Neutrality of a four-phase spherical inhomogeneity under an arbitrary uniform remote load

We achieve neutrality of a four-phase spherical inhomogeneity embedded in an infinite elastic matrix subjected to an arbitrary uniform remote load. In the four-phase composite, the internal spherical elastic inhomogeneity is bonded to the surrounding matrix through two concentric spherical annular interphase layers. The neutrality of the inhomogeneity is characterized by the fact that its introduction does not disturb the original uniform stress distribution in the matrix. As a consequence, we obtain exact representations of the effective shear modulus and effective bulk modulus of doubly coated sphere assemblages of various sizes which can completely replace the matrix.