Subject to compression, elastic materials may undergo bifurcation of various kinds. A homogeneous material forms creases, whereas a bilayer consisting of a stiff film and a compliant substrate forms wrinkles. Here, we show several new types of bifurcation behavior for bilayers consisting of films and substrates of comparable elastic moduli. Depending on the ratios of moduli and thicknesses of the two materials, the critical strain for the onset of creases can be either smaller or larger than that for the onset of wrinkles. When the critical strain for the onset of creases is lower than that of wrinkles, creases can be subcritical or supercritical. When the critical strain for the onset of wrinkles is lower than that of creases, wrinkles can further channel to creases at a strain much lower than the critical strain for the onset of creases in a homogeneous material. Experiments, conducted with bilayer polydimethylsiloxane (PDMS) structures subject to compressive loading, show that the different types of bifurcation behavior agree with the theoretical predictions.

References

1.
Genzer
,
J.
, and
Groenewold
,
J.
,
2006
, “
Soft Matter With Hard Skin: From Skin Wrinkles to Templating and Material Characterization
,”
Soft Matter
,
2
(
4
), pp.
310
323
.10.1039/b516741h
2.
Chen
,
X.
, and
Yin
,
J.
,
2010
, “
Buckling Patterns of Thin Films on Curved Compliant Substrates With Applications to Morphogenesis and Three-Dimensional Micro-Fabrication
,”
Soft Matter
,
6
(
22
), pp.
5667
5680
.10.1039/c0sm00401d
3.
Li
,
B.
,
Cao
,
Y. P.
,
Feng
,
X. Q.
, and
Gao
,
H. J.
,
2012
, “
Mechanics of Morphological Instabilities and Surface Wrinkling in Soft Materials: A Review
,”
Soft Matter
,
8
(
21
), pp.
5728
5745
.10.1039/c2sm00011c
4.
Khang
,
D. Y.
,
Jiang
,
H. Q.
,
Huang
,
Y.
, and
Rogers
,
J. A.
,
2006
, “
A Stretchable Form of Single-Crystal Silicon for High-Performance Electronics on Rubber Substrates
,”
Science
,
311
(
5758
), pp.
208
212
.10.1126/science.1121401
5.
Stafford
,
C. M.
,
Harrison
,
C.
,
Beers
,
K. L.
,
Karim
,
A.
,
Amis
,
E. J.
,
Vanlandingham
,
M. R.
,
Kim
,
H. C.
,
Volksen
,
W.
,
Miller
,
R. D.
, and
Simonyi
,
E. E.
,
2004
, “
A Buckling-Based Metrology for Measuring the Elastic Moduli of Polymeric Thin Films
,”
Nat. Mater.
,
3
(
8
), pp.
545
550
.10.1038/nmat1175
6.
Lin
,
P. C.
,
Vajpayee
,
S.
,
Jagota
,
A.
,
Hui
,
C. Y.
, and
Yang
,
S.
,
2008
, “
Mechanically Tunable Dry Adhesive From Wrinkled Elastomers
,”
Soft Matter
,
4
(
9
), pp.
1830
1835
.10.1039/b802848f
7.
Chan
,
E. P.
,
Smith
,
E. J.
,
Hayward
,
R. C.
, and
Crosby
,
A. J.
,
2008
, “
Surface Wrinkles for Smart Adhesion
,”
Adv. Mater.
,
20
(
4
), pp.
711
716
.10.1002/adma.200701530
8.
Lin
,
P. C.
, and
Yang
,
S.
,
2009
, “
Mechanically Switchable Wetting on Wrinkled Elastomers With Dual-Scale Roughness
,”
Soft Matter
,
5
(
5
), pp.
1011
1018
.10.1039/b814145b
9.
Kim
,
J.
,
Yoon
,
J.
, and
Hayward
,
R. C.
,
2010
, “
Dynamic Display of Biomolecular Patterns Through an Elastic Creasing Instability of Stimuli-Responsive Hydrogels
,”
Nat. Mater.
,
9
(
2
), pp.
159
164
.10.1038/nmat2606
10.
Kucken
,
M.
