Supplementary material
Tailoring the wrinkle pattern of a microstructured membrane
Dong Yan,1 Kai Zhang,1 Fujun Peng,2 and Gengkai Hu1
1) School of Aerospace Engineering, Beijing Institute of Technology, Beijing 100081, China
2) Aerospace System Engineering Shanghai, Shanghai 201108, China
Experiment
The membrane used in the experiment is a polyimide membrane with dimensions of 125 mm in
length 2L, 50 mm in width 2C and 0.025 mm in thickness t. Young modulus E is measured to be
3.0±0.1 GPa by uniaxial tensile test and Poisson ratio ν is assumed to be 0.31 according to Ref. [1].
A pair of holes with radius r0 is punched along the central line of the membrane, and the distance
x0 from the hole to the clamped end varies according to design purpose. Two ends of the specimen
are clamped between aluminum sheets and the specimen is stretched by a testing machine to 5%
strain.
Postbuckling analysis
The postbuckling analysis is carried out by using the commercial finite element software
ABAQUS to study the effect of holes on the wrinkle pattern of the membrane. The shell element
S4R and linear elastic constitutive model are used. The geometric parameters (L, C, t, r0, x0) and
the material properties (E, ν) used in the numerical simulation are the same as those in the
experiment. A mesh sensitivity study is performed to ensure that the element sizes are sufficiently
fine. The size of the element is about 0.1 mm × 0.1 mm near the holes and 0.5 mm × 0.5 mm in
the other regions. Geometric imperfections consisting of the superposition of several buckling
modes with a maximum value of 0.01t are imposed at the nodes of the original planar mesh. As
shown in Fig. S1, the numerical results by postbuckling analysis (right) agree well with the
experimental results (left).
Stress analysis
The wrinkling of a two-end clamped stretched membrane without holes has been explained by
former researchers [2-5]. The clamped boundaries, preventing the membrane from contraction due
to the Poisson effect, lead to compressive stress at some distance from the clamped ends, and this
compressive stress finally triggers wrinkling [2-4]. However the stress analysis on the membrane
is either qualitative [4] or calculated by FEM [5], and there are few reports on an analytical stress
solution, which will be helpful in parametric study and clearly understanding the mechanism of
wrinkling. According to Refs. [2-4], the deforming process of a stretched membrane under
clamped boundaries can be divided into two parts: a uniform uniaxial tension force is first
introduced to stretch the membrane and then a shear force is provided to prevent the contraction
due to the clamped end. For simplification, a pair of concentrated forces F is used to replace the
nonuniform distributed shear force on the clamped end, as shown in Fig. S2(a). According to the
theory of elasticity [6], by assuming the solution in the form of a Fourier series, the analytical
stress of a two-end clamped uniform membrane under stretched strain εx can be obtained as
1
 x  E x 
y 
2F
L
F 2F

L
L


m 1


B 
 m 
xr  ,
 cosh  m yr   Byr sinh  m yr   cos 
 


 
  A  m


m 1


B 
 m 
xr ,
 cosh  m yr   Byr sinh  m yr   cos 
 

 

  A  m
(S1)

 m 
 xy

 A sinh  m yr   Byr cosh  m yr   sin   xr ,


m 1
m cosh m
m sinh m
m
m
A   1
, B   1
,
cosh m sinh m  m
cosh m sinh m  m
2F

L
where xr=x/L and yr=y/C are the coordinates along the length 2L and width 2C direction,
respectively, with the origin at the center of the membrane. λ=C/L is the width-length ratio and
mλ=mπλ. The concentrated force F=ανεxEC, and α is a factor related to the equivalence between
the concentrated force and the nonuniform distributed shear force, which keeps nearly constant for
slender membranes with different sizes (α: 0.378-0.382 when λ: 0.01-0.45, calculated by FEM).
The results of FEM are calculated by using ABAQUS with plane stress assumption. The
geometric parameters (L, C, r0, x0) and the material properties (E, ν) used in the numerical
simulation are the same as those in the experiment. The element CPS4R and the linear elastic
constitutive model are adopted. The size of the element is about 0.1 mm × 0.1 mm near the holes
and 0.5 mm × 0.5 mm in the other regions. The analytical stress solution for a slender membrane
with different λ (λ≤0.45) has a good agreement with the results by FEM as shown in Fig. S3.
Similarly, when a hole is introduced into a slender membrane, the corresponding stress could be
obtained by considering a plate with a circular hole under a uniform tension (Part 1) and
concentrated forces (Part 2) as shown in Fig. S2(b). For further simplification, when the hole is
much smaller comparing with the size of membrane, the stress in Part 1 could be approximately
described by the stress solution of an infinite plate with a circular hole under a uniform tension [6],
written in polar coordinates r-θ as
r 
E x 
1  E x 
1 
1 
1  2  
 1  2  1  3 2  cos 2 ,
2  dr 
2  d r 
dr 
 
E x 
1  E x 
1 
1  2  
 1  3 4  cos 2 ,
2  dr 
2 
dr 
 r    r
dr 
(S2)
E 
1 
1 
  x 1  2 1  3 2  sin 2 ,
2  d r 
dr 
( x  xh ) 2  ( y  yh ) 2
r

