A comprehensive characterization of the novel class of anti-tetrachiral cellular solids, both considering the static and the dynamic response, is provided in the paper.The heterogeneous material is characterized by a periodic microstructure made of equi-spaced rings each interconnected by four ligaments. In the most general case, rings and ligaments are surrounded by a softer matrix and the rings can be filled by a different material.First, the first order linear elastic homogenized constitutive response is estimated resorting to two different microstructural models: a discrete model, in which the ligaments are modeled as beams and the presence of the matrix is neglected and the equivalent elastic properties are evaluated through a simplified analytical approach, and a more detailed continuous model, where the actual properties of matrix, ligaments and rings, varying in the 2D domain, are considered and the first order computational homogenization is adopted. Special attention is given to the dependence of the 2D overall Cauchy-type elastic constants on the mechanical and geometrical parameters characterizing the microstructure. The results, indeed, show the existence of large variations in the linear elastic constants and degree of anisotropy. A comparison with available experimental results confirms the validity of the analytical and numerical approaches adopted.Finally, the rigorous Floquet-Bloch approach is applied to the periodic cell of the cellular solid to evaluate the dispersion of propagation waves along the orthotropic axes in the framework of elasticity and to detect band gaps characterizing the material. A numerical approach, based on the first order computational homogenization, is also adopted and the rigorous and approximate solutions are compared.
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