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Scalar mobility counting rules and their symmetry extensions are reviewed for finite frameworks and also for infinite periodic frameworks of the bar-and-joint, body-joint and body-bar types. A recently published symmetry criterion for the existence of equiauxetic character of an infinite framework is applied to two long known but apparently little studied hinged-hexagon frameworks, and is shown to detect auxetic behaviour in both. In contrast, for double-link frameworks based on triangular and square tessellations, other affine deformations can mix with the isotropic expansion mode.

Use of counting rules in the study of rigidity and mobility of frameworks has a venerable history, going back to Maxwell's 1864 rule for bar-and-joint frameworks [

Direct extension of symmetric Maxwell and mobility rules from point groups and finite objects to unit-cell symmetry in infinite repetitive systems [

The new symmetry theorem for equiauxetics gives an opportunity to look back at some frameworks that were first discussed [

As the analysis with a periodic symmetry extension of the mobility criterion for body-bar frameworks will show, both frameworks have an auxetic expansion mode. Connection patterns with one and two bars were later used as the inspiration for the construction of families of expanding polyhedra [

In view of the interdisciplinary nature of the material in this paper, it seems useful to give a short lexicon of equivalences between mathematical and engineering/materials science terminology used here. “Mechanisms” and “states of self-stress” are vectors in the left-nullspace and nullspace, respectively, of the equilibrium matrix, a matrix which describes the relationship between internal and external forces in a structure [

Maxwell's 1864 rule for pin-jointed frameworks [_{object} (_{object} (

The scalar counting rule (1) is simply the character of the full symmetry equation under the identity operation. Each additional class of symmetry operations has the possibility of generating an extra counting relation, and extra information for the analysis. It has been said with some justification that each conventional counting rule is just the tip of a symmetry-counting iceberg.

For periodic pin-jointed frameworks, the behaviour of a representative unit cell is considered, and we look for mechanisms and states of self-stress that are propagated by translation of the unit cell,

The symmetry equivalent of _{T}_{R}_{T}_{R}_{T}_{T}_{T}_{R}

For a finite system consisting of _{i}_{i}_{i}

The symmetry extensions of _{‖} (_{0} is the totally symmetric representation, Γ_{freedoms} is the representation of the set of all relative freedoms of the joints, and the translational and rotational representations Γ_{T}_{R}

Contributions to Γ_{freedoms} are calculated according to the type of the joint, and its placement with respect to symmetry elements [

Finally, we deal briefly with body-bar frameworks. If

We can derive this counting rule directly from the body-joint

The adaptation of this formula to periodic symmetry does not appear to have been stated explicitly elsewhere, but it is easily seen to be
_{T}_{R}

The type of auxetic behaviour that can be detected by symmetry is, as mentioned earlier,

For an equiauxetic mode to exist in 2D or 3D, it is therefore sufficient that Γ(

The hinged-hexagon frameworks that are the subject of the present paper belong to plane group _{6}, so that the presence of an unblocked mechanism of either

The single-link version of the hinged-hexagon framework is shown in _{1} for this framework is given in _{1} are rigid hexagons. The pin-jointed bars, are associated with the edges of _{1} and connect adjacent hexagons to give a locally chiral 6-fold symmetry. The plane group of this 2D structure is therefore _{6}, as noted above.

A suitable unit cell is the rhombus shown in

A word about the method of calculation of the characters χ(_{object}(

The calculation of the net mobility Γ(_{6}
_{6}
_{3}

Γ(

_{T}

_{R}

^{2}1 1 1 4 1 1 3 2 0 −1 −1 0 2

−Γ(

_{T}

_{T}

_{T}

_{R}

^{2}2 0 −1 −4 0 2 −3

^{2}0 0 −3 0 0 0 1 −1 1 5 5 1 −1

= Γ(

_{1}for all

Experimentation with physical models of finite portions of the single-link hexagon framework give an impression of floppiness, suggesting that there may be extra flexibility to be identified, but the periodic analysis indicates that these are likely to be boundary effects not present in the infinite framework.

