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Proc. R. Soc. A (2012) 468, 932–953
doi:10.1098/rspa.2011.0567
Published online 23 November 2011
Folding wrinkles of a thin stiff layer
on a soft substrate
BY JEONG-YUN SUN1 , SHUMAN XIA3,† , MYOUNG-WOON MOON2 ,
KYU HWAN OH1 AND KYUNG-SUK KIM3, *
1 Department
of Materials Science and Engineering, Seoul National University,
Seoul 151-742, Republic of Korea
2 Interdisciplinary Fusion Technology Division, Korea Institute of Science
and Technology, Seoul 136-791, Republic of Korea
3 School of Engineering, Brown University, Providence, RI 02912, USA
We present the mechanics of folding surface-layer wrinkles on a soft substrate,
i.e. inter-touching of neighbouring wrinkle surfaces without forming a cusp. Upon
laterally compressing a stiff layer attached on a finite-elastic substrate, certain material
nonlinearities trigger a number of bifurcation processes to form multi-mode wrinkle
clusters. Some of these clusters eventually develop into folded wrinkles. The first
bifurcation of the multi-mode wrinkles is investigated by a perturbation analysis
of the surface-layer buckling on a pre-stretched neo-Hookean substrate. The postbuckling equilibrium configurations of the wrinkles are then trailed experimentally
and computationally until the wrinkles are folded. The folding process is observed
at various stages of wrinkling, by sectioning 20–80 nm thick gold films deposited on
a polydimethylsiloxane substrate at a stretch ratio of 2.1. Comparison between the
experimental observation and the finite-element analysis shows that the Ogden model
deformation of the substrate coupled with asymmetric bending of the film predicts the
folding process closely. In contrast, if the bending stiffness of the film is symmetric or
the substrate follows the neo-Hookean behaviour, then the wrinkles are hardly folded.
The wrinkle folding is applicable to construction of long parallel nano/micro-channels
and control of exposing functional surface areas.
Keywords: wrinkle folding; nonlinear deformation; bifurcation; post-buckling equilibrium;
gold film; PDMS
1. Introduction
Wrinkles of a thin stiff layer attached on a soft substrate, ranging from bio-cellular
to geological-crust wrinkles, have been observed widely in nature. The formation
process of such wrinkles has recently been recognized as an attractive means
of making two-dimensional self-organized patterns at multiple length scales for
various applications, such as flexible electronics, optics with pitch-controllable
*Author for correspondence ([email protected]).
† Present address: George W. Woodruff School of Mechanical Engineering, Georgia Institute of
Technology, Atlanta, GA 30332-0405, USA.
Received 17 September 2011
Accepted 26 October 2011
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Folding wrinkles
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gratings, and nanotechnology of adhesion, wetting and particle-sieving control
(Stafford et al. 2004; Genzer & Groenewold 2006; Chan et al. 2008; Rahmawan
et al. 2010). Most compression-induced wrinkles are formed by buckling of the
surface layer. In such buckling processes, very rich patterns of wrinkle clusters
can be produced by extensive material property combinations of the bi-material
system (Cerda & Mahadevan 2003; Efimenko et al. 2005; Moon et al. 2007).
Upon further compression, some of these clusters eventually develop into folded
wrinkles, i.e. inter-touching of neighbouring wrinkle surfaces without forming a
cusp. This wrinkle-folding process provides a mechanical means of concealing
and re-exposing partial surface areas, and can be used to construct long parallel
nano- and micro-channels. To date, the mechanics of such large-amplitude folded
wrinkles and the underlying formation mechanisms are still not well understood,
owing to the broad-scale nonlinear deformation characteristics entailed in the
wrinkle formation process.
Historically, interest in compression-induced wrinkles was motivated by the
buckling of the sandwich panels in aircraft structures (Gough et al. 1940; Hoff &
Mautner 1945). The buckling was then treated in the context of elastic stability
by Timoshenko & Gere (1961) and of incremental elasticity by Biot (1957, 1965).
Approximate structural analysis for the buckling of an elastic layer was well
summarized by Allen (1969). The simplified structural buckling analysis was
extensively used for various applications (Röll 1976; Bowden et al. 1998; Stafford
et al. 2004). A more complete analysis of elasticity for the buckling was provided
by Shield et al. (1994), which presented the exact formulation of the first-order
perturbation analysis for the wrinkle formation with incrementally small-strain
deformation of an isotropic substrate. A general incremental bifurcation theory
that incorporates the effects of boundary elasticity was presented by Steigmann &
Ogden (1997). Lee et al. (2008) applied the bifurcation analysis for determining
the critical load for the onset of wrinkling and the associated wavelength to
substrates in which the elastic modulus is an arbitrary function of depth. Recently,
a high-order perturbation analysis was carried out for stiff-layer wrinkles on a
neo-Hookean (Rivlin 1948) elastic substrate; however, the analysis was limited to
isotropic elastic perturbation with respect to the undeformed configuration of the
substrate, and to the single-mode wrinkle deformation with vanishing interface
shear tractions (Song et al. 2008).
While a variety of patterns have been observed under multi-axial compression
of the surface layer (Bowden et al. 1998; Chen & Hutchinson 2004; Huang et al.
2005; Moon et al. 2007), an array of parallel straight wrinkles formed by uniaxial
compression are regarded as generic wrinkles. Multi-scale clustering of smallamplitude generic wrinkles has been observed experimentally on a UV-cured
polydimethylsiloxane (PDMS) substrate (Efimenko et al. 2005; Huck 2005). The
UV treatment of PDMS produces a graded distribution of elastic stiffness near
the surface and the multi-mode wrinkles of small amplitude may be attributed
to graded stiffness effect. On the other hand, a thin film layer of distinct uniform
property on a soft substrate also produces multimode wrinkles as the amplitude
grows large. The multi-mode generic wrinkles of large amplitude are generated
by multiple bifurcations of the equilibrium wrinkle configuration. Formation of
such multi-mode generic wrinkles of large amplitude is needed to fold wrinkles.
However, the formation of the multi-mode generic wrinkles is insufficient to fold
wrinkles, and only some of the multi-mode generic wrinkles develop into folded
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J.-Y. Sun et al.
(a)
W
(b)
lpsW
h
(c)
(d)
lL
lW
Figure 1. Schematics of tracing the evolution of surface-layer wrinkles on a pre-stretched soft
substrate: (a) undeformed configuration of the substrate with width W ; (b) pre-stretched
configuration of the substrate with width lps W ; (c) deposition of the unstrained surface layer
on the pre-stretched substrate and (d) compression-induced buckling of the surface layer caused
by relaxing the substrate to the stretch ratio of l; the nominal compressive strain applied on the
surface layer is 3LC = (lps − l)/lps .
wrinkles as the amplitude grows larger. As will be shown in this paper, certain
material nonlinearities in large deformation kinematics are required to trigger the
wrinkle folding.
