Effects of tension–compression asymmetry on the surface wrinkling
of film–substrate systems
Xiao Huanga, Bo Li a, Wei Hongb, Yan-Ping Caoa, and Xi-Qiao Fenga,*
a
Institute of Biomechanics and Medical Engineering, AML, Department of Engineering
Mechanics, Tsinghua University, Beijing 100084, P. R. China
b
Department of Aerospace Engineering, Iowa State University, Ames, IA 50011, USA
Corresponding author. Tel.: +86 10 62772934; fax: +86 10 62781824.E-mail address:
fengxq@tsinghua.edu.cn (X. Q. Feng).
*
Abstract
Many soft materials and biological tissues are featured with the tension–compression
asymmetry of constitutive relations. The surface wrinkling of a stiff thin film lying on a
compliant substrate is investigated through theoretical analysis and numerical simulations. It is
found that the tension–compression asymmetry of the soft substrate not only affects the critical
strain of buckling but, more importantly, may also dictate the wrinkling pattern that occurs in the
film–substrate system under specified loading conditions. Due to this mechanism, the thin film
subjected to equi-biaxial compression will first buckle into a hexagonal array of dimples or
bulges, rather than the checkerboard pattern theoretically predicted in previous studies, and
consequently evolve into labyrinths with further loading. Under non-equi-biaxial compression,
the system may buckle either into a parallel bead-chain pattern or a stripe pattern, depending on
the substrate nonlinearity and the loading biaxiality. Phase diagrams are established for the
wrinkling patterns in a wide range of geometric and mechanical parameters, which facilitate the
design of surface patterns with desired properties and functions.
Keywords: Film–substrate system; Tension–compression asymmetry; Wrinkling; Biaxial
compression; Morphological evolution
1
1. Introduction
Surface wrinkles widely exist in both synthetic and natural systems, e.g., constrained
swelling hydrogels under environmental stimuli (e.g., pH, humidity, and temperature) (Basu et
al., 2005; Hong et al., 2008; Zhang et al., 2011; Ding et al., 2013), dried fruits and vegetables
(Chen and Yin, 2010; Li et al., 2011b), and anatomical structures (Li et al., 2011a; Ciarletta and
Amar, 2012; Budday et al., 2014). The wrinkling behavior of a system depends on its material
properties, geometric configurations, and boundary conditions. When subjected to an in-plane
compressive strain, a film–substrate structure may buckle into various surface morphologies, e.g.
sinusoidal or period-doubling wrinkles and folds under uniaxial compression (Hunt et al., 1989;
Huang, 2005; Huang et al., 2005; Jiang et al., 2007; Brau et al., 2011; Xie et al., 2014) and
checkerboard, hexagonal, herringbone, and labyrinth patterns under biaxial compression (Audoly
and Boudaoud, 2008; Breid and Crosby, 2011; Cai et al., 2011; Jia and Amar, 2013; Huang et al.,
2015). On one hand, wrinkling may limit the service performance of materials and structures and,
therefore, is usually thought of as a nuisance that should be avoided. On the other hand, it can
also be harnessed for many technologically significant applications in, for instance, advanced
functional materials with desired surface properties, stretchable or foldable electronics and
medical devices, and mechanical characterization techniques (Stafford et al., 2004; Chan and
Crosby, 2006; Chung et al., 2007; Kim and Rogers, 2008; Cao et al., 2009; Guvendiren et al.,
2009; Kim et al., 2010; Li et al., 2012; Huang et al., 2014; Rahmawan et al., 2014).
For a system consisting of a thin stiff film bonded to a compliant substrate, surface
wrinkling is generally dictated by the minimization of the total potential energy. The formation
2
of surface wrinkles releases a part of the elastic strain energy in the film but increases that in the
substrate. Therefore, both the mechanical properties of the film and the substrate contribute
significantly to the diversity of wrinkling patterns. The role of the linear elastic substrate in the
wrinkling of film–substrate systems has been explored recently (Huang et al., 2005; Genzer and
Groenewold, 2006; Song et al., 2008; Cai et al., 2011). In general, uniaxial compression beyond
a threshold triggers the occurrence of sinusoidal wrinkling in a film–substrate system. Under
biaxial compression, the system may buckle into the checkerboard, hexagonal, or herringbone
modes, depending on the geometric, mechanical, and boundary conditions. Cai et al. (2011)
conducted an upper-bound analysis to compare the energy states of these wrinkling patterns.
They claimed that in the case of equi-biaxial compression, the checkerboard wrinkling mode has
the lowest energy and should appear first with loading, and it would give way to the herringbone
mode during postbuckling. In contrary to this theoretical prediction, however, their experiments
found that the hexagonal mode occurred once the compressive strain reached the threshold. They
attributed this discrepancy to the small initial curvature of the film since a perfectly flat
configuration is difficult to be fabricated. In addition, the wrinkling pattern evolution in the
postbuckling process is modulated by the mechanical property and pre-deformation of the
substrate (Auguste et al., 2014). A substrate with prescribed compression favors the wrinkling
transition from a sinusoidal mode to a period-doubling mode and eventually to a spatial-chaos
mode (Zang et al., 2012; Brau et al., 2013; Auguste et al., 2014; Budday et al., 2015; Wang and
Zhao, 2015), while a prestretch of the substrate may lead to the ridge mode (Cao and Hutchinson,
2012; Zang et al., 2012; Cao et al., 2014; Chen and Crosby, 2014). These studies demonstrate
that the material nonlinearity of the substrate plays an important role in the buckling evolution of
the thin film atop.
3
Many soft materials (e.g., rubbers, hydrogels, and biological tissues) in film–substrate
systems possess nonlinear constitutive relations with the feature of tension–compression
asymmetry. Bimodular material is a typical example with tension–compression asymmetry
(Rizzi et al., 2000; Patel et al., 2005; Bertoldi et al., 2008; Destrade et al., 2010; Du and Guo,
2014). The stress–strain relationship of such materials is often modeled by a piecewise linear
function with slope discontinuity at the origin. In other words, a bimodular constitutive model
has different Young’s moduli under tension and compression. Bimodular materials can be found
in aramid–rubber, polyester–rubber carbon–carbon composites, and soft biological tissues. For
example, the tensile modulus of carbon–carbon composites can be two to five times greater than
the compressive modulus (Jones and Morgan, 1980). The tensile modulus of an articular
cartilage is about 20 times higher than its compressive modulus (Soltz and Ateshian, 2000). The
mechanical responses of such materials have attracted considerable attention (Bertoldi et al.,
2008; Du and Guo, 2014; Cai et al., 2015). To date, however, it remains unclear how the
wrinkling behavior of a film–substrate system depends on the property of tension–compression
asymmetry of its substrate.
In this paper, we investigate the surface wrinkling and morphological evolution of a film
lying on a bimodular substrate subjected to either uniaxial or biaxial compression. The layout of
this paper is as follows. In Section 2, an analytical method based on the energy principle and a
semi-implicit Fourier spectral numerical method are presented. In Section 3, the wrinkling
behavior of a film under uniaxial compression is examined, and the critical buckling and initial
postbuckling modes are obtained. In Section 4, we investigate how the substrate property affects
the wrinkling patterns in the cases of equi-biaxial and non-equi-biaxial compression. In Section 5,
the influence of the tension–compression asymmetry of the substrate on the morphological
4
transition during postbuckling is explored under a specified loading biaxiality. Phase diagrams
are established for the wrinkling patterns in terms of the mechanical property of the substrate and
the external compressive strains. Finally, Section 6 summarizes the main conclusions drawn
from this study.
2. Methods
2.1. Theoretical model
Consider a stiff thin film perfectly bonded to a compliant semi-infinite and flat substrate.
Refer to a Cartesian coordinate system ( x1 , x2 , x3 ), as shown in Fig. 1a, where the origin O is
located at the middle surface of the film in the initial state. The film is subjected to uniform
biaxial compressive strains 110 and  220 in the x1 and x2 directions, respectively, which may be
induced by thermal expansion, mechanical loading or other mechanisms. For simplicity, the
shear stresses at the film–substrate interface are ignored (Huang et al., 2005). The film is
assumed to be an isotropic and linear elastic material with Young’s modulus E and Poisson’s
ratio v . The constitutive relation for a material with tension–compression asymmetry is
illustrated as the dashed curve in Fig. 1b. Since the elastic strain energy density is determined by
the area bounded by the curve and the w-axis, such a nonlinear constitutive relation can also be
approximated, from the viewpoint of energy, by a bimodular stress–strain relation with different
elastic moduli under tension and compression (solid line). In our theoretical analysis, the
nonlinear substrate is modeled as a Winkler foundation, which is described by the tensile
stiffness K t and compressive stiffness K c (Fig. 1b). The tension–compression asymmetry of the
substrate is qualified by the ratio between its compressive and tensile stiffnesses, Kc / K t ,
hereinafter referred to as the stiffness ratio.
5
When the externally applied compressive strain reaches a certain condition, the film would
buckle. The nonlinear von Kármán plate theory is used to model the film. Its equilibrium
equations read
N
x
 0,
(1)
Eh 3 4

