{
  "id": "2505.05436",
  "version": "v1",
  "published": "2025-05-08T17:24:25.000Z",
  "updated": "2025-05-08T17:24:25.000Z",
  "title": "The effective energy of a lattice metamaterial",
  "authors": [
    "Xuenan Li",
    "Robert V. Kohn"
  ],
  "comment": "74 pages, 9 figures",
  "categories": [
    "math-ph",
    "math.MP"
  ],
  "abstract": "We study the sense in which the continuum limit of a broad class of discrete materials with periodic structures can be viewed as a nonlinear elastic material. While we are not the first to consider this question, our treatment is more general and more physical than those in the literature. Indeed, it applies to a broad class of systems, including ones that possess mechanisms; and we discuss how the degeneracy that plagues prior work in this area can be avoided by penalizing change of orientation. A key motivation for this work is its relevance to mechanism-based mechanical metamaterials. Such systems often have ``soft modes'', achieved in typical examples by modulating mechanisms. Our results permit the following more general definition of a soft mode: it is a macroscopic deformation whose effective energy vanishes -- in other words, one whose spatially-averaged elastic energy tends to zero in the continuum limit.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2025-05-08T17:24:25.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "49N99",
      "74Q05",
      "74B20"
    ],
    "keywords": [
      "effective energy",
      "lattice metamaterial",
      "soft mode",
      "continuum limit",
      "broad class"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 74,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}