{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,29]],"date-time":"2026-04-29T07:30:48Z","timestamp":1777447848777,"version":"3.51.4"},"reference-count":23,"publisher":"SAGE Publications","issue":"1-2","license":[{"start":{"date-parts":[[2017,3,15]],"date-time":"2017-03-15T00:00:00Z","timestamp":1489536000000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/journals.sagepub.com\/page\/policies\/text-and-data-mining-license"}],"content-domain":{"domain":["journals.sagepub.com"],"crossmark-restriction":true},"short-container-title":["Asymptotic Analysis"],"published-print":{"date-parts":[[2017,3,15]]},"abstract":"<jats:p>In this paper we properly justify the modeling of a thin piezoelectric shallow shell in unilateral contact with a rigid plane. Starting from the three-dimensional nonlinear Signorini problem, we establish the convergence of the displacement field and of the electric potential as the thickness of the shell goes to zero. More precisely we obtain that the transverse mechanical displacement field coupled with the in-plane components solve an obstacle problem described new piezoelectric characteristics. We also investigate the very popular case of cubic crystals and show that, for two-dimensional shallow shells, the coupling piezoelectric effect disappears.<\/jats:p>","DOI":"10.3233\/asy-171411","type":"journal-article","created":{"date-parts":[[2017,3,17]],"date-time":"2017-03-17T10:43:54Z","timestamp":1489747434000},"page":"71-97","update-policy":"https:\/\/doi.org\/10.1177\/sage-journals-update-policy","source":"Crossref","is-referenced-by-count":1,"title":["Mathematical justification of the obstacle problem for a piezoelectric shallow shell"],"prefix":"10.1177","volume":"102","author":[{"given":"Guan","family":"Yan","sequence":"first","affiliation":[{"name":"Mathematics & Science College, Shanghai Normal University, Shanghai 200234, China. E-mails:\u00a0,\u00a0"}]},{"given":"Bernadette","family":"Miara","sequence":"additional","affiliation":[{"name":"Mathematics & Science College, Shanghai Normal University, Shanghai 200234, China. E-mails:\u00a0,\u00a0"}]}],"member":"179","published-online":{"date-parts":[[2017,3,15]]},"reference":[{"key":"ref001","unstructured":"W.G.\u00a0Cady Piezoelectricity, An Introduction to the Theory and Applications of Electromechanical Phenomena in Crystals, McGraw-Hill, New York, 1946."},{"key":"ref002","unstructured":"P.G.\u00a0Ciarlet, Mathematical Elasticity, Vol II, Theory of Plates, North-Holland, Amsterdam, 1997."},{"key":"ref003","doi-asserted-by":"publisher","DOI":"10.1007\/s10659-005-4738-8"},{"key":"ref004","first-page":"315","volume":"18","author":"Ciarlet P.G.","year":"1979","journal-title":"J. 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J. of Numerical Analysis and Modeling, B"},{"key":"ref015","unstructured":"J.L.\u00a0Lions, Quelques M\u00e9thodes de R\u00e9solution des Probl\u00eames aux Limites Non Lin\u00e9aires, Dunod\u2013Gauthier-Villars, Paris, 1969."},{"key":"ref016","doi-asserted-by":"publisher","DOI":"10.1016\/S0895-7177(98)00105-8"},{"key":"ref017","doi-asserted-by":"publisher","DOI":"10.3233\/ASY-2012-1126"},{"key":"ref018","unstructured":"J.C.\u00a0Paumier, Mod\u00e9lisation asymptotique d\u2019un probl\u00e8me de plaque mince en contact unilat\u00e9ral avec frottement contre un obstacle rigide, Pr\u00e9publication L.M.C, 2002, http:\/\/www-lmc.imag.fr\/~paumier\/signoplaque.ps."},{"key":"ref019","unstructured":"M.\u00a0Rahmoune, Plaques intelligentes piezoelectriques, modelisation et application au contr\u00f4le sant\u00e9, Ph.D. Thesis, University of Paris VI, 1998."},{"key":"ref020","doi-asserted-by":"publisher","DOI":"10.3233\/ASY-2007-843"},{"key":"ref021","doi-asserted-by":"publisher","DOI":"10.3233\/ASY-2001-415"},{"key":"ref022","doi-asserted-by":"publisher","DOI":"10.3233\/ASY-151339"},{"key":"ref023","unstructured":"G.\u00a0Yan and B.\u00a0Miara, Mathmatical Jusitification of the Obstacle Problem for piezoelectric shallow, shell in curvilinear coordinate, manuscript."}],"container-title":["Asymptotic Analysis"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/journals.sagepub.com\/doi\/pdf\/10.3233\/ASY-171411","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/journals.sagepub.com\/doi\/full-xml\/10.3233\/ASY-171411","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/journals.sagepub.com\/doi\/pdf\/10.3233\/ASY-171411","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,28]],"date-time":"2026-04-28T12:36:00Z","timestamp":1777379760000},"score":1,"resource":{"primary":{"URL":"https:\/\/journals.sagepub.com\/doi\/10.3233\/ASY-171411"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2017,3,15]]},"references-count":23,"journal-issue":{"issue":"1-2","published-print":{"date-parts":[[2017,3,15]]}},"alternative-id":["10.3233\/ASY-171411"],"URL":"https:\/\/doi.org\/10.3233\/asy-171411","relation":{},"ISSN":["0921-7134","1875-8576"],"issn-type":[{"value":"0921-7134","type":"print"},{"value":"1875-8576","type":"electronic"}],"subject":[],"published":{"date-parts":[[2017,3,15]]}}}