{
  "id": "1701.01033",
  "version": "v1",
  "published": "2017-01-04T14:50:54.000Z",
  "updated": "2017-01-04T14:50:54.000Z",
  "title": "On the existence of minimisers for strain-gradient single-crystal plasticity",
  "authors": [
    "Keith Anguige",
    "Patrick Dondl",
    "Martin Kružík"
  ],
  "comment": "19 pages",
  "categories": [
    "math.AP"
  ],
  "abstract": "We prove the existence of minimisers for a family of models related to the single-slip-to-single-plane relaxation of single-crystal, strain-gradient elastoplasticity with $L^p$-hardening penalty. In these relaxed models, where only one slip-plane normal can be activated at each material point, the main challenge is to show that the energy of geometrically necessary dislocations is lower-semicontinuous along bounded-energy sequences which satisfy the single-plane condition, meaning precisely that this side condition should be preserved in the weak $L^p$-limit. This is done with the aid of an 'exclusion' lemma of Conti \\& Ortiz, which essentially allows one to put a lower bound on the dislocation energy at interfaces of (single-plane) slip patches, thus precluding fine phase-mixing in the limit. Furthermore, using div-curl techniques in the spirit of Mielke \\& M\\\"uller, we are able to show that the usual multiplicative decomposition of the deformation gradient into plastic and elastic parts interacts with weak convergence and the single-plane constraint in such a way as to guarantee lower-semicontinuity of the (polyconvex) elastic energy, and hence the total elasto-plastic energy, given sufficient ($p>2$) hardening, thus delivering the desired result.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-01-04T14:50:54.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "49J10",
      "49J45",
      "74G25",
      "74G65",
      "74N15"
    ],
    "keywords": [
      "strain-gradient single-crystal plasticity",
      "minimisers",
      "elastic parts interacts",
      "total elasto-plastic energy",
      "main challenge"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 19,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}