## Abstract

We studied the mechanical process of seed pods opening in *Bauhinia variegate* and found a chirality-creating mechanism, which turns an initially flat pod valve into a helix. We studied conﬁgurations of strips cut from pod valve tissue and from composite elastic materials that mimic its structure. The experiments reveal various helical conﬁgurations with sharp morphological transitions between them. Using the mathematical framework of “incompatible elasticity,” we modeled the pod as a thin strip with a flat intrinsic metric and a saddle-like intrinsic curvature. Our theoretical analysis quantitatively predicts all observed conﬁgurations, thus linking the pod’s microscopic structure and macroscopic conformation. We suggest that this type of incompatible strip is likely to play a role in the self-assembly of chiral macromolecules and could be used for the engineering of synthetic self-shaping devices.

Motion in plants is a highly nontrivial process. It is often based on anisotropic swelling and shrinkage, driven by the relatively slow variation of water content within the tissue (*1*). Swelling and shrinkage may lead to the accumulation of elastic energy and the buildup of stress. This stress can be released via mechanical instabilities, such as fracture, buckling, and snapping (*2*, *3*). The buildup of stress may also occur in sclerenchymal tissue (tissue made of dead cells) and is thus a process that can be analyzed from a purely mechanical point of view. Sclerenchymal tissue typically consists of fiber cells whose walls are made of layered cellulose fibrils with a preferred orientation. When absorbing/expelling water, the tissue expands/shrinks anisotropically, perpendicularly to the fibrils’ orientation [(*1*), p. 200]. Changes in air humidity induce such uniaxial swelling/shrinkage that drive, for example, the opening and closure of a pine cone (*4*) and the penetration of wheat seeds into soil (*5*).

Chiral pod opening is a nontrivial example of hygroscopic (humidity-driven) motion in sclerenchymal tissue. In this process, two initially flat pod valves curl into helical strips of opposite handedness. A detailed study of the structure of pod valves in over 300 species of Leguminosae shows a wealth of architectures (*6*). We studied pod opening in *Bauhinia variegate* (Fig. 1, A and B), whose pod valves are known to consist of two fibrous layers, oriented roughly at ±45° with respect to the pod’s longitudinal axis [supporting online material (SOM) text and fig. S1, x-ray scattering measurements and fig. S2]. It turns out that the mere presence of two layers that shrink in perpendicular directions is sufficient to drive the flat-to-helical transition in pod opening.

As a proof of concept, we constructed a mechanical analog of the Bauhinia pod that copies its geometry. We stretched two identical thin latex sheets uniaxially by the same elongation factor. We then glued one on top of the other in perpendicular directions (Fig. 1C). Once released from the external stretching, the composite object underwent the same shrinkage profile as the Bauhinia valve: The two layers shrunk uniaxially in perpendicular directions. Elongated strips cut out from the composite sheets were found to curl into helical conﬁgurations (Fig. 1F).

It is well known that uniaxial shrinkage of one layer in a bilayered sheet induces curvature, which accommodates the difference in length of the two layers. When the sheet is thin enough, it bends accordingly. Strips with a single intrinsic curvature have been studied extensively; large sheets adopt a cylindrical configuration, whereas narrow strips adopt helical conﬁgurations that are “cut from a cylinder” (*7*–*10*).

A qualitatively different situation occurs when two layers shrink perpendicularly. In such a case, the sheet “wants” to bend in two opposite and perpendicular directions—that is, they locally assume a saddle-like conﬁguration (Fig. 1, D and E). This tendency to bend into a saddle creates a metric incompatibility: On the one hand, the in-plane rest lengths between surface elements require the surface to be metrically flat, to have zero Gaussian curvature. On the other hand, a saddle-like conﬁguration has nonzero Gaussian curvature. Thus, the body cannot assume a stress-free conﬁguration and must contain residual stress. The key feature of residually stressed bodies is that their conﬁguration changes when they are dissected into smaller parts, reflecting the release of stored elastic energy.

Geometrically incompatible materials are well known in the mechanical literature and are described by the theory of “incompatible elasticity” (*11*, *12*). A mathematical theory for incompatible thin sheets was recently derived in (*13*, *14*). In this theory, thin sheets are modeled as two-dimensional surfaces. The conﬁguration of a surface is fully characterized by two tensors: a metric tensor, *a*, which contains all information about lateral distances between points, and a curvature tensor, *b*, which contains information about the local curvature. An elastic sheet is characterized by two additional tensors that are intrinsic to its structure: a reference metric, *t* is the thickness of the sheet, *dS* is the infinitesimal surface element, and *a* and *b*.

Naïvely, one would think that the sheet can eradicate its elastic energy by settling on a conﬁguration that satisfies *15*). If the local shrinkage is such that the resulting

The first step in modeling the shaping of Bauhinia pods (or the mechanical model) is to determine the reference tensors *x*, *y*) that are parallel to the directions of principal curvature, we obtain a saddle-like (hyperbolic) reference curvature tensor of the form_{0} is the magnitude of the reference curvature. In Bauhinia pods, the directions of principal curvature form an angle of 45° with respect to the longitudinal axis. In strips cut out from latex sheets, this angle can be determined at will. A strip is characterized by the following parameters: a thickness *t*, a width *w*, a length *L*, an induced curvature κ_{0}, and an angle θ between the strip’s longitudinal axis and a direction of principal curvature.

