Fourth-order phase field modeling of dynamic fracture in porous brittle materials using an adaptive isogeometric analysis

The fourth-order phase field modeling of dynamic fracture in porous brittle materials is performed via an adaptive isogeometric analysis. The hybrid phase field model is extended to dynamic fracture and fourth-order theory to capture the dynamic crackings. The proposed fourth-order phase field model can relax the mesh size requirements while accurately regularizing sharp cracks. The developed model is capable of flexibly constructing the C1 continuous basis functions that are required by the fourth-order model. An adaptive refinement scheme based on hierarchical T-meshes is performed to define the complex geometric shapes and automatically refine the meshes in the crack regions to improve the computational efficiency. To prevent possible crack healings, the history field at integration points is transferred from old meshes to new ones. Numerical examples show the accuracy and efficiency of the method to reproduce dynamic crack branching in the benchmark problem, as well as to capture complex dynamic cracking patterns in porous materials. It is also shown that the solutions are convergent with respect to the mesh refinement and the decreasing of time increment.