The Proof of Strain Discontinuity in Eshelby Problems with Rotational Symmetrical Inclusions

Abstract

One of the important approaches to study thermal stress in heterogeneous material/devices is the Eshelby equivalent inclusion method, which is based on the Eshelby uniform solution (the Eshelby property) for ellipsoid-like inclusions. Despite the non-uniformity in stresses fields, it has been proved \\citep{WangXu2006} that rotational symmetrical inclusions satisfy the arithmetic mean theorem (the quasi-Eshelby property). That is, for N-fold rotational symmetric inclusions, the average (more accurately, the arithmetic mean) of the strains inside the inclusion equals the strain of the circular inclusion and vanishes for the points outside the inclusion. Similar to the Eshelby property, the quasi-Eshelby property can be used to study induced internal stress in micromechanics via the Eshelby equivalent inclusion method. Consequently, the Eshelby equivalent inclusion method can be extended to study inclusions of rotational symmetrical shape. In this paper, the discontinuity relation of the average strains in the Eshelby problem of rotational symmetrical inclusions obtained by [11] is proved by virtue of the stress continuity and the discontinuity of the displacement gradient.