Interplay of gross and ne structures in strongly-curved sheets Mengfei He1 2Vincent D emery3 4and Joseph D. Paulsen1 2 1Department of Physics Syracuse University Syracuse NY 13244

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Interplay of gross and ﬁne structures in strongly-curved sheets

Mengfei He,1, 2, ∗Vincent D´emery,3, 4 and Joseph D. Paulsen1, 2

1Department of Physics, Syracuse University, Syracuse, NY 13244

2BioInspired Syracuse: Institute for Material and Living Systems, Syracuse University, Syracuse, NY 13244

3Gulliver, CNRS, ESPCI Paris, PSL Research University, 10 rue Vauquelin, 75005 Paris, France

4Univ Lyon, ENS de Lyon, Univ Claude Bernard Lyon 1,

CNRS, Laboratoire de Physique, F-69342 Lyon, France

Although thin ﬁlms are typically manufactured in planar sheets or rolls, they are often forced

into three-dimensional shapes, producing a plethora of structures across multiple length-scales.

Existing theoretical approaches have made progress by separating the behaviors at diﬀerent scales

and limiting their scope to one. Under large conﬁnement, a geometric model has been proposed

to predict the gross shape of the sheet, which averages out the ﬁne features. However, the actual

meaning of the gross shape, and how it constrains the ﬁne features, remains unclear. Here, we

study a thin-membraned balloon as a prototypical system that involves a doubly curved gross shape

with large amplitude undulations. By probing its proﬁles and cross sections, we discover that

the geometric model captures the mean behavior of the ﬁlm. We then propose a minimal model

for the balloon cross sections, as independent elastic ﬁlaments subjected to an eﬀective pinning

potential around the mean shape. This approach allows us to combine the global and local features

consistently. Despite the simplicity of our model, it reproduces a broad range of phenomena seen

in the experiments, from how the morphology changes with pressure to the detailed shape of the

wrinkles and folds. Our results establish a new route to understanding ﬁnite buckled structures

over an enclosed surface, which could aid the design of inﬂatable structures, or provide insight into

biological patterns.

Complex patterns of wrinkles, crumples and folds can

arise when a thin solid ﬁlm is stretched [1–3], com-

pacted [4–7], stamped [8–10], or twisted [11–13]. These

microstructures arise to solve a geometric problem: they

take up excess length at a small scale to facilitate changes

in length imposed at a larger scale, by boundary condi-

tions at the edges or by an imposed metric in the bulk.

When the conﬁning potential is suﬃciently soft, like that

presented by a liquid, the sheet can have signiﬁcant free-

dom to select the overall response that the small scale fea-

tures decorate [11,14–20]. Understanding how gross and

ﬁne structures are linked, especially in situations with

large curvatures and compression, remains a frontier in

the mechanics and geometry of thin ﬁlms.

To date, the dominant approach has been to treat gross

and ﬁne structure separately. For example, tension-ﬁeld

theory [21] accounts for the mechanical eﬀect of wrin-

kling on the stress and strain ﬁelds, while ignoring the

detailed deformations at the scale of individual buck-

les. Recent work [3] has established how to calculate

the energetically-favored wrinkle wavelength anywhere

within a buckled region. But owing to geometric non-

linearities, these approaches have been limited to situ-

ations with small slopes and a high degree of symme-

try, and they assume small-amplitude sinusoidal wrinkles

at the outset. A geometric model was recently devel-

oped [16,22] for situations where an energy external to

the sheet - like a liquid surface tension or gravity - ulti-

mately selects the gross shape. Although this model can

address situations with arbitrary slopes, it is typically ag-

∗mhe100@syr.edu

nostic to the exact form of the ﬁne structures. Moreover,

it is unclear what the gross shape precisely corresponds

to.

At the small scale, signiﬁcant progress has been made

on understanding oscillating buckled features by analyz-

ing an inextensible rod attached to a ﬂuid or solid founda-

tion [2,23]. This approach can predict the energetically-

favored wavelength of monochromatic wrinkles, but much

less is known about the selection of more complex or

evolving microstructures. Morphological transitions, like

the formation of localized folds, have been largely ana-

lyzed on planar substrates [24–27], and it is not known

how they are modiﬁed when the gross shape is curved

and can freely deform [14].

