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Journal of the Mechanics and Physics of Solids

journal homepage: www.elsevier.com/locate/jmps

Instabilities of soft films on compliant substrates

M.A. Hollanda, B. Lib, X.Q. Fengb, E. Kuhla,c,⁎

a Department of Mechanical Engineering, Stanford University, Stanford, CA 94305, USAb Institute of Biomechanics and Medical Engineering, Department of Engineering Mechanics, Tsinghua University, Beijing 100084, Chinac Department of Bioengineering, Stanford University, Stanford, CA 94305, USA

A R T I C L E I N F O

Keywords:InstabilitiesBifurcationBilayered systemSoft matterLiving matter

A B S T R A C T

Instabilities in bilayered systems can generate a wide variety of patterns ranging from simplefolds, wrinkles, and creases to complex checkerboards, hexagons, and herringbones. Physics-based theories traditionally model these systems as a thin film on a thick substrate underconfined compression and assume that the film is orders of magnitude stiffer than the substrate.However, instability phenomena in soft films on soft substrates remain insufficiently understood.Here we show that soft bilayered systems are highly sensitive to the stiffness ratio, boundaryconditions, and mode of compression. In a systematic analysis over a wide range of stiffnessratios, from β0.1 < < 1000, for eight different compression modes including whole-domaincompression, substrate prestretch, and film growth, we observe significantly different instabilitycharacteristics in the low-stiffness-contrast regime, for β < 10. While systems with inversestiffness ratios under whole-domain compression are unstable for a wide range of wrinklingmodes, under film-only compression, the same systems display distinct wrinklingmodes.Strikingly, these discrepancies disappear when using measures of effective strain,effective stiffness, and effective wavelength. Our study suggests that future instability studiesshould use these effective measures to standardize their findings. Our results have importantapplications in soft matter and living matter physics, where stiffness contrasts are low and smallenvironmental changes can have large effects on morphogenesis, pattern selection, and theevolution of shape.

1. Motivation

Bilayered systems, consisting of a thin film of one material on a thicker substrate of another, are found in a variety of applications(Li et al., 2012). Under compression, they can experience one of the several instability modes, including wrinkling (Cao andHutchinson, 2012; Budday et al., 2014a), folding (Sultan and Boudaoud, 2008; Sun et al., 2012), creasing (Cao and Hutchinson,2011; Hong et al., 2009; Jin et al., 2015), and cusping (Tallinen et al., 2014; Tallinen and Biggins, 2015). In some instances, theseinstabilities are undesired failure modes (Hong et al., 2009); in others, they are mechanical features that can be tuned for a desiredproperty or performance (Jiang et al., 2008). Traditionally, bilayered systems for engineering applications involved a high stiffnessratio between the stiff topfilm and the soft bottom substrate, β E E= /f s, as in sandwich panels (Allen, 1969) or silicon (Jiang et al.,2007; Song et al., 2008) and gold (Sun et al., 2012) films on polydimethylsiloxane substrates. These systems were heavily studied,analytically (Allen, 1969; Biot, 1937), experimentally (Jiang et al., 2007; Jin et al., 2015; Tallinen et al., 2014), and numerically(Javili et al., 2015; Jin et al., 2015; Tallinen and Biggins, 2015), as far back as the turn of the century. When researchers from the

http://dx.doi.org/10.1016/j.jmps.2016.09.012Received 17 June 2016; Received in revised form 8 August 2016; Accepted 14 September 2016

⁎ Corresponding author at: Department of Mechanical Engineering, Stanford University, Stanford, CA 94305, USA.E-mail addresses: mholla@stanford.edu (M.A. Holland), libome@tsinghua.edu.cn (B. Li), fengxq@tsinghua.edu.cn (X.Q. Feng), ekuhl@stanford.edu (E. Kuhl).

