Higher-order phase field fracture simulation in nearly incompressible viscoelasticity

The phase field modeling of viscoelastic fracture can effectively predict complex crack behavior and offer valuable insights for analyzing time-dependent failure mechanisms. However, its application in nearly incompressible viscoelastic materials presents several numerical challenges, including the volumetric locking and inhibition of crack opening. In addition, the low computational efficiency remains a significant challenge in the development of the phase field models (PFMs). Previous studies have shown that the fourth-order phase field theory can improve the convergence rate of the numerical solution with a smoother phase field. In this study, a fourth-order PFM is proposed, which incorporates the B-bar method and the material penalization to predict the fracture behavior of viscoelastic materials limited by the near compressibility. The additional C¹ continuity requirement on the phase field is ensured in the isogeometric framework. No extra degrees of freedom are introduced to resolve the locking problem, which is practical for the PFM, especially in three-dimensional (3D) applications. The loosening of the near incompressibility constraints in the fully damaged zone is achieved by the material penalization, which allows the crack to open freely while preserving the constraints in other regions. The accuracy of the fourth-order PFM is verified through one-dimensional (1D) simulations. Benchmark examples are performed to investigate the numerical behavior of the model. Comparisons between the predicted results and the existing data demonstrate the effectiveness and accuracy of the proposed PFM.