In the phase-field modeling of fracture, the search for a physically reasonable and computationally feasible criterion to split the elastic energy density into fractions that may or may not contribute to crack propagation has been the subject of many recent studies. Within this context, we propose an energy split – or energy decomposition – aimed at accurately representing the evolution of a crack under load reversal. To this purpose, two key assumptions are made. First, the damage gradient direction is interpreted as being representative of the normal-to-crack direction, as already assumed in previous works in the literature. The second assumption consists of considering the sign of the projection of the stress tensor onto the damage gradient direction at a point as an indicator of whether this point should behave as an opening or as a closing crack. We associate the latter case (crack closing) to both (a) a complete recovery of elastic energy density of the intact material (i.e., perfectly rough crack surfaces) and (b) a zero crack driving force at that point. The first case (crack opening) is treated classically as a damageable material point at which damage can increase. The implementation of the proposed approach turns out to be remarkably simple and computationally robust. For the evaluation of the displacements and damage gradients at nodes, the classical technique is used, and a new effective and computationally convenient iterative strategy is implemented to guarantee convergence of the staggered scheme. Four examples are presented in order to assess the suitability of the present model by using both AT1 and AT2 regularization models. Results show the desired effect of limiting crack propagation to prevailing tensile states, as well as of recovering the initial intact stiffness upon load reversal, even when two of the most common energy splits fail.

中文翻译：

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基于相场梯度的能量分割，用于模拟载荷反转下的脆性断裂


在断裂的相场建模中，寻找物理上合理且计算上可行的标准来将弹性能量密度分解为可能或可能不会促进裂纹扩展的分数一直是许多最近研究的主题。在此背景下，我们提出了一种能量分裂或能量分解，旨在准确地表示负载反转下裂纹的演变。为此，做出了两个关键假设。首先，损伤梯度方向被解释为代表法向裂纹方向，正如文献中之前的作品中已经假设的那样。第二个假设包括考虑应力张量在某一点损伤梯度方向上的投影的符号，作为该点是否应表现为开口裂纹或闭合裂纹的指示符。我们将后一种情况（裂纹闭合）与（a）完整材料的弹性能量密度完全恢复（即完全粗糙的裂纹表面）和（b）该点的零裂纹驱动力联系起来。第一种情况（裂纹开口）通常被视为可损坏材料点，在该点处损坏会增加。事实证明，所提出方法的实现非常简单且计算稳健。为了评估节点处的位移和损伤梯度，采用了经典技术，并采用了一种新的有效且计算方便的迭代策略来保证交错格式的收敛。为了通过使用 AT1 和 AT2 正则化模型来评估本模型的适用性，提出了四个示例。 结果表明，即使两种最常见的能量分裂失败，也能达到将裂纹扩展限制到主要拉伸状态以及在负载反转时恢复初始完整刚度的预期效果。