‘Ruban à godets’: an elastic model for ripples in plant leaves

The formation of ripples along the edge of plant leaves is studied using a model of an elastic strip with spontaneous curvature. The equations of equilibrium of the strip are established in an explicit form. A numerical method of solution is presented and carried out. Owing to the presence of geometric nonlinearities, several equilibrium configurations are found but we show that only one of them is physical. To our knowledge, this is the first investigation of ripples in plant leaves that is based on the equations of elasticity. To cite this article: B. Audoly, A. Boudaoud, C. R. Mecanique 330 (2002) 831–836.  2002 Académie des sciences/Éditions scientifiques et médicales Elsevier SAS solids and structures / elastic rods / growth in biology Ruban à godets : un modèle élastique pour les fronces des feuillages Résumé On étudie la formation des fronces au bord des feuillages grâce à un modèle de bande élastique à courbure spontanée. Les équations d’équilibre de la bande sont établies explicitement. Une méthode numérique de résolution est présentée puis mise en œuvre. À cause des non-linéarités géométriques, on trouve plusieurs configurations d’équilibre ; une seule peut prétendre décrire les feuillages. Ceci constitue la premiére étude des fronces de feuillages s’appuyant sur les équations de l’élasticité. Pour citer cet article : B. Audoly, A. Boudaoud, C. R. Mecanique 330 (2002) 831–836.  2002 Académie des sciences/Éditions scientifiques et médicales Elsevier SAS solides et structures / tiges élastiques / croissance en biologie Many plant leaves, such as lettuce (Fig. 1), have rippled edges. This is due to the fact that their boundaries have a larger natural length that if they were flat. A possible explanation, as suggested in [1], is that the rate of growth of the tissue is larger at the boundary than in the bulk. Ripples caused by distended edges of this kind are in fact observed in a variety of contexts, ranging from fashion (the ‘jupe à godets’ which inspired the title of reference [1] and ours) to the tearing of plates made of plastic materials [2]. A purely geometrical approach to this problem is to study the embedding of hyperbolic surfaces into the 3D Cartesian space. A talk on this subject has seemingly been presented in Paris as early as on 28 August 1878 by Tchebytchev [3] under the title ‘On the cut of clothes’ but this work has remained unpublished. E-mail addresses: [email protected] (B. Audoly); [email protected] (A. Boudaoud).  2002 Académie des sciences/Éditions scientifiques et médicales Elsevier SAS. Tous droits réservés S1631-0721(02)01545-0 /FLA 831 B. Audoly, A. Boudaoud / C. R. Mecanique 330 (2002) 831–836 In a recent paper, Nechaev and Voituriez [1] constructed the explicit embedding of a particular hyperbolic surface with distended edges, which they called a ‘surface à godets’. Its embedding features a cascade of ripples with smaller and smaller lengthscales going toward the edge. The existence of isometric embeddings for an arbitrary 2-manifold endowed with a ‘distended edge’like metric remains an open question in general. If such embeddings did exist, they would certainly be far from unique. Therefore, it seems that the framework that is best suited to this problem is the theory of thin elastic plates – in which isometric embeddings would simply show up as solutions with a very low energy. Recently, Sharon et al. [2] reported experiments in which plastic sheets are torn, leading to buckling patterns along the cut which are similar to those of leaves. In some situations, they even obtained a cascade of ripples with smaller and smaller wavelengths as the edge is approached. They also reproduced these patterns qualitatively in numerical simulations. The aim of the present Note is to provide a theoretical explanation for these rippled patterns based on the theory of elasticity. To do so, we solve a model of elastic strip describing the edge of a leaf (Fig. 1). Consider a small filament obtained by cutting the leaf parallel to its edge. This filament, shown in the insert of Fig. 1, has two important features. First, it has a greater natural length along one edge (formerly the edge of the leaf, shown using a solid line in the insert) than along the other (the cut line, shown using a dashed line), hence some spontaneous curvature. Second, it is flat, namely its section has a large aspect ratio – a flat filament of this kind is commonly called an elastic strip. Our goal is to determine the periodic equilibrium configurations of this elastic strip with spontaneous curvature under tension, and show that they can reproduce the rippled patterns observed in plant leaves. This strip model has recently been introduced in [4] but does not seem to have received any satisfactory analysis so far. Strips are a special case of elastic rods, whose equilibrium equations have been established by Kirchhoff [5] in 1859 and have been the subject of numerous investigations. The equations for rods of circular cross section are integrable; for arbitrary sections, however, they are not and spatial chaos can occur [6]. This is probably the reason why rods with noncircular cross sections have received little attention until recently. Van der Heiden et al. [7,8] computed some of the localized buckling shapes of such rods, while Goriely et al. [9,10] investigated the dynamic instabilities of initially straight or helical rods. The natural (stress-free) configuration of our strip with spontaneous curvature is a crown, as shown in the insert of Fig. 1 or, more accurately, its universal covering. In term of the coordinates (s, q), it is Figure 1. Edge of a green salad leaf (top), and experimental snapshot of a periodic configuration of a ‘ruban à godets’ (bottom). The insert shows the connection between the strip model and the original plate problem.