We first prove that the internal stress state within a double coated elastic inhomogeneity having an ( \(n+1\) )-fold axis of symmetry with \(n\ge 2\) can still remain uniform and hydrostatic when the matrix is subjected to uniform remote hydrostatic stresses. The three interfaces of the four-phase composite can be described by a three-term mapping function. The two coatings have a common shear modulus and distinct Poisson’s ratios. The plane-strain bulk modulus of the inhomogeneity and the Poisson’s ratio of the outer coating can be uniquely determined for given elastic properties of the inner coating and matrix and given geometry of the composite by iteratively solving two coupled non-linear equations. The hoop stress in the inner coating is found to be uniform along the entire inhomogeneity-inner coating interface. By using a similar approach, we also prove the existence of an internal uniform hydrostatic stress field within a triple (or quadruple) coated elastic inhomogeneity having an ( \(n+1\) )-fold axis of symmetry when subjected to uniform remote hydrostatic stresses. The four (or five) interfaces of the five-phase (or six-phase) composite are described by a four-term (or five-term) mapping function. A set of three (or four) coupled non-linear equations needs to be solved.
