A phase field formulation for modelling fracture of nearly incompressible hyperelastic materials

Flexible materials are integral to modern applications due to their unique properties, particularly their ability to stretch and resilience to fracture. However, predicting the fracture behaviour of these materials through simulations remains challenging, primarily due to the lack of numerical robustness. This study proposes a rate-independent phase field model to predict finite strain fracture in nearly incompressible hyperelastic materials. A Griffith-type criterion is used to predict the fracture behaviour, with a relaxation of the incompressibility constraint in damaged elements, thus allowing crack propagation without affecting the intact material. A novel mixed formulation is developed using a quadratic dissipation function originally proposed by Ambrosio and Tortorelli (AT2) for the phase field method, incorporating two history fields to prevent crack healing. Spatial discretisation is achieved using linear approximations for the displacement and damage fields, whereas the pressure field is treated as discontinuous across the element boundaries (Q1Q0Q1 elements). The numerical algorithm is implemented within a finite element framework using a user-defined element (UEL) subroutine in ABAQUS, with a monolithic solver based on the Broyden-Fletcher-Goldfarb-Shanno (BFGS) technique to solve the global problem. Numerical trials confirmed that this is a robust algorithm that avoids excessive distortion of damaged elements, eliminating the need for adaptive meshing and distorted mesh deletion techniques. The algorithm is tested using three examples and is compared with experimental data. Reproduction of load-displacement behaviour and crack paths confirm the effectiveness of the method. The results indicate that the approach effectively predicts the fracture behaviour of nearly incompressible materials under large stretch conditions while maintaining numerical robustness. Additionally, the method successfully predicts multiple crack initiations, propagation paths, and their merging, consistent with various experimental observations. Consequently, this robust numerical scheme, involving 3D finite elements, can be readily applied to simulate various devices made of rubber-like materials, facilitating faster optimal design development and offers a promising alternative to multiple experiments and prototype testings resulting in a significant cost reduction.