Twenty years in since their introduction, it is now plain that the regularized formulations dubbed as phase-field of the variational theory of brittle fracture of Francfort and Marigo (1998) provide a powerful macroscopic theory to describe and predict the propagation of cracks in linear elastic brittle materials under arbitrary quasistatic loading conditions. Over the past ten years, the ability of the phase-field approach to also possibly describe and predict crack nucleation has been under intense investigation. The first of two objectives of this paper is to establish that the existing phase-field approach to fracture at large — irrespectively of its particular version — cannot possibly model crack nucleation. This is so because it lacks one essential ingredient: the strength of the material.

The second objective is to amend the phase-field theory in a manner such that it can model crack nucleation, be it from large pre-existing cracks, small pre-existing cracks, smooth and non-smooth boundary points, or within the bulk of structures subjected to arbitrary quasistatic loadings, while keeping undisturbed the ability of the standard phase-field formulation to model crack propagation. The central idea is to implicitly account for the presence of the inherent microscopic defects in the material — whose defining macroscopic manifestation is precisely the strength of the material — through the addition of an external driving force in the equation governing the evolution of the phase field. To illustrate the descriptive and predictive capabilities of the proposed theory, the last part of this paper presents sample simulations of experiments spanning the full range of fracture nucleation settings.

自从引入二十年以来，现在很显然，被称为Francfort和Marigo（1998）的脆性断裂变分理论的相场的正则化公式提供了一个强大的宏观理论，来描述和预测线性弹性中的裂纹扩展。任意准静态载荷条件下的脆性材料。在过去的十年中，对相场方法也可能描述和预测裂纹成核的能力一直在深入研究。本文的两个目标中的第一个是要确定现有的大范围断裂的相场方法（无论其特定版本如何）都不可能模拟裂纹成核。之所以如此，是因为它缺少一种必不可少的成分：材料的强度。

第二个目标是以某种方式修改相场理论，以便可以模拟裂纹成核，无论是来自大的既有裂纹，小小的既有裂纹，光滑和非光滑的边界点，还是在大范围内结构承受任意准静态载荷，同时保持标准相场公式化模拟裂纹扩展的能力。中心思想是通过在控制相场演变的方程式中添加外部驱动力，隐式地考虑材料中固有的微观缺陷的存在（其定义的宏观表现恰好是材料的强度）。为了说明所提出理论的描述性和预测性，