Finite auxetic deformations of plane tessellations

Abstract

We systematically analyse the mechanical deformation behaviour, in particular Poisson's ratio, of floppy barand-joint frameworks based on periodic tessellations of the plane. For frameworks with more than one deformation mode, crystallographic symmetry constraints or minimization of an angular vertex energy functional are used to lift this ambiguity. Our analysis allows for systematic searches for auxetic mechanisms in archives of tessellations; applied to the class of one-or two-uniform tessellations by regular or star polygons, we find two auxetic structures of hexagonal symmetry and demonstrate that several other tessellations become auxetic when retaining symmetries during the deformation, in some cases with large negative Poisson ratios ν < −1 for a specific lattice direction. We often find a transition to negative Poisson ratios at finite deformations for several tessellations, even if the undeformed tessellation is infinitesimally non-auxetic. Our numerical scheme is based on a solution of the quadratic equations enforcing constant edge lengths by a Newton method, with periodicity enforced by boundary conditions.

Symmetry Perspectives on Some Auxetic Body-Bar Frameworks

“…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 [36]). …”

“…We note that a comprehensive catalogue of 2D periodic frameworks considered as candidates for auxetic behaviour has been compiled [37]. (See also [36].) Some of these can be considered as periodic arrays of rigid plates, variously hinged: Star Tilings A, B and C in the catalogue correspond to single-hinge versions of the triangular, square and hexagonal tessellations of the plane, respectively.…”

Geometry: The leading parameter for the Poisson’s ratio of bending-dominated cellular solids

“…We here choose the Newton-Raphson scheme described in Mitschke et al (2013a) to identify mechanisms. This scheme finds the roots of the left-hand side of Eq.…”

“…Indices used by authors in previous publications ν inst and ν SS are consolidated intoν ff and ν LE and ν FEM intoν cs (Mitschke et al, 2011(Mitschke et al, , 2013a.…”

Bi-stiffness property of motion structures transformed into square cells

“…In this respect, bar-and-joint frameworks are valid for representing the concise deformations of these materials by mathematical or numerical analysis: for example, the rigidity or flexibility of the frameworks modelled on repetitive cells [42], zeolites [43,44], auxetic materials [45] and the loop structures connected by revolute joints [46]. The aim of this study is to explore structural materials with bi-stiffness, using the abovementioned modelling approach.…”