An image-based approach for structure investigation and 3D numerical modelling of polymeric foams


Polymeric expanded materials are of great importance in many engineering applications. Despite this, as of today the development of models able to describe the mechanical behaviour of these material as a function of their microstructure is still an open challenge. In this study an image-based approach is proposed for both microstructure characterisation and 3D numerical mechanical simulations. Microstructure is investigated through different algorithms, such as Mean Intercept Length and Autocorrelation function, to determine synthetic parameters able to describe the internal structure. A novel algorithm has been developed to convert the images obtained from computed tomography into a finite element mesh with an optimized number of elements: this method preserves the original structure and can also be used to generate other fictitious structures that can be analysed. The investigation led to the identification of general relationships between foam microstructure and relevant macroscopic physical and mechanical properties. These relationships can serve as a tool to optimize foam morphology or product final properties for several different engineering applications.

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  1. 1.

    For each direction of the test lines, Wald computed a bone mean intercept length (BMIL) as the average of the distance between a bone entry point and the subsequent bone exit point. The BMIL is then an estimate of the average value of the foam wall thickness in the different directions.


  1. 1.

    ASTM D6226 (2015) Standard test method for open cell content of rigid cellular plastics

  2. 2.

    BS EN ISO 11357–3 (2018) Plastics – Differential scanning calorimetry (DSC) – Part 3: Determination of temperature and enthalpy of melting and crystallization

  3. 3.

    Andena L, Caimmi F, Leonardi L, Nacucchi M, De Pascalis F (2019) Compression of polystyrene and polypropylene foams for energy absorption applications: a combined mechanical and microstructural study, J. of Cell. Plast. 55,

  4. 4.

    Bocciarelli M, Carvelli V, Mariani S, Tenni M (2020) Assessment of the shock adsorption properties of bike helmets: a numerical/experimental approach, Computer Methods in Biomech. and Biomed. Eng. 23,

  5. 5.

    Cowin SC (1985) The relationship between the elasticity tensor and the fabric tensor. Mech of Mater 4(2):137–147

    Article  Google Scholar 

  6. 6.

    Cowin SC (1986) Wolff’s law of trabecular architecture at remodeling equilibrium. J of Biomech Eng 108(1):83–88

    CAS  Article  Google Scholar 

  7. 7.

    Cowin SC, Doty SB (2007) Tissue Mechanics. Springer, New York USA

    Google Scholar 

  8. 8.

    De Pascalis F, Nacucchi M, Scatto M, Albertoni R (2016) Quantitative characterisation of low density, high performance polymeric foams using high resolution X-ray computed tomography and laser confocal microscopy. NDT & E Int 83:123–133

    Article  Google Scholar 

  9. 9.

    De Pascalis F, Nacucchi M (2019) Relationship between the anisotropy tensor calculated through global and object measurements in high-resolution X-ray tomography on cellular and composite materials. J of Microsc 273:65–80.

    CAS  Article  Google Scholar 

  10. 10.

    Deshpande VS, Ashby MF, Fleck NA (2001) Foam topology bending versus stretching dominated architectures. Acta Mater 49:1035–1040

    CAS  Article  Google Scholar 

  11. 11.

    Fazekas A, Dendievel R, Salvo L, Brechet Y (2002) Effect of microstructural topology upon the stiffness and strength of 2D cellular structures. J of Mech Sci 44:2047–2066

    Article  Google Scholar 

  12. 12.

    Fischer F, Lim GT, Handge UA, Altstädt V (2009) Numerical simulation of mechanical properties of cellular materials using computed tomography analysis. J of Cell Plast 45

  13. 13.

    Gibson LJ, Ashby MF (2002) Cellular solids structure and properties, 2nd edition, Cambridge Solid State Science Series

  14. 14.

    Gong L, Kyriakides S, Jang WY (2005) Compressive response of open-cell foams. Part I: Morphology and elastic properties. Int J of Solids and Struct 42:1355–1379

    Article  Google Scholar 

  15. 15.

    Gong L, Kyriakides S (2005) Compressive response of open-cell foams. Part II: Initiation and evolution of crushing. Int J of Solids and Struct 42:1381–1399

    Article  Google Scholar 

  16. 16.

    Gong L, Kyriakides S, Triantafyllidis N (2005) On the stability of Kelvin cell foams under compressive loads. J Mech Phys Solids 53:771–794

    Article  Google Scholar 

  17. 17.

    Harrigan T, Mann RW (1984) Characterization of microstructural anisotropy in orthotropic materials using a second rank tensor. J of Mater Sci 19(2):761–767

    CAS  Article  Google Scholar 

  18. 18.

    Hildebrand T, Rüesgsegger P (1997) A new method for the model-independent assessment of thickness in three-dimensional images. J of Microsc 185(1):67–75

    Article  Google Scholar 

  19. 19.

    Jacobs LJM, Kemmere MF, Keurentjes JTF (2008) Sustainable polymer foaming using high pressure carbon dioxide: a review on fundamentals, processes and applications. Green Chem 10:731–738.

    CAS  Article  Google Scholar 

  20. 20.

    Ketcham RA, Ryan TM (2004) Quantification and visualization of anisotropy in trabecular bone. J of Microsc 213(2):158–171

    CAS  Article  Google Scholar 

  21. 21.

    Lorensen WE, Cline HE (1987) Marching cubes: a high-resolution 3D surface construction algorithm. Computer Graph 21(4):163–169

    Article  Google Scholar 

  22. 22.

    Moreno R, Borga M, Smedby Ö (2014) Techniques for computing fabric tensors: a review. In: Westin CF, Vilanova A, Burgeth, B. (eds) Visualization and Processing of Tensors and Higher Order Descriptors for Multi-Valued Data. Springer, Berlin, Heidelberg Germany

  23. 23.

