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

The strain field c(x) in an infinitely large, homogenous, and isotropic elastic medium induced by a uniform eigenstrain epsilon(0) in a domain omega depends linearly upon epsilon(0): epsilon(ij)(x) = S-ijkl(omega)(x)epsilon(0)(kl). It has been a longstanding conjecture that the Eshelby's tensor field S'(x) is uniform inside omega if and only if omega is ellipsoidally shaped. Because of the minor index symmetry S-ijkl(omega) = S-jikl(omega) = S', S' 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-omega, we show that the isotropic part S of S-omega vanishes outside omega and is uniform inside omega with the same value as the Eshelby's tensor So for 3D spherical or 2D circular domains. We further show that the anisotropic part A(omega) = S-omega - S of S' is characterized by a second- and a fourth-order deviatoric tensors and therefore have at maximum 14 or four independent components as characteristics of omega's geometry. Remarkably, the above irreducible structure of S omega is independent of omega'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-omega at the center of a 2D l(n)(n >= 3, n not equal 4)-symmetric or 3D icosahedral omega and the average value of S-omega over such a w are equal to So. (c) 2005 Elsevier Ltd. All rights reserved.