Within this article, we develop a residual type a posteriori error estimator for a time discrete quasi-static phase-field fracture model. Particular emphasize is given to the robustness of the error estimator for the variational inequality governing the phase-field evolution with respect to the phase-field regularization parameter $\epsilon$. The article concludes with numerical examples highlighting the performance of the proposed a posteriori error estimators on three standard test cases; the single edge notched tension and shear test as well as the L-shaped panel test.

中文翻译：

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具有不可逆约束的准静态裂缝相场模型自适应解的后验估计器

在本文中，我们为时间离散准静态相场断裂模型开发了一种残差型后验误差估计器。特别强调了误差估计器的鲁棒性，用于控制相场演化相对于相场正则化参数 $\epsilon$ 的变分不等式。文章最后用数值例子强调了所提出的后验误差估计器在三个标准测试用例上的性能；单边缺口拉伸和剪切测试以及 L 形面板测试。