Where Humpty Dumpty Breaks: Geometry-Driven Fracture in Ellipsoidal Shells

Abstract: Fracture networks are ubiquitous in nature, spanning scales from millimeter-sized cracks in botanical peels to hundred-kilometer-long lineae on planetary satellites. The propagation of a crack is a complex, nonlinear phenomenon governed by the interplay of mechanical properties, rheological behavior, and system geometry. While fracture mechanics has long addressed structural failure, the relationship among fracture, elasticity, and nonlinear geometry has recently revived as a focal point in condensed matter and biophysics. However, a unified framework that systematically explains how surface geometry prescribes the transition between disparate fracture morphologies remains elusive. Here we show that shell curvature provides a geometric blueprint for fracture, governing the evolution of complex crack networks through induced stress anisotropy. By internally pressurizing thin, bilayer spheroidal shells, we demonstrate that a rich diversity of crack morphologies across lateral, longitudinal, and random orientations depends on the curvature ratio between the pole and the equator. We find that these patterns arise from the nonlinear mechanics of the shell, which can be leveraged to effectively control crack growth. Our results establish a direct link between structural curvature and fractures, providing a predictive framework that integrates nonlinear geometry with the classical Griffith and von Mises criteria. Beyond our model system, we find that the disparate fracture patterns observed in ripening muskmelons and in the icy crust of Europa follow the same geometric principles. We expect that this unified understanding of crack morphogenesis will inform the design principles of novel functional materials that are resilient to fracture and provide insights into the mechanical performance of curved biological and geophysical architectures.