Filled elastomers: A theory of filler reinforcement based on hydrodynamic and interphasial effects

Experimental evidence has by now established that (i) the hydrodynamic effect and (ii) the presence of stiff interphases (commonly referred to as bound rubber) "bonding" the underlying elastomer to the fillers are the dominant microscopic mechanisms typically responsible for the enhanced macroscopic stiffness of filled elastomers. Yet, because of the technical difficulties of dealing with these fine-scale effects within the realm of finite deformations, the theoretical reproduction of the macroscopic mechanical response of filled elastomers has remained an open problem. The object of this paper is to put forward a microscopic field theory with the capability to describe, explain, and predict the macroscopic response of filled elastomers under arbitrarily large nonlinear elastic deformations directly in terms of: (i) the nonlinear elastic properties of the elastomeric matrix, (ii) the concentration of filler particles, and (iii) the thickness and stiffness of the surrounding interphases. Attention is restricted to the prominent case of isotropic incompressible elastomers filled with a random and isotropic distribution of comparatively rigid fillers. The central idea of the theory rests on the construction of a homogenization solution for the fundamental problem of a Gaussian elastomer filled with a dilute concentration of rigid spherical particles bonded through Gaussian interphases of constant thickness, and on the extension of this solution to non-Gaussian elastomers filled with finite concentrations of particles and interphases by means of a combination of iterative and variational techniques. For demonstration purposes, the theory is compared with full 3D finite-element simulations of the large-deformation response of Gaussian and non-Gaussian elastomers reinforced by isotropic distributions of rigid spherical particles bonded through interphases of various finite sizes and stiffnesses, as well as with experimental data available from the literature. Good agreement is found in all of these comparisons. The implications of this agreement are discussed.