2020/04/19 by Stefano Almi, Sandro Belz, Almi, Stefano +5 · 1 citation
Computer Science · Engineering · Mathematics · #Advanced Mathematical Modeling in Engineering #Advanced Numerical Methods in Computational Mathematics #Applied mathematics #Composite Material Mechanics #Composite material #Discretization #Estimator #FOS: Mathematics #Fracture (geology) #Fracture mechanics #Geometry #Materials science #Mathematical analysis #Mathematical optimization #Mathematics #Numerical Analysis (math.NA) #Phase (matter) #Phase field models #Physics #Reduction (mathematics) #cs.NA #math.NA
paper · pdf · doi:10.48550/arxiv.2004.08871
published in arXiv (Cornell University) (Cornell University)
arxiv created 2020/04/19 · openalex publication_date 2020/04/19 · arxiv updated 2020/04/21 · openalex created_date 2022/07/26 · openalex updated_date 2026/08/06
In this paper we derive a new two-dimensional brittle fracture model for thin\nshells via dimension reduction, where the admissible displacements are only\nnormal to the shell surface. The main steps include to endow the shell with a\nsmall thickness, to express the three-dimensional energy in terms of the\nvariational model of brittle fracture in linear elasticity, and to study the\n\Γ-limit of the functional as the thickness tends to zero. The numerical\ndiscretization is tackled by first approximating the fracture through a phase\nfield, following an Ambrosio-Tortorelli like approach, and then resorting to an\nalternating minimization procedure, where the irreversibility of the crack\npropagation is rigorously imposed via an inequality constraint. The\nminimization is enriched with an anisotropic mesh adaptation driven by an a\nposteriori error estimator, which allows us to sharply track the whole crack\npath by optimizing the shape, the size, and the orientation of the mesh\nelements. Finally, the overall algorithm is successfully assessed on two\nRiemannian settings and proves not to bias the crack propagation.\n