 w  N w,  P,
12
where w is the deflection, E =E / (1  v 2 ) is the plane-strain elastic modulus, N are the
membrane forces in the film, h is the film thickness, and  4 is the bi-harmonic operator. P
refers to the normal traction at the film–substrate interface. Einstein’s summation convention is
adopted for repeated Greek indices, which take the values of 1 and 2. For a thin plate, the
constitutive
relation
can
be
expressed
in
terms
of
the
internal
forces
as
N  Eh[(1  v)  v  ], where   is the Kronecker delta and   denote the middle-
plane strains in the film, which are related to in-plane displacements u and out-of-plane
0
displacement w by   
 (u ,  u , ) / 2  w, w, / 2 . For the bimodular substrate, the
normal interfacial stress per unit area takes the form:
 Kt w
P( w)  
 Kc w
w  0,
w  0.
(2)
In the wrinkled state, the elastic strain energy in the film contains two main parts, namely,
the membrane energy U m induced by the in-plane deformation and the bending energy U b
induced by the out-of-plane deformation. They are calculated by
1
N   dx1dx2 ,
2
(3)
Eh 3
[(1  v )w, w,  vw, w, ]dx1dx2 .
24 
(4)
Um 
Ub 
6
The elastic strain energy in the bimodular substrate is expressed as
1
U s   P( w) wdx1dx2 .
2
(5)
Under the condition of displacement-controlled loading, the total potential energy in the system
is U tot = U m + U b + U s . The occurrence of a specific wrinkling mode is dictated by the
minimization of the total potential energy U tot . From this condition, one can determine the
wrinkling mode and the critical strain of buckling.
It is noteworthy that the normal traction P at the film–substrate interface is selfequilibrating over the entire interface, that is,
 P( w)dx dx
1
2
 0.
(6)
For a film on a linear substrate, Eq. (6) is automatically satisfied when one assumes the
sinusoidal deflection w  A cos(kx1 ) , where A is the corrugation amplitude and k is the
wavenumber. However, it is not the case for a nonlinear substrate with tension–compression
asymmetry. To satisfy the equilibrium condition (6), a correction over the sinusoidal deflection
mode is needed. Here for simplicity, a deflection field consisting of a constant vertical shift 
and a sinusoidal component is assumed:
w  A cos(kx1 )  ,
(7)
The shift  will be determined by the force equilibrium condition in Eq. (6).
2.2. Fourier spectral method
We will use the Fourier spectral method (Huang et al., 2004) to validate our theoretical
solution and to track the postbuckling evolution of wrinkling patterns. A semi-implicit iterative
form is employed, in which the linear and the nonlinear parts in Eq. (1) are calculated in the
7
Fourier space and the real space, respectively. After rescaling all stresses by the plane-strain
modulus of the film E , all lengths by the film thickness h and introducing a pseudo viscous part
ŵ to Eq. (1), the normalized equilibrium equations are reformulated in the Fourier space as
ik  Nˆ   0,
wˆ 
1
S (k k )2 wˆ 
12
( SN w,  SP )  0,
(8)
where i is the imaginary unit, k are the coordinates in the Fourier space, and S  E / ( K t h) is a
dimensionless parameter describing the relative stiffness of the film to the substrate. Both the
hatted symbols and the operator
() stand for the Fourier transformation of the corresponding
fields. By adopting a semi-implicit Fourier spectral method, we can obtain the iterative scheme
for the deflection w as
1


