一大类有限黏弹性模型的Abaqus实现
Abaqus implementation of a large family of finite viscoelasticity models
AI总结：
本文提出一个Abaqus UMAT子程序，用于实现涵盖非高斯弹性与非线性黏性的有限黏弹性本构模型，通过高阶显式Runge-Kutta格式求解约束ODE，并验证了准确性与收敛性。
摘要：
本文介绍了一个Abaqus UMAT子程序，用于一族描述任意可压缩性（包括完全不可压缩）各向同性弹性体在有限变形下黏弹性响应的本构模型。这些模型可被选择以涵盖广泛的非高斯弹性，以及广泛的非线性黏性。从数学观点看，模型的结构使得黏性耗散由一个内部变量 $\textbf{C}^v$ 表征，该变量受基于物理的约束 $\det\textbf{C}^v=1$ 限制，且是关于时间的非线性一阶ODE的解。该ODE通过能够恒等保持约束 $\det\textbf{C}^v=1$ 的高阶显式Runge-Kutta格式求解。通过与几种Abaqus内置混合有限元（包括单纯形单元C3D4H和C3D10H以及六面体单元C3D8H和C3D20H）的精确解进行比较，从数值上证明了代码的准确性和收敛性。本文最后部分致力于展示代码的能力，将其部署用于计算双连续橡胶共混物的均匀化响应。
英文摘要：
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.
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