p s s On the auxetic properties of rotating rhombi and parallelograms: A preliminary investigation solidi physica status

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1 phys. stat. sol. () 45, No. 3, 5 59 (8) / DOI./pss On the auxetic properties of rotating rhomi and parallelograms: A preliminary investigation physica Joseph N. Grima *, Pierre-Sandre Farrugia, Ruen Gatt, and Daphne Attard Department of Chemistry, University of Malta, Msida MSD 8, Malta Received 7 July 7, revised Octoer 7, accepted 9 Decemer 7 Pulished online Feruary 8 PACS 6.. x, 8.5.Zx * Corresponding author: joseph.grima@um.edu.mt, p s s asic solid state physics Auxetics exhiit the unusual property of expanding when uniaxially stretched (negative Poisson s ratio), a property that is usually linked to particular geometric features and deformation mechanisms. One of the mechanisms which results in auxetic ehaviour is the one involving rotating rigid units, for which, systems made from triangles, suares or rectangles have already een considered. In this work we extend this study y considering systems which can e constructed from either connected rhomi or connected parallelograms. We show that various types of such systems can exist and we discuss in detail the properties of one type of rotating rhomi and one type of rotating parallelograms. We also show that the Poisson s ratio of these systems, which can e positive or negative, is anisotropic and dependent on the shape of the parallelograms/rhomi and the degree of openness of the system. 8 WILEY-VCH Verlag GmH & Co. KGaA, Weinheim Introduction The way in which a material or structure deforms when sujected to an applied force, is of great importance in estalishing its properties. Most of the commonly encountered systems contract in the lateral direction when stretched, and expand laterally when compressed, i.e. they have a positive Poisson s ratio. However, in recent years, several studies have een carried out on materials and structures [ 39] that ehave exactly in the opposite way, i.e., they expand in the lateral direction when stretched and contract laterally when compressed. These counter-intuitive systems exhiit a negative Poisson s ratio and are commonly referred to as auxetic [3]. Real interest in auxetics took off in the late 98s, following Lakes discovery that auxetic foams could e easily manufactured from conventional open-cell foam []. Since then, extensive research has een done to gain insight into what makes materials auxetic and how their properties compare with those of conventional ones. In fact, studies on auxetic materials show that they have various eneficial properties for example, a superior aility (compared to conventional materials) to resist indentation [4, 5], a natural aility to form synclastic curvatures [] and enhanced acoustic properties [6, 7]. Such ualities make auxetics potentially useful in various engineering and practical applications. It has een found that auxeticity can e descried in terms of the geometry of the system and the deformation mechanism taking place. This is a scale independent property, i.e. the same mechanism can operate at and lead to auxetic ehaviour at either the macro-, micro-, or nano- (atomic) level. Over the years, several geometrical models that exhiit negative Poisson s ratios have een proposed, studied and tested for their mechanical properties. Among the most important classes of such auxetic structures we find re-entrant honeycoms (that were the first systems to e studied in detail [8 ]), chiral honeycoms [ 3] and rotating rigid/semi-rigid units, in particular rotating two-dimensional units [4 3]. In this paper we will extend existing studies on rotating two-dimensional units to discuss the auxetic potential of systems constructed from rhomi or parallelograms which are connected together through hinges at their vertices. Such systems deform in such a way that when uniaxially stretched, they rotate relative to each other thus potentially forming more open structures. 8 WILEY-VCH Verlag GmH & Co. KGaA, Weinheim

