Multicoated elastic inhomogeneities of arbitrary shape neutral to multiple fields

We rigorously establish the interesting result that in anti-plane elasticity an elastic epitrochoidal inhomogeneity can be made neutral to multiple uniform fields applied in the matrix via the insertion of two intermediate coatings. Using a two-term conformal mapping function, the simply connected domain occupied by the epitrochoidal inhomogeneity and its surrounding inner and outer coatings is mapped onto the interior of the unit circle in the image plane. The mismatch parameters are determined in an analytical manner by solving a set of two non-linear equations. An elastic inhomogeneity of arbitrary shape can be made neutral to multiple fields through the insertion of N coatings when the proposed mapping function for the simply connected domain occupied by the multicoated inhomogeneity is described in terms of a polynomial of finite degree containing N non-constant terms. In this case, the mismatch parameters are determined by iteratively solving a set of N non-linear equations.