TY - JOUR

T1 - Finite auxetic deformations of plane tessellations

AU - Mitschke, Holger

AU - Robins, Vanessa

AU - Mecke, Klaus

AU - Schröder-Turk, Gerd E.

PY - 2013/1/8

Y1 - 2013/1/8

N2 - 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.

AB - 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.

KW - Cellular structures

KW - Isostaticity

KW - Poisson's ratio

KW - Skeletal frameworks

KW - Strain amplification

KW - Tilings and tessellations

UR - http://www.scopus.com/inward/record.url?scp=84872346068&partnerID=8YFLogxK

U2 - 10.1098/rspa.2012.0465

DO - 10.1098/rspa.2012.0465

M3 - Article

SN - 1364-5021

VL - 469

JO - Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences

JF - Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences

IS - 2149

M1 - 20120465

ER -