wˆ ( m1)  1  S (k k )2dt 
 12

1
wˆ
( m)
m)
 [ SN wˆ ,(
 SP( wˆ ( m) )]dt,
(9)
where dt is the iterative increment in each step, and wˆ ( m ) denotes the out-of-plane displacement
of the film at the iteration step m. In each step, the displacement field wˆ ( m ) is known, and
P( wˆ ( m ) ) can be calculated with checking the sign of wˆ ( m ) at each position.
We simulate the wrinkling pattern in a square region with periodic boundary conditions. The
square region is meshed into 256  256 elements with identical sizes. In all simulations, we set
the Poisson’s ratio of the stiff film v  0.3 and the relative stiffness of the film to the substrate
S  1000 . A random deflection field with magnitude smaller than 0.001 is prescribed at t  0 to
numerically trigger all possible modes of instability. The iteration will be stopped when the total
energy converges to a constant.
3. Uniaxial compression
8
0
 0.
The above model is first applied to the case of uniaxial compression with 110  0 and  22
In the initial buckling stage, the sinusoidal wrinkling mode is expressed as Eq. (7). The force
equilibrium condition within a period of the wrinkling pattern in Eq. (6) becomes

k 1 arccos(  / A)
0
 /k
K t wdx1   1
k
arccos(  / A)
Kc wdx1  0 ,
(10)
which can be simplified as
 Kc  
2 
  Kc 
1

1


arccos(

) 
  0.



K t  
A2 A
A  K t A

(11)
From Eq. (11), one can solve the value of the relative shift  / A . The elastic strain energy in the
substrate is calculated as
Us 
 /k
k  k 1 arccos(  / A) 1
1
 1
2
K
w
d
x

Kc w2dx1   A2 K t ,
t
1
1



0
k
arccos(


/
A
)

2
2
 4
(12)
where
2
1  Kc   
 
2 
  K 
2 

   1   3
1     1  2 2  arccos( )   c 1  2 2  ,
  Kt   A
A 
A  Kt 
A 
 A 


(13)
which depends only on the stiffness ratio Kc / K t . The total potential energy of the system is thus
derived as
U tot  U b  U m  Us 
1
1
Eh 4 4
Eh(110 )2  k 2 A2 (  Eh110  Ehf ) 
k A,
2
4
32
(14)
with
f 
(kh)2
 Kth

.
12
(kh)2 E
(15)
The total potential energy is a function of wavenumber k and wrinkling amplitude A . By
minimizing U tot with respect to k and A , i.e., U tot / k  0 and U tot / A  0 , one obtains
1/4
 E 
6
110
K h 
A

h
 1,
 cr   t  , cr  2 h 
,

3
 cr
 3E 
 12 K t h 
1/2
9
(16)
where  cr and cr are the critical strain of wrinkling and the corresponding critical wavelength,
respectively.
Eq. (16) reveals that both the critical strain and wavelength depend on the stiffness ratio
Kc / K t . For a linear substrate with Kc / Kt  1 , combining Eqs. (11) and (13) gives   1 , and
then Eq. (16) degrades into the solution for the critical buckling of a film on a linear Winkler
foundation (Koiter, 2009).
The variations of the critical strain  cr and the normalized wavelength cr / h with respect to
the stiffness ratio Kc / K t are illustrated in Fig. 2a and 2b, respectively. A good agreement is
found between the numerical results and the theoretical solution. With the decrease in the
stiffness ratio, the critical strain decreases whereas the wavelength increases. The stiffness ratio
of the substrate strongly affects the critical strain. For example, the critical strain  cr under
Kc / Kt  0.2 is about 60% lower than that under Kc / Kt  1 . It is found that the critical strain
determined by taking the winkling mode in Eq. (7) is smaller than that by neglecting the vertical
shift (  0 ). Their relative error can be as large as 30% when Kc / Kt  0.2 . This means that the
assumption of the pure sinusoidal mode w  A cos(kx1 ) in the previous studies is appropriate for
linear substrate but may cause a great overestimation of the critical strain of wrinkling. In
addition, Fig. 2c shows that the tension–compression asymmetry with a small stiffness ratio
Kc / K t strongly affects the geometric features of the morphology. When Kc / K t  1 , the
wrinkled film will have a downward shift of the undulation, i.e.,   0 , and vice versa. The
downward shift generates deep troughs but shallow crests, resembling the morphology with high
size ratio of crypt-to-villi observed at the inner surface of animal intestines (Hannezo et al.,
2011).
10
4. Biaxial compression
4.1 Theoretical analysis
Now we consider the case where the film is subjected to the external strains 110  0 and
 220  0 . Let    220 / 110 be the loading biaxiality to quantify the anisotropy of biaxial
compression. Without loss of generality, we take 0    1 , with   0 representing uniaxial
compression,   1 equi-biaxial compression, and 0    1 non-equi-biaxial compression. Due
to the diversity of possible wrinkling morphologies under biaxial loading, we take the following
mode of deflection
1

