An analysis of the symmetries characterizing the micro-architecture topologies and the elastic material properties is performed. The goal is to elucidate a systematic procedure that facilitates the design of elastic metamaterial with a prescribed target elasticity tensor via inverse homogenization methodologies. This systematic procedure, which is defined through a set of rules, is based on the relationship established between the elasticity tensor symmetries and the symmetry displayed by the micro-architecture topology.Following this procedure, it can be guaranteed that the designed composites, with the attained micro-structures, have effective elasticity tensors that possess the same or higher symmetries than those shown by the target elasticity tensors. Furthermore, the micro-architectures designed through this technique display simple topologies.Both properties that are supplied by the procedure, i.e., the accomplishment of the required symmetry of the composite homogenized elasticity tensor combined with the topology simplicity, are assessed through numerical simulations of several micro-architecture design problems. They are designed by formulating the inverse homogenization problem as a topology optimization problem which is solved with two different standard algorithms. The proposed procedure and the conclusions here obtained do not depend on the algorithm adopted for solving this problem.

Summary New tools for the design of metamaterials with periodic microarchitectures are presented. Initially, a two‐scale material design approach is adopted. At the structure scale, the material effective properties and their spatial distribution are obtained through a Free Material Optimization technique. At the microstructure scale, the material microarchitecture is designed by appealing to a Topology Optimization Problem (TOP). The TOP is based on the topological derivative and the level set function. The new proposed tools are used to facilitate the search of the optimal microarchitecture configuration. They consist of the following: (i) a procedure to choose an adequate shape of the unit cell domain where the TOP is formulated and shapes of Voronoi cells associated with Bravais lattices are adopted and (ii) a procedure to choose an initial material distribution within the Voronoi cell being utilized as the initial configuration for the iterative TOP.

The objective of this paper is the design of three-dimensional elastic metamaterials with periodic microarchitectures. The microarchitectures of these materials are attained by following an inverse design technique jointly with an homogenization-based topology optimization algorithm.In this context, we have particularly studied the connection between the symmetry of the material layout at the microscale of 3D periodic composites and the symmetry of the effective elastic properties. We have analyzed some possible Bravais lattices and space groups, which are typically associated with crystallography, to study the way in which the symmetries of these geometrical objects can be usefully used for the microarchitecture design of 3D elastic metamaterial.Following a previous work of the authors for two-dimensional problems, we suggest adopting the design domain of the topology optimization problem coincident with the Wigner-Seitz cells of specific Bravais lattices having the same point group to that of the target elasticity tensor.The numerical assessment described in this papers aims at the design of an extreme material. The solutions obtained with this procedure show that different composite microarchitectures emerge depending on the cell shape selection.