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A rigidity result for a perturbation of the geometrically linear three‐well problem
Author(s) -
Capella Antonio,
Otto Felix
Publication year - 2009
Publication title -
communications on pure and applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.12
H-Index - 115
eISSN - 1097-0312
pISSN - 0010-3640
DOI - 10.1002/cpa.20297
Subject(s) - scaling , martensite , upper and lower bounds , mathematics , tetragonal crystal system , rigidity (electromagnetism) , shape memory alloy , perturbation (astronomy) , volume fraction , mathematical analysis , elastic energy , metastability , statistical physics , microstructure , geometry , materials science , phase (matter) , physics , thermodynamics , composite material , quantum mechanics
We study a three‐dimensional model for alloys that undergo a cubic‐to‐tetragonal phase transition in the martensitic phase. Any pair of the three martensitic variants can form a stress‐free laminate. However, this laminate is only compatible on average with the remaining variant. The resulting local stresses favor a microstructure if all three variants are present (for instance, because of an externally imposed average strain). Next to the linearized elastic energy, the variational model features an interfacial energy between the three variants. This introduces a material length scale, which together with the sample size (which we mimic by periodic boundary conditions) gives rise to a nondimensional parameter η. We rigorously establish the scaling of the minimal energy e per volume in case of externally imposed volume fractions of the martensitic variants in η. More precisely, we show e ∼ η 2/3 . The upper bound construction is achieved by a few patches of branched laminates; the lower bound relies on suitable interpolation inequalities. This is in the spirit of a celebrated work by Kohn and Müller and relies on techniques developed for domain branching in micromagnetics. We also prove a rigidity result in the sense that if the energy per volume e of a configuration is much smaller than η 2/3 , the configuration is approximately a simple unbranched laminate. In particular, one of the three volume fractions has to be small. This is related to similar rigidity results by Dolzmann and Müller and by Kirchheim. © 2009 Wiley Periodicals, Inc.