Explicit expressions of the Eshelby tensor for an arbitrary 3D weakly non-spherical inclusion

The Eshelby tensor plays a crucial role in micromechanics of matrix-particle composites. Until now, explicit expressions of the Eshelby tensor for an arbitrary 3D shape inclusion are not yet available. Herein, we derive explicit expressions of the Eshelby tensor inside and outside an arbitrary 3D weakly non-spherical inclusion whose shape is described by the inclusion shape function. Then, we obtain the average Eshelby tensor in an arbitrary 3D weakly non-spherical inclusion. Finally, we verify our expressions of the Eshelby tensor by comparing some exact solutions and FEM results.