Elastic properties of 2D auxetic honeycomb structures-a review

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Introduction
Auxetics are defined as materials or structures with the elastic property of negative Poisson's ratio (NPR): when the material is stretched in one direction, it expands in one or more transverse directions as well.This differs from most materials / structures which, when stretched in one direction, typically react to the resulting increase of size in that direction by contracting transversally.Hence the term auxetic, first suggested by Evans et al. in 1991 [1], from the Greek word 'auxetikos' meaning 'what tends to increase'.This behaviour emerges at the level of the internal structure of the material or stems from the structural micro / macro architecture.In the case of auxetic 'materials', the phenomenon typically depends on the molecular patterns that, when subjected to a negative stress, can occupy the intrinsic free volume between molecules to contract also in the lateral direction, by allowing a substantial increase in the material density [1,2].There are several studies focused on structures resulting into auxetic behaviour on micro and molecular level, like the rigid 'free' molecules described by Wojcikowski [3], micropourous auxetic form of polytetrafluoroethylene [4] expanding to micropourous polymers whose auxetic behaviour stems from the nodul and fibruli mechanism [5], molecular rods [6], liquid crystalline polymers [7], and α-cristobalite [8,9]; with new auxetic materials continuously being identidfied [10].Similar mechanisms can be exploited configuring structures at micro scale and macro scale levels, like in the case of polymeric foams [11], or mainly macro scale structures like sandwich panels or stents [12,13], and can be created from non-auxetic materials.However, it is important to note that the constituting material still has an important role on the global mechanical behaviour [14,15].For many auxetic structures, the auxetic behaviour derives from the geometry and deformation mechanisms of an auxetic unit cell, and is scaled up by repetition cells [16].Cellular structures can be divided into two categories -2D structures, usually referred to as honeycombs, and 3D structures, referred to as foams [16,17].
Auxetic materials and structures have received strong attention, as they exhibit properties such as higher indentation resistance, thermal impact resistance, higher shear moduli as well as higher fracture toughness [1,18,19] that can be leveraged in a number applications.Their acoustic properties are very interesting as the Poisson's ratio influences the speed of wave propagation in materials [20].Another advantage of auxetic materials is the ability to assume a synclastic or dome-shaped curvature out-of-plane − conversely to regular materials which assume an anti-clastic or saddle shaped curvature [12,18] − a useful feature when manufacturing doubly curved sandwich panels, as explained by Evans [12].Moreover, auxetic cellular macrostructures maintain the same advantage as ordinary cellular material, with a lower density [18].This literature review will focus on the mechanical properties of 2D auxetic cellular structures, which find direct application in several sectors.In aeronautical and naval architecture, the use of sandwich panels with auxetic honeycomb cores can provide significantly increased flexural rigidity and buckling load, enhance the vibroacoustic design and ease the manufacturing of synclastic skin parts [12,21].In biomedical engineering, stents based on 2D auxetic patterns wrapped in a tubular shape are reported to enhance luminal patency and reduce migration.Moreover, these arrangements also allow to obtain structures with high circumferential strength when expanded, and low flexural rigidity when delivered into the patient in a crimped configuration [13,[22][23][24][25].The ability of these tubular structures to reduce kinking during bending has also been exploited to restore the morphology and dynamics of the mitral valve in annuloplasty rings devices [13,24].The use of auxetic 2D structures has also been explored for the fabrication of scaffolds in tissue engineering applications, due to their closer behaviour to several biological tissues, which appear to exhibit NPR [26].The features of 2D auxetics are also employed in textiles and components for sport and fashion applications, where their ability to conform to the user's body shape provides better comfort and new aesthetic opportunities, as well as the potential advantage to reduce the number of required sizes to fit the different customers [27][28][29].Another common general application is in the area of filters, where the 2D auxetic structure allows for control of the size of the pores by mechanical action [30].
In this review, auxetic honeycombs will be classified into three main groups, depending on the main mechanism controlling deformation: -Re-entrant honeycomb structures; -Chiral structures; and -Rotating plate structures.
For each group, the most typical configurations and their common variations will be reviewed.
Focus will be on the mechanical properties of the structures, without details of other properties associated with auxetic materials [31].Together with the Poisson's ratio, the nominal Young's modulus and shear modulus will be examined, and a unified and consistent formulation will be provided for all the different structures examined in this review, thus allowing direct comparison.
In particular, to allow for direct comparison, the equations are rewritten adopting the same nomenclature and dimensional parameters between the different models, namely the horizontal and vertical cell dimensions (B and H respectively) or, in case of hexagonal unit cells like the hexachiral the unit inscribed diameter D, the cell wall thickness t, and the depth of the cell structure d.The horizontal in-plane direction of orthotropy is indicated as direction 1, the vertical as direction 2 , and the out-of-plane direction as direction 3. The derived equations estimate, for the two in-plane directions of orthotropy, the Poisson's ratios ν 12 and ν 21 , the elastic moduli E 1 and E 2 and, where available, the shear modulus G 12 , with the subscripts defined as in the usual notation adopted for orthotropic materials.
A comparison of the responses predicted from the different equations summarised for the most investigated cases are represented graphically.These are derived assuming a unit cell of dimensions B × H = 1 × 1 unit lengths or D = 1 unit length, with a uniform wall thickness t = 0.05 unit lengths, wall depth d = 0.5 unit lengths and Poisson's ratio of the wall material ν s = 0.3.
By narrowing the scope of this review to 2D auxetic structures and their mechanical properties, we hope not only to provide an in-depth overview of the different methods that have been reported in the literature to evaluate the auxetic behaviours, but also facilitate comparison between different structures.The standardised nomenclature can guide the choice on the most suitable auxetic structure for each different application, according to the required elastic behaviour, range of Poisson's ratio and sensitivity to realistic manufacturing challenges.
The described structures can be easily expanded to a large range of different applications where a change of the micro / macro structure is the basis for the auxetic behaviour as well as the basis for many of the 3D adaptations.In fact, 2D honeycombs provide a simplified model that can directly be extended to 3D cellular structures.

