# Numerical Homogenization of Multi-Layered Corrugated Cardboard with Creasing or Perforation

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Corrugated Board—Material Definition

#### 2.2. Creases and Perforations—Numerical Study

#### 2.3. Homogenization Technique

**A**,

**B**,

**D**, and

**R**as follows:

**A**represents extensional and shear stiffnesses;

**B**represents extension-bending coupling stiffnesses; and

**D**represents bending and torsional stiffnesses, while

**R**represents transverse shear stiffness.

## 3. Results

#### 3.1. Validation of the Proposed Method

#### 3.2. Detailed Results

**A**,

**B**,

**D**, and

**R**are non-zero; in particular, matrix

**B**(see Table 5), which combines the bending effects with the membrane stiffness of the plate.

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Paperboard mechanical behavior. The stress–strain relationships in different material directions.

**Figure 3.**Perforation types: (

**a**) Type 2/6—model SW; (

**b**) Type 4/4—model SW; (

**c**) Type 6/2—model SW; (

**d**) Type 2/6—model DW; (

**e**) Type 4/4—model DW; (

**f**) Type 6/2—model DW.

**Figure 5.**Perforation orientation in sample SW-44-C: (

**a**) rotation by 15 degrees; (

**b**) rotation by 30 degrees; (

**c**) rotation by 45 degrees; (

**d**) rotation by 60 degrees; (

**e**) rotation by 75 degrees; (

**f**) rotation by 90 degrees.

**Figure 6.**Perforation orientation in sample SW-44-F: (

**a**) rotation by 15 degrees; (

**b**) rotation by 30 degrees; (

**c**) rotation by 45 degrees; (

**d**) rotation by 60 degrees; (

**e**) rotation by 75 degrees.

**Figure 7.**Perforation orientation in sample DW-44-F: (

**a**) rotation by 15 degrees; (

**b**) rotation by 30 degrees; (

**c**) rotation by 45 degrees; (

**d**) rotation by 60 degrees; (

**e**) rotation by 75 degrees.

**Figure 8.**Crushed samples: (

**a**–

**c**) Single-walled sample crushed by 10%, 20%, and 30%, respectively; (

**d**–

**f**) Double-walled sample crushed by 10%, 20%, and 30%, respectively.

**Figure 9.**Cross section of the corrugated board along the wave: (

**a**) the reference SW sample—no offset; (

**b**) SW sample—offset equal to 1/16 P; (

**c**) SW sample—offset equal to 2/16 P; (

**d**) SW sample—offset equal to 3/16 P; (

**e**) SW sample—offset equal to 4/16 P; (

**f**) the reference DW sample—no offset; (

**g**) DW sample—offset equal to 1/16 P; (

**h**) DW sample—offset equal to 2/16 P; (

**i**) DW sample—offset equal to 3/16 P; (

**j**) DW sample—offset equal to 4/16 P.

**Figure 10.**RVE—external (in red color) and internal nodes and finite elements: (

**a**) SW model; (

**b**) DW model.

**Figure 11.**Representative shell elements of saw tooth geometry with quadrilateral mesh (single period): (

**a**) model with a fine 4-node mesh; (

**b**) model with a coarse 3-node mesh; (

**c**) model geometry.

**Figure 12.**Stiffness degradation in sample: (

**a**) SW-26; (

**b**) SW-44; (

**c**) SW-62; (

**d**) DW-26; (

**e**) DW-44; (

**f**) DW-62.

**Figure 13.**Stiffness degradation in sample SW: (

**a**) F-15; (

**b**) F-30; (

**c**) F-45; (

**d**) F-60; (

**e**) F-75. Three types of perforations were analyzed (2/6 mm, 4/4 mm, or 6/2 mm).

**Figure 14.**Stiffness degradation in a sample DW: (

**a**) F-15; (

**b**) F-30; (

**c**) F-45; (

**d**) F-60; (

**e**) F-75. Three types of perforation were analyzed (2/6 mm, 4/4 mm, or 6/2 mm).

**Figure 15.**Stiffness degradation in sample C-0: (

**a**) SW-26; (

**b**) SW-44; (

**c**) SW-62; (

**d**) DW-26; (

**e**) DW-44; (

**f**) DW-62.

**Figure 16.**Stiffness degradation in sample: (

**a**) SW-26-C-0-R-xx; (

**b**) SW-44-C-0-R-xx; (

**c**) SW-62-C-0-R-xx; (

**d**) DW-26-C-0-R-xx; (

**e**) DW-44-C-0-R-xx; (

**f**) DW-62-C-0-R-xx. Here xx is a crush level (0%; 10%, 20%, and 30%).

**Table 1.**Material data of intact double wall corrugated cardboard used for modeling the paper layers according to orthotropic constitutive relation.

