Isometric Incompatibility in Growing Elastic Sheets

Abstract

Geometric incompatibility, the inability of a material's rest state to be realized in Euclidean space, underlies shape formation in natural and synthetic thin sheets. Classical Gauss and Mainardi-Codazzi-Peterson (MCP) incompatibilities explain many patterns in nature, but they do not exhaust the mechanisms that frustrate thin elastic sheets. We identify a new incompatibility that forbids any stretching-free configuration, even when the rest state of the elastic sheet locally satisfies the Gauss and MCP compatibility conditions. We demonstrate this principle in a model of surface growth with positive Gaussian curvature, where a geometric horizon forms, leading to the onset of frustration. Experiments, simulations, and theory show that the sheet responds by nucleating periodic d-cone-like dimples. We show that this obstruction to stretching-free configurations is topological, and we point to open questions concerning the origin of frustration.

Figures (4)

• Figure 1: Emergent frustration in an axisymmetric growing disc with positive Gaussian curvature. (a) Growth of a disc domain endowed with a positive reference Gaussian curvature field $\bar{K}(v)=K_0 v/v_l$. The total reference curvature $\bar{K}_T = 2\pi$. (b) Onset of a geometric horizon at $\bar{K}_T=4\pi$, where the boundary normals (arrows) collapse into a common direction, obstructing further smooth isometric extension. (c) For $\bar{K}_T>4\pi$, symmetry breaking occurs. (d) At the horizon $v_0=v_h$, the principal curvature $b_{vv}$ becomes singular.
• Figure 2: Pattern formation induced by the isometric frustration, and recovery of isometric embeddability. (a,b) Symmetry breaking during the increase of accumulated Gaussian curvature in (a) Simulations and (b) Experiments. Near $\bar{K}_T = 4\pi$, the edge reaches a geometric horizon; For $\bar{K}_T > 4\pi$, isometric embeddings break and periodic dimples with Pogorelov-like ridge and d-cone emerge. (c) Phase diagram showing the theoretical boundary $v_0=v_h$ (solid curve). Symbols denote simulations (open) and experiments (filled); circles and square indicate isometric and frustrated configurations, respectively. (d) Topological surgery restores smooth isometric embeddability. Cutting $iii$ (a) and III (b) along $v$ removes the incompatibility and yields overlapped spherical embeddings.
• Figure 3: Post-horizon analysis. (a) Gaussian curvature map showing a nearly flat inner edge preceded by curvature undulations. (b) Cross-section profile and curvature discrepancy, revealing a narrow band of Pogorelov ridges. (c) Accumulated Gaussian curvature along $v$: initially following $\bar{K}_T(v)$ in the interior, but collapses to $4\pi$ at the edge. (d) Perimeter versus $v$: deviations between actual ($p$), reference ($\bar{p}$), and projected ($\hat{p}$) perimeters reflect symmetry breaking, and highlight the tensile nature of the inner edge by $p(v_0) = \hat{p}(v_0)> \bar{p}(v_0)$.
• Figure 4: Embedding limitations and singularities in surfaces of constant Gaussian curvature $K_0$. (a) Wrinkling beyond the horizon in a hyperbolic surface ($K_0<0$) sharon2010mechanicsmarder2006geometry. (b) Onset of a horizon in an elliptic surface ($K_0>0$). For both hyperbolic and elliptic surfaces, the horizon forms a rigidifying curve with the normal (indicated by red arrows) constant on the edge. Note that at the horizon $v_0=v_h$, the total Gaussian curvature reaches $|\bar{K}_T|=4\pi$, while the principal curvatures are singular with $b_{uu}\to 0$ and $b_{vv}\to\infty$.