Re-entrant structures
One of the earliest re-entrant configurations identified for its auxetic behaviour and intentionally exploited in a structural design by Gibson et al. in 1982 [17] is the re-entrant hexagonal honeycomb structure.This is also one of the first auxetic configurations where a 3D adaptation of the 2D structure was first attempted in the configuration proposed by Almgren in 1985, that offered a Poisson's ratio of -1 in all three dimensions [32].Later, other re-entrant systems were identified and introduced, such as the double arrowhead shape [33] and the STAR-systems [11,34].
The auxetic behaviour of re-entrant structures mostly depends on the angles between the ribs defining the cells, which change with the cells deformation.Thus, the Poisson's ratio generally varies non-linearly with the nominal strain [20].Grima et al. investigated different ways to achieve linear negative compressibility through constrained angle stretching, instead of the more commonly studied modes of deformation like flexure and hinging [35].

Table 1
Equations derived by the different research groups for the determination of the Poisson's ratios, elastic moduli and, where available, shear modulus, in the two in-plane directions of orthotropy of hexagonal honeycombs.[16,36,17] Poisson's ratios:
Notes: Homogenisation model of simple formulation, providing similar results to previous models.

Inverted or re-entrant hexagonal honeycomb
The most extensively investigated auxetic re-entrant configurations is the inverted hexagonal honeycomb, represented in Fig. 1.
The structure exhibits orthotropic behaviour, and both the in-plane and out-of-plane properties have been widely studied using different analytical approaches, as well as numerically and experimentally.The first systematic study on this class of materials was published by Gibson et al. [17] in 1982.This analysed the mechanical response of two-dimensional cellular materials, including re-entrant hexagonal honeycombs, and proposing basic equations for the prediction of their behaviour, validated by experimental models.Later, Gibson and Ashby dedicated a chapter in their book 'Cellular Solids' [16] to the analytical study of honeycombs, particularly the hexagonal honeycomb structure, including the auxetic variation.This work investigated both the elastic in-plane and out-of-plane deformations of honeycomb structures, as well as their failure mechanisms.The approach proposed by Gibson and Ashby identifies an initial response of honeycomb structures to in-plane compression led by bending of the cell walls.This is analysed by applying the conventional beam theory to determine expressions of the nominal in-plane Poisson's ratios, Young's moduli and shear modulus (see Table 1).Hence, the mechanical response is assumed linear elastic up to the cell collapse, which, depending on the dimensions and material of the structure, occurs by elastic failure (i.e., buckling of compressed walls), ductile failure (i.e., plastic collapse due to plastic hinge formation) or brittle failure (fracture).Compressive collapse is associated with a plateau in the nominal stress level, eventually lost when the material densification caused by the phenomenon results in a substantial increase in the stiffness of the structure (see Fig. 2).
The expression of the shear modulus from the model proposed by Gibson and Ashby appears to underestimate the value observed experimentally [16,40].Moreover, it becomes inaccurate for configurations characterised by slanted walls about orthogonal to the longitudinal walls, for which unacceptable values of the nominal Poisson's ratios and Young's moduli tending to infinite may be obtained.This problem was addressed by an expanded version of the model proposed by Masters and Evans [37] that includes the description of other modes of elastic deformation of the honeycomb beyond bending, such as hinging and stretching.The latter takes over as principal mode of deformation when the angle between the longitudinal and slanted walls tends to 90 • .Despite being accurate for modelling honeycombs with a negligible thickness compared to the wall lengths, these models do not take into account the effect of the shape of the ligaments and the mode of connection on the deformation, which becomes significant as the ratio of the wall thickness to the wall in-plane lengths increases.In order to address this limitation, Grima et al. [38] have recently proposed a set of adjusted equations using the results from numerical analyses.A number of other studies include further details in the modelling of the connections between the walls.However, they are limited to periodic hexagonal honeycomb structures, and do not investigate their reliability in the case of re-entrant auxetic configurations.These include the work from Balawi and Abot [41] that takes into account the curved intersections of the honeycomb walls commonly present in commercially manufactured honeycombs [42] (although Gibson and Ashby had already introduced a model with double wall thickness in the vertical direction [16], this neglected the deformation in the vertical walls).Also, compared to the description from Masters and Evans, this model introduces the presence of curvature in the cell walls beyond the hinging region.This allows a better description of the reduction in in-plane nominal moduli observed numerically as the radius of curvature at the wall joints increases.Another study worth mentioning, although validated only for regular hexagonal honeycombs, was recently published by Malek and Gibson [40].This introduces a beam model taking into account the effective bending length of the cell walls, which provides better alignment with both numerical and experimental results found in earlier works by Gibson at high relative densities [16,43].
The studies described above model the mechanical response of honeycombs by applying the beam theory, but other approaches have been used as well, based on homogenisation to analyse the structure as an equivalent continuum, or on energy methods [36].
Gonella and Ruzzene [36] derived equations equivalent to those determined by Gibson and Ashby [17], but obtained through partial differential equations associated with the homogenised continuum models of the hexagonal and re-entrant hexagonal honeycomb lattices.Their model was also developed further to study analytically the wave propagation in the structure.
Berinskii [39] also used homogenisation to derive analytically the Poisson's ratios and elastic constants for the re-entrant honeycomb structure.This model takes into account the elastic deformation of the ribs, including deformation from flexure, stretching and shearing.Although the derived equations are relatively simple, the model leads to a behaviour very similar to that predicted by Masters and Evans [37].Beriskii also established a framework to extend the same approach to the estimation of the elastic constants for a few different auxetic structures, although he did not provide in-depth validation in the article.
An alternative approach to model the mechanical response of honeycombs, based on empirical models using dummy atoms (EMUDA), was proposed by Grima et al. [44].This was specifically applied to re-entrant hexagonal honeycomb structures due to the broad availability of data suitable for validation.
The equations derived by the various research groups to describe the in-plane mechanical behaviour of hexagonal honeycombs are summarised in Table 1.Using the standardised nomenclature defined above, adding the angle ϑ between vertical and inclined struts.The plane of the common wall of the neighbouring cells was aligned vertically and defined as direction 2 of orthotropy.The horizontal in-plane direction of orthotropy was indicated as direction 1.
Diagrams are represented in Fig. 3 for a range of the angle ϑ between vertical and inclined struts changing from 45 ∘ to 135 ∘ , thus covering the case of conventional (ϑ ≥ 90 ∘ ) and auxetic honeycombs (ϑ < 90 ∘ ).The diagrams confirm that all models provide close predictions of the Poisson's ratio ν 12 for angles between vertical and inclined struts far from 90 ∘ When ϑ approaches this value, the model proposed by Gibson and Ashby [16] leads to inacceptable values of ν 12 (indeterminate for configurations characterised by slanted walls about orthogonal to the longitudinal walls, for which values of the nominal Poisson's ratios and Young's moduli tending to infinite are determined).
Although the studies described above assume, for simplicity, uniformity of the walls defining the hexagonal honeycomb, different models have been developed to deal with variations, such as the use of ribs with different mechanical properties [35].Also, a number of

