Skip to main content

Normal and shear behaviours of the auxetic metamaterials: homogenisation and experimental approaches


The auxetic metamaterials exhibit attractive mechanical properties, including negative Poisson’s ratio and compressional resistance. Although auxetic metamaterials have been extensively investigated using experimental and computational approaches, the consistent estimation of shear properties has still not been clarified. According to Cauchy elasticity, the shear properties of an auxetic structure should be enhanced owing to the negative Poisson’s ratio. However, this study used homogenisation and experimental approaches to demonstrate that the shear elasticity is highly non-linear with respect to the characteristic geometrical parameters of a unit cell and that shear properties, particularly normal elasticity, are not always improved. Furthermore, the estimation of the shear elasticity based on the classical isotropic continuum elasticity relations can result in misleading values and should be avoided.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5


  1. 1.

    Astm d4255 / d4255m - 15a (2015) Standard test method for in-plane shear properties of polymer matrix composite materials by the rail shear method. ASTM International, West Conshohocken, PA

    Google Scholar 

  2. 2.

    Astm d3039 / d3039m-17 (2017) Standard test method for tensile properties of polymer matrix composite materials. ASTM International, West Conshohocken, PA

    Google Scholar 

  3. 3.

    Addessi D, De Bellis ML, Sacco E (2016) A micromechanical approach for the cosserat modeling of composites. Meccanica 51(3):569–592

    MathSciNet  Article  MATH  Google Scholar 

  4. 4.

    Alderson K, Fitzgerald A, Evans K (2000) The strain dependent indentation resilience of auxetic microporous polyethylene. J Mater Sci 35(16):4039–4047

    ADS  Article  Google Scholar 

  5. 5.

    Argatov II, Guinovart-Díaz R, Sabina FJ (2012) On local indentation and impact compliance of isotropic auxetic materials from the continuum mechanics viewpoint. Int J Eng Sci 54:42–57

    Article  MATH  Google Scholar 

  6. 6.

    Artioli E (2018) Asymptotic homogenization of fibre-reinforced composites: a virtual element method approach. Meccanica 53(6):1187–1201

    MathSciNet  Article  MATH  Google Scholar 

  7. 7.

    Assidi M, Ganghoffer JF (2012) Composites with auxetic inclusions showing both an auxetic behavior and enhancement of their mechanical properties. Compos Struct 94(8):2373–2382

    Article  Google Scholar 

  8. 8.

    Chan N, Evans K (1998) Indentation resilience of conventional and auxetic foams. J Cell Plast 34(3):231–260

    Article  Google Scholar 

  9. 9.

    Covezzi F, de Miranda S, Fritzen F, Marfia S, Sacco E (2018) Comparison of reduced order homogenization techniques: prbmor, nutfa and mxtfa. Meccanica 53(6):1291–1312

    Article  Google Scholar 

  10. 10.

    D’Alessandro L, Zega V, Ardito R, Corigliano A (2018) 3d auxetic single material periodic structure with ultra-wide tunable bandgap. Sci Rep 8(1):2262

    ADS  Article  Google Scholar 

  11. 11.

    Fu M, Xu O, Hu L, Yu T (2016) Nonlinear shear modulus of re-entrant hexagonal honeycombs under large deformation. Int J Solids Struct 80:284–296

    Article  Google Scholar 

  12. 12.

    Giambanco G, Ribolla ELM, Spada A (2018) Meshless meso-modeling of masonry in the computational homogenization framework. Meccanica 53(7):1673–1697

    MathSciNet  Article  Google Scholar 

  13. 13.

    Gibson LJ, Ashby MF (1999) Cellular solids: structure and properties. Cambridge University Press, Cambridge

    MATH  Google Scholar 

  14. 14.

    Gonella S, Ruzzene M (2008) Homogenization and equivalent in-plane properties of two-dimensional periodic lattices. Int J Solids Struct 45(10):2897–2915

    Article  MATH  Google Scholar 

  15. 15.

    Guild M, Walker C, Calvo D, Mott P, Roland C (2017) Deviation from classical elasticity in the acoustic response of auxetic foams. Rubber Chem Technol 90(2):381–386

    Article  Google Scholar 

  16. 16.