, and
Newell
,
A. C.
,
2004
, “
A Model for Fingerprint Formation
,”
Europhys. Lett.
,
68
(
1
), pp.
141
146
.10.1209/epl/i2004-10161-2
11.
Pauchard
,
L.
, and
Couder
,
Y.
,
2004
, “
Invagination During the Collapse of an Inhomogeneous Spheroidal Shell
,”
Europhys. Lett.
,
66
(
5
), pp.
667
673
.10.1209/epl/i2003-10242-8
12.
Savin
,
T.
,
Kurpios
,
N. A.
,
Shyer
,
A. E.
,
Florescu
,
P.
,
Liang
,
H. Y.
,
Mahadevan
,
L.
, and
Tabin
,
C. J.
,
2011
, “
On the Growth and Form of the Gut
,”
Nature
,
476
(
7358
), pp.
57
62
.10.1038/nature10277
13.
Richman
,
D. P.
,
Stewart
,
R. M.
,
Hutchinson
,
J. W.
, and
Caviness
,
V. S.
,
1975
, “
Mechanical Model of Brain Convolutional Development
,”
Science
,
189
(
4196
), pp.
18
21
.10.1126/science.1135626
14.
Bayly
,
P. V.
,
Taber
,
L. A.
, and
Kroenke
,
C. D.
,
2014
, “
Mechanical Forces in Cerebral Cortical Folding: A Review of Measurements and Models
,”
J. Mech. Behav. Biomed. Mater.
,
29
(S), pp.
568
581
.10.1016/j.jmbbm.2013.02.018
15.
Budday
,
S.
,
Raybaud
,
C.
, and
Kuhl
,
E.
,
2014
, “
A Mechanical Model Predicts Morphological Abnormalities in the Developing Human Brain
,”
Sci. Rep.
,
4
, p.
5644
.10.1038/srep05644
16.
Lidmar
,
J.
,
Mirny
,
L.
, and
Nelson
,
D. R.
,
2003
, “
Virus Shapes and Buckling Transitions in Spherical Shells
,”
Phys. Rev. E
,
68
(
5
), p.
051910
.10.1103/PhysRevE.68.051910
17.
Dumais
,
J.
, and
Steele
,
C. R.
,
2000
, “
New Evidence for the Role of Mechanical Forces in the Shoot Apical Meristem
,”
J. Plant Growth Regul.
,
19
(
1
), pp.
7
18
.10.1007/s003440000003
18.
Liang
,
H. Y.
, and
Mahadevan
,
L.
,
2009
, “
The Shape of a Long Leaf
,”
Proc. Natl. Acad. Sci. U.S.A.
,
106
(
52
), pp.
22049
22054
.10.1073/pnas.0911954106
19.
Liang
,
H. Y.
, and
Mahadevan
,
L.
,
2011
, “
Growth, Geometry, and Mechanics of a Blooming Lily
,”
Proc. Natl. Acad. Sci. U.S.A.
,
108
(
14
), pp.
5516
5521
.10.1073/pnas.1007808108
20.
Armon
,
S.
,
Efrati
,
E.
,
Kupferman
,
R.
, and
Sharon
,
E.
,
2011
, “
Geometry and Mechanics in the Opening of Chiral Seed Pods
,”
Science
,
333
(
6050
), pp.
1726
1730
.10.1126/science.1203874
21.
Shipman
,
P. D.
, and
Newell
,
A. C.
,
2004
, “
Phyllotactic Patterns on Plants
,”
Phys. Rev. Lett.
,
92
(
16
), p.
168102
.10.1103/PhysRevLett.92.168102
22.
Dervaux
,
J.
, and
Amar
,
M. B.
,
2008
, “
Morphogenesis of Growing Soft Tissues
,”
Phys. Rev. Lett.
,
101
(
6
), p.
068101
.10.1103/PhysRevLett.101.068101
23.
Katifori
,
E.
,
Alben
,
S.
,
Cerda
,
E.
,
Nelson
,
D. R.
, and
Dumais
,
J.
,
2010
, “
Foldable Structures and the Natural Design of Pollen Grains
,”
Proc. Natl. Acad. Sci. U.S.A.
,
107
(
17
), pp.
