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# Geometrically controlled snapping transitions in shells with curved creases

Edited by Tom C. Lubensky, University of Pennsylvania, Philadelphia, PA, and approved July 27, 2015 (received for review May 12, 2015)

## Significance

Shape-programmable structures have recently used origami to reconfigure using a smooth folding motion, but are hampered by slow speeds and complicated material assembly. Inspired by natural systems like the leaves of Venus flytraps and hummingbird beaks, we use curved creases to imbue elastic shells with programmable fast “snapping” motion. This deformation between preprogrammed states can be tuned to be either continuously foldable or snap discontinuously. Our results provide a purely geometrical mechanism for designing multistable structures, thus circumventing the need for complex materials or fabrication methods in creating structures with fast dynamics. This technique will find application in designing structures over a wide range of length scales, including self-folding materials, tunable optics, and switchable frictional surfaces for microfluidics.

## Abstract

Curvature and mechanics are intimately connected for thin materials, and this coupling between geometry and physical properties is readily seen in folded structures from intestinal villi and pollen grains to wrinkled membranes and programmable metamaterials. While the well-known rules and mechanisms behind folding a flat surface have been used to create deployable structures and shape transformable materials, folding of curved shells is still not fundamentally understood. Shells naturally deform by simultaneously bending and stretching, and while this coupling gives them great stability for engineering applications, it makes folding a surface of arbitrary curvature a nontrivial task. Here we discuss the geometry of folding a creased shell, and demonstrate theoretically the conditions under which it may fold smoothly. When these conditions are violated we show, using experiments and simulations, that shells undergo rapid snapping motion to fold from one stable configuration to another. Although material asymmetry is a proven mechanism for creating this bifurcation of stability, for the case of a creased shell, the inherent geometry itself serves as a barrier to folding. We discuss here how two fundamental geometric concepts, creases and curvature, combine to allow rapid transitions from one stable state to another. Independent of material system and length scale, the design rule that we introduce here explains how to generate snapping transitions in arbitrary surfaces, thus facilitating the creation of programmable multistable materials with fast actuation capabilities.

Curved shells are generally used to enhance structural stability (1⇓–3), because the coupling between bending and stretching makes them energetically costly to deform. The consequences of this coupling are seen in both naturally occurring scenarios, such as intestinal villi and pollen grains (4, 5), and find use in man-made structures such as programmable metamaterials (6⇓⇓–9). When these shells have multistable configurations, the transition between them is opposed by geometrically enhanced rigidity resulting from the dominant stretching energy. Often, even for relatively small range of deformation, stretching leads to the high forces and rapid acceleration associated with a “snap-through” transition in many natural and man-made phenomena (10⇓⇓⇓⇓⇓⇓–17). For example, Venus flytraps (*Dionaea muscipula*) use this mechanism to generate a snapping motion to close their leaves (11), hummingbirds (*Aves: Trochilidae*) twist and rotate their curved beaks to catch insect prey (14), and engineered microlenses use a combination of bending and stretching energy to rapidly switch from convex to concave shapes to tune their optical properties (12). Despite the ability to engineer bistability and snapping transitions in a variety of systems by using prestress or material anisotropy (18⇓⇓⇓⇓⇓–24), a general geometric design rule for creating a snap between stable states of arbitrary surfaces does not exist. This stands in stark contrast to the well-known rules and consequences for folding of a flat sheet, as shown in origami design (25⇓–27). In origami, weakening the material locally by introducing a crease allows the sheet to deform without stretching, and thus allows the sheet to access low-energy states without requiring nonlinear material strain.

## Geometrical Mechanics of Folding a Shell

Inspired by these ideas from origami, we consider the folding of curved surfaces with creases. Although this concept has been realized on rare occasions in art (27⇓–29), the continuum mechanics of a creased shell is far from fully understood. In particular, folding a curved surface along a crease often leads to large deformations of the shell. However, despite these nonlinear deformations, we show that the local geometry of the crease alone creates a large energy barrier that leads to a snapping transition in a sufficiently thin shell. Because our proposed design principle arises purely from geometry, it does not rely on special materials or anisotropy to generate rapid snap-through transitions; in practical applications, this enables one to harness the instability for fast actuations purely by design, thereby providing a simple method for the design of rapidly actuating structures from a wide range of elastic materials.

