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1. Auxetic Regions in Large Deformations of Periodic Frameworks

   https://doi.org/10.1007/978-3-030-16423-2_18  

   Borcea, Ciprian S.; Streinu, Ileana (May 2019, Interdisciplinary Applications of Kinematics. Mechanisms and Machine Science)

   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. 

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2. Saturation and periodic self-stress in geometric auxetics

   https://doi.org/10.1098/rsos.220765  

   Borcea, Ciprian S.; Streinu, Ileana (August 2022, Royal Society Open Science)

   The auxetic structures considered in this paper are three-dimensional periodic bar-and-joint frameworks. We start with the specific purpose of obtaining an auxetic design with underlying periodic graph of low valency. Adapting a general methodology, we produce an initial framework with valency seven and one degree of freedom. Then, we describe a saturation process, whereby edge orbits are added up to valency 16, with no alteration of the deformation path. This is reflected in a large dimension for the space of periodic self-stresses. The saturated version has higher crystallographic symmetry and allows a precise description of the deformation trajectory. Reducing saturation by adequate removal of edge orbits results in vast numbers of distinct auxetic designs which obey the same kinematics. 

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3. Infinitesimal Periodic Deformations and Quadrics

   https://doi.org/10.3390/sym13091719  

   Borcea, Ciprian S.; Streinu, Ileana (September 2021, Symmetry)

   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. 

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4. Auxetic interval determination and experimental validation for a three-dimensional periodic framework

   https://doi.org/10.1177/10812865211041390  

   Borcea, Ciprian S.; Broeren, Freek G.; Herder, Just L.; Streinu, Ileana; van der Wijk, Volkert (May 2022, Mathematics and Mechanics of Solids)

   Auxetic behavior refers to lateral widening upon stretching or, in reverse, lateral shrinking upon compression. When an initially auxetic structure is actuated by compression or extension, it will not necessarily remain auxetic for larger deformations. In this paper, we investigate the auxetic range in the deformation of a periodic framework with one degree of freedom. We use geometric criteria to identify the interval where the deformation is auxetic and validate these theoretical findings with compression experiments on sample structures with [Formula: see text] unit cells. 

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5. ABC ‐Auxetics: An Implicit Design Approach for Negative Poisson's Ratio Materials

   https://doi.org/10.1002/adem.202301359  

   Ebert, Matthew; Adhikari, Riddhi; Hasan, Md_Kamrul; Lupo, Kai; Akleman, Ergun; Pharr, Matt; Krishnamurthy, Vinayak_R (February 2024, Advanced Engineering Materials)

   A novel methodology is introduced for designing auxetic (negative Poisson's ratio) structures based on topological principles and is demonstrated by investigating a new class of auxetics based on two‐dimensional (2D) textile weave patterns. Conventional methodology for designing auxetic materials typically involves determining a single deformable block (a unit cell) of material whose shape results in auxetic behavior. Consequently, patterning such a unit cell in a 2D (or 3D) domain results in a larger structure that exhibits overall auxetic behavior. Such an approach naturally relies on some prior intuition and experience regarding which unit cells may be auxetic. Second, tuning the properties of the resulting structures is typically limited to parametric variations of the geometry of a specific type of unit cell. Thus, most of the currently known auxetic structures belong to a selected few classes of unit cell geometries that are explicitly defined in accordance with a specified topological (i.e., grid structure). Herein, a new class of auxetic structures is demonstrated that, while periodic, can be generated implicitly, i.e., without reference to a specific unit cell design. The approach leverages weave‐based parameters (A–B–C), resulting in a rich design space for auxetics that is previously unexplored. 

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