The shape of a long leaf

Abstract

Long leaves in terrestrial plants and their submarine counterparts, algal blades, have a typical, saddle-like midsurface and rippled edges. To understand the origin of these morphologies, we dissect leaves and differentially stretch foam ribbons to show that these shapes arise from a simple cause, the elastic relaxation via bending that follows either differential growth (in leaves) or differential stretching past the yield point (in ribbons). We quantify these different modalities in terms of a mathematical model for the shape of an initially flat elastic sheet with lateral gradients in longitudinal growth. By using a combination of scaling concepts, stability analysis, and numerical simulations, we map out the shape space for these growing ribbons and find that as the relative growth strain is increased, a long flat lamina deforms to a saddle shape and/or develops undulations that may lead to strongly localized ripples as the growth strain is localized to the edge of the leaf. Our theory delineates the geometric and growth control parameters that determine the shape space of finite laminae and thus allows for a comparative study of elongated leaf morphology.

Fig. 1.

Experiments and observations of long leaf and ribbon morphology. (A) Shape of a plantain lily Hosta lancifolia leaf showing the saddle-like shape of the midsurface and the rippled edges. Dissection along the midrib leads to a relief of the incompatible strain induced by differential longitudinal growth and causes the midrib to straighten, except near the tip, consistent with the notion that the shape is a result of elastic interactions of a growing plate. The dashed red line is the original position of the midrib. (B) A foam ribbon that is stretched beyond the elastic limit relaxes into a saddle shape when the edge strain is β ∼ 5%, but relaxes into a rippled shape when the edge strain is β ∼ 20%. (C) The observed lateral strain ɛ(y) is approximately parabolic for the saddle-shaped ribbon but is localized more strongly to the edge for the rippled ribbon.

Fig. 2.

Characterization of saddle-shaped laminae. (A) A saddle shaped lamina corresponding to a scaled width w = 10, growth exponent n = 10, scaled maximum growth strain βw ² = 6.5 and κ_x w ² = 1.9. (B) The postbuckling behavior shows the scaled curvature κ_x w ² vs. βw ² on a log-log plot; the results for different w collapse onto a single curve. The Inset shows that the onset of buckling occurs via a supercritical pitchfork bifurcation, with κ_x w ∼(β − β*)^1/2, with β*w ² ≃ 2.51 for all values of the growth exponent n; only the positive branch is meaningful here. (Notation: S-stable, U-unstable.) (C) Cross-sectional profile f(y) for different values of the maximum growth strain β, when the critical growth strain β*w ² = 2.51 and w = 100.

Fig. 3.

Characterization of rippled laminae. (A) Three types of periodic buckling modes arise depending on the persistence of an edge pinch, or equivalently the ratio of the wavelength of the mode 1/k to the width w. Filament-like buckling for kw ≪ 1, doubly-curved buckling for kw ∼1, and edge rippling for kw ≫ 1. (B) Rescaled critical strain wavenumber kw vs. β*w ², showing three distinct regimes associated with the above. The solid line corresponds to the case when the center-line of the ribbon is clamped. (C) The cross-sectional profile f(y) for different scaled wavenumbers, when w = 100, with the growth law ∈_g = β(y/w)¹⁰ shows how the periodic ripples localize to the edge as kw increases.

Fig. 4.

Numerical simulations yield a phase diagram for the different undulatory shapes of a long, growing ribbon as a function of the maximum edge growth strain β and the scaled width W. The boundaries that demarcate the different phases follow the scaling β* ∼ 1/w ², consistent with our scaling and analytic estimates (see Eq. 12 and SI Appendix). We use the power law ∈_g = β (y/w)¹⁰.