This paper outlines a rigorous variational-based multilevel Global–Local formulation for ductile fracture. Here, a phase-field formulation is used to resolve failure mechanisms by regularizing the sharp crack topology on the local state. The coupling of plasticity to the crack phase-field is realized by a constitutive work density function, which is characterized through a degraded stored elastic energy and the accumulated dissipated energy due to plasticity and damage. Two different Global–Local approaches based on the idea of multiplicative Schwarz’ alternating method are proposed: (i) A global constitutive model with an elastic–plastic behavior is first proposed, while it is enhanced with a single local domain, which, in turn, describes an elastic–plastic fracturing response. (ii) The main objective of the second approach is to introduce an adoption of the Global–Local model toward the multilevel local setting. In (ii), an elastic–plastic global constitutive model is augmented with two distinct local domains; in which, the first local domain behaves as an elastic–plastic material and the second local domain is modeled due to the fractured state. To further reduce the computational cost, predictor–corrector adaptivity within the Global–Local concept is introduced. An adaptive scheme is devised through the evolution of the effective global plastic flow (for only elastic–plastic adaptivity), and through the evolution of the local crack phase-field state (for only fracture adaptivity). Thus, two local domains are dynamically updated during the computation, resulting in a two-way adaptivity procedure. The overall response of the Global–Local approach in terms of accuracy/robustness and efficiency is verified using single-scale problems. The resulting framework is algorithmically described in detail and substantiated with numerical examples.

本文概述了韧性断裂的严格的基于变分的多级全局-局部公式。在这里，相场公式用于通过调整局部状态上的尖锐裂纹拓扑来解决失效机制。塑性与裂纹相场的耦合是通过本构功密度函数实现的，其特征在于储存的弹性能量退化，以及由于塑性和损伤而累积的耗散能量。提出了基于乘法 Schwarz 交替方法思想的两种不同的全局-局部方法：（i）首先提出了具有弹塑性行为的全局本构模型，同时通过单个局部域，反过来，它描述了弹塑性压裂响应。(ii) 第二种方法的主要目标是在多级本地环境中采用全球-本地模型。在（ii）中，弹塑性全局本构模型增加了两个不同的局部域；其中，第一个局部域表现为弹塑性材料，第二个局部域因断裂状态而建模。为了进一步降低计算成本，引入了全局-局部概念中的预测器-校正器自适应性。通过有效全局塑性流动的演变设计了一种自适应方案（仅适用于弹塑性适应性），并通过局部裂纹相场状态的演变（仅适用于断裂适应性）。因此，两个局部域在计算过程中动态更新，导致双向自适应过程。Global-Local 方法在准确性/鲁棒性和效率方面的整体响应使用单尺度问题进行验证。由此产生的框架在算法上进行了详细的描述，并用数值例子加以证实。