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The governing equations of the variational approach to brittle and ductilefracture emerge from the minimization of a non-convex energy functional subjectto irreversibility constraints. This results in a multifield problem governedby a mechanical balance equation and evolution equations for the internalvariables. While the balance equation is subject to kinematic admissibility ofthe displacement field, the evolution equations for the internal variables aresubject to irreversibility conditions, and take the form of variationalinequalities, which are typically solved in a relaxed or penalized way that canlead to deviations of the actual solution. This paper presents aninterior-point method that allows to rigorously solve the system of variationalinequalities. With this method, a sequence of perturbed constraints isconsidered, which, in the limit, recovers the original constrained problem. Assuch, no penalty parameters or modifications of the governing equations areinvolved. The interior-point method is applied in both a staggered and amonolithic scheme for both brittle and ductile fracture models. In order tostabilize the monolithic scheme, a perturbation is applied to the Hessianmatrix of the energy functional. The presented algorithms are applied to threebenchmark problems and compared to conventional methods, where irreversibilityof the crack phase-field is imposed using a history field or an augmentedLagrangian.

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