# Modeling Warp in Corrugated Cardboard Based on Homogenization Techniques for In-Process Measurement Applications

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## Abstract

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## 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

## 1. Introduction

## 2. Modeling

#### 2.1. General Model Assumptions

#### 2.2. Effects of Internal Stress

#### 2.3. Eliminating Excess Degrees of Freedom

^{T}(see Figure 3). Thus, it is important to note that the strain components (and therefore also the stress components) are not independent from each other, but can be derived by:

#### 2.4. A Condition for Warp-Free Corrugated Board

## 3. Applying the Model on Measured Surface Data

## 4. Comparison with Measured Surfaces

## 5. Discussion

## 6. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Typical production defects of corrugated board. The overall board is warped. The two magnified details show a damaged top liner (

**I**) and washboarding (

**II**). The aim of the presented model is to separate large-scale warping from those (and similar) small-scale effects.

**Figure 2.**Possible example structure of corrugated board with two (different) flutings (

**a**) and the equivalent orthotropic plate model (

**b**).

**Figure 3.**Model assumption for the deformation of the individual paper sheets during production (dashed line) compared to the later equilibrium state (solid line). This causes a position-dependent displacement (u,v)

^{T}.

**Figure 4.**Application of the model to interpret measured data. A least-square fit is used to find the effective material parameters and displacement fields that provide the best explanation for the observed data. Note that this diagram only displays the principal data flow between separate logical modules needed to apply the model to measured data. The optimization algorithm (“Optimizer”) implements the main control flow (including, e.g., termination conditions when the desired level of precision is needed), utilizing the bending model to evaluate its cost function.

**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.

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**MDPI and ACS Style**

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

**AMA Style**

Beck M, Fischerauer G.
Modeling Warp in Corrugated Cardboard Based on Homogenization Techniques for In-Process Measurement Applications. *Applied Sciences*. 2022; 12(3):1684.
https://doi.org/10.3390/app12031684

**Chicago/Turabian Style**

Beck, Markus, and Gerhard Fischerauer.
2022. "Modeling Warp in Corrugated Cardboard Based on Homogenization Techniques for In-Process Measurement Applications" *Applied Sciences* 12, no. 3: 1684.
https://doi.org/10.3390/app12031684