Layers | ${\mathit{E}}_{1}\text{}$ | ${\mathit{E}}_{2}\text{}$ | ${\mathit{\nu}}_{12}\text{}$ | ${\mathit{G}}_{12}\text{}$ | ${\mathit{G}}_{13}\text{}$ | ${\mathit{G}}_{23}\text{}$ |
---|---|---|---|---|---|---|

(MPa) | (MPa) | (-) | (MPa) | (MPa) | (MPa) | |

liners | 3326 | 1694 | 0.34 | 859 | 429.5 | 429.5 |

fluting | 2614 | 1532 | 0.32 | 724 | 362 | 362 |

Perforation Type | Model SW | Model DW |
---|---|---|

4 mm cut, 4 mm gap | SW-44-Y ^{1}-xx ^{2} | DW-44-Y-xx |

2 mm cut, 6 mm gap | SW-26-Y-xx | DW-26-Y-xx |

6 mm cut, 2 mm gap | SW-62-Y-xx | DW-62-Y-xx |

^{1}Y means model type and can be: F-flute or C-cut.

^{2}xx is the cut or crease orientation and can be: 00, 15, 30, 45, 60, 75, or 90.

**Table 3.**The stiffnesses of representative shell element computed for a different approach of modeling confronted with data from [39] for saw tooth geometry.

Stiffness | Ref. [39] | Corse Model | Fine Model |
---|---|---|---|

${A}_{11},\text{}\left(\mathrm{N}/\mathrm{mm}\right)$ | 1.108 10^{6} | 1.118 10^{6} | 1.118 10^{6} |

${A}_{22},\text{}\left(\mathrm{N}/\mathrm{mm}\right)$ | 1.358 10^{6} | 1.380 10^{6} | 1.378 10^{6} |

${A}_{12},\text{}\left(\mathrm{N}/\mathrm{mm}\right)$ | 3.324 10^{5} | 3.449 10^{5} | 3.448 10^{5} |

${A}_{33},\text{}\left(\mathrm{N}/\mathrm{mm}\right)$ | 4.168 10^{5} | 4.115 10^{5} | 4.115 10^{5} |

${D}_{11},\text{}\left(\mathrm{N}\xb7\mathrm{mm}\right)$ | 9.195 10^{8} | 9.211 10^{8} | 9.210 10^{8} |

${D}_{22},\text{}\left(\mathrm{N}\xb7\mathrm{mm}\right)$ | 9.822 10^{8} | 9.926 10^{8} | 9.925 10^{8} |

${D}_{12},\text{}\left(\mathrm{N}\xb7\mathrm{mm}\right)$ | 2.758 10^{8} | 2.777 10^{8} | 2.777 10^{8} |

${D}_{33},\text{}\left(\mathrm{N}\xb7\mathrm{mm}\right)$ | 3.220 10^{8} | 3.269 10^{8} | 3.268 10^{8} |

${A}_{44}$, $\text{}\left(\mathrm{N}/\mathrm{mm}\right)$ | - | 5.194 10^{4} | 5.184 10^{4} |

${A}_{55}$,$\text{}\left(\mathrm{N}/\mathrm{mm}\right)$ | - | 7.408 10^{4} | 7.376 10^{4} |

A & B | B & D | R | |||||||
---|---|---|---|---|---|---|---|---|---|

1 | 2 | 3 | 1 | 2 | 3 | 4 | 5 | ||

A & B | 1 | 2184.4 | 388.92 | 0 | 0 | 0 | 0 | ||

2 | 388.92 | 1756.9 | 0 | 0 | 0 | 0 | |||

3 | 0 | 0 | 667.81 | 0 | 0 | 0 | |||

B & D | 1 | 0 | 0 | 0 | 8628.2 | 1506.5 | 0 | ||

2 | 0 | 0 | 0 | 1506.5 | 5469.3 | 0 | |||

3 | 0 | 0 | 0 | 0 | 0 | 2300.2 | |||

R | 4 | 105.08 | 0 | ||||||

5 | 0 | 130.91 |

A & B | B & D | R | |||||||
---|---|---|---|---|---|---|---|---|---|

1 | 2 | 3 | 1 | 2 | 3 | 4 | 5 | ||

A & B | 1 | 3313.8 | 593.33 | 0 | 1117.1 | 195.90 | 0 | ||

2 | 593.33 | 2967.5 | 0 | 196.36 | 1200.6 | 0 | |||

3 | 0 | 0 | 1077.8 | 0 | 0 | 409.89 | |||

B & D | 1 | 1117.1 | 196.36 | 0 | 20 619 | 3620.8 | 0 | ||

2 | 195.90 | 1200.6 | 0 | 3620.8 | 15 042 | 0 | |||

3 | 0 | 0.0 | 409.89 | 0 | 0 | 5934.5 | |||

R | 4 | 233.13 | 0 | ||||||

5 | 0 | 242.28 |

**Table 6.**Selected stiffnesses in SW samples with no perforation and with different flute orientations.

SW-0-F-00 | SW-0-F-15 | SW-0-F-30 | SW-0-F-45 | SW-0-F-60 | SW-0-F-75 | SW-0-F-90 | |
---|---|---|---|---|---|---|---|

${A}_{11}$ (MPa mm) | 2184.4 | 2127.2 | 1990.3 | 1854.2 | 1774.2 | 1751.5 | 1756.9 |

