Two-scale model for the effect of physical aging in elastomers filled with hard nanoparticles

A two-scale model is developed, and solved numerically, to describe the mechanical behavior of elastomers filled with hard nanoparticles. Of particular interest is the slow recovery of the elastic modulus after large-amplitude oscillatory deformation. To account for this effect, the physical aging of the glassy bridges between the filler particles is captured with two thermal degrees of freedom for the matrix material, namely a kinetic and a configurational one. Formulating the two-scale model enriched with aging in a nonequilibrium thermodynamics context, first results in a constitutive relation for the Cauchy stress tensor. Second, the dynamics of physical aging is described, which eventually results in the slow recovery of the elastic modulus with waiting time. The proposed model is investigated numerically under large amplitude oscillatory shear deformation. Of particular interest in this respect is the coupling of the micro-scale dynamics with the physical aging on the macroscopic scale. This coupling is examined in detail, both in an approximate way using a Gaussian approximation, as well as numerically, under specific conditions. It turns out that the CONNFFESSIT approach (Laso and Öttinger 1993 [46]) can not be employed for the numerical solution of the model under arbitrary loading conditions because of the novel structure of the two-level coupling term. While a procedure for solving the model numerically for the case of strong applied deformation is presented in this paper, other solution methodologies need to be sought for the cases of weak and no applied deformation.