Acta Mechanica

, Volume 230, Issue 6, pp 2171–2185 | Cite as

Analytical solution and finite element approach to the dense re-entrant unit cells of auxetic structures

  • M. Shokri Rad
  • H. HatamiEmail author
  • Z. Ahmad
  • A. Karimdoost Yasuri
Original Paper


Chiral, star honeycomb, and re-entrant structures are among the most important structures of auxetic materials. In this study, a dense re-entrant unit cell is introduced for making a 3D auxetic structure to be used in high stiffness applications. A re-entrant structure is chosen due to its fundamental characteristics underlying the main characteristics of auxetic structures. The energy methods of solid mechanics along with numerical methods are used to study the fundamental concept of auxetic structures. Understanding the characteristics of the re-entrant structure will lead to the better comprehension of other structures of auxetic materials, which will eventually contribute to the advance of research in this new class of materials.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.



  1. 1.
    Arago, F.: Oeuvres complètes. Number v. 2 in Oeuvres complètes (1854)Google Scholar
  2. 2.
    Greaves, G.: Poisson’s ratio and modern materials. Nat. Mater. 10, 723–806 (2011)CrossRefGoogle Scholar
  3. 3.
    Callister, W.: Materials Science and Engineering: An Introduction, 7th edn. Wiley, New York (2006)Google Scholar
  4. 4.
    Lakes, R.: Deformation mechanics in negative Poisson’s ratio materials. J. Mater. Sci. 26, 2287–2292 (1991)CrossRefGoogle Scholar
  5. 5.
    Lakes, R.: Advances in negative Poisson’s ratio materials. Adv. Mater. 5, 293–296 (1993)CrossRefGoogle Scholar
  6. 6.
    Lakes, R.: Extreme damping in composite material with negative stiffness inclusions. Nature 410, 565–567 (2001). CrossRefGoogle Scholar
  7. 7.
    Prawoto, Y.: Seeing auxetic materials from the mechanics point of view: a structural review on the negative Poisson’s ratio. Comput. Mater. Sci. 58, 140–153 (2012)CrossRefGoogle Scholar
  8. 8.
    Reis, F., Ganghoffer, J.: Equivalent mechanical properties of auxetic lattices from discrete homogenization. Comput. Mater. Sci. 51, 314–321 (2012)CrossRefGoogle Scholar
  9. 9.
    Blumenfold, R.: Auxetic strains-insight from iso-auxetic materials. Mol. Simul. 31, 867–871 (2005)CrossRefGoogle Scholar
  10. 10.
    Grima, J., Gatt, R., Ravirala, N., Alderson, A., Evans, K.: Negative Poisson’s ratios in cellular foam materials. Mater. Sci. Eng. 423, 214–218 (2006)CrossRefGoogle Scholar
  11. 11.
    Gibson, L., Ashby, M., Schajer, G., Robertson, C.: The mechanics of two dimensional cellular materials. J. Proc. Lond. R. Soc. 382, 25–42 (1982)CrossRefGoogle Scholar
  12. 12.
    Masters, I., Evans, K.: Models for the elastic deformation of honeycombs. Compos. Struct. 35, 403–408 (1996)CrossRefGoogle Scholar
  13. 13.
    Evans, K., Alderson, A., Christian, F.: Auxetic two-dimensional polymer networks. An example of tailoring geometry for specific mechanical properties. J. Chem. Soc. Faraday Trans. 91, 2671–2680 (1995)CrossRefGoogle Scholar
  14. 14.
    Lu, Z.-X., Liu, Q., Yang, Z.-Y.: Predictions of Young’s modulus and negative Poisson’s ratio of auxetic foams. J. Basic Solid State Phys. 248, 167–174 (2011)CrossRefGoogle Scholar
  15. 15.
    Chan, N., Evans, E.: Microscopic examination of the microstructure and deformation of conventional and auxetic foams. J. Mater. Sci. 32, 5725–5736 (1997)CrossRefGoogle Scholar
  16. 16.
    Subramani, P., Sohel Rana, D.V., Oliveira, R.F., Xavier, J.: Development of novel auxetic structures based on braided composites. Mater. Des. 61, 286–295 (2014)CrossRefGoogle Scholar
  17. 17.
    Wang, X.T., Wang, B., Li, X.W., Ma, L.: Mechanical properties of 3D re-entrant auxetic cellular structures. Int. J. Mech. Sci. 131–132, 396–407 (2017)CrossRefGoogle Scholar
  18. 18.
