LETTERS
PUBLISHED ONLINE: 23 OCTOBER 2011 | DOI: 10.1038/NMAT3144
Hierarchical folding of elastic membranes under
biaxial compressive stress
Pilnam Kim1†‡ , Manouk Abkarian2,3‡ and Howard A. Stone1 *
Mechanical instabilities that cause periodic wrinkling during
compression of layered materials find applications in stretchable electronics1–3 and microfabrication4–7 , but can also limit an
application’s performance owing to delamination or cracking
under loading8 and surface inhomogeneities during swelling9 .
In particular, because of curvature localization, finite deformations can cause wrinkles to evolve into folds. The wrinkleto-fold transition has been documented in several systems,
mostly under uniaxial stress10–13 . However, the nucleation, the
spatial structure and the dynamics of the invasion of folds in
two-dimensional stress configurations remain elusive. Here,
using a two-layer polymeric system under biaxial compressive
stress, we show that a repetitive wrinkle-to-fold transition
generates a hierarchical network of folds during reorganization
of the stress field. The folds delineate individual domains, and
each domain subdivides into smaller ones over multiple generations. By modifying the boundary conditions and geometry,
we demonstrate control over the final network morphology. The
ideas introduced here should find application in the many situations where stress impacts two-dimensional pattern formation.
Thin, layered materials develop surface undulations or wrinkles
when they experience small compressive strain. This response is
the result of a complex interplay between deformation of the
top layer and its foundation14 . Indeed, wrinkles release in-plane
compression of the film, as bending is energetically less costly
than compression, and this is accompanied by bulk deformation
of the foundation material. The prediction of the wavelength
and the distribution of elastic energy of the wrinkles is therefore
non-trivial and depends on the mechanical properties, both of the
film and the viscoelastic substrate15 , as well as on the geometric
features of the stress field16,17 . However, on further compression, the
wrinkles become unstable and new morphological phases emerge
depending on the elastic nature of the foundation. For example,
for an elastic film floating on a liquid layer a gradual curvature
localization occurs, forming a single linear structure called a fold,
whereas for an elastic foundation13 self-similar wrinkling patterns5
as well as a cascade of period-doubling wrinkles11 have been
reported. Similarly, wrinkles can also evolve to yield cracks, as
observed in some layered semiconductor materials18 . However,
a systematic approach to elucidate the effect of the elasticity of
the foundation on the nucleation of such phase transitions is
not available. Moreover, to our knowledge, the dynamics of fold
proliferation is poorly understood, and in the case of a twodimensional system is unknown. Such systems are found in layered
geological strata19 , epithelial tissue20 , and functional and flexible
electrical devices1 . Therefore, to understand the morphological
transitions in two-dimensional configurations, we study an elastic
membrane on a viscoelastic substrate under a biaxial compressive
stress, visualize the dynamics of the formation of a network of folds,
which illustrates a distinct kind of pattern formation, and show how
it can be controlled.
The experimental system consists of a polymeric bilayer, with a
lower layer composed of a viscoelastic, partially cured pre-polymer
(20 µm thick), which is covered with a thin elastic crust cross-linked
by a plasma treatment (see Supplementary Fig. S1 and Methods).
Ion irradiation produced by the treatment isotropically expands the
newly formed crust, submitting it to an equi-biaxial compression,
with in-plane compressive (areal) strain ε (<0). We estimate the
strain in the crust using atomic force microscopy (AFM). We
observe that the thin film buckles into a sinusoidal deformation (a
wrinkle, Supplementary Fig. S2) of wavelength λ0 for strains larger
than a critical strain εc , that is |ε| > εc = 0.52(Es /Ef )2/3 , where Ef
and Es are the elastic moduli of the viscoelastic substrate and of the
thin crust, respectively7 . In Fig. 1a, we summarize the evolution of
the wrinkles as the strain increases. The wavelength of the initial
wrinkles, λ0 , is approximately7
λ0 ≈ 2πh(Ef /3Es )1/3
where h is the thickness of the crust. We find that the amplitude A of
the wrinkles gradually increases with increasing strain proportional
to |ε|1/2 as predicted theoretically17 :
√
1+ν p
A
≈
|ε| − εc
λ0
π
By further compression of the crust, for strain values |ε| >
∼ 0.06,
we observe non-uniformity in the surface deformation in that
the wrinkle amplitude varies markedly over the surface. However,
in contrast to the previous studies for uniaxial deformation that
observed a progressive evolution of wrinkles to folds11,13 , we observe
an abrupt transition for strain values |ε| >
∼ 0.2: the wrinkles become
unstable and the strain localizes into folds in which growing
wrinkles make self-contact (Supplementary Movie S1). Such a sharp
wrinkle-to-fold transition has been observed in other thin films
floating on water, such as gold nanoparticle layers under uniaxial
stress10 and thin elastic membranes under axisymmetric loading21 .