Here we elucidate the interplay of the gross and ﬁne

structures of a strongly deformed thin sheet by studying

a water balloon made by unstretchable thin membranes

(Fig. 1). We obtain a wide range of deformations by vary-

ing the internal air pressure and the volume of liquid in

the balloon. Despite the complex surface arrangements,

a simple geometric model omitting the bending cost cap-

tures accurately the mean, or the gross shape of the bal-

loon membrane measured from extracting the cross sec-

tions. The observed side proﬁle of a balloon, on the other

hand, may diﬀer from the mean shape due to surface ﬂuc-

tuations. We show how the ﬁnite wrinkle amplitude can

be accounted for and combined with the mean behav-

ior to predict the outer envelope of the wrinkled balloon,

in agreement with our experiments. To understand the

buckled microstructure in more detail, we model a cross

section of the balloon as an eﬀective ﬁlament pinned to

the prediction of the geometric model. Remarkably, our

parsimonious model quantitatively captures various ob-

arXiv:2210.00099v1 [cond-mat.soft] 30 Sep 2022

FIG. 1. (a) We make partially-ﬁlled water balloons by sealing

two circular sheets of radius rdisk that bend easily but strongly

resist stretching. We vary the air pressure inside the balloon,

which we measure using the height hwof a water column

in a U-shaped tube. We use cylindrical coordinates (r, z),

and we denote the arclength from the top of the balloon as

ls. (b) Side-view photograph of the setup with water volume

Vwater = 277.9 mL and air pressure p= 276 Pa.

served surface morphologies of the balloon, while retain-

ing the agreement between the mean shape and the pre-

diction of the geometric model. We measure the size

of the self-contacting loops at the tips of the folds that

form at high pressures, which further corroborates the

two-dimensional behavior of the cross sections as elastic

ﬁlaments. These results provide a paradigmatic example

of how to analyze gross shape and ﬁne structures in a

uniﬁed approach, for strongly-curved sheets.

I. EXPERIMENTAL SETUP

We construct closed membranes by sealing together

disks of initially-planar plastic sheets. The disks are cut

from polyethylene produce bags of thickness t= 8.0±0.8

µm and Young’s moduli varying from E= 315 MPa to

E= 1103 MPa as a function of the angle, or from plastic

food wraps with t= 10.2±0.8µm and Young’s moduli

varying from E= 133 MPa to E= 195 MPa as a function

of the angle. The measurements are detailed in SI. An

air-tight seal is formed by pressing a heated iron ring

of diameter 166 mm. The fused circular double layers

are then cut out, and a nylon washer of radius rtop is

glued to the center of the top layer for hanging the bag

and connecting a tube that can supply air and water

into the bag. The same tube is connected to a custom

manometer made from two plastic cylinders connected

by a U shaped tube; the diﬀerence in water height hw

allows us to measure the overpressure via the hydrostatic

pressure diﬀerence ρghw[Fig. 1(a)].

Once inﬂated, the balloon transforms from a ﬂat initial

state to a strongly curved global shape with a complex

arrangement of surface structures [Fig. 1(b)]. Side views

are photographed with a Nikon DLSR with back light-

ing from an LED white screen. Despite the anisotropy

of the Young’s modulus, we do not see large systematic

variations in the morphological behaviors as a function

of angle of loading.

We also measure horizontal cross sections of the bal-

loons by scattering a laser light sheet into the system

[Fig. 3a]. We photograph the scattered laser light from

the side at an oblique angle. A calibrated perspective

transformation is then applied to produce the ﬁnal hori-

zontal image. To better capture the back of the balloon,

we reﬂect a portion of the light sheet onto the back of the

system with two vertical mirrors. The above procedure is

sometimes repeated from diﬀerent azimuthal angles and

the results are superimposed, to reduce noise and better

capture the back of the balloon. The images captured

from diﬀerent angles collapse well on one another, in-

dicating that the cross sections are obtained accurately

without geometric distortion.

II. GEOMETRIC MODEL FOR GROSS SHAPE

To capture the side-view proﬁles of the balloons, we

use a simple geometric model [16,22,28] that idealizes

the balloon as a smooth, axisymmetric surface r(z) with

no surface ﬂuctuations [Fig. 1(a)]. This eﬀective surface

is free to bend and cannot carry compressional stresses

as they are relieved by the wrinkles and folds around

the balloon. Assuming the entire balloon is wrinkled so

that the circumferential tension vanishes everywhere, the

only remaining tension is the eﬀective longitudinal stress

Teﬀ =Tsls/r, where Tsis the physical longitudinal stress

in the balloon membrane, and lsthe meridian arc-length

from the top center to the point in question [Fig. 1(a)].

Force balance [21,29–33] for the eﬀective surface gives:

κT eﬀ =p(1)

∂r(rT eﬀ)=0,(2)

where κ=−r00/(1+r02)3/2is the curvature of the proﬁle,

and p=ρghwor ρg(hw−z) is the local pressure drop

across the membrane, above or below water. We set z= 0

as the water level.

