Inclusion problem in second gradient elasticity

The Green's function and Eshelby tensors of an infinite linear isotropic second gradient continuum are derived for an inclusion of arbitrary shape. Particularly for spherical, cylindrical and ellipsoidal inclusions, Eshelby tensors and their volume averages are obtained in an analytical form. It is found that the Eshelby tensors are not uniform inside the inclusion even for a spherical inclusion, and their variations depend on the two characteristic lengths of second gradient theory. When size of inclusion is large enough compared to the characteristic lengths, the Eshelby tensor of the second gradient medium is reduced to the classical one, as expected. It is also demonstrated that the existing Green's functions and Eshelby tensors of couple stress theory, Aifantis, Kleinert and Wei-Hutchinson special strain gradient theories could be recovered as special cases. This work paves the way for constructing micromechanical method to predict size effect of composite materials, as shown for the effective modulus of particulate composite derived with the proposed theory.