Phase-Field Modelling of Damage and Fracture—Convergence and Local Mesh Refinement

In this contribution, we outline the combination of a phase-field model of brittle fracture with adaptive spline-based approximations. The phase-field method provides a convenient way to model crack propagation without topological updates of the used discretisation as the crack is represented implicitly in terms of an order parameter field that can be interpreted as damage variable. For the accurate approximation of the order parameter field that may exhibit steep gradients, we utilise locally refined hierarchical B-splines in conjunction with Bezier extraction. The latter allows for the implementation of the approach in any standard finite element code. Moreover, standard procedures of adaptive finite element analysis for error estimation and marking of elements are directly applicable due to the strict use of an element viewpoint. Two different demonstration problems are considered. At first we examine the convergence properties of the phase-field approach and explain the influence of the domain size and the discretisation for the one-dimensional problem of a bar. Afterwards, results of the adaptive local refinement are compared with uniformly refined Lagrangian and spline-based discretisations. In the second example, the developed algorithms are applied to simulate crack propagation in a two-dimensional single-edge notched, shear loaded plate.