Large-amplitude wrinkles of a stiff surface layer are formed on a soft substrate
generally pre-strained in finite deformation. The finite deformation of the
substrate is highly nonlinear and induces elastic anisotropy for incremental
deformation. The large-amplitude wrinkling often bends the surface layer beyond
its elastic limit. In this paper, we address the effects of such nonlinear deformation
on the wrinkle formation from its inception to final folding. Figure 1 shows a
series of schematics for the wrinkling process of an initially un-stressed thin stiff
layer on a pre-stretched soft substrate. In these schematics, a long strip of a soft
polymeric (PDMS) substrate of width W (figure 1a) is pre-stretched in the width
direction to a stretch ratio lps (figure 1b) in plane-strain deformation. The prestretching is followed by deposition of a thin metallic (gold) film of thickness h
on the surface (figure 1c). Then, the substrate is released to a stretch ratio of
l < lps , inducing thin film buckling to form post-buckling wrinkles (figure 1d). In
our experimental study reported in §3, we measure the wavelengths and shapes of
the wrinkle patterns, at various levels of l, with atomic force microscope (AFM)
topography and scanning electron microscope (SEM) imaging of the wrinkle cross
sections exposed by focused ion beam (FIB) sectioning. In the experiment, we
employ the notion of nominal compressive strain 3LC = (lps − l)/lps applied to the
surface layer by variation of the substrate stretch from the pre-stretch lps to a
released state of stretch l.
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Folding wrinkles
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2. Bifurcation of a generic wrinkle on a neo-Hookean substrate
Buckling of a thin stiff layer on a pre-stretched soft substrate is analysed in this
section. We show how eigenmodes of the substrate deformation are involved in
buckling the surface layer under compression to form bifurcated wrinkle patterns
with two different wavelengths, and how the bifurcated states evolve as the
film is further compressed by shrinking the pre-stretched substrate. The soft
substrate is typically made of a non-porous rubbery material such as PDMS which
exhibits characteristics of incompressible finite deformation at room temperature
(Yoo et al. 2006). The nature of such incompressible finite deformation is well
epitomized by the neo-Hookean constitutive relation (Rivlin 1948), for example,
for uniaxial tension of a typical PDMS in the range of stretch ratio from 0.7 to
1.5. The eigenmodes of perturbed substrate deformations in a uniformly stretched
neo-Hookean nonlinear finite deformation are analysed first in the following.
(a) Analysis of the substrate deformation
We consider a plane-strain deformation of the substrate for the half space
of −∞ < X1 < ∞ and −∞ < X2 ≤ 0 in which vertical surface displacement is
perturbed by dL cos kX1 on a pre-stretched configuration of lateral stretch
vx1 /vX1 = l and vertical stretch vx2 /vX2 = 1/l. Here, the wrinkle wave number
k = 2p/L and the perturbation amplitude parameter d 1 are employed with L
the undeformed-configurational wavelength of the perturbation. Then, the generic
form of the mapping of the configuration x2 (X1 , X2 ) is assumed to be
X2
(2.1)
+ dL cos kX1 eakX2 ,
l
where a > 0 is a parameter to be determined from the equilibrium condition. The
incompressibility condition, up to O(d2 ), requires the mapping x1 (X1 , X2 ) as
x2 =
x1 = lX1 − adl2 L sin kX1 eakX2 .
(2.2)
Here, we assume these functional forms of perturbation since the finite prestretching of the substrate generates an incrementally anisotropic elastic state
(Steigmann & Ogden 1997) and the anisotropic linear elastic field in a twodimensional half space loaded by a sinusoidal traction can be expressed by
superposition of two elastic fields that are the same as the assumed perturbation
fields with two distinct a values (A. C. Pipkin 1978, unpublished data). Furthermore, as the hydrostatic pressure for incompressible neo-Hookean solids has
to be determined independently from the deformation kinematics, the pressure
distribution is assumed to be
p = p0 + dp1 cos kX1 eakX2
(2.3)
with p0 and p1 to be determined from boundary conditions.
The first Piola–Kirchhoff stress is given by
Tij = JFik−1 skj = QFijT − pFij−1 ,
(2.4a)
for the incompressible neo-Hookean deformation energy potential of
UN−H = Q(l12 + l22 + l32 − 3),
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(2.4b)
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936
J.-Y. Sun et al.
where li are the principal stretches, Q = E0 /3 with E0 the elastic modulus of
the neo-Hookean solid at li = 1, Fij = vxi /vXj the deformation gradient, and the
Jacobian J = det[Fij ] = 1 for incompressibility. Then, the deformation gradient
obtained explicitly from (2.1) and (2.2) is inserted into (2.4a) to have
p1 1
− d Qkal2 L + p0 kaL +
cos kX1 eakX2 + O(d2 ), (2.5a)
T11 = Ql − p0
l
l
T12 = −d(QkL + p0 ka2 l2 L) sin kX1 eakX2 + O(d2 ),
(2.5b)
T21 = −d(Qka2 l2 L + p0 kL) sin kX1 eakX2 + O(d2 ) and
Q
T22 =
− p0 l + d(QkaL + p0 kal2 L − lp1 ) cos kX1 eakX2 + O(d2 ).
l
(2.5c)
(2.5d)
0
The zeroth order (d = 0) solution is then obtained from (2.5) and T22
= 0 as
2
p0 = Q/l and
1
(0)
T11 = Q l − 3 .
(2.6)
l
With the above stress representations, the equilibrium equations up to the first
order are expressed as
p1 dk sin kX1 e akX2 + O(d2 ) = 0
(2.7a)
T11,1 + T21,2 = 2pQal2 (1 − a2 ) +
l
and
T12,1 + T22,2 = (−2pQ + 2pQa2 − lap1 )dk cos kX1 eakX2 + O(d2 ) = 0.
Then, equation (2.7) implies an eigenvalue problem of a as
2
al (1 − a2 ) 1/l 2pQ
O(d)
=
.
