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Fig. 1: Young’s, shear and bulk moduli.
Fig. 2: Total stiffness.


  1. 1.

    Milton, G. W., Nature 564, (2018).

  2. 2.

    Hashin, Z. & Shtrikman, S. A variational approach to the theory of the elastic behaviour of multiphase materials. J. Mech. Phys. Solids 11, 127–140 (1963).

    ADS  MathSciNet  Article  Google Scholar 

  3. 3.

    Berger, J. B., Wadley, H. N. G. & McMeeking, R. M. Mechanical metamaterials at the theoretical limit of isotropic elastic stiffness. Nature 543, 533–537 (2017).

    CAS  ADS  Article  Google Scholar 

  4. 4.

    Norris, A. N. A differential scheme for the effective moduli of composites. Mech. Mater. 4, 1–16 (1985).

    Article  Google Scholar 

  5. 5.

    Milton, G. W. in Homogenization and Effective Moduli of Materials and Media (eds Ericksen, J. L. et al.) 150–174 (Springer-Verlag, New York, 1986).

  6. 6.

    Francfort, G. A. & Murat, F. Homogenization and optimal bounds in linear elasticity. Arch. Ration. Mech. Anal. 94, 307–334 (1986).

    MathSciNet  Article  Google Scholar 

  7. 7.

    Cherkaev, A. Variational Methods for Structural Optimization (Springer-Verlag, New York, 2000).

    Book  Google Scholar 

  8. 8.

    Milton, G. W. The Theory of Composites (Cambridge Univ. Press, Cambridge, 2002).

    Book  Google Scholar 

  9. 9.

    Allaire, G. Shape Optimization by the Homogenization Method (Springer-Verlag, New York, 2012).

    MATH  Google Scholar 

  10. 10.

    Torquato, S. Random Heterogeneous Materials: Microstructure and Macroscopic Properties (Springer Science & Business Media, New York, 2002).

    Book  Google Scholar 

  11. 11.

    Bourdin, B. & Kohn, R. V. Optimization of structural topology in the high-porosity regime. J. Mech. Phys. Solids 56, 1043–1064 (2008).

    ADS  MathSciNet  Article  Google Scholar 

  12. 12.

    Berryman, J. G. & Milton, G. W. Microgeometry of random composites and porous media. J. Phys. D 21, 87–94 (1988).

    ADS  Article  Google Scholar 

  13. 13.

    Cherkaev, A. V. & Gibiansky, L. V. Coupled estimates for the bulk and shear moduli of a two-dimensional isotropic elastic composite. J. Mech. Phys. Solids 41, 937–980 (1993).

    ADS  MathSciNet  Article  Google Scholar 

  14. 14.

    Sigmund, O. Materials with prescribed constitutive parameters: an inverse homogenization problem. Int. J. Solids Struct. 31, 2313–2329 (1994).

    MathSciNet  Article  Google Scholar 

  15. 15.

    Milton, G. W. & Cherkaev, A. V. Which elasticity tensors are realizable? J. Eng. Mater. Technol. 117, 483–493 (1995).

    Article  Google Scholar 

  16. 16.

    Milton, G. W., Briane, M. & Harutyunyan, D. On the possible effective elasticity tensors of 2-dimensional and 3-dimensional printed materials. Math. Mech. Complex Syst. 5, 41–94 (2017).

    MathSciNet  Article  Google Scholar 

  17. 17.

    Camar-Eddine, M. & Seppecher, P. Determination of the closure of the set of elasticity functionals. Arch. Ration. Mech. Anal. 170, 211–245 (2003).

    MathSciNet  Article  Google Scholar 

  18. 18.

    Seppecher, P., Alibert, J.-J. & dell’Isola, F. Linear elastic trusses leading to continua with exotic mechanical interactions. J. Phys. 319, 012018 (2011).

    Google Scholar 

  19. 19.

    Andreassen, E., Lazarov, B. S. & Sigmund, O. Design of manufacturable 3D extremal elastic microstructure. Mech. Mater. 69, 1–10 (2014).

    Article  Google Scholar 

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Author information




J.B.B. created the ideas, conceived and designed the new material geometries and performed the structural analysis. R.M.M. developed the analytical models for the strain energy and moduli and, with H.N.G.W., contributed to refining the concepts, contextualizing the results and providing critiques and assessments.

Corresponding author

Correspondence to J. B. Berger.

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Competing interests

The material geometry identified in this work to achieve the theoretical bounds in performance has been included in a Patent Cooperation Treaty (PCT/US2015/010458) by Nama Development, LLC (DE), which is majority-owned by J.B.B.

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Berger, J.B., Wadley, H.N.G. & McMeeking, R.M. Berger et al. reply. Nature 564, E2–E4 (2018).

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  • Hashin Shtrikman
  • Material Geometry
  • Moderate Relative Density
  • Zener Anisotropy Ratio
  • Systems Materials Engineering


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