2 physica pss 5 J. N. Grima et al.: On the auxetic properties of rotating rhomi and parallelograms (a) () (c) Figure (a) The rotating suares geometry and ( c) the rotating rectangles of () Type I and (c) Type II. Rotating parallelograms and rhomi In recent years, there have een various investigations on rotating rigid polygons, in particular rotating suares [4, 5] (see Fig. a), rotating euilateral triangles [8] and rotating rectangles of two different types (Type I, see Fig., and Type II, see Fig. c) [9, 3]. Mathematical analysis of rotating suares and euilateral triangles models suggests an isotropic in-plane Poisson s ratio of [4, 8], irrespective of the dimensions of the suares or triangles. On the other hand, analysis of the rotating rectangles model shows that the mechanical properties of the system depend on the way that the rectangular units are connected to each other, i.e. either as Type I or Type II. Type I systems, which are systems where the empty loops etween the rectangles are always rhomi, have een found to e anisotropic with a strain-dependent Poisson s ratio that depends on (i) the aspect ratio of the rectangles, (ii) the angles etween the rectangles (i.e. the degree of openness), and (iii) the direction of loading. For example, it was shown that the Type I rotating rectangles systems ehave auxetically as they are stretched from the fully closed configuration until the angle etween the rectangles reaches a critical value (defined y the dimensions of the rectangles), after which the structure starts to ehave conventionally until a second critical value is reached, whereupon the structure starts to ehave auxetically again until the system is once again fully closed. In contrast, the Type II systems were found to e isotropic with a Poisson s ratio of, irrespective of the rectangle dimensions, the angles etween the rectangles and the direction of loading. It is important to note that although rotating rectangles models represent a more general version of the rotating suares model (suares can e considered as a particular case of a rectangles where a, i.e. suares are defined y a gle variale, a, whilst rectangles are defined y two variales, a and ), this is not the only generalisation that can e made. In particular we note that: () the suares can e generalised to rhomi which are characterised y two variales: a, the length of the sides, and φ, the internal angle etween the sides; () the rhomi can e further generalised to parallelograms (which can also e considered as generalised rectangles) which are defined y three variales: a and, the two non-eual sides of the parallelograms and φ, the internal angle etween the sides. Type α rhomi (a) Type I α parallelograms () Type II α parallelograms (c) Type β rhomi Type I β parallelograms Type II β parallelograms (d) (e) Figure New rotating rhomi and rotating parallelograms systems: (a) rotating rhomi of Type α; ( c) rotating parallelograms of Type I α and Type II α respectively, (d) rotating rhomi of Type β; (e f) rotating parallelograms of Type I β and Type II β respectively. (f) 8 WILEY-VCH Verlag GmH & Co. KGaA, Weinheim

3 Original Paper phys. stat. sol. () 45, No. 3 (8) 53 We note that as was the case with the rectangles, where two non-euivalent networks could e constructed depending on the connectivity of the rectangles together (the Type I and Type II, see Fig. and c), when rhomi are connected together at their vertices, one can also otain two non-euivalent networks (see Fig. a and d) which we shall henceforth refer to as Type α and Type β where: Type α can e descried as a structure made from rhomi having their smaller angle attached with the larger angle of adjacent rhomi (see Fig. a); (a) () (c) (d) (e) (f) Figure 3 Different configurations of the new rotating rhomi and rotating parallelograms systems at different degree of openness: (a) rotating rhomi of Type α; () rotating rhomi of Type β; (c d) rotating parallelograms of Type I α and Type II α respectively, (e f) rotating parallelograms of Type I β and Type II β respectively. 8 WILEY-VCH Verlag GmH & Co. KGaA, Weinheim