O(d)
p1
(a2 − 1)
−la
(2.7b)
(2.8)
The positive eigenvalues, a = 1 and 1/l2 , of equation (2.8) for non-diverging
solution and the corresponding eigenvector solutions provide p1 = 0 and p1 =
2p(1/l4 − 1)lQ, respectively. The two distinct eigenvalues, for l = 1, stand for
two distinct eigenstates of the perturbed deformation on the incrementally
anisotropic elastic state generated by the pre-stretched finite deformation. In
neo-Hookean solids the incremental modulus softens in the stretching direction
while it stiffens in the shrinking direction. For the degenerate case of l = 1, the
two eigenstates are rearranged by lima→1 (eakX2 , veakX2 /va) to those of decaying
in −X2 direction with ekX2 and X2 ekX2 , respectively, which have been used in
conventional analysis of isotropic linear elasticity (A. C. Pipkin 1978, unpublished
data). The two eigensolutions provide the first-order functional forms of the first
Piola–Kirchhoff stress components as
(1)
T11
1
1
2B
2
= l − 3 − 2pd A l2 + 2 ekX2 + 4 ekX2 /l cos kX1 ,
(2.9a)
Q
l
l
l
(1)
T12
1
2
= −2pd 2AekX2 + B 1 + 4 ekX2 /l sin kX1 ,
(2.9b)
Q
l
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937
Folding wrinkles
(1)
T21
1
B kX2 /l2
2
kX2
= −2pd A 2 + l e + 2 2 e
sin kX1
Q
l
l
(1)
T22
1
kX2
2
kX2 /l2
= 2pd 2Ae + B 2 + l e
cos kX1 ,
Q
l
and
(2.9c)
(2.9d)
where A and B are coefficients to be determined by the boundary conditions
at the interface between the surface layer and the substrate. The Cauchy
stress components sij = J −1 Fik Tkj , then, provide the expression of the traction
components at the interface as
(1)
s21 (x1 , 0)
1
kx1
= −2pd 2lA + l + 3 B sin
Q
l
l
(2.10a)
(1)
s22 (x1 , 0)
2A
1
kx1
= 2pd
+ 3 + l B cos
.
Q
l
l
l
(2.10b)
and
Corresponding displacement components are obtained from (2.1), (2.2) and
(2.8) as
(1)
u1 (x1 , 0) = −(Al2 + B)dL sin
kx1
l
(2.11a)
and
(1)
u2 (x1 , 0) = (A + B)dL cos
kx1
.
l
(2.11b)
(b) Plate-theory analysis of the surface-layer deformation
The expressions for the tractions and the displacements at the interface
in (2.10) and (2.11) have to be compatible with the deformation of the surface
layer. Although the compatibility relationship can be expressed exactly up to the
accuracy of the linear elastic description of the layer deformation (Shield et al.
1994), the simple theory of plate stretching and bending with single-axis curvature
is accurate enough and more conveniently applicable to layer deformations of
asymmetric bending to be discussed in the following sections. The compatibility
relationships of the interface tractions and displacements with respect to the
linearized layer deformation consistent with the first-order perturbation of the
substrate deformation are expressed as (Shield et al. 1994):
(1)
(1)
2 (1)
h 2 v4 u2 (x1 , 0) h v3 u1 (x1 , 0)
EL h
(1)
C v u2 (x1 , 0)
−
+ 3L
+ s22 (x1 , 0) = 0
2
(1 − nL2 ) 3
vx14
vx13
vx12
(2.12a)
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J.-Y. Sun et al.
and
EL h
(1 − nL2 )
(1)
(1)
v2 u1 (x1 , 0) h v3 u2 (x1 , 0)
−
2
vx12
vx13
(1)
− s21 (x1 , 0) = 0,
(2.12b)
where 3LC is the cross-sectional average strain of axial compression in the layer, and
EL the Young’s modulus and nL the Poisson’s ratio of the surface layer. Then,
as (2.10) and (2.11) are inserted into (2.12), we will have a highly nonlinear
eigenvalue problem of the normalized wave number k̄ for buckling as
⎡ 3
⎤
3LC k̄
3LC k̄
k̄ 2
2Q̂ k̄ 3
k̄ 2
1
k̄
−
− 2 + 3 + l Q̂ ⎥ ⎢ 3l4 − 2l − l2 + l
l 3l4 2l3
l ⎥ A = 0 (2.13)
⎢
2
2
⎦ B
⎣
0
k̄
1 k̄
1
1 k̄
Q̂
+
2l
Q̂
−
+
l
+
k̄ −
2 l3
l2 2 l3
l3
with k̄ = 2p/(L/h) and Q̂ = Q(1 − nL2 )/EL , where the characteristic equation gives
the buckling strain 3B
3LC → 3B =
lk̄ 4 + 4(1 + l2 )Q̂ k̄ 3 + 6l(1 − l2 )2 Q̂ k̄ 2
+12l4 (1 + l2 )Q̂ k̄ + 24l3 (1 + l4 )Q̂ 2
12l2 k̄{lk̄ + (1 + l2 )Q̂}
.
(2.14)
Since the stretch ratio l of the substrate and the stiffness ratio Q̂ between the
layer and the substrate are positive, all the coefficients of the k̄ polynomials
in the numerator of (2.14) are also positive, and thus the numerator is a
monotonically increasing fourth-order positive polynomial function of k̄, for
0 < k̄. The denominator of (2.14) makes 3B singular at k̄ = 0 and −(1 + l2 )Q̂/l,
having a single positive minimum in the domain of 0 < k̄. The minimum 3B is
the compressive strain at the onset of buckling, 3Bcr , and the corresponding k̄ B
determines the onset wave number of the wrinkle k̄ cr
B . By making the derivative
of (2.14) with respect to k̄ vanish, i.e.
7l 4 8(1 + l2 ) 3 6l 4
2l2
k̄ 5 +
k̄ −
2l + (l2 − 1)Q̂ k̄ 2
k̄ +
Q̂ 3 (1 + l2 )
Q̂ 2
Q̂
Q̂ 2
(2.15)
48l4 (1 + l4 )
3
4
k̄ − 24l (1 + l ) = 0,
−
Q̂(1 + l2 )
we have the unique positive root of (2.15) as the critical buckling (minimum)
wave number k̄ cr
B.