1

w  A cos(kx1 )  B cos  kx1  cos  qkx2    ,
2

2

(17)
where A and B are two amplitude coefficients, and q is a dimensionless parameter. The
parameters k and qk are the wavenumbers in the x1 and x2 directions, respectively. The various
wrinkling patterns expressed by Eq. (17) are illustrated in Fig. 3, where we define the parameter
p  B / A when A  0 . Some representative patterns are as follows:
(i)
the sinusoidal checkerboard mode when A  0 and q  1 ,
(ii) the one-dimensional (1D) sinusoidal mode when p  0 ,
(iii) the hexagonal mode when p  2 and q  3 , and
(iv) the parallel bead-chain mode when p  1 and q  3 .
In patterns (iii) and (iv), the film will bulge outwards as convex bumps when A  0 and sink
inwards as dimple arrays when A  0 , as shown in Fig. 3.
11
For the wrinkling mode described by Eq. (17), the in-plane displacements u1 and u2 of the
film can be obtained by solving Eq. (1) as
1

1

3 
1

u1  c1 sin  kx1  cos  qkx2   c2 sin  kx1  cos  qkx2 
2
2
2
2








 c3 sin( kx1 ) cos( qkx2 )  c4 sin(2kx1 )  c5 sin( kx1 ),
1
 1

3  1

u2  c6 cos  kx1  sin  qkx2   c7 cos  kx1  sin  qkx2 
2
 2

2
 2

 c8 cos(kx1 )sin( qkx2 )  c9 sin( qkx2 ),
(18)
where
c1 
ABk (q 4  2q 2  2q 2 v  1)
ABk (q 4  6q 2  6q 2v  27)
,
c

,
2
2(1  q 2 ) 2
2(9  q 2 ) 2
B2k
A2 k
B 2 k (1  q 2v)
, c4 
, c5 
,
32
8
32
ABkq(1  q 2 v)
ABkq(9  q 2v)
c6  
,
c


,
7
(1  q 2 ) 2
(9  q 2 ) 2
c3 
c8 
(19)
B 2 kq
B 2 k (q 2  v )
, c9 
.
32
32q
The averaged bending energy in the film per unit area is calculated from Eq. (4) as
Ub 
1
Eh3k 4 32 A2  B 2 (1  q 2 )2  .
1536
(20)
By ignoring the fourth- and higher-order terms of A and B , the membrane energy in the film is
obtained as
Um 
1
Eh110 8 A2k 2 (1   v )  B 2k 2 [1  q 2v   ( q 2  v )]  16(1   2  2  v )110 .
32
(21)
To calculate the elastic strain energy stored in the substrate, we should first determine the
vertical shift  from the self-equilibrating equation of the normal traction. Due to the
complexity of analytical solution, we resort to a dichotomy numerical method. Once the stiffness
ratio Kc / K t and the wrinkling mode ( p , q ) have been given, the vertical shifts can be acquired
as  A0 ,  A , and  A under the condition A  0 , A  0 , and A  0 respectively. Accordingly,
12
the regions with positive or negative values of w can be determined. Finally, the energy in the
substrate can be expressed as
1 2
 4 B K t A 0

1
U s   A2 K t  A +
4
1 2
 4 A K t A

if A  0,
if A  0,
(22)
if A  0.
Here the parameters  A0   A0 ( q, Kc / Kt ) ,  A+   A+ ( p, q, Kc / Kt ) , and  A   A ( p, q, Kc / Kt )
correspond to the checkerboard mode, the outward-bulging modes, and the inward-sinking
modes, respectively. They can be achieved numerically since the corresponding vertical shifts
have been obtained.
By minimizing U tot with respect to k and A (or B if A  0 ), i.e., U tot / k  0 and
U tot / A  0 (or U tot / B  0 ), the critical buckling strains and the wavelengths for the
different modes are solved as
1/2

2(1  q 2 )
  A0 K t h 



2
2
 1  q v   ( q  v )  3E 
1/2

2[32  p 2 (1  q 2 ) 2 ]1/2
  A+ K t h 
 cr  


2
2
2
 8  8 v  p [1  q v   ( q  v )]  3E 
1/2

2[32  p 2 (1  q 2 ) 2 ]1/2
  A K t h 



2
2
2
 8  8 v  p [1  q v   ( q  v )]  3E 
1/4


E
2 
 h 1  q 


 24 A0 K t h 

1/4
1/4


1 2
E


2 2
cr  2 h 1  p (1  q )  