For the double-link hexagonal framework (shown in _{6}
_{6}
_{3}

Γ(

_{T}

_{R}

^{2}1 1 1 4 1 1 3 2 0 −1 −1 0 2

−Γ(

_{T}

_{T}

_{T}

_{R}

^{2}2 0 −1 −4 0 2 −6

^{2}0 0 0 0 0 0 1 −1 1 5 5 1 −1

= Γ(

^{2}1 1 4 1 1 1Here, the freedoms grow as 3

^{2}but the constraints grow as 6

^{2}, leading to a heavily over-constrained system for all

_{1}, indicating an uncancelled totally symmetric mechanism. The general case can be formulated in terms of the regular representation Γ

_{reg}(which has χ

_{reg}(

_{reg}(

With _{2} − (_{1} + _{2}) = _{1}, as before. The case _{6} occur in Γ(

For completeness, we briefly consider the linked frameworks that can be based on the other regular tessellations of the plane, by equilateral triangles and by squares. Simple counting shows that a single-link framework based on either of these tessellations will be highly under-constrained. However, the double-link cases are potentially more interesting. In the case of the double-link triangular framework, each body has six constraining bars, with each bar joined to one neighbour, and hence this framework will be locally isostatic [

The arrangement of bodies and bars in this framework and the corresponding contact polyhedron and unit cell are shown in _{6}. The usual tabular calculation for Γ(_{6}
_{6}
_{3}
_{2}

Γ(

_{T}

_{R}

^{2}0 2 0 2 0 3 2 0 −1 0 2

−Γ(

_{T}

_{T}

_{T}

_{R}

^{2}0 0 0 0 0 −6

^{2}0 0 0 0 0 1 −1 1 5 1 −1

= Γ(

An interesting feature of the calculation for this framework is that Γ(_{T}_{T}_{t}_{r},_{a}

Specifically, the calculation shows here that, for all _{2} − _{1}, indicating a constant set of three mechanisms consisting of an isotropic expansion mode and a pair of shear deformations, which is perhaps best understood as a set of independent dilations across three lines at 120° to each other, and a pair of states of self-stress. It is straightforward to find the forms of the _{1} pair of states of self-stress from the fact that they transform under the operations of the group in the same way as the pair of translations {_{x}, T_{y}

For this framework, isotropic expansion is possible, but is not the only allowed affine deformation, and therefore a general motion of the framework will mix isotropic expansion and shear deformation.

The arrangement of bodies and bars in this framework and the corresponding contact polyhedron and unit cell are shown in _{4}. The usual tabular calculation gives
_{4}

Γ(

_{T}

_{R}

^{2}1 2 1 4 1 2 3 1 1 −1 −1 1 1

−Γ(

_{T}

_{T}

_{T}

_{R}

^{2}1 2 −1 −4 1 2 −4

^{2}0 0 0 0 0 0 1 −1 −1 5 5 −1 −1

= Γ (

^{2}0 1 4 1 0 1

The mobility for general

As with the triangular framework, isotropic expansion here is not the only allowed affine deformation of the double-link square tessellation.

Auxetic materials and substances have a huge variety of actual and proposed applications, from shock absorbers and self-cleaning filters to tunable photonics and strain amplifiers on the molecular scale (to take just the selection of proposals cited in the introduction of one recent paper [

Analysis of some examples of hinged frameworks using periodic symmetry reveal the essential equiauxetic mechanism shared by these systems. We note that a comprehensive catalogue of 2D periodic frameworks considered as candidates for auxetic behaviour has been compiled [

In periodic structures found in Nature, the system can often be considered to be free from boundary effects, in that the number of unit cells is of the order of Avogadro's Number which for practical purposes is effectively infinite, and the system is considered to obey strict 2D or 3D toroidal boundary conditions. Engineered structures consist of much numbers of units that are smaller by many orders of magnitude, and edge/perimeter/boundary effects may be significant, especially in small physical models as we noted above for single-link hexagon frameworks. Analysis using ideal periodic symmetry gives a criterion for identifying what is a bulk property and what a boundary effect in these cases.

Finally, we note that, although we have taken an “analytical” approach here, using essentially only pencil-and-paper calculations and human reasoning about the interplay in the counting between stresses and mechanisms, it is clear that there is scope for building automated algorithms that incorporate finite and periodic point-group symmetry to reach these conclusions.

P.W. Fowler acknowledges support from the Royal Society/Leverhulme Trust in the form of a Senior Research Fellowship for 2013. T. Tarnai is grateful for financial support under OKTA grant K81146.

The authors declare no conflict of interest.

Two infinite periodic frameworks composed of linked rigid regular hexagons. (_{1} for this single-link case; (_{2} has a digon replacing every edge of _{1}; (_{6}, with one 6-fold axis, two additional 3-fold axes and three additional 2-fold axes (at cell centre and centres of left/right and top/bottom edges).

An infinite periodic framework composed of double-linked rigid equilateral triangles. (_{6}, with one 6-fold axis, two additional 3-fold axes and three additional 2-fold axes (at cell centre and centres of left/right and top/bottom edges).

An infinite periodic framework composed of double-linked rigid squares. (_{4}, with two 4-fold axes (at cell centre and corners) and two additional 2-fold axes (at centres of left/right and top/bottom edges).