Stiff competition

arising from J. B. Berger, H. N. G. Wadley & R. M. McMeeking, Nature 543, 533–537 (2017); https://doi.org/10.1038/nature21075

The paper of Berger, Wadley & McMeeking¹ presents beautiful results on structured composites near the edge of maximal stiffness for a given porosity. However, it appears that the authors were unaware of the large body of work on this subject, much of it summarized in refs ^2,3,4,5,6. In particular, their claim that “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” is not accurate. Multiscale elastically isotropic composites with simultaneously maximal bulk and shear modulus—and hence maximal stiffness and energy storage—were identified independently in refs ^7,8,9. There is a Reply to this Comment by J. B. Berger et al., Nature 564, https://doi.org/10.1038/s41586-018-0725-7 (2018).

Moreover, the simple argument made in ref. ⁸—that the Hashin–Shtrikman bounds are attained if the actual field in the material matches the trial field, which is constant in one phase—shows that any hierarchical laminate, in which layers of the stiffer phase are sequentially added to the composite in different orientations, necessarily achieves these upper bounds if layering is done so that the final material is elastically isotropic. Later it was established by Bourdin and Kohn¹⁰ that no separation of length scales is needed if the volume fraction of the stiffer phase is small. These geometries are formed by the union of families of parallel plates, with each family having a different orientation, and include the cubic foam, octet foam and cubic + octet foam described in ref. ¹. The novelty of ref. ¹ is that it shows that this class of microstructure also works well if the volume fraction is moderate. Other porous three-dimensional microgeometries with very large bulk and shear moduli, at a moderate volume fraction of 0.338, have been found using topology optimization methods¹¹ (see, in particular, point e in figure 9 of ref. ¹¹). Yet it is still not known if a single-scale geometry can exactly attain the shear bounds away from the low-density limit. Single-scale geometries can achieve the bulk modulus bounds^12,13,14.

To finish, we briefly mention important results that cover more general questions than those addressed in ref. ¹ to bring readers up to speed on current developments. If the second material is not void, there are improved bounds that couple the possible bulk and shear moduli^15,16, and the range of possible (bulk, shear) pairs has been explored numerically^11,17. A recent paper¹⁸ goes a long way to completely characterizing the possible elasticity tensors of three-dimensional printed, possibly anisotropic, materials constructed from a given isotropic material with given porosity. These materials include elastically isotropic microstructures that asymptotically attain the Hashin–Shtrikman upper bulk modulus bound for any given volume fraction, yet have an arbitrarily small shear modulus. If one allows the starting material to be as stiff as one likes, and replaces the void material by a material that is as compliant as one likes, then one can get any desired elasticity tensor¹⁹—a result also suggested by numerics¹⁷. In fact, non-local effective behaviours are possible too and, remarkably, these have also been completely characterized for linear elasticity²⁰. In principle, one can obtain composites for which uniform strains cost little energy, but gradients in the strains (double gradients of the displacement) cost considerable energy (see, for example, ref. ²¹ for some interesting examples).