In materials science and engineering, auxetic behavior refers to deformations of flexible structures where stretching in some direction involves lateral widening, rather than lateral shrinking. We address the problem of detecting auxetic behavior for flexible periodic bar-and-joint frameworks. Currently, the only known algorithmic solution 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 three-dimensional periodic bar-and-joint frameworks with three 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. Related algorithmic alternatives are also considered.

在材料科学和工程学中，膨胀行为是指柔性结构的变形，其中在某个方向上的拉伸涉及横向扩展而不是横向收缩。我们解决了在灵活的周期性棒-关节框架中检测拉力行为的问题。当前，唯一已知的算法解决方案是基于固定维半定编程的相当繁琐的机制。在本文中，我们提出了一种新的，更简单的算法方法，该方法适用于具有三个自由度的自然的三维周期杆和关节框架系列。此类包含大多数沸石结构，这对于计算材料科学中的应用很重要。我们表明，膨胀变形的存在与相关的椭圆曲线的性质有关。通过定义曲线的三次形式的经典Aronhold不变量，获得了一种用于识别促发能力的快速算法。还考虑了相关的算法选择。