Mechanics and band gaps in hierarchical auxetic rectangular perforated composite metamaterials.

We describe in this work a composite metamaterial with a hierarchical topology made by tessellating perforations that exhibit an auxetic (negative Poisson’s ratio) behaviour. We perform an analysis of the hierarchical structure by evaluating the fractal order of the topologies associated to the per-forared composites.The periodic hierarchical lattice conﬁguration shows negative Poisson’s ratio characteristics at higher levels of hierarchy, even when the baseline conﬁguration has a topology not exhibiting an auxetic behaviour. We investigate the wave propagation characteristics of these particular hierarchical lattices by using a Bloch Wave approach applied to detailed Finite Element geometries of the unit cell conﬁgurations. We show that the level of hierarchy creates new band gaps with large relative widths, and it also shifts the same bandgaps towards lower frequencies. We correlate the mechanical properties, fractal order and the dispersion characteristics of the multiscale auxetic perforated metamaterial with the parameters deﬁning the geometry of the lattice and the hierarchy levels, and discuss the results in a nondimensional form to provide a performance map of the mechanical and dynamic properties.


Introduction
Mechanical metamaterials have been recently hailed as a new class of structural concepts able to bring novel multifunctionalities [1] by changes of compliance,shapes, or by embedding oscillators or smart materials inserts.
Negative Poisson's ratio [11] is a mechanical feature of auxetic [12] or dilational [13] materials, and indicates an unusual large volumetric deformation that corresponds to a transverse dilatation with a uniaxial tensile loading. Auxetic structures and solids have been extensively evaluated for their mechanical wave propagation behaviour, because of their strong acoustic signature and potential phononic applications [14,15,16]. By applying patterns of perforations it is possible to generate negative Poisson's ratio effects 2 in continuum planar structures. The presence of perforations with specific geometry and spacing in a planar continuum structure creates an in-plane negative Poisson's ratio behaviour, whether one can use an elliptical [17] or rhomboidal architecture [18,19]. The use of perforations is quite instrumental to create hierarchical configurations by tessellating in a self-similar way the perforated pattern and obtain auxetic configurations in planar and cylindrical domains [20]. A similar approach has also been taken with the engineering of patterns of slits in fractal order [21], or following Kagome-types and various centresymmetric tessellations [22,23]. The introduction of hierarchy in porous solids has been long recognised as a way to design enhanced specific buckling and stiffness performance [24,25,26,27], as well as the transport properties of cellular and porous materials [28]. Recent work has also examined the use of cut hinges topologies in a hierarchical tessellation both from the static mechanical and in-plane wave propagation behaviour [29]. Waves in self-similar domains have several appealing features, like localization phenomena in fluid-filled periodic fractal inclusion acoustic band gap crystals or filters [30], Sierpinsky or quasi-fractal arrangements [31], and the creation of large bandgaps at lower dimensionless frequencies in beam lattices [32]. Reference [29] suggests that the general use of perforations could constitute a quite interesting strategy to design extremely tailorable bandgap materials especially at lower frequencies, due to the ease of producing these 2D metamaterials by simple automatic cutting/CNC machining.
In this work we describe a configuration of hierarchical 2D metamaterial that is produced by a self-similar generation of a rectangular perforated topology with in-plane negative Poisson's ratio ratio behaviour. The topology is derived by configurations identified by Sigmund when developing through Topological Optimization cellular configurations with in-plane weak shear stiffness [33]. The in-plane stiffness, negative Poisson's ratios and shear of this perforated rectangular configuration has been evaluated by Slann and co-workers both from the experimental and numerical point of view [34].
The original rectangular perforated topology maintains an in-plane auxetic behaviour, although for some specific types of pores, aspect ratios and thickness of the cut vertical side the cellular structure switches to a positive inplane Poisson's ratio behaviour. We will show that by using a hierarchical structure of this perforation we obtain a cellular 2D composite material that is always auxetic, even when the baseline self-similar cell is not. Quite significantly, the use of different hierarchical levels with fractal dimensions leads to tailoring and enlarging full and partial bandgaps in a way that could be used to design 2D metamaterials with multiple filtering capabilities.