Figure 2a shows plots of (2.15) for the substrate stretch ratio l at which the
surface layer critically buckles with the wave number k̄ cr
B for various stiffness-ratio
Q̂ values. Equations (2.14) and (2.15) asymptotically converges, for a very high
stiffness surface layer on a very soft substrate, to
1/3
2 2 2
)
3
Q̂
(1
+
l
for Q̂ 1
(2.16a)
3Bcr →
2
6
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(a)
5
stretch ratio of the substrate, l
Folding wrinkles
4
Qˆ1 = 0.00001
Qˆ2 = 0.0001
3
Qˆ 3 = 0.001
2
Qˆ 4 = 0.01
Qˆ 5 = 0.1
1
0
0
0.6
0.8
1.0
0.2
0.4
cr
normalized critical buckling wave number, k¯ B
wavelength in deformed configuration, L*/h
(b) 1000
Q = 0.15 MPa
EL = 20 GPa
800
v = 0.3
Qˆ = 6.8 × 10–6 lps = 1.5
600
400
200
0
0
0.001
0.002
0.003
0.004
0.005
nominal compressive strain, e CL (l ps − l))l ps
Figure 2. Buckling points and the first-order perturbation prediction of wrinkle wavelengths of
a stiff thin surface layer on a stretched neo-Hookean substrate: (a) the substrate stretch ratio
l at which the surface layer critically buckles with the normalized wave number k̄ cr
B for various
stiffness-ratio Q̂ values; (b) solid curve: the first-order-predicted wavelengths of the post-buckling
wrinkles, measured in deformed configurations, with respect to the nominal compressive strain
3LC = (lps − l)/lps applied on the surface layer; dotted curve: trace of the critical buckling points
on the stretched substrate.
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940
J.-Y. Sun et al.
and
1
2
3
k̄ cr
B → l{6(1 + l )Q̂}
forQ̂ 1.
(2.16b)
Equations (2.16a) and (2.16b) show that the critical compressive strain 3Bcr for the
surface layer to be critically buckled at the stretch ratio l of the substrate scales
with {(1 + l2 )/2}2/3 , and the wavelength in the deformed configuration (L∗ = lL)
of the wrinkle with {(1 + l2 )/2}−1/3 . As an example, the roots of equations (2.14)
(solid line) and (2.15) (dashed line) are plotted in figure 2b for elastic buckling of
a porous gold film with nominal Young’s modulus of 20 GPa and Poisson’s ratio
of 0.3, attached on a PDMS substrate. Since we use PDMS in our experiment,
the deformation characteristics of which is close to those of the neo-Hookean
solid with Q = 0.15 MPa approximately up to l = 1.5, we plot the wavelength
L∗ in the deformed configuration as a function of the nominal compressive
strain 3LC = (lps − l)/lps near the buckling point. The dashed line shows the roots
of (2.15), which stands for the trace of the critical buckling points for different l.
The critical buckling wavelength looks constant in figure 2b; however, it is slightly
increasing for decreasing l (or increasing 3LC = (lps − l)/lps ). On the other hand,
the solid curve indicates the roots of (2.14), meaning the trajectory of the buckling
wavelength versus the nominal compressive strain during release of the substrate
strain from the prestretched state of lps = 1.5. As shown by the solid curve, the
buckling state bifurcates beyond the critical nominal compressive strain of 0.065
per cent. The wavelength of the primary buckling mode decreases as the nominal
compressive strain increases, while the long wavelength of the secondary buckling
mode grows rapidly as the compressive strain increases. As the wavelength of the
secondary mode diverges so quickly that most of the laboratory observations in
a single microscopic scale are limited to the primary buckling mode. However, an
observation of the bifurcation phenomenon in computational simulation of a thin
layer buckling on a neo-Hookean solid substrate was reported by Efimenko et al.
(2005) and Huck (2005). Such a bifurcation was also observed in experiments
by Shield et al. (1994). As shown in figure 2b, the wavelength of the primary
wrinkle quickly reduces as soon as the layer buckles and strains a little further
within a fraction of the critical compressive strain. Therefore, the measurement
of the critical buckling wavelength may not be accurate, if we rely only on
direct experimental observations in using (2.16b) to measure the stiffness of a
surface layer. However, the prediction of the post-buckling wavelength by the
first-order perturbation has limited accuracy beyond the buckling point. Highorder perturbation analysis is expected to improve the accuracy of the evaluation
of the post-buckling wavelength. In the following sections, evolution of the postbuckling large-amplitude wrinkles towards folding is investigated by the hybrid
method of experiment and finite-element analysis.
3. Experiments on evolution of large-amplitude wrinkles
The post-buckling wrinkle patterns were generated experimentally in a thin gold
film attached on a PDMS substrate by the following procedures. We prepared the
PDMS by mixing dimethylsiloxane monomers with Sylgrad 184 silicone elastomer
curing agent in a weight ratio of 15 : 1, followed by degassing in a vacuum chamber
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Folding wrinkles
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and curing at 80◦ C over 1 h. The PDMS samples were prepared in the size of
40.0 × 10.0 × 3.0 mm. As depicted in figure 1b, the PDMS sample was prestretched to lps = 2.1, using a screw-driven tensile device which could apply
the nominal compressive strain, 3LC = (lps − l)/lps , more than 50 per cent in
the resolution of 0.5 per cent. The gold film was then ion-coated on the prestrained PDMS using a plasma-ion coater (IB-3, Eiko) with 6 mA current, as
illustrated in figure 1c. The transmission electron microscope (TEM) image
shown in figure 5a reveals that the gold film was porous in the nanometre scale
and the grain size was in the range 10–20 nm. In our experiments, we used
samples with various film thicknesses ranging from 20 to 80 nm. The gold film
coated on the pre-stretched PDMS was compressed as the pre-stretched strain was
relaxed, generating periodic wrinkles grooved perpendicular to the compression
direction. The wrinkled morphology was observed and measured in situ with
optical and AFM (Digital Instrument), while the pre-stretch of the PDMS was
released, as shown in figure 1d. After the pre-stretch was completely released,
the PDMS was re-stretched back to the original pre-stretch, during which the
gold film returns to the strain-free state. While the gold film was compressed and
the amplitude of the wrinkles grew, the cross-sectional shapes of the wrinkles
were observed by sectioning the film with FIB (NOVA200, FEI) milling. The
cross-sectional shapes were recorded by the high-resolution scanning electron
microscope (HR-SEM) built in the FIB-system, at four different stages of the
nominal compressive strains, 12, 35, 41 and 46 per cent. Before being sectioned
by the FIB, the surface of the wrinkled thin film was pre-coated with carbon in
the thickness range 200–500 nm, followed by a platinum coating of 500–2500 nm
thickness to hold the shape of the wrinkle from relaxation and to prevent potential
damages of the film from gallium ion irradiation.