We consider a crease to be a long but narrow region of locally weak material introduced, for example, through a local thinning of the shell. This local weakening behaves as a foldable hinge in the shell, but the curvature of the rest of the shell limits the deformation of this hinge, because the shell and hinge itself must deform to accommodate folding along the creased area. When the entire shell is sufficiently thin, this deformation will be approximately isometric, meaning it is devoid of in-plane strain. Geometry and the condition of isometry combine to allow us to relate the shape of the crease to the deformation of the shell in the vicinity of the fold. To proceed, imagine an unfolded shell upon which a hinge, parametrized by arc length *s* and having tangent vector *A*). The perpendicular vectors *ψ* with respect to the shell (Fig. 1*B*), we can define the geodesic curvature, *B*) and normal (*B*) of the shell’s midsurface.

An advantage of this decomposition is that, if in-plane strains in the shell vanish, the geodesic curvature of the fold must remain unchanged after folding (30, 31), yielding the relationship *Supporting Information*) obtains the mean curvature of the shell near the fold,*τ* is the torsion of the fold, which measures the rate that the osculating plane twists around the hinge and, hence, the nonplanarity of the hinge (32).

A special role is played by angles along which a fold does not change its space curvature, *κ*, after folding. In this case we may use the definition of the geodesic curvature to solve for the angle *ψ*, which yields *ψ*, one of which may be physically understood as a local reflection of the shell through the osculating plane of the fold. A surface can be folded by an angle *B*). This “mirror reflection” is naturally an isometry of the surface in the vicinity of the fold, although the mean curvature *H* must switch signs on one side relative to the other. The mirror reflection isometry was noted in the seminal monograph on bending of surfaces by Pogorelov (33), whose work on spherical isometries we discuss below.

The bending energy density of a folded surface, in the vicinity of the fold, is *B* is a bending modulus and *B*) is not zero because *H* generically diverges (Eq. **1**, see the *Supporting Information*). Whereas the bending energy between the folded and unfolded states is infinite for isometric deformations, in any real material as the shell bends the energy will reach a scale where stretching becomes favorable. As a result, our assumption that the shell does not stretch must have been flawed, and in-plane stresses must have developed near the fold similar to stress-focusing phenomena seen in other curved surfaces (1, 34⇓⇓⇓⇓–39). Beyond this special stressed configuration, however, in-plane stresses are no longer necessary and the surface can, at least in principle, accommodate the folding through bending deformations alone.

An important exception to the previous analysis arises when the crease has zero normal curvature everywhere. Curves with vanishing *C*). This occurs trivially on flat paper inscribed with a curved fold because

## General Design Principle

These considerations provide a geometrical design rule: simply by introducing a crease with finite normal curvature (*C*.

To test the applicability of this design rule we crease elastomeric and plastic helicoids, cylinders, and spheres, whose *C* and *Materials and Methods*).

## Negative Gaussian Curvature: Helicoid

To explore snapping behavior and the lack thereof in negative Gaussian curvature surfaces, we specifically choose the helicoid. The helicoid is the only ruled minimal surface (such that *Supporting Information* for more details). The coexistence of this set of curves makes the helicoid an ideal surface on which to validate the design rule.

Thus, we fabricate plastic helicoids with creases along three kinds of curve: (*i*) a *ii*) the first *iii*) the second *A*, *B*, and *C*, respectively). Deforming these shells along the *A*). Moreover, for the other two planar and nonplanar creases, we observe the continuous “smooth” motion characteristic of a simple hinge as predicted for curves with vanishing normal curvature (Movie S1 and Fig. 2 *B* and *C*). Hence, the purely geometric nature of our design rule offers a way to understand the continuity or discontinuity of folding even without a detailed understanding of the complex shell mechanics.

## Zero Gaussian Curvature: Cylinder

As a singly curved surface, a cylinder has only one set of planar curves with *θ*) at a distance (*d*) from the apex, as shown in Fig. 3*A* and *Materials and Methods*. According to our hypothesis, we expect the cylinder to undergo a snapping transition when deformed along this crease. Remarkably, despite the introduction of a crease, the free cylinder displays a global bending deformation instead of snapping (Movie S2). Such global deformations arise because cylinders have

These pathways can be eliminated by imposing fixed boundary conditions on one of the free ends of the cylinder by inserting a rigid cylindrical plug. We use two representative results from the parameter space consisting of *B* (Movie S2), a transition to an antisymmetric mode is observed before snapping completely into the mirror reflection isometry. Notably, stability of the isometric state and the presence of an antisymmetric mode in our experiments is consistent with finite-element analysis (FEA, performed using ABAQUS, Dassault Systemes).

