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.

对有限准静态变形下橡胶中刚性夹杂物悬浮液的宏观或均匀粘弹性响应进行了数值和分析研究。重点是等轴夹杂物的随机各向同性悬浮体牢固嵌入具有恒定粘度的各向同性不可压缩高斯橡胶中的原型案例。从数值的角度来看，基于一致的 Crouzeix-Raviart 空间有限元离散化和时间的高阶精确显式 Runge-Kutta 离散化，引入了一种鲁棒方案来解决控制初始边界值问题，其中特别适合应对有限变形和橡胶的不可压缩性约束带来的挑战。该方案用于在各种负载条件下为相同（单分散）尺寸的球形夹杂物悬浮的基本情况生成样本解决方案。从互补的角度来看，解析解是在极限范围内得出的：（$\mathrm{一世}$) 的小变形， ($\mathrm{一世}\mathrm{一世}$) 无限缓慢或无限快地施加的有限变形，以及 ($\mathrm{一世}\mathrm{一世}\mathrm{一世}$) 当橡胶失去储存弹性能量的能力并还原为牛顿流体时。引人注目的是，尽管下面的橡胶基质具有恒定的粘度，但解决方案表明悬浮液的粘弹性响应表现出有效的剪切稀化型非线性粘度。解决方案进一步表明，悬架的粘弹性响应具有与底层橡胶相同类型的短程记忆行为 - 与普遍预期的长程记忆行为相反。在渐近分析结果和数值解的指导下，提出了一种简单而准确的悬浮体宏观粘弹性响应的近似解析解。