(a) Observations on evolution of multi-mode wrinkles
The optical-image sequence (a1)–(a6) in figure 3 shows evolving wrinkles
of a 66 nm thick gold film during a compression–decompression cycle. As the
pre-stretch of the PDMS was released a wrinkle pattern emerged as shown in
(a2), implying that the film already buckled at a nominal compressive strain
3LC less than 0.5 per cent. Non-uniformity of the wrinkles in (a2) indicates
that nucleation sites of buckling were randomly distributed and incoherent
phases of the buckles were merging together at the small nominal compressive
strain, presumably owing to fluctuations of the structural properties such as film
thickness, roughness of the substrate and variations of the stiffness of the film
and the substrate. Once the nominal compressive strain was increased to 7 per
cent, all the pitches of the wrinkles were coherently aligned as shown in (a3). At
this nominal compressive strain two bifurcated modes of wrinkles were observed;
the primary-mode wrinkles of 2.5 mm wavelength were nested in the enveloping
secondary-mode wrinkles of 30–40 mm wavelength. Such bifurcation was predicted
by the analysis in §2, and also observed experimentally and in computational
simulations by Huck (2005) and Efimenko et al. (2005). The aspect ratio of the
PDMS substrate width in stretching direction to its lateral span was too large
to produce plane-strain deformation, and the Poisson expansion of the 7 per
cent compression cracked the film at some places (not shown in the frame). In
(a4), the compressive strain was 40 per cent and the primary-mode wrinkles
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J.-Y. Sun et al.
5 µm
(a1)
Au
prestretched PDMS
(a2)
(a3)
(a4)
(a5)
(a6)
(b1)
(b2)
0.4
0.4
0
(b3)
1 2 3 4
0 1 2 3 4 5
0.4
0 2 4 6 8 10
Figure 3. Post-buckling wrinkle evolution of a 66 nm thick gold film attached on a PDMS
substrate initially pre-stretched to lps = 2.1 (optical micrographs): (a1) flat film at 3LC = 0 (i.e.
substrate l = lps ); (a2) wrinkles at 3LC = 0.005; (a3) primary (small wavelength) and secondary
(long wavelength) wrinkles at 3LC = 0.07; (a4) doubly paired primary wrinkles at 3LC = 0.40;
(a5) quadruply paired primary wrinkles at 3LC = 0.50; (a6) plastically deformed residual wrinkles at
3LC = 0: AFM topography of paired gold film wrinkles with (b1) 23, (b2) 30 and (b3) 44 nm thickness,
respectively, on a PDMS substrate at 3LC = 0.40 − 0.50 (units of the inset scales are in mm).
of a single-period evolved into paired wrinkles of dual period. As the film was
further compressed to 50 per cent, secondary pairing of paired wrinkles occurred,
activating the quadruple-wavelength mode as shown in (a5). After this point, the
compression of the film was released by re-stretching the PDMS substrate. When
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Folding wrinkles
(a) (i)
(b) 12
wavelength (mm)
Pt
C
Au
PDMS
(ii)
(iii)
943
10
single-period mode
dual-period mode
8
quad-period mode
L* L(1–e CL )
6
4
2
(iv)
5 mm
0
10
20
30
40
50
60
nominal compressive strain (%), e CL (lps − l))lps
Figure 4. Folding of wrinkles: (a) SEM images of the FIB-sectioned wrinkle cross sections at
(i) 3LC = 0.12, (ii) 3LC = 0.35, (iii) 3LC = 0.41 (wrinkle folding) and (iv) 3LC = 0.46 (partial unfolding),
of a 74 nm thickness gold film; (b) variation of the wrinkle wavelength with respect to the applied
compressive strain in the film of 74 nm thickness; the wavelength of the primary wrinkle varies
with L∗ = L(1 − 3LC ), implying wavelength locking in the undeformed configuration, caused by
plastic bending.
the applied compression was completely removed, the film was not fully recovered
to its original flatness. Instead, plastically deformed wrinkle patterns remained
as shown in (a6). The plastically deformed wrinkle patterns could not be totally
removed even after the film was further stretched to 10 per cent tensile strain.
The three frames of figure 3(b1), (b2) and (b3) show triple, double and
quadruple wrinkle pairs observed in gold films of 23, 30 and 44 nm thicknesses,
respectively, at nominal compressive strains in the range 40–50%. In the in situ
AFM measurements of wrinkle topography for a 30 nm thick gold film, a singleperiod wavy pattern of 1.3 mm wavelength was distinctly observed at 5 per cent
compressive strain, while the paired wrinkles of dual period emerged at 41 per
cent compressive strain. Fast Fourier transformation of the AFM topography
showed a clear spectrum split of the wavelength into 0.9 and 1.8 mm for the
paired wrinkles. At this point, the total arc length of the wrinkle measured
by the AFM topography was reduced substantially. The reduction of the AFMtraced arc length implied folding of the wrinkle which did not allow access of the
AFM tip to trace the arc length at the folded region. This observation prompted
us to section-wrinkled films with FIB at different stages of compression. Upon
complete removal of the applied nominal compressive strain in the film, the
residual deformation of the paired wavy pattern remained permanently owing
to irreversible plastic deformation.
Figure 4a shows the cross-sectional shapes of a buckled gold film of 74 nm
thickness cut along the loading axis by FIB at four different places for four
different levels of the nominal compressive strain (12, 35, 41 and 46%). The
exposed cross sections were imaged by SEM at an angle of 38◦ tilt with respect
to the normal of the cross-section, towards the surface normal of the undeformed
PDMS substrate. Therefore, actual amplitudes of the wrinkles are larger than
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J.-Y. Sun et al.
(a)
Pt
carbon
gold
PDMS
0.05 mm
(b)
(c) 1.6
M
M–
k
nominal stress (MPa)
O
uniaxial test
second-order Ogden model
neo-Hookean model
1.4
M+
1.2
1.0
0.8
0.6
0.4
0.2
0
0.2
0.4 0.6 0.8
nominal strain
1.0
1.2
Figure 5. (a) An S-TEM image of the cross section of a 50 nm thickness gold film showing an
asymmetric distribution of grain grooves, which creates asymmetric bending stiffness of the film.
(b) A schematic of the relationship between bending moment and curvature of the thin gold film.
(c) Experimentally measured nominal stress–strain curve of the PDMS substrate.
those shown in the figure by factor of 1/cos 38◦ . As shown in (i) of figure 4a,
the primary wrinkle at the nominal compressive strain of 12 per cent exhibits
single-period waviness. However, the curvature at the valley of the wrinkle is
considerably larger than that at the peak, insinuating that the film might have
asymmetric bending stiffness. This observation brought us about imaging the
cross-sectional configuration of the nano-grains in the gold film with a TEM,
which is shown later in figure 5a. As the compressive strain reached 35 per cent,
the curvature contrast was even enhanced as shown in (ii), and repeated pairs
of the wrinkles begin to bunch, activating the paired wrinkle pattern of dual
period. A close look reveals that the platinum coating layer applied prior to FIB
sectioning was cracked open in every other valley of the wrinkle. Implication
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Folding wrinkles
945
of this phenomenon will be treated in §4 in comparison with computational
results. The cross-sectional configuration of the wrinkles formed at the nominal
compressive strain of 41 per cent, shown in (iii), unveils that indeed the bunching
pairs of wrinkles touched each other to create periodic folding of wrinkles,
for which the AFM tip could not follow the whole cord of the film. As the
nominal compressive strain was further increased up to 46 per cent in (iv),
the energy-release capacity of the primary and secondary bending modes in the
folded film with dual period seemed to be saturated, and the increment of the
compressive strain was rather accommodated by bending the film with quadruplewavelength period. In this stage, every other fold of the film was opened by the
period-doubling mode of deformation.