 32
  12 A+ K t h 

1/4

1/4

E
2 h 1  1 p 2 (1  q 2 ) 2  
 32
  12 K h 

A t 


13
if A  0,
if A  0,
(23)
if A  0,
if A  0,
if A  0,
if A  0.
(24)
Once the loading biaxiality  , the stiffness ratio Kc / K t and the relative stiffness of the film to
the substrate S  E / ( K t h) are specified, the system selects the wrinkling mode ( p , q ) that
minimizes the total potential energy. Equivalently, the selected mode can be determined by
minimizing  cr with respect to p and q , i.e.,  cr / p  0 and  cr / q  0 .
4.2 Equi-biaxial compression
In the case of equi-biaxial loading (   1 ), the critical strains for the occurrence of different
wrinkling patterns can be determined from Eq. (23), as shown in Fig. 4a, where we take the
stiffness ratio Kc / Kt  0.6 . It is seen that among all wrinkling modes, the critical strain at
p  2 , q  3 , and A  0 is minimal, indicating that the inward hexagonal mode will appear in
the system subjected to equi-biaxial compression. Fig. 4b compares the critical strains for the 1D
mode, checkerboard, and outward-bulging hexagonal mode (marked as outward hexagon in the
figure), and inward-sinking hexagonal mode (inward hexagon). As shown by Cai et al. (2011),
the critical strains of these wrinkling modes are all the same for a linear substrate with
Kc / Kt  1 . However, for a nonlinear substrate with Kc / K t  1 , the inward hexagonal mode
attains the lowest critical strain among the four modes and hence may be observed in real film–
substrate systems, while the outward-bulging hexagonal mode has the largest critical strain. In
the case when the substrate stiffness under compression is lower than that under tension
( Kc / K t  1 ), the inward hexagonal deformation mode has a lower elastic strain energy in the
substrate than the outward hexagonal mode. In the opposite case of Kc / K t  1 , the outward
hexagonal mode may be observed in reality.
14
The above results are further confirmed by the Fourier spectral method presented in Section
2. The wrinkling patterns at the initial postbuckling state are simulated under different stiffness
ratios Kc / K t . The numerical results for the normalized deflection w in the middle plane are
0
 1.02 cr . The
shown in Fig. 5, where the equi-biaxial compression strains are taken as 110   22
numerical results agree well with the above theoretical prediction. It can be seen that with
decreasing stiffness ratio, a transition from the checkerboard mode to the hexagonal mode takes
place. When the substrate is linear ( Kc / Kt  1 ), the system favors the checkerboard mode, as
predicted by Cai et al. (2011). With the decrease in Kc / K t , the material nonlinearity of the
substrate will play a significant role in the mode selection, making the hexagonal mode more
energy-favorable than the checkerboard mode.
For a film resting on a linear substrate subjected to equi-biaxial loading, the theoretical
analysis of Cai et al. (2011) predicted that the checkerboard mode will occur at the first
bifurcation of the system. However, their experiment showed that the system consisting of a thin
film supported by a PDMS substrate preferred the hexagonal mode. This discrepancy between
the theoretical prediction and the experimental observation was attributed to the initial curvature
of the film which may probably exist in the initial state or develop upon the swelling of the
PDMS substrate. In the present paper, it is shown that the tension–compression asymmetry of the
nonlinear substrate might be another mechanism that may dictate the selection of wrinkling
patterns. Due to this mechanism, the hexagonal mode has the minimal potential energy and the
minimal strain among all possible wrinkling modes. Thus the system will buckle into the
hexagonal mode, rather than the checkerboard mode. Our theoretical and numerical results
qualitatively agree with the elegant experiments of Cai et al. (2011). In addition, our analysis
shows that during buckling, the film will sink in the substrate, leading to a hexagonal array of
15
dimples, in consistency with the experimental observations of Breid and Crosby (2011).
Therefore, both the initial curvature and the material nonlinearity of the substrate should be
important for the selection of wrinkling patterns.
4.3 Non-equi-biaxial compression
Now we examine the effect of tension–compression asymmetry on the wrinkling features of
a film–substrate system subjected to non-equi-biaxial compression. Take the loading biaxiality
  0.7 and the stiffness ratio Kc / Kt  0.25 as an example. The critical strains for the
occurrence of the various modes are determined from Eq. (23), as shown in Fig. 6a. It is found
that among all different wrinkling modes, the critical strain  cr at p  1.5 , q  1.5 , and A  0 is
minimal, indicating that the parallel bead-chain pattern will occur in such a film–substrate
system. For the 1D mode ( p  0 ) and the parallel bead-chain modes ( p  2 ), Fig. 6b compares
the variations of the critical strains  cr with respect to the stiffness ratio Kc / K t . It can be seen
that there is a threshold value of Kc / K t at which the mode transition occurs ( p  0 ). For a
highly nonlinear substrate with Kc / K t  0.38 , the critical strain for the parallel bead-chain
wrinkling is smaller than that for the 1D mode and thus the film favors a parallel bead-chain
wrinkling pattern. For a substrate with low nonlinearity ( Kc / K t  0.38 ), the critical strain for the
1D mode is lower. The threshold value of Kc / K t for the mode transition is highlighted in Fig.
6b. Therefore, when Kc / K t is larger or smaller than this critical value, the system will buckle
into the 1D mode or the parallel bead-chain mode, respectively.
To validate the above theoretical solution, we also simulate the wrinkling pattern evolution
under non-equi-biaxial compression by using the numerical method presented in Section 2. In the