    Jiang Y, Li Y (2018) 3d printed auxetic mechanical metamaterial with chiral cells and re-entrant cores. Sci Rep 8(1):2397

    ADS  Article  Google Scholar 

  17. 17.

    Krödel S, Delpero T, Bergamini A, Ermanni P, Kochmann DM (2014) 3d auxetic microlattices with independently controllable acoustic band gaps and quasi-static elastic moduli. Adv Eng Mater 16(4):357–363

    Article  Google Scholar 

  18. 18.

    Lee J, Choi J, Choi K (1996) Application of homogenization fem analysis to regular and re-entrant honeycomb structures. J Mater Sci 31(15):4105–4110

    ADS  Article  Google Scholar 

  19. 19.

    Logg A, Mardal KA, Wells G (2012) Automated solution of differential equations by the finite element method: the FEniCS book, vol 84. Springer, Berlin

    Book  MATH  Google Scholar 

  20. 20.

    Mott PH, Roland CM (2009) Limits to poisson’s ratio in isotropic materials. Phys Rev B 80:132104

    ADS  Article  Google Scholar 

  21. 21.

    Nguyen VD, Béchet E, Geuzaine C, Noels L (2012) Imposing periodic boundary condition on arbitrary meshes by polynomial interpolation. Comput Mater Sci 55:390–406

    Article  Google Scholar 

  22. 22.

    Penta R, Merodio J (2017) Homogenized modeling for vascularized poroelastic materials. Meccanica 52(14):3321–3343

    MathSciNet  Article  MATH  Google Scholar 

  23. 23.

    Rayneau-Kirkhope D (2018) Stiff auxetics: Hierarchy as a route to stiff, strong lattice based auxetic meta-materials. Sci Rep 8(1):12437

    ADS  Article  Google Scholar 

  24. 24.

    Reis F, Pires FA (2014) A mortar based approach for the enforcement of periodic boundary conditions on arbitrarily generated meshes. Comput Methods Appl Mech Eng 274:168–191

    ADS  MathSciNet  Article  MATH  Google Scholar 

  25. 25.

    Roh JH, Giller CB, Mott PH, Roland CM (2013) Failure of classical elasticity in auxetic foams. AIP Adv 3(4):042126

    ADS  Article  Google Scholar 

  26. 26.

    Scarpa F, Ciffo L, Yates J (2003) Dynamic properties of high structural integrity auxetic open cell foam. Smart Mater Struct 13(1):49

    ADS  Article  Google Scholar 

  27. 27.

    Smith C, Lehman F, Wootton R, Evans K (1999) Strain dependent densification during indentation in auxetic foams. Cell Polym 18(2):79–101

    Google Scholar 

  28. 28.

    Yang L, Harrysson O, West H, Cormier D (2015) Mechanical properties of 3d re-entrant honeycomb auxetic structures realized via additive manufacturing. Int J Solids Struct 69–70:475–490

    Article  Google Scholar 

  29. 29.

    Yang S, Qi C, Wang D, Gao R, Hu H, Shu J (2013) A comparative study of ballistic resistance of sandwich panels with aluminum foam and auxetic honeycomb cores. Adv Mech Eng 5:589216

    Article  Google Scholar 

  30. 30.

    Yang W, Li ZM, Shi W, Xie BH, Yang MB (2004) Review on auxetic materials. J Mater Sci 39(10):3269–3279

    ADS  Article  Google Scholar 

Download references


This work was supported by the Ministry of Education, Youth and Sports of the Czech Republic and the European Union - European Structural and Investment Funds in the frames of Operational Programme Research, Development and Education - project Hybrid Materials for Hierarchical Structures (HyHi, Reg. No. CZ.02.1.01/0.0/0.0/16_019/0000843).

Author information



Corresponding author

Correspondence to L. Čapek.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Appendix: Parameterisation of auxetic structures

Appendix: Parameterisation of auxetic structures

See Figs. 6 and 7.

Fig. 6

Parametrisation of structure A

Fig. 7

Parametrisation of structure B

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Henyš, P., Vomáčko, V., Ackermann, M. et al. Normal and shear behaviours of the auxetic metamaterials: homogenisation and experimental approaches. Meccanica 54, 831–839 (2019).

Download citation


  • Auxetic metamaterial
  • Linear elasticity
  • Computational homogenization
  • Finite element
  • Optical measurement
  • Shear modulus