Equation 1is the normal force balance, involving only

the curvature of the proﬁle due to the absence of az-

imuthal tension. Equation 2is the horizontal force bal-

ance, which immediately leads to

Teﬀ =f

r,(3)

where fhas the dimension of force. Substituting Eq. 3

into Eq. 1gives

−f

ρg

r00

r(1 + r02)3/2=ßhw0< z < ztop

hw−z zbot < z < 0.

(4a)

(4b)

We denote ztop,zbot as the top and bottom coordinates

of the balloon. Equations 4are then two second-order

ODE’s with three parameters f,ztop and zbot unknown

FIG. 2. (a-d) Side-view photographs of a balloon ﬁlled with Vwater = 100.4 mL of water at various internal pressures from

p= 128 Pa to 0 Pa. Image grayscale inverted for clarity. Green curves: Predictions from the geometric model (Eqs. 4) at

the corresponding pressures with no free parameters. The agreement is very good at high pressures, but there is a signiﬁcant

discrepancy at p= 0. (e) Pink curve: Adding the amplitude of the wrinkled envelope (Eq. 5) to the geometric model gives

good agreement with the side-view proﬁle of panel (d).

a priori, and hence should be supplemented with seven

boundary conditions: r(0+) = r(0−) and r0(0+) = r0(0−)

at the water level, r(ztop) = rtop,r(zbot) = 0 and

r0(zbot)→ ∞ at the top and the bottom of the bal-

loon, together with the sheet inextensibility constraint

rtop +Rztop

zbot √1 + r02dz= 2rdisk and a prescribed water

volume R0

zbot πr2dz=Vwater.

Equations 4are integrated numerically using odeint

implemented in the SciPy package integrate, and the so-

lutions, which we denote as rgm(z), are plotted in Fig. 2

over the corresponding images. This comparison is done

with no free parameters. There is an excellent agree-

ment between the numerical predictions and the appar-

ent shape of the balloon at high pressure, despite the

complex surface arrangements. However, the agreement

is poor at low pressure, as shown by the p= 0 case in

Fig. 2(d). To understand this apparent discrepancy at

low air pressures we turn to the investigation of the ﬁne

structure of the balloon.

III. CROSS SECTIONS

Cross sections at a ﬁxed vertical distance from the top

are colorized, and superimposed to highlight the morpho-

logical change as a function of pressure. [Fig. 3(a)]. The

image shows a transition in the ﬁne structure, starting

from folds at large pressure to wrinkles at lower pressure,

with an increase of the wavenumber.

For each contour, we identify the center of the balloon

with the centroid of the contour, and then measure rmean:

the mean distance between the contour and the center of

the balloon. Figure 3(c) compares the measured rmean

versus the predicted radius of the gross shape rgm in the

same plane. The agreement shows that the geometric

model accurately predicts the mean shape of the balloon

surface. This agreement is robust; the data include cross

sections at multiple vertical locations in the deﬂated bal-

loon, and diﬀerent pressures in the inﬂated balloons, and

we have also varied the membrane material and water

volume.

To quantify the diﬀerence between the mean shape of

the balloon and the apparent proﬁle, we measure the av-

erage protrusion dmean of each cross section, which we

deﬁne to be the average amplitude of the local maxima

of each contour. Figure 3(d) shows that dmean decays

rapidly with increasing pressure. Taken together, the

results in Figs. 3(c,d) resolve the apparent discrepency

at low pressure between the geometric model and the

side-view proﬁles in Fig. 2. Namely, the side-view pro-

ﬁle is given by the mean shape plus the amplitude of the

wrinkly undulations, rgm +dmean. Figure 3(d) shows that

these undulations can be rather large at zero pressure —

more than 10% of the mean — but they become much

smaller when there is an internal pressure. To quantify

the deviation dmean, and understand how it arises from

the particular curve shapes, we now study the p= 0 case

in more detail.

IV. WRINKLE SHAPE AT ZERO PRESSURE

Cross sections of a deﬂated water balloon measured

at equal vertical intervals above the seam are shown in

Fig. 4(a). Noting that the apparent proﬁle of the top

half of the balloon looks to be conical, we extrapolate

an apex of the apparent proﬁle from the side-view im-

age. We then rescale the cross sections by the distance

to this apex. This simple rescaling nearly collapses all

the cross sections [Fig. 4(b)], indicating that not only

the gross shape but also the ﬁne structure has the shape

of a generalized cone.

A consequence of a conical structure is that the wrin-

kle wavenumber does not vary signiﬁcantly with height z.

