An extension of the Eshelby conjecture to domains of general shape in anti-plane elasticity

Choi, Doosung; Kim, Kyoungsun; Lim, Mikyoung

doi:10.48550/arxiv.1807.09981

Cited by 1 publication

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“…Indeed, for the general case, besides the transmission condition regarding the displacements, one should also consider the one concerning the continuity of tractions at the boundary of the inclusion. This would put forward another research avenue-the solution of the so-called E-inclusion problem for the plane elastostatic case-thus extending the results found in [11] for the conductivity problem. Such a problem involves the determination of the shape of the inclusion that provides uniform fields inside the inclusion for any or some applied far-field loadings.…”

Section: Discussionmentioning

confidence: 68%

“…Moreover, the two sets of interior and exterior basis functions have explicit relations on the boundary of the inclusion so that the interior and exterior values of the solution can be matched by using the transmission condition on the boundary of the inclusion (see also [10]). This geometric series solution method was successfully applied to the study of inclusion problems in anti-plane elasticity and the spectral property of the Neumann-Poincaré operator [9,11,24]. Analogous results for the elastic problem have not been found yet.…”

Section: Introductionmentioning

confidence: 99%

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Explicit analytic solution for the plane elastostatic problem with a rigid inclusion of arbitrary shape subject to arbitrary far-field loadings

Mattei¹,

Lim²

2020

Preprint

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We provide an analytical solution for the elastic fields in a two-dimensional unbounded isotropic body with a rigid isotropic inclusion. Our analysis is based on the boundary integral formulation of the elastostatic problem and geometric function theory. Specifically, we use the coordinate system provided by the exterior conformal mapping of the inclusion to define a density basis functions on the boundary of the inclusion, and we use the Faber polynomials associated with the inclusion for a basis inside the inclusion. The latter, which constitutes the main novelty of our approach, allows us to obtain an explicit series solution for the plane elastostatic problem for an inclusion of arbitrary shape in terms of the given arbitrary far-field loading.

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Section: Discussionmentioning

confidence: 68%