Rate-dependent phase field fracture simulation in polymers with adaptive mixed isogeometric approach

•A new mixed formulation of the phase field fracture model for nearly incompressible viscoelastic polymers is presented under finite strain.•The Sargado-type non-uniform energy degradation suppresses the incompressibility loosening before fracture.•The developed stable hierarchical spline space combination resolves the instability issue of pressure oscillations and maintains the high continuity advantage of isogeometric analysis.•An adaptive refinement algorithm of displacement-pressure hierarchical meshes improves computational efficiency. The fracture of polymers involves viscous dissipation and finite deformation, which poses difficulties for theoretical and numerical analysis. The phase field model (PFM) is a promising tool for fracture simulation, but the near incompressibility nature of polymers presents challenges. This study proposes a fourth-order PFM to deal with the rate-dependent fracture of nearly incompressible polymers within the adaptive mixed isogeometric framework that integrates the hierarchical B-splines. The higher-order terms improve the regularity of the phase field solution. The penalty formulation in PFM incorporates the non-uniform energy degradation scheme that integrates the Sargado-type degradation function to address the contradiction between the incompressibility constraint and crack opening. This scheme preserves the undamaged response before fracture while narrowing the incompressibility loosening band. Based on the weighted residual method, a new mixed formulation is derived and serves as the weak form of the governing equations of PFM. Using the robust staggered scheme, the established multi-field problem is decoupled, and the mixed displacement-pressure (u-p) formulation of the deformation sub-problem is discretized by the presented stable hierarchical spline space combination, maintaining the high continuity advantage of isogeometric analysis. To enhance the computational efficiency, an adaptive local refinement algorithm based on a level-by-level marking strategy is proposed for the u-p hierarchical meshes. The performance of the developed mixed u-p elements is assessed in the benchmark example by comparing them with the standard elements. Representative numerical examples are performed to demonstrate the effectiveness and accuracy of the proposed phase field fracture model.