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.

翻译：软质和软质骨折的调节方程式,其治理方程式来自受不可逆限制的非软质能源功能的最小化。这导致由机械平衡方程式和内部变量进化方程式管理的多领域问题。虽然平衡方程式受运动性可接受迁移场的制约,但内部变量的演进方程式受不可逆条件的制约,其形式为变异性方程式,通常以宽松或惩罚的方式解决,可能导致实际解决方案的偏差。本文介绍了一种内部点方法,以便能够严格解决变异性不平等体系。采用这种方法,可以考虑周遭制约的制约顺序,在极限内恢复最初的受限方程式问题。因此,没有涉及调整方程式的罚款参数或修改。内位公式既适用于交错,又适用于可导致实际解决方案偏差的单一性办法。为了稳定单一性办法,对赫斯海斯的矩阵矩阵矩阵应用了一种穿透度方法,在极限内位模型中,将一个功能性模型应用了三度,将功能性模型用于常规水平的模型。