Using scanning electron microscopy (SEM), we show that folds
invaginate the viscoelastic supporting layer (Fig. 1b).
1 Department
of Mechanical and Aerospace Engineering, Princeton University, Princeton, New Jersey 08544, USA, 2 Université Montpellier 2, Laboratoire
Charles Coulomb UMR 5221, F-34095, Montpellier, France, 3 CNRS, Laboratoire Charles Coulomb UMR 5221, F-34095, Montpellier, France. † Present
address: Department of Materials Science and Engineering, Korea Advanced Institute of Science and Technology, Daejeon 305-701, Republic of Korea.
‡ These authors contributed equally to this work. *e-mail: [email protected].
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NATURE MATERIALS DOI: 10.1038/NMAT3144
LETTERS
a
b
Actual area of wrinkled surface: S0
Fold
Projected area: S
0.30
ε ≈ 2 (S ¬ S0)/S0 = 2ΔS/S0
0.25
5 μm
⎪ε ⎪ ≈ 0.14
A /λ0
0.20
2A
⎪ε ⎪ ≈ 0.05
0.15
c
1.4 μm
Realignment of wrinkles
30 μm
0.5 μm
2A
30 μm
0.10
0.05
0
d
Fold
Non-uniform growth
Uniform growth
0
0.05
Ef /Es ≈ 104
10 μm
e
0.10
⎪ε ⎪
0.15
Ef /Es ≈ 103
10 μm
0.20
f
25 μm
Ef /Es ≈ 300
5 μm
g
Ef /Es ≈ 100
5 μm
Figure 1 | Morphological transition through fold localization. a, Experimental measurements of the aspect ratio of a wrinkle (A/λ0 ) as a function of the
√
strain ε. The solid line represents A/λ0 ≈ 1 + ν/π (|ε| − εc )1/2 . Inset: the line profiles obtained by AFM analysis show uniform (|ε| ≈ 0.05) and
non-uniform (|ε| ≈ 0.14) amplitude of wrinkles. Note that we estimated the strain in the crust using AFM to measure the actual area, S0 , and change in
area, 1S, of the wrinkled surface, 1S = S − S0 < 0, with S the projected area of the measured surface (inset). Because of material expansion, the relative
change of area 1S/So ≈ (1 − ν)ε for small values of the strain, with ν the Poisson ratio of the cross-linked film. Compliant materials such as that used here
are almost incompressible, with Poisson ratio ν ≈ 1/2, and therefore ε ≈ 21S/So . b, Representative cross-sectional SEM image of a localized fold. c, SEM
image of co-existing wrinkles and a fold (|ε| >
∼ 0.2). The fold evolves in a region of disordered wrinkles. As the fold grows, the stress becomes re-oriented
perpendicular to the direction of folding within a region around the fold tip. d–g, AFM images showing the effect of the ratio of Young’s moduli (Ef /Es ) on
2
< 3
the morphological transition during fold localization (|ε| >
∼ 0.2). For 10 < Ef /Es ∼ 10 , wrinkles remain throughout the deformation process and a
4
multi-periodic pattern is observed (e,f), whereas in the case of Ef /Es ≈ 10 , the strain localizes into a single fold, which relaxes wrinkles in its
2
neighbourhood (d). In the case of Ef /Es <
∼ 10 (g). Almost every wrinkle is transformed in one fold.
