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# Programming curvilinear paths of flat inflatables

Edited by Thomas A. Witten, Physics Department and James Franck Institute, University of Chicago, Chicago, IL, and accepted by Editorial Board Member Herbert Levine July 11, 2019 (received for review March 19, 2019)

## Significance

Inflatable structures are flat and foldable when empty and both lightweight and stiff when pressurized and deployed. They are easy to manufacture by fusing 2 inextensible sheets together along a defined pattern of lines. However, the prediction of their deployed shape remains a mathematical challenge, which results from the coupling of geometrical constraints and the strongly nonlinear and asymmetric mechanical properties of their composing material: thin sheets are very stiff on extensional loads, while they easily shrink by buckling or wrinkling when compressed. We discuss the outline shape, local cross-section, and state of stress of any curvilinear open path. We provide a reverse model to design any desired curved 2-dimensional shape from initially flat tubes.

## Abstract

Inflatable structures offer a path for light deployable structures in medicine, architecture, and aerospace. In this study, we address the challenge of programming the shape of thin sheets of high-stretching modulus cut and sealed along their edges. Internal pressure induces the inflation of the structure into a deployed shape that maximizes its volume. We focus on the shape and nonlinear mechanics of inflated rings and more generally, of any sealed curvilinear path. We rationalize the stress state of the sheet and infer the counterintuitive increase of curvature observed on inflation. In addition to the change of curvature, wrinkles patterns are observed in the region under compression in agreement with our minimal model. We finally develop a simple numerical tool to solve the inverse problem of programming any 2-dimensional (2D) curve on inflation and illustrate the application potential by moving an object along an intricate target path with a simple pressure input.

A domain of application for pneumatic structures has emerged with the current development of soft robotics actuators (1). Unidirectional bending of elastomeric pneumatic structures can be easily controlled by internal pressure (2), and recently, more general complex shape morphing was achieved (3). As they rely on large material strains, these structures are based on elastomers and therefore, have a relatively low stiffness, which makes them unsuitable for large-scale structures and heavy loads. In contrast, stiff inflatables may be obtained by stitching flat pieces of thin but nearly inextensible material. As a first example, sky lanterns were invented during the 3rd century in China (4); then, they were rediscovered and scaled up by the Montgolfier brothers for ballooning in the 18th century. Since then, stiff inflatables have been widely used in engineering (5), medicine (6), architecture, and aerospace (7⇓–9). Here, we show how to shape-program slender “flat-inflatable” structures, which are extremely easy to manufacture: 2 identical patches are cut in thin sheets and sealed along their boundaries (10). Common examples from everyday life are Mylar balloons. Although they are easy to manufacture, predicting the 3-dimensional shape of such flat-inflatable structures (i.e., maximizing a volume that a thin inextensible sheet can encompass) remains a challenge due to geometrical constraints. Indeed, changing the Gaussian curvature (i.e., the product of both principal curvatures of a surface) implies a distortion of the distances within the surface. In the case of thick elastic plates, local stretching or compression may accommodate changes in metrics. However, inextensible sheets behave nonlinearly: they can accommodate compression by forming wrinkles but cannot be stretched. Tension field theory, the minimal mathematical framework to address this problem, has been developed to predict the general shape of initially flat structures. While solutions have been found for axisymmetric convex surfaces (11⇓–13) and polyhedral structures (14, 15), predictions in a general case remain an open issue and have been addressed numerically in the computer graphics community (16). In a seminal paper, Taylor (17) described the shape of an axisymmetric parachute with an unstretchable sail, a solution also appearing in recent studies on the wrapping of droplets with thin polymeric sheets (18⇓–20).

We study macroscopic structures made of thin quasiinextensible planar sheets heat sealed along a desired path using a soldering iron with controllable temperature mounted on the tracing head of a 2-directional plotter (10) (*Materials and Methods* and Fig. 1*A*). We focus on simple configurations where pairs of identical flat patches forming curvilinear paths of constant width are bonded along their edges. When inflating a straight ribbon, we trivially obtain, far from the extremities, a perfect cylinder of circular cross-section. In contrast, inflating a flat ring results into complex features. We observe, for instance, an out-of-plane instability in the case of closed paths and the presence of radial wrinkles and folds (Fig. 1 *B* and *C*). We show in this article that inflation induces an overcurvature of the outline through a detailed study of its cross-section. We first describe the cross-section of axisymmetric annuli. We then extend our analysis to open rings to predict the position of compressive zones and the change in intrinsic curvature. We finally devise an inverse method for programming the outline of any arbitrary inflated curved flat path and illustrate the strong workload capacity of these actuators by displacing an object along a complex path with a simple pressure input.

## Results and Discussion

### Closed Rings.

We first consider a swim ring configuration: a planar axisymmetric annulus of inner radius R and width w. We describe the cross-section of the inflated annulus in the *SI Appendix*). Here, we choose to derive this shape by considering the balance of tension along the membrane path and applied pressure in the *SI Appendix*, Fig. S1), and we consider the force balance along a surface element of extent **1** shows that the shape of the section is the solution of the classical nonlinear oscillator ordinary differential equation for **2**. A second imposed condition is that the inner seam remains under tension, which leads to **1**) provides the corresponding condition for φ: **1** and **2** may be found in *SI Appendix* using variational techniques. The equation is solved with standard shooting methods, which determines the constant *B*, solid lines and compare them with experimental profiles (triangles in Fig. 3*B* and image in Fig. 3*A*) for values of the aspect ratio *SI Appendix*). The agreement between calculated and measured profiles is remarkable without any adjustable parameter. The toroidal structure is, as predicted by the geometrical model, decorated with alternating wrinkles and crumples (21, 25) everywhere except at the inner edge of the structure (Fig. 3*A* and *SI Appendix*, Fig. S2). However, we observe that the global structure does not remain in plane on inflation and tends to buckle out of plane, exhibiting either diametrically opposed localized kinks for very thin sheets and *SI Appendix*, Fig. S2, and Movie S1).

