The phase field paradigm, in combination with a suitable variational structure, has opened a path for using Griffith’s energy balance to predict the fracture of solids. These so-called phase field fracture methods have gained significant popularity over the past decade, and are now part of commercial finite element packages and engineering fitness- for-service assessments. Crack paths can be predicted, in arbitrary geometries and dimensions, based on a global energy minimization—without the need for ad hoc criteria. In this work, we review the fundamentals of phase field fracture methods and examine their capabilities in delivering predictions in agreement with the classical fracture mechanics theory pioneered by Griffith. The two most widely used phase field fracture models are implemented in the context of the finite element method, and several paradigmatic boundary value problems are addressed to gain insight into their predictive abilities across all cracking stages; both the initiation of growth and stable crack propagation are investigated. In addition, we examine the effectiveness of phase field models with an internal material length scale in capturing size effects and the transition flaw size concept. Our results show that phase field fracture methods satisfactorily approximate classical fracture mechanics predictions and can also reconcile stress and toughness criteria for fracture. The accuracy of the approximation is however dependent on modelling and constitutive choices; we provide a rationale for these differences and identify suitable approaches for delivering phase field fracture predictions that are in good agreement with well-established fracture mechanics paradigms.

This article is part of a discussion meeting issue ‘A cracking approach to inventing new tough materials: fracture stranger than friction’.

相场范式与合适的变分结构相结合，为使用格里菲斯的能量平衡预测固体断裂开辟了道路。这些所谓的相场断裂方法在过去十年中获得了极大的普及，现在已成为商业有限元包和工程适用性评估的一部分。可以基于全局能量最小化以任意几何形状和尺寸预测裂纹路径，而无需特别标准。在这项工作中，我们回顾了相场断裂方法的基础知识，并检查了它们在提供与格里菲斯开创的经典断裂力学理论一致的预测方面的能力。两种最广泛使用的相场断裂模型是在有限元方法的背景下实现的，并且解决了几个典型的边界值问题，以深入了解它们在所有裂解阶段的预测能力；研究了裂纹扩展的开始和稳定的裂纹扩展。此外，我们检查了具有内部材料长度尺度的相场模型在捕捉尺寸效应和过渡缺陷尺寸概念方面的有效性。我们的结果表明，相场断裂方法令人满意地近似于经典断裂力学预测，并且还可以协调断裂的应力和韧性标准。然而，近似的准确性取决于建模和本构选择；

这篇文章是讨论会问题“发明新的坚韧材料的破解方法：断裂比摩擦更奇怪”的一部分。