EXPONENTIAL FINITE ELEMENT SHAPE FUNCTIONS FOR A PHASE FIELD MODEL OF BRITTLE FRACTURE

In phase field models for fracture a continuous scalar field variable is used to indicate cracks, i.e. the value 1 of the phase field variable is assigned to sound material, while the value 0 indicates fully broken material. The width of the transition zone where the phase field parameter changes between 1 and 0 is controlled by a regularization parameter. As a finite element discretization of the model needs to be fine enough to resolve the crack field and its gradient, the numerical results are sensitive to the choice of the regularization parameter in conjunction with the mesh size. This is the main challenge and the computational limit of the finite element implementation of phase field fracture models. To overcome this limitation a finite element technique using special shape functions is introduced. These special shape functions take into account the exponential character of the crack field as well as its dependence on the regularization length. Numerical examples show that the exponential shape functions allow a coarser discretization than standard linear shape functions without compromise on the accuracy of the results. This is due to the fact, that using exponential shape functions, the approximation of the surface energy of the phase field cracks is impressively precise, even if the regularization length is rather small compared to the mesh size. Thus, these shape functions provide an alternative to a numerically expensive mesh refinement.