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
(-) plane
lying below the parabola is mapped (conformally) onto the lower half of the image
(-)
plane. The corresponding boundary value problem is then analyzed in the
-plane.
A second conformal mapping, which maps the exterior of the
region occupied by the (simply-connected) inclusion in the
-plane
onto the exterior of the unit circle, is then used to construct an auxiliary function of
which, when used together with analytic continuation, allows us to extend our
analysis to an inclusion of arbitrary shape.
Keywords
Eshelby inclusion, parabolic boundary, conformal mapping,
analytic continuation, auxiliary function
Department of Mechanical
Engineering
University of Alberta
10-203 Donadeo Innovation Center for Engineering
9211-116 Street NW
Edmonton AB T6G 1H9
Canada