Fig. 3. Comparison of the in-plane
Poisson's ratios, Young's moduli and shear modulus determined with the different models proposed for the analysis of hexagonal honeycombs (see Table 1) in conventional and auxetic configurations.re-entrant hexagonal honeycomb structures based on rounded walls has been investigated [45].
A main limitation of the described analytical models is their inability to correctly predict the non-linear behaviour associated with potentially large deformations.An analytical attempt to address this issue was proposed by Wan et al. [46].Another important aspect to consider when adopting these models to predict the behaviour of real cases, or when validating them versus experimental data and numerical simulations, is the fact that they do not account for the effect of the size of the specimen nor, in most cases, of its depth [47].In reality, due to the presence of stress-free cut cell edges at borders of the surface of a specimen, as well as constrains applied at the boundaries, the behaviour of physical specimens is expected to depart from that described by analytical models, that are derived for single cells or infinite size specimens.As the size of the sample increases, a plateau is typically observed, and the properties of the specimen converge to those of an infinite specimen.In

Table 2
Equations of the in-plane Poisson's ratios and elastic moduli for double arrowhead structures.Berinskii [39] Poisson's ratios: Young's moduli: )) Notes: Homogenisation model of simple formulation, providing similar results to previous models.
the case where numerical models are used for validation, implementing periodic boundary conditions has been suggested as a possible approach to model the properties of infinite plates [48].
The out-of-plane properties have been generally studied less than the in-plane properties for the re-entrant hexagonal honeycomb.Main studies include the work from Gibson and Ashby [16] and Zhang and Ashby [49], that provide some description of the out-of-plane behaviour of the honeycomb lattices and the associated failure mechanisms, essentially consisting in linear buckling and fracture [49].Smith et al. [31] and Scarpa et al. [50] report increased out-of-plane elastic and shear moduli and larger collapse stresses for auxetic configurations, when compared with analogous hexagonal honeycombs with the same relative density.

Double arrowhead
Another common auxetic structure based on the unfolding of reentrant cells was first identified by Larsen et al. [33], and is represented in Fig. 4.This can be found in the literature under different names, such as double-headed arrow [27], double arrow [51] or double   arrowhead structure [39], here preferred because more commonly used.
Although the double arrowhead configurations can be adapted to a number of practical applications, such as knitted fabrics [28], their mechanical properties have been investigated far less than the re-entrant hexagonal honeycomb structure.The analytical continuum model proposed by Berinskii [39], derived as a part of a generalised study for comparison with other auxetic structures, provides one of the most complete analytical predictions of the in-plane mechanical response of double arrowhead structures.As described in the previous section, this model uses homogenisation and takes into account the elastic deformation of the ribs, including deformation from flexure, stretching and shearing.The resulting equations are summarised in Table 2, using the standardised nomenclature defined above, adding the angles ϑ and φ between the direction 1 of orthotropy and the short and long walls, respectively.Direction 1 of orthotropy is defined as the in-plane direction orthogonal to the in-plane axis of symmetry of the cells, which is the direction 2 of orthotropy.Diagrams are represented in Fig. 5 for the possible range of the angle ϑ between the direction of orthotropy 1 and the short walls.The angle φ between the direction 1 of orthotropy and the long walls is univocally defined from the previous parameters, as φ = arctan(2H /B + tanϑ).
The diagrams indicate a highly different mechanical response compared to the re-entrant hexagonal honeycombs, which can result in much larger values of the negative Poisson's ratio ν 21 , increasing the auxetic behaviour of the structure.

