Please use this identifier to cite or link to this item:
doi:10.22028/D291-26297
Title: | A link between the shape of the austenite-martensite interface and the behaviour of the surface energy |
Author(s): | Elfanni, Abdellah Fuchs, Martin |
Language: | English |
Year of Publication: | 2003 |
Free key words: | microstructure martensitic phase transformation |
DDC notations: | 510 Mathematics |
Publikation type: | Other |
Abstract: | 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}. |
Link to this record: | urn:nbn:de:bsz:291-scidok-44468 hdl:20.500.11880/26353 http://dx.doi.org/10.22028/D291-26297 |
Series name: | Preprint / Fachrichtung Mathematik, Universität des Saarlandes |
Series volume: | 100 |
Date of registration: | 10-Feb-2012 |
Faculty: | MI - Fakultät für Mathematik und Informatik |
Department: | MI - Mathematik |
Collections: | SciDok - Der Wissenschaftsserver der Universität des Saarlandes |
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preprint_100_03.pdf | 250,88 kB | Adobe PDF | View/Open |
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