Eshelby's tensors for plane strain and cylindrical inclusions based on a simplified strain gradient elasticity theory

The Eshelby tensor for a plane strain inclusion of arbitrary cross-sectional shape is first presented in a general form, which has 15 independent non-zero components (as opposed to 36 such components for a three-dimensional inclusion of arbitrary shape). It is based on a simplified strain gradient elasticity theory that involves one material length scale parameter. The Eshelby tensor for an infinitely long cylindrical inclusion is then derived using the general form, with its components obtained in explicit (closed-form) expressions for the two regions inside and outside the inclusion for the first time based on a higher-order elasticity theory. This Eshelby tensor is separated into a classical part and a gradient part. The latter depends on the position, the inclusion size, the length scale parameter, and Poisson's ratio. As a result, the new Eshelby tensor is non-uniform even inside the cylindrical inclusion and captures the size effect. When the strain gradient effect is not considered, the gradient part vanishes and the newly obtained Eshelby tensor reduces to its counterpart based on classical elasticity. The numerical results quantitatively show that the components of the new Eshelby tensor vary with the position, the inclusion size, and the material length scale parameter, unlike their classical elasticity-based counterparts. When the inclusion radius is comparable to the material length scale parameter, it is found that the gradient part is too large to be ignored. In view of the need for homogenization analyses of fiber-reinforced composites, the volume average of the newly derived Eshelby tensor over the cylindrical inclusion is obtained in a closed form. The components of the average Eshelby tensor are observed to depend on the inclusion size: the smaller the inclusion radius, the smaller the components. However, as the inclusion size becomes sufficiently large, these components are seen to approach from below the values of their classical elasticity-based counterparts. © 2009 Springer-Verlag.