The Relation Between Similarity Fields and Invariant Path Integrals in Nonlinear Fracture Mechanics

非线性断裂力学中相似场和不变路径积分之间的关系伊利诺伊大学香槟分校的Gustavo Gioia计算模拟表明，扩展裂纹的塑性区场通过单个相似变量依赖于应力强度因子K和t-应力T，这意味着我们可以通过对单个变量的一系列值进行实验来确定与K和T的任意组合相关的断裂类型(脆性或延性)。然而，相似场的起源仍不清楚。在这里，我们将追踪这些相似场的存在不变的路径积分。我们将把相似变量与裂纹尖端能流、粘度、内聚力和三轴性联系起来。对于几个破坏准则，我们将在K-T空间中构建识别延性、过渡区和脆性区域的图。我们还将研究可分离的裂纹尖端相似场，并建立它们与临界现象的联系。最后，我们将研究静止裂纹的情况，其中场不是相似场，但存在不变的路径积分。

The Relation Between Similarity Fields and Invariant Path Integrals in Nonlinear Fracture MechanicsGustavo Gioia, University of Illinois at Urbana-Champaign Computational simulations indicate that the plastic-zone fields of growing cracks depend on the stress intensity factor K and the t-stress T through a single similarity variable, implying that we can ascertain the type of fracture (brittle or ductile) associated with any combination of K and T by performing experiments over a range of values of a single variable. Yet the origin of similiarity fields remains unknown. Here we will trace these similarity fields to the existence of invariant path integrals. We will relate the similarity variable to the crack-tip energy flux, viscosity, cohesion, and triaxiality. For several failure criteria, we will construct diagrams identifying the ductile, transitional, and brittle regions in K-T space. We will also study separable crack-tip similarity fields and establish their connection with critical phenomena. Last, we will study the case of stationary cracks, where the fields are not similarity ields but there exists an invariant path integral.