  Auxetic materials, or negative-Poisson's-ratio materials, are important
technologically and fascinating theoretically. When loaded by external
stresses, their internal strains are governed by correlated motion of internal
structural degrees of freedom. The modelling of such materials is mainly based
on ordered structures, despite existence of auxetic behaviour in disordered
structures and the advantage in manufacturing disordered structures for most
applications. We describe here a first-principles expression for strains in
disordered such materials, based on insight from a family of 'iso-auxetic'
structures. These are structures, consisting of internal structural elements,
which we name 'auxetons', whose inter-element forces can be computed from
statics alone. Iso-auxetic structures make it possible not only to identify the
mechanisms that give rise to auxeticity, but also to write down the explicit
dependence of the strain rate on the local structure, which is valid to all
auxetic materials. It is argued that stresses give rise to strains via two
mechanisms: auxeton rotations and auxeton expansion / contraction. The former
depends on the stress via a local fabric tensor, which we define explicitly for
2D systems. The latter depends on the stress via an expansion tensor. Whether a
material exhibits auxetic behaviour or not depends on the interplay between
these two fields. This description has two major advantages: it applies to any
auxeton-based system, however disordered, and it goes beyond conventional
elasticity theory, providing an explicit expression for general auxetic strains
and outlining the relevant equations.
