# Estimation of the Edge Crush Resistance of Corrugated Board Using Artificial Intelligence

^{1}

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Paperboard and Cardboard Laboratory Tests

**Figure 2.**Laboratory machines and samples used in the research: (

**a**) short-span compression tester for paperboard; (

**b**) tensile stiffness and strength tester for paperboard.

^{2}is marked BE-590.

#### 2.2. Artificial Neural Networks–Training Data

**I**is the identity matrix.

#### 2.3. Gaussian Processes

- Be symmetrical. That means that $k\left({x}_{i},\text{}{x}_{j}\right)=k\left({x}_{j},{x}_{i}\right)$;

- Be positively defined. That means that the kernel matrix ${K}_{xx}$ induced by $k$ for any set of inputs should be a positive definite matrix.

## 3. Results

^{th}and 75

^{th}percentiles of the obtained errors, respectively. The interquartile range is represented by the area included between the bottom and top of each box. The red line represents the median of the errors. One can notice the results skewness if the median is not centered within the box. The lines extending above and below each box are called the whiskers, and they go from the end of the interquartile range to the furthest observation within the whisker length (the adjacent value). The observations out the whisker length, the outliers, are marked with red +.

^{2}is also shown.

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**ECT samples cut at different angles: (

**a**) 0 degrees (CD); (

**b**) 15 degrees; (

**c**) 30 degrees; (

**d**) 45 degrees; (

**e**) 60 degrees; (

**f**) 75 degrees.

**Figure 4.**Laboratory machines and samples used in the research: (

**a**) edge crush tester for corrugated board; (

**b**) device to cut corrugated samples for ECT.

**Figure 5.**Estimation error of three different methods: (

**a**) 3-layer corrugated board model; (

**b**) 5-layer cardboard model.

**Figure 6.**Linear regression of trained 3-ply corrugated model; (

**a**) based on feedforward neural network; (

**b**) based on deep neural network; (

**c**) based on Gaussian process.

**Figure 7.**Linear regression of trained 5-ply corrugated model; (

**a**) based on feedforward neural network; (

**b**) based on deep neural network; (

**c**) based on Gaussian process.

**Figure 8.**The ECT estimates using two GP-based models; (

**a**) 3-ply cardboard model; (

**b**) 5-ply cardboard model; the black dots represent the mean value of the prediction, while the vertical black lines represent the ±3 standard deviation for each sample. Experimental ECT values are shown as cyan dots.

**Figure 9.**Normalized sensitivity of the GP based model (3-ply cardboard) to all 28 input parameters.

**Table 1.**Component papers and geometry of the corrugated boards [72].

Corrugated Board | Component Papers | Corrugated Layers | ||||||
---|---|---|---|---|---|---|---|---|

Wave | Grammage | Height | Paper ID | Grammage | Thickness | Height | Period | Take-Up |