### Coiling of Open Rings.

These observations suggest the existence of geometrical frustration in closed inflated rings, which is reminiscent of the buckling of rings with incompatible intrinsic curvature (26) or of the warping of curved folds (27). This constraint is readily assessed when a cut is performed on the annuli (and both ends are sealed), thus removing the closing condition. With this additional degree of freedom, the structures remain in plane, but the curvature of their outline increases, which results into an overlapping angle *A*). Considering a cut in the *D*). Counterintuitively, pressurizing curved structures increases their curvature.

We show in *SI Appendix* that overcoiling is associated with an increase of the enclosed volume and assume that the optimal coiling is determined by the inextensibility condition. For a quantitative description, we consider an open annulus of inner radius R and width w in the flat configuration. The overlap results into a new inner radius *B* the normalized difference*B*). As *B*). Beyond this point, **3**, the profile

If we approximate this profile as the closed axisymmetric profile before additional curving [that is, **3**], the position of the line under orthoradial tension can be readily determined. Searching for the condition when the maximum of

In Fig. 4*C*, we compare the experimental measurement of **4** and find a very good agreement with experimental data for *D*) also matches the observed region free from wrinkles. Nevertheless, this region is actually not limited to a line but presents a finite width. We interpret this difference as a consequence of the finite stiffness of the sheet as described in a seminal paper by King et al. (21) in a simpler geometry and of the simplifying assumption that the profile of the structure is strictly similar to the axisymmetric closed configuration.

### Inverse Problem.

Having rationalized the change in curvature on inflation, we propose to use our geometrical model to inverse the problem (i.e., to determine the path of width w leading to an inflated structure of an arbitrary desired 2D shape with free ends). For a given target curve (Fig. 5 *A* and *E*), we first numerically calculate the curvature *B* and *F*). The same parametrization may be used in the flat state, since the inner edge does not stretch or contract on inflation, and the tube is chosen slender (*C* (Fig. 5 *B* and *F*). This relationship is rigorously valid only in the case of slowly varying curvatures (*C* and *G*). If overlap occurs, as in the case of the “Hello” curve, the path is printed in several nonoverlapping distinct parts that are bonded together using tape. On inflation, we do obtain with great precision the target shape (Movies S3 and S4). Depending on the initial curves, the inflated structures may expand (Hello) or conversely, contract (waving man). The programming of a simpler and smoother shape, a lemniscate, is shown in *SI Appendix*, Fig. S4. This offers a path for a kind of strong lightweight actuators with programmable shapes. Harnessing geometrical nonlinearities, one can predict the complex deformation path to displace objects with mere pressure input. In Fig. 5*I* and Movie S5, the octopus-like arm lifts a mug weighing several times its own weight and carries it to a platform behind an obstacle. Large workload with particularly large stroke may thus be reached with a very simple object.

### Concluding Remarks.

In this report, we have shown that the physics and geometry of apparently mundane flat sealed inflatables, such as “Mylar balloons,” are far richer than expected: the shape of their section includes singularities and a nontrivial distribution of wrinkles; the outline of an inflated curved balloon with free ends overcurves under inflation. Commercially available Mylar balloon letters are empirically designed to compensate for this overcurvature. For example, the letter O has, before inflation, a missing angular sector and rather looks like a C (14) (*SI Appendix*, Fig. S3). Our model based on the assumption of perfectly inextensible and infinitely bendable membranes does capture quantitatively this coiling for aspect ratio

## Materials and Methods

We fabricate the curved balloons by displaying 2 thin sheets made of the same thermosealable material (thermoplasticurethane-impregnated nylon fabric, Mylar, polypropylene) covered by a sheet of greaseproof paper in the working area of an XY plotter (from Makeblock). A soldering iron with controllable temperature (PU81 from Weller) is then mounted on the tracing head of the plotter (Fig. 1*A*). Using the dedicated software mDraw, we “print” the desired path designed with any vector graphics software. Playing with both temperature and displacement speed of the head, one can simply seal or additionally cut along the path. The envelopes obtained are then connected to the compressed air of the laboratory and inflated. The pressure is then set at typically 0.1 bar to ensure that we remain in the regime of interest (quasiinextensible, compression modulus negligible) for our structures with a width on the order of 10 cm, thickness t of typically 10 μm, and Young modulus E of the order of the gigapascals. Cross-sections are measured by drawing a radial line on a transparent Mylar balloon; a photograph from the side is then taken, and the line is extracted.

## Acknowledgments

This work was partially funded by the French Agence Nationale de Recherche (ANR), project SMART. We thank Corrado Maurini for enlightening discussions.

## Footnotes

- ↵
^{1}To whom correspondence may be addressed. Email: emmanuel.siefert{at}espci.fr.

Author contributions: E.S. and B.R. designed research; E.S. performed research; E.S., E.R., J.B., and B.R. developed theory; E.S. analyzed data; and E.S., E.R., J.B., and B.R. wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission. T.A.W. is a guest editor invited by the Editorial Board.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1904544116/-/DCSupplemental.

Published under the PNAS license.

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