An Enriched Phase-Field Method (XPFM) for the Efficient Simulation of Fracture Processes

Complex crack processes like crack initiation, branching, and coalescence can be simulated with the phase-field method for fracture. Here, the criterion for crack propagation is considered within the formulation of the partial differential equation. Its solution, the scalar phase-field, indicates the crack in a smeared manner. This makes the evaluation of a separate criterion for crack propagation and the tracking of the crack geometry obsolete. A challenge for the phase-field method for fracture, however, is the accurate and efficient reproduction of the phase-field, the displacement field, and their gradients. For this, the use of extremely fine meshes is often required within the standard phase-field method. In this contribution, an enhancement of the phase-field method through the adjustment of the ansatz functions is presented. A transformation of the phase-field ansatz by embedding a quadratic ansatz into an exponential function is utilised to improve the approximation of the phase-field even with coarse meshes. The crack is not additionally restricted by the ansatz and can still develop independently of the mesh geometry. Furthermore, the displacement ansatz is extended by a term with adjusted shape-functions carrying information about the crack geometry from the phase-field. It can reproduce the high displacement gradients across the crack. The modified shape-functions are calculated for each enriched element on a submesh by considering the local reduction of stiffness due to the phase-field. Since these shape-functions are directly coupled to the phase-field ansatz, no additional discretisation of the crack geometry is required.