An arc-length control technique for solving quasi-static fracture problems with phase field models and a staggered scheme

This paper describes a new arc-length control procedure for tracing the equilibrium curve of brittle fracture problems modeled with a phase field approach. The balance equations of this model are solved with a staggered strategy. The control equation of the arc-length procedure determines the displacement increments during the mechanical stage. The arc-length parameter is interpreted as imposing a given increment of the driving force appearing into the micro-force balance equation. The innovative technique consisting of applying the control equation to the displacement degrees of freedoms (DOFs) of the mechanical stage offers an enhancement over earlier arc-length strategies that focused on controlling the damage DOFs in the micro-force balance equation stage. This advancement enables the phase field approach to handle and simulate a broader range of problems, as demonstrated in this paper. The arc-length parameter is stepwise adjusted to yield a pre-established maximum damage increment in each staggered scheme step. As a consequence, the crack tip advance can be strictly controlled in every step holding bounded the pseudo-time integration error, even using an explicit staggered strategy. This procedure entails moderate computational costs for tracing the complete equilibrium curve, including unstable responses, limit points, snap-backs, etc., with the subsidiary advantage that lack of convergence has never been detected in the tests presented in this paper. Additionally, line search techniques have not been necessary. The proposed arc-length procedure is easily implemented in standard finite element codes, and according to our numerical experiments, it does not significantly increase the computational burden of the original explicit staggered strategy.