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Mechanical metamaterials at the theoretical limit of isotropic elastic stiffness

Nature volume 543, pages 533537 (23 March 2017) | Download Citation


A wide variety of high-performance applications1 require materials for which shape control is maintained under substantial stress, and that have minimal density. Bio-inspired hexagonal and square honeycomb structures and lattice materials based on repeating unit cells composed of webs or trusses2, when made from materials of high elastic stiffness and low density3, represent some of the lightest, stiffest and strongest materials available today4. Recent advances in 3D printing and automated assembly have enabled such complicated material geometries to be fabricated at low (and declining) cost. These mechanical metamaterials have properties that are a function of their mesoscale geometry as well as their constituents3,5,6,7,8,9,10,11,12, leading to combinations of properties that are unobtainable in solid materials; however, a material geometry that achieves the theoretical upper bounds for isotropic elasticity and strain energy storage (the Hashin–Shtrikman upper bounds) has yet to be identified. Here we evaluate the manner in which strain energy distributes under load in a representative selection of material geometries, to identify the morphological features associated with high elastic performance. Using finite-element models, supported by analytical methods, and a heuristic optimization scheme, we identify a material geometry that achieves the Hashin–Shtrikman upper bounds on isotropic elastic stiffness. Previous work has focused on truss networks and anisotropic honeycombs, neither of which can achieve this theoretical limit13. We find that stiff but well distributed networks of plates are required to transfer loads efficiently between neighbouring members. The resulting low-density mechanical metamaterials have many advantageous properties: their mesoscale geometry can facilitate large crushing strains with high energy absorption2,14,15, optical bandgaps16,17,18,19 and mechanically tunable acoustic bandgaps20, high thermal insulation21, buoyancy, and fluid storage and transport. Our relatively simple design can be manufactured using origami-like sheet folding22 and bonding methods.

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H.N.G.W. is grateful for support for this work by the ONR (grant number N00014-15-1-2933), managed by D. Shifler, and the DARPA MCMA programme (grant number W91CRB-10-1-005), managed by J. Goldwasser.

Author information


  1. Materials Department, University of California, Santa Barbara, California 93106-5050, USA

    • J. B. Berger
    •  & R. M. McMeeking
  2. Department of Mechanical Engineering, University of California, Santa Barbara, California 93106-5050, USA

    • J. B. Berger
    •  & R. M. McMeeking
  3. Department of Materials Science and Engineering, School of Engineering and Applied Science, University of Virginia, Charlottesville, Virginia 22904, USA

    • H. N. G. Wadley
  4. School of Engineering, University of Aberdeen, King’s College, Aberdeen AB24 3UE, UK

    • R. M. McMeeking
  5. INM-Leibniz Institute for New Materials, Campus D22, 66123 Saarbrücken, Germany

    • R. M. McMeeking


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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 strain energy and moduli, and, with H.N.G.W., contributed to refining the concepts, contextualizing the results, and providing critiques and assessments.

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.

Corresponding author

Correspondence to J. B. Berger.

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  1. 1.

    Supplementary Information

    This file contains Supplementary Equations and Methods. It contains derivations for elastic strain energy and stiffness of octet-foam, cubic-foam, and octet-foam and cubic-foam combined, and methods for finite element verification.

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