Type | (g/m^{2}) | (mm) | (g/m^{2}) | (mm) | (mm) | (mm) | Factor | |

B | 410 | 2.912 | TL3125 | 124 ± 6 | 0.27 ± 0.1 | - | - | - |

WS120 | 118 ± 6 | 0.25 ± 0.1 | 2.55 | 6.34 | 1.337 | |||

TL3125 | 126 ± 6 | 0.27 ± 0.1 | - | - | - | |||

C | 590 | 4.110 | KLB170 | 168 ± 8 | 0.36 ± 0.2 | - | - | - |

S.C.175 | 176 ± 8 | 0.36 ± 0.2 | 3.63 | 7.95 | 1.427 | |||

KLB170 | 169 ± 8 | 0.36 ± 0.2 | - | - | - | |||

E | 480 | 1.586 | TLWC160 | 158 ± 8 | 0.17 ± 0.1 | - | - | - |

WS135 | 133 ± 6 | 0.13 ± 0.1 | 1.16 | 3.50 | 1.236 | |||

TLW160 | 159 ± 8 | 0.17 ± 0.1 | - | - | - | |||

BC | 790 | 6.740 | KLB170 | 168 ± 8 | 0.28 ± 0.2 | - | - | - |

W135 | 136 ± 6 | 0.25 ± 0.1 | 2.55 | 6.34 | 1.337 | |||

WS80 | 79 ± 4 | 0.22 ± 0.1 | - | - | - | |||

WS135 | 133 ± 6 | 0.25 ± 0.1 | 3.63 | 7.95 | 1.427 | |||

KLB170 | 172 ± 8 | 0.28 ± 0.2 | - | - | - | |||

BE | 600 | 4.150 | TLW140 | 141 ± 7 | 0.28 ± 0.2 | - | - | - |

WS95 | 94 ± 5 | 0.22 ± 0.1 | 2.55 | 6.34 | 1.337 | |||

WS80 | 81 ± 4 | 0.22 ± 0.1 | - | - | - | |||

WS95 | 94 ± 5 | 0.22 ± 0.1 | 1.16 | 3.50 | 1.236 | |||

TL3125 | 124 ± 6 | 0.28 ± 0.2 | - | - | - | |||

BE | 590 | 4.120 | TL3125 | 125 ± 6 | 0.32 ± 0.2 | - | - | - |

WS95 | 94 ± 5 | 0.24 ± 0.1 | 2.55 | 6.34 | 1.337 | |||

W80 | 79 ± 5 | 0.22 ± 0.1 | - | - | - | |||

WS95 | 96 ± 5 | 0.24 ± 0.1 | 1.16 | 3.50 | 1.236 | |||

TL3125 | 124 ± 6 | 0.29 ± 0.2 | - | - | - |

Layer | SCT | Tensile Stiffness | Flute | GRM ^{*} | THK ^{*} | ANG ^{*} | ||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

CD | MD | 45 d | CD | MD | 45 d | Width | Height | TUF ^{*} | ||||

Liner | 1 | 2 | 3 | 4 | 5 | 6 | - | - | - | 7 | 8 | - |

Flute | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | - |

Liner | 20 | 21 | 22 | 23 | 24 | 25 | - | - | - | 26 | 27 | - |

- | - | - | - | - | - | - | - | - | - | - | - | 28 |

Layer | SCT | Tensile Stiffness | Flute | GRM ^{*} | THK ^{*} | ANG ^{*} | ||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

CD | MD | 45 d | CD | MD | 45 d | Width | Height | TUF ^{*} | ||||

Liner | 1 | 2 | 3 | 4 | 5 | 6 | - | - | - | 7 | 8 | - |

Flute | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | - |

Liner | 20 | 21 | 22 | 23 | 24 | 25 | - | - | - | 26 | 27 | - |

Flute | 28 | 29 | 30 | 31 | 32 | 33 | 34 | 35 | 36 | 37 | 38 | |

Liner | 39 | 40 | 41 | 42 | 43 | 44 | - | - | - | 45 | 46 | |

- | - | - | - | - | - | - | - | - | - | - | - | 47 |

**Table 4.**The input training data for all models. Short-span compression strength as well as tensile stiffness are presented as mean values supplemented with corresponding standard deviations.

Grade ID | Paper ID | Short-Span Compression Strength | Tensile Stiffness | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

