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

Autor:	Thomas Blesgen, Anja Schlömerkemper
Rok vydání:	2014
Předmět:	Shear (sheet metal) Physics Class (set theory) Explicit formulae Computer Science::Information Retrieval General Mathematics Linear elasticity Mathematical analysis Elastic energy Uniqueness Microscopic scale Ansatz
Zdroj:	Proceedings of the Royal Society of Edinburgh: Section A Mathematics. 144:241-266
ISSN:	1473-7124 0308-2105
DOI:	10.1017/s030821051200203x
Popis:	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.
Databáze:	OpenAIRE
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