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.

Figures (4)

• Figure 1: Cracks on Europa, melon surface, and model bilayer shells (a) Surface image of the moon of Jupiter, Europa, made from images taken by NASA's Galileo spacecraft in the late 1990s. (b) Photograph of a cantaloupe, (c) Model experimental system of fracture on shells. Cross-section of bilayer spheroidal shell of the inner-elastic and outer-brittle layers (scale bar: 10 mm). The inset shows a close-up view of the cross-section of the bilayer shell (scale bar: 1 mm). (d) Top view of the post-fractured surface of one of the shells (as Fig. \ref{['fig:2']}(c-iii)). (e) Simulation results from the phase-field-combined FEM simulation.
• Figure 2: Crack pattern of bilayer spheroidal shells (a) Experimental setup of our model experiments, where our shell is pressurized from inside by a syringe pump. Inset: photograph of the inner mold of a spheroid ($b/a = 1.5$). The lateral curvature, $a$, and thickness, $h$, are fixed throughout as $(a,h) = (20, 0.4)$ mm. (b) Reconstructed volumetric image through $\mu$CT scan. The black curves represent the crack paths on the shell. Crack morphology of bilayer shells: (c) top-view and (d) side-view. The photographs of shells with curvature ratios of (c-i)(d-i) $b/a = 0.5$, (c-ii)(d-ii) $b/a = 1.0$, (c-iii)(d-iii) $b/a = 1.5$, and (c-iv)(d-iv) $b/a = 2.0$. Scale bars: (a) 20mm, (b)-(d) 10mm.
• Figure 3: Shape morphology and stress profile obtained from the simulation results. (a) Typical FEM snapshots for (a-i) $b/a = 0.6$, (a-ii) $b/a = 1.0$, and (a-iii) $b/a = 2.0$ (See (d) for the corresponding values of internal pressure). Red colored regions represent the cracked area. (b) Schematics of spheroid geometry. The meridian and lateral radii of curvatures, $(r_{\varphi},r_{\theta})$, defined through the surface normal vector, $\hat{\bm{n}}$, characterize the local geometry. The latitude of a spheroid is defined as an angle between $\hat{\bm{n}}$ and the $z$-axis. (c-i) Profiles of the longitudinal, $\sigma_{\varphi\varphi}$ ($\circ$), and lateral components, $\sigma_{\theta\theta}$ ($\times$), of the stress tensors obtained from FEM in the absence of cracks are compared with the analytical formulas (solid lines). Red, black, and blue data points and curves represent simulation results and theory for $b/a = 0.8, 1.0$, and $2.0$. The origin of latitude $\varphi=0$ corresponds to the north pole. (c-ii) The ratio of lateral and longitudinal stress, $\sigma_{\theta\theta}/\sigma_{\varphi\varphi}$, is summarized as a heatmap on the $\varphi$-$(b/a)$ plane. Red and blue denote that the lateral and longitudinal components are more dominant, respectively. (c-iii) The analytical profile of the von Mises stress, $\sigma(\varphi, b/a)$, normalized by the maximum, $\sigma^*(b/a)$, is compared for several $b/a$ and is summarized as a heatmap. The bright color represents the latitude of the largest von Mises stress for each $b/a$. (d) Phase diagram classifying whether the line cracks are visible ($\triangle$) or not ($\circ$). The dashed line represents the prediction from Eq. \ref{['eq:smax']} with a single fitting parameter, $\gamma = 0.65$, as $\sigma^* = \gamma \sqrt{\sigma_cE}$.
• Figure 4: Crack direction for model bi-layer shells, linea of Europa, and young muskmelon (a) Crack angle distribution, $\mathcal{P}(\beta)$, for several $b/a$ of experiments. (b-i) Snapshots of FEM simulation results for developed crack patterns for each shell geometry $b/a = 0.5, 1.0, 1.6, 2.0$. The corresponding crack angle distribution is plotted in (b-ii). (c-i) Schematics of the idealized crack initiation, where a single circular crack is invoked. Snapshots of FEM simulation results with an initial single defect in the vicinity of the critical pressure for $b/a = 0.8$ and 2.0 are shown in (c-ii) and (c-iii), respectively. (d)Averaged crack direction $\bar{\beta}$ as a function of the curvature ratio $b/a$. The empty and filled circular data represent the experimental and numerical results, respectively. The empty triangle is an average crack angle of the linea of Europa, and the empty squares indicate the average crack angles in three regions of the muskmelon surface. The error bars represent the standard deviations. The insets are a photograph of a young muskmelon (scale bar: 5 cm) and the 3D reconstruction of the linea of Europa (normalized by its radius).