Journal of Materials Science

, Volume 53, Issue 14, pp 10194–10208 | Cite as

Minimal surface designs for porous materials: from microstructures to mechanical properties

  • Xiaoyang Zheng
  • Zhibing Fu
  • Kai Du
  • Chaoyang Wang
  • Yong Yi


In this work, we present four types of topological bicontinuous porous structures, namely Gyroid (G), Schwarz Diamond (D), Schwarz Primitive (P), and iWp (W), which are generated from mathematically defined triply periodic minimal surfaces. A systematic semi-theoretical investigation is performed to analyze the relations between the microstructures and the macroscopic mechanical behavior. Benefiting from the straightforward controllability on parameters, the scaling laws of the geometrical properties and mechanical properties are determined as functions of the relative density according to numerical analysis and computational simulation. An application to additive manufacturing accompanying with uniaxial compression testing is also performed, and the results show a highly agreement with the above scaling laws. Moreover, the simulation results indicate that the mechanical properties are highly dependent on topological architectures, which affect the deformation behavior of porous materials. It is shown that P topology has the highest stiffness and strength with stretching-dominated mode, while the rest exhibit a flexibly bending-dominated deformation behavior. The present study provides not only new insights into the structure–property relations of such topologies, but also a practical guide for their fabrication and application.



This work was supported by Longshan academic talent research supporting program of SWUST (17LZX408). The authors gratefully acknowledge Dr. Li Bo and Mr. Zhong Shengyuan for equipment and technique support.

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.


