A micromorphic computational homogenization framework for auxetic tetra-chiral structures

Auxetic chiral structures exhibit many unique and enhanced mechanical properties, which emerge from the coupling of stretching and rotational mechanisms within the underlying unit cell. The standard first-order Computational Homogenization (CH) approach is inadequate in capturing this coupling effect. Moreover, a size effect arising from the interactions between micro and macro length scales cannot be accounted for. In this contribution, the micromorphic CH framework developed in Biswas and Poh (2017), Biswas et al. (2019), is reformulated for 2D tetra-chiral solids to address these limitations. Together with the standard macro-distortion tensor, an additional kinematic variable is introduced to characterize the deformation and rotation of the central ring. These two kinematic fields, each characterizing a particular aspect of the unit cell deformation, enable the micromorphic framework to adequately characterize the underlying dominant deformation modes. The predictive capability of the micromorphic CH framework is demonstrated through three benchmark problems in plane strain condition. Considering an axial loading problem, it is shown that the proposed micromorphic framework accurately captures the coupling between internal ring rotation and stretching of tangential ligaments. The predictive capability of the micromorphic approach is next demonstrated with a flat punch indentation problem in a regime without a clear separation of length scales. The homogenized micromorphic model predicts the correct global force-displacement response, as well as the underlying deformation mechanisms. Finally, the capability of the micromorphic framework in capturing both distortional and rotational modes of the central ring is demonstrated through the axial loading of a tapering plate.