A Multiscale Mixed Three‐Field Finite Element Formulation Coupled With Phase Field Fracture for Incompressible Rubber‐Like Materials

ABSTRACT Rubber‐like materials possess remarkable properties such as high stretchability, low modulus, and impressive toughness, making them ideal for applications across various emerging domains. Given their growing relevance, understanding the fracture behavior of these materials is crucial for designing applications against failure. Although numerous models have been developed to simulate fracture propagation in these materials, few accurately account for their incompressible nature, which often leads to numerical challenges. This study introduces a multiscale polymer model integrated with the phase field approach, specifically designed to capture the fracture behavior in incompressible rubber‐like materials using mixed finite elements. At the microscale, the entropic chain behavior is modeled using non‐Gaussian statistics while also accounting for an internal energy due to molecular bond distortions. The non‐affine microsphere network model, adapted for damaged systems, is employed to bridge the microscale deformations with those at the macroscale. Furthermore, the phase field approach is utilized at the macroscale to model damage, which is primarily assumed to be caused by the rupture of chain segments. This is coupled with a three‐field mixed formulation, and augmented Lagrangian iterations are performed to strongly enforce the incompressibility constraint. The major advantage of this proposed formulation lies in the fact that it does not lead to an increase in the global system size. The numerical implementation of the proposed model using a monolithic scheme is detailed, and three‐dimensional simulations are performed to validate the model's performance. The predictions are compared with experimental data to evaluate the potential and reliability of the framework in accurately predicting the fracture behavior of rubber‐like materials. Furthermore, the volumetric deformation contours are visualized to demonstrate the effectiveness of the proposed model in strictly enforcing the incompressibility constraint. Moreover, the effect of the phase field regularization parameter on the predicted fracture behavior in specimens having geometric features of different sizes is investigated.