16
0
/   1.02 cr , the results are given in Fig. 7. The numerical method predicts the
case of 110   22
same changing tendency as the theoretical solution. It is found that in the case of non-equibiaxial compression, the parallel bead-chain mode will appear at the critical buckling of the film
lying on a substrate with distinct tension–compression asymmetry. For an approximately linear
substrate, however, the 1D wrinkling mode will occur at the critical buckling of the system
subjected to non-equi-biaxial compression, in consistency with the theoretical prediction of
Audoly and Boudaoud (2008), who did not account for the effect of tension–compression
asymmetry.
On the basis of the theoretical analysis and numerical simulations, a phase diagram is
established for the wrinkling patterns that occur in the critical buckling state, as shown in Fig. 8.
It clearly illustrates the dependence of the wrinkling modes on the stiffness ratio Kc / K t and the
loading biaxiality  . For example, when  is smaller than 0.45 or, in other words, the external
strain 110 is significantly larger than  220 , the system will always buckle into parallel stripes for
Kc / Kt  0.1 . Compared to the stripe pattern, the parallel bead-chain mode occurs in a system
with a smaller stiffness ratio Kc / K t and subjected to biaxial compression with a larger  .
Under a fixed stiffness ratio Kc / K t and a fixed strain 110 , increasing the value of  (or  220 )
may trigger a transition from the 1D mode into the parallel bead-chain mode. A similar transition
of wrinkling modes was observed in experiments by Breid and Crosby (2011), as shown in Fig. 9.
In their experiments, the biaxial compressive strain state varied in the specimen, corresponding
to the increase of  from the left side to the right. In region A, the film is subjected to non-equibiaxial swelling strains, triggering the formation of parallel stripes. In region B, the biaxial
compressive strains have a lower anisotropy (or a larger  ), resulting in the parallel bead-chain
17
wrinkling mode. In region C, the compressive strains are nearly equi-biaxial (   1 ) and thus the
hexagonal mode was observed. Our numerical simulations show a similar continuous transition
of wrinkling modes to that observed in the experiments of Breid and Crosby (2011) and highlight
the role of substrate nonlinearity in the selection of wrinkling patterns.
5. Morphological transition during postbuckling
5.1 Effect of loading state
The Fourier spectral method is further used to track the postbuckling of the film–substrate
system. For a linear substrate with Kc / Kt  1 and a nonlinear substrate with Kc / Kt  0.6 , the
morphological evolutions with increasing equi-biaxial compressive strain are shown in Figs.
10(a–d) and 10(e–h), respectively. It is seen that under equi-biaxial compression, the film on a
linear substrate prefers a checkerboard wrinkling pattern at lower overstrains. Nevertheless, the
checkerboard mode can exist only in a narrow range of overstrains, and it will soon transform to
labyrinths with an increase in the external compressive strains. In contrary, when the substrate
has a distinct feature of tension–compression asymmetry, the system will favor a hexagonal
pattern. In this case, the hexagonal mode stably exists in a wide range of overstrains and
transforms to the labyrinth morphology only when the compressive strains are sufficiently high
(e.g., 110 /  cr  1.5 ). During this pattern transition process, hybrid patterns consisting of two or
more modes can be observed.
In the case of non-equi-biaxial compression, the system can be destabilized into hybrid
patterns or herringbone patterns, depending on the stiffness ratio Kc / K t and the externally
applied strains 110 and  220 . For a linear substrate with Kc / Kt  1 and a nonlinear substrate with
Kc / K t  0.25 , the morphological evolutions with the proportional increase in the compressive
18
0
 110 are shown in Figs. 11(a–d) and 11(e–h), respectively, where we take
strains 110 and  22
the loading biaxiality   0.7 . The simulations show that, with the increase in the external
strains, the film on the linear substrate first buckles into stripes, then shifts to herringbones, and
finally transforms to labyrinths. For a substrate with Kc / K t  0.25 , the parallel bead-chain
pattern emerges first, and then it gives way to a hybrid pattern consisting of hexagons and stripes,
which further evolves into labyrinths with continuous compression.
The simulation results in Fig. 12 show the effect of loading biaxiality  on the
morphological transition, where we set Kc / Kt  0.2 and the overstrain 110 /  cr  3.5 . Three
distinct wrinkling morphologies are observed with varying loading biaxiality: labyrinth in the
range of   0.9 , hybrid of stripes and hexagons under 0.35    0.9 , and stripes for   0.35 .
The hybrid morphology consisting of parallel stripes and hexagonal patterns occurs in a
moderate range of  while the individual mode occurs in a relatively narrow range of  . When
the loading biaxiality is low (e.g.,   0 ), the parallel stripe mode is favorable. The energy
induced by the increase in the compressive strain is mainly released by the magnification of the
amplitude of sinusoidal wrinkles. When the compressive strains are nearly equi-biaxial (   1 ),
sufficiently large overstrains will lead to the formation labyrinths. With the increase in the
loading biaxiality  , (or  220 ), the wrinkling morphology switches from parallel stripes to
hexagon–stripe hybrids and then to labyrinths.
5.2 Phase diagrams
According to our numerical simulations, a film–substrate system subjected to biaxial
compression may buckle into various wrinkling patterns, e.g., checkerboard, hexagon, parallel
19
bead-chain, herringbone, labyrinth, and their combinations. The selection of a specific pattern by
