Asymptotic Rigidity and Homogenization of Auxetic Metamaterials with Checkerboard Structure
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In this talk, we discuss homogenization of elastic composites with stiff rotating squares without self-interpenetration. We establish an asymptotic rigidity result, showing that, under suitable scaling assumptions, the admissible macroscopic deformations are affine conformal contractions. This identifies the composite as a mechanical metamaterial with a negative Poisson's ratio. Our strategy for deriving our result is to tackle first an idealized model with full rigidity on the stiff tiles to acquire insight into the mechanics of the model and then transfer the findings and methodology to the model with diverging elastic constants. The latter requires a new quantitative geometric rigidity estimate for non-connected touching squares. The homogenization limit is obtained by variational methods. References: Wolf-Patrick Düll, Dominik Engl, Carolin Kreisbeck, A variational perspective on auxetic metamaterials of checkerboard-type, Arch. Ration. Mech. Anal. 248, no. 3, Paper No. 46, 55 pp., 2024.