Once a fold appears, the stress field is locally modified within
a zone-of-influence that extends a distance several times further
than the length of the fold (Fig. 1c). Within this lobed region,
wrinkles are parallel to each other and perpendicular to the
growing fold. The modified wrinkles in the zone-of-influence are
a strong function of the properties of the supporting layers, that
is, the ratio of Young’s moduli (Ef /Es ). In fact, in the case of
3
a relatively rigid supporting layer, 102 < Ef /Es <
∼ 10 , wrinkles
remain throughout the deformation and a multi-periodic pattern
is observed at large values of the strain (|ε| >
∼ 0.2), as shown
in Fig. 1e,f. This response is reminiscent of the multiple-lengthscale elastic instability observed in the case of bilayer systems
under uniaxial stress5,11 . In contrast, by using a viscoelastic
4
polymeric layer (Ef /Es >
∼ 10 ), we do not observe a regime where
wrinkles evolve into a multi-periodic pattern, but rather the
strain localizes into single folds, which relax wrinkles in their
neighbourhood, as shown in Fig. 1d. Such local behaviour is similar
to the finite deformations of thin floating films12,13 . This feature
results from the softness of the underlying foundation, allowing
dynamical reorganization of stress fields in the film. Finally,
2
when Ef /Es <
∼ 10 (Fig. 1g), most of the wrinkles are individually
transformed into short folds, which is similar to the surface creasing
2
instability observed at the surface of bulk elastic materials22,23 .
Therefore, we observed that the viscoelasticity plays a crucial role
both in the morphological and dynamical features observed for
large deformations (see further discussion in the Supplementary
Information). In the rest of the paper we focus on the regime
where Ef /Es ≈ 104 .
Folds nucleate simultaneously over the entire surface during
continuous compression (Fig. 2a). They propagate along the surface
and connect to other folds (Fig. 2b), thus producing a first
generation of a network with well-identified closed domains
(Fig. 2c). At a location where the orientations of wrinkles intersect,
that is, a disclination defect, we observe that during the propagation
of an individual fold a tip bifurcates into two daughter folds
(Fig. 2d–g). In other words, topological defects of the wrinkle field
interrupt the longitudinal propagation of folds, which temporally
stagnate and/or split. We observe that the bifurcation angle
peaks near 60◦ (see Supplementary Fig. S3). Such defect-triggered
bifurcations produce a first generation of a network with irregular
polygonal domains.
Once an initial closed network is created, a second generation
of aligned wrinkles appears in individual domains (Fig. 3a), which
indicates that the stress field has been reorganized. As it is well
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NATURE MATERIALS DOI: 10.1038/NMAT3144
a
LETTERS
b
Nucleation of folds
t=9s
100 µm
c
Growth of folds
t = 14 s
100 µm
Formation of network
t = 34 s
100 µm
Time
d
e
f
Fold
g
Reorganization
of wrinkles
Wrinkle fields
Bifurcation
Disclination
defect
Disclination
defect
Fold
Bifurcation
Fold
Fold
t=1s
t=3s
20 µm
t=6s
20 µm
20 µm
Figure 2 | Formation of the initial network of folds and bifurcation of a fold. a–c, The evolution of folds including a, nucleation, b, growth, and c, the
formation of a network. Eventually, closed domains are created on the film (Supplementary Movie S2). d, Schematic representation showing the
mechanism of fold bifurcation. When a fold tip meets a disclination defect, which corresponds to the intersection point of different orientations of the
wrinkle field, the tip often bifurcates into new directions. e–g, Optical images show one sequence of the bifurcation of a fold.