Other re-entrant shapes 2.3.1. Star-shapes
Star configurations were identified as potentially auxetic by Theocaris et al. in 1997 [34], and more recently described and expanded in terms of configurations by Grima et al. [11].As in the inverted honeycomb and double arrowhead structures, their main principle of deformation consists in the unfolding of re-entrant cells.Grima et al. [11] applied empirical modelling using dummy atoms (EMUDA) to investigate the mechanical response of STAR-3, STAR-4 and STAR-6 configurations (these are described in Fig. 6).Their analysis indicates that the STAR-3 configuration displays auxetic behaviour only for a few combinations of hinging force constants, while the STAR-4 and STAR-6 configurations are auxetic for most of them.STAR-4 configurations are generally characterised by higher negative Poisson's ratios than STAR-6 (for corresponding values of the hinging force constants) [11], but still present less auxetic behaviour then re-entrant hexagonal honeycombs.The STAR-4 configuration was also studied by Theocaris et al., using homogenisation, examining it as a beam structure as well as star shaped inclusions in a continuum [34].The use of star-shaped pores to achieve auxetic behaviour was further investigated by Mizzi et al. [52].
Other variations of the connected star configurations were analytically analysed by Ai and Gao [53] using Castigliano's theorem.
Another auxetic structure investigated in the literature and based on the unfolding deformation mechanism inspired by the double arrowhead configuration, is the Milton lattice [54] represented in Fig. 7.This was developed to explain the mechanism of a laminate showing a negative Poisson's ratio.

Chiral structures
A different approach to achieve structural auxetic behaviour consists in enforcing the wrapping and unwrapping of ligaments around specific nodes.Structures exploiting this principle are called chiral structures and are classified based on the number of ligaments (e.g., trichiral, tetrachiral, hexachiral, etc. as shown in Fig. 8) and on the way each ligament wraps around the nodes at its ends.In particular, if ligaments are fastened on the same side of both nodes at their ends, thus forcing these to rotate in opposite orientations during the deformation, antichiral configurations are obtained, as shown in Fig. 9 (denoted by the prefix 'anti-').
The use of chiral configurations as potential auxetic structures was   first suggested by Lakes in 1991 [55,56], who described a hexachiral configuration.These structures exhibit hexagonal symmetry, which results in mechanical in-plane isotropy [56], and are reported to offer high shear rigidity and a deformation mechanism which allows high strains in the elastic range of the wall material [57].Numerical and experimental investigations of the hexachiral configuration suggested by Lakes, as well as of tetrachiral, anti-tetra, trichiral and anti-trichiral configurations presented by Alderson [58] indicate that the in-plane deformation of chiral and anti-chiral structures is predominantly led by cylinder rotation and ligament bending.The Poisson's ratio is close to -1 for the analysed hexachiral, tetrachiral and anti-tetrachiral configurations, and does not appear to be affected by the ligament length.However, the auxetic behaviour decreases with the ligament thickness, especially for the hexachiral model.On the contrary, the anti-trichiral model displays negative Poisson's ratios only for shorter ligaments, and the trichiral model does not show auxetic behaviour.This was explained by the presence of competing deformation mechanisms, between the bending of the ligaments due to the rotation of the nodes and the direct bending of off-axis ligaments.
Hexachiral and tetrachiral models exhibit higher Young's moduli than the trichiral models, and chiral models has a higher in-plane compressive modulus than the anti-chiral equivalents, for any given number of ligaments.This was realistically attributed to the fact that ligaments of chiral structures bend producing two buckles, with a change of curvature at their midspan, while in anti-chiral structures they deform in a single buckle.This appears clearer when comparing the deformed configurations in Figs. 8 and 9. To correctly interpret the results described by Alderson et al. [58] we need to consider the fact that they applied 1-2 % compressive strain to their specimen, which also had much thicker walls than those previously tested by Prall and Lakes.
The majority of the analytical studies of the elastic constants for chiral structures focuses on hexachiral configurations.The first attempt to describe their in-plane elastic properties was presented by Prall and Lakes, who applied the beam theory and an energy approach to predict a constant in-plane Poisson's ratio of -1 [56].This approach, similarly to Gibson and Ashby's analysis on the re-entrant hexagonal honeycomb, assumes that deformations are small, the main deformation mode is bending of the cell walls, and axial deformation and shear within the ligaments can be neglected.The bending of the cell walls and the equal rotation of all cell nodes were confirmed and validated with experimental tests, although the node wall thickness in the physical specimens was larger than the ligament thickness, contributing to the validity of a rigid node model.Tests also confirmed that the structures maintained the predicted Poisson's ratio of about -1 up to 25% of nominal strain [56].
This analytical approach is reported to be only valid for slender

Table 3
Equations derived by the different research groups for the determination of the in-plane Poisson's ratio, elastic modulus and, where available, shear modulus, for hexachiral honeycombs.
Shear Modulus: Notes: Micropolar continuum model, removing the indeterminant of ν = − 1, with the assumption of ridgid rings

