An Eshelby inclusion of arbitrary shape undergoing uniform antiplane eigenstrains in an anisotropic elastic plane
摘要
We utilize the techniques of one-to-one mapping and analytic continuation to first derive a general solution to the antiplane strain problem of an Eshelby inclusion of arbitrary shape undergoing uniform antiplane eigenstrains embedded in an infinite homogeneous monoclinic material with symmetry plane at x3 = 0 in a Cartesian coordinate system. The elastic field of strains and stresses within the inclusion is obtained once the polynomial representing the principal part of the remote asymptotic behavior of an auxiliary function is determined. We then derive an explicit solution to the problem of an inclusion having an (n + 1)-fold axis of quasi-symmetry (with n ≥ 1) in an infinite monoclinic material. The inclusion boundary has an (n + 1)-fold axis of symmetry in the z-plane, where z is the single complex variable appearing in the complex variable formulation. The inclusion boundary itself is described by a mapping function containing an arbitrary number M + 1 of terms. The non-uniform total antiplane strains and Eshelby tensor within the inclusion are determined. We further prove the quasi Eshelby property that when n ≥ 2, The arithmetic mean of the Eshelby tensors at n + 1 rotational symmetric points within the inclusion in the z-plane and the Eshelby tensor at the center of the inclusion in the z-plane are equal to the constant Eshelby tensor within a special elliptical inclusion, the boundary of which is circular in the z-plane; furthermore, we show that they are both independent of the elastic property of the monoclinic material and the orientation of the inclusion boundary in the z-plane.