a
b
200
160
Elliptical
Buckled
Length (μm)
⎪σ xx⎪<⎪σ yy⎪
σ xx
σ xx
σ yy
σ xx
⎪σ xx⎪>>⎪σ yy⎪
(v)
20 μm
40 μm
(v)
120
Tip B
80
(i)
200 μm
(ii)
Tip A
0
100
(iv)
(ii)
100 μm
Tip A
40
(iii)
(i)
(iii)
Tip B
150
(iv)
200
250
Time (s)
Figure 3 | Dynamic reorganization of stress fields and tip–tip interactions in a domain. a, The second generation of wrinkles arises from the stress field
due to the initial fold, which reflects the stress distribution at the level of an individual domain. Inset: sketch defining the stress-relaxation mechanism for
the fold propagation in a domain level. Note that the x-direction is along the fold (red arrow), and the y-direction is transverse. When the stage of the stress
field in a domain has |σxx | < |σyy |, a new fold propagates in the x-direction, releasing the stress parallel to the fold direction (σyy ). Thus, the local stress field
is dynamically modified such that |σxx | |σ yy |. b, An example of the interaction between an elliptical-shaped tip and the buckled tip. The velocity of the
buckled tip A (ii) is decreased until the active tip B propagates and releases the dominant strain in a domain (see Supplementary Movie S3).
known that the orientation of wrinkles is determined by the local
stress distribution17,24 , we can rationalize the dynamical stress
field by observing the wrinkle field. The new wrinkles appear
perpendicular to the pre-existing folds that outline the border of
each domain. We denote the axis of the fold as the x-direction,
with y the perpendicular direction. When the state of the stress
field in front of a fold has |σxx | < |σyy |, then the fold propagates
along the x-direction to release the stress (inset in Fig. 3a). This
second wrinkle-to-fold transition further subdivides each domain.
At this stage, a well-ordered evolution of multiple folds frequently
occurs within individual domains. We observe two different types
of tips of a growing fold (inset in Fig. 3b (top)): a propagating
elliptical-shaped tip (tip A), which has parallel wrinkles aligned
on either side of the tip, and a static buckled-shape tip (tip
B). The velocity of tip A eventually decreases, at which time
tip B begins to propagate and relaxes the local wrinkle field
(see Fig. 3b and also Supplementary Movie S3). We conclude
that the folds communicate with each other, which we consider
a form of self-regulation, as they produce local modifications
of the stress field.
This self-regulation process creates a sequential and orderly
dynamics for domain division by repetitive wrinkle-to-fold transitions, and so produces a hierarchical pattern. We illustrate in
Fig. 4a how a single domain is successively divided into five smaller
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NATURE MATERIALS DOI: 10.1038/NMAT3144
LETTERS
a
1
b
2
500
TB
400
20 µm
3
20 µm
4
Number of domains
SB
Time
300
200
First generation
100
0
20 µm
20 µm
Second and
higher generations
0
50
100
150
200
Time (s)
250
300
350
Figure 4 | Hierarchical partitioning of space accompanied with terminal and segmental branching. a, An example of the sequence of the subdivision
process that characterizes the second and higher generations of domain formation. The time interval between each image is 1.5 s. A terminal branch,
having a Y-shape, crosses a domain, and subdivides it into three. Subsequently, a new segmental branch appears from the left domain, dividing it into two.
The right-hand domain division is initiated by the tip connecting to the central fold, which indicates that neighbour domains are mechanically correlated.
Finally, new folds are connected to the pre-existing folds of the initial domain, which completes the subdivision (also see Supplementary Movie S4). b, The
total number of domains as a function of time. In the initial stage, the increase in the number of domains is not significant because the folds are growing
and forming a closed network (1st generation). After t ≈ 50 s, the number of domains increases linearly in time.
a
b
200 µm
c
200 µm
200 µm
Figure 5 | Geometry-dependent networking process. a,b, Isotropic wrinkle pattern gives rise to irregular-shaped domains in a hierarchical network,
whereas folds localize into regular, almost-rectangular domains from an initial well-oriented wrinkle pattern (also see Supplementary Movie S5). c, The
striated pattern of the network obtained when the experiment is performed on a curved surface. An initial fold propagates along the axial direction, and
then the next generation of folds is aligned perpendicular to the initial folds.
ones. An initial fold splits at its tip into a terminal branch (TB),
which refers to a Y-shaped bifurcation from one into two folds.
