Journal
PROCEEDINGS OF THE ROYAL SOCIETY A-MATHEMATICAL PHYSICAL AND ENGINEERING SCIENCES
Volume 469, Issue 2149, Pages -Publisher
ROYAL SOC
DOI: 10.1098/rspa.2012.0465
Keywords
Poisson's ratio; strain amplification; tilings and tessellations; isostaticity; skeletal frameworks; cellular structures
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Funding
- German Science Foundation (DFG) through the Engineering of Advanced Materials Cluster of Excellence (EAM) [SCHR1148/3-1, ME1361/12-1]
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We systematically analyse the mechanical deformation behaviour, in particular Poisson's ratio, of floppy bar-and-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 nu < -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.
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