We derive the variational formulation of a gradient damage model by applying the energetic formulation of rate-independent processes and obtain a regularized formulation of fracture. The model exhibits different behavior at traction and compression and has a state-dependent dissipation potential which induces a path-independent work. We will show how such formulation provides the natural framework for setting up a consistent numerical scheme with the underlying variational structure and for the derivation of additional necessary conditions of global optimality in the form of a two-sided energetic inequality. These conditions will form our criteria for making a better choice of the starting guess in the application of the alternating minimization scheme to describe crack propagation as quasistatic evolution of global minimizers of the underlying incremental functional. We will apply the procedure for two- and three-dimensional benchmark problems and we will compare the results with the solution of the weak form of the Euler-Lagrange equations. We will observe that by including the two-sided energetic inequality in our solution method, we describe, for some of the benchmark problems, an equilibrium path when the damage starts to manifest, which is different from the one obtained by solving simply the stationarity conditions of the underlying functional.

中文翻译：

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准脆性断裂的变分非对称相场模型：能量解及其计算。

我们通过应用与速率无关的过程的能量公式推导出梯度损伤模型的变分公式，并获得断裂的正则化公式。该模型在牵引和压缩时表现出不同的行为，并且具有与状态相关的耗散势，这会导致与路径无关的功。我们将展示这种公式如何提供自然框架，以建立具有潜在变分结构的一致数值方案，并以双边能量不等式的形式推导全局最优性的其他必要条件。这些条件将形成我们的标准，以便在交替最小化方案的应用中更好地选择起始猜测，以将裂纹扩展描述为基础增量函数的全局极小值的准静态演化。我们将应用该程序解决二维和三维基准问题，并将结果与​​欧拉-拉格朗日方程的弱形式的解进行比较。我们将观察到，通过在我们的求解方法中包含双边能量不等式，我们描述了一些基准问题，当损害开始显现时，平衡路径与通过简单求解平稳条件获得的路径不同的基础功能。我们将应用该程序解决二维和三维基准问题，并将结果与​​欧拉-拉格朗日方程的弱形式的解进行比较。我们将观察到，通过在我们的求解方法中包含双边能量不等式，我们描述了一些基准问题，当损害开始显现时，平衡路径与通过简单求解平稳条件获得的路径不同的基础功能。我们将应用该程序解决二维和三维基准问题，并将结果与​​欧拉-拉格朗日方程的弱形式的解进行比较。我们将观察到，通过在我们的求解方法中包含双边能量不等式，我们描述了一些基准问题，当损害开始显现时，平衡路径与通过简单求解平稳条件获得的路径不同的基础功能。