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. growing surfaces | edge actuation | leaves | buckling | rippling L aminae, or leaf-like structures, are thin, i.e., they have one dimension much smaller than the other two. They arise in biology in a variety of situations, ranging from the gracefully undulating submarine avascular algal blades (1) to the saddle-shaped, coiled or edge-rippled leaves of many terrestrial plants (2). The variety of their planforms and three-dimensional shapes reflects both their growth history and their mechanical properties and poses many physico-chemical questions that may be broadly classified into two kinds: (i) How does inhomogeneous growth at the molecular and cellular level lead to the observed complex shapes at the mesoscopic/macroscopic level? and (ii) how does the resulting mesoscopic shape influence the underlying molecular growth processes? At the molecular level, mutants responsible for differential cell proliferation (3) lead to a range of leaf shapes. At the macroscopic level, stresses induced by external loads lead to phenotypic plasticity in algal blades that switch between long, narrow, blade-like shapes in rapid flow to broader undulating shapes in slow flow (1). Understanding the origin of these morphological variants requires a mathematical theory that accounts for the process by which shape is generated by inhomogeneous growth in a tissue. Recent work has focused on some of these questions by highlighting the self-similar structures that form near the edge because of variations in a prescribed intrinsic metric of a surface (4, 5), and also on the case of a circular disk with edge-localized growth (6-8), but does not consider the subtle role of the boundary conditions at the free edge, the effect of the finite width of a leaf, or the phase space of different shapes that quantify the diversity in leaf morphology. Motivated by our experimental observations of long leaves and artificial mimics thereof, here we address the question of the morphology of a long leaf or lamina of finite dimensions (length 2L, width 2W , thickness H, with H W < L). In particular, we pose mathematically a nonlinear boundary-value problem that accounts for the coupling of growth to the shape of a lamina. We analyze the resulting equations by using a combination of scaling concepts, asymptotics, and stability analysis to deduce the various morphologies that arise and show that the finite width of a long leaf leads to a qualitatively new class of shapes, such as saddles and rippled surfaces. We corroborate these by using numerical simulations to construct a simple phase diagram for the classification of the long leaf morphology. Shape of a Plantain Lily Leaf. In Fig. 1A , we show a typical long leaf of the plantain lily Hosta lancifolia. We see that the midvein curvature is largest near the distal end and monotonically decreases towards the base, whereas the lamina attached to the midvein is bent transversely so that the leaf is shaped roughly like a saddle, with negative Gaussian curvature. However, the edges of the leaf show a localized undulatory rippling pattern superposed on the global saddle-shape that is most prominent where the lamina is the widest. This morphology is relatively common in leaves and petals of vascular and avascular plants that grow in air and water, i.e., it is relatively independent of the effects of gravity. To focus on the relative role of inhomogeneous growth and the intrinsic elastic nature of the thin leaf, we cut the lamina into thin strips parallel to the midvein. We see that the relatively stiff midvein unbends from its naturally curved state (when the lamina is attached to it) to a straight state, except near its tip, whereas the relaxed length of the strips after they have straightened is different from that when they are part of the whole leaf. The strips further from the midvein extend more, i.e., the nonuniform distribution of growth-induced strain is such that it generates compressive stresses along the leaf edges, which can lead to buckling instabilities. These observations are consistent with prior measurements in vascular and avascular blades (1, 2), which show similar trends. Shape of a Stretched Ribbon. To mimic growth differentially in a thin lamina, we stretch a naturally flat thin foam ribbon (2 mm × 4 cm × 11 cm) inhomogeneously past its elastic limit and then unload it. The inhomogeneous residual plastic strains that remain cause the sheet to relax to a bent state. Moderate stretching strains (∼5%) lead to a saddle-like shape, and large stretching strains (∼20 %) lead to ripples along its edge, shown in Fig. 1B , along with the lateral strain distributions shown in Fig. 1C . This gradient in plastic strain is introduced easily by pulling the edges of the foam with fingers and it leads to a plastic strain that peaks along the edge and decreases toward the midline. We see that the effect of the inhomogeneous residual strain is equivalent to that of inhomogeneous, growth-induced strain * . Analogous phenomena arise elsewhere and perhaps afford easier ways to understand the basic mechanism of shaping a lamina. For example, when knitting or crocheting a scarf, if the number of stitches or knots per unit length is increased as one moves away from the center line, the scarf first forms a saddle shape because such a shape easily accommodates the small excess length of the edges relative to the middle. Eventually, the edge length becomes so large that the edge itself starts to ripple. A similar phenomenon is seen in potato chips made by frying circular disks of soft, wet
A long flat lamina deforms to a saddle shape and/or develops undulations that may lead to local ripples as the growth strain is localized to the contour of the leaf REF .
The shape of a long leaf
Proceedings of the National Academy of Sciences of the United States of America
Proceedings of the National Academy of Sciences of the United States of America
Materials Science Medicine