Meanwhile, the lower-left subdomain in the figure is divided into
two by a segmental branch (SB or T-shaped branch), which refers
to the nucleation of a fold from the border of a domain. The
intersection of this last fold with a border initiates a subsequent fold
that divides the upper-right subdomain. After completion of the
initial (first generation) network (t ≈ 50 s), in the next branching
generations the number of domains in the network increases
approximately linearly with time (Fig. 4b), which is consistent with
the self-regulation process described above.
The space-division process observed here resembles the growth
of crack patterns in thin films of desiccating mud25 or thin layers
of colloidal suspensions26,27 . Both cracks and folds release the
stress locally and anisotropically, and intersect with each other
because of the tensorial nature of the stress field, which results
in a hierarchical space-dividing pattern28 . We note that recent
numerical simulations tested the validity of such a mechanism of
pattern formation under compression and obtained hierarchical
reticulated networks29 , which is consistent with our observations.
Nevertheless, folding differs from cracking in several aspects.
For example, a crack remains unchanged after being formed
4
and is therefore not affected by cracks formed at later times.
In contrast, the folding pattern reorganizes in time because
the domains are mechanically linked. For example, the angles
between folds are the result of a balance of forces, which
changes slowly in time both because of material expansion and
the connection, or nucleation, of new folds (see Supplementary
Fig. S4). Also, two folds that approach tip to tip can coalesce
because they can modify and share wrinkles in front of them
(see Supplementary Fig. S5), whereas facing cracks are repelled
because the local stress intensity at the crack tip is high enough to
inhibit propagation30 .
We suggest that the morphology of the final network is
influenced by the initial conditions. Here we examine two different
types of initial conditions: randomly oriented regions of a few
parallel wrinkles, and aligned wrinkles coming from a few preexisting folds (see Supplementary Movie S5 and Methods). In the
former case, after several subdivisions, folds form irregular domains
(Fig. 5a), whereas the latter case produces a reticulated pattern with
a large fraction of rectangular domains (Fig. 5b).
To interpret such a dependence on the initial state, we recall the
wrinkle-to-fold transition that produces the initial network. In the
case of the equi-biaxial stress, folds nucleate widely over the entire
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NATURE MATERIALS DOI: 10.1038/NMAT3144
surface and bifurcate at disclination defects of the wrinkle field,
which produce many irregular polygonal domains. We measure
the distribution of angles between the folds, which is very broad,
spanning 40◦ –120◦ (Supplementary Fig. S6). In contrast, the aligned
wrinkle field triggered by the original folds localizes into a new
generation of folds that grow at right angles to the border, resulting
in T-shaped connections (that is, segmental branches). Thus, the
angle of the initial polygonal domains is near 90◦ . As a further
example of the importance of geometry on the final pattern, we
deposited a polymeric film on a curved poly(dimethylsiloxane)
(PDMS) slab (see Methods). Under compressive stress, we observe
the development of a striated pattern of folds in which the major
longitudinal folds lie parallel to the axis of the cylindrical geometry
and are connected transversally by minor commissural folds
(Fig. 5c). These observations indicate that the final morphology of
the network pattern formed after many generations of subdivisions
may reflect the geometrical features of the initial folds as the
partitioning process yields a hierarchy in space and time.
Our findings could influence the study of the dynamics of
elastic instabilities and bring a better understanding to other
layered geological, biological and material systems where strain
localization may play a crucial role. For instance, it has not
escaped our notice that the hierarchical pattern we obtain has
similarities to the venation network patterns of leaves. This
observation indicates the possible role stress-release mechanisms
could play to provide signalling at the cellular level, guiding
further differentiation and growth of veins in specific directions.
Nevertheless, we recognize also the role played by chemical
signalling (the growth hormone auxin) in the development of
veins, and at this time it is unknown whether both chemical and
mechanical signals could play coupled and complementary roles in
hierarchical patterning during venation.
