{
  "aliases": null,
  "backlinks": [
    {
      "field": "about",
      "target": "269C7M56G5"
    },
    {
      "field": "uses",
      "target": "269C7M56G5"
    },
    {
      "field": "defines",
      "target": "S34G7679AK"
    }
  ],
  "body": "\n## Summary\n\nA function of an $m \\times n$ matrix is polyconvex if it is a convex function of the matrix together with all its minors.\nFor $3 \\times 3$ matrices the minors are the entries, the entries of the [[cofactor|0XPKR18APA]] matrix and the [[determinant|00KDZRW4C0]].\nPolyconvexity implies [[Morrey quasiconvexity|DVJ17KVWXP]], and it is much easier to check.\nIt allows energies that blow up as the determinant tends to $0$, so it is the standard hypothesis on stored energies in nonlinear elasticity.\n\n## Explanations\n\n### L1 human\n\nConvexity in the full gradient is too strong for elasticity, because it is incompatible with the fact that compressing a material to zero volume costs infinite energy.\nPolyconvexity is a way around this.\nOne allows the energy to depend convexly on lengths, areas and volumes separately, that is, on the gradient, its cofactor and its determinant.\nFor example, for $2 \\times 2$ matrices, $f(\\xi) = \\abs{\\xi}^2 + (\\det \\xi)^2$ is polyconvex, since it is a convex function of $\\xi$ and $\\det \\xi$.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "kind": "concept",
    "msc": [
      "49J45",
      "26B25",
      "74B20"
    ],
    "name": "Polyconvexity",
    "same_as": {
      "wikidata": "Q7226241",
      "wikipedia": "Polyconvex function"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Polyconvex_function"
      }
    ],
    "tag": "8VYYWQ5CMB"
  },
  "kind": "concept",
  "link": "https://tamnd.github.io/math2/node/8VYYWQ5CMB/",
  "links": null,
  "markdown": "https://tamnd.github.io/math2/node/8VYYWQ5CMB.md",
  "msc": [
    "49J45",
    "26B25",
    "74B20"
  ],
  "name": "Polyconvexity",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/8VY/8VYYWQ5CMB.md",
  "status": null,
  "tag": "8VYYWQ5CMB",
  "uri": "m2://8VYYWQ5CMB",
  "url": "https://tamnd.github.io/math2/wiki/polyconvexity/",
  "writing": {
    "tag": "8VYYWQ5CMB",
    "kind": "concept",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      },
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}