Size-dependent Eshelby's ellipsoidal inclusion problem based on generalized first strain gradient elasticity theory

An analytical solution of Eshelby's ellipsoidal inclusion problem is derived in this paper using the generalized first strain gradient theory (GFSGT) of elasticity. The GFSGT employed here is a reformulation of Mindlin's first-order strain gradient theory, retaining only three independent material length scales. The Eshelby's tensor for an ellipsoidal inclusion embedded in a strain gradient elastic continuum is obtained from the result for an arbitrary inclusion, which is available in the literature. The fourth-order Eshelby's tensor depicts size-dependent behaviour, and a position-dependence is noted even within the domain of the inclusion. The volume average of this position-dependent tensor is determined inside and outside the inclusion, for the evaluation of size-dependent homogenized properties of composites. As expected, the results of this study asymptotically approach those of classical elasticity when the geometric dimensions of the inclusion are increased. For inclusions having sizes on the order of the material length scale parameters, these volume averages are strongly influenced by gradient effects. A comparison of the current model based on the GFSGT with that built on the simplified theories of strain gradient elasticity is made to illustrate the complete strain gradient effects. This comparison, along with the suitability of a generalized model involving different length scales for crystals and polymers, illustrates the importance of using GFSGT for determining the effective properties of micro- and nanocomposites.