  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.

Numerical Analysis;Numerical Analysis