Numerical Homogenization of Multi-Layered Corrugated Cardboard with Creasing or Perforation
Abstract
:1. Introduction
2. Materials and Methods
2.1. Corrugated Board—Material Definition
2.2. Creases and Perforations—Numerical Study
2.3. Homogenization Technique
3. Results
3.1. Validation of the Proposed Method
3.2. Detailed Results
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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Layers | ||||||
---|---|---|---|---|---|---|
(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 |
Stiffness | Ref. [39] | Corse Model | Fine Model |
---|---|---|---|
1.108 106 | 1.118 106 | 1.118 106 | |
1.358 106 | 1.380 106 | 1.378 106 | |
3.324 105 | 3.449 105 | 3.448 105 | |
4.168 105 | 4.115 105 | 4.115 105 | |
9.195 108 | 9.211 108 | 9.210 108 | |
9.822 108 | 9.926 108 | 9.925 108 | |
2.758 108 | 2.777 108 | 2.777 108 | |
3.220 108 | 3.269 108 | 3.268 108 | |
, | - | 5.194 104 | 5.184 104 |
, | - | 7.408 104 | 7.376 104 |
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 |
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 | |
---|---|---|---|---|---|---|---|
(MPa mm) | 2184.4 | 2127.2 | 1990.3 | 1854.2 | 1774.2 | 1751.5 | 1756.9 |
(MPa mm) | 1756.9 | 1751.5 | 1774.2 | 1854.2 | 1990.3 | 2127.2 | 2184.4 |
(MPa mm) | 667.81 | 699.26 | 760.50 | 792.80 | 760.50 | 699.30 | 667.80 |
(MPa mm3) | 8628.2 | 8313.5 | 7480.9 | 6521.5 | 5897.3 | 5575.8 | 5469.3 |
(MPa mm3) | 5469.3 | 5575.8 | 5897.3 | 6520.4 | 7480.9 | 8313.5 | 8628.2 |
(MPa mm3) | 2300.2 | 2425.2 | 2650.1 | 2755.4 | 2650.1 | 2425.2 | 2300.2 |
(MPa mm) | 105.08 | 108.15 | 119.80 | 132.90 | 127.20 | 126.20 | 130.90 |
(MPa mm) | 130.91 | 126.16 | 127.20 | 132.80 | 119.80 | 108.10 | 105.10 |
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 | |
---|---|---|---|---|---|---|---|
(MPa mm) | 3313.8 | 3250.6 | 3090.4 | 2955.2 | 2912.0 | 2939.7 | 2967.5 |
(MPa mm) | 2967.5 | 2939.7 | 2912.0 | 2955.3 | 3090.4 | 3250.6 | 3313.8 |
(MPa mm) | 1077.8 | 1127.5 | 1225.3 | 1275.9 | 1225.3 | 1127.5 | 1077.8 |
(MPa mm3) | 20,242 | 19,610 | 17,980 | 16,221 | 15,123 | 14,662 | 14,556 |
(MPa mm3) | 14,556 | 14,662 | 15,123 | 16,220 | 17,980 | 19,610 | 20,242 |
(MPa mm3) | 5778.6 | 6071.8 | 6634.3 | 6910.6 | 6634.3 | 6071.8 | 5778.6 |
(MPa mm) | 233.13 | 240.21 | 246.71 | 257.56 | 247.51 | 242.88 | 242.28 |
(MPa mm) | 242.28 | 242.88 | 247.51 | 257.43 | 246.71 | 240.21 | 233.13 |
Stiffness | SW-0-F-15 | SW-26-F-15 | SW-44-F-15 | SW-62-F-15 |
---|---|---|---|---|
(MPa mm) | 2127.2 | 2116.1 | 2082.1 | 2052.3 |
(MPa mm) | 1751.6 | 1609.1 | 1267.7 | 885.12 |
(MPa mm) | 699.26 | 681.92 | 608.30 | 524.18 |
(MPa mm3) | 8313.4 | 8276.1 | 8166.4 | 8048.5 |
(MPa mm3) | 5575.8 | 5290.9 | 4291.8 | 2877.2 |
(MPa mm3) | 2425.2 | 2384.5 | 2216.7 | 1968.9 |
(MPa mm) | 108.15 | 107.68 | 106.48 | 106.77 |
(MPa mm) | 126.16 | 120.04 | 94.100 | 83.465 |
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 (%) |
---|---|---|---|---|---|---|
0.523 | 2.121 | 3.519 | 0.508 | 1.903 | 3.364 | |
8.133 | 27.66 | 49.46 | 7.852 | 27.77 | 50.98 | |
2.480 | 13.01 | 25.04 | 2.735 | 12.66 | 24.50 | |
0.449 | 1.769 | 3.187 | 0.467 | 1.786 | 3.247 | |
5.110 | 23.03 | 48.40 | 6.377 | 25.41 | 49.18 | |
1.677 | 8.598 | 18.81 | 2.171 | 10.25 | 20.88 | |
0.435 | 1.545 | 1.273 | −0.349 | 1.032 | 1.177 | |
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 (%) |
---|---|---|---|---|
−0.023 | −0.121 | −1.061 | −0.055 | |
−0.018 | −0.061 | −0.086 | −0.003 | |
−0.035 | −0.089 | −0.062 | 0.038 | |
0.023 | 0.099 | −0.687 | 0.059 | |
0.018 | 0.053 | −0.007 | 0.050 | |
0.124 | 0.495 | 1.102 | 1.720 | |
3.533 | 13.41 | 10.63 | 1.771 | |
1.286 | 4.036 | 8.186 | 8.956 |
Stiffness Reduction | 1/16 P (%) | 2/16 P (%) | 3/16 P (%) | 4/16 P (%) |
---|---|---|---|---|
−0.018 | −0.094 | −1.052 | −0.037 | |
−0.013 | −0.044 | −0.075 | −0.003 | |
−0.032 | −0.082 | −0.056 | 0.039 | |
0.012 | 0.029 | −1.048 | −0.012 | |
0.011 | 0.009 | −0.062 | 0.021 | |
−0.029 | 0.110 | 0.459 | 0.880 | |
2.706 | 9.932 | 8.977 | 1.396 | |
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
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 StyleGarbowski, 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