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On the Allen—Cahn/Cahn—Hilliard system with a geometrically linear elastic energy

Published online by Cambridge University Press:  20 March 2014

Thomas Blesgen  and
Anja Schlömerkemper
Thomas Blesgen*
Affiliation:
Max Planck Institute for Mathematics in the Sciences, Inselstraße 22, 04103 Leipzig, Germany
Anja Schlömerkemper
Affiliation:
University of Würzburg, Institute for Mathematics, Emil-Fischer-Straße 40, 97074 Würzburg, Germany, (anja.schloemerkemper@mathematik.uni-wuerzburg.de)
*
*Present address: Bingen University, Berlinstraße 109, 55411 Bingen, Germany, (t.blesgen@fh-bingen.de)
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Abstract

We present an extension of the Allen-Cahn/Cahn-Hilliard system that incorporates a geometrically linear ansatz for the elastic energy of the precipitates. The model contains both the elastic Allen-Cahn system and the elastic Cahn-Hilliard system as special cases, and accounts for the microstructures on the microscopic scale. We prove the existence of weak solutions to the new model for a general class of energy functionals. We then give several examples of functionals that belong to this class. This includes the energy of geometrically linear elastic materials for dimensions D < 3. Moreover, we show this for D = 3 in the setting of scalar-valued deformations, which corresponds to the case of anti-plane shear. All this is based on explicit formulae for relaxed energy functionals newly derived in this article for D = 1 and D = 3. In these cases we can also prove the uniqueness of the weak solutions.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 144 , Issue 2 , April 2014 , pp. 241 - 266
Copyright
Copyright © Royal Society of Edinburgh 2014 

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