 1,
r0
r0
where r0, xh, and yh are the radius, x-coordinate, and y-coordinate of the hole’s center, respectively.
More accurately, stress solution of a finite plate with a circular hole under a uniform tension could
be used. The influence of holes on the stress of the membrane is checked by FEM. We first
introduce the holes into Part 2 and then compare its stress distribution with that in Part 2. We find
that there is little difference between them, especially in the central region of the membrane away
from holes. However, the difference on the stress caused by the holes in Part 1 is significant.
Therefore, we neglect the holes in Part 2. The analytical stress of a stretched membrane with holes
could finally be obtained by superposing the stress solution of an infinite plate with a circular hole
2
under a uniform tension [Eq. (S2)] and the stress solution in Eq. (S1) with a zero uniform tension.
The analytical stress for the membrane with holes at different positions is verified by FEM as
shown in Fig. S4.
Furthermore, in order to have a clearer understanding of the superposition stress, two effective
forces Fy1 and Fy2 are defined, which are the integrals of σy in Part 1 and Part 2 [Fig. S2(b)],
respectively, with respect to y from -C/2 to C/2. The domain of integral is determined by
considering the distribution region of wrinkles and holes in the membrane. Then the effective
force Fy1 is written as
Fy1 
C
2


x x
4  12rr2
16rr2 
1
2 

dy

2
E

C
r


, t  r hr , t  rr ,
x
r
2
2
3
C y1
1  4t 1  4t 2  1  4t 2  




2
(S3)
where xhr=xh/L and rr=r0/C represent the x-coordinate and radius of the hole’s center, respectively.
The interaction of the two holes is neglected because of the long distance between them. The
effective force Fy2 is written as
Fy 2 
C
2

C

2
y2


m
m

dy   x EC 1  8  1   m  cos   xr   ,



m 1

sinh 2
where  (m ) 
(S4)
m 
m
m 
4cosh   m sinh  

2 
2
2 
.
m  2m  sinh 2m 
Based on the analysis above, the effect of holes on the stress distribution and wrinkle pattern
could be described by Fy=Fy1L+Fy1R+Fy2, where Fy1L and Fy1R are the effective forces caused by the
left hole and the right hole, respectively. As shown in Fig. S2(b), the plot of Fy2, effective force of
the stretched uniform membrane with two clamped ends, behaves as compressive in the center of
the membrane and tensile near the clamped ends. But the trend of Fy1L+Fy1R, effective force caused
by the left hole and the right hole, is reversed, i.e., a remarkable compressive region near the hole
and then a tensile region relatively far from the hole. So the moving of Fy1L and Fy1R by varying
the position of holes will lead to different superposition results. It should be mentioned that,
effective force Fy, reflected a global mechanical state, can be used to analyze the mechanism of
buckling together with local stress field because of the high compressive stress near the holes.
References
[1]
[2]
[3]
[4]
[5]
[6]
Y. W. Wong and S. Pellegrino, J. Mech. Mater. Struct. 1, 1 (2006).
E. Cerda and L. Mahadevan, Phys. Rev. Lett. 90, 074302 (2003).
E. Cerda, K. Ravi-Chandar, and L. Mahadevan, Nature 419, 579 (2002).
N. Friedl, F. G. Rammerstorfer, and F. D. Fischer, Comput. Struct. 78, 185 (2000).
V. Nayyar, K. Ravi-Chandar, and R. Huang, Int. J. Solids Struct. 48, 3471 (2011).
S. P. Timoshenko and J. N. Goodier, Theory of Elasticity (McGraw-Hill Book Company,
Tokyo, 1970).
3
FIG. S1. Wrinkle patterns of a two-end clamped stretched membrane at 5% strain observed in
experiment (left) and numerical postbuckling analysis (right): (a) membrane without holes,
membrane with holes of (b) x0/L=0.05 r0=1.5 mm, (c) x0/L=0.15 r0=1.5 mm, (d) x0/L=0.40 r0=1.5
mm, (e) x0/L=0.05 r0=2.0 mm, (f) x0/L=0.05 r0=3.0 mm, and (g) x0/L=0.15 r0=5.0 mm.
4
FIG. S2. Schematic of the approximate mechanical model for a two-end clamped stretched
membrane (a) without holes and (b) with a pair of holes.
FIG. S3. σy of a two-end clamped stretched uniform membrane with different width-length ratio λ
at 5% strain calculated by the proposed analytical solution (left) and FEM (right): (a) λ=0.2, (b)
λ=0.25, (c) λ=0.4, (d) λ=0.45, and (e) λ=0.6.
5
FIG. S4. σy of a two-end clamped stretched membrane (λ=0.4) with holes (r0=1.5 mm) at different
positions at 5% strain calculated by the proposed analytical solution (left) and FEM (right): (a)
membrane without holes, membrane with holes at (b) x0/L=0.05, (c) x0/L=0.15, and (d) x0/L=0.40.
6