Liu et al. [61]
Poisson's ratios: Young's moduli: Shear Modulus: Not calculated Notes: Based on micropolar continuum method.Verified by comparison to the exact solution of the corresponding discrete models.
Bacigalupo & Gambarotta [62] Poisson's ratios: Young's moduli: Shear Modulus: Notes: Based on micropolar continuum method introducing the concept of effective beam length for the ligaments.beams with a reference value of the relative density of the structure below 0.29 [16,56].Moreover, the predicted Poisson's ratio, which is constant for any change in geometric parameters and equal to -1, results in a structure with a theoretically infinite shear modulus [56,59,60].In an attempt to remove this indeterminacy, Spadoni and Ruzzene [59] used micropolar continuum methods with both rigid and deformable nodes to calculate the elastic constants of a hexachiral auxetic structure.The rigid node model leads to a Poisson's ratio dependant on the ligament thickness, and equal to -1 only for the ideal and unattainable condition that the ligament thickness is zero.The deformable node analysis relies on numerical simulations by means of finite element analysis.Comparison of the rigid node analytical model with the deformable node numerical model shows substantial discrepancies in the estimated Youngs's modulus.Importantly, with the analytical approach suggested by Spadoni and Ruzzene, the estimated shear modulus, indeterminate (infinite) in Prall and Lakes model, becomes finite and determinable.Their analysis indicates that the shear modulus is equivalent to that of a lattice consisting of regular triangles, and much lower if the contribution of deformable rings is considered.Liu et al.
[61] applied a similar approach, still based on the micropolar theory through a continuum theory model with reinterpretation of in-plane isotropic tensors.Their results were matching the exact solution of corresponding discrete models.
A more recent study presented by Bacigalupo and Gambarotta [62] applies a micropolar homogenisation derived from Spadoni and Ruzzene [59] and Liu et al. [61], while introducing an additional parameter defining the deformable portion of the ligaments.In the same work, they propose an alternative approach based on a second gradient Fig. 10.Comparison of the in-plane Poisson's ratio, Young's modulus and shear modulus determined with the different models proposed for the analysis of hexachiral honeycombs (see Table 3).homogenisation, developed to study periodic cells consisting of deformable portions like the ligaments, nodes, and eventual filling material in between the ligaments and in the nodes.Interestingly, the study indicates that the presence of filling material between the walls of the auxetic structure (even if very soft) strongly reduces and eventually reverts the auxetic behaviour [62].
The equations derived by the various research groups to describe the in-plane mechanical behaviour of hexachiral honeycombs are summarised in Table 3.The standardised nomenclature presented above is used, with the addition of the angle ϑ between radial direction and the inclined struts.
The responses predicted from the different equations summarised in Table 3 are represented in Fig. 10 according to the standardised unit diameters presented above.Diagrams are represented for the theoretical range of the angle ϑ between 0 ∘ (corresponding to the case where the circles at the nodes degenerate into points and the structure, therefore, becomes made of triangles) and 90 ∘ (where the circles at the nodes occupy the entire cell diameter and the structure, therefore, becomes made of circles).The diagrams confirm that the Poisson's ratio of -1 predicted by Prall and Lakes is a theoretical limit practically unattainable, and approached for an angle ϑ = 45 ∘ .In fact, the auxetic behaviour reduces at smaller and larger angles, reverting at angles close to 0 ∘ and 90 ∘ .All three models based on micropolar homogenisation predict a similar behaviour, with the descriptions of Spadoni and Ruzzene [59] Table 4 Equations derived by Mousanezhad et al. [63] for the determination of the in-plane Poisson's ratio, elastic modulus and, where available, shear modulus, for tetrachiral, anti-tetrachiral, trichiral and anti-trichiral honeycombs.As above, the material properties of the cell walls are denoted by ν s (Poisson's ratio), E s (Young's modulus) and, G s (shear modulus).
Tetrachiral [63] Poisson's ratios: Young's moduli: Shear modulus: Mousanezhad et al. [63] the Poisson's ratio is indicated as equal to zero, but it is here assumed that it was a typographical error.[63] Poisson's ratios:
and Liu et al. [61] providing practically identical results for both Poisson's ratio and Young's modulus.All models fail to give realistic estimates of the structure moduli at large values of ϑ.
Other chiral configurations have received less attention in terms of analytical description.Recently, Mousanezhad et al. [63] applied energy-methods based on Castigliano's second theorem to derive analytical expressions for the in-plane mechanical response of tetrachiral, anti-tetrachiral, trichiral and anti-trichiral configurations.Since the tetrachiral and anti-tetrachiral systems are stretching dominated, both the stretching and the bending terms of Castigliano's theorem were included.Instead, in the case of trichiral and anti-trichiral, structures were assumed to deform in a bending dominated manner and their analysis only included the bending terms.
Comparison of the analytical predictions with numerical solutions revealed a number of discrepancies.In particular, the tetrachiral and trichiral configurations did not exhibit a negative Poisson's ratio in the computational models.This behaviour, in the case of the tetrachiral model, is in contrast with that described by Alderson et al. [58], and is attributed by Mousanezhad et al. to the different boundary conditions imposed in the two analyses.In the case of the trichiral configuration, the different behaviour between the analytical and numerical predictions is attributed by the authors to the fact that, in the numerical case, the main form of deformation shifted from the ligaments to the nodes, assumed rigid in the equations.This also justifies the Young's modulus estimated analytically.
The equations derived by Mousanezhad et al. for the different configurations are summarised in Table 4, using the notations reported in the figures represented in the table.The responses predicted for the different configurations in Table 4 are represented in Fig. 11.All geometric parameters and the Poisson's ratio of the wall material are kept consistent with previous studies.Diagrams are represented for ratios between the radius of the node and the dimension B of the unit cell ranging from zero (ligaments only) to 1 (node circles only).The diagrams confirm that the trichiral and anti-trichiral structures can exhibit auxetic behaviour only for a limited range of geometrical configurations.
As described above, one of the main features of chiral structures is their ability to offer the same mechanical characteristics in the different in-plane directions.However, chiral configurations can be readapted to provide orthotropic responses.In particular, the anti-tetrachiral structure is the most suitable for this application, and a range of variations on this structure have been suggested and analysed, that differentiate the responses in the two directions of orthotropy by allowing for different length of the ligaments (Chen et al., 2013) [55]; for different length and thickness of the ligaments (Gatt et al., 2013) [56]; for different length of the ligaments and elliptical nodes (Wu et al., 2017) [64].Also, hybrid tetrachiral -anti-tetrachiral structures with rectangular nodes have been suggested and analysed (Grima et al., 2008) [65].Recently, attempts have been made to adapt similar approaches to hexachiral structures, by introducing irregularities [66].
The anti-tetrachiral orthotropic configurations described above and the analytical expressions of the derived mechanical properties are summarised in Table 5.
The out-of-plane behaviour of chiral structures has been analysed for hexachiral honeycombs by Spadoni et al. [57], who studied the problem with analytical approaches based on linear buckling for thin plates, as well as shells and bifurcation numerical simulations by means of finite element analysis.This study identifies the geometric parameters defining the structure that can be altered to increase the flat-wise buckling response, such as the diameter to length ratio for the cylinder nodes, or the wall thickness, which increases the global and local buckling loads.On the contrary, the ratio between the ligaments length and the cylinder nodes diameter (or the angle between the radial direction and the ligaments), defining the level of chirality, decreases the   buckling load even when normalised for relative density.Scarpa et al. [69] studied the same configuration by means of finite element analyses and experimental tests.Flat-wise buckling and the anelastic buckling behaviour of the hexachiral structure was further explored by Miller et al. [70], by means of numerical and experimental approaches, and expanded to tetrachiral and anti-tetrachiral structures.Lorato et al. [71] studied the out-of-plane properties of the hexachiral, tetrachiral, anti-tetrachiral, trichiral and anti-tetrachiral configurations applying analytical, numerical and experimental approaches.Both the transverse Young's modulus and the transverse shear modulus were studied: the transversal nominal Young's modulus was reported to increase when moving from the trichiral to the anti-trichiral, to the tetrachiral, to the anti-tetrachiral and, finally, to the hexachiral configurations, although some minor changes in this order were observed in experiments with thicker ligaments.
Another interesting configuration, proposed in year 2000 by Smith et al. [72] to describe the behaviour of auxetic foams, is represented by the missing rib model.This is based on a lozenge grid with missing rib