J. Mech. Phys. Solids 98 (2017) 350–365

0022-5096/ © 2016 Elsevier Ltd. All rights reserved.Available online 08 October 2016

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mechanics community became interested in complex biological folding patterns in layered soft tissues, they applied these findings todevelopment (Bayly et al., 2013; Budday et al., 2014a), morphogenesis (Dervaux et al., 2009; Hohlfeld and Mahadevan, 2011), andpattern formation (Ciarletta et al., 2014; Li et al., 2011). It was easy to assume, then, based on morphological similarities, that thestiffness ratio between the thin outer layer and the thick substrate was of the same order as these traditional systems. Thus earlystudies of brain development used stiffness ratios around β = 101 or 102 (Richman et al., 1975). Yet, recent experiments have shownthat, on the contrary, the stiffness ratio in ultrasoft tissues like the brain is close to or even less than unity (Budday et al., 2015;Weickenmeier et al., 2016; Xu et al., 2010).

Much less work has been done on the regime of stiff substrates with soft films – bilayered materials with ‘inverse’ stiffness ratiosof β < 1. Recent studies have used the Föppl–von Kármán equations (Föppl, 1907; von Kármán, 1910) to study instabilities ofultrasoft materials (Bayly et al., 2013; Budday et al., 2014a). The Föppl–von Kármán equations describe the deflections of large thinplates, and are based on the fundamental assumption that the thickness of the plate does not change significantly duringcompression (Dervaux et al., 2009). This assumption is not valid in the case of soft materials experiencing compressive strains up to50%. One group has examined the bifurcation of the energy functional in the range of β0 < < 5 for an incompressible growing layer,or rather a prestretched substrate (Tallinen and Biggins, 2015), while a few other groups have studied the instabilities of ahomogenous system with β = 1 under compression (Biot, 1963; Cao and Hutchinson, 2011; Fu and Ciarletta, 2015). An elegant wayto model systems in the low-stiffness-contrast regime is to represent the structure as a multi-layered system with weak intermediatelayers (Lejeune et al., 2016). This approach seems particularly suitable for structures that are actually composed of multiple layerslike the human cerebellum (Lejeune et al., 2016).

Strikingly, a comparison of the existing models for systems close to the homogeneity limit with β ≈ 1 reveals that the analyticallypredicted wavelengths and critical strains vary significantly depending on the type of compressive strains (Jin et al., 2015; Tallinenand Biggins, 2015). Namely, the profile of compressive strains across the material varies depending on the mode of loading: whenonly the substrate is prestretched (Song et al., 2008) or only the film grows while attached to the substrate (Budday et al., 2014a),compressive strains will arise almost exclusively in the film while the substrate is loaded neutrally or under tension. Compressivestrains will arise in both layers, however, when compression is applied to the sides of the whole domain as common in manygeophysical applications (Schmalholz and Schmid, 2012).

The objective of this paper is to explore the instabilities of systems with low stiffness ratios β E E= /f s. In Section 2, we describethe model problem and derive the generalized set of eight governing equations for the bifurcation problem in bilayered systems. InSection 3, we tailor the general case to two particular cases, whole-domain compression and film-only compression, and solve theeigenvalue problem numerically in Section 4. In Section 4.3, we introduce effective measures of strain, stiffness, and wavelength toaccount for the finite deformations during soft matter instabilities. In Section 4.6, we investigate the applicability of the Föppl–vonKármán equations to instabilities in soft tissue. We conclude by discussing the limitations of our study in Section 5 and summarizingthe major findings in Section 6.

2. Governing equations

We begin by summarizing the governing equations for instabilities in bilayered systems. We first illustrate the generalformulation for the case of uniform compression in Section 2.1 to which we superpose a small perturbation in Section 2.2. We thenillustrate the Euler–Lagrange equations in Section 2.3 and highlight the essential and natural boundary conditions in Sections 2.4and 2.5. This results in the generalized set of governing equations in Section 2.6 that form the basis of our analysis of eight differentmodes of compression in Section 3. In all eight cases, we analyze systems that consist of a thin film with a shear modulus μf on top ofa thick substrate with a shear modulus μs. We study a wide range of stiffness ratios β μ μ= /f s, from β < 1 via β ≈ 1 to β ≫ 1. Fig. 1illustrates the bilayered system in its undeformed and deformed configurations, with the undeformed and deformed film thicknessesHf and hf and substrate stiffnesses Hs and hs.

