Mechanical Response of 3D Printed Bending-Dominated Ligament-Based Triply Periodic Cellular Polymeric Solids

  • Aliaa M. Abou-Ali
  • Oraib Al-Ketan
  • Reza Rowshan
  • Rashid Abu Al-RubEmail author


Lightweight materials with complex structures such as cellular solids (or foams) have proven to possess desirable properties, specifically in terms of its stiffness, strength, and thermal conductivity, among other mechanical and thermal performance aspects while the density is reduced. The fabrication of such attractive yet complex materials has become possible due to the witnessed advancements in fabrication techniques. However, a major challenge in adapting cellular solids in mechanical design is choosing the appropriate lattice design. Therefore, this paper focuses on studying the compressive mechanical behavior of four different types of cellular solids with topologies based on the mathematically known triply periodic minimal surfaces (TPMS); namely, Diamond (D), I-WP (IWP), Gyroid (G), and Fisher-Koch C(Y) (CY). These cellular materials are 3D printed using the powder bed fusion selective laser sintering technique out of Nylon thermoplastic polymer at various relative densities. The effects of the number of unit cells, type of the ligament-based TPMS architecture, and relative density on the stiffness, yield strength, ultimate strength, and toughness are thoroughly investigated. The results indicated that the effect of the architecture is stronger when the relative density is decreased. Also, the analyses showed that all the tested architectures were bending dominated implying that it could be best applied in shock absorbing and vibration mitigation applications.


additive manufacturing advanced characterization static mechanical 



Experimental parts were printed using Core Technology Platform resources at NYU Abu Dhabi. We thank Khulood Alawadi and Jumaanah Elhashemi from NYU Abu Dhabi for assistance with 3D printing.


