# Full-Field Measurements in the Edge Crush Test of a Corrugated Board—Analytical and Numerical Predictive Models

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Corugated Cardboard

^{2}. Both corrugated layers (E and B flutes) and the flat layer in between are made from lightweight recycled fluting WB with a grammage of 100 g/m

^{2}. As an internal layer again the white test liner with a grammage of 120 g/m

^{2}was used. The arrangement of individual layers and the geometry of the cardboard cross-section are shown in Figure 1.

#### 2.2. Measurements

#### 2.3. Optical Measurements of Sample Deformation

#### 2.4. Predictive Models

^{®}[34] software (version 2020, Dassault Systemes SIMULIA Corp., Johnston, IA, USA), which uses a linear elastic orthotropic material model with von Mises plasticity. Shell elements used in the calculations are quadrilaterals with four nodes, named S4, which use the full integration scheme with built-in techniques to prevent locking phenomena. The approximate size of a single element was 1 mm, which gives in total 17,825 elements, 18,668 nodes, and 112,008 degrees of freedom. In order to provide all the required material constants, the empirical equations provided by Baum [35] were used. First the ${E}_{MD}^{i}$ and ${E}_{CD}^{i}$ stiffness indexes (given in Table 2) were transformed to stiffness coefficients ${E}_{1}^{i}$ and ${E}_{2}^{i}$, respectively, by the equation:

## 3. Results

#### 3.1. Edge Crush Test Results

#### 3.2. Optical Measurements Results

#### 3.3. Predictive Analytical Model

#### 3.4. Predictive Numerical Model

^{®}software. This imperfect geometry was used in nonlinear analysis where the standard Newton–Raphson algorithm was used to find convergence in the subsequent iterations.

#### 3.5. Compilation of All Results

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 5.**Specimen: (

**a**) back and front face of the specimen; (

**b**) virtual extensometers on the front face of the specimen.

**Figure 6.**Setup of the optical measurements: (

**a**) camera recording the front face; (

**b**) camera recording the back face.

**Figure 8.**Compressive strength vs. deformation. (

**a**) the case where the SCT is lower than the critical load of the i-th layer; (

**b**) the case where the critical load is lower than the SCT of the i-th layer.

**Figure 11.**Analytical model: (

**a**) Visualization of first buckling mode for the i-th layer; (

**b**) constitutive relationships for all corrugated board layers.

**Figure 12.**Numerical model. Visualization of the first buckling mode for the whole corrugated board (front view).

**Figure 13.**Equivalent plastic strains on both sides of the ECT sample in the last iteration of the nonlinear analysis.

**Figure 14.**Displacements on both sides of the ECT sample in the last iteration of the nonlinear analysis.

**Figure 15.**Experimental and numerical results: (

**a**) load–displacement curves from experimental studies; (

**b**) load–displacement curves from numerical studies.

**Figure 17.**Stress-strain curves: (

**a**) TLW120 exchanged with TLW130; (

**b**) TLW120 exchanged with TLW145.

Wave (Flute) | Pitch [mm] | Height [mm] | Take-Up Ratio [–] |
---|---|---|---|

E | 3.50 | 1.18 | 1.242 |

B | 6.48 | 2.5 | 1.315 |

Layer Name | Thickness [$\mathsf{\mu}\mathbf{m}$] | ${\mathit{E}}_{\mathit{M}\mathit{D}}$ [kN/m] | ${\mathit{E}}_{\mathit{C}\mathit{D}}$ [kN/m] | $\mathit{S}\mathit{C}{\mathit{T}}_{\mathit{C}\mathit{D}}$ [kN/m] |
---|---|---|---|---|

TLWC 140 | 180 | 725 | 323 | 2.32 |

W 100 | 160 | 886 | 328 | 1.76 |

TLW 120 | 170 | 907 | 313 | 1.81 |

Layer Name | ${\mathit{E}}_{1}$ [MPa] | ${\mathit{E}}_{2}$ [MPa] | ${\mathit{\nu}}_{12}$ [–] | ${\mathit{G}}_{12}$ [MPa] | ${\mathit{G}}_{13}$ [MPa] | ${\mathit{G}}_{23}$ [MPa] |
---|---|---|---|---|---|---|