7635
7639
.10.1073/pnas.0911223107
24.
Yin
,
J.
,
Cao
,
Z. X.
,
Li
,
C. R.
,
Sheinman
,
I.
, and
Chen
,
X.
,
2008
, “
Stress-Driven Buckling Patterns in Spheroidal Core/Shell Structures
,”
Proc. Natl. Acad. Sci. U.S.A.
,
105
(
49
), pp.
19132
19135
.10.1073/pnas.0810443105
25.
Hohlfeld
,
E.
, and
Mahadevan
,
L.
,
2011
, “
Unfolding the Sulcus
,”
Phys. Rev. Lett.
,
106
(
10
), p.
105702
.10.1103/PhysRevLett.106.105702
26.
Biot
,
M. A.
,
1963
, “
Surface Instability of Rubber in Compression
,”
Appl. Sci. Res.
,
12
(
2
), pp.
168
182
.
27.
Chen
,
X.
, and
Hutchinson
,
J. W.
,
2004
, “
Herringbone Buckling Patterns of Compressed Thin Films on Compliant Substrates
,”
ASME J. Appl. Mech.
,
71
(
5
), pp.
597
603
.10.1115/1.1756141
28.
Audoly
,
B.
, and
Boudaoud
,
A.
,
2008
, “
Buckling of a Stiff Film Bound to a Compliant Substrate—Part I: Formulation, Linear Stability of Cylindrical Patterns, Secondary Bifurcations
,”
J. Mech. Phys. Solids
,
56
(
7
), pp.
2401
2421
.10.1016/j.jmps.2008.03.003
29.
Song
,
J.
,
Jiang
,
H.
,
Liu
,
Z. J.
,
Khang
,
D. Y.
,
Huang
,
Y.
,
Rogers
,
J. A.
,
Lu
,
C.
, and
Koh
,
C. G.
,
2008
, “
Buckling of a Stiff Thin Film on a Compliant Substrate in Large Deformation
,”
Int. J. Solids Struct.
,
45
(
10
), pp.
3107
3121
.10.1016/j.ijsolstr.2008.01.023
30.
Kim
,
D. H.
,
Lu
,
N. S.
,
Ma
,
R.
,
Kim
,
Y. S.
,
Kim
,
R. H.
,
Wang
,
S. D.
,
Wu
,
J.
,
Won
,
S. M.
,
Tao
,
H.
,
Islam
,
A.
,
Yu
,
K. J.
,
Kim
,
T. I.
,
Chowdhury
,
R.
,
Ying
,
M.
,
Xu
,
L. Z.
,
Li
,
M.
,
Chung
,
H. J.
,
Keum
,
H.
,
McCormick
,
M.
,
Liu
,
P.
,
Zhang
,
Y. W.
,
Omenetto
,
F. G.
,
Huang
,
Y. G.
,
Coleman
,
T.
, and
Rogers
,
J. A.
,
2011
, “
Epidermal Electronics
,”
Science
,
333
(
6044
), pp.
838
843
.10.1126/science.1206157
31.
Keplinger
,
C.
,
Sun
,
J. Y.
,
Foo
,
C. C.
,
Rothemund
,
P.
,
Whitesides
,
G. M.
, and
Suo
,
Z. G.
,
2013
, “
Stretchable, Transparent, Ionic Conductors
,”
Science
,
341
(
6149
), pp.
984
987
.10.1126/science.1240228
32.
Kolle
,
M.
,
Lethbridge
,
A.
,
Kreysing
,
M.
,
Baumberg
,
J. J.
,
Aizenberg
,
J.
, and
Vukusic
,
P.
,
2013
, “
Bio-Inspired Band-Gap Tunable Elastic Optical Multilayer Fibers
,”
Adv. Mater.
,
25
(
15
), pp.
2239
2245
.10.1002/adma.201203529
33.
Bowden
,
N.
,
Brittain
,
S.
,
Evans
,
A. G.
,
Hutchinson
,
J. W.
, and
Whitesides
,
G. M.
,
1998
, “
Spontaneous Formation of Ordered Structures in Thin Films of Metals Supported on an Elastomeric Polymer
,”
Nature
,
393
(
6681
), pp.
