In this paper, we derive a new 2D brittle fracture model for thin shells via dimension reduction, where the admissible displacements are only normal to the shell surface. The main steps include to endow the shell with a small thickness, to express the three-dimensional energy in terms of the variational model of brittle fracture in linear elasticity, and to study the [Formula: see text]-limit of the functional as the thickness tends to zero.The numerical discretization is tackled by first approximating the fracture through a phase field, following an Ambrosio–Tortorelli like approach, and then resorting to an alternating minimization procedure, where the irreversibility of the crack propagation is rigorously imposed via an inequality constraint. The minimization is enriched with an anisotropic mesh adaptation driven by an a posteriori error estimator, which allows us to sharply track the whole crack path by optimizing the shape, the size, and the orientation of the mesh elements.Finally, the overall algorithm is successfully assessed on two Riemannian settings and proves not to bias the crack propagation.

中文翻译：

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具有网格自适应性的薄壳脆性断裂降维模型

在本文中，我们通过降维推导出了一种新的薄壳二维脆性断裂模型，其中允许的位移仅垂直于壳表面。主要步骤包括赋予壳厚度小，用线弹性脆性断裂变分模型表达三维能量，研究泛函的[公式：见正文]-极限厚度趋于零。数值离散化的解决方法是首先通过相场近似断裂，遵循类似 Ambrosio-Tortorelli 的方法，然后采用交替最小化程序，其中裂纹扩展的不可逆性通过不等式严格施加约束。