Table 5
Schematic representation of different orthotropic chiral configurations and equations derived by different groups for the determination of the Poisson's ratios and, where available, elastic moduli for the two in-plane directions of orthotropy.As above, the material properties of the cell walls are denoted by ν s (Poisson's ratio), E s (Young's modulus) and, G s (shear modulus).
portions, which results in a configuration which shares the features of both tetrachiral and re-entrant structures (see Fig. 12.a).In fact, the deformation of missing rib models is associated with rotations of the hub of crossed-ligaments, as well as hinging at the ligaments joints.As for the chiral configurations previously described, anti-chiral missing rib arrangements can also be designed (see Fig. 12.b), which would still retain an auxetic behaviour [73].Analytical models of this type of structures are still highly simplified, and mostly referred to chiral missing rib configurations.The equations derived by Smith et al. [72] to describe the in-plane mechanical behaviour of these structures are summarised in Table 6.The reference dimensions used in the equations are described in the figure in the table, and the joints with angle ϑ are spring hinges, with spring constant equal to k.

Rotating plates
At the beginning of the new millennium, various research groups identified auxetic behaviour in the rotational degree of freedom of plates or crystals, interconnected through hinges at their vertices in such a way that, when they are compressed / expanded in one direction, they rotate the plates so that they compress / expand in the other direction as well, thus producing a negative Poisson's ratio.An example of this mechanism is described in Fig. 13.
Grima and Evans investigated the auxetic behaviour of rigid rotating squares and triangular plates showing that, in idealised settings with rigid components, their Poisson's ratio is equal to -1 [74].They later expanded their investigation into rotating triangles [75], rectangles with different connectivity [76,77], rhombi [78,79], parallelograms [79] and non-regular plates.Since these rigid models are highly idealised structures and typically overestimate the auxeticity of the systems [80], semi-rigid and stretching connected plates were also investigated [81].The same main deformation mechanism was shown to be applicable to crystal structures [82,83] and justify the natural negative Poison's observed in materials such as some silicates like α-cristobalite [8,84] and zeolites like natrolite [85].

Table 6
Equations derived for the determination of the in-plane Poisson's ratio and elastic modulus for missing rib structures, with the spring constant k.

