Topology optimization of 2D chiral metamaterials with dilatation-shear and shear-shear coupling capabilities

Metamaterials with chiral microstructures exhibit unique mechanical coupling among various deformation modes. Traditional approaches for designing such materials rely heavily on discrete models and unit cells with predefined architectures, and hence it has been challenging to develop methodologies that can explore a broad range of chiral configurations and optimize the mechanical coupling behavior without being constrained by specific unit cell geometries. In the current study, a new multi-objective topology optimization (TO) method is developed for designing 2D chiral metamaterials with prescribed mechanical coupling among dilatation and two distinct shear deformation modes. The new method incorporates material symmetry constraints (including the $C_2$ and $C_4$ symmetries) into the TO process. A strain energy-based homogenization approach is adopted to determine the effective elastic stiffness matrix for each periodic chiral metamaterial. The TO process begins with maximizing the trace of the stiffness matrix to avoid cases with vanishing bulk or shear moduli, which is followed by maximizing/minimizing a selected off-diagonal component to optimize the dilatation-shear or shear-shear coupling. The proposed method successfully identifies optimal topologies that exhibit chiral layouts consistent with the imposed material symmetry constraints, and it maximizes mechanical coupling among dilatation and shear deformation modes. This newly developed method enables the exploration of diverse chiral material configurations, achieving optimized mechanical coupling without relying on a specific unit cell architecture.