Incompatible Sets of Gradients and Metastability

J. M. Ball, R. D. James

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10 Scopus citations


We give a mathematical analysis of a concept of metastability induced by incompatibility. The physical setting is a single parent phase, just about to undergo transformation to a product phase of lower energy density. Under certain conditions of incompatibility of the energy wells of this energy density, we show that the parent phase is metastable in a strong sense, namely it is a local minimizer of the free energy in an L1 neighbourhood of its deformation. The reason behind this result is that, due to the incompatibility of the energy wells, a small nucleus of the product phase is necessarily accompanied by a stressed transition layer whose energetic cost exceeds the energy lowering capacity of the nucleus. We define and characterize incompatible sets of matrices, in terms of which the transition layer estimate at the heart of the proof of metastability is expressed. Finally we discuss connections with experiments and place this concept of metastability in the wider context of recent theoretical and experimental research on metastability and hysteresis.

Original languageEnglish (US)
Pages (from-to)1363-1416
Number of pages54
JournalArchive For Rational Mechanics And Analysis
Issue number3
StatePublished - Dec 30 2015
Externally publishedYes

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Publisher Copyright:
© 2015, Springer-Verlag Berlin Heidelberg.


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