2.1. Uniform compression

We first subject the bilayered system to a uniform compression and denote its deformation gradient as

Fig. 1. Bilayered system with a thin film on top of a thick substrate in undeformed and deformed configurations. The geometry is characterized through theundeformed and deformed film thicknesses Hf and hf , the substrate stiffnesses Hs and hs, the wavelength l, the upper and lower boundaries of the film and substrate Γfand Γs, the interface Γ0, the upper and lower normals to film and substrate nf and ns, and the film and substrate normals at interface n0f and n0s.

M.A. Holland et al. J. Mech. Phys. Solids 98 (2017) 350–365

351

⎡

⎣⎢⎢

⎤

⎦⎥⎥F

λλ

λ=

0 00 00 0

,0e

1

2

3 (1)

where λ1, λ2, and λ3 denote the stretches in the direction of compression, in the thickness direction, and in the out-of-plane direction.For this uniform compression, the trace of the right Cauchy Green deformation tensor C F F= ·0

e0et

0e and the Jacobian FJ = det( )0

e0e

become

C λ λ λ J λ λ λtr( ) = + + and = .0e

12

22

32

0e

1 2 3 (2)

We model both the film and the substrate as incompressible neo-Hookean materials characterized through the free energy density

C Iψ μ μP J= 12

[tr( ) − tr( )] − [ − 1],0 0e

0 0e

(3)

where Itr( ) = 3 is the trace of the unit tensor I , P0 is a Lagrange multiplier that enforces incompressibility, and μ is the shearmodulus. We then introduce the overall energy Ψ0 by integrating the energy density ψ0 over the domain of the bilayered system ,

∫ ∫Ψ ψ V μ λ λ λ μP λ λ λ V= d = 12

[ + + − 3] − [ − 1] d ,0 0 12

22

32

0 1 2 3 (4)

and evaluate its stationarity,

∫δΨ Ψλ

δλ ΨP

δP λ P λ λ δλ λ λ λ δP V= ∂∂

+ ∂∂

= [[ − ] − [ − 1 ] ] d ≐ 0.00

22

0

00 2 0 1 3 2 1 2 3 0

(5)

Finally, we use the incompressibility condition, λ λ λ= 1/2 1 3, to obtain an explicit expression for the Lagrange multiplier,

P λ λ λ λ λ= / = 1/ .0 2 1 3 12

32 (6)

2.2. Superposed perturbation

To explore instabilities of the bilayered system, we consider the system under uniform compression according to Section 2.1, andsuperpose a small perturbation u X X X Xu u u( ) = [ ( ), ( ), ( )]1 2 3

t at every point X X X X= [ , , ]1 2 3t. We consider the special case with

Xu ( ) = 03 , and calculate the deformation gradient, F I u X= + ∂ /∂∼, of this perturbation,

⎡

⎣⎢⎢⎢

⎤

⎦⎥⎥⎥

Fu u

u u=1 + 0

1 + 00 0 1

.∼ 1,1 1,2

2,1 2,2

(7)

Next, we introduce the total elastic deformation gradient Fe as the superposition of the initial uniform deformation F0e from Eq. (1)

and the perturbation from Eq. (7),

⎡

⎣⎢⎢⎢

⎤

⎦⎥⎥⎥

F F Fλ u u λ λ

λ u u λ λλ

= · =[1 + ] /[ ] 0

[1 + ]/[ ] 00 0

.∼e0e

1 1,1 1,2 1 3

1 2,1 2,2 1 3

3 (8)

For this total deformation, the trace of the elastic right Cauchy Green deformation tensor, C F F= ·e et e, and the Jacobian of the totalelastic deformation, FJ = det( )e e , become

C λ u λ uλ λ

uλ λ

u λ J u u u utr( ) = [1 + ] + + 1 [1 + ] + 1 + and = [1 + ][1 + ] − .e12

1,12

12

2,12

12

32 2,2

2

12

32 1,2

232 e

1,1 2,2 1,2 2,1(9)

We calculate the free energy density of this total deformation using the neo-Hookean model of Eq. (3),