The stability of this isometric state can be explained by considering the concentration of bending and stretching energy during the deformation. Indenting an uncreased cylinder near the free edge produces a deformation pattern that is composed of two parts: a region of mirror isometry that contains only bending, and a localized ridge (1, 33, 37, 42). The localized region, which acts as an elastic boundary layer, contains all of the stretching energy of the deformation. By weakening the material through creasing, the stretching cost for the ridge can be dramatically lowered, and the folded state may become stable. Here, we observe that for lower values of

## Positive Gaussian Curvature: Sphere

Given their axisymmetry, spherical shells are well suited to quantitative analytical, computational, and experimental analysis. Moreover, mechanisms involving pure bending are avoided in spherical shells, because surfaces with doubly curved shells naturally require stretching for many deformations of the surface (1, 43, 44). Because the spherical geometry is devoid of any *A*). Upon indentation, for an uncreased shell (*B*). In a similar fashion to the cylinder, we observe a local minimum in force for lower values of α (=0.5) devoid of a stable folded state, but indenting a creased shell with higher α (=0.6) leads first to an unstable, nonaxisymmetric snap, soon followed by a well-defined stable snap (Fig. 4*B*, *Insets* and Movie S3).

Along the lines of argument we presented for stability of creased cylinders, a spherical shell poses a system which can be solved analytically. There is a well-known nearly isometric deformation of a sphere seen for displacements larger than the thickness but smaller than the crease size (33, 46). This deformation regime is characterized by an inverted bulge of radius *r* and bounded by a ridge of size *C*). The energy for this state has a bending energy contribution from the inverted bulge that scales as *Y* as the Young’s modulus of the material. Hence the total energy (*B* can be expressed as*γ* is the Föppl–von Kármán number *A*). For a thin shell *ν* is Poisson’s ratio, such that

In the case of a creased sphere, we assume that the deformation of the shell retains the same structure as this classical solution, but now the thickness of the sphere in the Pogorelov ridge is a function of the bulge radius *r*, such that *A*, for a creased spherical shell there is a local minimum in the Pogorelov energy centered at *α* the monotonically increasing bending energy overcomes the energy gain from thinning the shell at the crease, and the folded state remains unstable. Evidently, for larger values of *α*, this gain surpasses the bending energy of the shell, resulting in bistability due to the presence of a local minimum in total energy. Thus, we infer that the stability of creased shells is governed by the competition between bending energy of the undeformed shell and stretching energy contained in the creased region.

To confirm this simple model, we again use FEA to determine the conditions under which there is a stable snap. For linear elastic materials this system is fully characterized by two dimensionless numbers, the reduced crease radius *α* and the Föppl–von Kármán number *γ*. We report the total energy for axisymmetric solutions with *h*) and the normalized crease radius (*α*) in Fig. 5*B*. We find that, beyond a critical crease radius, there is a bifurcation of stability and the energy curves develop a well-defined local minimum (solid) and maximum (dashed), with the region between these curves denoting a basin of attraction for the folded state.

By examining creased hemispherical shells over a range of *γ*, we construct a phase diagram for stability of creased spherical shells (Fig. 5*C*). Through numerical simulations, we find that for increasing thickness, larger values of the crease radius are required to create a stable snap. Moreover, we conduct a series of experiments on spherical shells with a range of *γ* and *α*, and identify the stability of the folded state. These reveal a boundary between bistability and monostability that is in excellent agreement with our numerical calculations. Further bolstering this, for some samples we observe the presence of folded states that are temporarily stable (for times on the order of seconds)––the proximity of these samples to the predicted phase boundary further demonstrates the agreement between experiments and simulation.

