Design of hierarchical microstructures with isotropic elastic stiffness

Elastic stiffness is one of the most fundamental properties of materials. Design of the microstructures with isotropic stiffness has been an attractive area in the field of metamaterials for over three decades. Despite many classes of isotropic microstructures, exploring novel isotropic microstructures based on innovative mechanics principles has attracted great and continuing interests. This paper presents a novel family of isotropic hierarchical microstructures (Iso-HMs). These hierarchical microstructures are modeled by replacing the solid parts of prescribed single-level microstructures with arrayed microstructures in the second level, where the key task is to identify the correct geometries of the second-level microstructures by conducting parameter space exploring. These Iso-HMs realize isotropic stiffness based on synergistic deformations of the members in the two levels, which is essentially different from existing isotropic microstructures replying on deformations of the members in a single level. Two categories of Iso-HMs with rectangular holes and Vidergauze-type struts are designed. Considering the large size difference in the designed Iso-HMs, additive manufacturing becomes a unique technique for manufacturing the designed Iso-HMs, where the size ratio between the 3D-printed specimens and the minimal features reaches 400:1. Both numerical and experimental results validate the isotropic stiffness of the designed Iso-HMs. Furthermore, the results of a microstructural instability analysis show that the designed Iso-HMs can gain improved buckling strength up to a hundred times higher than their single-level counterparts. The hierarchical design provides a new way to identify novel functional microstructures for applications, and the hierarchical configurations expand the space of the already-known families of isotropic microstructures.