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The Relaxation of Two-well Energies with Possibly Unequal Moduli

Chenchiah, Isaac V. and Bhattacharya, Kaushik (2008) The Relaxation of Two-well Energies with Possibly Unequal Moduli. Archive for Rational Mechanics and Analysis, 187 (3). pp. 409-479. ISSN 0003-9527. https://resolver.caltech.edu/CaltechAUTHORS:20131008-143908773

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Abstract

The elastic energy of a multiphase solid is a function of its microstructure. Determining the infimum of the energy of such a solid and characterizing the associated optimal microstructures is an important problem that arises in the modeling of the shape memory effect, microstructure evolution, and optimal design. Mathematically, the problem is to determine the relaxation under fixed phase fraction of a multiwell energy. This paper addresses two such problems in the geometrically linear setting. First, in two dimensions, we compute the relaxation under fixed phase fraction for a two-well elastic energy with arbitrary elastic moduli and transformation strains, and provide a characterization of the optimal microstructures and the associated strain. Second, in three dimensions, we compute the relaxation under fixed phase fraction for a two-well elastic energy when either (1) both elastic moduli are isotropic, or (2) the elastic moduli are well ordered and the smaller elastic modulus is isotropic. In both cases we impose no restrictions on the transformation strains. We provide a characterization of the optimal microstructures and the associated strain. We also compute a lower bound that is optimal except possibly in one regime when either (1) both elastic moduli are cubic, or (2) the elastic moduli are well ordered and the smaller elastic modulus is cubic; for moduli with arbitrary symmetry we obtain a lower bound that is sometimes optimal. In all these cases we impose no restrictions on the transformation strains and whenever the bound is optimal we provide a characterization of the optimal microstructures and the associated strain. In both two and three dimensions the quasiconvex envelope of the energy can be obtained by minimizing over the phase fraction. We also characterize optimal microstructures under applied stress.


Item Type:Article
Related URLs:
URLURL TypeDescription
http://dx.doi.org/10.1007/s00205-007-0075-3DOIArticle
http://link.springer.com/article/10.1007/s00205-007-0075-3PublisherArticle
ORCID:
AuthorORCID
Bhattacharya, Kaushik0000-0003-2908-5469
Additional Information:© 2008 Springer-Verlag. Received July 2, 2007. Accepted July 17, 2007. Published online October 5, 2007. Communicated by R. D. James. This work draws on IVC’s doctoral thesis at the California Institute of Technology [13] and was completed and written up during his stay at the Max Planck Institute for Mathematics in the Sciences, Leipzig. We gratefully acknowledge the partial financial support of the the US Army Research Office (MURI grant DAAD19-01-1-0517) and the US Office of Naval Research (N00014-01-1-0937). We thank Robert Kohn for various discussions and suggestions.
Funders:
Funding AgencyGrant Number
Army Research Office (ARO)DAAD19-01-1-0517
Office of Naval Research (ONR)N00014-01-1-0937
Issue or Number:3
Record Number:CaltechAUTHORS:20131008-143908773
Persistent URL:https://resolver.caltech.edu/CaltechAUTHORS:20131008-143908773
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:41750
Collection:CaltechAUTHORS
Deposited By: Ruth Sustaita
Deposited On:08 Oct 2013 22:51
Last Modified:03 Oct 2019 05:51

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