${A}_{22}$ (MPa mm) | 1756.9 | 1751.5 | 1774.2 | 1854.2 | 1990.3 | 2127.2 | 2184.4 |

${A}_{33}$ (MPa mm) | 667.81 | 699.26 | 760.50 | 792.80 | 760.50 | 699.30 | 667.80 |

${D}_{11}$ (MPa mm^{3}) | 8628.2 | 8313.5 | 7480.9 | 6521.5 | 5897.3 | 5575.8 | 5469.3 |

${D}_{22}$ (MPa mm^{3}) | 5469.3 | 5575.8 | 5897.3 | 6520.4 | 7480.9 | 8313.5 | 8628.2 |

${D}_{33}$ (MPa mm^{3}) | 2300.2 | 2425.2 | 2650.1 | 2755.4 | 2650.1 | 2425.2 | 2300.2 |

${R}_{44}$ (MPa mm) | 105.08 | 108.15 | 119.80 | 132.90 | 127.20 | 126.20 | 130.90 |

${R}_{55}$ (MPa mm) | 130.91 | 126.16 | 127.20 | 132.80 | 119.80 | 108.10 | 105.10 |

**Table 7.**Selected stiffnesses in DW samples with no perforation and with different flute orientations.

DW-0-F-00 | DW-0-F-15 | DW-0-F-30 | DW-0-F-45 | DW-0-F-60 | DW-0-F-75 | DW-0-F-90 | |
---|---|---|---|---|---|---|---|

${A}_{11}$ (MPa mm) | 3313.8 | 3250.6 | 3090.4 | 2955.2 | 2912.0 | 2939.7 | 2967.5 |

${A}_{22}$ (MPa mm) | 2967.5 | 2939.7 | 2912.0 | 2955.3 | 3090.4 | 3250.6 | 3313.8 |

${A}_{33}$ (MPa mm) | 1077.8 | 1127.5 | 1225.3 | 1275.9 | 1225.3 | 1127.5 | 1077.8 |

${D}_{11}$ (MPa mm^{3}) | 20,242 | 19,610 | 17,980 | 16,221 | 15,123 | 14,662 | 14,556 |

${D}_{22}$ (MPa mm^{3}) | 14,556 | 14,662 | 15,123 | 16,220 | 17,980 | 19,610 | 20,242 |

${D}_{33}$ (MPa mm^{3}) | 5778.6 | 6071.8 | 6634.3 | 6910.6 | 6634.3 | 6071.8 | 5778.6 |

${R}_{44}$ (MPa mm) | 233.13 | 240.21 | 246.71 | 257.56 | 247.51 | 242.88 | 242.28 |

${R}_{55}$ (MPa mm) | 242.28 | 242.88 | 247.51 | 257.43 | 246.71 | 240.21 | 233.13 |

**Table 8.**The selected stiffnesses in SW models for different perforations and flute rotated by 15 degrees.

Stiffness | SW-0-F-15 | SW-26-F-15 | SW-44-F-15 | SW-62-F-15 |
---|---|---|---|---|

${A}_{11}$ (MPa mm) | 2127.2 | 2116.1 | 2082.1 | 2052.3 |

${A}_{22}$ (MPa mm) | 1751.6 | 1609.1 | 1267.7 | 885.12 |

${A}_{33}$ (MPa mm) | 699.26 | 681.92 | 608.30 | 524.18 |

${D}_{11}$ (MPa mm^{3}) | 8313.4 | 8276.1 | 8166.4 | 8048.5 |

${D}_{22}$ (MPa mm^{3}) | 5575.8 | 5290.9 | 4291.8 | 2877.2 |

${D}_{33}$ (MPa mm^{3}) | 2425.2 | 2384.5 | 2216.7 | 1968.9 |

${R}_{44}$ (MPa mm) | 108.15 | 107.68 | 106.48 | 106.77 |

${R}_{55}$ (MPa mm) | 126.16 | 120.04 | 94.100 | 83.465 |

**Table 9.**Stiffness reduction for both SW and DW samples with flute rotated by 15 degrees for three cases of perforation.