Figure 4b shows the variation of the wrinkle wavelength in the 74 nm thick film
with respect to the nominal compressive strain. The variation was monitored
by in situ AFM measurements. The wavelength variation nearly followed the
substrate shrinkage rate, L∗ L(1 − 3LC ), implying that the primary wavelength
of the wrinkle in the undeformed configuration might be locked by plastic bending.
When the nominal compressive strain reached 35 per cent, the paired wrinkles
with dual period were produced; however, this dual-period mode of deformation
did not affect the variation trend of the primary period. In the same manner, when
the quadruple-period mode was set off, the wavelength variations of the primary
mode seemed to be hardly influenced, strongly suggesting that the primary
wavelength was set by plastic bending at an early stage of wrinkle development.
(b) Material characterizations
As discussed above, nonlinear deformation characteristics as well as nonuniformity and anisotropy of the substrate stiffness are critically related to
clustering and folding of surface-layer wrinkles. In addition, asymmetric bending
stiffness and/or asymmetric yield moment of the surface layer produce
bending configurations with high contrast of curvatures at the valley and the
peak of the wrinkle, which seem to promote folding of the wrinkle. The cause
of such asymmetry in the gold film with nanometre scale thickness was found to
be an asymmetric distribution of grain grooves on the top and bottom surfaces.
The cross section of a gold film of nominally 50 nm thickness, imaged by an STEM and shown in figure 5a, was porous and had asymmetric distribution of
grain grooves. It had more and deeper grain grooves on the gold/PDMS interface
side, presumably owing to growth history of the nano-grains during the deposition
process. When the film is bent to close the deep grain grooves the effective bending
stiffness is relatively higher than the bending stiffness to open the grooves, since
the groove surfaces contact each other as they are closed. In addition, the bending
to open the grooves will readily induce plastic yielding.
A schematic of the bending moment–curvature relationship is depicted in
figure 5b as a dashed curve. However, direct experimental measurement of the
moment–curvature relationship is difficult for such a thin film. We employed
a hybrid method of experiment and computation to evaluate the bending
characteristics by matching the experimentally observed shapes of the wrinkles
with computationally simulated ones. The effective Young’s modulus EL of
the porous gold film at the onset of buckling was determined as 20 GPa,
and the Poisson ratio nL was adopted as 0.3, by matching the shapes of
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946
J.-Y. Sun et al.
relatively small-amplitude wrinkles. On the other hand, the effective bending
stiffness for large bending curvature was set as K ± = x± EL h 3 /{12(1 − nL2 )}, where
x± is the asymmetry factor of bending, and the superscripts ± stand for
bending in the directions of closing and opening the interface grain grooves,
respectively. We identified x+ = 1 and x− = 0.1 for the ion-deposited gold film,
with finite-element simulations.
Figure 5c shows the experimental uniaxial stress–strain curve of the PDMS
substrate, together with the best fits of two hyperelasticity models. The secondorder Ogden (1972, 1998) model curve is fitted to the experimental data
with a Poisson’s ratio of 0.48. For the incompressible neo-Hookean model, the
experimental data points of nominal strain between 0 and 0.5 are used for the
data fitting. Both models are used in the present study. For the neo-Hookean
model, we represented the hyperelastic energy potential as expressed in (2.4b).
The stress–strain curve of a PDMS uniaxial tension test closely matches the neoHookean curve up to 50 per cent stretching with Q = 0.1487 MPa, as shown in
figure 5c. However, the stress–strain curve substantially deviates from the neoHookean curve beyond the nominal tensile strain of 0.5, and we employed the
Ogden (1972, 1998) model to match the deformation behaviour all the way to
the experimentally adopted value of pre-stretch, lps = 2.1. The general form of
the Ogden (1998) strain energy potential is given by
UOgden =
N
2mi
a2i
i=1
(l̄a1 i + l̄a2 i + l̄a3 i − 3) +
1
(J − 1)2 ,
D
(3.1)
where J = l1 l2 l3 is the volume ratio, l̄i = J −1/3 li are the deviatoric stretches,
and mi , ai and D are material parameters. By taking N = 2 and fitting the Ogden
potential to the uniaxial stress–strain curve over the whole strain range, we
obtain m1 = 0.0001440 MPa, a1 = 14.24, D = 1.263 MPa−1 , m2 = 0.2674 MPa and
a2 = 2.743.
4. Computational simulations of wrinkle folding: finite-element analysis
While the bifurcation of wavelengths at the first buckling point and the postbuckling evolution of relatively small amplitude wrinkles could be understood
with the perturbation analysis described in §2, we analyse folding of highly
nonlinear large-amplitude wrinkles of gold thin films on a PDMS substrate with
finite-element method in this section. The finite-element simulations are carried
out as a two-dimensional plane strain problem in ABAQUS/Standard using the
stabilized Newton–Raphson method. Figure 6 shows a schematic of the finiteelement model in its initial configuration. The film is modelled using 2-node
linear two-dimensional beam (B21) elements, and the substrate is meshed with
4-node bilinear generalized plane-strain quadrilateral (CPEG4) elements. In the
simulations, the substrate is first pre-stretched to lps = 2.1, with respect to the
original substrate width. In the mean time, the film undergoes an artificial thermal
expansion in the lateral direction, in order to maintain stress-free state while
deforming together with the substrate. Then, the film and the substrate are
compressed together as done for the experiments. During the first 0.5 per cent
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947
Folding wrinkles
gold thin film
h
PDMS substrate
H
2L
Figure 6. Schematic of the finite-element model, with a close-up view of finite-element mesh at
two different locations. In consideration of the symmetry and periodicity of the problem, a twodimensional unit cell with a period of 2L is used to simulate primary, doubly paired and quadruply
paired wrinkling modes. Here, L = 2.14 mm is the wavelength of the primary wrinkling mode in
the undeformed configuration. The PDMS substrate in the simulations has a height of H = 60 mm,
and the gold thin film has a thickness of h = 74 nm. The substrate is meshed with a non-uniform
rectangular grid, which has a close-to-square shape and a fine grid size (approx. L/150) near the
top wrinkling surface.
compressive strain, the film is guided by displacement boundary conditions to
deform it into a buckling mode. After the system is relaxed by removing the
boundary conditions on the film, the system is further compressed to desired
compressive strain of 3LC . During compressive loading, small damping forces are
added to the global equilibrium equations for stabilizing the wrinkling process.
