Midveins regulate the shape formation of drying leaves

TL;DR

This work shows that the diverse shapes observed in drying leaves arise from a mechanical constraint imposed by the midvein on a shrinking lamina. Using a non-Euclidean elasticity framework coupled with FEM simulations, the authors identify two primary morphologies—curling-dominated and folding-dominated—whose realization depends on the bending stiffness ratio between the midvein and lamina. They derive scaling laws linking midvein curvature to shrinkage strain and lamina thickness, and establish a phase diagram that maps morphologies as a function of material and geometric parameters. The findings offer a mechanistic understanding of natural leaf morphogenesis and supply principles for designing bio-inspired, morphable thin structures that exploit internal constraints for programmable shape change.

Abstract

Dried leaves in nature often exhibit curled and crumpled morphologies, typically attributed to internal strain gradients that produce dome-like shapes. However, the origin of these strain gradients remains poorly understood. Although leaf veins--particularly the midvein--have been suggested to influence shape formation, their mechanical role has not been systematically investigated. Here, we demonstrate that mechanical constraints imposed by the midvein play a crucial role in generating the diverse morphologies that emerge during leaf drying. Combining numerical simulations and theoretical analysis, we show that a uniformly shrinking leaf lamina constrained by a non-shrinking midvein gives rise to two distinct types of configurations: curling-dominated and folding-dominated morphologies. In the curling-dominated regime, both S-curled and C-curled shapes emerge, with C-curled configurations more commonly observed due to their lower elastic energy. In contrast, the folding-dominated regime features folding accompanied by edge waviness. Theoretical modeling reveals a linear relationship between midvein curvature and mismatch strain, consistent with simulation results. Moreover, we find that the morphological outcome is governed by the ratio of bending stiffnesses between the lamina and the midvein. We construct a comprehensive phase diagram for the transitions between different configurations. These findings provide a mechanical framework for understanding shape formation in drying leaves, offering new insights into natural morphing processes and informing the design of bio-inspired morphable structures.

Figures (9)

• Figure 1: Drying leaves display diverse shape transformations influenced by midvein constraints.A. Representative plant species and their fresh leaves: (i)Gardenia jasminoides, (ii)Cyrtophyllum fragrans, (iii)Plumeria obtusa, and (iv)Excoecaria cochinchinensis. B. Fallen leaves collected from the ground display a variety of morphologies, including curling, folding, and combined folding with edge waviness. C. Typical dried leaves from the corresponding species shown in A, labeled i–iv. D. Simulation results of drying leaf morphologies with elliptical laminae constrained by a thickened midvein, based on the model proposed in this study detailed in later sections. The four configurations correspond to the respective real leaf morphologies for the species in C. Color maps indicate out-of-plane displacement in the $z$-direction $w$, with red denoting the maximum magnitude.
• Figure 2: Simplified model capturing shape transformation in drying leaves.A. A frangipani (Plumeria obtusa) leaf that is flat in the fresh state (top) transforms into a folded and wavy 3D shape when dried (bottom), with a visible structural distinction between the lamina and midvein. B. Schematic of a simplified leaf structure: the lamina is modeled as a thin plate (yellow) with width $W$, length $L$, and thickness $h$, embedded with a central midvein represented as a cylindrical beam (brown) of radius $R$. Upon drying, the lamina undergoes uniform isotropic shrinking, while the midvein remains undeformed.
• Figure 3: FEM simulation of a simplified drying leaf structure.A. FEM model consisting of a central beam (midvein) embedded in a thin rectangular plate (lamina), where the lamina is subjected to a prescribed shrinkage strain $\epsilon_0 = 0.1$. Contour colors represent the out-of-plane displacement in the $z$-direction $w$. B. Deformed profiles of the lamina (top; cross-section along the $yz$-plane) and the midvein (bottom; cross-section along the $xz$-plane) corresponding to A. The color bar indicates varying magnitudes of shrinkage strain $\epsilon_0$. Key geometric measures $w_{\mathrm{max}}$ and $\kappa_x$ are labeled. C. Distributions of the shrinkage strain in the $x$-direction: in the lamina ($\epsilon_x^{\text{plate}}$, top) and in the midvein ($\epsilon_x^{\text{vein}}$, bottom). D. Stress components: longitudinal stress $\sigma_{xx}$ (top) and transverse stress $\sigma_{yy}$ (bottom). Negative values indicate compressive stress. E. Quantification of the deformation response as a function of shrinkage strain $\epsilon_0$: normalized maximum deflection $w_{\mathrm{max}}/W$ (left), and midvein curvature $\kappa_x$ (right). The model geometry is defined by $W = 10$, $L = 20$, $h = 0.075$, and $R = 0.25$.
• Figure 4: Evolution of morphology with increasing shrinkage strain in the lamina.A–C. Simulations of leaf curling with a smaller midvein-to-lamina thickness ratio ($R/h = 0.5$). D–F. Simulations of leaf folding with a larger thickness ratio ($R/h = 2.5$). In both cases, the plate dimensions are fixed as $L/W = 2$, $h/W = 1\times10^{-3}$. A, D. Deformed configurations at shrinkage strains, $\epsilon_0$: (i) 1%, (ii) 2%, (iii) 6%, and (iv) 10%. Color contours represent the out-of-plane displacement $w$. B, E. Edge profiles extracted at $y = W/2$, corresponding to configurations (i–iv). C, F. Normalized maximum deflection of the $yz$-cross section, $w_{\mathrm{max}}/W$, plotted as a function of the shrinkage strain $\epsilon_0$.
• Figure 5: Illustration of theoretical predictions of shape formation.A. Flat configuration. B. Curling configuration with invisible transition layer under small strain mismatch. C. Curling configuration with appreciable transition layer under relatively large strain mismatch. D. Folding configuration under small strain mismatch.The small folding angle is denoted as $\beta$. E. Waving configuration associated with folding under relatively large strain mismatch. The function of the wavy edges are approximated by $f(s) =A \sin(2\pi/\lambda s)$. Under mild shrinkage strain, the shape-morphing follows the path "A-B-C" when the midveins are relatively soft, but follows the path "A-D-E" when the midveins are relatively stiff. In all configurations, we assume that the corner angles remain $\pi/2$ during the drying process.