AI 中文总结
本文提出反平面情形下的粘聚断裂强度退化相场正则化模型，整合多断裂理论为统一变分框架，推导闭式解、数值格式并验证等效粘聚律等特性。
AI 中文摘要
相场断裂方法最初设计用于正则化脆性断裂的Griffith模型，如今通常被视为梯度损伤模型，其正则化长度成为驱动裂纹形核的材料属性。该方法的一个缺陷是强度曲面无法任意设定：其形状由弹性能决定，大小由正则化长度决定。本文聚焦Bourdin、Marigo、Maurini和Zolesi（arXiv:2506.22558）提出的模型的反平面版本，该版本通过退化强度而非刚度，处理沿未知路径的裂纹扩展以及由任意凸强度曲面控制的形核。它可被解释为软化塑性的正则化，其中局部化带服从由强度域和韧性设定的等效粘聚律；当正则化长度远小于弹-粘聚长度时，其作用仅为数值层面的。因此，强度、刚度和韧性成为独立的材料数据，极限分析、理想塑性、粘聚断裂和脆性断裂被整合到单一变分框架中。本文推导了简单剪切问题的闭式解，提出了结合交替最小化和圆锥规划的数值格式，并数值验证了等效粘聚律、其对正则化的独立性以及由弹-粘聚长度控制的尺寸效应。“冲浪”模拟凸显了扩展裂纹的结构，而凹入V型缺口被用于展示该模型如何无需先验假设即可衔接小尺度屈服、粘聚断裂和脆性断裂。
英文摘要
Phase-field approaches to fracture, initially designed as regularization of the Griffith model of brittle fracture, are now commonly viewed as gradient-damage models whose regularization length becomes a material property driving crack nucleation. One weakness of this approach is that the strength surface cannot be arbitrary: its shape is dictated by the elastic energy, and its magnitude by the regularization length. We focus on the antiplane version of the model introduced by Bourdin, Marigo, Maurini and Zolesi (arXiv:2506.22558), which handles crack propagation along unknown paths and nucleation governed by an arbitrary convex strength surface by degrading the strength instead of the stiffness. It can be interpreted as a regularization of softening plasticity in which localization bands obey an equivalent cohesive law set by the strength domain and the toughness, while the role of the regularization length, when small compared to the elasto-cohesive length, is purely numerical. Strength, stiffness, and toughness thus become independent material data, and limit analysis, perfect plasticity, cohesive fracture, and brittle fracture merge into a single variational framework. We derive closed-form solutions for a simple shear problem, propose a numerical scheme combining alternate minimization and conic programming, and numerically verify the equivalent cohesive law, its independence of the regularization, and the size effect governed by the elasto-cohesive length. A "surfing" simulation highlights the structure of the propagating crack while a re-entrant V-notch is used to show how the model bridges small-scale yielding, cohesive fracture, and brittle fracture without a priori hypotheses.
Comments43 pages, 18 figures, 1 table. Submitted to Comptes Rendus. Mecanique
