Hierarchical auxetic and isotropic porous medium with extremely negative Poisson’s ratio

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Abstract

We propose a novel two-dimensional hierarchical auxetic structure consisting of a porous medium in which a homogeneous matrix includes a rank-two set of cuts characterised by different scales. The six-fold symmetry of the perforations makes the medium isotropic in the plane. Remarkably, the mesoscale interaction between the first- and second-level cuts enables the attainment of a value of the Poisson’s ratio close to the minimum reachable limit of -1. The effective properties of the hierarchical auxetic structure are determined numerically, considering both a unit cell with periodic boundary conditions and a finite structure containing a large number of repeating cells. Further, results of the numerical study are validated experimentally on a polymeric specimen with appropriately arranged rank-two cuts, tested under uniaxial tension. We envisage that the proposed hierarchical design can be useful in numerous engineering applications exploiting an extreme auxetic effect.

Introduction

Auxetic materials are characterised by the unconventional property of possessing an effective negative Poisson’s ratio, so that they expand (contract) transversally when stretched (compressed) longitudinally.

Named “auxetic” after Evans [1], these media owe their special behaviour mainly to their microstructure rather than to their chemical composition. Hence, auxeticity has been observed at different scales, from macro- to nano-dimensions. Apart from examples of natural materials with an intrinsic negative Poisson’s coefficient [2], [3], [4], [5], auxetic media are generally artificially-made systems, whose microstructure is designed by exploiting different geometries and mechanisms: re-entrant unit cells [6], [7], [8], [9], star-shaped inclusions [10], [11], chiral configurations [12], [13], [14], [15], double arrowhead honeycombs [16], perforations and cuttings [17], [18], [19], [20], [21], rotating rigid units [22], lattices [23], [24], [25] and elastic instabilities [26], [27]. Alternative approaches to design auxetic media are presented in the reviews [28], [29], [30], [31], [32]. Some auxetic systems are also characterised by a negative value of the coefficient of thermal expansion, implying that they shrink when subjected to an increase in temperature [33], [34], [35]. Additionally, it has been shown that it is also possible to achieve a smooth transition through a wide range of negative and positive Poisson’s ratios by using an origami cell that morphs continuously between a Miura mode and an eggbox mode [36], [37], [38].

The increasing interest of the scientific community in auxetic metamaterials is due to their enhanced mechanical properties with respect to those of conventional materials, including higher indentation resistance [39] and impact energy absorption abilities [40], [41], improved fatigue performance [42] and pull-out strength [43], as well as the possibility to bend with synclastic curvature [44]. For these reasons, auxetic metamaterials have tremendous potential in many fields, particularly in the aviation industry, e.g. for aircraft design [43], [45], [46], in sports applications for enhanced comfort and protection [47], in electronics to increase electric power output [48] and in biomedical engineering for the design of novel types of stents [49], [50], [51] and orthopedic implants [52]. On the other hand, zero Poisson’s ratio materials have been shown to be useful in other applications, for instance, in the design of morphing aircraft [53], [54], [55].

Hierarchical structures are widely exploited in natural [56] and in bioinspired artificial [57] materials to enhance mechanical properties. The simultaneous presence of multiple length scales, together with material heterogeneity, has been shown to allow the simultaneous optimisation​ of strength and toughness [58], [59], but also to enable the improvement of other mechanical characteristics such as adhesion [60], [61], friction [62], or to achieve band gap engineering in dynamics [63], [64].

In this paper, we investigate the auxetic behaviour of a porous hierarchical medium, consisting of a homogeneous elastic material containing two classes of perforations, characterised by two different length scales and exhibiting a six-fold symmetry, which makes the medium isotropic [65], [66]. First, by numerically studying the periodic elementary cell, we demonstrate that, for some ratios between the lengths of the two classes of cuts, the effective Poisson’s ratio approaches the limit of –1. Second, we verify the results of the periodic analysis by determining the response of a finite model, containing a large number of periodic cells, under uniaxial loading. Subsequently, we validate the numerical results by experimentally testing a specimen with hierarchical cuts in uniaxial tension. Finally, we provide some analysis and concluding remarks.

Section snippets

Numerical model

We introduce a 2-D hexagonal periodic unit cell with oriented cuts, as shown in Fig. 1a. The cuts are all rotated by the same relative angle θ (θ-π/6 with respect to the intersected edge of the hexagonal unit cell of size l). Each cut is of length a and width b, which is also the diameter of the rounded extremities (in consideration of a practical realisation). The proposed design leads to a periodic pattern with either six-fold or three-fold symmetry, similar to previously considered designs

Experimental testing and validation

Experiments were performed to validate the extreme auxetic behaviour of the described hierarchical plate. A thin 280.2 × 599.8 × 2 mm3 PolyCarbonate (PC) specimen with the designed hierarchical distribution of traversing cuts was produced by a Roland EGX-600 engraving machine (accuracy 0.01 mm). The chosen geometry corresponds to a hierarchical ratio s = 1/4. The length of the rank-two cuts was reduced by 13% with respect to the geometry considered in Fig. 3, Fig. 4, to avoid exceedingly thin

Conclusions

In conclusion, we have designed and tested a simple 2-D structure consisting of a hierarchical arrangement of rotated thin cuts, which displays extreme auxetic properties along each of its 6-fold symmetry directions, providing an isotropic architecture. The novelty of this structure is that it effectively expands uniformly in the plane when loaded uniaxially in any direction.

The design procedure starts from a previously considered single scale (non-hierarchical) design, adding smaller

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

MM, FB, ASG, DM and NMP are supported by the European Commission under the H2020 FET Open (“Boheme”) grant No. 863179. MB acknowledges the financial support of Regione Autonoma della Sardegna, Italy, project ADVANCING.

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