Geometry and mechanics of inextensible curvilinear balloons

Abstract

Mylar balloons are popular in funfairs or birthday parties. Their conception is very simple: two pieces of flat thin sheets are cut and sealed together along their edges to form a flat envelope. Inflation tends to deform this envelope in order to maximize its inner volume. However, although thin sheets are easy to bend and hardly resist compressive loads, they barely stretch, which imposes non-trivial geometrical constraints. Such thin sheets are generally described under the framework of “tension field theory” where their stiffness is considered as infinite under stretching and vanishes under compression or bending. In this study, we focus on the shape after inflation of flat, curved templates of constant width. Counter-intuitively, the curvatures of the paths tend to increase upon inflation, which leads to out of plane buckling of non-confined closed structures. After determining the optimal cross section of axisymmetric annuli, we predict the change in local curvature induced in open paths. We finally describe the location of wrinkled and smooth areas observed in inflated structures that correspond to compression and tension, respectively.

Introduction

Inflated structures present several advantages such as high stiffness-to-weight ratio, efficient storage, quick deployment and cost efficiency. They are widely used in a broad range of applications, such as lightweight tents, in which rigid poles are replaced by air beams, scientific ballooning (Pagitz, Pellegrino, 2007, Pagitz, Pellegrino, 2007), stent deployment in angioplasty (Serruys et al., 1994), ultralight deployable structures for space exploration (Pellegrino, 2014) and soft robotics (Hawkes, Blumenschein, Greer, Okamura, 2017, Rus, Tolley, 2015). Thin-walled inflated structures may be separated into two mechanical regimes depending on the magnitude of membrane strains induced by pressure. In a first regime, when p ~ Et/L (where p is the inflating pressure, E the Young modulus of the envelope material, t the typical thickness of the envelope and L the smallest typical size of a structure unit), the envelope is strongly stretched and typical strains are finite. This configuration is relevant to elastomers that can accommodate finite stretching without failure. This hyperelastic regime has been recently investigated in numerous studies concerning soft robotics (Shepherd, Ilievski, Choi, Morin, Stokes, Mazzeo, Chen, Wang, Whitesides, 2011, Siéfert, Reyssat, Bico, Roman, 2019), the propagation of bulges (Chater, Hutchinson, 1984, Kyriakides, Yu-Chung, 1991), or the shape and stability of toroidal membranes (Roychowdhury, DasGupta, 2015, Roychowdhury, DasGupta, 2018, Tamadapu, DasGupta, 2013). We note that such elastomer structures remain however intrinsically soft and cannot sustain their own weight at large scale, since the typical Young modulus of the material used is of the order of a few MPa. Large inflated structures thus belong to a second regime p ≪ Et/L, and can be separated in two other subcases depending on the ability of pressure forces to bend the envelope of the structure. In the first regime, stiff engineering structures, such as tanks and shells, are designed to sustain pressure without large deflections (Kiefner, 1973). In this case the typical deflection scales as δ ~ pL/(Et³/L³), where pL is the force (per unit width) applied by pressure, and Et³/L³ is the typical bending stiffness of a structure with length L (per unit width). This regime therefored holds when δ ≪ L, or equivalently p ≪ Et³/L³. In the other limit, p ≫ Et³/L³, the bending stiffness of the structure cannot sustain pressure loads.

The present study focuses on the doubly asymptotic regime Et³/L³ ≪ p ≪ Et/L, where the envelope is quasi-inextensible but infinitely bendable. In this regime, the envelope can accommodate compression at almost no cost by forming wrinkles or folds that may be viewed as a “decoration” (King, Schroll, Davidovitch, Menon, 2012, Steigmann, 1990). Following the seminal study of the shape of parachutes by Taylor (1963), theoretical work has been dedicated to the shape of inflated circular disks (Ligaro, Barsotti, 2008, Paulsen, 1994), polyhedral surfaces (Pak, 2006), and instabilities in scientific balloons (Deng, 2012, Pagitz, Pellegrino, 2007). Recent progress has been achieved in solving numerically the inverse problem, that is programming the envelope made of a collection of flat panels seamed together, in order to morph onto a target shape upon inflation (Skouras et al., 2014). Here, we experimentally and theoretically investigate the shapes of initially flat annuli and, more generally, of any curvilinear path such as balloon numbers or letters (Fig. 1a). This study generalizes a preliminary work presented in reference (Siéfert et al., 2019b), where the families of cross section in open and closed configurations were assumed to be identical.

Mylar balloons exhibit non-intuitive features that we wish to rationalize. As a striking result, the curvature of the outline of the structures tends to increase upon inflation: the initially open template of a number “9” balloon closes when inflated (Fig. 1). Radial wrinkles present in most of the envelope indicate compressive strains, which may appear counter-intuitive for a structure under pressure. Nonetheless, wrinkles are absent along two up-down symmetric bands that follow the outline of the structure and in the vicinity of the inner seam (red regions highlighted in Fig. 1b).

In this article, we aim at understanding the inflation-induced coiling of the outline, the shape of the cross sections, and the location of radial wrinkles (i.e. the strain distribution along the envelope). The structures are made from two superimposed identical curved, flat templates, seam-welded together along their boundary. We first derive the equations for the equilibrium shape of an inflated axisymmetric flat ring, using two methods : a variational volume maximization, or a direct force balance. The predicted cross-sections are quantitatively compared with the experiments on closed rings. We then extend our model to open circular rings and to arbitrary curved paths, where coiling is allowed, and relate the orientation and extension of wrinkles to the direction and location of compression. We finally solve the inverse problem, i. e. we find the flat reference state that will deform onto a target outline upon inflation, and test it on a few examples.