Erratum: Finding auxetic frameworks in periodic tessellations - Mitschke, Holger; Schwerdtfeger, Jan; Schury, Fabian; Stingl, Michael; Körner, Carolin; Singer, Robert; Robins, Vanessa; Mecke, Klaus; Schröder-Turk, Gerd (Advanced Materials, 2011)

Finding Auxetic Frameworks in Periodic Tessellations - Mitschke, Holger; Schwerdtfeger, Jan; Schury, Fabian; Stingl, Michael; Körner, Carolin; Singer, Robert; Robins, Vanessa; Mecke, Klaus; Schröder-Turk, Gerd (Advanced Materials, 2011)
It appears that most models for micro-structured materials with auxetic deformations were found by clever intuition, possibly combined with optimization tools, rather than by systematic searches of existing structure archives. Here we review our recent approach of finding micro-structured materials with auxetic mechanisms within the vast repositories of planar tessellations. This approach has produced two previously unknown auxetic mechanisms, which have Poisson's ratio v(ss) = -1 when realized as a skeletal structure of stiff incompressible struts pivoting freely at common vertices. One of these, baptized Triangle-Square Wheels, has been produced as a linear-elastic cellular structure from Ti-6Al-4V alloy by selective electron beam melting. Its linear-elastic properties were measured by tensile experiments and yield an effective Poisson's ratio v(LE) approximate to -0.75, also in agreement with finite element modeling. The similarity between the Poisson's ratios v(SS) of the skeletal structure and v(LE) of the linear-elastic cellular structure emphasizes the fundamental role of geometry for deformation behavior, regardless of the mechanical details of the system. The approach of exploiting structure archives as candidate geometries for auxetic materials also applies to spatial networks and tessellations and can aid the quest for inherently three-dimensional auxetic mechanisms.

Finite auxetic deformations of plane tessellations - Mitschke, Holger; Robins, Vanessa; Mecke, Klaus; Schröder-Turk, Gerd (Proceedings of the Royal Society A-Mathematical Physical and Engineering Sciences, 2013)
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.
Keywords: Poisson's ratio; strain amplification; tilings and tessellations; isostaticity; skeletal frameworks; cellular structures

Geometry: The leading parameter for the Poisson's ratio of bending-dominated cellular solids - Mitschke, Holger; Schury, Fabian; Mecke, Klaus; Wein, Fabian; Stingl, Michael; Schroder-Turk, Gerd E. (International Journal of Solids and Structures, 2016)
Control over the deformation behaviour that a cellular structure shows in response to imposed external forces is a requirement for the effective design of mechanical metamaterials, in particular those with negative Poisson's ratio. This article sheds light on the old question of the relationship between geometric microstructure and mechanical response, by comparison of the deformation properties of bar-and-joint frameworks with those of their realisation as a cellular solid made from linear-elastic material. For ordered planar tessellation models, we find a classification in terms of the number of degrees of freedom of the framework model: first, in cases where the geometry uniquely prescribes a single deformation mode of the framework model, the mechanical deformation and Poisson's ratio of the linearly-elastic cellular solid closely follow those of the unique deformation mode; the result is a bending-dominated deformation with negligible dependence of the effective Poisson's ratio on the underlying material's Poisson's ratio and small values of the effective Young's modulus. Second, in the case of rigid structures or when geometric degeneracy prevents the bending-dominated deformation mode, the effective Poisson's ratio is material-dependent and the Young's modulus (E) over tilde (cs) large. All analysed structures of this type have positive values of the Poisson's ratio and large values of (E) over tilde (cs). Third, in the case, where the framework has multiple deformation modes, geometry alone does not suffice to determine the mechanical deformation. These results clarify the relationship between mechanical properties of a linear-elastic cellular solid and its corresponding bar-and-joint framework abstraction. They also raise the question if, in essence, auxetic behaviour is restricted to the geometry-guided class of bending-dominated structures corresponding to unique mechanisms, with inherently low values of the Young's modulus. (C) 2016 Elsevier Ltd. All rights reserved.
Keywords: Cellular structures; Homogenisation; Floppy frameworks; Auxeticity; Mechanical metamaterial

Symmetry detection of auxetic behaviour in 2D frameworks - Mitschke, Holger; Schröder-Turk, Gerd; Mecke, Klaus; Fowler, P. W.; Guest, S. D. (EPL - Europhysics Letters, 2013)
A symmetry-extended Maxwell treatment of the net mobility of periodic bar-and-joint frameworks is used to derive a sufficient condition for auxetic behaviour of a 2D material. The type of auxetic behaviour that can be detected by symmetry has Poisson's ratio -1, with equal expansion/contraction in all directions, and is here termed equiauxetic. A framework may have a symmetry-detectable equiauxetic mechanism if it belongs to a plane group that includes rotational axes of order n = 6, 4, or 3. If the reducible representation for the net mobility contains mechanisms that preserve full rotational symmetry (A modes), these are equiauxetic. In addition, for n = 6, mechanisms that halve rotational symmetry (B modes) are also equiauxetic. Copyright (c) EPLA, 2013