⎡⎣⎢

⎤⎦⎥ψ μ λ u λ u

λ λu

λ λu λ μ P P u u u u= 1

2[1 + ] + + 1 [1 + ] + 1 + − 3 − [ + ][[1 + ][ 1 + ] − − 1],1

21,1

212

2,12

12

32 2,2

2

12

32 1,2

232

0 1,1 2,2 1,2 2,1(10)

where p P P= +0 is an additive Lagrange multiplier to enforce the incompressibility constraint, J − 1 ≐ 0e (Cao and Hutchinson,2011). We introduce the total energy Ψ by integrating the energy density ψ over the entire domain of the bilayered system . Byusing the periodicity condition, substituting the value of the Lagrangian multiplier P0 from Eq. (6), and collecting the termsindependent of the wrinkling perturbation, c λ λ μ λ λ λ λ( , ) = [ + + − 3]/21 3 1

232

1−2

3−2 , we simplify the total energy to

∫ ∫Ψ ψ V μ λ u uλ λ

u uλ λ

u u u u= d = 12

[ + ] + 12

[ + ] − 1 [ − ]12

1,12

2,12

12

32 1,2

22,22

12

32 1,1 2,2 1,2 2,1

(11)

P u u u u u u c λ λ V− [ + + − ] + ( , ) d .1,1 2,2 1,1 2,2 1,2 2,1 1 3 (12)

We evaluate the stationarity of this total energy,

M.A. Holland et al. J. Mech. Phys. Solids 98 (2017) 350–365

352

δΨ Ψu

δu Ψu

δu ΨP

δP= ∂∂

+ ∂∂

+ ∂∂

≐ 0.1

12

2(13)

with respect to the wrinkling displacements u1 and u2 and the Lagrange multiplier P,

⎡⎣⎢

⎤⎦⎥

⎡⎣⎢

⎤⎦⎥

⎡⎣⎢⎢

⎡⎣⎢

⎤⎦⎥

⎤⎦⎥⎥

⎡⎣⎢⎢

⎡⎣⎢

⎤⎦⎥

⎡⎣⎢

⎤⎦⎥

⎤⎦⎥⎥

∫

∫

∫

μ λ u Pλ λ

u δu μ λ uλ λ

u P δu μ u u δP V

μ λ uλ λ

u P nλ λ

u u n δu S

μ λ uλ λ

u nλ λ

u u P n δu S

− + 1 + + 1 − − [ + ] d

+ − 1 − + 1 [ + ] d

+ + 1 + 1 [ − ] − d ≐ 0,

Λ

Λ

12

1,11 ,112

32 1,22 1 1

22,11

12

32 2,22 ,2 2 2,2 1,1

12

1,112

32 2,2 1

12

32 1,2 2,1 2 1

12

2,112

32 1,2 1

12

32 2,2 1,1 2 2

(14)

where Γ denotes the boundary of the film and substrate subdomains and n n n n= [ , , ]1 2 3t denotes their outward normal according to

Fig. 1.

2.3. Euler–Lagrange equations

The volume integrals in Eq. (14) must vanish for all admissible variations δu1, δu2, and δP. This results in the Euler–Lagrangeequations,

λ uλ λ

u P λ uλ λ

u P u u0 = + 1 − 0 = + 1 − 0 = + ,12

1,1112

32 1,22 ,1 1

22,11

12

32 2,22 ,2 1,1 2,2

(15)

where the third equation represents the incompressibility condition. We assume a periodic wrinkling displacement in the x x{ , }1 2plane of the form

xu a x kx( ) = ( )sin( ),1 2 1 (16)

where we assume that u1 is periodic with wavelength l π k= 2 / . From the Euler–Lagrange equations,

x xu b x kx P q x kx( ) = ( )cos( ) and ( ) = ( )cos( ).2 2 1 2 1 (17)

Substituting Eqs. (16) and (17) into the Euler–Lagrange equations in Eq. (15), we find

λ k aλ λ

a kq λ k bλ λ

b q b ka0 = − + 1 ″ + 0 = − + 1 ″ − ′0 = ′ + ,12 2

12

32 1

2 2

12

32 (18)

or, via substitution,

b k λ λ b k λ λ b⁗ − [ + 1] ″ + = 0.214

32 4

14

32 (19)

The characteristic equation,

t k λ λ t k λ λ− [ + 1] + = 0,4 214

32 2 4

14

22 (20)

has characteristic roots

t k t kλ λ= ± and = ± .12

3 (21)

