Irreducible structure, symmetry and average of Eshelby's tensor fields in isotropic elasticity

The strain field ?(x) in an infinitely large, homogenous, and isotropic elastic medium induced by a uniform eigenstrain ?⁰ in a domain ω depends linearly upon . It has been a long-standing conjecture that the Eshelby's tensor field S^ω(x) is uniform inside ω if and only if ω is ellipsoidally shaped. Because of the minor index symmetry , S^ω might have a maximum of 36 or nine independent components in three or two dimensions, respectively. In this paper, using the irreducible decomposition of S^ω, we show that the isotropic part S of S^ω vanishes outside ω and is uniform inside ω with the same value as the Eshelby's tensor S⁰ for 3D spherical or 2D circular domains. We further show that the anisotropic part A^ω=S^ω-S of S^ω is characterized by a second- and a fourth-order deviatoric tensors and therefore have at maximum 14 or four independent components as characteristics of ω's geometry. Remarkably, the above irreducible structure of S^ω is independent of ω's geometry (e.g., shape, orientation, connectedness, convexity, boundary smoothness, etc.). Interesting consequences have implication for a number of recently findings that, for example, both the values of S^ω at the center of a 2D C_n(n?3,n≠4)-symmetric or 3D icosahedral ω and the average value of S^ω over such a ω are equal to S⁰.