  1. 1.
    L.J. Gibson and M.F. Ashby, Cellular Solids: Structure And Properties, Cambridge University Press, Cambridge, 1999Google Scholar
  2. 2.
    M. Ashby, The Properties of Foams and Lattices, Philos. Trans. R. Soc. Lond. A: Math. Phys. Eng. Sci., 2006, 364(1838), p 15–30CrossRefGoogle Scholar
  3. 3.
    V. Deshpande, M. Ashby, and N. Fleck, Foam Topology: Bending Versus Stretching Dominated Architectures, Acta Mater., 2001, 49(6), p 1035–1040CrossRefGoogle Scholar
  4. 4.
    S. Guessasma, P. Babin, G. Della Valle, and R. Dendievel, Relating Cellular Structure of Open Solid Food Foams to Their Young’s Modulus: Finite Element Calculation, Int. J. Solids Struct., 2008, 45(10), p 2881–2896CrossRefGoogle Scholar
  5. 5.
    W. Lee, Cellular Solids, Structure and Properties, Mater. Sci. Technol., 2000, 16(2), p 233Google Scholar
  6. 6.
    M.K. Ravari, M. Kadkhodaei, M. Badrossamay, and R. Rezaei, Numerical Investigation on Mechanical Properties of Cellular Lattice Structures Fabricated by Fused Deposition Modeling, Int. J. Mech. Sci., 2014, 88, p 154–161CrossRefGoogle Scholar
  7. 7.
    V. Valuiskikh, Method of Stochastic Simulation Modeling of the Structure, Calculation, and Optimization of the Physicomechanical Characteristics of Foam Plastics, Mech. Compos. Mater., 1990, 25(4), p 429–435CrossRefGoogle Scholar
  8. 8.
    V. Yakushin and U. Stirna, Physicomechanical Characteristics of Spray-on Rigid Polyurethane Foams at Normal and Low Temperatures, Mech. Compos. Mater., 2002, 38(3), p 273–280CrossRefGoogle Scholar
  9. 9.
    R. Gümrük, R. Mines, and S. Karadeniz, Determination of Strain Rate Sensitivity of Micro-struts Manufactured Using the Selective Laser Melting Method, J. Mater. Eng. Perform., 2018, 27(3), p 1016–1032CrossRefGoogle Scholar
  10. 10.
    M.K. Ravari and M. Kadkhodaei, A Computationally Efficient Modeling Approach for Predicting Mechanical Behavior of Cellular Lattice Structures, J. Mater. Eng. Perform., 2015, 24(1), p 245–252CrossRefGoogle Scholar
  11. 11.
    T. Lu, H. Stone, and M. Ashby, Heat Transfer in Open-Cell Metal Foams, Acta Mater., 1998, 46(10), p 3619–3635CrossRefGoogle Scholar
  12. 12.
    L.R. Meza, S. Das, and J.R. Greer, Strong, Lightweight, and Recoverable Three-Dimensional Ceramic Nanolattices, Science, 2014, 345(6202), p 1322–1326CrossRefGoogle Scholar
  13. 13.
    O. Al-Ketan, R. Rezgui, R. Rowshan, H. Du, N.X. Fang, and R.K. Abu Al-Rub, Microarchitected Stretching-Dominated Mechanical Metamaterials with Minimal Surface Topologies, Adv. Eng. Mater., 2018, 20(9), p 1800029CrossRefGoogle Scholar
  14. 14.
    X. Zheng, W. Smith, J. Jackson, B. Moran, H. Cui, D. Chen, J. Ye, N. Fang, N. Rodriguez, T. Weisgraber, and C.M. Spadaccini, Multiscale Metallic Metamaterials, Nat. Mater., 2016, 15, p 1100CrossRefGoogle Scholar
  15. 15.
    A.H. Schoen, Infinite Periodic Minimal Surfaces Without Self-Intersections, NASA Report D5541, 1970Google Scholar
  16. 16.
    D. Cvijović and J. Klinowski, The Computation of the Triply Periodic I-WP Minimal Surface, Chem. Phys. Lett., 1994, 226(1), p 93–99CrossRefGoogle Scholar
  17. 17.
    S.C. Kapfer, S.T. Hyde, K. Mecke, C.H. Arns, and G.E. Schröder-Turk, Minimal Surface Scaffold Designs for Tissue Engineering, Biomaterials, 2011, 32(29), p 6875–6882CrossRefGoogle Scholar
  18. 18.
    M. Afshar, A.P. Anaraki, H. Montazerian, and J. Kadkhodapour, Additive Manufacturing and Mechanical Characterization of Graded Porosity Scaffolds Designed Based on Triply Periodic Minimal Surface Architectures, J. Mech. Behav. Biomed. Mater., 2016, 62, p 481–494CrossRefGoogle Scholar
  19. 19.