## Conclusion

The ability to introduce tunable bistability into a curved shell via structural inhomogeneity represents a major step in generating programmable materials with rapid actuation capabilities. While inhomogeneous shells have already been predicted to serve as a template for constructing tunable shapes (49), and used to design next-generation substances such as lock-and-key colloids (50) or controllably collapsible capsules (39), our geometric design principle adds further insight into controlling the mechanics of thin shells. Because the speed of the snap arises from stretching in the shell, inertia mediates the transition at the speed of sound in the material (Movies S1–S3), and crucially, the snap is unimpeded by poroelasticity or hydraulic damping as displayed in many natural snapping systems (51). Our work lays the foundation for developing non-Euclidean origami, in which multiple folds and vertices combine to create new structures. Indeed, smoothly deployable structures built from non-Euclidean surfaces could be engineered using origami-like principles that build upon the isometric design rules for negative Gaussian curvature surfaces that we derive here. Finally, because the principles and methods we describe are purely geometric, they open the door for developing design paradigms independent of length scale and material system.

## Materials and Methods

### Shell Fabrication.

Three-dimensional models of different geometries were designed in a CAD software. The non-Euclidean geometries (helicoid and hemisphere) were fabricated using a commercial 3D printer (Stratys Inc., uDimensions) to obtain two-part molds with embossed features to generate creases (Fig. 1*C*). The hemispherical shells were fabricated using poly(vinyl siloxane) by curing a commercially available two-part base–catalyst mixture [Zhermack SpA Elite Double 32, Elastic modulus *θ*). The scored sine wave was scaled to different amplitudes to obtain the combinations of

### Helicoid Characterization.

Helicoids with different creases were clamped on one edge, and deformed along the crease using a rigid indenter by hand. Composite images using frames at equal time intervals from these movies were created by using alpha blending. For the sample with a snap-through, frames were chosen to be 300 ms apart. For the sample with a planar crease, frames are 1 and 6 s apart for deformation on either side of the torsional hinge. Lastly, for the sample with helical crease, frames were 1.5 and 1.5 s apart for deformation on either side of the torsional hinge.

### Load Displacement Characterization.

A custom-built force displacement device, combining a linear translation stage (Zaber Technologies Inc., T-LSM 100) and a load cell (Loadstar Sensors Inc., RPG-10), was used to perform strain-controlled force measurements. For both cylindrical and hemispherical samples, 3D printed point indenters (radius ratio of indenter with respect to shell

## Differential Geometry of Curves and Surfaces

A crease placed on a surface is parametrized by a space curve *s* an arc length variable that runs along the curve. At each point on the crease we define the orthonormal Frenet frame *κ* is the curvature of the crease and *τ* is the torsion. The surface of the shell is composed of two regions that are divided by the crease, each parametrized by a local orthonormal frame. In a frame of reference where one surface is fixed in space a local orthonormal frame *ψ* measures the difference between the surface tangent

The angle *ψ* is particularly important if we wish to consider folding the shells about this crease. The equation for *ψ* in terms of normal (or geodesic) curvature has two solutions, that is,

To determine the stability of the folded state we first write the energy for deformation of a thin shell:*Y* is Young’s modulus, *ν* is Poisson’s ratio, and *t* is the thickness of the shell. To describe the strain and bending tensors of the surface, we need the first and second fundamental forms, given by

We define a coordinate system on the shell *s* is an arc length along the crease and *v* is measured orthogonal to the crease, such that*s* is an arc-length parametrization, the first fundamental form may be written as *s* is an arc-length variable we know that

Finding the bending energy requires that we find the mean curvature of the surface, which requires us to compute the components of the curvature tensor, **r** is a parametrization of the surface. With the geometric definitions given above, we have the fairly simple results that *Theorema Egregium*, which states that *L* is not constrained by the *Theorema Egregium*, and thus the shell is not constrained isometrically by the crease.

The bending energy density *ψ:*

The energy, proportional to *ψ* passes through zero, indicating that our isometric model cannot accurately describe the transition between folded shell states for creases that have finite normal curvature. The existence of an infinite barrier in this singular limit indicates that the angle *ψ* may not be folded continuously from the folded state to the unfolded state.

