#### Vol. 13, No. 2, 2018

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Eshelby inclusion of arbitrary shape in isotropic elastic materials with a parabolic boundary

### Xu Wang, Liang Chen and Peter Schiavone

Vol. 13 (2018), No. 2, 191–202
##### Abstract

We employ analytic continuation and conformal mapping techniques to derive analytic solutions for Eshelby’s problem of an elastic inclusion of arbitrary shape in an isotropic elastic plane with parabolic boundary. The region of the physical ($z$-) plane lying below the parabola is mapped (conformally) onto the lower half of the image ($\xi$-) plane. The corresponding boundary value problem is then analyzed in the $\xi$-plane. A second conformal mapping, which maps the exterior of the region occupied by the (simply-connected) inclusion in the $\xi$-plane onto the exterior of the unit circle, is then used to construct an auxiliary function of $\xi$ which, when used together with analytic continuation, allows us to extend our analysis to an inclusion of arbitrary shape.

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