Modelling the elastic stiffness of nanocomposites using the Mori-Tanaka method

This report describes mathematical modelling of the elastic stiffness of nanocomposites, which in this context is referred to as particles of nano-size included in a polymer matrix, i.e. particles with one dimension of nanometre size. The main motivation for this work was to establish mathematical models for calculating the elastic properties of different nanocomposites, which then can be included in a “model toolbox” for future applications and for improved understanding of this type of materials. In this study, it is assumed that micromechanics models and continuum mechanics theory can be applied in modelling. In this report, the Mori-Tanaka method is considered, where the particles are described as having a spheroidal shape. From this assumption, the Eshelby tensor can be applied to calculate the influence of the particles to the matrix, and the overall elastic stiffness of the composite due to the inclusions. The particle shape and orientation will affect the macroscopic elastic stiffness of the composite. Thus, different spheroidal shapes (e.g. spheres, prolate and oblate) are considered, as well as both aligned and random particle orientation. The current study is, however, restricted to two-phase composites, i.e. composites with one particle inclusion phase. When searching the literature, different models based on the Mori-Tanaka method are found. Expressions are available for specific geometric shapes and particle orientations. A more general multi-phase Mori-Tanaka model, which is applicable to several shapes and different orientations, is also found. The different models are implemented in Matlab, and the calculated model results are compared. Furthermore, the general Mori-Tanaka model is compared with experimental data found in the literature for some relevant nanoparticle/epoxy systems. The model calculations agree very well. Moreover, the model results for the general two-phase Mori-Tanaka model agree with most of the experimental results, but the model is not able to predict the improved stiffness for low volume fractions very well. Additional studies should therefore consider other effects that will influence the elastic stiffness of the nanocomposites. First of all, more than one inclusion phase, e.g. voids, agglomerates or other particles, should be included as part of the model toolbox. Second, it is relevant to establish models that consider the effect the nanoparticle interphase, which may be modelled as a region surrounding the particles with different elastic properties compared to the neat matrix.