The seminal paper of Francfort and Marigo (1998) introduced a variational formulation for Griffith fracture () that has resulted in substantial theoretical and practical progress in modeling and simulating fracture. In particular, it led to the phase-field approximation proposed in Bourdin et al. (2000), which has been widely implemented. However, the formulation in Francfort and Marigo (1998) is known to have limitations, including its inability to treat applied loads and its reliance on global minimization. In addition, the phase-field model (Bourdin et al., 2000) and its extensions, as implemented, are not generally approximations of the global minimizers in Francfort and Marigo (1998). In this paper, we show that there is a local variational principle satisfied by global and local minimizers of the energy introduced in Francfort and Marigo (1998), which is compatible with loads, and which is a generalization of the stress intensity factor. We use this principle to reformulate variational fracture, including formulations that, for the first time, can include all forms of applied loads. We conclude by showing the connection between phase-field models, as implemented, and our formulations.

中文翻译：

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断裂的局部变分原理

Francfort 和 Marigo (1998) 的开创性论文介绍了 Griffith 断裂 () 的变分公式，该公式在断裂建模和模拟方面取得了实质性的理论和实践进展。特别是，它导致了 Bourdin 等人提出的相场近似。 （2000），已被广泛实施。然而，众所周知，Francfort 和 Marigo（1998）中的公式有局限性，包括它无法处理施加的载荷以及它对全局最小化的依赖。此外，相场模型（Bourdin et al., 2000）及其扩展（如实施）一般不是 Francfort 和 Marigo (1998) 中全局最小化器的近似值。在本文中，我们证明存在由 Francfort 和 Marigo (1998) 中引入的能量的全局和局部最小化器满足的局部变分原理，该原理与载荷兼容，并且是应力强度因子的推广。我们利用这一原理重新公式化了变分断裂，包括首次可以包含所有形式的施加载荷的公式。最后，我们展示了所实施的相场模型与我们的公式之间的联系。