Green's Function and Eshelby's Tensor Based on Mindlin's 2nd Gradient Model: An Explicit Study of Cylindrical Inclusion Case

Based on Mindlin's 2nd gradient model that involves two length-scale parameters, Green's function, Eshelby tensor and Eshelby-like tensor for an inclusion of arbitrary shape are derived. It is proved that the Eshelby tensor consists of two parts: the classical Eshelby tensor and a gradient part including the length-scale parameters, which enable the interpretation of the size effect. When the strain gradient is not taken into account, the obtained Green's function and Eshelby tensor reduce to its analogue based on the classical elasticity. For the cylindrical inclusion case, the Eshelby tensor in and outside the inclusion, the volume average of the gradient part and the Eshelby-like tensor are explicitly obtained. Unlike the classical Eshelby tensor, the results show that the components of the new Eshelby tensor vary with the position and the inclusion dimensions. It is demonstrated that the contribution of the gradient part should not be neglected.