In this paper, we study the wrinkling instability of two layers embedded in a homogeneous matrix of infinite size. Using a linear stability analysis, we characterize the wrinkling of the two layers as a function of the layer spacing and the shear moduli ratio between the two materials. When the layers are stiffer than the surrounding matrix, stiffness contrast largely determines the stability behavior of the system. When the layers are softer than the surrounding matrix, stiffness contrast and layer spacing interact to determine critical threshold strain and wavelength, and result in striking discontinuities in wavelength between regimes. When the layers are close to each other, the system has a strong preference for the symmetric wrinkling mode, but as the distance between the two layers increases, the anti-symmetric mode may emerge.

References

References
1.
Biot
,
M. A.
,
1963
, “
Surface Instability of Rubber in Compression
,”
Appl. Sci. Res.
,
12
(
2
), pp.
168
182
.
2.
Allen
,
H. G.
,
1969
,
Analysis and Design of Sandwich Panels
,
Pergamon
,
New York
.
3.
Crosby
,
A. J.
, ed.,
2010
, “
Themed Issue: The Physics of Buckling
,”
Soft Matter
,
6
, pp.
5647
5878
.https://pubs.rsc.org/en/Journals/JournalIssues/SM#!issueid=sm006022
4.
Ambrosi
,
D.
,
Ateshian
,
G. A.
,
Arruda
,
E. M.
,
Cowin
,
S. C.
,
Dumais
,
J.
,
Goriely
,
A.
,
Holzapfel
,
G. A.
,
Humphrey
,
J. D.
,
Kemkemer
,
R.
,
Kuhl
,
E.
,
Olberding
,
J. E.
,
Taber
,
L. A.
, and
Garikipati
,
K.
,
2011
, “
Perspectives on Biological Growth and Remodeling
,”
J. Mech. Phys.
,
59
(
4
), pp.
863
883
.
5.
Li
,
B.
,
Cao
,
Y. P.
,
Feng
,
X. Q.
, and
Gao
,
H.
,
2012
, “
Mechanics of Morphological Instabilities and Surface Wrinkling in Soft Materials: A Review
,”
Soft Matter
,
8
(
21
), pp.
5728
5745
.
6.
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
.
7.
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
.
8.
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
.
9.
Andrianov
,
I. V.
,
Kalamkarov
,
A. L.
, and
Weichert
,
D.
,
2012
, “
Buckling of Fibers in Fiber-Reinforced Composites
,”
Comp. B: Eng.
,
43
(
4
), pp.
2058
2062
.
10.
Douville
,
M.
, and
Le Grognec
,
P.
,
2013
, “
Exact Analytical Solutions for the Local and Global Buckling of Sandwich Beam-Columns Under Various Loadings
,”
Intl. J. Solids Struct.
,
50
(
16–17
), pp.
2597
2609
.
11.
Parnes
,
R.
, and
Chiskis
,
A.
,
2002
, “
Buckling of Nano-Fibre Reinforced Composites: A Re-Examination of Elastic Buckling
,”
J. Mech. Phys. Solids
,
50
(
4
), pp.
855
879
.
12.
Price
,
N. J.
, and
Cosgrove
,
J. W.
,
1990
,
Analysis of Geological Structures
,
Cambridge University Press
,
Cambridge, UK
.
13.
Cloetingh
,
S.
, and
Burov
,
E.
,
2011
, “
Lithospheric Folding and Sedimentary Basin Evolution: A Review and Analysis of Formation Mechanisms
,”
Basin Res.
,
23
(
3
), pp.
257
290
.
14.
Schmalholz
,
S. M.
, and
Schmid
,
D.
,
2012
, “
Folding in Power-Law Viscous Multi-Layers
,”
Phil. Trans. R. Soc. A
,
370
(
1965
), pp.
1798
1826
.
15.
Pauchard
,
L.
, and
Couder
,
Y.
,
2004
, “
Invagination During the Collapse of an Inhomogeneous Spheroidal Shell
,”
Europhys. Lett.
,
66
(
5
), pp.
667
673
.
16.
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
.
17.
Ben Amar
,
M.
, and
Goriely
,
A.
,
2005
, “
Growth and Instability Inelastic Tissues
,”
J. Mech. Phys. Solids
,
53
(
10
), pp.
2284
2319
.
18.
Goriely
,
A.
, and
Ben Amar
,
M.
,
2005
, “
Differential Growth and Instability in Elastic Shells
,”
Phys. Rev. Lett.
,
94
(
19
), p.
198103
.
19.
Moulton
,
D. E.
, and
Goriely
,
A.
,
2011
, “
Circumferential Buckling Instability of a Growing Cylindrical Tube
,”
J. Mech. Phys. Solids
,
59
(
3
), pp.
525
537
.
20.
Ciarletta
,
P.
,
Balbi
,
V.
, and
Kuhl
,
E.
,
2014
, “
Pattern Selection in Growing Tubular Tissues
,”
Phys. Rev. Lett.
,
113
(
24
), p.
248101
.
21.
Ben Amar
,
M.
, and
Borden
,
A.
,
2017
, “
Mimicking Cortex Convolutions Through the Wrinkling of Growing Soft Bilayers
,”
J. Elasticity
,
129
(
1–2
), pp.
213
238
.
22.
Cao
,
Y. P.
, and
Hutchinson
,
J. W.
,
2012
, “
Wrinkling Phenomena in Neo-Hookean Film/Substrate Bilayers
,”
ASME J. Appl. Mech.
,
79
(
3
), p.
031019
.