a system depends on the nonlinear property of the substrate and the loading states. To reveal the
effects of Kc / K t and 110 /  cr , we here construct the phase diagrams for the wrinkling patterns in
two representative cases.
Fig. 13 shows the phase diagram for the wrinkling modes with respect to Kc / K t and
110 /  cr in the case of equi-biaxial loading (   1 ). Typical wrinkling patterns include,
checkerboard (II), hexagon (III), checkerboard–stripe hybrid (IV), hexagon–stripe hybrid (V),
and labyrinth (VI). In the case of equi-biaxial compression, the checkerboard pattern appears
only in a very narrow range of Kc / K t and 110 /  cr . Therefore, it is hardly observed in real
systems under equi-biaxial compression. When Kc / K t is around 1, the initial postbuckling
manifests the checkerboard mode, while the hexagonal pattern emerges when the tension–
compression asymmetry of the substrate is distinct ( Kc / K t  0.93) . With the increase in the
compressive strains, both the checkerboard and hexagonal patterns will transform into a hybrid
state and finally into a labyrinth. In addition, it is seen that with the increase in Kc / K t , the
critical strains at the III–V and V–VI boundaries decline monotonically.
In the case of non-equi-biaxial loading with   0.7 , Fig. 14 provides the phase diagram for
the wrinkling modes. The morphological evolution involves rich wrinkling patterns, e.g., stripe
(II), parallel bead-chain (III), herringbone (IV), hexagon–stripe hybrid (V), and labyrinth (VI).
Akin to the case of equi-biaxial loading, the substrate nonlinearity plays a crucial role in the
pattern selection during postbuckling. Two typical pattern evolution paths with increasing
compressive strains are as follows. First, when the substrate nonlinearity is weak ( Kc / K t  1),
the initial flat film buckles into parallel stripes, which will deform into herringbones and finally
20
become labyrinths. Second, when the substrate nonlinearity is strong (e.g., Kc / K t  0.4 ), the
parallel bead-chain pattern will emerge first, followed by the transition to a hybrid pattern of
hexagons and stripes and the final formation of labyrinths at high overstrains.
6. Conclusions
The buckling and postbuckling behavior of a stiff thin film bonded to a compliant substrate
has been investigated through theoretical analysis and numerical simulations. The attention has
been focused on the effects of tension–compression asymmetry of the soft substrate,
characterized by the bimodular constitutive model. Under biaxial compression, the system may
buckle into a diversity of wrinkling morphologies, e.g., checkerboard, hexagon, stripe, and
parallel bead-chain. The critical conditions for the onset of these wrinkling modes are all
established in terms of the stiffness ratio of the substrate and the loading biaxiality. It is found
that the nonlinearity of the soft substrate not only affects the critical strain of buckling but, more
importantly, may also govern the wrinkling pattern of the film–substrate system. In the case of
equi-biaxial compression, for example, the hexagonal mode corresponds to a lower critical strain
(or lower elastic strain energy) than the checkerboard mode when the substrate has a distinct
tension–compression asymmetry (e.g., Kc / Kt  0.93 ). This provides a novel mechanism to
elucidate why hexagonal patterns are much more often observed than checkerboard patterns in
reality since most soft materials (e.g., hydrogels and biological tissues) have a pronounced
difference in their mechanical properties under tension and compression. The feature of tension–
compression asymmetry also dictates whether the hexagonal mode will be outward bulges (when
Kc / K t  1) or inward dimples (when Kc / K t  1 ). In the case of non-equi-biaxial compression,
our analysis shows that the loading biaxiality and the stiffness ratio control the mode transition
21
from stripes to the parallel bead-chain patterns. Both the large substrate nonlinearity and the high
loading biaxiality favor the appearance of the parallel bead-chain mode.
The Fourier spectral method has been used to explore the morphological evolution of film–
substrate systems during postbuckling. It is found that the substrate nonlinearity, along with
loading biaxiality of the externally applied strains, regulate pattern evolution. In the case of equibiaxial compression, a film on a linear elastic substrate first buckles into the checkerboard
pattern and then evolves into labyrinths with increasing strain, while a film on a nonlinear
substrate prefers the hexagonal mode followed by the formation of labyrinths. Under non-equibiaxial compression, the film–substrate systems have two distinct paths, which involve the
pattern evolutions (i) from stripes, herringbones to labyrinths and (ii) from parallel bead-chains
to labyrinths, respectively, depending on the stiffness ratio and the loading biaxiality. For the
wrinkling patterns in the critical buckling state and their evolutions during postbuckling, phase
diagrams have been provided in terms of the stiffness ratio, loading biaxiality, and external
strains.
This work has highlighted the role of substrate nonlinearity in the buckling and postbuckling
behavior of film–substrate systems. Our results show that besides such factors as initial curvature
that have been revealed in previous studies (Cai et al., 2011), the tension–compression
asymmetry of the nonlinear substrate might be another important mechanism affecting the
selection of wrinkling patterns. Our method and results also suggest a novel route for fabricating
and regulating surface patterns in film–substrate systems.
Acknowledgments
22
Supports from the National Natural Science Foundation of China (Grant Nos. 11432008,
11172155, and 11542005), the 973 Program of MOST (2012CB934101), Tsinghua University
