Energy Scaling and Domain Branching in Solid-Solid Phase Transitions

Allan Chan () and Sergio Conti ()
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Allan Chan: Rheinische Friedrich-Wilhelms-Universität Bonn, Institut für Angewandte Mathematik
Sergio Conti: Rheinische Friedrich-Wilhelms-Universität Bonn, Institut für Angewandte Mathematik

A chapter in Singular Phenomena and Scaling in Mathematical Models, 2014, pp 243-260 from Springer

Abstract: Abstract We consider a vectorial model for solid-solid phase transformations, namely, $$\displaystyle{E_{\varepsilon }[u] =\int _{\varOmega }W(Du) +\varepsilon \vert {D}^{2}u\vert \,\mathit{dx}\,,}$$ where $$u:\varOmega \subset {\mathbb{R}}^{2} \rightarrow {\mathbb{R}}^{2}$$ and W vanishes on a set of the form $$K = SO(2)A \cup SO(2)B$$ , with A, B two rank-one connected matrices representing the eigenstrains of two martensitic variants. We study the scaling of the minimal energy under Dirichlet boundary conditions corresponding to the average of A and B. In the case that A and B have two rank-one connections we show that the minimum of $$E_{\varepsilon }$$ scales, for small $$\varepsilon$$ , as $${\varepsilon }^{2/3}$$ , in agreement with previous results on the scalar version of the model. In the case that the two matrices have a single rank-one connection instead we show that a different scaling appears, with energy proportional to $${\varepsilon }^{4/5}$$ . Both results correspond to a self-similar refinement of the microstructure around the boundary, with a different period-doubling pattern. Our results extend to a vectorial, properly frame-indifferent framework previous results on a scalar model by Kohn and Müller.

Keywords: Domain Branching; Solid-Solid Phase Transformations; Eigenstrain; Martensite Variants; Generating Error Terms (search for similar items in EconPapers)
Date: 2014
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-319-00786-1_11

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DOI: 10.1007/978-3-319-00786-1_11

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