Green’s function and Eshelby’s tensor based on a simplified strain gradient elasticity theory

Abstract

The Eshelby problem of an infinite homogeneous isotropic elastic material containing an inclusion is analytically solved using a simplified strain gradient elasticity theory that involves one material length scale parameter in addition to two classical elastic constants. The Green’s function in the simplified strain gradient elasticity theory is first obtained in terms of elementary functions by applying Fourier transforms, which reduce to the Green’s function in classical elasticity when the strain gradient effect is not considered. The Eshelby tensor is then derived in a general form for an inclusion of arbitrary shape, which consists of a classical part and a gradient part. The former contains Poisson’s ratio only, while the latter includes the length scale parameter additionally, thereby enabling the interpretation of the size effect. By applying the general form of the Eshelby tensor derived, the explicit expressions of the Eshelby tensor for the special case of a spherical inclusion are obtained. The numerical results quantitatively show that the components of the new Eshelby tensor for the spherical inclusion vary with both the position and the inclusion size, unlike their counterparts based on classical elasticity. It is found that when the inclusion radius is small, the contribution of the gradient part is significantly large and thus should not be ignored. For homogenization applications, the volume average of this newly obtained Eshelby tensor over the spherical inclusion is derived in a closed form. It is observed that the components of the averaged Eshelby tensor change with the inclusion size: the smaller the inclusion radius, the smaller the components. Also, these components are seen to approach from below the values of their counterparts based on classical elasticity when the inclusion size becomes sufficiently large.