Eshelby's inhomogeneity model within Mindlin's first strain gradient elasticity theory and its applications in composite materials

Eshelby's inhomogeneity problem is solved within the second form of Mindlin's first strain gradient elasticity theory for the prediction of the effective elastic properties of composites. Considering Green's function technique, an integral equation is established for an ellipsoidal inhomogeneity embedded in a homogeneous elastic medium and subjected to non-uniform boundary conditions. Within isotropic elasticity, the mean strain inside a spherical inhomogeneity is detailed to provide analytical results. In addition to the elastic properties of the inhomogeneity and the matrix, the strain localization depends on five gradient elastic constants, introduced by the first strain gradient elasticity theory. The effective bulk and shear moduli of a two-phase composite are predicted through Mori-Tanaka's scheme. The strain localization and the effective elastic moduli are then expressed within some simplified gradient elasticity theories. To test the relevance of the developed model, its predictions are compared with those of some investigations and the effective elastic properties are analyzed for a metal matrix composite. Finally, some comparisons with experimental data are performed to estimate the characteristic length scale parameters and gradient elastic constants of local phases.