Rotating quadrangular plates
The idealised structure of rotating rigid squares has been found isotropic, with a Poisson's ratio of -1 [74], independently of the strain level, for any angle characterising the configuration [80].The Young's modulus is dependant on the stiffness of the hinges and on the strain, approaching infinity when the structure becomes fully closed and open [81] (given the assumption of perfectly rigid squares and hinges).Similarly, the shear modulus is constant and equal to infinite for any configuration.In reality, the Poisson's ratio would be dependant on whether hinging or deformation of the plates would be the dominant type of deformation; hence, it would be dependant on the relative   rigidity of the squares with respect to the rigidity of the hinges [76].
When considering real semi-rigid structures, the Poisson's ratio is expected to be less negative than the ideal value of -1 estimated for rigid units, with some further reduction also deriving from misalignment of deformation with the major axes of the plane.The nominal shear modulus also becomes finite, as effect of the hinges compliance and the material shear modulus, which also results in shape changes in the single plates, that depart from the perfectly square shape to become rhombohedral, rectangular or parallelogram shapes.To account for these changes, a model was introduced by Grima et al. [80] allowing the diagonals of the squares to deform independently.This, together with the expected increase in Poisson's ratio compared to theoretical ideal value of -1, shows a loss of the properties of isotropy, independency on the scale and on the nominal strain.Similarly, the Young's and shear moduli return finite (although the latter still appears to approach infinity when the structure is fully open).
When using rectangular plates, two alternative connectivity arrangements can be achieved, usually indicated as Type I, which defines two orthogonal axes of symmetry (resulting into rhombic empty shapes), and Type II, where symmetry is lost (and the empty shapes appear as parallelograms).These are described in Fig. 14.
Rectangular Type I configurations exhibit anisotropic mechanical behaviour and Poisson's ratios variable with the level of strain, controlled by the proportions of the sides of the plates.Otherwise, Type II configurations are characterised by isotropy, with a constant Poisson's ratio of − 1, similar to the rotating squares system [77].
Rotating rhombohedral plate configurations were presented as an alternative to rotating squares and as a way to generalise the model further.Similarly to rectangular plates, two different connecting arrangements can be used, usually indicated as Type α, where the obtuse angle of one rhombus is connected to the acute angle of its adjacent plate, and Type β, where the connecting angles of the rhombi at the same connection are identical.Type α is a space filling means to connect the rhombi, while Type β leaves gaps even when fully compressed.The mechanical behaviour of the two arrangements is substantially different, with Type α being highly anisotropic and exhibiting both positive and negative Poisson's ratios, depending on the shape of the rhombi and on the strain (it is dependant on the angles between the plates).Type β configurations, instead, are isotropic with a constant Poisson's ratio of -1, independently of the strain, and cannot shear [78] (Fig. 15).These concepts can be further generalised by expanding into parallelograms, which can combine the connection arrangements described for rhombohedral and rectangular plates, resulting into four potential configurations Type I α, Type II α, Type I β and Type II β, as described in Fig. 16 [79].
Type II α configuration has been studied in depth by Grima et al. [79], who found that the Poisson's ratio varies with the nominal strain level and can be either positive or negative.Interestingly, this behaviour is very different from that observed for Type I rectangular plates (that exhibits a constant negative Poisson's ratio equal to -1), of which Type II α can be regarded as a generalisation.Instead, the variation law of the Poisson's ratio with the nominal strain is actually very close to that observed in Type I rectangular plate systems and Type α rhombohedral plates.
The equations derived by Grima and collaborators [74,[76][77][78][79] to describe the in-plane mechanical behaviour of different quadrangular rigid plate configurations are summarised in Table 7.A figure describing the reference dimensions used in the equations is included on the left side of each set of equations.
For all cases, K h is the stiffness constant of the hinges.The notation adopted for the directions of orthotropy, the Poisson's ratios, the elastic moduli and, where available, the shear modulus, are the same as in previous tables.
A comparison of the responses predicted from the different equations summarised in Table 7 is represented in Fig. 17.These were derived assuming plates of area l × l = 1 × 1 square unit lengths for the square and rhombohedral configurations, a × b = 1.5 × 0.666 = 1 square unit lengths for the rectangular and parallelogram plates, and an angle φ = 60 ∘ for the rhombohedral and parallelogram plates.Diagrams are represented for a range of the angle ϑ changing from 0 to 90 • .Rigidity is described through a relative adimensional modulus, defined as E 1,2 = A K h , where A corresponds to the area of the plate.The diagrams confirm the mentioned similarity between the behaviour of Type I rectangular, Type α rhombohedral and Type II α parallelepiped plate systems.In particular, equations provide equivalent values for Type α rhombohedral and Type II α parallelepiped plate systems, while the similarity is only qualitative for the case of Type I rectangular plates.

Rotating triangular plates
The first proposed rotating triangular plate system was based on equilateral triangles which are connected and deform as described in Fig. 18.a.For an ideal structure with perfectly rigid plates and hinges, where all deformations are due to rotation of the triangles, the system is isotropic with constant negative Poisson's equal to -1.The Young's modulus approaches infinity when the structure is fully collapsed and fully expanded, and the shear modulus is infinite for any configuration [75].