  1. 1.
    Giannitelli S, Accoto D, Trombetta M, Rainer A (2014) Current trends in the design of scaffolds for computer-aided tissue engineering. Acta Biomater 10(2):580CrossRefGoogle Scholar
  2. 2.
    Kantaros A, Chatzidai N, Karalekas D (2016) 3D printing-assisted design of scaffold structures. Int J Adv Manuf Technol 82(1–4):559CrossRefGoogle Scholar
  3. 3.
    Wang X, Xu S, Zhou S, Xu W, Leary M, Choong P, Qian M, Brandt M, Xie YM (2016) Topological design and additive manufacturing of porous metals for bone scaffolds and orthopaedic implants: a review. Biomaterials 83:127CrossRefGoogle Scholar
  4. 4.
    Choren JA, Heinrich SM, Silver-Thorn MB (2013) Young’s modulus and volume porosity relationships for additive manufacturing applications. J Mater Sci 48(15):5103. CrossRefGoogle Scholar
  5. 5.
    Shbeh MM, Goodall R (2017) Open celled porous titanium. Adv Eng Mater 19(11):1600664CrossRefGoogle Scholar
  6. 6.
    Gu D, Meiners W, Wissenbach K, Poprawe R (2012) Laser additive manufacturing of metallic components: materials, processes and mechanisms. Int Mater Rev 57(3):133CrossRefGoogle Scholar
  7. 7.
    Olhero SM, Fernandes HR, Marques CF, Silva BC, Ferreira JM (2017) Additive manufacturing of 3D porous alkali-free bioactive glass scaffolds for healthcare applications. J Mater Sci 52(20):12079. CrossRefGoogle Scholar
  8. 8.
    Lurie S, Solyaev Y, Rabinskiy L, Polyakov P, Sevostianov I (2018) Mechanical behavior of porous Si3N4 ceramics manufactured with 3D printing technology. J Mater Sci 53(7):4796. CrossRefGoogle Scholar
  9. 9.
    Compton BG, Lewis JA (2014) 3D-printing of lightweight cellular composites. Adv Mater 26(34):5930CrossRefGoogle Scholar
  10. 10.
    Schaedler TA, Jacobsen AJ, Torrents A, Sorensen AE, Lian J, Greer JR, Valdevit L, Carter WB (2011) Ultralight metallic microlattices. Science 334(6058):962CrossRefGoogle Scholar
  11. 11.
    Maiti A, Small W, Lewicki J, Weisgraber T, Duoss E, Chinn S, Pearson M, Spadaccini C, Maxwell R, Wilson T (2016) 3D printed cellular solid outperforms traditional stochastic foam in long-term mechanical response. Sci Rep 6:24871CrossRefGoogle Scholar
  12. 12.
    Papazetis G, Vosniakos GC (2016) Direct porous structure generation of tissue engineering scaffolds for layer-based additive manufacturing. Int J Adv Manuf Technol 86(1–4):871CrossRefGoogle Scholar
  13. 13.
    e Sá AM, Mello VM, Echavarria KR, Covill D (2015) Adaptive voids. Vis Comput 31(6):799Google Scholar
  14. 14.
    Hedayati R, Sadighi M, Mohammadi-Aghdam M, Zadpoor A (2017) Analytical relationships for the mechanical properties of additively manufactured porous biomaterials based on octahedral unit cells. Appl Math Model 46:408CrossRefGoogle Scholar
  15. 15.
    Hedayati R, Sadighi M, Mohammadi-Aghdam M, Zadpoor A (2016) Mechanical behavior of additively manufactured porous biomaterials made from truncated cuboctahedron unit cells. Int J Mech Sci 106:19CrossRefGoogle Scholar
  16. 16.
    Dementjev A, Tarakanov O (1970) Influence of the cellular structure of foams on their mechanical properties. Mech Polym 4:594Google Scholar
  17. 17.
    Hedayati R, Sadighi M, Mohammadi-Aghdam M, Zadpoor A (2016) Mechanics of additively manufactured porous biomaterials based on the rhombicuboctahedron unit cell. J Mech Behav Biomed Mater 53:272CrossRefGoogle Scholar
  18. 18.
    Babaee S, Jahromi BH, Ajdari A, Nayeb-Hashemi H, Vaziri A (2012) Mechanical properties of open-cell rhombic dodecahedron cellular structures. Acta Mater 60(6–7):2873CrossRefGoogle Scholar
  19. 19.
    Lord EA, Mackay AL (2003) Periodic minimal surfaces of cubic symmetry. Curr Sci 85(3):346–362Google Scholar
  20. 20.
    Wohlgemuth M, Yufa N, Hoffman J, Thomas EL (2001) Triply periodic bicontinuous cubic microdomain morphologies by symmetries. Macromolecules 34(17):6083CrossRefGoogle Scholar
  21. 21.
    Mille C, Tyrode EC, Corkery RW (2011) Inorganic chiral 3-D photonic crystals with bicontinuous gyroid structure replicated from butterfly wing scales. Chem Commun 47(35):9873CrossRefGoogle Scholar
  22. 22.
    Gan Z, Turner MD, Gu M (2016) Biomimetic gyroid nanostructures exceeding their natural origins. Sci Adv 2(5):e1600084CrossRefGoogle Scholar
  23. 23.
    Michielsen K, Stavenga DG (2008) Gyroid cuticular structures in butterfly wing scales: biological photonic crystals. J R Soc Interface 5(18):85CrossRefGoogle Scholar
  24. 24.
    Kapfer SC, Hyde ST, Mecke K, Arns CH, Schröder-Turk GE (2011) Minimal surface scaffold designs for tissue engineering. Biomaterials 32(29):6875CrossRefGoogle Scholar
  25. 25.
    Abueidda DW, Bakir M, Al-Rub RKA, Bergström JS, Sobh NA, Jasiuk I (2017) Mechanical properties of 3D printed polymeric cellular materials with triply periodic minimal surface architectures. Mater Des 122:255CrossRefGoogle Scholar
  26. 26.
    Al-Ketan O, Rowshan R, Al-Rub RKA (2018) Topology-mechanical property relationship of 3D printed strut, skeletal, and sheet based periodic metallic cellular materials. Addit Manuf 19:167CrossRefGoogle Scholar
  27. 27.