4 physica pss 54 J. N. Grima et al.: On the auxetic properties of rotating rhomi and parallelograms Type β can e descried as a structure made from rhomi having their smaller angle attached with the smaller angle of adjacent rhomi and the larger angle eing attached with the larger angle of adjacent rhomi (see Fig. d). We also note that when parallelograms are connected together at their vertices, one can otain four noneuivalent networks (see Fig. ), ce these systems exhiit oth Type I and II as well as Type α and β features thus otaining the Type I α (Fig. ), Type I β (Fig. e), Type II α (Fig. c) and Type II β (Fig. f). It is interesting to note that despite the differences etween the various systems (for example, we note that the fully closed conformation of the Type α rotating rhomi is space filling whilst that of Type β is not), these networks have an important property in common: oth noneuivalent rotating rhomi networks and the four noneuivalent rotating parallelograms may deform through relative rotations of the polygons, a deformation mechanism which in some cases may result in auxetic ehaviour (see Fig. 3). In view of this, it is important to investigate analytically, numerically and experimentally the ehaviour of these configurations. In this work we analyse in detail the properties of the Type α rotating rhomi and the Type II α rotating parallelograms. In particular, analytical expressions relating on-axis and off-axis mechanical properties to the geometrical features of the parallelograms and rhomi are derived, thus otaining a etter understanding of the auxetic potential of these systems. 3 Analytical model for the Type II a rotating parallelograms and the Type a rotating rhomi 3. The on-axis properties of the Type II a rotating parallelograms In this section we shall first derive analytical expressions for the Type II α rotating parallelograms illustrated in Fig. c. This system can e descried y a rectangular unit cell, made up of two hinged parallelograms at an angle θ to each other that are translated mirror images of each other as illustrated in Fig. 4. Each of these parallelograms has side lengths a and, and internal angles φ and π - φ. We shall e assuming that the parallelograms are perfectly rigid (i.e. a, and φ do not change when the structure is sujected to a load), and that they are connected together y hinges having a stiffness constant K h (i.e. idealised rotating rigid parallelograms model ) where K h relates the induced moment M to the change in the angle dθ through the euation: M K dθ. () h Thus, ce the parameters a,, φ and θ completely define the in-plane dimensions of the unit cell shown in Fig. 4, and a, and φ are assumed to e constants in this derivation, the shape and size of the unit cell of this system is a function of the gle variale θ. X a φ θ X Figure 4 Parameters that define the parallelograms and the unit cell. This system has an infinite on-axis shear modulus and also zero on-axis shear coupling coefficients. Thus, its mechanical properties are completely descried y the 3 3 compliance matrix S which has 4 non-zero elements, and is therefore defined y Ê ν - ˆ Á E E Ês s ˆ Á ν S Ás s Á-, () Á Á E E Ë Á Á Ë where E and E are the on-axis Young s moduli whilst ν and ν are the on-axis Poisson s ratios. Thus, to derive the mechanical properties of the Type II α rotating parallelograms system, it suffices to derive analytical expressions for the on-axis Poisson s ratios and Young s moduli. Referring again to Fig. 4, and assuming a thickness z in the third dimension, the projections of the unit cell in the Ox, Ox and Ox 3 directions (which