The composite Eshelby Tensors and their applications to homogenization

In recent studies, the exact solutions of the Eshelby tensors for a spherical inclusion in a finite, spherical domain have been obtained for both the Dirichletand Neumann boundary value problems, and they have been further applied to the homogenization of composite materials [15], [16]. The present work is an extension to a more general boundary condition, which allows for the continuity of both the displacement and traction field across the interface between RVE (representative volume element) and surrounding composite. A new class of Eshelby tensors is obtained, which depend explicitly on the material properties of the composite, and are therefore termed ‘the Composite Eshelby Tensors’. These include the Dirichletand the NeumannEshelby tensors as special cases. We apply the new Eshelby tensors to the homogenization of composite materials, and it is shown that several classical homogenization methods can be unified under a novel method termed the ‘Dual Eigenstrain Method’. We further propose a modified Hashin-Shtrikman variational principle, and show that the corresponding modified Hashin-Shtrikman bounds, like the Composite Eshelby Tensors, depend explicitly on the composite properties.