|Abstract:|| In this paper we derive a new two-dimensional 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 ????-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
Finally, the overall algorithm is successfully assessed on two Riemannian
settings and proves not to bias the crack propagation.|