(2012Z02103 and 20121087991), and the Thousand Young Talents Program of China are
acknowledged.
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26
Figure 1. (a) A stiff thin film bonded to a compliant substrate. (b) The nonlinear constitutive
relation of the substrate with the feature of tension–compression asymmetry (dashed curve)
is simplified as a bilinear elastic model with tensile modulus K t and compressive modulus
K c (solid curve).
Figure 2. Comparison between theoretical and numerical solutions. Effects of the stiffness ratio
Kc / K t on (a) the critical strain  cr , (b) the normalized wavelength cr / h , and (c) the
wrinkling morphology w / h . The theoretical model without vertical shift (  0 ) shows a
deviation from the numerical solution. In (c), we take Kc / Kt  0.2 and 110 /  cr  1.5 .
Figure 3. Various wrinkling patterns expressed by the deflection in Eq. (16).
Figure 4. (a) The critical strains for different modes with   1 and Kc / Kt  0.6 . The lowest
point corresponds to the critical strain  cr for the mode ( p, q ) that actually appears. (b) The
critical strains for the occurrence of different wrinkling modes with respect to the stiffness
ratio under equi-biaxial compression.
Figure 5. Effect of the stiffness ratio Kc / K t on the critical buckling pattern under equi-biaxial
0
 1.02 cr .
compression, where we take 110   22
Figure 6. (a) The critical strains for the occurrence of different wrinkling modes under   0.7
and Kc / Kt  0.25 . The critical strain  cr at the lowest point corresponds to the mode ( p, q )
that actually appears. (b) The critical strains for various modes with respect to the stiffness
ratio under non-equi-biaxial compression with   0.7 . The inset shows the p versus
Kc / K t relation at the smallest critical strain.
Figure 7. Effects of the stiffness ratio Kc / K t on the critical wrinkling pattern under non-equi0
/   1.02 cr .
biaxial compression, where we set   0.7 and 110   22
27
Figure 8. Phase diagram for the wrinkling modes at the critical buckling under non-equi-biaxial
compression.
Figure 9. Numerical simulations produce similar patterns observed in the experiments of Breid
and Crosby (2011). In region A, the swelling stress is non-equi-biaxial, resulting in the 1D
mode. In region B, the non-equi-biaxial swelling stress has a higher loading biaxiality,
leading to the parallel bead-chain mode. In region C, the stress state is nearly equi-biaxial
and it triggers the hexagonal mode.
Figure 10. Postbuckling pattern evolution of a thin film on (a)–(d) a linear substrate with
Kc / Kt  1.0 and (e)–(h) a bimodular substrate with Kc / Kt  0.6 . Here, the system is
0
subjected to equi-biaxial compressive strains 110 = 22
.
Figure 11. Postbuckling pattern evolution of a thin film on (a)–(d) a linear substrate with
Kc / Kt  1.0 and (e)–(h) a bimodular substrate with Kc / Kt  0.25 . Here, the system is
0
=0.7110 .
subjected to non-equi-biaxial compressive strains 110 and  22
Figure 12. Effects of loading biaxiality on the postbuckling patterns, where we take
0
/   3.5 cr .
Kc / Kt  0.2 and 110   22
Figure 13. Phase diagram for various wrinkling patterns during postbuckling under equi-biaxial
compression (   1 ). Under different conditions, the surface morphology of the thin film
may be flat (I), checkerboard (II), hexagon (III), checkerboard–stripe hybrid (IV), hexagon–
stripe hybrid (V), or labyrinth (VI).
Figure 14. Phase diagram for various wrinkling patterns during postbuckling under equi-biaxial
compression (   0.7 ). Under different conditions, the surface morphology of the thin film
28
may be flat (I), stripe (II), parallel bead-chain (III), herringbone (IV), hexagon–stripe hybrid
(V), or labyrinth (VI).
Graphical Abstract
Figures
29
Stiff film
x2
x3
x1
o
Bimodular substrate
(a)
(b)
Figure 1
30
Critical strain  cr
0.020
Numerical solutions
Present theory w  A cos(kx1 )  
Previous theory w  A cos(kx1 )
0.017
0.014
0.011
0.008
0.0
0.2
0.4
0.6
0.8
1.0
Stiffness ratio Kc / K t
(a)
Normalized wavelength cr / h
30
Numerical solutions
Present theory w  A cos(kx1 )  
Previous theory w  A cos(kx1 )
28
26
24
22
20
18
0.0
0.2
0.4
0.6
Stiffness ratio Kc / K t
(b)
Figure 2
31
0.8
1.0
Normalized deflection w / h
1.0
Numerical solution
Present theory w  A cos(kx1 )  
Previous theory w  A cos(kx1 )
0.5
0.0
-0.5
-1.0
0
50
100
Coordinate x1 / h
(c)
Figure 2
32
150
checkerboard mode
A0
q 1
p  0 (1D mode)
p2
p2
(hexagonal
mode)
(parallel bead-chain mode)
p2
A0
p  1, q  3
q 3
p  4, q  3
p  1, q  3
q 3
p  4, q  3
A 0
Figure 3
33
0.015
Solid line:
Dash line:
Dot line:
Critical strain  cr
p3
0.014
A0
A0
A0
p2
0.013
0.012
 cr
0.011
0.5
p 1
p0
1.0
1.5
3
2.0
2.5
q
(a)
0.015
Critical strain  cr
Outward hexagon
0.012
Checkerboard
0.009
0.006
0.003
0.0
1D mode
Inward hexagon
0.2
0.4
Simulation
Line: Theory
0.6
Stiffness ratio Kc / K t
(b)
Figure 4
34
0.8
1.0
Figure 5
35
0.013
Solid line:
Dash line:
Dot line:
Critical strain  cr
0.012
A0
A0
A0
0.011
p 1
0.010
0.009
 cr
p2
p  1.5
p0
0.008
0.5
1.0
1.5
2.0
2.5
q
(a)
2.0
0.019
Critical strain  cr
p  0.8, q  1.5
1.0
p
0.016
1.5
0.5
0.013
0.0
-0.5
0.1
0.38
0.2
0.3
0.4
1D mode
0.5
Kc / K t
Bifurcation
0.01
Simulation
Line: Theory
0.007
p  1.6, q  1.5
0.38
0.1
0.4
Stiffness ratio Kc / K t
(b)
Figure 6
36
0.7
1
Figure 7
37
Figure 8
38
Increasing 
A
B
C
Experimental
observations
Numerical
results
1D mode
Parallel
Hexagonal mode
bead-chain mode
Figure 9
39
Figure 10
40
Figure 11
41
Figure 12
42
Figure 13
43
Figure 14
Highlights

Buckling and postbuckling of a stiff film on a compliant bimodular substrate.

Both theoretical analysis and Fourier spectral numerical method are performed.

Substrate nonlinearity greatly affects the selection of surface wrinkling modes.

Non-equi-biaxial compression may engender a diversity of surface morphologies.

Phase diagrams for the wrinkling patterns and their evolutions during postbuckling.
44