  In this manuscript, the phase-field fracture irreversibility constraint is
transformed into an equality-based constraint using the slack variable
approach. The equality-based fracture irreversibility constraint is then
introduced in the phase-field fracture energy functional using the Lagrange
Multiplier Method and the Penalty method. Both methods are variationally
consistent with the conventional variational inequality phase-field fracture
problem, unlike the history-variable approach. Thereafter, numerical
experiments are carried out on benchmark problems in brittle and quasi-brittle
fracture to demonstrate the efficacy of the proposed method.
