Thermal Expansion Induced Neutrality of a Circular and an Annular Elastic Inhomogeneity

An elastic inhomogeneity is termed neutral if its introduction does not disturb the original stress field in the initially uncut elastic body. Neutrality in this sense is often achieved by appropriate design criteria such as a careful choice of the shape of the inhomogeneity and the properties of the interfacial layer between the inhomogeneity and its surrounding matrix. Unfortunately, mismatched stress and strain fields in the resulting composite structure make it difficult to simultaneously control both the shape of the inhomogeneity and its interfacial properties to achieve the desired neutrality property. We assert that the associated temperature field can be used to adjust the stress and strain fields within the inhomogeneity via thermal expansion, thus allowing us to control the properties of the interfacial layer for a given shape of inhomogeneity. Our theoretical results show that the design of a neutral circular or annular elastic inhomogeneity requires an accompanying internal uniform temperature field when the elastic body is in equi-biaxial tension and an internal temperature field which is quadratic if the body is subjected to uniaxial tension or shear force. More importantly, in contrast to the well-established result in the literature for a purely elastic inhomogeneity, under certain conditions, a neutral elastic inhomogeneity can be designed via thermal expansion despite the assumption of a perfectly bonded interface between the inhomogeneity and the surrounding matrix.