Allowing for the possibility that λ λ = 112

3 , in which case there would be only two distinct roots, we include redundant variables so thatthe solution of Eq. (19) reads

b x c c kx kx c c kx kx c kλ λ x c kλ λ x( ) = [ + ]exp(− ) + [ + ]exp( ) + exp(− ) + exp( ),2 1 2 2 2 3 4 2 2 5 12

3 2 6 12

3 2 (22)

with

a x c c kx kx c c kx kx c λ λ kλ λ x c λ λ kλ λ x( ) = [ + [ − 1]]exp(− ) − [ + [ + 1 ]]exp( ) + exp(− ) − exp( )2 1 2 2 2 3 4 2 2 5 12

3 12

3 2 6 12

3 12

3 2 (23)

⎡⎣⎢⎢

⎡⎣⎢

⎤⎦⎥

⎤⎦⎥⎥

⎡⎣⎢⎢

⎡⎣⎢

⎤⎦⎥

⎤⎦⎥⎥q x c Λ c kx Λ

λ λλ k kx c Λ c kx Λ

λ λλ k kx( ) = + + 3 − exp(− ) − + − 3 + exp( )2 1 − 2 2 −

12

32 1

22 3 − 4 2 −

12

32 1

22

(24)

with the abbreviation Λ λ λ λ= − 1/[ ]− 12

12

32 .

2.4. Essential boundary conditions

The essential boundary conditions ensure that the solution is continuous at the film–substrate interface,

u u u u x− = 0 and − = 0 on = 0,1f

1s

2f

2s

2 (25)

and that the perturbation disappears at the lower boundary,

M.A. Holland et al. J. Mech. Phys. Solids 98 (2017) 350–365

353

u u x h= 0 and = 0 on = − ,1s

2s

2 s (26)

where the superscripts f and s refer to the film and substrate domains. We could also use a traction-free lower boundary instead of adisplacement-free lower boundary (Jin et al., 2015). However, for sufficiently large substrate thicknesses hs, the boundary conditionat the lower boundary should not affect the folding pattern of the film.

2.5. Natural boundary conditions

The natural boundary conditions follow from vanishing coefficients of the variations in the surface integrals in Eq. (14), using therelation u u= −2,2 1,1 from the incompressibility condition in Eq. (15),

⎡⎣⎢⎢

⎡⎣⎢⎢

⎡⎣⎢

⎤⎦⎥

⎤⎦⎥⎥

⎡⎣⎢

⎤⎦⎥

⎤⎦⎥⎥

⎡⎣⎢⎢

⎡⎣⎢

⎤⎦⎥

⎡⎣⎢

⎤⎦⎥

⎤⎦⎥⎥∫ ∫μ λ

λ λu P n

λ λu u n δu S μ λ u

λ λu n

λ λu P n δu S+ 1 − + 1 [ + ] d + + 1 − 2 + d ≐ 0.

Γ Γ12

12

32 1,1 1

12

32 1,2 2,1 2 1 1

22,1

12

32 1,2 1

12

32 1,1 2 2

(27)

At the film–substrate interface, at x = 02 , the natural boundary conditions in Eq. (27) imply that

λ λu u β

λ λu u

λ λu P β

λ λu βP x1 [ + ] − [ + ] = 0 and 2 + − 2 − = 0 on = 0.

1s2

3s2 1,2

s2,1s

1f2

3f2 1,2

f2,1f

1s2

3s2 1,1

s s

1f2

3f2 1,1

f f2

(28)

At the free surface of the film, at x h=2 f , the natural boundary conditions in Eq. (27) imply that

u uλ λ

u P x h+ = 0 and 2 + = 0 on = .1,2f

2,1f

1f2

3f2 1,1

f f2 f

(29)

2.6. Governing equations

Substituting the forms of the wrinkle displacements from Eqs. (22)–(24) into the four essential boundary conditions in Eqs. (25)and (26) and the four natural boundary conditions in Eqs. (28) and (29), we formulate the general set of eight governing equationsfor instabilities in bilayered systems,

