Analytical Solution of the Dilute Strain Concentration Tensor for Coated Cylindrical Inclusions, and Applications for Polymer Nanocomposites

Recently nanoparticle-reinforced polymer nanocomposite materials, comprised of the inclusion, (non-bulk polymer) interphase and the bulk polymer matrix, have received considerable interest. Because of interaction between the nanoinclusion and surrounding polymer matrix, the non-bulk polymer in the vicinity of the nanoinclusion has different properties than the bulk polymer. With tremendous amount of surface area, the interphase may have a large influence on the overall nanocomposite properties and complicate micromechanical predictions of effective properties. Although several micromechanical approaches can provide approximations of the effective elastic modulus, they require one to calculate the dilute concentration tensor using the well-known Eshelby tensor that treat interphase as separate, physically distinct inclusions. However, their elegant solutions are no longer available when the real geometry of the annular interphase must be considered. This work analytically determined the components of the dilute strain concentration tensors for both the inclusion and the interphase by addressing four auxiliary loading cases, which can be directly implemented within standard micromechanical approaches, such as the Mori-Tanaka model, to predict the effective properties of polymer nanocomposites with cylindrical/fibrous nanoinclusions. Comparison of the predictions of the proposed model with predictions based on the traditional Multiphase Mori-Tanaka approach show that differences between the models are largest when the annular interphase region is softer than the matrix material, attributed to the ability of the proposed model to capture the “stress-shielding effect” in the case of the softer annular interphase. In addition, we have examined several sets of experimental data from the literature for both stiff and soft interphase systems to shed further insight on the utility of the proposed model. The model proposed here would provide an important guideline to evaluate the impact of chemical functionalization techniques and other strategies that seek to tailor the properties of the interphase region in nanocomposite materials.