{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,12]],"date-time":"2025-12-12T09:00:02Z","timestamp":1765530002948,"version":"3.48.0"},"reference-count":19,"publisher":"MDPI AG","issue":"24","license":[{"start":{"date-parts":[[2025,12,11]],"date-time":"2025-12-11T00:00:00Z","timestamp":1765411200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001665","name":"French National Research Agency","doi-asserted-by":"publisher","award":["ANR-23-CE29-0003"],"award-info":[{"award-number":["ANR-23-CE29-0003"]}],"id":[{"id":"10.13039\/501100001665","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001665","name":"French National Research Agency","doi-asserted-by":"publisher","award":["ANR-17-EURE-0002"],"award-info":[{"award-number":["ANR-17-EURE-0002"]}],"id":[{"id":"10.13039\/501100001665","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Nanomaterials"],"abstract":"<jats:p>A numerical approach based on the Rayleigh-Ritz method and using a modification of the so-called xyz algorithm is introduced to calculate the acoustic vibrations of clamped objects whose shape is delimited by superquadrics. It is then used to improve the convergence for the free vibrations of core\u2013shell objects. The issue in this case is first illustrated in the simpler one-dimensional case of the thickness breathing vibration of an infinite \u201ccore-shell\u201d plate. Functions suitable for solving the clamped vibrations of the core are added to the original xyz basis of functions to improve the convergence for core\u2013shell superquadrics. The new basis obeys the same symmetry rules as the original one, which allows calculating vibrations for individual irreducible representations when the objects are made of cubic, tetragonal, or orthorhombic materials whose principal axes are aligned with those of the superquadrics. This method is validated for an isotropic spherical core\u2013shell system for which analytic solutions exist.<\/jats:p>","DOI":"10.3390\/nano15241865","type":"journal-article","created":{"date-parts":[[2025,12,11]],"date-time":"2025-12-11T15:43:43Z","timestamp":1765467823000},"page":"1865","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Rayleigh\u2013Ritz Approximation of the Acoustic Vibrations of Clamped Superquadrics\u2014Application to Free Core\u2013Shell Objects"],"prefix":"10.3390","volume":"15","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-3734-9794","authenticated-orcid":false,"given":"Sajana","family":"S","sequence":"first","affiliation":[{"name":"Universit\u00e9 Bourgogne Europe, CNRS, Laboratoire Interdisciplinaire Carnot de Bourgogne ICB UMR 6303, 21000 Dijon, France"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5135-4026","authenticated-orcid":false,"given":"Mar\u00eda del Carmen","family":"Marco de Lucas","sequence":"additional","affiliation":[{"name":"Universit\u00e9 Bourgogne Europe, CNRS, Laboratoire Interdisciplinaire Carnot de Bourgogne ICB UMR 6303, 21000 Dijon, France"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1249-2730","authenticated-orcid":false,"given":"Lucien","family":"Saviot","sequence":"additional","affiliation":[{"name":"Universit\u00e9 Bourgogne Europe, CNRS, Laboratoire Interdisciplinaire Carnot de Bourgogne ICB UMR 6303, 21000 Dijon, France"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2025,12,11]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"26","DOI":"10.1063\/1.881483","article-title":"Resonant Ultrasound Spectroscopy","volume":"49","author":"Maynard","year":"1996","journal-title":"Phys. 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Sound Vib."}],"container-title":["Nanomaterials"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2079-4991\/15\/24\/1865\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,12,12]],"date-time":"2025-12-12T08:55:03Z","timestamp":1765529703000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2079-4991\/15\/24\/1865"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,12,11]]},"references-count":19,"journal-issue":{"issue":"24","published-online":{"date-parts":[[2025,12]]}},"alternative-id":["nano15241865"],"URL":"https:\/\/doi.org\/10.3390\/nano15241865","relation":{},"ISSN":["2079-4991"],"issn-type":[{"type":"electronic","value":"2079-4991"}],"subject":[],"published":{"date-parts":[[2025,12,11]]}}}