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Rigorous analysis of shape transitions in frustrated elastic ribbons

Cy Maor, Maria Giovanna Mora

TL;DR

The paper rigorously analyzes shape transitions in frustrated elastic ribbons by developing a full three-dimensional non-Euclidean elasticity model and performing Γ-convergence to one-dimensional theories along the midline. It distinguishes Gauss- and Codazzi-incompatibilities via deficits $\delta^G$ and $\delta^C$, and derives sharp energy scalings in narrow regimes: for Gauss-incompatible ribbons, $\inf E_{t,w}$ scales like $\min\{t^2,w^4\}$, while for Codazzi-incompatible ribbons it scales like $t^2 w^2$ under a nondegeneracy condition; otherwise more complex behavior can arise. In the wide-ribbon regime with flat mid-surfaces, the plate limit yields a higher-level description, where Gauss incompatibility drives a $t^2$-type energy and Codazzi-incompatible cases depend on the transversal nonvanishing of $\mathbb{II}_{11}$ along the midline, yielding a mesoscopic transition. The work combines geometric constructions, Γ-convergence, and recovery sequences to provide a rigorous foundation for the observed microscopic and mesoscopic transitions and connects to prior non-Euclidean plate/ribbon theories, while outlining several open directions for general geometries and non-smooth metrics.

Abstract

Ribbons are elastic bodies of thickness $t$ and width $w$ with $t\ll w\ll 1$ (after appropriate nondimensionalization). Many ribbons in nature have a non-trivial internal geometry, making them incompatible with Euclidean space. This incompatibility -- expressed mathematically as a failure of the Gauss-Codazzi equations for surfaces -- can trigger shape transitions between narrow and wide ribbons. These transitions depend on the internal geometry: ribbons whose incompatibility arises from failure of the Gauss equation always exhibit a transition, whereas those whose incompatibility arises from failure of the Codazzi equations, may or may not. We give the first rigorous analysis of this phenomenon, mainly for ribbons whose first fundamental form is flat. For Gauss-incompatible ribbons we identify the natural energy scaling of the problem and prove the existence of a shape transition. For Codazzi-incompatible ribbons we give a necessary condition for a transition to occur. Furthermore, our study reveals a fundamental distinction: the transition is "microscopic" for Gauss-incompatible ribbons, persisting as the width tends to $0$, whereas it is "mesoscopic" for Codazzi-incompatible ribbons, observable only at small but finite width. The results are obtained by calculating the $Γ$-limits, as $t,w\to 0$, for narrow ribbons ($w^2 \ll t$), and wide ribbons (taking $t$ to zero and then $w$), in the natural energy scalings dictated by the internal geometry.

Rigorous analysis of shape transitions in frustrated elastic ribbons

TL;DR

The paper rigorously analyzes shape transitions in frustrated elastic ribbons by developing a full three-dimensional non-Euclidean elasticity model and performing Γ-convergence to one-dimensional theories along the midline. It distinguishes Gauss- and Codazzi-incompatibilities via deficits and , and derives sharp energy scalings in narrow regimes: for Gauss-incompatible ribbons, scales like , while for Codazzi-incompatible ribbons it scales like under a nondegeneracy condition; otherwise more complex behavior can arise. In the wide-ribbon regime with flat mid-surfaces, the plate limit yields a higher-level description, where Gauss incompatibility drives a -type energy and Codazzi-incompatible cases depend on the transversal nonvanishing of along the midline, yielding a mesoscopic transition. The work combines geometric constructions, Γ-convergence, and recovery sequences to provide a rigorous foundation for the observed microscopic and mesoscopic transitions and connects to prior non-Euclidean plate/ribbon theories, while outlining several open directions for general geometries and non-smooth metrics.

Abstract

Ribbons are elastic bodies of thickness and width with (after appropriate nondimensionalization). Many ribbons in nature have a non-trivial internal geometry, making them incompatible with Euclidean space. This incompatibility -- expressed mathematically as a failure of the Gauss-Codazzi equations for surfaces -- can trigger shape transitions between narrow and wide ribbons. These transitions depend on the internal geometry: ribbons whose incompatibility arises from failure of the Gauss equation always exhibit a transition, whereas those whose incompatibility arises from failure of the Codazzi equations, may or may not. We give the first rigorous analysis of this phenomenon, mainly for ribbons whose first fundamental form is flat. For Gauss-incompatible ribbons we identify the natural energy scaling of the problem and prove the existence of a shape transition. For Codazzi-incompatible ribbons we give a necessary condition for a transition to occur. Furthermore, our study reveals a fundamental distinction: the transition is "microscopic" for Gauss-incompatible ribbons, persisting as the width tends to , whereas it is "mesoscopic" for Codazzi-incompatible ribbons, observable only at small but finite width. The results are obtained by calculating the -limits, as , for narrow ribbons (), and wide ribbons (taking to zero and then ), in the natural energy scalings dictated by the internal geometry.

Paper Structure

This paper contains 40 sections, 33 theorems, 293 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The reduced shell model, Gauss and Codazzi incompatibilities.
  3. Shape transitions in non-Euclidean ribbons.
  4. The three-dimensional model.
  5. Main results.
  6. Necessary condition for Codazzi shape transition and its geometrical meaning.
  7. "Microscopic" vs. "mesoscopic": The different nature of Gauss and Codazzi shape transitions.
  8. Relation to previous works.
  9. Future directions.
  10. Structure of the paper.
  11. Acknowledgements.
  12. Statements and declarations.
  13. The geometry of ribbons
  14. Metrics on ribbons
  15. Flat mid-surfaces
  16. ...and 25 more sections

Key Result

Theorem 1.2

Consider a narrow ribbon in the sense that is, We then have

Figures (1)

  • Figure 1: Examples of shape transitions in ribbons (figure adapted from LSSM21). The first column depicts the reference forms $\mathfrak{a}$ (flat figure) and $\textup{II}$ (curved), that appear in the next two columns. Case (a) is Gauss-incompatible, (b)--(e) are Codazzi-incompatible. The fourth and fifth columns display experimental pictures of elastic ribbons with these reference forms, in the wide and narrow regimes (in case (c) the right end of the wide ribbon is cut to reveal the second form the body wants to achieve). The sixth column indicates whether there is a (first order) shape transition between narrow and wide ribbons, and the scaling at which it occurs, as predicted by formal asymptotics and confirmed by experiments; for (b) and (d), the predicted scalings are only approximate and the exact scaling is unknown, see §\ref{['sec:wide_examples']} for details. The present work rigorously proves the existence and scaling of shape transitions in case (a) (as well as in all Gauss-incompatible ribbons with a flat midsurface), confirms the presence of a transition in (d), and shows that no transition occurs in (c) and (e).

Theorems & Definitions (38)

  • Definition 1.1
  • Theorem 1.2: Narrow ribbons
  • Theorem 1.3: Wide ribbons
  • Corollary 1.4
  • Lemma 2.1
  • Lemma 2.2
  • Proposition 2.3
  • Remark 2.4
  • Theorem 4.1: Narrow ribbons, compactness
  • Theorem 4.2: Narrow ribbons, $\Gamma$-convergence -- Gauss-incompatibility
  • ...and 28 more