TY - GEN
T1 - A link between the shape of the austenite-martensite interface and the behaviour of the surface energy
A1 - Elfanni,Abdellah
A1 - Fuchs,Martin
Y1 - 2012/02/10
N2 - 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}.
CY - Saarbrücken
PB - Saarländische Universitäts- und Landesbibliothek
AD - Postfach 151141, 66041 Saarbrücken
UR - http://scidok.sulb.uni-saarland.de/volltexte/2012/4446
ER -