Strain gradient solution for Eshelby’s ellipsoidal inclusion problem

Abstract

Eshelby’s problem of an ellipsoidal inclusion embedded in an infinite homogeneous isotropic elastic material and prescribed with a uniform eigenstrain and a uniform eigenstrain gradient is analytically solved. The solution is based on a simplified strain gradient elasticity theory (SSGET) that contains one material length scale parameter in addition to two classical elastic constants. The fourth-order Eshelby tensor is obtained in analytical expressions for both the regions inside and outside the inclusion in terms of two line integrals and two surface integrals. This non-classical Eshelby tensor consists of a classical part and a gradient part. The former involves Poisson’s ratio only, while the latter includes the length scale parameter additionally, which enables the newly obtained Eshelby tensor to capture the inclusion size effect, unlike its counterpart based on classical elasticity. The accompanying fifth-order Eshelby-like tensor relates the prescribed eigenstrain gradient to the disturbed strain and has only a gradient part. When the strain gradient effect is not considered, the new Eshelby tensor reduces to the classical Eshelby tensor, and the Eshelby-like tensor vanishes. In addition, the current Eshelby tensor for the ellipsoidal inclusion problem includes those for the spherical and cylindrical inclusion problems based on the SSGET as two limiting cases. The non-classical Eshelby tensor depends on the position and is non-uniform even inside the inclusion, which differ from its classical counterpart. For homogenization applications, the volume average of the new Eshelby tensor over the ellipsoidal inclusion is analytically obtained. The numerical results quantitatively show that the components of the newly derived Eshelby tensor vary with both the position and the inclusion size, unlike their classical counterparts. When the inclusion size is small, it is found that the contribution of the gradient part is significantly large. It is also seen that the components of the averaged Eshelby tensor change with the inclusion size: the smaller the inclusion, the smaller the components. Moreover, these components are observed to approach the values of their classical counterparts from below when the inclusion size becomes sufficiently large.