    Abueidda DW, Al-Rub RKA, Dalaq AS, Lee DW, Khan KA, Jasiuk I (2016) Effective conductivities and elastic moduli of novel foams with triply periodic minimal surfaces. Mech Mater 95:102CrossRefGoogle Scholar
  28. 28.
    Abueidda DW, Al-Rub RKA, Dalaq AS, Younes HA, Al Ghaferi AA, Shah TK (2015) Electrical conductivity of 3D periodic architectured interpenetrating phase composites with carbon nanostructured-epoxy reinforcements. Compos Sci Technol 118:127CrossRefGoogle Scholar
  29. 29.
    Abueidda DW, Dalaq AS, Al-Rub RKA, Jasiuk I (2015) Micromechanical finite element predictions of a reduced coefficient of thermal expansion for 3D periodic architectured interpenetrating phase composites. Compos Struct 133:85CrossRefGoogle Scholar
  30. 30.
    Lee DW, Khan KA, Al-Rub RKA (2017) Stiffness and yield strength of architectured foams based on the Schwarz primitive triply periodic minimal surface. Int J Plast 95:1CrossRefGoogle Scholar
  31. 31.
    Dalaq AS, Abueidda DW, Al-Rub RKA, Jasiuk IM (2016) Finite element prediction of effective elastic properties of interpenetrating phase composites with architectured 3D sheet reinforcements. Int J Solids Struct 83:169CrossRefGoogle Scholar
  32. 32.
    Khaderi S, Deshpande V, Fleck N (2014) The stiffness and strength of the gyroid lattice. Int J Solids Struct 51(23–24):3866CrossRefGoogle Scholar
  33. 33.
    Eymard R, Gallouët T, Herbin R (2000) Finite volume methods. Handb Numer Anal 7:713Google Scholar
  34. 34.
    Kabel M, Andrä H (2012) Fast numerical computation of precise bounds of effective elastic moduli. Berichte des Fraunhofer ITWM 224(224):1Google Scholar
  35. 35.
    Rutka V, Wiegmann A (2006) Explicit jump immersed interface method for virtual material design of the effective elastic moduli of composite materials. Numer Algorithms 43(4):309CrossRefGoogle Scholar
  36. 36.
    Yi Y, Zheng X, Fu Z, Wang C, Xu X, Tan X (2018) Multi-scale modeling for predicting the stiffness and strength of hollow-structured metal foams with structural hierarchy. Materials 11(3):380CrossRefGoogle Scholar
  37. 37.
    Von Schnering H, Nesper R (1991) Nodal surfaces of Fourier series: fundamental invariants of structured matter. Zeitschrift für Physik B Condens Matter 83(3):407CrossRefGoogle Scholar
  38. 38. (2018). Accessed 1 Jan 2018
  39. 39.
    Onck P, Andrews E, Gibson L (2001) Size effects in ductile cellular solids. Part I: modeling. Int J Mech Sci 43(3):681CrossRefGoogle Scholar
  40. 40.
    Andrews E, Gioux G, Onck P, Gibson L (2001) Size effects in ductile cellular solids. Part II: experimental results. Int J Mech Sci 43(3):701CrossRefGoogle Scholar
  41. 41. (2018). Accessed 1 Jan 2018
  42. 42. (2018). Accessed 1 Jan 2018
  43. 43.
    ASTM International (2015) Standard test method for compressive properties of rigid plastics. ASTM International, West ConshohockenGoogle Scholar
  44. 44.
    Danesh G, Lippold C, Ziebura T, Reinhardt KJ, Schäfer E, Ehmer U (2006) In-vitro Investigation on suitability of light-cured resins for interocclusal splints. J Orofac Orthop/Fortschritte der Kieferorthopädie 67(2):138CrossRefGoogle Scholar
  45. 45.
    Gibson LJ, Ashby MF (1999) Cellular solids: structure and properties. Cambridge University Press, CambridgeGoogle Scholar
  46. 46.
    Rouquerol J, Avnir D, Fairbridge C, Everett D, Haynes J, Pernicone N, Ramsay J, Sing K, Unger K (1994) Recommendations for the characterization of porous solids (technical report). Pure Appl Chem 66(8):1739CrossRefGoogle Scholar
  47. 47.
    S.S.S. of America (2008) Glossary of soil science terms 2008. ASA-CSSA-SSSA, MadisonGoogle Scholar
  48. 48.
    Deshpande VS, Fleck NA, Ashby MF (2001) Effective properties of the octet-truss lattice material. J Mech Phys Solids 49(8):1747CrossRefGoogle Scholar
  49. 49.
    Ashby M (2006) The properties of foams and lattices. Philos Trans R Soc Lond A Math Phys Eng Sci 364(1838):15CrossRefGoogle Scholar
  50. 50.
    Deshpande V, Ashby M, Fleck N (2001) Foam topology: bending versus stretching dominated architectures. Acta Mater 49(6):1035CrossRefGoogle Scholar
  51. 51.
    Uchic MD, Dimiduk DM, Florando JN, Nix WD (2004) Sample dimensions influence strength and crystal plasticity. Science 305(5686):986CrossRefGoogle Scholar
  52. 52.
    Hedayati R, Sadighi M, Mohammadi-Aghdam M, Hosseini-Toudeshky H (2018) Comparison of elastic properties of open-cell metallic biomaterials with different unit cell types. J Biomed Mater Res Part B Appl Biomater 106(1):386CrossRefGoogle Scholar
  53. 53.
    Kadkhodapour J, Montazerian H, Raeisi S (2014) Investigating internal architecture effect in plastic deformation and failure for TPMS-based scaffolds using simulation methods and experimental procedure. Mater Sci Eng C 43:587CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Joint Laboratory for Extreme Conditions Matter PropertiesSouthwest University of Science and TechnologyMianyangChina
  2. 2.Research Center of Laser FusionChina Academy of Engineering PhysicsMianyangChina

Personalised recommendations