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

The strain field ɛ(x)ɛ(x) in an infinitely large, homogenous, and isotropic elastic medium induced by a uniform eigenstrain ɛ0ɛ0 in a domain ωω depends linearly upon ɛ0:ɛij(x)=Sijklω(x)ɛkl0. It has been a long-standing conjecture that the Eshelby's tensor field Sω(x)Sω(x) is uniform inside ωω if and only if ωω is ellipsoidally shaped. Because of the minor index symmetry Sijklω=Sjiklω=Sijlkω, SωSω might have a maximum of 36 or nine independent components in three or two dimensions, respectively. In this paper, using the irreducible decomposition of SωSω, we show that the isotropic part SS of SωSω vanishes outside ωω and is uniform inside ωω with the same value as the Eshelby's tensor S0S0 for 3D spherical or 2D circular domains. We further show that the anisotropic part Aω=Sω-SAω=Sω-S of SωSω is characterized by a second- and a fourth-order deviatoric tensors and therefore have at maximum 14 or four independent components as characteristics of ωω's geometry. Remarkably, the above irreducible structure of SωSω is independent of ωω'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 SωSω at the center of a 2D Cn(n⩾3,n≠4)Cn(n⩾3,n≠4)-symmetric or 3D icosahedral ωω and the average value of SωSω over such a ωω are equal to S0S0.