are eual to the unit cell dimensions) are given y θ + φ X a, θ + ( π -φ) θ -φ X È Í, (3) Î z. X3 3.. The on-axis Poisson s ratios Since we are assuming that the parallelograms are rigid, it follows that a, and φ are constants, and infinitesimally small strains dε i in Note that the two Young s moduli and the two Poisson s ratios are not independent of each other ce s s whilst ν ν - as discussed elow. This suggests that the mechanical properties of this system are solely descried y two independent constants, a property which is a direct conseuence of the fact that the model geometry has only one degree of freedom. 8 WILEY-VCH Verlag GmH & Co. KGaA, Weinheim

5 Original Paper phys. stat. sol. () 45, No. 3 (8) 55 the Ox i directions can e otained in terms of an infinitesimal change dθ in the angle θ, i.e. dxi dxi dεi dθ. (4) Xi Xi dθ Thus, the Poisson s ratios ν and ν can e otained after differentiating Es. (3) and sustituting the resulting expressions into E. (4) to otain: Êθ - φˆ Êθ + φˆ - dε Ë ν ( ν) - dε θ φ θ φ Ê - ˆ + tan θ - φ tan θ + Ê φ ˆ. (5) 3.. The on-axis Young s moduli The Young s modulus, E along the Ox direction can e derived from an energy approach, ug the procedure outlined elsewhere [9]. In particular, for loading in the Ox direction, recog- nig that the energy stored in a unit cell due to a small strain dε must e eual to the work done y the four hinges per unit cell of the system, the Young s modulus E for loading in the Ox direction is given y 4KX h dx E Ê dθ ˆ X X Ë dθ 3 - ( dθ) 4Kh Êθ + φˆ, (6) θ φ θ φ az Ê - ˆ + where Kh is the spring constant associated with the θ -hinges and z is the thickness in the third dimension. Similarly, the Young s modulus E along the Ox direction is given y E 4K Êθ -φˆ h. (7) θ φ θ φ az Ê - ˆ + Region of negative Poisson s ratio n E Region of positive Poisson s ratio n E 3 o f 75 Poisson s ratio (n) 5 o 45 o 9 o 35 o 8 o -5 - Young s Modulus (E) o 45 o 9 o 35 o 8 o (a) () 5 o f 6 Poisson s ratio (n) 5 o 45 o 9 o 35 o 8 o -5 - Young s Modulus (E) o 45 o 9 o 35 o 8 o (c) (d) 3 5 o f 3 Poisson s ratio (n) 5 o 45 o 9 o 35 o 8 o -5-3 Young s Modulus (E) o 45 o 9 o 35 o 8 o (e) (f) Figure 5 On axis Poisson s ratios and Young s moduli plotted against θ for different values of φ in the case of rotating parallelograms of Type II α. The sides of the parallelograms are taken to e a 3 and in all cases. 8 WILEY-VCH Verlag GmH & Co. KGaA, Weinheim

6 physica pss 56 J. N. Grima et al.: On the auxetic properties of rotating rhomi and parallelograms Typical plots of the on-axis Poisson s ratios and Young s moduli against θ for this system are shown in Fig. 5. These clearly suggest that such systems can exhiit auxetic ehaviour for some, ut not all, values of θ. 3. The on-axis properties of the Type a rotating rhomi Type α rotating rhomi may e considered as a special case of the Type II α rotating parallelograms, where the sides are of eual length, i.e. a. In this case, the Poisson s ratios are given y the same expression as for the rotating parallelograms, ce these are independent of oth a and (see E. (5)) whilst the Young s moduli E and E are given y Es. (6) and (7) respectively, with eing replaced y a. In other words, the moduli differ only y a scaling factor. 3.3 The off-axis mechanical properties Ug standard techniues [39], the off-axis values for the Poisson s ratio, Young s and shear moduli can e shown to e eual to Êθ - φˆ Êθ + φˆ ζ 4K h E, az È θ φ θ φ θ φ θ φ Ê - ˆ Ê + ˆ ( ζ) Ê - ˆ + ( ζ Í - ) Î ν ζ G Êθ - φˆ Êθ + φˆ θ - φ θ + φ ( ζ) - ( ζ), (8) θ φ θ φ θ φ θ φ Ê - ˆ Ê + ˆ - + ( ζ) - ( ζ) Êθ + φˆ Êθ -φˆ 4K h, az ( ζ) ( φ) ζ ζ eing the angle etween the original Ox i axis and the transformed axis. 4 Discussion The plots illustrated in Fig. 5 show that the on-axis Poisson s ratio for Type II α parallelograms can e oth negative and positive depending on the value of θ that can range in the interval [, π ]. Provided that φ π π/ (a