23.
Sun
,
J. Y.
,
Xia
,
S.
,
Moon
,
M. W.
,
Oh
,
K. H.
, and
Kim
,
K. S.
,
2012
, “
Folding Wrinkles of a Thin Stiff Layer on a Soft Substrate
,”
Proc. Roy. Soc. A.
,
468
(
2140
), pp.
932
953
.
24.
Hong
,
W.
,
Zhao
,
X.
, and
Suo
,
Z.
,
2009
, “
Formation of Creases on the Surfaces of Elastomers and Gels
,”
Appl. Phys. Lett.
,
95
, pp.
2009
2011
.
25.
Jin
,
L.
,
Auguste
,
A.
,
Hayward
,
R. C.
, and
Suo
,
Z.
,
2015
, “
Bifurcation Diagrams for the Formation of Wrinkles or Creases in Soft Bilayers
,”
ASME J. Appl. Mech.
,
82
(
6
), p.
061008
.
26.
Tallinen
,
T.
, and
Biggins
,
J. S.
,
2015
, “
Mechanics of Invagination and Folding: Hybridized Instabilities When One Soft Tissue Grows on Another
,”
Phys. Rev. E
,
92
, p. 022720.
27.
Zang
,
J.
,
Zhao
,
X.
,
Cao
,
Y.
, and
Hutchinson
,
J. W.
,
2012
, “
Localized Ridge Wrinkling of Stiff Films on Compliant Substrates
,”
J. Mech. Phys. Solids
,
60
(
7
), pp.
1265
1279
.
28.
Budday
,
S.
,
Kuhl
,
E.
, and
Hutchinson
,
J. W.
,
2015
, “
Period-Doubling and Period Tripling in Growing Bilayered Systems
,”
Philos. Mag.
,
95
(
28–30
), pp.
3208
3224
.
29.
Huang
,
Z.
,
Hong
,
W.
, and
Suo
,
Z.
,
2004
, “
Evolution of Wrinkles in Hard Films on Soft Substrates
,”
Phys. Rev. E
,
70
(3), p.
030601
.
30.
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
.
31.
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
.
32.
Audoly
,
B.
, and
Boudaoud
,
A.
, “
Buckling of a Stiff Film Bound to a Compliant substrate—Part II: A Global Scenario for the Formation of Herringbone Pattern
,”
J. Mech. Phys. Solids
,
56
(
7
), pp.
2422
2443
.
33.
Holland
,
M. A.
,
Li
,
B.
,
Feng
,
X. Q.
, and
Kuhl
,
E.
,
2017
, “
Instabilities of Soft Films on Compliant Substrates
,”
J. Mech. Phys. Solids
,
98
, pp.
350
365
.
34.
Xu
,
F.
,
Potier-Ferry
,
M.
,
Belouettar
,
S.
, and
Cong
,
Y.
,
2014
, “
3D Finite Element Modeling for Instabilities in Thin Films on Soft Substrates
,”
Int. J. Solids Struct.
,
51
(
21–22
), pp.
3619
3632
.
35.
Xu
,
F.
,
Potier-Ferry
,
M.
,
Belouettar
,
S.
, and
Hu
,
H.
,
2015
, “
Multiple Bifurcations in Wrinkling Analysis of Thin Films on Compliant Substrates
,”
Int. J. Non-Linear Mech.
,
76
, pp.
203
222
.
36.
Xu
,
F.
, and
Potier-Ferry
,
M.
,
2016
, “
A Multi-Scale Modeling Framework for Instabilities of Film/Substrate Systems
,”
J. Mech. Phys. Solids
,
86
, pp.
150
172
.
37.
Flynn
,
C.
, and
Mccormack
,
B. A. O.
,
2010
, “
Simulating the Wrinkling and Ageing of Skin With a Multi-Layer Finite Element Model
,”
J. Biomech.
,
43
(
3
), pp.
442
448
.
38.
Eskandari
,
M.
,
Pfaller
,
M. R.
, and
Kuhl
,
E.
,
2013
, “
On the Role of Mechanics in Chronic Lung Disease
,”
Materials
,
6
(
12
), pp.
5639
5658
.
39.
Lejeune
,
E.
,
Javili
,
A.
, and
Linder
,
C.
,
2016
, “An Algorithmic Approach to Multi-Layer Wrinkling,”
Extreme Mech. Lett.
,
7
, pp.
10
17
.
40.
Gao
,
C.
, and
Li
,
C.
,
2017
, “
Tuning the Wrinkling Patterns of an Interfacial/Coating Layer Via a Regulation Interphase
,”
Int. J. Solids Struct.
,
104–105
, pp.
92
102
.
41.
Bertoldi
,
K.
, and
Lopez-Pamies
,
O.
,
2012
, “
Some Remarks on the Effect of Interphases on the Mechanical Response and Stability of Fiber-Reinforced Elastomers
,”
ASME J. Appl. Mech.
,
79
(
3
), p.
031023
.
42.
Colin
,
J.
, and
Holland
,
M. A.
,
2019
, “
Layer Wrinkling in an Inhomogeneous Matrix
,”
Intl. J. Solids Struct.
,
156–157
, pp.
119
125
.
43.
Wolfram Research, Inc.
,
2015
,
Mathematica, Version 10.1.0
,
Wolfram Research
,
Champaign, IL
.
44.
Ridders
,
C.
,
1979
, “
A New Algorithm for Computing a Single Root of a Real Continuous Function
,”
IEEE Trans. Circuits Syst.
,
26
(
11
), pp.
979
980
.
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