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.
Highlights
Despite ubiquity in daily life and high production volume (45 × 109 m2 with 1.6%annual growth in Europe as of 2019 [1]), corrugated cardboard and especially its production receive only a little attention in scientific research
In [4,5], the effect of partially crushing the corrugated structure—a problem that routinely occurs while printing or otherwise processing corrugated 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 present in the standardized testing procedures applied in corrugated board industry
In [7], the authors provide both a numerical and an analytical model to predict compressive strength, edge crush test, and overall stiffness of corrugated board, and could prove that the much more evaluated analytical model reaches the same accuracy as the numerical approach
Summary
Despite ubiquity in daily life and high production volume (45 × 109 m2 with 1.6%. annual growth in Europe as of 2019 [1]), corrugated cardboard and especially its production receive only a little attention in scientific research. Combining numerical finite-element calculations with lab tests, those articles derived improved analytical formulas for estimating the compressive strength of such modified boxes compared to classical and widely used formulas Production defects such as warping of the board see less research activity. The knowledge about the physical properties of corrugated board embedded in this model reduces the degrees of freedom which need to be determined from measured data, and, as a consequence, the number of sensors required This allows automated, cost-efficient, and robust (ignoring small-scale surface defects) measurement of warp in corrugated board manufacturing, providing a previously unattained level of information about the warping of the produced board. The situation in corrugated board is more complex, which forces this article to take a more general approach
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