Finite auxetic deformations of plane tessellations

Holger Mitschke, Vanessa Robins, Klaus Mecke, Gerd E. Schröder-Turk*

*Corresponding author for this work

    Research output: Contribution to journalArticlepeer-review

    29 Citations (Scopus)


    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.

    Original languageEnglish
    Article number20120465
    JournalProceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
    Issue number2149
    Publication statusPublished - 8 Jan 2013


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