{
  "abstract": "We analyze the unusual microstructure resulting from the cubic to orthorhombic phase transformation in the shape-memory alloy ${\\rm Ti}_{76}{\\rm Nb}_{22}{\\rm Al}_{2}$. This seems to represent an approximate experimental realization of a deformation having a finite number of incompatible deformation gradients, which has been known to exist only as a theoretical possibility. Because the middle eigenvalue $γ$ of the transformation strain is very close to 1, the phase transformation in this alloy takes place via nucleation of exact austenite-martensite interfaces rather than, as usual for other alloys, by classical austenite-martensite interfaces separating the austenite from a finely-twinned laminate of martensite. According to the moving mask approximation of Della Porta this leads to the question whether a microstructure can be formed from the set $H$ of the twelve matrices ${\\mathbf V}_i$ on the martensite energy wells that are rank-one connected to the identity, even though these matrices are mutually incompatible. We show that when $γ=1$ and the transformation is volume preserving, such a microstructure is possible that does not deform the grain boundary. This is a consequence of $H$ forming a degenerate $T_{12}$ configuration in the sense of Kirchheim. In fact there are such microstructures supported on just 8 of the 12 matrices in $H$. However there is no deformation satisfying the same boundary condition whose gradient belongs to $H$ almost everywhere. An additional prediction, confirmed experimentally, is that the macroscopic deformation gradient equals the identity everywhere. We also analyse the approximate compatibility of the ${\\mathbf V}_i$. Although mutually incompatible, they are nearly compatible, with some pairs much more nearly compatible than others, these pairs corresponding to interfaces that are more frequently observed.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.06383",
  "author_total": 3,
  "authors": [
    {
      "name": "John M. Ball"
    },
    {
      "name": "Francesco Della Porta"
    },
    {
      "name": "Tomonari Inamura"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/3fevn8dr4ft1s.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.06383",
  "fulltext_url": null,
  "id": "3fevn8dr4ft1s",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.06383",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cond-mat.mtrl-sci"
    ],
    [
      "arxiv",
      "math.AP"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2611.06383",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-04",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/3fevn8dr4ft1s.ris",
  "sources": [
    {
      "first_seen": "2026-11-05T15:20:56Z",
      "native_id": "2611.06383",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.06383"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    },
    {
      "field": "physics",
      "origin": "arxiv:cond-mat.mtrl-sci",
      "subfield": "condensed-matter"
    }
  ],
  "title": "Complex Martensitic Microstructures in ${\\rm Ti}_{76}{\\rm Nb}_{22}{\\rm Al}_{2}$",
  "title_text": "Complex Martensitic Microstructures in Ti_76Nb_22Al_2",
  "type": "preprint",
  "updated_at": "2026-11-09T00:20:30Z",
  "url": "https://research.draconitic.com/paper/3fevn8dr4ft1s",
  "venue": "arXiv",
  "versions": []
}