When the desired compressive strain is reached, an additional relaxation step is
performed by removing the damping forces to ensure static equilibrium.
The material properties given in §3b are used for the finite-element analysis.
For the simulations with incompressible neo-Hookean model, the material
incompressibility of the substrate is approximated by setting a Poisson’s ratio
of 0.4999, since perfect incompressibility is not supported by CPEG4 elements.
(a) Effects of substrate material nonlinearity
Beyond the first bifurcation point for neo-Hookean substrates, we searched
for the second bifurcation possibility of the primary wrinkle when the amplitude
becomes larger under further compression. However, in our limited computational
search range of 1 ≤ l ≤ lps = 2.1 for Q̂ = 0.84 × 10−5 , the single mode of the
primary wrinkle persisted and the second bifurcation was not observed in the
model of a symmetric bending-stiffness layer on the neo-Hookean substrate.
On the other hand, the second bifurcation occurred at a nominal compressive
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J.-Y. Sun et al.
(nH1)
(Ex1)
(Og1)
(nH2)
(Ex2)
(Og2)
(nH3)
(Ex3)
(Og3)
0
(Og4*)
1.0
2.0
(Ex4)
3.0
4.0 19.1
(Og4**)
Figure 7. Equilibrium wrinkle configurations of a surface layer with asymmetric bending stiffness
on a neo-Hookean substrate (nH1–3), on a PDMS substrate (Ex1–3) and on an Ogden model
substrate (Og1–3). The far-field compression of 3LC = 0.12, 0.35 and 0.41 by the compression
control device produced local nominal compressive strain of 3̂C
L = 0.17, 0.42 and 0.47, respectively.
The local nominal compressive strains were estimated by measuring the cord length of the film
in the cross-sectional view: computational equilibrium wrinkle configuration of a surface layer
with symmetric bending stiffness, on an Ogden model substrate (Og4∗ ), experimental equilibrium
wrinkle configuration of a 74 nm thickness gold film on a PDMS substrate (Ex4) and computational
equilibrium wrinkle configurations of a surface layer with asymmetric bending stiffness, on an
Ogden model substrate (Og4∗∗ ), at 3̂C
L = 0.51.
strain 3LC between 12 and 35 per cent in the symmetric bending-stiffness layer
on the second-order Ogden model substrate. However, when the film was
further compressed beyond 41 per cent nominal strain, the bi-modal primary
wrinkle created by the second transition returned back to the single mode
primary wrinkle as shown in figure 7(Og4∗ ).
Noticing that the substrate material nonlinearity could induce high-order
bifurcations of the primary wrinkles, we compared the wrinkle shapes of the
same asymmetric-stiffness layers on neo-Hookean and Ogden model substrates.
Figure 7(nH1), (nH2) and (nH3) is the result for the neo-Hookean model, while
(Og1), (Og2) and (Og3) for the Ogden model. These results were also compared
with experiments. In close examination of the experimental figures, it was found
that the nominal compressive strain imposed by the experimental apparatus
was different from the local values evaluated by measuring the cord length of
the wrinkle in the cross-sectional view. While the imposed nominal compressive
strains were 12, 35 and 41 per cent for the cross-sectional images shown in (Ex1),
(Ex2) and (Ex3), the local measurements gave 17, 42 and 47 per cent, respectively.
We used the local nominal compressive strains for the simulations. Although the
cross-sectional shapes of the single-mode primary wrinkles of the neo-Hookean
and the Ogden models were close to each other at the 17 per cent compression as
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Folding wrinkles
949
shown in (nH1) and (Og1), the stress-concentration patterns were very different.
The cross-sectional configurations became substantially different as well at the
42 per cent compression. The second bifurcation transition was just starting in the
neo-Hookean model shown in (nH1); on the other hand, (Og2) indicates that
the transition in the Ogden model occurred at a lower compression level. Finally,
the Ogden model in (Og3) shows folding of wrinkles at 47 per cent compression
as observed in (Ex3), while the neo-Hookean model in (nH3) did not.
(b) Effects of asymmetric bending: bending nonlinearity of the surface layer
As mentioned above, a layer of symmetric bending stiffness on a neo-Hookean
substrate hardly produced the second bifurcation transition. Although the layer
of symmetric bending stiffness attached on an Ogden substrate generated the
second transition, it did not show any folding of wrinkles. It did not exhibit the
third transition to the quadruple-wavelength mode observed in the experiment.
Furthermore, the maximum bending curvature was observed at different locations
in the layer. The symmetric bending stiffness of the layer always gave the
maximum curvature at the peak of the wrinkle, while the maximum curvature was
spotted at the valley of the wrinkles in the experiment. As mentioned earlier, these
observations directed us to consider asymmetric bending stiffness of the gold films.
When we tuned the asymmetric bending stiffness factor x− down to 0.1 from unity,
we could match the third transition to the quadruple mode at 3̂C
L = 0.51, as shown
in figure 7(Og4∗∗ ). In this simulation, we allowed inter-penetration of folding
surfaces of the wrinkles. With the asymmetric bending stiffness factor x− = 0.1,
not only the third transition mode but also the maximum curvatures at the right
locations at different compression stages could be produced. Nevertheless, still
some subtle differences between the experiment and simulation were observed.
In figure 7(Og2), every other valley was much deeper than those observed in the
experiment (Ex2). However, close examination reveals that the platinum coating
in every other valley was cracked in the experiment, and the SEM images of the
valleys far away from the FIB sectioned region indicate that the valleys with
coating cracks were deeper than the valleys without coating cracks. Presumably,
the stress relaxation during FIB sectioning open the coating crack near the
sectioned region. The simulations were carried out by searching for the minimum
energy configurations, floating the primary wavelength with symmetric boundary
conditions for the simulation zone size twice the primary wavelength, except for
(Og4∗∗ ). The symmetry boundary conditions were imposed for the simulation
region of (Og4∗∗ ) with quadruple length of the primary wavelength.
When a stiff layer is buckled on a compliant substrate, delamination of
interface may occur even without a noticeable growth of wrinkle amplitude if the
interface bonding is weak (Shield et al. 1994). Therefore, we investigated the stress
state produced by a folded large-amplitude wrinkle. Figure 8 shows the simulated
wrinkling profile compared with experiment (figure 8a) and the von Mises stress
distribution (figure 8b) in (Og3) of figure 7. The von Mises stress distribution was
limited to 4.54 MPa. Figure 8c shows that the compressive interface traction at
the bottom of the closed valley was very large down to −9.46 MPa, while the
tension was limited to 0.87 MPa. The maximum shear traction was 2.53 MPa.
