Small volume asymptotics for anisotropic elastic inclusions

Elena Beretta, Eric Bonnetier, Elisa Francini, Anna L. Mazzucato

    Research output: Contribution to journalArticle

    Abstract

    We derive asymptotic expansions for the displacement at the boundary of a smooth, elastic body in the presence of small inhomogeneities. Both the body and the inclusions are allowed to be anisotropic. This work extends prior work of Capdeboscq and Vogelius (Math. Modeling Num. Anal. 37, 2003) for the conductivity case. In particular, we obtain an asymptotic expansion of the diérence between the displacements at the boundary with and without inclusions, under Neumann boundary conditions, to first order in the measure of the inclusions. We impose no geometric conditions on the inclusions, which need only be measurable sets. The first-order correction contains a moment or polarization tensor M that encodes the eéct of the inclusions. We also derive some basic properties of this tensor. In the case of thin, strip-like, planar inhomogeneities we obtain a formula for only in terms of the elasticity tensors, which we assume strongly convex, their inverses, and a frame on the curve that supports the inclusion. We prove uniqueness of in this setting and recover the formula previously obtained by Beretta and Francini.

    Original languageEnglish (US)
    Pages (from-to)1-23
    Number of pages23
    JournalInverse Problems and Imaging
    Volume6
    Issue number1
    DOIs
    StatePublished - Feb 1 2012

    Fingerprint

    Inclusion
    Tensors
    Tensor
    Inhomogeneity
    Asymptotic Expansion
    First-order
    Measurable set
    Elastic body
    Neumann Boundary Conditions
    Conductivity
    Strip
    Elasticity
    Polarization
    Uniqueness
    Boundary conditions
    Moment
    Curve
    Modeling

    Keywords

    • Anisotropic elasticity
    • Homogenization
    • Inclusions
    • Polarization tensor
    • Small volume asymptotics

    ASJC Scopus subject areas

    • Analysis
    • Modeling and Simulation
    • Discrete Mathematics and Combinatorics
    • Control and Optimization

    Cite this

    Beretta, E., Bonnetier, E., Francini, E., & Mazzucato, A. L. (2012). Small volume asymptotics for anisotropic elastic inclusions. Inverse Problems and Imaging, 6(1), 1-23. https://doi.org/10.3934/ipi.2012.6.1

    Small volume asymptotics for anisotropic elastic inclusions. / Beretta, Elena; Bonnetier, Eric; Francini, Elisa; Mazzucato, Anna L.

    In: Inverse Problems and Imaging, Vol. 6, No. 1, 01.02.2012, p. 1-23.

    Research output: Contribution to journalArticle

    Beretta, E, Bonnetier, E, Francini, E & Mazzucato, AL 2012, 'Small volume asymptotics for anisotropic elastic inclusions', Inverse Problems and Imaging, vol. 6, no. 1, pp. 1-23. https://doi.org/10.3934/ipi.2012.6.1
    Beretta, Elena ; Bonnetier, Eric ; Francini, Elisa ; Mazzucato, Anna L. / Small volume asymptotics for anisotropic elastic inclusions. In: Inverse Problems and Imaging. 2012 ; Vol. 6, No. 1. pp. 1-23.
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