Periodic Auxetics: Structure and Design

Abstract

SummaryMaterials science has adopted the term of auxetic behavior for structural deformations where stretching in some direction entails lateral widening, rather than lateral shrinking. Most studies, in the last three decades, have explored repetitive or cellular structures and used the notion of negative Poisson’s ratio as the hallmark of auxetic behavior. However, no general auxetic principle has been established from this perspective. In the present article, we show that a purely geometric approach to periodic auxetics is apt to identify essential characteristics of frameworks with auxetic deformations and can generate a systematic and endless series of periodic auxetic designs. The critical features refer to convexity properties expressed through families of homothetic ellipsoids.

Transition mechanism for a periodic bar-and-joint framework with limited degrees of freedom controlled by uniaxial load and internal stiffness

“…The proposed periodic structure transforms under three rotational degrees of freedom. In the following, details of the mechanism are explained in a visual way based on geometry [ 58 , 59 ]. The green circle in figure 1 b shows the possible range of rotation of the bars pinned at the centre.…”

Auxetic Regions in Large Deformations of Periodic Frameworks

“…If we use a general structural result established in [12], we can state an equivalent characterization.…”

“…Thus, for large deformations, auxetic behavior can be expected only over restricted regions. The main purpose of the present contribution is to offer an introduction to this type of kinematic inquiry, based on our geometric theory of auxetic deformations and recent fundamental results on the structure and design of auxetic periodic frameworks [8,11,12].…”

Auxetic deformations and elliptic curves

“…As reviewed after this introduction, the hallmark of an auxetic trajectory is that the Gram matrix of a basis of periods traces a curve of symmetric matrices with all tangent directions contained in the positive semidefinite cone. This purely geometric approach has uncovered new principles for auxetic design [11, 12] and has opened the door to rigorous mathematical techniques. Design and recognition of auxetic features may be considered the obverse and reverse of the auxetic coinage.…”

“…This leads, in particular, to an infinite family of planar periodic “auxetic mechanisms”. For arbitrary dimensions, our design principles [11, 12] and connections with convex algebraic geometry bring new significance to classification problems for spectrahedra, as pursued in [30]. Also related are classical results on ternary quartic forms [32].…”