Effect of an homogeneous interphase zone on the bulk modulus of particulate composite containing inclusions

Nanoparticle reinforced materials, hollow particle filled composites as well as concrete are only some examples of particulate composites where the effective elastic properties are strongly affected not only by the elastic moduli of the two phases and the volume fraction of the particles, but also by the geometric characteristics and elastic properties of the interphase zone. As an example, due to manufacturing errors, flaws and defects can easily form at the interface between the matrix and the inclusions embedded to improve the effective stiffness of the composite material. As consequence, a transition zone around the inclusions forms and its thickness can be comparable with the inclusion characteristic size. In such cases, models based on the assumption of a sharp interface are not adequate. This work deals with the derivation of the effective properties of particulate composites modelled as a suspension of elastic homogeneous spherical particles in a continuous matrix, taking into account the effects of an inhomogeneous interphase zone around the inclusions. In particular, the attention is focused on the derivation of explicit expressions for the bulk modulus. In order to do this, the analysis employs the closed form analytical solution for the problem of a single hollow or solid sphere embedded in a matrix with an inhomogeneous interphase around the inclusion with elastic properties in the form of power-law in radial direction [1]. Assuming hydrostatic pressure the elastic problem is solved in the framework of elasticity theory and the effective elastic bulk modulus is obtained using the energy method [2]. These results, recently obtained by the first author in [3], are investigated in this paper for very small and very large concentrations of inclusions. In addition to the case of dilute suspension conditions, the non dilute case is also considered and, in order to do this, well-known approximation schemes for effective properties of particulate composites are studied and compared. As an example, the differential method and the generalized self consistent method are employed. The cases of voids or solid/hollow inclusions in a homogeneous matrix as special cases of the solution proposed are compared and a parametric analysis is performed to investigate the influence of the geometric characteristics and elastic properties of the interphase zone on the effective bulk modulus of the composite material.