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.

在工程实践中，非常需要具有极限各向异性刚度的机械超材料。特别地，已经证明，由6组平板组成的板微结构（PM）在理论上可达到任何极高的刚度。在本文中，我们解决了设计最佳PM的两个遗留问题。一方面，我们研究了三个PM的刚度优化，这些PM涉及少于6个承受任意规定多重载荷（通常是准周期性）的自由定向板组。由于它们的几何形状更简单，板组更少，因此在实际应用中是首选。另一方面，我们确定了两个最优周期板由7个板组组成的晶格结构，证明它们能够在低体积分数极限内的任何多载荷下达到接近最佳的刚度（理论上超过极限刚度的97％和99％）。为了确保有足够大的装载空间来验证这些PM的刚度最优性，首先使用了成千上万的随机多次装载，然后系统地为每个PM确定了产生最高刚度缺陷的最坏的多次装载。数值结果不仅说明了这些粉末冶金材料的刚度最优，而且还为选择应用最少的板组的最简单的粉末冶金材料提供了建议。