Table 7 (continued )
Rotating Square Plates [74] Rotating Parallelograms [79] Type II α Poisson's ratios: Young's moduli*: Pairs of irregular triangles have also been studies for applications in rotating plate auxetic structures (see Fig. 18.b).In this case, the system highlights behaviours that are closer to those observed in honeycombs auxetic systems, which exhibit anisotropy and Poisson's ratios depending on the nominal strain, that can be negative at small strains, but return positive at larger deformations [86].
The equations derived by Grima et al. [75,86] to describe the in-plane mechanical behaviour of equilateral and irregular triangular rotating rigid plate configurations are summarised in Table 8.A figure describing the reference dimensions used in the equations is included on the left side of each set of equations.

General afternotes and conclusion of literature review
There is a large number of two-dimensional auxetic structures that can provide a range of different auxetic behaviours.Several of these structures have been investigated thoroughly in the literature, providing indications on the elastic behaviours, ranges of Poisson's ratio Fig. 17.Diagrams of the in-plane Poisson's ratios and Young's moduli determined with the equations for the analysis of rotating quadrangular rigid plates (see Table 7) in conventional and auxetic configurations.
theoretically achievable with each of the designs, as well as the sensitivity to more realistic manufacturing and usage circumstances.However, no clear comparisons have been made that provides a more direct indication on which structures would be more suited to each specific practical application.There are a few papers that mention the implications of exploiting auxetic configurations, stressing the fact that auxetic solutions require design considerations that may not be present in more conventional manufactures, like for example auxetic nails by Ren et al. [87].Although some of the problems encountered by Ren et al. in creating an auxetic might not be as prominent for cases that already use cellular structures, it is worth to remember that most of the studies published on the topic have been mainly theoretical.The unification of the nomenclature, as far as possible, and partial evaluation with standardised parameters, as provided here, may be of some guidance on

Table 8
Equations derived for the determination of the in-plane Poisson's ratio, elastic modulus shear modulus, for triangular plate configurations.The stiffness constant is denoted as K h .

Equilateral
Poisson's ratios: Young's moduli: which elastic behaviours to expect for each in comparison to others.For example, one observation is the more stable Poisson's ratio w.r.t.angle for the chiral configurations (the hexachiral has shown experimentally to retain it for deformations up to 25%) in comparison to the rapidly changing Poisson's ratio w.r.t.angle of the re-entrant structures, where the angle itself changes with deformation.Some of the structures suitable for cases where a larger NPR would be desirable would be the double arrowhead or the re-entrant hexagonal.For isotropic continuous materials, the Poisson's ratio is equal in all directions and is limited to the range -1 < ν < 0.5 (expanded to -1 < ν < 1 for two-dimensional isotropic systems [76,88]) while for anisotropic structures no such limits exists [1].Theoretically, isotropic systems like the hexachiral system and some of the rotating plates hence have a limit on maximum NPR achievable in any direction.For anisotropic systems like some of the re-entrant systems it can be observed that although they can achieve higher NPR's in one direction, this is generally linked to a lower NPR in the other in-plane direction.So, even though re-entrant configurations like the re-entrant hexagonal or double arrowhead systems can achieve very high NPR in one direction, there is a trade-off in the other.Further comparable studies with unified methodology, analytical, numerical and experimental, would help provide guidance for selecting the most appropriate structure.The configurations reported in this review only describe the most common basic structures analysed in the literature.A number of alternative 2D auxetic arrangements have been proposed in the literature.Still, most of these structures exploit the same principles and mechanisms discussed above.As an example, hybrid auxetic materials such as re-entrant and chiral arrangements (see Fig. 19a) are based on combinations of different basic configurations [89,29].Other hybrid materials exploit hierarchical organisations, such as the multi-level hierarchical rotating squares structure described in Fig. 19b, where the basic auxetic pattern is 'layered' [64,90].
Finally, the concepts described above can directly be expanded to 3D auxetic configurations.These are achieved in different ways, with the most common being compressed foams, where the cellular bubble takes the shape of an auxetic kinematic configuration similar to the re-entrant hexagonal structure [16,91,92].More complex configurations, based on three-dimensional well-defined unit cells, include different 3D versions of the cathegories presented above, such as re-entrant, chiral and ridgid block structures [32,[93][94][95][96]. Considerations made on the achievable NPR can also be extended to these configurations, with isotropic 3D arrangements characterised by a limit of -1 [88], which can be exceeded for orthotropic systems, for which no limits [97] or a factorial upper limit [98] is reported.

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.

Fig. 4 .
Fig. 4. Example of double arrowhead structure before and after uniaxial compressive deformation.

Fig. 5 .
Fig. 5. Diagrams of the in-plane Poisson's ratios and Young's moduli determined with the equations for the analysis of double arrowhead structures (see Table2) in conventional and auxetic configurations.
Fig. 5. Diagrams of the in-plane Poisson's ratios and Young's moduli determined with the equations for the analysis of double arrowhead structures (see Table2) in conventional and auxetic configurations.

Fig. 7 .Fig. 8 .
Fig. 7. Schematic representation of a Milton lattice and description of the identified mechanism providing internally auxetic behaviour.

Fig. 11 .
Fig. 11.Comparison of the in-plane Poisson's ratio and Young's modulus determined for the different chiral and anti-chiral configurations summarised in Table4.
Fig. 11.Comparison of the in-plane Poisson's ratio and Young's modulus determined for the different chiral and anti-chiral configurations summarised in Table4.

Fig. 14 .
Fig. 14.Deformation mechanism of rotating rectangular plates, connected with Type I (a) and Type II (b) arrangements.

Fig. 19 .
Fig. 19.Example of a re-entrant trichiral honeycomb structure (a) and of a two-level hierarchical rotating squares structure (b).