted that those tests are designed to destroy the analyzed samples. In [7], the authors provide both a numerical and an analytical model to predict compressive strength, edge crush test, and overall stiffness of
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Appl. Sci. 2022 , 12 , 1684. https://doi.org/10.3390/app12031684
https://www.mdpi.com/journal/applsci
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