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The understanding of wave propagation in a metamaterial with hierarchical, auxetic rectangular perforations is presented in this work. The metamaterial is a 2D structure with chaining horizontal and vertical perforations exhibiting auxetic in-plane behaviour. The unit cell of this lattice is identified as the reference level 0. Hierarchical structures are composed of structural elements which themselves have structure. At level 0, 4 rigid squares are present in the unit cell. In each square, the reference structure is used by applying a scale ratio to obtain the level 1. The same strategy is used to reach the upper level in each subunit. A geometric parametric investigation of these rectangular perforations using a numerical asymptotic homogenisation finite element approach is done. Some numerical eigenvalue tools are used for the dispersion analysis of this structure. It is first observed that the total width of Band gaps increases with the hierarchy. The porosity induced by the perforations is taken into account in the mechanical properties. The symmetry of the geometry in the x-y plane allow to define the entire geometry of the unit cell using only 2 parameters: the void aspect ratio, the intercell spacing and the hierarchy level. When decreasing the intercell spacing, the total width of Band gaps increases and the effective stiffness in x and y directions decrease, allowing for increased rotations of the rigid squares, so auxetic behaviour is greater. Hierarchical levels shift from isotropic to orthotropic, hierarchical levels are always auxetic.

Parametric study of wave propagation in hierarchical auxetic perforated metamaterials