Interior-point methods for the phase-field approach to brittle and ductile fracture

• Published in: Computer methods in applied mechanics and engineering 2021-03, Vol.375, p.113612, Article 113612
• Main Authors: Wambacq, J., Ulloa, J., Lombaert, G., François, S.
• Format: Article
• Language: English
• Subjects: Algorithms; Brittle fracture; Constraints; Ductile fracture; Ductile-brittle transition; Evolution; Heat treating; Hessian matrices; Inequalities; Interior-point methods; Monolithic scheme; Parameter modification; Perturbation; Phase-field models; Staggered scheme
• ISSN: 0045-7825; 1879-2138
• DOI: 10.1016/j.cma.2020.113612

The governing equations of the variational approach to brittle and ductile fracture emerge from the minimization of a non-convex energy functional subject to irreversibility constraints. This results in a multifield problem governed by a mechanical balance equation and evolution equations for the internal variables. While the balance equation is subject to kinematic admissibility of the displacement field, the evolution equations for the internal variables are subject to irreversibility conditions, and take the form of variational inequalities, which are typically solved in a relaxed or penalized way that can lead to deviations of the actual solution. This paper presents an interior-point method that allows to rigorously solve the system of variational inequalities. With this method, a sequence of perturbed constraints is considered, which, in the limit, recovers the original constrained problem. As such, no penalty parameters or modifications of the governing equations are involved. The interior-point method is applied in both a staggered and a monolithic scheme for both brittle and ductile fracture models. In order to stabilize the monolithic scheme, a perturbation is applied to the Hessian matrix of the energy functional. The presented algorithms are applied to three benchmark problems and compared to conventional methods, where irreversibility of the crack phase-field is imposed using a history field or an augmented Lagrangian. •The governing equations of brittle and ductile fracture are obtained variationally.•An interior-point method is used to rigorously solve the constrained problem.•The presented method is applied in both a staggered and a monolithic scheme.•The presented schemes are validated and compared to a benchmark staggered scheme.