On ‘strange’ properties of some symmetric inhomogeneities

The paper presents an analysis of elasticity problems involving a single inhomogeneity which possesses certain types of symmetries. As observed earlier, isotropic problems of that kind exhibit some ‘strange’ and remarkable properties. Under the action of uniform far-field stresses, the averages of the fields inside the inhomogeneities preserve the structure of the far-field loads. Here, it is shown that these properties are exhibited for a wider class of problems, which include anisotropic and non-uniform materials subjected to either far-field loads or constant transformational strains within the inhomogeneity. The proposed modified Eshelby technique facilitates a straightforward analysis of these problems, which is based entirely on the assumed symmetry. It is also shown that some remarkable properties of symmetric inhomogeneities discovered here are related to the so-called ‘strange’ properties of the Eshelby inclusions extensively covered in the literature. Some implications of these findings are discussed.