Methods
The multi-layer system was fabricated from a ultraviolet-curable polyurethane
polymer (NoA 71, Norland Products). A drop of polymer solution was dispensed
on a cleaned glass substrate and a PDMS elastomeric blanket was placed on top
of the polymer film, followed by ultraviolet exposure (315–400 nm, 72 W) for
15 s (Supplementary Fig. S1). The PDMS blanket/slab is brought into contact to
yield a flat and uniform liquid film approximately 20 µm thick. When the film was
covered with a highly permeable thick PDMS slab, oxygen was continually diffused
into the liquid film interface from the PDMS because of the high permeability to
air (∼3.6 × 10−2 mm2 Pa−1 s−1 ; ref. 31). It is well known that a photo-cross-linking
process is inhibited by oxygen. Hence, the top of the liquid film could be partially
cured by controlling the ultraviolet exposure time, which allows the film to form a
partially cured viscoelastic layer.
Next, we used a corona discharge on the surface, not only to form a thin elastic
crust, but also to swell the constrained crust layer biaxially, which results from the
cross-linking of pre-polymer by the plasma treatment. Unless stated otherwise,
the corona discharge is maintained for the entire time of each experiment, which
keeps the top layer in an approximately uniformly increasing state of strain. For
the corona discharge experiments, a hand-held discharge unit (Electro-Technic
Products, model BD-20AC) was used, with an output voltage of ∼25 kV. If a
rubber-like polymer is placed under the corona discharge, the charged particles
that are generated deposit on the surface, where they initiate radical reactions
that modify the rubber surface by creating functional groups (such as –COH,
–COOH, –CO, –C–O–C, and so on; ref. 32). Thus, we believe that ion irradiation
by corona discharge of the prepared viscous/viscoelastic film slightly modifies its
mechanical properties both by introducing new cross-linking on the top of the
film and by annealing, which generates an elastic crust. Moreover, the continuous
ion irradiation tends to expand the film, which produces the biaxial compressive
stress in the film layer, as it is known to do in other materials at higher ion
energies33 . By varying the sample exposure time to initial ultraviolet radiation,
the degree of cross-linking of the pre-polymer can be precisely controlled, which
enables us to control the elastic modulus of the supporting layer. Hence, we obtain
wrinkles with various ratios of Young’s moduli (Ef /Es ) from 102 −104 . Ef /Es can be
estimated from the measured critical strain εc for the onset of wrinkles, according
to εc = 0.52(Es /Ef )2/3 .
To produce an initially aligned wrinkle pattern, we carry out a further step
before applying the uniform plasma exposure over the entire surface. If the
exposed surface is far from the source of the plasma (corona discharge), the stress
field is no longer uniform and thus creates a few radial and quasi-parallel linear
LETTERS
folds. Once we observe these folds, we stop the plasma and bring the source close
into the region with linear folds, which results in well-aligned wrinkles over the
entire surface (Fig. 5b).
To introduce the effect of curvature on folding, we modified our set-up
by producing the film on a curved thin PDMS slab (width (w) × length (l):
2 × 2 cm). The PDMS slab was coated with a thin partially cured pre-polymer
and then modified into a cylindrical geometry by bending (bending ratio,
4 = (w0 − w)/w0 = 0.5). After fixing the sample on a holder, we applied a plasma
treatment on the surface of the thin polymer film inside the cylindrical PDMS slab.
Received 22 June 2011; accepted 13 September 2011;
published online 23 October 2011
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Author contributions
P.K., M.A. and H.A.S. designed the concepts. P.K. and M.A. carried out the
experiments. P.K., M.A. and H.A.S. discussed and interpreted results. P.K., M.A. and
H.A.S. wrote the paper.
Additional information
Acknowledgements
This work was supported by the Multidisciplinary University Research Initiative (MURI)
program project funded through the Army Research Office, the National Research
Foundation of Korea Grant (NRF-2009-352-D00034) and the Centre National de
6
la Recherche Scientifique. M.A. is particularly grateful for the support offered by
H.A.S. during his stay.
The authors declare no competing financial interests. Supplementary information
accompanies this paper on www.nature.com/naturematerials. Reprints and permissions
information is available online at http://www.nature.com/reprints. Correspondence and
requests for materials should be addressed to H.A.S.
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© 2011 Macmillan Publishers Limited. All rights reserved.