A polyconvex strain‐energy split for a high‐order phase‐field approach to fracture

This contribution focuses on a novel phase‐field model for a high‐order phase‐field approach to brittle fracture in the range finite deformation. In particular, two different challenges are tackled in this study: First, we want to establish a polyconvex free energy density to ensure the existence of a minimizer for the coupled problem, second, we have to deal with a fourth‐order Cahn‐Hilliard type equation for the approximation of the phase‐field. Phase‐field methods employ a variational framework for brittle fracture and have proven to predict complex fracture patterns in two and three dimensional examples. Basis of the model are the conjugate stresses of the three strain measures deformation gradient (line map), its cofactor (area map) and its determinant (volume map). The introduction of the tensor cross product simplifies the presentation of the first Piola‐Kirchhoff stress tensor and its derivatives in elegant manner. The proposed Cahn‐Hilliard type equation requires global ${\cal C}^1$ ‐continuity. Therefore, we apply an isogeometric framework using NURBS basis functions. Moreover, a general hierarchical refinement scheme based on subdivision projection is used here for one, two and three dimensional simulations. This technique allows to enhance the approximation space using finer splines on each level but preserves the partition of unity as well as the continuity properties of the original discretization. We finally demonstrate the accuracy and the robustness with a series of benchmark problems. (© 2016 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)