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 displa...

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Bibliographic Details
Published in:Computational mechanics Vol. 73; no. 4; pp. 751 - 772
Main Authors: Zambrano, J., Toro, S., Sánchez, P. J., Duda, F. P., Méndez, C. G., Huespe, A. E.
Format: Journal Article
Language:English
Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.04.2024
Springer
Springer Nature B.V
Subjects:
Classical and Continuum Physics
Computational Science and Engineering
Computing costs
Crack propagation
Crack tips
Damage
Energy dissipation
Engineering
Load
Mathematical models
Original Paper
Parameters
Theoretical and Applied Mechanics
Time integration
Tracing
Variables
Arc-length control strategy
Quasi-static brittle fracture
Phase field models
Staggered methodology
Path-following algorithm
Unstable crack growth
ISSN:0178-7675, 1432-0924
Online Access:Get full text
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Summary: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.
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ISSN:0178-7675
1432-0924
DOI:10.1007/s00466-023-02388-7