    Odgaard A, Kabel J, Van Rietbergen B, Dalstra M, Huiskes R (1997) Fabric and principal directions of cancellous bone are closely related. J Biomech 30:487–496

    CAS  Article  Google Scholar 

  24. 24.

    Odgaard A (2001) Quantification of cancellous bone architecture. In: Cowin SC editor. Bone Mechanics Handbook, Boca Raton, FL: CRC Press LLC, chapter 14

  25. 25.

    Roberts AP, Garboczi EJ (2001) Elastic moduli of model random three-dimensional closed-cell cellular solids. Acta Mater 49:189–197

    CAS  Article  Google Scholar 

  26. 26.

    Rodriguez-Perez MA, Alvarez-Lainez M, de Saja JA (2009) Microstructure and physical properties of open-cell polyolefin foams. J Appl Polym Sci 114:1176–1186.

    CAS  Article  Google Scholar 

  27. 27.

    Sauceau M, Fages J, Common A, Nikitine C, Rodier E (2011) New challenges in polymer foaming: a review of extrusion processes assisted by supercritical carbon dioxide. Prog in Polym Sci 36:749–766.

    CAS  Article  Google Scholar 

  28. 28.

    Schroeder W, Martin K, Lorensen B (2006) Visualization Toolkit, An Object-Oriented Approach to 3D Graphics. 4th ed. Kitware Inc. ISBN: 1–930934–19-X, New York

  29. 29.

    Stock SR (2008) Recent advances in x-ray microtomography applied to materials. Int Mater Rev 53:129–181.

    CAS  Article  Google Scholar 

  30. 30.

    Tabor Z (2009) On the equivalence of two methods of determining fabric tensor. Med Eng & Phys 31(10):1313–1322

    Article  Google Scholar 

  31. 31.

    Tagliabue S, Rossi E, Baino F, Vitale-Brovarone C, Gastaldi D, Vena P (2017) Micro-CT based finite element models for elastic properties of glass-ceramic scaffolds, J of the Mech Behav of Biomed Mater 65:248–255.

    CAS  Article  Google Scholar 

  32. 32.

    Trofa M, Di Maio E, Maffettone PL (2019) Multi-graded foams upon time-dependent exposition to blowing agent. Chem Eng J 362:812–817.

    CAS  Article  Google Scholar 

  33. 33.

    Wald MJ, Vasilic B, Saha PK, Wehrli FW (2007) Spatial autocorrelation and mean intercept length analysis of trabecular bone anisotropy applied to in vivo magnetic resonance imaging. Med Phys 34(3):1110–1120

    Article  Google Scholar 

  34. 34.

    Whitehouse WJ (1974) The quantitative morphology of anisotropic trabecular bone. J of Microsc 101(2):153–168

    CAS  Article  Google Scholar 

  35. 35.

    Wismans JGF, Goveart LE, van Dommelen JAW (2010) X-Ray computed tomography-based modelling of polymeric foams: the effect of finite element model size on the large strain response. J Polym Sci Part B: Polym Phys 48:1526–1534.

    CAS  Article  Google Scholar 

  36. 36.

    Wunderlich B (1990) Thermal Analysis, Academic Press, pp.417–431

  37. 37.

    Zhu HX, Knott JF, Mills NJ (1996) Analysis of the elastic properties of open-cell foams with tetrakaidecahedral cells. J Mech Phys Solids 45:319–343

    Article  Google Scholar 

  38. 38.

    Zysset PK, Goulet RW, Hollister SJ (1998) A global relationship between trabecular bone morphology and homogenized elastic properties. J Biomech Eng 120(5):640–646

    CAS  Article  Google Scholar 

  39. 39.

    Zysset PK (2003) A review of morphology-elasticity relationships in human trabecular bone: theories and experiments. J Biomech 36(10):1469–1485

    Article  Google Scholar 

  40. 40.

    Morton DT, Reyes A, Clausen AH, Hopperstad OS (2020) Mechanical response of low density expanded polypropylene foams in compression and tension at different loading rates and temperatures. Mater Today Comm 23

  41. 41.

    Rodriguez-Perez MA, Lobos J, Perez-Munoz CA, de Saja JA (2009) Mechanical response of polyethylene foams with high densities and cell sizes in the microcellular ranges. J of Cell Plast 45:389–403

  42. 42.

    Huber N (2018) Connections between topology and macroscopic mechanical properties of three-dimensional open-pore materials, Materials Front Mater 5(69)

  43. 43.

    Chen W, Hao H, Hughes D, Shi Y, Cui J, Li ZX (2015) Static and dynamic mechanical properties of expanded polystyrene. Mater and Design 69:170–180.

    CAS  Article  Google Scholar 

  44. 44.

    Harper CA (2002) Handbook of plastics, elastomers and composites, McGraw Hill Professional 4th edition

  45. 45.

    The Scientist and Engineer’s Guide to Digital Signal Processing, 2nd edition, by Steven W. Smith, ISBN 0–9660176–6–8

  46. 46.

    Tang Q, Fang L, Guo W (2019) Effects of bamboo fiber length and loading on mechanical, thermal and pulverization properties of phenolic foam composites, J of Biores and Bioprod 4:51–59

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Correspondence to Stefano Tagliabue.

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Tagliabue, S., Andena, L., Nacucchi, M. et al. An image-based approach for structure investigation and 3D numerical modelling of polymeric foams. J Polym Res 28, 75 (2021).

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  • Polymeric foam
  • Microstructure
  • X-ray computed micro-tomography (µ-CT)
  • Mean Intercept Length (MIL)
  • Autocorrelation Function (ACF)
  • Finite Element Modelling (FEM)