On coated inclusions neutral to bulk strain fields in two dimensions

The neutral inclusion problem in two dimensional isotropic elasticity is considered. The neutral inclusion, when inserted in a matrix having a uniform applied field, does not disturb the field outside the inclusion. The inclusion consists of the core and shell of arbitrary shapes, and their elasticity tensors are isotropic. We show that if the coated inclusion is neutral to a uniform bulk field, then the core and shell must be concentric disks, provided that the shear and bulk moduli satisfy certain conditions.