In this paper, a novel hyperelastic constitutive model for soft elastomers is developed based on the concept of the tortuous tube. This model incorporates the finite extensibility of the polymer chain, the entanglement contribution to elasticity and the non-affine micro-to-macro scale transition in a unified way. To reflect the entanglement effect and its influence on the deformation of soft elastomers, the tortuous tube concept is introduced. The finite extensibility and conformational statistics of an entangled polymer chain in such a tortuous tube are clarified. By embedding the tortuous tube into the microsphere and employing the principle of minimum averaged free energy, a new non-affine scale transition rule is proposed to establish the relationship between the local deformation of a polymer chain at the microscopic scale and the overall deformation at the macroscopic scale. Based on the probability density function related to the conformational statistics, the Helmholtz free energy is established and further decoupled into a volumetric part and an isochoric part. The spatial Kirchhoff stress tensor and spatial elasticity tensor are derived from the newly established Helmholtz free energy. The proposed model is further implemented into the finite element program ABAQUS by writing a user-defined material subroutine. The prediction capability of the proposed model is verified by simulating the homogeneous and inhomogeneous deformations of soft elastomers under various loading modes, including uniaxial tension, uniaxial compression, pure shear, equi-biaxial tension, general biaxial tensile loadings, inflation and indentation. Moreover, the influence of entanglement concentration on the stretchability and stiffness of soft elastomers is predicted and discussed using the proposed model.

中文翻译：

---

考虑纠缠依赖性有限延展性的软弹性体超弹性本构模型


本文基于曲折管的概念，开发了一种新的软弹性体超弹性本构模型。该模型以统一的方式结合了聚合物链的有限延展性、对弹性的纠缠贡献和非仿射微观到宏观尺度的过渡。为了反映纠缠效应及其对软弹性体变形的影响，引入了曲折管概念。阐明了这种曲折管中纠缠聚合物链的有限延展性和构象统计。通过将曲折管嵌入微球中，并采用最小平均自由能原理，提出了一种新的非仿射尺度过渡规则，以建立聚合物链在微观尺度上的局部变形与宏观尺度上整体变形之间的关系。基于与构象统计相关的概率密度函数，建立亥姆霍兹自由能并进一步解耦为体积部分和等容部分。空间基尔霍夫应力张量和空间弹性张量由新建立的亥姆霍兹自由能推导而来。通过编写用户定义的材料子程序，将所提出的模型进一步实现到有限元程序 ABAQUS 中。通过模拟软弹性体在各种加载模式下的均匀和非均匀变形，包括单轴拉伸、单轴压缩、纯剪切、等双轴拉伸、一般双轴拉伸载荷、膨胀和压痕，验证了所提模型的预测能力。 此外，使用所提出的模型预测并讨论了纠缠浓度对软弹性体拉伸性和刚度的影响。