Fragmentation of inflated elastic brittle rings: Emergence of the quasi-equidistant spacing of cracks

Abstract

The order of appearance and the position of meridional cracks in brittle domes is a delicate question of mechanics. This paper investigates a dimension-reduced model, a pressurized brittle ring constrained to the plane, to show that a simple, deterministic approach based on the Griffith theory of fracture predicts quasi-equidistant, i.e., close to an equal spacing of the emerging cracks. It is also demonstrated that the order of emergence is significantly affected by the variation of the in-plane elastic support as well the bending rigidity of the ring. In the model, the energy minimization is first recast into a sequence of coupled boundary value problems. The relationship between the mechanical model and geometric properties of the emerging pattern is studied numerically. The pattern is shown to be driven by elaborate co-dimension one bifurcations and by the requirement of global energy minimization.