Inclusion problem for a generalized strain gradient elastic continuum

This paper treats an inclusion problem in the framework of a generalized strain gradient theory (GSGT) of elasticity. The GSGT used here consists of three material characteristic length scales for the evaluation of the elastic response. The Green's function for the governing differential equations, involving stresses and higher-order stresses, is obtained considering the strain gradient model mentioned above. Then, Eshelby's tensor is derived in the current framework for an inclusion of any arbitrary shape, under uniform eigenfields. This general form of the Eshelby's tensor is used to derive closed-form expressions for a spherical inclusion and an infinitely long cylindrical inclusion, each embedded in an unbounded matrix to ignore the boundary effects. To determine the homogenized properties of the composites considering size-effects, the volume averages of the position-dependent Eshelby's tensor are evaluated for both the inclusions. All the results from the current study match the corresponding results from classical elasticity, when the strain gradient effects are ignored. The homogenized properties of composites with either of the inclusions are strongly influenced by the gradient effects when the dimensions of the inclusions are comparable to the characteristic length scales. They approach the volume averages evaluated using classical elasticity, when the dimensions of the inclusion are very high in comparison with the characteristic lengths. The current study presents the importance of considering the size-effects using GSGT, without any assumption, for the evaluation of the effective properties of the micro- and nano-composites.