146
149
.10.1038/30193
34.
Cerda
,
E.
, and
Mahadevan
,
L.
,
2003
, “
Geometry and Physics of Wrinkling
,”
Phys. Rev. Lett.
,
90
(
7
), p.
074302
.10.1103/PhysRevLett.90.074302
35.
Tanaka
,
T.
,
Sun
,
S. T.
,
Hirokawa
,
Y.
,
Katayama
,
S.
,
Kucera
,
J.
,
Hirose
,
Y.
, and
Amiya
,
T.
,
1987
, “
Mechanical Instability of Gels at the Phase-Transition
,”
Nature
,
325
(
6107
), pp.
796
798
.10.1038/325796a0
36.
Gent
,
A. N.
, and
Cho
,
I. S.
,
1999
, “
Surface Instabilities in Compressed or Bent Rubber Blocks
,”
Rubber Chem. Technol.
,
72
(
2
), pp.
253
262
.10.5254/1.3538798
37.
Ghatak
,
A.
, and
Das
,
A. L.
,
2007
, “
Kink Instability of a Highly Deformable Elastic Cylinder
,”
Phys. Rev. Lett.
,
99
(
7
), p.
076101
.10.1103/PhysRevLett.99.076101
38.
Trujillo
,
V.
,
Kim
,
J.
, and
Hayward
,
R. C.
,
2008
, “
Creasing Instability of Surface-Attached Hydrogels
,”
Soft Matter
,
4
(
3
), pp.
564
569
.10.1039/b713263h
39.
Hong
,
W.
,
Zhao
,
X. H.
, and
Suo
,
Z. G.
,
2009
, “
Formation of Creases on the Surfaces of Elastomers and Gels
,”
Appl. Phys. Lett.
,
95
(
11
), p.
111901
.10.1063/1.3211917
40.
Wang
,
Q. M.
,
Zhang
,
L.
, and
Zhao
,
X. H.
,
2011
, “
Creasing to Cratering Instability in Polymers Under Ultrahigh Electric Fields
,”
Phys. Rev. Lett.
,
106
(
11
), p.
118301
.10.1103/PhysRevLett.106.118301
41.
Cai
,
S. Q.
,
Chen
,
D. Y.
,
Suo
,
Z. G.
, and
Hayward
,
R. C.
,
2012
, “
Creasing Instability of Elastomer Films
,”
Soft Matter
,
8
(
5
), pp.
1301
1304
.10.1039/C2SM06844C
42.
Weiss
,
F.
,
Cai
,
S.
,
Hu
,
Y.
,
Kang
,
M. K.
,
Huang
,
R.
, and
Suo
,
Z.
,
2013
, “
Creases and Wrinkles on the Surface of a Swollen Gel
,”
J. Appl. Phys.
,
114
(7), p.
073507
.10.1063/1.4818943
43.
Dervaux
,
J.
,
Couder
,
Y.
,
Guedeau-Boudeville
,
M. A.
, and
Amar
,
M. B.
,
2011
, “
Shape Transition in Artificial Tumors: From Smooth Buckles to Singular Creases
,”
Phys. Rev. Lett.
,
107
(
1
), p.
018103
.10.1103/PhysRevLett.107.018103
44.
Diab
,
M. Z.
,
Zhang
,
T.
,
Zhao
,
R. K.
,
Gao
,
H. J.
, and
Kim
,
K. S.
,
2013
, “
Ruga Mechanics of Creasing: From Instantaneous to Setback Creases
,”
Proc. R. Soc. London, Ser. A
,
469
(
2157
), p.
20120753
.10.1098/rspa.2012.0753
45.
Hutchinson
,
J. W.
,
2013
, “
The Role of Nonlinear Substrate Elasticity in the Wrinkling of Thin Films
,”
Philos. Trans. R. Soc., A
,
371
(
1993
), p.
20120422
.10.1098/rsta.2012.0422
46.
Wu
,
Z.
,
Bouklas
,
N.
, and
Huang
,
R.
,
2013
, “
Swell-Induced Surface Instability of Hydrogel Layers With Material Properties Varying in Thickness Direction
,”
Int. J. Solids Struct.
,
50
(
3–4
), pp.