situation which is treated later on), we note that ce the Poisson s ratio may e written as ν - dε Ê dxˆ Ê dxˆ ( ν ) - - dε Á Ë X dθ Á Ë X dθ - d dx X X -, (9) Ë dθ Ë dθ X then for negative Poisson s ratios we reuire that - d ÊdX ˆ Ê X ˆ X Á dθ Á > dθ. () Ë Ë X However, ce the unit cell lengths are always positive, i.e. X i >, this condition reduces to the reuirement that the derivatives with respect to θ e such that they have the same sign, i.e. either or dx d θ > and d dx d θ < and d - dx θ > (a) dx θ <, () from which it follows that either Êθ + φˆ θ -φ > and - > (a) or Êθ + φˆ θ -φ < and - <. () In terms of angles, these conditions reduce to either < θ + φ < π and - π< θ - φ < (3a) or π< θ + φ < π and < θ - φ < π. (3) Since the allowed values of θ must lie etween and π to avoid overlap of the parallelograms, it turns out that the Poisson s ratio is negative in the regions, < θ < min ( φ,π- φ) and max ( φ, π- φ) < θ < π, i.e. the Poisson s ratio is negative in the region < θ < θ, positive in the region θ < θ < θ, and negative in the region θ < θ < π, where θ min ( φ, π - φ) and θ max ( φ, π - φ). (4) This suggests that the structure is initially auxetic as it is stretched from its first fully closed configuration until the angle of deformation reaches a value of θ min ( φ, π - φ). Beyond this value, the structure ehaves conventionally up to θ max ( φ, π - φ), after which the structure starts to 8 WILEY-VCH Verlag GmH & Co. KGaA, Weinheim

7 Original Paper phys. stat. sol. () 45, No. 3 (8) 57 ehave auxetically again until the second fully closed configuration is reached at θ π. The points at which the Poisson s ratio changes sign are of interest. One of these points occurs when θ φ, at which point d X /dθ. Since this term is found in the numerator of ν, this Poisson s ratio changes sign y going smoothly through zero. Furthermore, ce it is also in the denominator of ν, this Poisson s ratio changes sign discontinuously (y diverging to - on one side and to + on the other side). In terms of the structure, this case corresponds to the sides of length eing all aligned in the vertical direction. In such a configuration, the Young s modulus E along the vertical direction ecomes infinite ( d X /dθ is also found in the denominator of E ) as in this conformation, any force applied along the Ox direction would e acting along an inextensile line and would not e ale to produce any moments that are necessary for the parallelograms to rotate (Fig. 6a). It is important to note that conformations where the Poisson s ratio changes sign asymptotically are of particular practical interest ce such configurations exhiit giant negative or positive Poisson s ratios. The other point at which the Poisson s ratio changes sign occurs when θ π - φ. In this case, d X/dθ, and ce this term is found in the denominator of the expression for ν, this Poisson s ratio tends to infinity as it changes sign discontinuously, diverging to - on one side and to + on the other side. For the structural configuration corresponding to this value of θ, all sides of length a are aligned horizontally to form an incompressile line so that a force exerted along the Ox does not generate any moments (Fig. 6). Similar to the previous case, this makes the Young s modulus E diverge to infinity ( d X/dθ is also found in the denominator of E, see E. (6)). The case when φ π/, i.e. the point when the Type II α rotating parallelograms structure reduces to the Type II rotating rectangles, is also of particular interest ce it can lead to a situation when oth d X/dθ and d X /dθ resulting in the condition where in E. (5) we have a zero divided y zero. This situation occurs when θ φ π- φ π/, i.e. when the Type II rotating rectangles structure is fully open, a conformation when oth sides of the rectangles are aligned either vertically or horizontally as shown in Fig. 6c. In this case oth on-axis Young s moduli are infinite ce a force