Phase‐field models have already been proven to predict complex fracture patterns for brittle fracture at small strains. In this paper we discuss a model for phase‐field fracture at finite deformations in more detail. Among the identification of crack location and projection of crack growth the numerical stability is one of the main challenges in solid mechanics. Here we present a phase‐field model at finite strains, which takes into account the anisotropy of damage by applying an anisotropic split of the modified invariants of the right Cauchy‐Green strain tensor. We introduce a suitable weak notion of solution that also allows for a spatial and temporal discretization of the model. In this framework we study the existence of solutions and we show that the time‐discrete solutions converge in a weak sense to a solution of the time‐continuous formulation of the model. Numerical examples in two and three space dimensions illustrate the range of validity of the analytical results.

中文翻译：

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基于修正不变量的有限应变相场断裂模型的分析与仿真

已经证明，相场模型可以预测小应变下脆性断裂的复杂断裂模式。在本文中，我们将更详细地讨论有限变形下的相场断裂模型。在确定裂纹位置和裂纹扩展预测中，数值稳定性是固体力学的主要挑战之一。在这里，我们提出了一个有限应变的相场模型，该模型通过应用右柯西-格林应变张量的修正不变量的各向异性分裂考虑了损伤的各向异性。我们介绍了一种合适的弱解决方案概念，该概念也允许模型在空间和时间上离散化。在这个框架中，我们研究了解的存在性，并且表明时间离散解在某种意义上收敛于模型的时间连续公式的解。在两个和三个空间维度上的数值示例说明了分析结果的有效性范围。