Plate microstructures with extreme stiffness for arbitrary multi-loadings

Mechanical metamaterials that achieve ultimate anisotropic stiffness are highly desired in engineering practice. Particularly, the plate microstructures (PM) that are comprised of 6 sets of flat plates have been proved to attain any extreme stiffness in theory. In this paper, we solve two remaining issues for design of optimal PMs. On one hand, we investigate the stiffness optimality of three PMs that involve fewer than 6 freely-oriented plate sets subjected to any prescribed multi-loadings, which are typically quasiperiodic. Because they have simpler geometries with fewer plate sets, they are preferred in practical applications. On the other hand, we identify two optimal periodic plate lattice structures which are comprised of 7 plate sets, and demonstrate that they are able to attain near-optimal stiffness (over 97% and 99% of the extreme stiffness in theory) for any multi-loadings in the low volume fraction limit. In order to ensure a sufficiently large loading space for verification of the stiffness optimality of these PMs, tens of thousands of random multi-loadings are first used and further the worst multi-loading that yields the highest stiffness deficiency is systematically identified for each PM. The numerical results not only illustrate the stiffness optimality of these PMs, but also provide suggestions on selection of the simplest PMs with the fewest plate sets in applications.