Three-phase model of particulate composites in second gradient elasticity

Generalized self-consistent method for particulate composites is realized in the framework of Mindlin's second gradient elasticity theory (SGET). Effective properties of the macroscopically isotropic medium containing spherical inclusions are determined based on the three-phase sphere model, in which the spherical inclusion surrounded by matrix shell is embedded in the effective medium. Analytical solutions for the hydrostatic pressure and pure shear problems for the considered model are found based on the generalized Papkovich-Neuber potentials in the spherical coordinate system. Eshelby integral formula generalized for SGET is used as a closing energetic equation to estimate the effective properties of the composite material. Positive size effects for the effective elastic moduli of composites with small size inclusions are predicted based on obtained analytical solutions. Strain concentration in the matrix phase is investigated and found to be a non-monotonic function of the inclusion size. Obtained results are verified by the finite element simulations realized for the special case of SGET with simplified constitutive equations.