A compatible multiscale model for nanocomposites incorporating interface effect

The asymptotic homogenization theory is usually employed for analysis of composites with hierarchical microstructures, and the interface effect has to be considered for nanocomposites due to the fact that the surface-to-volume ratio of the nanofillers becomes significant. However, commonly-used two-dimensional surface theories of elasticity cannot be compatibly applied in conjunction with the asymptotic homogenization theory for analysis of nanocomposites due to traction discontinuity and the difficulty in specifying positive definiteness of elasticity tensor in terms of two-dimensional interface parameters. This study is focused on solving the incompatibility problem. A simple but useful three-dimensional interface theory is proposed, in which stress discontinuity conditions across two-dimensional interfaces are replaced by traction continuity on its two sides, and the requirement for positive definiteness of elasticity tensor can be easily specified. This three-dimensional interface theory is linked to the well-accepted two-dimensional models through energy-equivalence and kinematics of its middle surface. It is found that if the mechanical behavior of an interface can be well captured by commonly-used two-dimensional surface theories of elasticity, the three-dimensional interface can be characterized by constant elastic moduli as well as a size-independent thickness. Based on the proposed three-dimensional interface theory and the asymptotic homogenization theory, a compatible multiscale model is then developed and provides a practical way to examine the scaling law of effective moduli of typical nanocomposites. It is examined using numerical results that two-dimensional interface parameters can be easily extracted from the scaling law, and the way we link three-dimensional interfaces to two-dimensional counterparts is justified.