A closed-form, hierarchical, multi-interphase model for composites—Derivation, verification and application to nanocomposites

A closed form Hierarchical Multi-interphase Model (HMM) based on the classical elasticity theory is proposed to study the influence of the interphase around inclusions on the enhancement mechanism of composites in the elastic regime. The HMM is verified by three-dimensional Finite Element simulations and highly consistent results are obtained for the cases with relatively low stiffness ratios (SR) between the inclusions and the matrix (SR<100). For cases with large SRs (up to 10,000), the HMM with the assumption of ellipsoidal inclusions provides a lower bound for the stiffnesses of composites enhanced by non-ellipsoidal particles with the same aspect ratio of inclusions. The Modified Hierarchical Multi-interphase Model (MHMM) is developed by introducing morphology parameters to the HMM, to capture the high morphology sensitivity of composites at high SRs with the non-uniform stress–strain fields. In addition, one important feature of the HMM and the MHMM is the particle-size dependency. As an application of this model to predict size effects and shape effects, the enhancement efficiencies of three typical inclusions – sphere, fiber-like particle and platelet – at different scales, are studied and compared, producing useful information about the morphology optimization at the nano-scale.