A quantitative study of strips cut out from Bauhinia pods and from latex sheets at an angle of θ = 45° exhibits two different regimes: Wide strips adopt a conﬁguration close to a cut from a cylindrical envelope (“cylindrical helices”), whereas narrow strips exhibit a “pure twist,” where the strip’s centerline is straight (“twisted helices”). At the coarsest level, we may characterize the shape of the strips by two parameters: the radius *r* and the pitch *p* of the strip’s mid-curve. Figure 2 shows results for a collection of strips cut from both Bauhinia pods and latex sheets. For each sample, we measured the parameters *t*, *w*, *r*, *p*, and κ_{0} (κ_{0} was estimated by measuring the curvature of very thin strips cut along principal directions). We plotted a dimensionless pitch

The theoretical study consists of minimizing the energy (Eq. 1) of a strip with reference tensors *tw*(κ_{0}*w*)^{4}, whereas the bending energy scales like *w* and a bending-dominated regime for small *w*. The transition between the two regimes is expected to occur when the two energies are of comparable magnitude, when

When *b* aims to optimally approximate*v*), whereas the second principal curvature is 0,

The two solutions are equally stable. The conﬁguration looks like strips cut out from the envelope of a cylinder, with a centerline that is a helix of radius and pitch (Fig. 2, right inset):

Such cylindrical helices are similar to configurations observed in strips with a single intrinsic curvature (*7*–*10*). Yet, there is a fundamental difference between the two cases because in the case of a single curvature,

When *a* that best approximates (in a mean square sense) the reference metric

The conﬁgurations in the narrow regime are twisted helices and are similar to conﬁgurations found in (stretch-free) liquid membranes (*16*, *17*). Once again, there is a fundamental difference between the two cases because twisted helices in liquid membranes are stress-free. In the present study, they result from a compromise between stretching and bending energies.

The radius and pitch in the narrow regime are *a* and *b*. The calculation of a global minimizer can be done numerically, as described in the SOM text. The predicted dimensionless radius and pitch are displayed as functions of

The transition between the two limiting regimes—wide versus narrow strips—is also manifested in the geometry of the surface. One way to characterize a surface is via the local values of its mean curvature, *H*, and Gaussian curvature, *K*, which are the respective mean and product of the two principal curvatures. In Fig. 3, we plot *H*^{2} and *K* as function of the distance from the edge for a collection of latex strips differing only in width. These data were extracted by measuring the surface topography of the latex strips with optical profilometry. Each scan provides us with a map of the surface height, from which we computed the local values of *H* and *K* by discrete differentiation. We then averaged *H* and *K* over points that are at equal distance from the edge of the strip. Narrow strips, *H* is no longer uniformly 0. For wide strips, *H* reaches the predicted value of (1 – *v*)κ_{0} far enough from the edge (dotted line) while remaining 0 at the edge, with a boundary layer connecting the two regions (*14*). For wide strips, *K* ≈ 0 as determined by (the Euclidean)*K* tends toward the asymptotic value of *w* → 0. Thus, our model is capable of predicting not only the global shape of the helical strips but their surface topography as well.

A natural question is whether the transition between the narrow and wide regimes is continuous or not; more precisely, is the change in equilibrium conﬁguration continuous as

Our theoretical and mechanical models capture well the behavior of Bauhinia valves and link quantitatively between the microscopic structure and the macroscopic conﬁgurations. Yet, they have an even wider scope of applicability. We next studied the dependence of the equilibrium conﬁguration on the angle θ. Conﬁgurations of latex, Bauhinia, and theoretically predicted strips, respectively, are shown in Fig. 4, A to C, for fixed width and varying θ. The dependence of the radius and the pitch on the angle for latex strips are shown in Fig. 4D. The parameters correspond to the “narrow” regime,

We have also conducted an experimental study of the dependence of the equilibrium conﬁguration on the angle θ in the “wide” regime, *18*), for an example.

The only inputs in our model are *6*).

The present study is potentially applicable to systems of different scales and contexts. For example, many organic and inorganic macromolecules are found in different helical conﬁgurations (*9*, *16*, *17*, *19*). In our model, the only nontrivial property of the sheet is its saddle-like reference curvature tensor. It can therefore be relevant to any sheet with such spontaneous curvatures, including monolayers and sheets with different microstructures. Macromolecules made of chiral molecules are often modeled by introducing a term in the bending energy that accounts for a spontaneous twist. It can be shown that geometrically, a sheet with a spontaneous twist is equivalent to one with hyperbolic reference curvature tensor (SOM text). Indeed, a transition qualitatively similar to the one presented in Fig. 2 was predicted in (*20*, *21*). After translating the parameters of (*21*), we found their predicted value for *19*). Lastly, mechanisms such as the one studied here can be implemented by using inducible responsive materials. Such bio-mimicking designs have an applicative potential as soft actuators (for a proof of concept, see movies S1 and S2).

## Supporting Online Material

www.sciencemag.org/cgi/content/full/333/6050/1726/DC1

SOM Text

Figs. S1 to S4

References

Movies S1 and S2

## References and Notes

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**Acknowledgments:**We are grateful to Y. Abraham and U. Raviv for performing the small-angle x-ray scattering measurements. R.K. was partially supported by the Israeli Science Foundation. E.S. was partially supported by the European Research Council SoftGrowth project. S.A. was supported by the Eshkol Scholarship sponsored by the Israeli Ministry of Science.