From segmentation to shattering: Structural transitions in the breakup of brittle rings

The explosive breakup of rings exhibits a transition from segmentation to fragmentation and ultimately to shattering as the loading strain rate increases. In this work, we investigate how this transition is governed by the evolving structure of the underlying crack pattern. Using a two-dimensional discrete element model of ring-shaped shells, we uncover a rich structural evolution of the crack ensemble characterized by increasing complexity and connectivity. At low strain rates, segmentation arises from straight, radially oriented cracks that span the ring. As the strain rate increases, these cracks develop branching structures with a growing fractal dimension, leading to a connected crack network. This network eventually spans the entire ring and induces fragmentation. Quantitative analysis reveals that the fractal dimension of spanning cracks increases logarithmically with strain rate, approaching the embedding dimension at the onset of fragmentation. The orientation of cracks becomes increasingly isotropic, reflecting the loss of memory of the radial loading direction. The transition to fragmentation is signaled by the emergence of a dominant, system-spanning crack resembling percolation-like critical phenomena. Our results demonstrate that the structural transition of crack patterns provides a fundamental mechanism driving the breakup of shells under dynamic loading.