The nonlinear viscoelastic response of suspensions of rigid inclusions in rubber: I—Gaussian rubber with constant viscosity

Abstract

A numerical and analytical study is made of the macroscopic or homogenized viscoelastic response of suspensions of rigid inclusions in rubber under finite quasistatic deformations. The focus is on the prototypical case of random isotropic suspensions of equiaxed inclusions firmly embedded in an isotropic incompressible Gaussian rubber with constant viscosity. From a numerical point of view, a robust scheme is introduced to solve the governing initial–boundary-value problem based on a conforming Crouzeix–Raviart finite-element discretization of space and a high-order accurate explicit Runge–Kutta discretization of time, which are particularly well suited to deal with the challenges posed by finite deformations and the incompressibility constraint of the rubber. The scheme is deployed to generate sample solutions for the basic case of suspensions of spherical inclusions of the same (monodisperse) size under a variety of loading conditions. From a complementary point of view, analytical solutions are worked out in the limits: (i) of small deformations, (ii) of finite deformations that are applied either infinitesimally slowly or infinitely fast, and (iii) when the rubber loses its ability to store elastic energy and reduces to a Newtonian fluid. Strikingly, in spite of the fact that the underlying rubber matrix has constant viscosity, the solutions reveal that the viscoelastic response of the suspensions exhibits an effective nonlinear viscosity of shear-thinning type. The solutions further indicate that the viscoelastic response of the suspensions features the same type of short-range-memory behavior — as opposed to the generally expected long-range-memory behavior — as that of the underlying rubber. Guided by the asymptotic analytical results and the numerical solutions, a simple yet accurate approximate analytical solution for the macroscopic viscoelastic response of the suspensions is proposed.