Auxetic strains—insight from iso-auxetic materials

Abstract

Auxeticity is the result of internal structural degrees of freedom that get in the way of affine deformations. This paper proposes a new understanding of strains in disordered auxetic materials. A class of iso-auxetic structures is identified, which are auxetic structures that are also isostatic, and these are distinguished from conventional elasto-auxetic materials. It is then argued that the mechanisms that give rise to auxeticity are the same in both classes of materials and the implications of this observation on the equations that govern the strain are explored. Next, the compatibility conditions of Saint Venant are demonstrated to be irrelevant for the determination of stresses in iso-auxetic materials, which are governed by balance conditions alone. This leads to the conclusion that elasticity theory is not essential for the general description of auxetic behaviour. One consequence of this is that characterisation in terms of negative Poisson's ratio may be of limited utility.

A new equation is then proposed for the dependence of the strain on local rotational and expansive fields. Central to the characterization of the geometry of the structure, to the iso-auxetic stress field equations, and to the strain-rotation relation is a specific fabric tensor. This tensor is defined here explicitly for two-dimensional systems, however, disordered. It is argued that, while the proposed dependence of the strain on the local rotational and expansive fields is common to all auxetic materials, iso-auxetic and elasto-auxetic materials may exhibit significantly different macroscopic behaviours.

Keywords:

• Iso-auxetic
• Elasto-auxetic
• Fabric tensor
• Disordered structures
• Stress–strain

Notes

^† Dilatancy is not mentioned explicitly in Ref. [15], but this phenomenon arises from grain rotations exactly as auxeticity originates in element rotations in auxetic materials. Thus, equation (4) describes both auxeticity and dilatancy in systems of rotating rigid elements. Equation (4) is paralleled by the second term on the right hand side of equation (5.1) in Ref. [15]. The equivalence can be observed by noting that: (i) their relation is for the strain rate rather than strain, i.e. ω _kl is the time derivative of my θ, (ii) their p is my Q, and (iii) their relation is given for three-dimensional systems while mine is for two,

^¶ Effects of the boundary are ignored because for these effects are of order These effects can be included for finite systems without loss off generality.

^‡ The terms and the notations are intended to make contact with a related analysis in solid open-cell foams, where vertices and cells are key concepts [11].

^† In fact, the following discussion holds for more general systems where the mean number of contacts per unit is three.