Equivalent Inclusion Approach for Micromechanics Estimates of Nanocomposite Elastic Properties

Abstract

Classical micromechanics approaches for heterogeneous media assume perfect bonding between phases, implying that both displacement and stress vectors are continuous across the interface between the phases. When nanoinclusions are involved, a stress vector discontinuity in the local equilibrium has to be accounted for. In this framework, this paper derives an approximate solution of the Lippmann-Schwinger (L-S) equation, which accounts for these surface stresses. This approach suggests introducing the concept of an equivalent particle that combines the particle with the surrounding interface, which can be directly implemented in any standard homogenization procedure, such as the Mori-Tanaka scheme. Analytical expressions for the stiffness tensor of the equivalent particle is derived for spheroidal inclusions, accounting for a wide range of nanoinclusion shapes and dimensions. Finally, an energy-based analysis proves how the dramatic increase of the elastic properties is controlled, for a given volume fraction, by the smallest size of the nanoinclusions.