578
587
.10.1016/j.ijsolstr.2012.10.022
47.
Wang
,
Q. M.
, and
Zhao
,
X. H.
,
2014
, “
Phase Diagrams of Instabilities in Compressed Film-Substrate Systems
,”
ASME J. Appl. Mech.
,
81
(
5
), p.
051004
.10.1115/1.4025828
48.
Cao
,
Y. P.
, and
Hutchinson
,
J. W.
,
2012
, “
Wrinkling Phenomena in Neo-Hookean Film/Substrate Bilayers
,”
ASME J. Appl. Mech.
,
79
(
3
), p.
031019
.10.1115/1.4005960
49.
Jin
,
L.
,
Chen
,
D.
,
Hayward
,
R. C.
, and
Suo
,
Z.
,
2014
, “
Creases on the Interface Between Two Soft Materials
,”
Soft Matter
,
10
(
2
), pp.
303
311
.10.1039/C3SM51512E
50.
Chen
,
D. Y.
,
Cai
,
S. Q.
,
Suo
,
Z. G.
, and
Hayward
,
R. C.
,
2012
, “
Surface Energy as a Barrier to Creasing of Elastomer Films: An Elastic Analogy to Classical Nucleation
,”
Phys. Rev. Lett.
,
109
(
3
), p.
038001
.10.1103/PhysRevLett.109.038001
51.
Hohlfeld
,
E.
, and
Mahadevan
,
L.
,
2012
, “
Scale and Nature of Sulcification Patterns
,”
Phys. Rev. Lett.
,
109
(
2
), p.
025701
.10.1103/PhysRevLett.109.025701
52.
Chen
,
D.
,
Jin
,
L.
,
Suo
,
Z.
, and
Hayward
,
R. C.
,
2014
, “
Controlled Formation and Disappearance of Creases
,”
Mater. Horiz.
,
1
(
2
), pp.
207
213
.10.1039/C3MH00107E
53.
Jin
,
L.
,
Takei
,
A.
, and
Hutchinson
,
J. W.
, “
Mechanics of Wrinkle/Ridge Transitions in Thin Film/Substrate Systems
,” J. Mech. Phys. Solids (submitted).
54.
Riks
,
E.
,
1972
, “
The Application of Newton's Method to the Problem of Elastic Stability
,”
ASME J. Appl. Mech.
,
39
(
4
), p.
1060
.10.1115/1.3422829
55.
Pocivavsek
,
L.
,
Dellsy
,
R.
,
Kern
,
A.
,
Johnson
,
S.
,
Lin
,
B. H.
,
Lee
,
K. Y. C.
, and
Cerda
,
E.
,
2008
, “
Stress and Fold Localization in Thin Elastic Membranes
,”
Science
,
320
(
5878
), pp.
912
916
.10.1126/science.1154069
56.
Cai
,
S. Q.
,
Bertoldi
,
K.
,
Wang
,
H. M.
, and
Suo
,
Z. G.
,
2010
, “
Osmotic Collapse of a Void in an Elastomer: Breathing, Buckling and Creasing
,”
Soft Matter
,
6
(
22
), pp.
5770
5777
.10.1039/c0sm00451k
57.
Ogden
,
R. W.
,
1997
,
Non-Linear Elastic Deformation
,
Dover Publications
,
New York
.
58.
Huang
,
Z. Y.
,
Hong
,
W.
, and
Suo
,
Z.
,
2005
, “
Nonlinear Analyses of Wrinkles in a Film Bonded to a Compliant Substrate
,”
J. Mech. Phys. Solids
,
53
(
9
), pp.
2101
2118
.10.1016/j.jmps.2005.03.007
59.
Jiang
,
H. Q.
,
Khang
,
D. Y.
,
Song
,
J. Z.
,
Sun
,
Y. G.
,
Huang
,
Y. G.
, and
Rogers
,
J. A.
,
2007
, “
Finite Deformation Mechanics in Buckled Thin Films on Compliant Supports
,”
Proc. Natl. Acad. Sci. U.S.A.
,
104
(
40
), pp.
15607
15612
.10.1073/pnas.0702927104
You do not currently have access to this content.