Auxetic strains—insight from iso-auxetic materials

Equally probable positive and negative Poisson's ratios in disordered planar systems

Auxetic materials, characterised by a negative Poisson's ratio, have properties that are different from most conventional materials. These are a result of the constraints on the kinematics of the material's basic structural components, and have important technological implications. Models of these materials have been studied extensively, but theoretical descriptions have remained largely limited to materials with an ordered microstructure. Here we investigate whether negative Poisson's ratios can arise spontaneously in disordered systems. To this end, we develop a quantitative description of the structure in systems of connected basic elements, which enables us to analyse the local and global responses to small external tensile forces. We find that the Poisson's ratios in these disordered systems are equally likely to be positive or negative on both the element and system scales. Separating the strain into translational, rotational and expansive components, we find that the translational strains of neighbouring basic structural elements are positively correlated, while their rotations are negatively correlated. There is no correlation in this type of system between the local auxeticity and local structural characteristics. Our results suggest that auxeticity is more common in disordered structures than the ubiquity of positive Poisson's ratios in macroscopic materials would suggest.

Auxetic two-dimensional lattices with Poisson's ratio arbitrarily close to −1

In this paper, we propose a class of lattice structures with macroscopic Poisson's ratio arbitrarily close to the stability limit −1. We tested experimentally the effective Poisson's ratio of the microstructured medium; the uniaxial test was performed on a thermoplastic lattice produced with a three-dimensional printing technology. A theoretical analysis of the effective properties was performed, and the expression of the macroscopic constitutive properties is given in full analytical form as a function of the constitutive properties of the elements of the lattice and on the geometry of the microstructure. The analysis was performed on three microgeometries leading to an isotropic behaviour for the cases of three- and sixfold symmetries and to a cubic behaviour for the case of fourfold symmetry.

A continuum hyperelastic model for auxetic materials

We propose a simple mathematical model to describe isotropic auxetic materials in the framework of the classical theory of nonlinear elasticity. The model is derived from the Blatz–Ko constitutive equation for compressible foams and makes use of a non-monotonic Poisson function. An application to the modelling of auxetic foams is considered and it is shown that the material behaviour is adequately described with only three constitutive parameters.

Finite auxetic deformations of plane tessellations

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 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.

Finding auxetic frameworks in periodic tessellations

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 νss=-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 νLE≈-0.75, also in agreement with finite element modeling. The similarity between the Poisson's ratios νSS of the skeletal structure and νLE 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.