The nonlinear elastic response of suspensions of rigid inclusions in rubber: II - A simple explicit approximation for finite-concentration suspensions

In Part I, an exact solution was determined for the problem of the overall nonlinear elastic response of Gaussian (or Neo-Hookean) rubber reinforced by a dilute isotropic distribution of rigid particles. Here, this fundamental result is utilized to construct an approximate solution for non-Gaussian rubber reinforced by an isotropic distribution of rigid particles at finite concentration. This is accomplished by means of two different techniques in two successive steps. First, the dilute solution is utilized together with a differential scheme in finite elasticity to generate a solution for Neo-Hookean rubber filled with an isotropic distribution of rigid particles of polydisperse sizes and finite concentration. This non-dilute result is then employed within the context of a new comparison medium method - derived as an extension of Talbot-Willis (1985) variational framework to the non-convex realm of finite elasticity - to generate in turn a corresponding solution for filled non-Gaussian rubber wherein the underlying elastomeric matrix is characterized by any ^I1-based stored-energy function Ψ( ^I1) of choice. The solution is fully explicit and remarkably simple. Its key theoretical and practical merits are discussed in detail. Additionally, the constructed analytical solution is confronted to 3D finite-element simulations of the large-deformation response of Neo-Hookean and non-Gaussian rubber reinforced by isotropic distributions of rigid spherical particles with the same size, as well as with different sizes. Good agreement is found among all three sets of results. The implications of this agreement are discussed.