CD | MD | 45 Deg | CD | MD | 45 Deg | ||||||||

(kN/m) | (kN/m) | ||||||||||||

TL3125 | 2.14 | ±0.12 | 3.97 | ±0.05 | 2.92 | ±0.05 | 373.3 | ±2.1 | 1012.7 | ±6.3 | 572.7 | ±10.8 | |

B-410 | WS120 | 2.09 | ±0.07 | 4.09 | ±0.15 | 3.14 | ±0.08 | 365.1 | ±8.8 | 1024.6 | ±8.5 | 516.5 | ±9.7 |

TL3125 | 2.09 | ±0.11 | 4.05 | ±0.11 | 3.11 | ±0.14 | 381.2 | ±11.2 | 1058.3 | ±9.8 | 595.1 | ±10.5 | |

KLB170 | 3.28 | ±0.18 | 5.96 | ±0.23 | 4.43 | ±0.19 | 527.8 | ±9.1 | 1472.1 | ±20.1 | 929.1 | ±20.3 | |

C-590 | S.C.175 | 4.18 | ±0.19 | 7.47 | ±0.18 | 5.84 | ±0.07 | 686.1 | ±13.9 | 1476.1 | ±10.9 | 924.7 | ±48.9 |

KLB170 | 3.19 | ±0.06 | 5.51 | ±0.18 | 4.65 | ±0.13 | 568.1 | ±14.8 | 1445.1 | ±31.7 | 956.2 | ±26.6 | |

TLWC160 | 2.75 | ±0.20 | 4.20 | ±0.15 | 3.49 | ±0.13 | 412.1 | ±6.8 | 1043.6 | ±11.1 | 635.0 | ±11.2 | |

E-480 | WS135 | 2.13 | ±0.10 | 4.25 | ±0.15 | 2.95 | ±0.07 | 365.0 | ±9.8 | 1067.5 | ±14.5 | 533.5 | ±8.5 |

TLW160 | 2.43 | ±0.11 | 4.09 | ±0.13 | 3.21 | ±0.11 | 443.9 | ±1.5 | 1102.1 | ±38.6 | 667.0 | ±5.4 | |

KLB170 | 3.39 | ±0.12 | 6.14 | ±0.19 | 4.69 | ±0.13 | 618.9 | ±17.6 | 1534.2 | ±6.7 | 990.0 | ±17.5 | |

W135 | 2.19 | ±0.09 | 4.27 | ±0.10 | 3.09 | ±0.13 | 369.1 | ±11.2 | 1113.5 | ±9.8 | 572.4 | ±22.6 | |

BC-790 | WS80 | 1.50 | ±0.08 | 2.30 | ±0.09 | 1.91 | ±0.04 | 317.1 | ±4.2 | 699.1 | ±3.3 | 445.0 | ±5.8 |

WS135 | 2.23 | ±0.04 | 4.37 | ±0.15 | 3.18 | ±0.06 | 385.9 | ±12.0 | 1147.4 | ±7.6 | 623.5 | ±8.0 | |

KLB170 | 3.30 | ±0.18 | 5.98 | ±0.32 | 4.66 | ±0.20 | 592.7 | ±8.3 | 1418.9 | ±23.6 | 838.2 | ±12.7 | |

TLW140 | 2.61 | ±0.13 | 3.92 | ±0.09 | 3.08 | ±0.11 | 506.0 | ±7.1 | 1000.0 | ±12.6 | 622.6 | ±11.7 | |

WS95 | 1.69 | ±0.09 | 2.99 | ±0.16 | 2.26 | ±0.09 | 331.9 | ±5.5 | 872.7 | ±6.0 | 498.6 | ±10.3 | |

BE-600 | WS80 | 1.42 | ±0.03 | 2.56 | ±0.17 | 1.88 | ±0.08 | 273.2 | ±3.8 | 812.8 | ±12.3 | 424.8 | ±7.2 |

WS95 | 1.52 | ±0.08 | 3.16 | ±0.13 | 2.46 | ±0.06 | 290.7 | ±6.7 | 885.5 | ±18.9 | 508.4 | ±14.7 | |

TL3125 | 2.13 | ±0.09 | 3.83 | ±0.13 | 2.93 | ±0.10 | 440.6 | ±3.4 | 1082.2 | ±13.3 | 623.3 | ±35.4 | |

TL3125 | 2.26 | ±0.13 | 3.64 | ±0.08 | 3.06 | ±0.11 | 412.9 | ±11.6 | 961.3 | ±10.0 | 587.0 | ±3.4 | |

WS95 | 1.50 | ±0.06 | 2.69 | ±0.13 | 2.01 | ±0.09 | 294.3 | ±8.6 | 756.4 | ±13.8 | 427.4 | ±8.8 | |

BE-590 | W80 | 1.47 | ±0.08 | 2.34 | ±0.06 | 1.94 | ±0.08 | 343.5 | ±5.8 | 696.5 | ±14.5 | 459.8 | ±5.0 |

WS95 | 1.75 | ±0.10 | 2.96 | ±0.18 | 2.15 | ±0.06 | 332.0 | ±4.0 | 854.7 | ±4.2 | 474.4 | ±11.0 | |

