In this paper, we introduce an Abaqus UMAT subroutine for a family of constitutive models for the viscoelastic response of isotropic elastomers of any compressibility -- including fully incompressible elastomers -- undergoing finite deformations. The models can be chosen to account for a wide range of non-Gaussian elasticities, as well as for a wide range of nonlinear viscosities. From a mathematical point of view, the structure of the models is such that the viscous dissipation is characterized by an internal variable $\textbf{C}^v$, subject to the physically-based constraint $\det\textbf{C}^v=1$, that is solution of a nonlinear first-order ODE in time. This ODE is solved by means of an explicit Runge-Kutta scheme of high order capable of preserving the constraint $\det\textbf{C}^v=1$ identically. The accuracy and convergence of the code is demonstrated numerically by comparison with an exact solution for several of the Abaqus built-in hybrid finite elements, including the simplicial elements C3D4H and C3D10H and the hexahedral elements C3D8H and C3D20H. The last part of this paper is devoted to showcasing the capabilities of the code by deploying it to compute the homogenized response of a bicontinuous rubber blend.

翻译：本文介绍了一种用于各向同性弹性体（包括完全不可压缩弹性体）在有限变形下粘弹性响应的本构模型族的Abaqus UMAT子程序。所选模型可涵盖广泛的高斯弹性与非线性粘性行为。从数学角度来看，模型结构通过满足物理约束$\det\textbf{C}^v=1$的内变量$\textbf{C}^v$来表征粘性耗散，该内变量由非线性一阶常微分方程（ODE）的时间演化决定。此ODE采用高阶显式龙格-库塔格式求解，该方法能严格保持$\det\textbf{C}^v=1$的约束条件。通过对比Abaqus内置混合有限元（包括C3D4H和C3D10H四面体单元以及C3D8H和C3D20H六面体单元）的精确解，数值验证了代码的精度与收敛性。论文最后部分通过计算双连续橡胶共混物的均匀化响应，展示了该子程序的应用能力。