A review on phase-field models of brittle fracture and a new fast hybrid formulation

In this contribution we address the issue of efficient finite element treatment for phase-field modeling of brittle fracture. We start by providing an overview of the existing quasi-static and dynamic phase-field fracture formulations from the physics and the mechanics communities. Within the formulations stemming from Griffith's theory, we focus on quasi-static models featuring a tension-compression split, which prevent cracking in compression and interpenetration of the crack faces upon closure, and on the staggered algorithmic implementation due to its proved robustness. In this paper, we establish an appropriate stopping criterion for the staggered scheme. Moreover, we propose and test the so-called hybrid formulation, which leads within a staggered implementation to an incrementally linear problem. This enables a significant reduction of computational cost-about one order of magnitude-with respect to the available (non-linear) models. The conceptual and structural similarities of the hybrid formulation to gradient-enhanced continuum damage mechanics are outlined as well. Several benchmark problems are solved, including one with own experimental verification.