If the crease has zero normal curvature, however, the component of the second fundamental form *L* determined entirely by bending away from the crease that is unconstrained by the condition of isometry. The folding angle and torsion, however, are constrained by

Written another way, we have that

Together, these results can be used to infer a number of things. First, finite normal curvature implies that there is an energy barrier, which implies that a subcritical bifurcation may occur. Second, zero normal curvature implies that locally, the shell may deform without stretching, and thus the angle *ψ* may be varied continuously without fear of approaching a stretching barrier. Furthermore, zero normal curvature explicitly means that one of the components of the curvature tensor vanishes identically; specifically, the curvature of the surface in the direction of the crease is always zero.

## Example: The Helicoid

A parametrization for the helicoid is

For a general surface with nonpositive Gaussian curvature, at every point there exists a pair of asymptotic curves such that the normal curvature along these curves is zero. For a curve parametrized by an arc length t, *v* (straight lines) or curves of constant *u* (helices). Asymptotic curves of constant *v* are particularly simple examples, because

Alternatively, we could write this as

Choosing curves of constant *ψ* will lead to locally isometric deformations that do not necessarily result in an energy barrier to folding. These arguments are all local, and there may be global constraints that lead to an energetic barrier, but this depends specifically on the type of surface and shape of the crease.

## Acknowledgments

The authors thank Jesse L. Silverberg, Thomas C. Hull, Douglas P. Holmes, and Dominic Vella for illuminating discussions. We are grateful to Michael J. Imburgia, Alfred J. Crosby, Mindy Dai, and Sam R. Nugen for assistance with both the 3D printer and laser cutter, and to Pedro Reis for discussions regarding the fundamentals of shell mechanics and insight on elastomer shells. This work was funded by the National Science Foundation through Emerging Frontiers in Research and Innovation Origami Design for Integration of Self-assembling Systems for Engineering Innovation (ODISSEI)-1240441 with additional support to S.I.-G. through the University of Massachusetts Materials Research Science and Engineering Center Division of Materials Research (DMR)-0820506 Research Experience for Undergraduates program.

## Footnotes

↵

^{1}N.P.B. and A.A.E. contributed equally to this work.- ↵
^{2}To whom correspondence may be addressed. Email: hayward{at}umass.edu or csantang{at}physics.umass.edu.

Author contributions: N.P.B., A.A.E., R.C.H., and C.D.S. designed research; N.P.B., A.A.E., S.I.-G., and L.A.M. performed research; N.P.B. and A.A.E. analyzed data; and N.P.B., A.A.E., S.I.-G., L.A.M., I.C., R.C.H., and C.D.S. wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1509228112/-/DCSupplemental.

## References

- ↵.
- Vaziri A,
- Mahadevan L

- ↵
- ↵
- ↵.
- Shyer AE, et al.

- ↵.
- Katifori E,
- Alben S,
- Cerda E,
- Nelson DR,
- Dumais J

- ↵.
- Schenk M,
- Guest SD

- ↵
- ↵.
- Silverberg JL, et al.

- ↵
- ↵
- ↵
- ↵
- ↵.
- Hayashi M,
- Feilich KL,
- Ellerby DJ

- ↵
- ↵.
- Shankar MR, et al.

- ↵
- ↵
- ↵.
- Guest SD,
- Pellegrino S

- ↵.
- Norman AD,
- Seffen KA,
- Guest SD

- ↵
- ↵
- ↵
- ↵
- ↵.
- Giomi L,
- Mahadevan L

- ↵.
- Guest SD

- ↵.
- Tachi T

- ↵.
- Demaine E,
- Demaine M,
- Koschitz D

*Origami*(CRC Press, Boca Raton, FL), pp 39–52^{5}: Proceedings of the 5th International Conference on Origami in Science, Mathematics and Education - ↵
- ↵
- ↵
- ↵
- ↵.
- Spivak M

- ↵.
- Pogorelov AV

- ↵
- ↵
- ↵
- ↵.
- Vaziri A

- ↵
- ↵
- ↵
- ↵.
- Struik DJ

- ↵
- ↵.
- Calladine CR

- ↵.
- Niordson FI

- ↵.
- Gupta NK,
- Easwara Prasad GL,
- Gupta SK

- ↵.
- Landau LD,
- Lifshitz EM

- ↵.
- Fung YC,
- Wittrick WH

- ↵
- ↵
- ↵
- ↵.
- Skotheim JM,
- Mahadevan L

- ↵.
- Hofer D

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