A nonlinear homogenization-based perspective on the soft modes and effective energies of some conformal metamaterials

2025/09/21 by Xuenan Li, Robert V. Kohn, Li, Xuenan +1 · 1 citation
Engineering · Physics and Astronomy · #49Nxx #Acoustic Wave Phenomena Research #Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Nonlinear Photonic Systems #Numerical methods in engineering

paper · pdf · doi:10.48550/arxiv.2509.16907

openalex publication_date 2025/09/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

There is a growing mechanics literature concerning the macroscopic properties of mechanism-based mechanical metamaterials. This amounts mathematically to a homogenization problem involving nonlinear elasticity. A key goal is to identify the "soft modes" of the metamaterial. We achieve this goal using methods from homogenization for some specific 2D examples -- including discrete models of the Rotating Squares metamaterial and the Kagome metamaterial -- whose soft modes are compressive conformal maps. The innovation behind this achievement is a new technique for bounding the effective energy from below, which takes advantage of the metamaterial's structure and symmetry.

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