Concurrent two-scale model for the viscoelastic behavior of elastomers filled with hard nanoparticles

A dynamic two-scale model is developed for describing the mechanical behavior of elastomers filled with hard nanoparticles. Using nonequilibrium thermodynamics, a closed system of evolution equations is derived, coupling continuum mechanics with a fine-scale description on the level of filler particles. So doing, a constitutive stress-strain relation emerges that is applicable to transient situations. In addition to the number density of filler particles, the particle arrangement is captured by the distribution of the difference vector between two representative interacting particles, which makes this model efficient in comparison with many-particle models. The two-particle model presented here is analyzed numerically in oscillatory deformation, for two purposes. First, the nonlinearity of the model is studied in detail, in terms of the Payne effect, that compares favorably with the literature. And second, the two-particle model is compared with a corresponding many-particle model in the literature.