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<p>We present two applications of the volume integral equation method in the context of linear isotropic plane strain elasticity. In the static case, building on Eshelby’s inclusion method, we derive a uniquely solvable integro-differential equation for the eigenstrain. For the time-harmonic elastodynamic case, we reformulate the boundary value problem as a Lippmann–Schwinger integral equation. Our focus then shifts to the inverse problem of reconstructing the elastic properties of an inclusion from the far-field pattern of scattered waves, using a limited number of incident fields. We derive the Fréchet derivative of the forward operator and formulate the linearized far-field equation, which is solved using a regularized Newton-type method. Numerical examples are provided to demonstrate the applicability of the proposed approach.</p>

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Direct and inverse problems in 2D isotropic elasticity using Eshelby and Lippmann–Schwinger integral formulations

  • 期刊论文

  • 发表日期：

    2025年9月21日

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  • D. Gintides,
  • L. Mindrinos

摘要

We present two applications of the volume integral equation method in the context of linear isotropic plane strain elasticity. In the static case, building on Eshelby’s inclusion method, we derive a uniquely solvable integro-differential equation for the eigenstrain. For the time-harmonic elastodynamic case, we reformulate the boundary value problem as a Lippmann–Schwinger integral equation. Our focus then shifts to the inverse problem of reconstructing the elastic properties of an inclusion from the far-field pattern of scattered waves, using a limited number of incident fields. We derive the Fréchet derivative of the forward operator and formulate the linearized far-field equation, which is solved using a regularized Newton-type method. Numerical examples are provided to demonstrate the applicability of the proposed approach.