TLW 120 | 5669 | 2050 | 0.176 | 1319 | 103 | 59 |

W 100 | 5537 | 2050 | 0.209 | 1112 | 101 | 59 |

TLWC 140 | 4028 | 1794 | 0.196 | 1040 | 73 | 51 |

Layer Name | $\mathit{b}$ (mm) | $\mathit{L}$ (m) | $\mathit{S}\mathit{C}{\mathit{T}}_{\mathit{C}\mathit{D}}$ (kN/m) | ${\mathit{P}}_{\mathit{c}\mathit{r}}$ (kN/m) |
---|---|---|---|---|

TLWC 140 | 3.50 | 25 | 2.32 | 4.212 |

W 100 (E) | 2.17 | 25 | 1.76 | 9.573 |

W 100 | - | 25 | 1.76 | - |

W 100 (B) | 4.26 | 25 | 1.76 | 2.444 |

TLW 120 | 6.48 | 25 | 1.81 | 1.237 |

**Table 5.**The measured/calculated compressive stiffness and strength in CD of the corrugated board 5EB650C3.

Test/Model | $\mathit{E}\mathit{C}{\mathit{T}}_{\mathit{C}\mathit{D}}$ (kN/m) | ${\mathit{E}}_{\mathit{C}\mathit{D}}$ (kN/m) |
---|---|---|

Producer specification | 7.60 | - |

FEMat—crosshead | 7.51 | 1991 |

Instron—crosshead | 7.03 | 2142 |

Instron—opt. extensometry | 7.03 | 6442 |

Numerical model | 7.94 | 5920 |

Analytical model | 8.08 | 7063 |

**Table 6.**The effect of improving individual corrugated board layers by 10 and 20 percent, respectively, on the changes in the ECT.

Reference | Single Layer Improved by 10% | Single Layer Improved by 20% | |||||
---|---|---|---|---|---|---|---|

Paperboard Symbol | ECT (kN/m) | Paperboard Symbol | ECT (kN/m) | Diff. (%) | Paperboard Symbol | ECT (kN/m) | Diff. (%) |

TLWC 140 | 8.08 | TLWC 155 | 8.261 | 2.24 | TLWC 170 | 8.442 | 4.48 |

W 100 (E) | 8.08 | W 110 (E) | 8.297 | 2.69 | W 120 (E) | 8.515 | 5.38 |

W 100 | 8.08 | W 110 | 8.255 | 2.17 | W 120 | 8.430 | 4.33 |

W 100 (B) | 8.08 | W 110 (B) | 8.310 | 2.85 | W 120 (B) | 8.541 | 5.71 |

TLW 120 | 8.08 | TLW 130 | 8.940 | 10.64 | TLW 145 | 10.046 | 24.33 |

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**MDPI and ACS Style**

Garbowski, T.; Grabski, J.K.; Marek, A. Full-Field Measurements in the Edge Crush Test of a Corrugated Board—Analytical and Numerical Predictive Models. *Materials* **2021**, *14*, 2840.
https://doi.org/10.3390/ma14112840

**AMA Style**

Garbowski T, Grabski JK, Marek A. Full-Field Measurements in the Edge Crush Test of a Corrugated Board—Analytical and Numerical Predictive Models. *Materials*. 2021; 14(11):2840.
https://doi.org/10.3390/ma14112840

**Chicago/Turabian Style**

Garbowski, Tomasz, Jakub Krzysztof Grabski, and Aleksander Marek. 2021. "Full-Field Measurements in the Edge Crush Test of a Corrugated Board—Analytical and Numerical Predictive Models" *Materials* 14, no. 11: 2840.
https://doi.org/10.3390/ma14112840