Making use of symmetries in the three-dimensional elastic inverse homogenization problem

Abstract

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 paper 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.