Stiffness Reduction | SW-26-F-15 (%) | SW-44-F-15 (%) | SW-62-F-15 (%) | DW-26-F-15 (%) | DW-44-F-15 (%) | DW-62-F-15 (%) |
---|---|---|---|---|---|---|

$1-{A}_{11}$$/{A}_{11}^{*}$ | 0.523 | 2.121 | 3.519 | 0.508 | 1.903 | 3.364 |

$1-{A}_{22}$$/{A}_{22}^{*}$ | 8.133 | 27.66 | 49.46 | 7.852 | 27.77 | 50.98 |

$1-{A}_{33}$$/{A}_{33}^{*}$ | 2.480 | 13.01 | 25.04 | 2.735 | 12.66 | 24.50 |

$1-{D}_{11}$$/{D}_{11}^{*}$ | 0.449 | 1.769 | 3.187 | 0.467 | 1.786 | 3.247 |

$1-{D}_{22}$$/{D}_{22}^{*}$ | 5.110 | 23.03 | 48.40 | 6.377 | 25.41 | 49.18 |

$1-{D}_{33}$$/{D}_{33}^{*}$ | 1.677 | 8.598 | 18.81 | 2.171 | 10.25 | 20.88 |

$1-{R}_{44}$$/{R}_{44}^{*}$ | 0.435 | 1.545 | 1.273 | −0.349 | 1.032 | 1.177 |

$1-{R}_{55}$$/{R}_{55}^{*}$ | 4.851 | 25.41 | 33.84 | 4.060 | 18.48 | 30.95 |

Stiffness Reduction | 1/16 P (%) | 2/16 P (%) | 3/16 P (%) | 4/16 P (%) |
---|---|---|---|---|

$1-{A}_{11}$$/{A}_{11}^{*}$ | −0.023 | −0.121 | −1.061 | −0.055 |

$1-{A}_{22}$$/{A}_{22}^{*}$ | −0.018 | −0.061 | −0.086 | −0.003 |

$1-{A}_{33}$$/{A}_{33}^{*}$ | −0.035 | −0.089 | −0.062 | 0.038 |

$1-{D}_{11}$$/{D}_{11}^{*}$ | 0.023 | 0.099 | −0.687 | 0.059 |

$1-{D}_{22}$$/{D}_{22}^{*}$ | 0.018 | 0.053 | −0.007 | 0.050 |

$1-{D}_{33}$$/{D}_{33}^{*}$ | 0.124 | 0.495 | 1.102 | 1.720 |

$1-{R}_{44}$$/{R}_{44}^{*}$ | 3.533 | 13.41 | 10.63 | 1.771 |

$1-{R}_{55}$$/{R}_{55}^{*}$ | 1.286 | 4.036 | 8.186 | 8.956 |

Stiffness Reduction | 1/16 P (%) | 2/16 P (%) | 3/16 P (%) | 4/16 P (%) |
---|---|---|---|---|

$1-{A}_{11}$$/{A}_{11}^{*}$ | −0.018 | −0.094 | −1.052 | −0.037 |

$1-{A}_{22}$$/{A}_{22}^{*}$ | −0.013 | −0.044 | −0.075 | −0.003 |

$1-{A}_{33}$$/{A}_{33}^{*}$ | −0.032 | −0.082 | −0.056 | 0.039 |

$1-{D}_{11}$$/{D}_{11}^{*}$ | 0.012 | 0.029 | −1.048 | −0.012 |

$1-{D}_{22}$$/{D}_{22}^{*}$ | 0.011 | 0.009 | −0.062 | 0.021 |

$1-{D}_{33}$$/{D}_{33}^{*}$ | −0.029 | 0.110 | 0.459 | 0.880 |

$1-{R}_{44}$$/{R}_{44}^{*}$ | 2.706 | 9.932 | 8.977 | 1.396 |

$1-{R}_{55}$$/{R}_{55}^{*}$ | 2.378 | 6.572 | 11.88 | 15.28 |

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Garbowski, T.; Knitter-Piątkowska, A.; Mrówczyński, D. Numerical Homogenization of Multi-Layered Corrugated Cardboard with Creasing or Perforation. *Materials* **2021**, *14*, 3786.
https://doi.org/10.3390/ma14143786

**AMA Style**

Garbowski T, Knitter-Piątkowska A, Mrówczyński D. Numerical Homogenization of Multi-Layered Corrugated Cardboard with Creasing or Perforation. *Materials*. 2021; 14(14):3786.
https://doi.org/10.3390/ma14143786

**Chicago/Turabian Style**

Garbowski, Tomasz, Anna Knitter-Piątkowska, and Damian Mrówczyński. 2021. "Numerical Homogenization of Multi-Layered Corrugated Cardboard with Creasing or Perforation" *Materials* 14, no. 14: 3786.
https://doi.org/10.3390/ma14143786