A 3D micromechanical model for hyperelastic rubber-like materials and its numerical resolution by the Asymptotic Numerical Method (ANM)

In this work, a $3D$ micromechanical model is developed to describe the behavior of macromolecular chains and to reflect the hyperelastic behavior of rubber-like materials. This model generalizes the $2D$ model recently developed in Ouardi (2023). The behavior law is defined by the minimization of a potential energy, each macromolecular chain has been represented by elastic segments linked by nonlinear elastic spiral nodes. We thus obtain a model with only three characteristic parameters. We investigate, in the $3D$ case, the effect of the number of macro-chain segments and the shape of the Representative Volume Element (RVE) using a high-order algorithm of the family of the Asymptotic Numerical Method (ANM) (Cochelin, 2007). In the ANM algorithm, the solution of the nonlinear problem is sought branch by branch, each branch being represented by a Taylor series. In this way, this high-order algorithm makes it easier to continuously investigate the solution curves. Numerical simulations are presented on different RVEs, four and eight chains models (Arruda and Boyce, 1993), under three types of boundary conditions: uniaxial tension, pure shear and equibiaxial tension. These numerical simulations are compared with experimental data from Treloar (1944) to identify the parameters material and to demonstrate the robustness of the proposed model. The studied chains models show a slight influence of the number of macro-chains and the number of segments in the RVE.