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.

中文翻译：

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脆性和延性断裂相场方法的内点法

易变脆性和脆性断裂的变分方法的控制方程是由受不可逆约束的非凸能量函数的最小化产生的。这导致了由机械平衡方程和内部变量的演化方程控制的多场问题。平衡方程受位移场的运动学容许性影响，而内部变量的演化方程则受不可逆条件的影响，并采取变分不等式的形式，通常以松弛或罚分的方式求解，这可能导致偏差实际解决方案。本文提出了一种内点方法，可以严格解决变分不等式系统。使用这种方法，考虑了一系列扰动约束，在极限内，恢复原始约束问题。这样，不涉及惩罚参数或控制方程式的修改。对于脆性和延性断裂模型，内点法既适用于交错方案，也适用于整体方案。为了稳定整体方案，将扰动应用于能量泛函的Hessian矩阵。所提出的算法适用于三个基准问题，并与传统方法进行了比较，在传统方法中，使用历史场或增强拉格朗日法施加了裂纹相场的不可逆性。对于脆性和延性断裂模型，内点法既适用于交错方案，也适用于整体方案。为了稳定整体方案，将扰动应用于能量泛函的Hessian矩阵。所提出的算法适用于三个基准问题，并与传统方法进行了比较，在传统方法中，使用历史场或增强拉格朗日法施加了裂纹相场的不可逆性。对于脆性和延性断裂模型，内点法既适用于交错方案，也适用于整体方案。为了稳定整体方案，将扰动应用于能量泛函的Hessian矩阵。提出的算法适用于三个基准问题，并与传统方法进行比较，在传统方法中，使用历史场或增强拉格朗日法强加了裂纹相场的不可逆性。