Finessing the fracture energy barrier in ballistic seed dispersal

Abstract

Fracture is a highly dissipative process in which much of the stored elastic energy is consumed in the creation of new surfaces. Surprisingly, many plants use fracture to launch their seeds despite its seemingly prohibitive energy cost. Here we use Impatiens glandulifera as model case to study the impact of fracture on a plant's throwing capacity. I. glandulifera launches its seeds with speeds up to 4 m/s using cracks to trigger an explosive release of stored elastic energy. We find that the seed pod is optimally designed to minimize the cost of fracture. These characteristics may account for its success at invading Europe and North America.

Fig. 1.

(A) Impatiens glandulifera seedpod with two seams indicated and top/bottom convention marked. (B) An unconstrained pod valve. The relaxed state is curled. (C) Top and side view of a valve glued to a glass slide. Each valve is approximately 20 mm long, and consists of a rigid section approximately 15 mm long with a triangular cross-section which widens and thickens along the length and a thin-walled and floppy 5 mm long membrane. The (red) dashed line demarcates the boundary between the floppy and the rigid parts. The scale bar corresponds to 5.6 mm. (D) Photographic sequence of the pod opening extracted from a high speed video. From left to right t = 0; 263; 789; 1,578; 2,104 μs. (see also Movie S1). Note that only the upper rigid section curls and accelerates the load.

Fig. 2.

(A) Force vs. displacement for a 26 mm long seedpod pulled from a relaxed state to its original length. Bottom right inset: cartoon of force-displacement apparatus showing a pod in an almost fully relaxed state. Top inset: Same data as in main figure but with the end-to-end distance normalized by the length. The solid line shows the result of a finite element calculation for the same geometry with a Mooney-Rivlin constitutive relation. (B) Results of finite element calculation. Elastic energy per valve (top) and energy release rate (bottom) for a seedpod (17.5 mm rigid section length) with a crack of length L - s extending from the stem downward. When the pod is fully sealed the elastic energy is 0.2 mJ per valve. As the crack extends virtually no energy is released until the crack is longer than 12 mm. Conversely, the pod is able to maintain its structural integrity even if the seams are split along 70% of their length. The energy release rates were calculated by differentiating the fit to the data for the elastic energy per valve (shown as solid line in top). Dashed lines shown the threshold value for unstable crack growth for a seam with 25 μm and 250 μm thickness. Inset: finite element model with arc length s shown as the distance from the bottom. The z = 0 plane is impenetrable and represents a neighboring valve.

Fig. 3.

Results of elastica computation for beams of varying cross section. The dashed curve shows the relaxed state of the beam. The solid curves show the shape of the beam for various tapers for the boundary conditions y = 0 at s = 0 and dy/ds = 0 at s = L corresponding physically to the attachment of the valve to its neighbor at a point (at s = 0) and along a line (at s = L). The result for the uniform and tapered width cases indicate that the beginning part of the valve will relax towards its preferred curvature κ_o. Conversely, the beam crosses the y = 0 plane for both cases in which the thickness is tapered. This result occurs because it is energetically favorable to allow the thicker portions of the beam relax towards the preferred curvature and use the thinner portions to satisfy the y = 0 boundary condition. However, in a pod the valve would be prevented from crossing the y = 0 plane by its neighbor, and thus the valve will do the next best thing which is to press flat against its neighbor.

Fig. 4.

Illustration of negative curvature (indicated by dashed segment) produced when part of the pod relaxes to its natural curvature κ_o while maintaining the boundary conditions dictated by bonding to neighboring valves. In the absence of a taper the decrease in energy from the relaxed portion outweighs the increase due to the negatively curved portion. With a taper in the thickness of the valve the energy balance is reversed: the energy increase due to the negatively curved segment outweighs the energy decrease due to relaxed segment because the negative curvature is on a thicker—and thus more rigid—segment.

Fig. 5.

Schematic illustrating the coupling between valves as viewed down the symmetry axis of the pod. When the bond between adjacent valves cracks, the valves begin to curl which in turn causes all the valves to rotate about the unbroken seams. The original uncracked arrangement of valves is shown in light gray. The mechanical model constructed from paper is given in Fig. S1 and its mechanical response to bending is given in Fig. S2.