Geometry of the hierarchical perforated auxetic lattice
The fundamental unit cell of the perforated lattice is shown in Figure 2.
The rectangular perforation is described by the parameters S, a and b, which represent the width of the vertical rib, and the length and vertical thickness of the perforation respectively. The overall width of the cell is r = a+b+2S and the aspect ratio AR is defined as a/b. The unit cell has a double symmetry around the central horizontal and vertical axis. The different hierarchical levels of the perforated structure are produced by repeating the fundamental unit cell on each quarter unit as shown in Figure 2. For a given aspect ratio of the perforation the equivalent volume fraction (φ) of the cellular structure 4 (defined as the ratio between the region on the unit filled with material and the total surface of the unit cell 2) varies significantly with the vertical rib S parameter. At low S levels the volume fraction tends to increase even more between the first and second hierarchical levels, with an increase for example of 1.48 times between levels 1 and levels 2 of the unit at AR = a/b = 4. The increment of the S parameter for a given hierarchical level is also significant, with a 140% decrease of the volume fraction between S = 0.2 and S = 0.8.
It is worth of notice that for the purpose of this work the volume fraction is directly proportional to the relative density ρ/ρ c , in which ρ and ρ c are the densities of the cellular structure and the solid material constituting the structure itself, respectively. The hierarchical tessellation repeats itself only in the central quarters of the cell belonging to the previous level ( Figure 2). As it will be evident from the analysis of the in-plane mechanical properties, for a constant aspect ratio the variation of the spacing parameter S may lead to a topology with discontinuous tessellation because of the particular strategy chosen for the construction of the hierarchy. We have therefore analysed the fractal order of the topologies at different hierarchical levels to assess the degree of selfsimilarity in the hierarchical perforated lattice. One of the most commonly accepted fractal dimension estimate is the Hausdorff-Besicovitch dimension D, which is defined as the logarithmic ratio between the number N of the internal homotheties of an object and the reciprocal of the common ratio R of this homothety [35]: where r = 1/R. We have used in this analysis one of the most popular techniques to estimate the fractal dimension, the box-counting approach [35,36]. In this case the common homotheteis consists in boxes of size R. Our algorithmic analysis was initially performed with the images converted into a binary format, via thresholding, where pixels with value 0 correspond to the background and with value 1 to the object of interest, in order to be successfully implemented for the box-counting algorithm [35,37]. The box counting algorithm applied on the binary images [38] initially ensured that the images had an even number of pixels via adding background (0s) pixels.
Boxes of various size R were overlaid on each image and the boxes that included object pixels (1s) were counted, so the number N of such boxes, at each size R, was determined. The slope obtained from the linear regression of the resulting plot of log N versus log (1/R) corresponded to the fractal dimension value for each structure. The analysis of the fractal dimension of the lattices has been carried out for the three different levels and varying aspect ratios between 3 and 6. The S spacing has been varied between 0.2 and 0.8. 3 is however small, on average less than 1 %. It is worth noticing that for high aspect ratios and spacing values the three hierarchical levels tend to provide quite similar fractal dimensions (between 1.73 and 1.70). The closeness of the fractal orders associated to the second and third hierarchy levels suggests a potential similarity in their in-plane mechanical properties, and a difference with the ones belonging to Level 1. As it will be evident in the next paragraph, hierarchy levels higher than 1 do actually exhibit a set of in-plane stiffness and Poisson's ratios values that difference them from the original baseline rectangular perforation.

in-plane mechanics
The in-plane mechanical properties have been computed using a Finite Element approach applied to a quarter unit cell. The FE models have been   The results have been initially benchmarked against experimental data on the in-plane rectangular perforations described in [34], and we obtained discrepancies between 1% and 2 % with the in-plane Young's moduli and Poisson's ratios of perforated plates. All the numerical results have been nondimensionalised against the Young's modulus E c of the solid material and the relative density ρ/ρ c (i.e., the volume fraction described above).

Wave propagation
Understanding the behavior of the perforated composite metamaterial in an extended frequency range requires a finer description of the geometry, and this is particularly important for the particular hierarchical metamaterial configuration evaluated in this work. The wave propagation analysis related to a fully detailed geometry model has been carried out by applying the Floquet-Bloch method [43]. To this end a plain stress Finite Element mesh has been created using the COMSOL platform. The models for all levels were composed by triangular elements (quadratic interpolation), with varying mesh size based on the level of the topology. As an example, for a configuration of AR = 4 and S = 0.2 a Level 1 model has 506 elements that were increasing to 1770 and 7944 when passing to Levels 2 and 3 respectively.
According to Floquet-Bloch theorem, the boundary conditions applied along the 1 and 2 directions at the cell edges can be represented as u R = e −jkxr u L and v R = e −jk 2 r v L , where u R (resp. v R ) is the displacement on the right edge and u L (resp. v L ) is the displacement on the left edge in x (resp. y) directions, k x and k y are respectively the wavenumbers in the x and y directions.
The equations of motion of the system assuming an harmonic solution can be described as: where u ∃ R 2 and σ, C and are the stress, second order elastic and strain tensors respectively. The associated undamped eigenvalue problem of Equation (2)

Conclusion
We have presented a hierarchical configuration of a rectangular perforation pattern that allows to create auxetic deformation behaviors in a 2D material composite planar structure. The use of the hierarchical configuration proposed allows to generate a negative Poisson's ratio and controlled orthotropy from a baseline perforated unit that not necessarily is auxetic, but it is always isotropic. The hierarchical construction leads to fractal dimensions of the lattices that tend to vary noticeably between the first and the second level, and with lower differences between the second and third