Finite-element models are used to identify a material geometry that achieves the theoretical bounds on isotropic elastic stiffness—a combination closed-cell cubic and octet foam.
A wide variety of high-performance applications1require 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 bandgaps16,17,18,19and mechanically tunable acoustic bandgaps20, high thermal insulation21, buoyancy, and fluid storage and transport. Our relatively simple design can be manufactured using origami-like sheet folding22and bonding methods.