Adaptive fourth-order phase field analysis for brittle fracture

Abstract

We present an adaptive h-refined fourth-order phase field model for studying fracture using a hybrid-staggered solution scheme. The conventional fourth-order phase field model involves the computation of higher-order derivatives of the phase field, which requires continuously differentiable shape functions. The proposed locally-refinable approximation scheme is devised in the framework of isogeometric analysis which provides a smooth C1 continuous discretization throughout the domain. With the fourth-order model, the cracked surface is captured more accurately with fewer degrees of freedom. The efficiency of the formulation is enhanced by employing an adaptive h-refinement scheme using polynomial splines over hierarchical T-meshes (PHT-Splines), which allows the hierarchical refinement to resolve the local quantities of interest. The inherent localization property of PHT-splines is exploited to develop an efficient data transfer algorithm for bridging the coarse and fine meshes. To enhance the accuracy of the proposed approach, we use a cubic stress-degradation function instead of the commonly used quadratic stress-degradation function. Moreover, we use a Taylor series expansion to approximate the stress-degradation function which improves the robustness of the non-linear solver. The improvement in the solution and the accuracy of the proposed approach have been presented by comparing the obtained results with the analytical solution available in the literature for a one-dimensional example. We also demonstrate the robustness of the proposed approach in capturing complex crack patterns through several two and three-dimensional examples. Code and data necessary for replicating the results of the examples in the article will be made available through a GitHub repository.