The Nonlinear Viscoelastic Response of Suspensions of Vacuous Bubbles in Rubber: I — Gaussian Rubber with Constant Viscosity

This paper presents an analytical and numerical study of the homogenization problem of suspensions of vacuous bubbles in viscoelastic rubber subject to finite quasistatic deformations. The focus is on the elementary case of bubbles that are initially equiaxed in shape and isotropically distributed in space and on isotropic incompressible rubber with Gaussian elasticity and constant viscosity. From an analytical point of view, asymptotic 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. From a numerical point of view, making use of a recently developed scheme based on a conforming Crouzeix-Raviart finite-element discretization of space and a high-order accurate explicit Runge-Kutta discretization of time, sample solutions are worked out for suspensions of initially spherical bubbles of the same (monodisperse) size under a variety of loading conditions. Consistent with a recent conjecture of Ghosh et al. (J. Mech. Phys. Solids 155:104544, 2021), the various asymptotic and numerical solutions indicate that the viscoelastic response of the suspensions features the same type of short-range-memory behavior — in contrast with the generally expected long-range-memory behavior — as that of the underlying rubber, with the distinctive differences that their effective elasticity is compressible and their effective viscosity is compressible and nonlinear. By the same token, the various solutions reveal a simple yet accurate analytical approximation for the macroscopic viscoelastic response of the suspensions under arbitrary finite quasistatic deformations.