applied along the Ox or Ox axis would e once again acting along an inextensile line with no moment eing induced. This means that the structure is locked in oth directions. We also note that for all other values of θ, the Poisson s ratios in E. (5) reduces to when φ π/, as expected from the Type II rotating rectangles model [3]. Another interesting property of the Type II α rotating parallelograms systems that can e noted from the on axis Figs. (Fig. 5) is the symmetry aout θ π/ that exists etween v and ν. Effectively, if in Es. (5) (7) we sustitute θ y π - θ, we find that v is changed to ν while E is changed to E and vice versa. This means that replacing θ y π - θ is euivalent to applying a rotation of π/ to the structure. A similar thing happens if φ is replaced y π - φ. In this way changing oth θ y π - θ and φ y π - φ would e euivalent to rotating the structure y π around the Ox 3 axis. Contrary to this, nothing changes if we interchange a and. The analysis carried out so far indicates that the ehaviour of the Type II α rotating parallelograms is much different from that oserved for Type II rotating rectangles, in that the on-axis Poisson s ratio of Type II rotating rectangles are always eual to. In fact, it is interesting to note that the ehaviour of Type II α rotating parallelograms is much more similar to that encountered with rotating rectangles of Type I, despite the fact that the Type II α can e considered as a general case of the Type II rotating rectangles. This similarity etween Type II α rotating parallelograms and the structurally different Type I rotating rectangles goes eyond just eing ualitative. In fact taking the euations for the Poisson s ratio for the Type I rotating rectangles given y Grima et al. [9] as ν ( a ) Á - ( ) - Á ( ν ) Êθ ˆ Êθ ˆ θ ( a ) - ( ) Á Á θ (5) φ φ π φ θ π φ π θ φ θ φ π φ (a) () (c) Figure 6 Configuration for the Type II α rotating parallelograms at which the Young s moduli along the (a) Ox and () Ox direction are infinite. Note that in (c), the case where the parallelograms ecome rectangles (the Type II rectangles), the structure will have infinite Young s moduli simultaneously in the Ox and Ox directions. 8 WILEY-VCH Verlag GmH & Co. KGaA, Weinheim

8 physica pss 58 J. N. Grima et al.: On the auxetic properties of rotating rhomi and parallelograms with a and eing the length of the sides of the rectangle and θ the angle etween the rectangles, they can e rewritten in a form that is mathematically similar to the euations otained for rotating parallelograms of Type II α, i.e. as - θ - χ θ + χ ν ( ν ) tan Á tan Á, (6) where χ is given y - Êa ˆ χ tan Á Ë. (7) The angle χ can e related to the structure as shown in Fig. 7. Thus, we see that mathematically, to go from the euation for the Poisson s ratio of the Type II α rotating parallelograms to that of the Type I rotating rectangles, we just have to sustitute φ with χ. Then, all the on-axis properties shown y ν for the Type II α rotating parallelograms will also e shown y ν for the Type I rotating rectangles. In particular, the Type I rotating rectangles also show a similar θ dependence of the Poisson s ratio, where we find that the system is auxetic for < θ < θ, conventional for θ < θ < θ and again auxetic for θ < θ < π, where θ min ( χ, π- χ) and θ max ( χ, π- χ). The only difference is found in the geometric configurations which correspond to the angles where the change of sign of the Poisson s ratios takes place ce: In the case of Type I rotating rectangles the angles θ and θ correspond to situations when the diagonals of the rectangles are aligned parallel to one of the Ox i axis as indicated in Fig. 8. In the case of the Type II α rotating parallelograms the angles θ and θ correspond to situations where the sides of the parallelograms are aligned to one of the Ox i axis as indicated in Fig. 6. It is important to note that the analysis carried out so far on the rotating