A hyperelastic constitutive model for soft elastomers considering the entanglement-dependent finite extensibility

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.