    Wang, Z.P., Poh, L.H., Dirrenberger, J., Zhu, Y., Forest, S.: Isogeometric shape optimization of smoothed petal auxetic structures via computational periodic homogenization. Comput. Methods Appl. Mech. Eng. 323, 250–271 (2017)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Wang, Z.P., Poh, L.H.: Optimal form and size characterization of planar isotropic petal-shaped auxetics with tunable effective properties using IGA. Compos. Struct. 201, 486–502 (2018)CrossRefGoogle Scholar
  20. 20.
    Qi, C., Remennikov, A., Pei, L.Z., Yang, S., Yu, Z.H., Ngo, T.D.: Impact and close-in blast response of auxetic honeycomb-cored sandwich panels: experimental tests and numerical simulations. Compos. Struct. 180, 161–178 (2017)CrossRefGoogle Scholar
  21. 21.
    Harkati, E., Daoudi, N., Bezazi, A., Haddad, A., Scarpa, F.: In-plane elasticity of a multi re-entrant auxetic honeycomb. Compos. Struct. 180, 130–139 (2017)CrossRefGoogle Scholar
  22. 22.
    El Nady, K., Reis, F.D., Ganghoffer, J.F.: Computation of the homogenized nonlinear elastic response of 2D and 3D auxetic structures based on micropolar continuum models. Compos. Struct. 170, 271–290 (2017)CrossRefGoogle Scholar
  23. 23.
    Fu, M.H., Chen, Y., Hu, L.L.: Bilinear elastic characteristic of enhanced auxetic honeycombs. Compos. Struct. 175, 101–110 (2017)CrossRefGoogle Scholar
  24. 24.
    Zheng, Q.S., Chen, T.: New perspective on Poisson’s ratios of elastic solids. Acta Mech. 150, 191–195 (2001)CrossRefzbMATHGoogle Scholar
  25. 25.
    Zhang, S., Guan, M., Wu, G., Gao, S., Chen, X.: An ellipsoidal yield criterion for porous metals with accurate descriptions of theoretical strength and Poisson’s ratio. Acta Mech. 228, 4199–4210 (2017)CrossRefGoogle Scholar
  26. 26.
    Karathanasopoulos, N., Reda, H., Ganghoffer, J.: Designing two-dimensional metamaterials of controlled static and dynamic properties. Comput. Mater. Sci. 138, 323–332 (2017)CrossRefGoogle Scholar
  27. 27.
    Ganghoffer, J.F., Goda, I., Novotny, A.A., Rahouadj, R., Sokolowski, J.: Homogenized couple stress model of optimal auxetic microstructures computed by topology optimization. J. Appl. Math. Mech. 98, 696–717 (2017)MathSciNetGoogle Scholar
  28. 28.
    Karathanasopoulos, N., Dos Reis, F., Reda, H., Ganghoffer, J.-F.: Computing the effective bulk and normal to shear properties of common two-dimensional architectured materials. Comput. Mater. Sci. 154, 284–294 (2018)CrossRefGoogle Scholar
  29. 29.
    Reda, H., Karathanasopoulos, N., Elnady, K., Ganghoffer, J.F., Lakiss, H.: The role of anisotropy on the static and wave propagation characteristics of two-dimensional architectured materials under finite strains. Mater. Des. 147, 134–145 (2018)CrossRefGoogle Scholar
  30. 30.
    Gilat, R., Aboudi, J.: Behavior of elastoplastic auxetic microstructural arrays. Materials 6, 726–737 (2013)CrossRefGoogle Scholar
  31. 31.
    Hu, L.L., Zhou, M.Z., Deng, H.: Dynamic indentation of auxetic and non-auxetic honeycombs under large deformation. Compos. Struct. 207, 323–330 (2019)CrossRefGoogle Scholar
  32. 32.
    Dirrenberger, J., Forest, S., Jeulin, D.: Elastoplasticity of auxetic materials. Comput. Mater. Sci. 64, 57–61 (2012)CrossRefGoogle Scholar
  33. 33.
    Zhu, Y., Wang, Z.-P., Poh, L.H.: Auxetic hexachiral structures with wavy ligaments for large elasto-plastic deformation. Smart Mater. Struct. 27, 055001 (2018)CrossRefGoogle Scholar
  34. 34.
    Shokri Rad, M., Prawoto, Y., Ahmad, Z.: Analytical solution and finite element approach to the 3D re-entrant structures of auxetic materials. Mech. Mater. 74, 76–87 (2014)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Austria, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Mechanical Engineering, Engineering FacultyLorestan UniversityKhorram abadIran
  2. 2.School of Mechanical Engineering, Faculty of EngineeringUTMJohor BahruMalaysia

Personalised recommendations