Path-following methods for phase-field simulation of quasi-static crack propagation

The variational approach to fracture, particularly through its regularization as a phase-field model, has become a widely used tool for simulating the quasi-static propagation of cracks in structures. However, classic incremental loading can induce unstable crack growth, violating the quasi-static assumption, and in some cases, leads to a loss of force balance, preventing the estimation of dissipated energy during snapback instabilities. To address this challenge, path-following methods are investigated. Their aim is to adjust the applied load so that it stays at the propagation threshold, thereby preserving the quasi-static assumption and ensuring equilibrium solutions. In this work, we apply and evaluate multiple path-following methods within the framework of variational phase-field fracture models, which are developed to regularize linear elastic variational sharp crack evolution problems. Our study pursues two objectives. First, we identify several path-following methods that are both constitutive model-independent and problem-independent, while remaining straightforward to implement. To achieve this, we focus on partitioned path-following methods based on the displacement field, which decouple the path-following control equation from the rest of the system, providing an easier integration into staggered solvers. In addition, we also introduce a new path-following method which limit the maximum strain increment outside the cracked regions. Second, through three crack propagation problems of increasing complexity, we assess the equilibrium path predictions of the variational phase-field model by comparison with the associated sharp crack model. The comparison demonstrates that the proposed path-following method offers a simple yet highly effective approach to capture the equilibrium path in phase-field fracture simulations. This method robustly maintains the quasi-static assumption, ensuring physically meaningful results. By enabling accurate estimation of the energy dissipated during snapback instabilities, it paves the way for the rational design of more resistant heterogeneous materials.