On the inverse elastic problem for isotropic media using Eshelby and Lippmann-Schwinger integral formulations
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| Format: | Preprint |
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2025
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| _version_ | 1866916659488358400 |
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| author | Gintides, Drossos Mindrinos, Leonidas |
| author_facet | Gintides, Drossos Mindrinos, Leonidas |
| contents | We present two applications of the integro-differential volume equation for the eigenstrain, building on Eshelby's inclusion method [15,16], in the contexts of both static and dynamic linear elasticity. The primary objective is to address the inverse problem of recovering the elastic moduli of the inhomogeneity using a limited number of incident fields. In the static case, we adopt an efficient reformulation of Eshelby's equation proposed by Bonnet [7]. By employing a first-order approximation in addition with a limited number of incident loadings and measurements, we numerically determine the material coefficients of the inclusion. In elastodynamics, we focus on the inverse scattering problem, utilizing the Lippmann-Schwinger integral equation to reconstruct the elastic properties of the inclusion through a Newton-type iterative scheme. We construct the Frechet derivative and we formulate the linearized far-field equation. Additionally, the corresponding plane strain problems are analyzed in both static and dynamic elasticity. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2503_17508 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the inverse elastic problem for isotropic media using Eshelby and Lippmann-Schwinger integral formulations Gintides, Drossos Mindrinos, Leonidas Analysis of PDEs Mathematical Physics We present two applications of the integro-differential volume equation for the eigenstrain, building on Eshelby's inclusion method [15,16], in the contexts of both static and dynamic linear elasticity. The primary objective is to address the inverse problem of recovering the elastic moduli of the inhomogeneity using a limited number of incident fields. In the static case, we adopt an efficient reformulation of Eshelby's equation proposed by Bonnet [7]. By employing a first-order approximation in addition with a limited number of incident loadings and measurements, we numerically determine the material coefficients of the inclusion. In elastodynamics, we focus on the inverse scattering problem, utilizing the Lippmann-Schwinger integral equation to reconstruct the elastic properties of the inclusion through a Newton-type iterative scheme. We construct the Frechet derivative and we formulate the linearized far-field equation. Additionally, the corresponding plane strain problems are analyzed in both static and dynamic elasticity. |
| title | On the inverse elastic problem for isotropic media using Eshelby and Lippmann-Schwinger integral formulations |
| topic | Analysis of PDEs Mathematical Physics |
| url | https://arxiv.org/abs/2503.17508 |