To understand this observation, we ﬁrst assume that the

selection of the wavelength is local [23] and dominated

FIG. 3. Cross-section model captures the morphological features of the balloon cross sections. (a) Cross-section shapes

measured by scattering a sheet of laser at a ﬁxed distance ztop −z= 39.4 mm below the top of a balloon with Vwater = 100.0

mL, which is gradually deﬂating from p= 278 Pa to 0 Pa. The photographs are tinted and superimposed to compare the

diﬀerent morphologies. (b) Shapes from the cross-section model. All physical parameters (pressure, bending modulus, length

of cross sections, volume of water) were set to match the conditions in panel (a), and a single η= 15 in Eq. 7was selected

by matching the wavenumber with experiment at high pressures. (c) The average distance of a cross section to its centroid,

rmean, versus the prediction of the geometric model, rgm , in the same z−plane. The data span various cross sections in diﬀerent

balloons made by produce bags (markers with black edges) or food wrap. Blue markers: experimental measurements. Red

markers: cross-section model. Filled markers: p= 0. Open markers: p > 0. Inset: wavenumber qversus pressure, showing

that the cross-section model captures the wrinkle-fold transition. (d) The average fractional protrusion of a cross section,

dmean/rmean, versus pressure. The cross-section model reproduces the trend in the experiments, where the data decay quickly

upon inﬂation. This trend explains how the geometric model can match the side-view proﬁles at high pressures [Fig. 2(a-c)].

by the tensional substrate stiﬀness [2]. Then the wave-

length would scale as: λ∼(B/T s)1/4rdisk1/2. On the

other hand, Tsvaries with z, and so should λ, leading to

a mismatching wavelength along the balloon meridian.

The resulting length scale `associated with a wavenum-

ber change scales as `∼λ2pTs/B [34]. Taken together,

`∼rdisk, which contradicts our assumption that the se-

lection of wavelength is local, and explains the invariance

of the wavenumber within the size of the balloon.

To study the shape of the individual cross sections,

we investigate the relations among the observables of

the mean maximum amplitude dmean, the wrinkle wave-

length and how much the cross section is compressed.

We deﬁne the material wavelength λs≡2πls/q, which

divides the total arclength of the cross section by the

number of undulations qmeasured by counting peaks.

We quantify the amount of compression by the eﬀec-

tive strain ≡(ls−rmean)/ls(to be contrasted with

the local material strain, which is vanishingly small for

our buckled ﬁlms). Note that λsapproaches the “usual

wavelength”, λ, when approaches 0. The particular

shape of the undulations tells how dmean,λs, and are

related. For example, small amplitude sinusoidal waves

obey dmean ∝√λsand for square waves dmean =λs/4.

To examine the relation in our system, we plot dmean ver-

sus λsin the inset to Fig. 4(c). Remarkably, the data

collapse onto a line that is ﬁt well by:

dmean ≈βλs,(5)

where βis the linear coeﬃcient with a best-ﬁt value of

0.30.

The simple relation of Eq. 5shows how to readily esti-

mate the amplitude of wrinkles dmean as a function of z.

Given a volume of water, a washer radius, and a bag size,

one may compute the arclength ls(z) and the mean radius

rmean ≈rgm(z) for the gross shape. The crucial ingre-

dient from the experiment is the observed wavenumber

q(which is approximately independent of z). With just

these parameters, one may then obtain (z) and λs(z),

which combine via Eq. 5to give dmean(z). This wrinkled

envelope can be added to the gross shape from the ge-

ometric model to yield a predicted apparent shape. We

do this in Fig. 2(e); the result matches the experimental

proﬁle very well, especially when compared to the geo-

metric model without the wrinkled envelope for the same

balloon, in Fig. 2(d). The correction continues to be fa-

vorable past the seam through the bottom of the balloon,

which oﬀers an explanation for the kink in the apparent

proﬁle at the location of the seam where the two disks

are heat-sealed together. Namely, this kink can be under-

stood as arising from the triangular peak in the amount

of material to be packed into the conﬁned gross shape,

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Interplayofgrossandnestructuresinstrongly-curvedsheetsMengfeiHe,1,2,VincentDemery,3,4andJosephD.Paulsen1,21DepartmentofPhysics,SyracuseUniversity,Syracuse,NY132442BioInspiredSyracuse:InstituteforMaterialandLivingSystems,SyracuseUniversity,Syracuse,NY132443Gulliver,CNRS,ESPCIParis,PSLResearchUnive...

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Interplay of gross and ne structures in strongly-curved sheets Mengfei He1 2Vincent D emery3 4and Joseph D. Paulsen1 2 1Department of Physics Syracuse University Syracuse NY 13244.pdf

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Interplay of gross and ne structures in strongly-curved sheets Mengfei He1 2Vincent D emery3 4and Joseph D. Paulsen1 2 1Department of Physics Syracuse University Syracuse NY 13244

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