parallelograms of Type II α applies also to the rotating rhomi of Type α. The reason for this is that the euation for the Poisson s ratio of Type α rotating rhomi is the same as that of Type II α rotating parallelograms, while the euations for the Young s and shear moduli of these two structures defer only y a multiplicative factor and this has no significant effect on the structural ehaviour. a χ Figure 7 Relation etween χ and the sides of the rectangles of Type I rotating rectangles. Figure 8 Two configurations where the rotating rectangles of Type I have infinite Young s modulus along (a) the Ox direction and () the Ox direction. This occurs ecause one of the diagonals of the rectangles is aligned along one of the Ox i axes. Note that in the special case when rectangles ecome suares, these two configurations reduce to one at θ π/. Furthermore, it is also important to note that the off ζ axis Poisson s ratio ν (E. (8)) suggests that like the onaxis Poisson s ratios, ν ζ is independent of the lengths of the sides of the parallelogram i.e. on a and ut depends solely on the angles θ and φ and also on the direction of loading (i.e. ζ ). This is very significant as it highlights the fact that, like other models that result in negative Poisson s ratios, the auxeticity is a scale independent property and thus, these systems can e constructed at either the macroscale, the microscale or even the nanoscale. We also note that these models provide us with new tools that can help us gain further insight on how real materials function. 5 Conclusion This work considers a generalisation of the rotating suares and rotating rectangles models to uadrilaterals having the shape of rhomi or parallelograms. We showed that two different types of structures can e constructed when the suares are replaced y rhomi of the same size, depending on the connectivity. It was also shown that if the rhomi are attached such that the small angle is connected with the ig angle of adjacent rhomi and vice versa, a space filling structure is attained, and this configuration was termed Type α rotating rhomi. On the other hand, if the rhomi are attached such that like angles of adjacent rhomi are connected to each other, a non space filling structure termed Type β rotating rhomi is otained. We also considered systems made from connected parallelograms of the same size (which can e considered as generalised rotating Type α or Type β rhomi or generalised rotating Type I or Type II rectangles. We found out that it is possile to otain four possile configurations in the case of rotating parallelograms, the Type I α, Type II α, Type I β and Type II β. A detailed mathematical model descriing the ehaviour of the Type α rotating rhomi and Type II α rotating parallelograms when deforming through hinging (i.e. relative rotations of the polygons) was derived, and this showed that the euation for the Poisson s ratio for these two structures has the same form, while those for the Young s and shear moduli differ 8 WILEY-VCH Verlag GmH & Co. KGaA, Weinheim

9 Original Paper phys. stat. sol. () 45, No. 3 (8) 59 only y a multiplicative factor. Further analyses indicates that these structures can have oth negative and positive Poisson s ratios and the region of transition etween negative and positive can show giant Poisson s ratio ce its value can diverge to + on one side and to - on the other. It was also shown that the magnitude of the Poisson s ratio is dependent on the shape of the parallelograms/rhomi, in particular on the internal angle of the parallelograms/rhomi, the angle etween the parallelograms/rhomi (i.e. Poisson s ratio is strain dependent) and the direction of loading. It was also shown that the ehaviour of the Type II α rotating parallelograms is similar to that of the Type I rotating rectangles even though they are structurally diverse while, on the other hand, it is very different from Type II rotating rectangles