Green's functions and Eshelby tensors for an ellipsoidal inclusion in a non-centrosymmetric and anisotropic micropolar medium

We derive simple integral solutions for the Green's tensors and Eshelby tensor for a generally anisotropic and non-centrosymmetric micropolar material. The material properties of a micropolar medium are characterized by three fourth-order tensors, C, B and D, in which C relates the stress to the strain, D connects the couple stress to the curvature tensor, and B is the coupling tensor, linking the stress to the curvature tensor or the couple stress to the strain tensor. Here we find that when C and B tensors possess minor symmetry conditions, the Green's tensors for a general anisotropic micropolar medium can be derived in simple integral expressions. We note however that in our formulation there is no any restriction on the D tensor. Specifically, we will show that, under the conditions on the C and B tensors, the Green's tensors can be exactly expressed as a simple line integral over a unit circle, and that the Eshelby's tensors for a general ellipsoidal inclusion can be derived as surface integrals over a unit sphere, entirely analogous to those of the classical elasticity. The exact integral form of the tensors can be implemented with numerical integration procedures. We will demonstrate that our numerical solutions are in good accuracy compared with the existing solutions for simple situations. In the literature analytic Green's tensors are existed only for isotropic and non-centrosymmetric micropolar medium. The solutions derived here can serve as benchmark solutions for certain classes of anisotropic micropolar medium, and also can be used as an approximate solution for a micropolar medium with general material properties without any constraint conditions.