Geometric auxetics

We formulate a mathematical theory of auxetic behavior based on one-parameter deformations of periodic frameworks. Our approach is purely geometric, relies on the evolution of the periodicity lattice and works in any dimension. We demonstrate its usefulness by predicting or recognizing, without experiment, computer simulations or numerical approximations, the auxetic capabilities of several well-known structures available in the literature. We propose new principles of auxetic design and rely on the stronger notion of expansive behavior to provide an infinite supply of planar auxetic mechanisms and several new three-dimensional structures

Expansive periodic mechanisms

A one-parameter deformation of a periodic bar-and-joint framework is expansive when all distances between joints increase or stay the same. In dimension two, expansive behavior can be fully explained through our theory of periodic pseudo-triangulations. However, higher dimensions present new challenges. In this paper we study a number of periodic frameworks with expansive capabilities in dimension $d\geq 3$ and register both similarities and contrasts with the two-dimensional case

Auxetic regions in large deformations of periodic frameworks

In materials science, auxetic behavior refers to lateral widening upon stretching. We investigate the problem of finding domains of auxeticity in global deformation spaces of periodic frameworks. Case studies include planar periodic mechanisms constructed from quadrilaterals with diagonals as periods and other frameworks with two vertex orbits. We relate several geometric and kinematic descriptions.Comment: Presented at the International Conference on "Interdisciplinary Applications of Kinematics" (IAK18), Lima, Peru, March 201

Auxetics Abounding

Auxetic behavior refers to lateral widening upon stretching. Although a structural origin for this rather counterintuitive type of deformation was often suggested, a theoretical understanding of the role of geometry in auxetic behavior has been a challenge for a long time. However, for structures modeled as periodic bar-and-joint frameworks, including atom-and-bond frameworks in crystalline materials, there is a complete geometric solution which opens endless possibilities for new auxetic designs. We construct a large family of three-dimensional auxetic periodic mechanisms and discuss the ideas involved in their design

Infinitesimal Periodic Deformations and Quadrics

We describe a correspondence between the infinitesimal deformations of a periodic bar-and-joint framework and periodic arrangements of quadrics. This intrinsic correlation provides useful geometric characteristics. A direct consequence is a method for detecting auxetic deformations, identified by a pattern consisting of homothetic ellipsoids. Examples include frameworks with higher crystallographic symmetry

New Principles for Auxetic Periodic Design

We show that, for any given dimension d ≥ 2, the range of distinct possible designs for periodic frameworks with auxetic capabilities is infinite. We rely on a purely geometric approach to auxetic trajectories developed within our general theory of deformations of periodic frameworks

Auxetic Deformations and Elliptic Curves

The problem of detecting auxetic behavior, originating in materials science and mathematical crystallography, refers to the property of a flexible periodic bar-and-joint framework to widen, rather than shrink, when stretched in some direction. The only known algorithmic solution for detecting infinitesimal auxeticity is based on the rather heavy machinery of fixed-dimension semi-definite programming. In this paper we present a new, simpler algorithmic approach which is applicable to a natural family of 3D periodic bar-and-joint frameworks with 3 degrees-of-freedom. This class includes most zeolite structures, which are important for applications in computational materials science. We show that the existence of auxetic deformations is related to properties of an associated elliptic curve. A fast algorithm for recognizing auxetic capabilities is obtained via the classical Aronhold invariants of the cubic form defining the curve

Periodic Tilings and Auxetic Deployments

We investigate geometric characteristics of a specific planar periodic framework with three degrees of freedom. While several avatars of this structural design have been considered in materials science under the name of chiral or missing rib models, all previous studies have addressed only local properties and limited deployment scenarios. We describe the global configuration space of the framework and emphasize the geometric underpinnings of auxetic deformations. Analogous structures may be considered in arbitrary dimension

Geometric Deformations of Sodalite Frameworks

In mathematical crystallography and computational materials science, it is important to infer flexibility properties of framework materials from their geometric representation. We study combinatorial, geometric and kinematic properties for frameworks modeled on sodalite

New Principles for Auxetic Periodic Design

We show that, for any given dimension d ≥ 2, the range of distinct possible designs for periodic frameworks with auxetic capabilities is infinite. We rely on a purely geometric approach to auxetic trajectories developed within our general theory of deformations of periodic frameworks