that are structurally similar to the rotating parallelograms of Type II α. All this is very significant, not only ecause we have extended the existing corpus of knowledge on auxetic systems made from rotating D rigid units, ut also ecause this work can e of help to other scientists working in the field who may make use of the work descried here to synthesise or manufacture new auxetic structures or materials which mimic the ehaviour of these novel systems, or to help explain the auxetic ehaviour of naturally occurring auxetics. Acknowledgments The work of Victor Zammit of the University of Malta is gratefully acknowledged. We also gratefully acknowledge the financial support of the Malta Council for Science and Technology (MCST) through their National RTDI funding programme. References [] R. S. Lakes, Science 35, 38 (987). [] B. D. Caddock and K. E. Evans, J. Phys. D, Appl. Phys., 877 (989). [3] N. R. Keskar and J. R. Chelikowsky, Nature 358, (99). [4] A. Yeganeh-Haeri, D. J. Weidner, and J. B. Parise, Science 57, 65 (99). [5] K. L. Alderson and K. E. Evans, J. Mater. Sci. 8, 49 (993). [6] A. P. Pickles, K. L. Alderson, and K. E. Evans, Polym. Eng. Sci. 36, 636 (996). [7] C. B. He, P. W. Liu, and A. C. Griffin, Macromolecules 3, 345 (998). [8] R. H. Baughman, J. M. Shacklette, A. A. Zakhidov, and S. Stafstrom, Nature 39, 36 (998). [9] C. W. Smith, J. N. Grima, and K. E. Evans, Acta Mater. 48, 4349 (). [] C. B. He, P. W. Liu, P. J. McMullan, and A. C. Griffin, phys. stat. sol. () 4, 576 (5). [] N. Gaspar, C. W. Smith, E. A. Miller, G. T. Seidler, and K. E. Evans, phys. stat. sol. () 4, 55 (5). [] C. Sanchez-Valle, S. V. Sinogeikin, Z. A. D. Lethridge, R. I. Walton, C. W. Smith, K. E. Evans, and J. D. Bass, J. App. Phys. 98, 5358 (5). [3] K. E. Evans, M. A. Nkansah, I. J. Hutchinson, and S. C. Rogers, Nature 353, 4 (99). [4] R. S. Lakes and K. Elms, J. Compos. Mater. 7, 93 (993). [5] A. Alderson, Chem. Ind., 384 (999). [6] F. Scarpa and F. C. Smith, J. Intell. Mater. Syst. Struct. 5, 973 (4). [7] F. Scarpa, W. A. Bullough, and P. Lumley, Proc. Inst. Mech. Eng. C, J. Mech. Eng. Sci. 8, 4 (4). [8] F. K. Ad el-sayed, R. Jones, and I. W. Burgens, Composites, 9 (979). [9] L. J. Gison and M. F. Ashy, Cellular Solids: Structure and Properties (Pergamon, Oxford, 988). [] I. G. Masters and K. E. Evans, Compos. Struct. 35, 43 (996). [] D. Prall and R. S. Lakes, Int. J. Mech. Sci. 39, 35 (997). [] A. Spadoni, M. Ruzzene, and F. Scarpa, phys. stat. sol. () 4, 695 (5). [3] N. Gaspar, X. J. Ren, C. W. Smith, J. N. Grima, and K. E. Evans, Acta Mater. 53, 439 (5). [4] J. N. Grima and K. E. Evans, J. Mater. Sci. Lett. 9, 563 (). [5] Y. Ishiashi and M. J. Iwata, J. Phys. Soc. Jpn. 69, 7 (). [6] A. A. Vasiliev, S. V. Dimitriev, Y. Ishiashi, and T. Shinegari, Phys. Rev. B 65, 94 (). [7] J. N. Grima, V. Zammit, R. Gatt, A. Alderson, and K. E. Evans, phys. stat. sol. () 44, 866 (7). [8] J. N. Grima and K. E. Evans, J. Mater. Sci. Lett. 4, 393 (6). [9] J. N. Grima, A. Alderson, and K. E. Evans, phys. stat. sol. () 4, 56 (5). [3] J. N. Grima, R. Gatt, A. Alderson, and K. E. Evans, J. Phys. Soc. Jpn. 74, 866 (5). [3] K. W. Wojciechowski, Mol. Phys. 6, 47 (987). [3] K. W. Wojciechowski and A. C. Branka, Phys. Rev. A 4, 7 (989). [33] K. W. Wojciechowski, J. Phys. A, Math. Gen. 36, 765 (3). [34] A. Alderson and K. E. Evans, Phys. Rev. Lett. 89, 553 (). [35] A. Alderson and K. E. Evans, J. Mater. Sci. 3, 797 (997). [36] H. Kimizuka, H. Kauraki, and Y. Kogure, Phys. Rev. B 67, 45 (3). [37] N. Gaspar, C. W. Smith, E. A. Behne, G. T. Seidler, and K. E. Evans, phys. stat. sol. () 4, 55 (5). [38] N. Gaspar, C. W. Smith, and K. E. Evans, J. Appl. Phys. 94, 643 (6). [39] K. W. Wojciechowski, A. Alderson, K. L. Alderson, B. Maruszewski, and F. Scarpa, phys. stat. sol. () 44, 83 (7). [4] J. F. Nye, Physical Properties of Crystals (Clarendon Press, Oxford, 957). 8 WILEY-VCH Verlag GmH & Co. KGaA, Weinheim