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

ZAMBRANO, J.; S. TORO; P.J SÁNCHEZ; DUDA, F.P.; MENDEZ, CARLOS GUSTAVO; A E. HUESPE

COMPUTATIONAL MECHANICS

SPRINGER

Lugar: Berlin; Año: 2023

0178-7675

This paper describes a new arc-length control procedure for tracing the equilibrium curveof brittle fracture problems modeled with a phase field approach. The balance equationsof this model are solved with a staggered strategy. The control equation of the arc-lengthprocedure determines the displacement increments during the mechanical stage. The arclength 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 displacementdegrees of freedoms (DOFs) of the mechanical stage offers an enhancement over earlier arclength strategies that focused on controlling the damage DOFs in the micro-force balanceequation stage. This advancement enables the phase field approach to handle and simulatea 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 canbe strictly controlled in every step holding bounded the pseudo-time integration error, evenusing an explicit staggered strategy. This procedure entails moderate computational costsfor tracing the complete equilibrium curve, including unstable responses, limit points, snapbacks, 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 elementcodes, and according to our numerical experiments, it does not significantly increase thecomputational burden of the original explicit staggered strategy.