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On the variational formulation and implementation of Allen‐Cahn and Cahn‐Hilliard‐type phase field theories
Author(s) -
Bartels Alexander,
Mosler Jörn
Publication year - 2015
Publication title -
pamm
Language(s) - English
Resource type - Journals
ISSN - 1617-7061
DOI - 10.1002/pamm.201510127
Subject(s) - allen–cahn equation , cahn–hilliard equation , phase (matter) , field (mathematics) , type (biology) , energy (signal processing) , energy functional , materials science , thermodynamics , statistical physics , mathematics , physics , mathematical analysis , geology , pure mathematics , paleontology , quantum mechanics , partial differential equation , statistics
Phase field theory is a promising framework for analyzing evolving microstructures in materials. Phenomena like those related to microstructures in Ni‐based superalloys, twin structures in martensites or precipitation in Al‐alloys can be predicted by phase field theory. While phase transformations such as those characterizing twinning are captured by an Allen‐Cahn‐type approach, a Cahn‐Hilliard‐type formulation is used, if the respective interface motion is driven by the concentration of the species. Although the Allen‐Cahn and the Cahn‐Hilliard formulation are indeed different, they do share some similarities. To be more precise, a Cahn‐Hilliard model is obtained by enforcing balance of mass in the Allen‐Cahn approach. Within an energy‐based formulation this can be implemented by adding additional energy terms to the underlying Allen‐Cahn energy. Such a universal energy‐based framework is elaborated in this presentation. (© 2015 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)