Encyclopedia of Continuum Mechanics

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| Editors: Holm Altenbach, Andreas Öchsner

Relation Between Eshelbyan Mechanics and Topological Derivative Concept

  • M. XavierEmail author
  • A. A. Novotny
  • J. Sokołowski
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DOI: https://doi.org/10.1007/978-3-662-53605-6_251-2



The Pólya-Szegö polarization tensor, introduced in 1951, naturally appears on the famous Eshelby problem, also referred to as Eshelby theorem. This problem, formulated by Eshelby in 1957 and 1959, forms the basis to the theory of elasticity involving the determination of effective elastic properties of materials with multiple inhomogeneities (inclusions, pores, defects, cracks, etc.). This important result represents one of the major advances in the continuum mechanics theory of the twentieth century. In this work, a relation between Eshelby Mechanics and the modern concept of topological derivative is discussed. The topological derivative is defined as the first term (correction) of the asymptotic expansion of a given shape functional with respect to a small parameter that measures the size of singular domain perturbations, such as holes, cavities, inclusions,...

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© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Applied MathematicsLaboratório Nacional de Computação Científica LNCC/MCTPetrópolisBrazil
  2. 2.Institut Élie Cartan, UMR 7502 Laboratoire de MathématiquesUniversité de LorraineVandoeuvre Lès Nancy CedexFrance
  3. 3.Systems Research Institute of the Polish Academy of SciencesWarszawaPoland

Section editors and affiliations

  • Pablo Andres Munoz Rojas
    • 1
  1. 1.Universidade do Estado de Santa Catarina – UDESCJoinvilleBrazil