    J. Kadkhodapour, H. Montazerian, A.C. Darabi, A. Zargarian, and S. Schmauder, The Relationships Between Deformation Mechanisms and Mechanical Properties of Additively Manufactured Porous Biomaterials, J. Mech. Behav. Biomed. Mater., 2017, 70, p 28–42CrossRefGoogle Scholar
  20. 20.
    I. Maskery, A.O. Aremu, L. Parry, R.D. Wildman, C.J. Tuck, and I.A. Ashcroft, Effective Design and Simulation of Surface-Based Lattice Structures Featuring Volume Fraction and Cell Type Grading, Mater. Des., 2018, 155, p 220–232CrossRefGoogle Scholar
  21. 21.
    I. Maskery, L. Sturm, A.O. Aremu, A. Panesar, C.B. Williams, C.J. Tuck, R.D. Wildman, I.A. Ashcroft, and R.J.M. Hague, Insights into the Mechanical Properties of Several Triply Periodic Minimal Surface Lattice Structures Made by Polymer Additive Manufacturing, Polymer, 2018, 152, p 62–71CrossRefGoogle Scholar
  22. 22.
    D.W. Abueidda, M. Bakir, R.K. Abu Al-Rub, J.S. Bergström, N.A. Sobh, and I. Jasiuk, Mechanical Properties of 3D Printed Polymeric Cellular Materials with Triply Periodic Minimal Surface Architectures, Mater. Des., 2017, 122, p 255–267CrossRefGoogle Scholar
  23. 23.
    O. Al-Ketan, R. Rowshan, and R.K. Abu Al-Rub, Topology-Mechanical Property Relationship of 3D Printed Strut, Skeletal, and Sheet Based Periodic Metallic Cellular Materials, Addit. Manuf., 2018, 19, p 167–183CrossRefGoogle Scholar
  24. 24.
    O. Al-Ketan, R.K. Abu Al-Rub, and R. Rowshan, The Effect of Architecture on the Mechanical Properties of Cellular Structures Based on the IWP Minimal Surface, J. Mater. Res., 2018, 33(03), p 343–359CrossRefGoogle Scholar
  25. 25.
    C. Yan, L. Hao, A. Hussein, and P. Young, Ti-6Al-4 V Triply Periodic Minimal Surface Structures for Bone Implants Fabricated via Selective Laser Melting, J. Mech. Behav. Biomed. Mater., 2015, 51, p 61–73CrossRefGoogle Scholar
  26. 26.
    D.W. Abueidda, R.K. Abu Al-Rub, A.S. Dalaq, D.-W. Lee, K.A. Khan, and I. Jasiuk, Effective Conductivities and Elastic Moduli of Novel Foams with Triply Periodic Minimal Surfaces, Mech. Mater., 2016, 95, p 102–115CrossRefGoogle Scholar
  27. 27.
    D.W. Abueidda, A.S. Dalaq, R.K. Abu Al-Rub, and H.A. Younes, Finite Element Predictions of Effective Multifunctional Properties of Interpenetrating Phase Composites with Novel Triply Periodic Solid Shell Architectured Reinforcements, Int. J. Mech. Sci., 2015, 92, p 80–89CrossRefGoogle Scholar
  28. 28.
    A.S. Dalaq, D.W. Abueidda, and R.K. Abu Al-Rub, Mechanical Properties of 3D Printed Interpenetrating Phase Composites with Novel Architectured 3D Solid-Sheet Reinforcements, Compos. A Appl. Sci. Manuf., 2016, 84, p 266–280CrossRefGoogle Scholar
  29. 29.
    O. Al-Ketan, M. Adel Assad, and R.K. Abu Al-Rub, Mechanical Properties of Periodic Interpenetrating Phase Composites with Novel Architected Microstructures, Compos. Struct., 2017, 176, p 9–19CrossRefGoogle Scholar
  30. 30.
    O. Al-Ketan, R.K. Abu Al-Rub, and R. Rowshan, Mechanical Properties of a New Type of Architected Interpenetrating Phase Composite Materials, Adv. Mater. Technol., 2017, 2(2), p 1600235CrossRefGoogle Scholar
  31. 31.
    O. Al-Ketan, A. Soliman, A.M. AlQubaisi, and R.K. Abu Al-Rub, Nature-Inspired Lightweight Cellular Co-Continuous Composites with Architected Periodic Gyroidal Structures, Adv. Eng. Mater., 2018, 20(2), p 1700549CrossRefGoogle Scholar
  32. 32.
    K.A. Khan and R.K. Abu Al-Rub, Time Dependent Response of Architectured Neovius Foams, Int. J. Mech. Sci., 2017, 126, p 106–119CrossRefGoogle Scholar
  33. 33.
    K.A. Khan and R.K. Abu Al-Rub, Modeling Time and Frequency Domain Viscoelastic Behavior of Architectured Foams, J. Eng. Mech., 2018, 144(6), p 04018029CrossRefGoogle Scholar
  34. 34.
    D.-W. Lee, K.A. Khan, and R.K. Abu Al-Rub, Stiffness and Yield Strength of Architectured Foams Based on the Schwarz Primitive Triply Periodic Minimal Surface, Int. J. Plast., 2017, 95, p 1–20CrossRefGoogle Scholar
