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Mechanical metamaterials at the theoretical limit of isotropic elastic stiffness


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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Figure 1: Material geometries.
Figure 2: Young’s, shear and bulk moduli.
Figure 3: Total stiffness.
Figure 4: The Zener anisotropy ratio.
Figure 5: Strain energy distributions.


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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

Authors and Affiliations



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.

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.

Extended data figures and tables

Extended Data Figure 1 Property space of isotropic and nearly isotropic materials.

Metamaterial geometries with suboptimal performance have been omitted. A theoretical bound (Ω = 1) limits the performance of all material systems and is defined by the highest performance possible for a two-phase system, which is achieved by the single-crystal diamond and void system. The parameter Ω is defined in the text. The cubic + octet material can bound property space when composed of materials with maximal properties, such as diamond, beryllium, boron carbide, fibre composites and lightweight alloys. Fabrication techniques limit our ability to achieve a wide and otherwise unoccupied region of property space (labelled ‘Metamaterials’).

Extended Data Figure 2 Strain energy distributions from axial stress.

af, Strain energy distributions in the geometries from Fig. 1, when subject to uniaxial stress; Uε is the local strain energy density and is the average solid-phase strain energy density. Macroscopic loads are transmitted through stiff networks of members aligned with the principal stress direction. Strains are small, but scaled to reveal the nature of the deformations. The two-dimensional connectedness of material in closed-cell geometries enables effective transmission of loads between neighbouring members, facilitating materials that can achieve the theoretical bounds (a). Open-cell and stochastic materials (df) have large strain energy concentrations.

Extended Data Figure 3 Strain energy distributions from shear loading.

af, Strains are scaled to highlight the nature of the deformations. The displacements in stiff closed-cell materials (ac) are largely affine and absent of bending. Despite the identical alignment of material in the octet truss (f) and octet foam (c), the absence of membrane stress enables substantial bending to take place in the former, open-cell configuration. All geometries have .

Extended Data Figure 4 Strain energy distributions from hydrostatic loading.

af, Strains are scaled to highlight the nature of the deformations. In maximally stiff materials, the deformations are limited to the filling of void space through member swelling. The displacements are primarily affine and strain energy distributions are nearly uniform. Poor alignment of neighbouring cell walls in the quasi-random material allows some bending to occur (d). All geometries have .

Extended Data Figure 5 Moduli of quasi-random and stochastic foams.

a, The normalized average Young’s, shear and bulk moduli of quasi-random foam, determined using finite-element models, are plotted against relative density. The coloured data (from this study) are fitted to third-order polynomials that are forced to go through the origin (0, 0) and the point (1, 1), corresponding to empty space and a dense solid, respectively. Data are also fitted to the model of ref. 23 (‘G-A’) for the stiffness of isotropic cellular materials, using ϕ = 2/3 (dashed line). (ϕ is the fraction of material subject to bending, and (1−ϕ) is the fraction of material subject to stretching.) Experimental data for Young’s modulus (open circles) are taken from ref. 23. A similar, but more extensive, finite-element study31 (open squares) produced similar results. b, Close-up of the grey shaded region in a.

Source data

Extended Data Table 1 Material properties used to populate property space

Supplementary information

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. (PDF 1296 kb)

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

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