TL3125 | 2.32 | ±0.07 | 3.75 | ±0.06 | 2.87 | ±0.17 | 413.8 | ±4.7 | 883.1 | ±20.9 | 588.3 | ±14.5 |

**Table 5.**The output training data for all models. The edge crush strength a corrugated board loaded at different angles is presented as mean values supplemented with corresponding standard deviations.

Board ID | Edge Crush Resistance | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

CD (0 Deg) | 15 Deg | 30 Deg | 45 Deg | 60 Deg | 75 Deg | |||||||

(kN/m) | ||||||||||||

B-410 | 5.48 | ±0.09 | 5.17 | ±0.12 | 4.40 | ±0.03 | 3.88 | ±0.10 | 3.05 | ±0.10 | 2.29 | ±0.13 |

C-590 | 9.68 | ±0.10 | 9.07 | ±0.14 | 7.60 | ±0.02 | 6.24 | ±0.12 | 4.83 | ±0.10 | 3.49 | ±0.23 |

E-480 | 6.37 | ±0.17 | 5.92 | ±0.08 | 5.99 | ±0.26 | 5.47 | ±0.27 | 5.16 | ±0.17 | 4.56 | ±0.18 |

BC-790 | 10.41 | ±0.13 | 9.56 | ±0.37 | 8.38 | ±0.29 | 6.96 | ±0.10 | 6.00 | ±0.17 | 4.31 | ±0.16 |

BE-600 | 8.95 | ±0.14 | 8.39 | ±0.06 | 7.76 | ±0.14 | 6.46 | ±0.18 | 5.66 | ±0.32 | 4.30 | ±0.35 |

BE-590 | 9.68 | ±0.10 | 9.07 | ±0.14 | 7.60 | ±0.02 | 6.24 | ±0.12 | 4.83 | ±0.10 | 3.49 | ±0.22 |

**Table 6.**Mean absolute estimation error of both models based on FF, DL and GP methods trained with full-length input training vector (full) and truncated input training vector (small).

Model | Mean Absolute Error (%) | ||

Method | 3-Layer Cardboard | 5-Layer Cardboard | |

FF (full) | 3.163 | 1.575 | |

FF (small) | 2.273 | 1.921 | |

DL (full) | 2.985 | 3.093 | |

DL (small) | 3.948 | 2.916 | |

GP (full) | 2.484 | 1.317 | |

GP (small) | 2.260 | 2.150 | |

Analytical [78] | 1.760 | 2.987 | |

FEM-1 [78] | 1.640 | 2.260 | |

FEM-2 [78] | 4.010 | 4.227 | |

Empirical-1 [78] | 3.547 | 10.60 | |

Empirical-2 [78] | 1.997 | 9.800 |

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## Share and Cite

**MDPI and ACS Style**

Garbowski, T.; Knitter-Piątkowska, A.; Grabski, J.K. Estimation of the Edge Crush Resistance of Corrugated Board Using Artificial Intelligence. *Materials* **2023**, *16*, 1631.
https://doi.org/10.3390/ma16041631

**AMA Style**

Garbowski T, Knitter-Piątkowska A, Grabski JK. Estimation of the Edge Crush Resistance of Corrugated Board Using Artificial Intelligence. *Materials*. 2023; 16(4):1631.
https://doi.org/10.3390/ma16041631

**Chicago/Turabian Style**

Garbowski, Tomasz, Anna Knitter-Piątkowska, and Jakub Krzysztof Grabski. 2023. "Estimation of the Edge Crush Resistance of Corrugated Board Using Artificial Intelligence" *Materials* 16, no. 4: 1631.
https://doi.org/10.3390/ma16041631