  A novel computational framework for designing metamaterials with negative
Poisson's ratio over a large strain range is presented in this work by
combining the density-based topology optimization together with a mixed
stress/deformation driven nonlinear homogenization method. A measure of
Poisson's ratio based on the macro deformations is proposed, which is further
validated through direct numerical simulations. With the consistent
optimization formulations based on nonlinear homogenization, auxetic
metamaterial designs with respect to different loading orientations and with
different unit cell domains are systematically explored. In addition, the
extension to multimaterial auxetic metamaterial designs is also considered, and
stable optimization formulations are presented to obtain discrete metamaterial
topologies under finite strains. Various new auxetic designs are obtained based
on the proposed framework. To validate the performance of optimized designs, a
multiscale stability analysis is carried out using the Bloch wave method and
rank-one convexity check. As demonstrated, short and/or long wavelength
instabilities can occur during the loading process, leading to a change of
periodicity of the microstructure, which can affect the performance of an
optimized design.
