A Multiscale Fractal Approach for Determining Cushioning Curves of Low-Density Polymer Foams
This study investigates the impact response of polymer foams commonly used in protective packaging, considering the fractal nature of their material microstructure. The research begins with static material characterization and impact tests on two low-density polyethylene foams. To capture the multis...
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| Format: | Article |
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MDPI AG
2025-01-01
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| Series: | Fractal and Fractional |
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| Online Access: | https://www.mdpi.com/2504-3110/9/1/32 |
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| author | Mariela C. Bravo-Sánchez Luis M. Palacios-Pineda José L. Gómez-Color Oscar Martínez-Romero Imperio A. Perales-Martínez Daniel Olvera-Trejo Jorge A. Estrada-Díaz Alex Elías-Zúñiga |
| author_facet | Mariela C. Bravo-Sánchez Luis M. Palacios-Pineda José L. Gómez-Color Oscar Martínez-Romero Imperio A. Perales-Martínez Daniel Olvera-Trejo Jorge A. Estrada-Díaz Alex Elías-Zúñiga |
| author_sort | Mariela C. Bravo-Sánchez |
| collection | DOAJ |
| description | This study investigates the impact response of polymer foams commonly used in protective packaging, considering the fractal nature of their material microstructure. The research begins with static material characterization and impact tests on two low-density polyethylene foams. To capture the multiscale nature of the dynamic response behavior of two low-density foams to sustain impact loads, fractional differential equations of motion are used to qualitatively and quantitatively describe the dynamic response behavior, assuming restoring forces for each foam characterized, respectively, by a polynomial of heptic degree and by a trigonometric tangential function. A two-scale transform is employed to solve the mathematical model and predict the material’s behavior under impact loads, accounting for the fractal structure of the material’s molecular configuration. To assess the accuracy of the mathematical model, we performed impact tests considering eight dropping heights and two plate weights. We found good predictions from the mathematical models compared to experimental data when the fractal derivatives were between 1.86 and 1.9, depending on the cushioning material used. The accuracy of the theoretical predictions achieved using fractal calculus elucidates how to predict multiscale phenomena associated with foam heterogeneity across space, density, and average pore size, which influence the foam chain’s molecular motion during impact loading conditions. |
| format | Article |
| id | doaj-art-93291900cece48da9b7dec7b1a13dacf |
| institution | Kabale University |
| issn | 2504-3110 |
| language | English |
| publishDate | 2025-01-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Fractal and Fractional |
| spelling | doaj-art-93291900cece48da9b7dec7b1a13dacf2025-01-24T13:33:26ZengMDPI AGFractal and Fractional2504-31102025-01-01913210.3390/fractalfract9010032A Multiscale Fractal Approach for Determining Cushioning Curves of Low-Density Polymer FoamsMariela C. Bravo-Sánchez0Luis M. Palacios-Pineda1José L. Gómez-Color2Oscar Martínez-Romero3Imperio A. Perales-Martínez4Daniel Olvera-Trejo5Jorge A. Estrada-Díaz6Alex Elías-Zúñiga7División de Estudios de Posgrado e Investigación, Tecnológico Nacional de México/Instituto Tecnológico de Pachuca, Carr. México-Pachuca Km 87.5, Pachuca 42080, Hidalgo, MexicoDivisión de Estudios de Posgrado e Investigación, Tecnológico Nacional de México/Instituto Tecnológico de Pachuca, Carr. México-Pachuca Km 87.5, Pachuca 42080, Hidalgo, MexicoDivisión de Estudios de Posgrado e Investigación, Tecnológico Nacional de México/Instituto Tecnológico de Pachuca, Carr. México-Pachuca Km 87.5, Pachuca 42080, Hidalgo, MexicoTecnologico de Monterrey, Institute of Advanced Materials for Sustainable Manufacturing, Ave. Eugenio Garza Sada 2501 Sur, Col. Tecnologico, Monterrey 64700, Nuevo Leon, MexicoTecnologico de Monterrey, Institute of Advanced Materials for Sustainable Manufacturing, Ave. Eugenio Garza Sada 2501 Sur, Col. Tecnologico, Monterrey 64700, Nuevo Leon, MexicoTecnologico de Monterrey, Institute of Advanced Materials for Sustainable Manufacturing, Ave. Eugenio Garza Sada 2501 Sur, Col. Tecnologico, Monterrey 64700, Nuevo Leon, MexicoTecnologico de Monterrey, Institute of Advanced Materials for Sustainable Manufacturing, Ave. Eugenio Garza Sada 2501 Sur, Col. Tecnologico, Monterrey 64700, Nuevo Leon, MexicoTecnologico de Monterrey, Institute of Advanced Materials for Sustainable Manufacturing, Ave. Eugenio Garza Sada 2501 Sur, Col. Tecnologico, Monterrey 64700, Nuevo Leon, MexicoThis study investigates the impact response of polymer foams commonly used in protective packaging, considering the fractal nature of their material microstructure. The research begins with static material characterization and impact tests on two low-density polyethylene foams. To capture the multiscale nature of the dynamic response behavior of two low-density foams to sustain impact loads, fractional differential equations of motion are used to qualitatively and quantitatively describe the dynamic response behavior, assuming restoring forces for each foam characterized, respectively, by a polynomial of heptic degree and by a trigonometric tangential function. A two-scale transform is employed to solve the mathematical model and predict the material’s behavior under impact loads, accounting for the fractal structure of the material’s molecular configuration. To assess the accuracy of the mathematical model, we performed impact tests considering eight dropping heights and two plate weights. We found good predictions from the mathematical models compared to experimental data when the fractal derivatives were between 1.86 and 1.9, depending on the cushioning material used. The accuracy of the theoretical predictions achieved using fractal calculus elucidates how to predict multiscale phenomena associated with foam heterogeneity across space, density, and average pore size, which influence the foam chain’s molecular motion during impact loading conditions.https://www.mdpi.com/2504-3110/9/1/32fractal cushioning packaging modelHe’s formulationtwo-scale dimension transformcushioning performance curvepackaging of fragile productsporous materials |
| spellingShingle | Mariela C. Bravo-Sánchez Luis M. Palacios-Pineda José L. Gómez-Color Oscar Martínez-Romero Imperio A. Perales-Martínez Daniel Olvera-Trejo Jorge A. Estrada-Díaz Alex Elías-Zúñiga A Multiscale Fractal Approach for Determining Cushioning Curves of Low-Density Polymer Foams Fractal and Fractional fractal cushioning packaging model He’s formulation two-scale dimension transform cushioning performance curve packaging of fragile products porous materials |
| title | A Multiscale Fractal Approach for Determining Cushioning Curves of Low-Density Polymer Foams |
| title_full | A Multiscale Fractal Approach for Determining Cushioning Curves of Low-Density Polymer Foams |
| title_fullStr | A Multiscale Fractal Approach for Determining Cushioning Curves of Low-Density Polymer Foams |
| title_full_unstemmed | A Multiscale Fractal Approach for Determining Cushioning Curves of Low-Density Polymer Foams |
| title_short | A Multiscale Fractal Approach for Determining Cushioning Curves of Low-Density Polymer Foams |
| title_sort | multiscale fractal approach for determining cushioning curves of low density polymer foams |
| topic | fractal cushioning packaging model He’s formulation two-scale dimension transform cushioning performance curve packaging of fragile products porous materials |
| url | https://www.mdpi.com/2504-3110/9/1/32 |
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