Abstract
We formulate a mathematical theory of auxetic behaviour based on one-parameter deformations of periodic frameworks. Our approach is purely geome- tric, 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 behaviour to provide an infinite supply of planar auxetic mechanisms and several new three-dimensional structures.
Highlights
The notion of auxetic behaviour emerged from renewed interest in materials with negative Poisson’s ratios [1,2,3]
We have shown above the far-reaching role of the notions of pseudo-triangulation and expansive behaviour for planar periodic frameworks and auxetic investigations
We have introduced a geometric theory of auxetic one-parameter deformations of periodic barand-joint frameworks, applicable in arbitrary dimension
Summary
The notion of auxetic behaviour emerged from renewed interest in materials with negative Poisson’s ratios [1,2,3]. Considerations related to composite materials appear in [23] (see [24,25,26,27]) In spite of this diversity of examples, no geometric principle underlying the auxetic properties of periodic structures has been proposed so far. We define a general notion of auxetic path or auxetic one-parameter deformation for periodic bar-and-joint frameworks in Euclidean spaces of arbitrary dimension d. We emphasize in this context the far-reaching role of the stronger notion of expansive behaviour and its correlate, the concept of periodic pseudotriangulation.
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