  35. 35.
    F. Bobbert, K. Lietaert, A. Eftekhari, B. Pouran, S. Ahmadi, H. Weinans, and A. Zadpoor, Additively Manufactured Metallic Porous Biomaterials Based on Minimal Surfaces: A Unique Combination of Topological, Mechanical, and Mass Transport Properties, Acta Biomater., 2017, 53, p 572–584CrossRefGoogle Scholar
  36. 36.
    I. Maskery, N.T. Aboulkhair, A.O. Aremu, C.J. Tuck, and I.A. Ashcroft, Compressive Failure Modes and Energy Absorption in Additively Manufactured Double Gyroid Lattices, Addit. Manuf., 2017, 16, p 24–29CrossRefGoogle Scholar
  37. 37.
    L. Zhang, S. Feih, S. Daynes, S. Chang, M.Y. Wang, J. Wei, and W.F. Lu, Energy Absorption Characteristics of Metallic Triply Periodic Minimal Surface Sheet Structures Under Compressive Loading, Addit. Manuf., 2018, 23, p 505–515CrossRefGoogle Scholar
  38. 38.
    A. Ataee, Y. Li, D. Fraser, G. Song, and C. Wen, Anisotropic Ti-6Al-4 V Gyroid Scaffolds Manufactured by Electron Beam Melting (EBM) for Bone Implant Applications, Mater. Des., 2018, 137, p 345–354CrossRefGoogle Scholar
  39. 39.
    C. Han, Y. Li, Q. Wang, S. Wen, Q. Wei, C. Yan, L. Hao, J. Liu, and Y. Shi, Continuous Functionally Graded Porous Titanium Scaffolds Manufactured by Selective Laser Melting for Bone Implants, J. Mech. Behav. Biomed. Mater., 2018, 80, p 119–127CrossRefGoogle Scholar
  40. 40.
    C. Yan, L. Hao, A. Hussein, S.L. Bubb, P. Young, and D. Raymont, Evaluation of Light-Weight AlSi10 Mg Periodic Cellular Lattice Structures Fabricated via Direct Metal Laser Sintering, J. Mater. Process. Technol., 2014, 214(4), p 856–864CrossRefGoogle Scholar
  41. 41.
    C. Yan, L. Hao, A. Hussein, and D. Raymont, Evaluations of Cellular Lattice Structures Manufactured Using Selective Laser Melting, Int. J. Mach. Tools Manuf, 2012, 62, p 32–38CrossRefGoogle Scholar
  42. 42.
    C. Yan, L. Hao, A. Hussein, P. Young, and D. Raymont, Advanced Lightweight 316L Stainless Steel Cellular Lattice Structures Fabricated via Selective Laser Melting, Mater. Des., 2014, 55, p 533–541CrossRefGoogle Scholar
  43. 43.
    A. Yánez, A. Cuadrado, O. Martel, H. Afonso, and D. Monopoli, Gyroid Porous Titanium Structures: A Versatile Solution to be Used as Scaffolds in Bone Defect Reconstruction, Mater. Des., 2018, 140, p 21–29CrossRefGoogle Scholar
  44. 44.
    K. Michielsen and J. Kole, Photonic Band Gaps in Materials with Triply Periodic Surfaces and Related Tubular Structures, Phys. Rev. B, 2003, 68(11), p 115107CrossRefGoogle Scholar
  45. 45.
    S. Van Bael, G. Kerckhofs, M. Moesen, G. Pyka, J. Schrooten, and J.-P. Kruth, Micro-CT-Based Improvement of Geometrical and Mechanical Controllability of Selective Laser Melted Ti-6Al-4 V Porous Structures, Mater. Sci. Eng. A, 2011, 528(24), p 7423–7431CrossRefGoogle Scholar
  46. 46.
    M.E. Ashby, A.G. Evans, N.A. Fleck, L.J. Gibson, J.W. Hutchinson, H.N.G. Wadley, Chapter 3: Characterization Methods 2000, Metal Foams, p 24–39Google Scholar
  47. 47.
    I. Maskery, N.T. Aboulkhair, A.O. Aremu, C.J. Tuck, I.A. Ashcroft, R.D. Wildman, and R.J.M. Hague, A Mechanical Property Evaluation of Graded Density Al-Si10-Mg Lattice Structures Manufactured by Selective Laser Melting, Mater. Sci. Eng. A, 2016, 670, p 264–274CrossRefGoogle Scholar

Copyright information

© ASM International 2019

Authors and Affiliations

  1. 1.Mechanical Engineering Department, Masdar InstituteKhalifa University of Science and TechnologyAbu DhabiUAE
  2. 2.Core Technology Platforms DivisionNew York University Abu DhabiAbu DhabiUAE
  3. 3.Aerospace Engineering DepartmentKhalifa University of Science and TechnologyAbu DhabiUAE

Personalised recommendations