  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
$\Gamma$-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.
