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Preprint (Vorabdruck) zugänglich unter
URN: urn:nbn:de:bsz:291-scidok-44468
URL: http://scidok.sulb.uni-saarland.de/volltexte/2012/4446/


A link between the shape of the austenite-martensite interface and the behaviour of the surface energy

Elfanni, Abdellah ; Fuchs, Martin

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Freie Schlagwörter (Englisch): microstructure , martensitic phase transformation
Institut: Fachrichtung 6.1 - Mathematik
DDC-Sachgruppe: Mathematik
Dokumentart: Preprint (Vorabdruck)
Schriftenreihe: Preprint / Fachrichtung Mathematik, Universit├Ąt des Saarlandes
Bandnummer: 100
Sprache: Englisch
Erstellungsjahr: 2003
Publikationsdatum: 10.02.2012
Kurzfassung auf Englisch: Let \Omega\subset\mathbb{R}^{2} denote a bounded Lipschitz domain and consider some portion \Gamma_{0} of \partial\Omega representing the austenite-twinned martensite interface which is not assumed to be a straight segment. We prove
\underset{u\inmathcal{W}(\Omega)}{\mbox{inf}}\int_{\Omega}\varphi(\nabla u(x,y))dxdy=0 (*)
for an elastic energy density \varphi:\mathbb{R}^{2}\rightarrow[0,\infty) such that \varphi(0,\pm1)=0. Here \mathcal{W}(\Omega) consists of all functions u from the Sobolev class W^{1,\infty}(\Omega) such that \left|u_{y}\right|=0 a.e. on \Omega together with u=0 on \Gamma_{0}. We will first show that for \Gamma_{0} having a vertical tangent one cannot always expect a finite surface energy, i.e. in the above problem the condition
u_{yy} is a Radon measure such that \int_{\Omega}\left|u_{yy}(x,y)\right|dxdy<+\infty
in general cannot be included. This generalizes a result of [W.] where \Gamma_{0} is a vertical straight line. Property (*) is established by constructing some minimizing sequences vanishing on the whole boundary \partial\Omega, that is, one can even take \Gamma_{0}=\partial\Omega. We also show that the existence or nonexistence of minimizers depends on the shape of the austenitetwinned martensite interface \Gamma_{0}.
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