The Arithmetic Mean Theorem of Eshelby Tensor for Exterior Points Outside the Rotational Symmetrical Inclusion

In 1957, Eshelby proved that the strain field within a homogeneous ellipsoidal inclusion embedded in an infinite isotropic media is uniform, when the eigenstrain prescribed in the inclusion is uniform. This property is usually referred to as the Eshelby property. Although the Eshelby property does not hold for the non-ellipsoidal inclusions, in recent studies we have successfully proved that the arithmetic mean of Eshelby tensors at $N$ rotational symmetrical points inside an $N$-fold rotational symmetrical inclusion is constant and equals the Eshelby tensor for a circular inclusion, when $N⩾3$ and $N≠4$⁠. The property is named the quasi-Eshelby property or the arithmetic mean theorem of Eshelby tensors for interior points. In this paper, we investigate the elastic field outside the inclusion. By the Green formula and the knowledge of complex variable functions, we prove that the arithmetic mean of Eshelby tensors at $N$ rotational symmetrical points outside an $N$-fold rotational symmetrical inclusion is equal to zero, when $N⩾3$ and $N≠4$⁠. The property is referred to as the arithmetic mean theorem of Eshelby tensors for exterior points. Due to the quality of the Green function for plane strain problems, the fourfold rotational symmetrical inclusions are excluded from possessing the arithmetic mean theorem. At the same time, by the method proposed in this paper, we verify the quasi-Eshelby property which has been obtained in our previous work. As corollaries, two more special properties of Eshelby tensor for $N$-fold rotational symmetrical inclusions are presented which may be beneficial to the evaluation of effective material properties of composites. Finally, the circular inclusion is used to test the validity of the arithmetic mean theorem for exterior points by using the known solutions.