{
  "id": "2209.04532",
  "version": "v1",
  "published": "2022-09-09T21:23:49.000Z",
  "updated": "2022-09-09T21:23:49.000Z",
  "title": "Derivation of the 1-D Groma-Balogh equations from the Peierls-Nabarro model",
  "authors": [
    "Stefania Patrizi",
    "Tharathep Sangsawang"
  ],
  "comment": "arXiv admin note: text overlap with arXiv:2006.15073",
  "categories": [
    "math.AP"
  ],
  "abstract": "We consider a semi-linear integro-differential equation in dimension one associated to the half Laplacian whose solution represents the atom dislocation in a crystal. The equation comprises the evolutive version of the classical Peierls-Nabarro model. We show that for a large number of dislocations, the solution, properly rescaled, converges to the solution of a fully nonlinear integro-differential equation which is a model for the macroscopic crystal plasticity with density of dislocations. This leads to the formal derivation of the 1-D Groma-Balogh equations \\cite{groma}, a popular model describing the evolution of the density of positive and negative oriented parallel straight dislocation lines. This paper completes the work of \\cite{patsan}. The main novelty here is that we allow dislocations to have different orientation and so we have to deal with collisions of them.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-09-09T21:23:49.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "groma-balogh equations",
      "peierls-nabarro model",
      "derivation",
      "oriented parallel straight dislocation lines",
      "semi-linear integro-differential equation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}