The results imply that the interface is well protected from tensile delamination
in large amplitude wrinkles on a soft substrate such as PDMS.
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J.-Y. Sun et al.
(a)
(b)
S, Mises
(avg: 75%)
+4.537e+01
+2.000e+00
+1.833e+00
+1.667e+00
+1.500e+00
+1.333e+00
+1.667e+00
+1.000e+00
+8.333e+01
+6.667e+01
+5.000e+01
+3.333e+01
+1.667e+01
+0.000e+00
(c)
4
traction (MPa)
2
0
–2
–4
–6
–8
normal
–10
–5 –4 –3 –2 –1 0 1 2 3
distance along the interface (mm)
4
5
Figure 8. Stress distribution in a folding configuration of a wrinkle pair: (a) enlarged configuration
of the folded wrinkle of figure 7(Ex3); (b) distribution of von Mises stress in the configuration
of figure 7(Og3); (c) distribution of normal and shear tractions along the interface between the
surface layer and the substrate in the folded configuration of figure 7(Og3).
5. Discussion and conclusions
As introduced in §2, the bifurcation of the post-buckling equilibrium wrinkle
configuration on a pre-stretched neo-Hookean substrate involves four eigenstates
of incremental substrate deformation. The incremental elastic anisotropy of the
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Folding wrinkles
951
pre-stretched substrate determines the −X2 -direction decay rate of the eigenstate.
The amplitude sharing between the eigenstates is decided by the interface
boundary conditions compatible with the deformation of the surface layer. The
interface compatibility condition provides the bifurcation eigenvalue problem for
the wave number of the sinusoidal buckling. Similar to the solution procedure
of the first bifurcation point provided by the first-order perturbation analysis
in §2, high-order asymptotic analysis is believed to be possible to follow the postbifurcation equilibrium paths for the growth of wrinkles. The asymptotic analysis
is also considered to be possibly extended beyond the neo-Hookean model of the
substrate and the symmetric-bending-stiffness model of the surface layer.
When we followed the post-buckling equilibrium configurations of the primary
wrinkles in 20–80 nm thin gold films attached on a pre-stretched PDMS substrate,
we experimentally observed the second and the third bifurcation points within
the stretch ratio of 1 ≤ l ≤ lps = 2.1. The second and the third bifurcations
generated periodic clustering or bunching of wrinkles. FIB sectioning revealed
that the clustering led to folding of the wrinkles when the surface layer was
highly compressed. While multi-scale clustering of small-amplitude wrinkles may
be possible on a substrate with graded material properties, folding of the wrinkles
requires multiple symmetry-breaking nonlinear material characteristics of both
the substrate and the surface layer. Our computational tracing of the equilibrium
wrinkle configurations revealed that the neo-Hookean substrate could not fold the
wrinkles regardless of the elastic surface-layer nonlinearity. In contrast, the Ogden
substrate could fold the wrinkles as observed in the experiments, if the surfacelayer stiffness was asymmetric in bending. However, when the bending stiffness of
the surface layer was symmetric, the Ogden model could not produce the folded
wrinkles either.
The folding of the wrinkles is believed to be retarded or promoted by inelastic
behaviour of the film and the substrate. Viscoelastic behaviour of the substrate
(Huang 2005) may provide time-dependent folding behaviour of the wrinkles.
Plastic bending and unbending processes of the surface layer (Kim & Kim
1988) can also generate different wrinkle patterns than predicted by the elastic
analysis. Our in situ AFM measurement of the wrinkle wavelengths strongly
suggests that the phase of the wrinkle curvature was locked in the undeformed
configuration of the film. The wavelength of the primary wrinkle conformally
followed the stretch of the substrate when the gold film was compressed to be
folded. In addition, the film flatness was not recovered when it was fully relaxed;
instead, plastically deformed wrinkle grooves remained. Once the wrinkle phase is
locked in the undeformed configuration by the surface-layer plasticity, incremental
configuration forces (Kim et al. 2010) caused by the external compression loading
may unlock the phase by unbending the plastically deformed surface layer.
Further studies on inelastic effects of the materials on wrinkle folding are in
progress. The multiple folds of thin films with nanometre-scale thickness are
expected to have extensive applications such as nano-fluidics in the parallel nanochannels of the folding wrinkles, self-cleaning adhesion control with concealing
and re-exposing surfaces, and creation of nanometre-scale solid reaction surfaces
for catalysts and battery electrodes.
S.X. and K.S.K. were supported by the U.S. National Science Foundation (DMR-0520651) through
MRSEC at Brown University, and by Korea Institute of Science and Technology (KIST); M.W.M.
by KIST and in part by the Ministry of Knowledge Economy of Korea through the Fundamental
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952
J.-Y. Sun et al.
R&D Program for Core Technology of Materials; J.Y.S. and K.H.O. were supported by Basic
Science Research Program through the National Research Foundation of Korea (NRF) funded by
the Ministry of Education, Science and Technology (R11-2005-065). We also thank M. Diab for
valuable discussions and the late H. M. Rho for his support for the hybrid computational analysis.
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Proc. R. Soc. A (2012)
Folding wrinkles of a thin film
stiff layer on a soft substrate
Jeong-Yun Sun, Shuman Xia, Myoung-Woon Moon,
rspa.royalsocietypublishing.org
Kyu Hwan Oh and Kyung-Suk Kim
Proc. R. Soc. A 468, 932–953 (8 April 2012; Published
online November 23 2011) (doi:10.1098/rspa.2011.0567)
Correction
Cite this article: Sun J-Y, Xia S, Moon M-W,
Oh KH, Kim K-S. 2013 Folding wrinkles of a thin
film stiff layer on a soft substrate. Proc R Soc A
469: 20120742.
http://dx.doi.org/10.1098/rspa.2012.0742
There is a typo in equation (2.4b) of Jeong-Yun Sun,
Shuman Xia, Myoung-Woon Moon, Kyu Hwan Oh and
Kyung-Suk Kim, ‘Folding wrinkles of a thin film stiff
layer on a soft substrate’.
It is currently published as UN−H = Q(λ21 + λ22 + λ23 − 3);
but it should read as
UN−H =
Q 2
2 (λ1
+ λ22 + λ23 − 3) (the factor 1/2 was missing).
c 2013 The Author(s) Published by the Royal Society. All rights reserved.