βλ λ

c c c c βΛ c cλ λ

c c c c Λ c c

βΛ c c βΛ c c βλ

c c Λ c c Λ c cλ

c c

c c kh kh c c kh kh λ λ Λ c kh λ λ c kh λ λ

c Λ c kh Λ Λ kh c Λ c kh Λ Λ khλ

c kh λ λ c kh λ λ

c c c c λ λ c c c c c c λ λ c c c c c c c c c c

c c kh kh c c kh kh λ λ c kh λ λ c kh λ λ

c c kh kh c c kh kh c kh λ λ c kh λ λ

0 = 2 [ − + + ] + [ + ] − 2 [ − + + ] − [ + ]0

= [ − ] − [ + ] + 2 [ − ] − [ − ] + [ + ] − 2 [ − ]0

= 2[ + [ − 1]]exp(− ) + 2[ + [ + 1]]exp( ) + [ exp(− ) + exp( )]0

= −[ + [ − ]]exp(− ) + [ + [ + ]]exp( ) − 2 [ exp(− ) − exp( )]0

= − + + + − [ − ] + − − − + [ − ]0 = − − − − + + + + 0

= [ − [ + 1]]exp( ) − [ + [− + 1]]exp(− ) + [ exp( ) − exp(− )]0

= [ − ]exp( ) + [ − ]exp (− ) + exp( ) + exp(− ).

1f2

3f2 1

f2f

3f

4f

+f

5f

6f

1s2

3s2 1

s2s

3s

4s

+s

5s

6s

+f

1f

3f

−f

2f

4f

3f 5

f6f

+s

1s

3s

−s

2s

4s

3s 5

s6s

1f

2f

f f 3f

4f

f f 1f2

3f2

+f

5f

f 1f2

3f

6f

f 1f2

3f

1f

+f

2f

f +f

−f

f 3f

+f

4f

f +f

−f

f3f 5

ff 1

f23f

6f

f 1f2

3f

1f

2f

3f

4f

1f2

3f

5f

6f

1s

2s

3s

4s

1s2

3s

5s

6s

1f

3f

5f

6f

1s

3s

5s

6s

1s

2s

s s 3s

4s

s s 1s2

3s

5s

s 1s2

3s

6s

s 1s2

3s

1s

2s

s s 3s

4s

s s 5s

s 1s2

3s

6s

s 1s2

3s (30)

where we have used the abbreviations Λ λ λ λ= + 1/[ ]+ 12

12

32 and Λ λ λ λ= − 1/[ ]− 1

212

32 .

3. Modes of compression

We now specialize the generalized set of equations of Eq. (30) to two modes of compression, whole-domain compression andfilm-only compression, induced by either external deformation or internal growth, as illustrated in Fig. 2. For simplicity, we consider

Fig. 2. Different modes of compression in soft bilayered materials. Whole-domain compression (left), and film-only compression introduced either by substrateprestretch (middle) or film growth (right).

M.A. Holland et al. J. Mech. Phys. Solids 98 (2017) 350–365

354

the substrate as infinitely thick, assuming that the substrate thickness does not affect solutions as long as H H> 10s f (Huang et al.,2005; Jin et al., 2015; Li et al., 2009). This eliminates the essential boundary conditions on the bottom surface as well as theunknowns c c c= = = 01

s2s

5s , and results in a system of six equations with nine unknowns, of which three are redundant,

βλ λ

c c c c βΛ c cλ λ

c c Λ c βΛ c c βΛ c c βλ

c c Λ c

Λ cλ

c c c kh kh c c kh kh λ λ Λ c kh λ λ

c kh λ λ c Λ c kh Λ Λ kh c Λ c kh Λ Λ kh

λc kh λ λ c kh λ λ c c c c λ λ c c c c λ λ c

c c c c c c

0 = 2 [ − + + ] + [ + ] − 2 [ + ] − 0 = [ − ] − [ + ] + 2 [ − ] +

+ + 2 0 = 2[ + [ − 1]]exp(− ) + 2[ + [ + 1]]exp( ) + [ exp(− )

+ exp( )]0 = −[ + [ − ]]exp(− ) + [ + [ + ]]exp( )

− 2 [ exp(− ) − exp( )]0 = − + + + − [ − ] − − − 0

= − − − − + + .

1f2

3f2 1

f2f

3f

4f

+f

5f

6f

1s2

3s2 3

s4s

+s

6s

+f

1f

3f

−f

2f

4f

3f 5

f6f

+s

3s

−s

4s

3s 6

s1f

2f

f f 3f

4f

f f 1f2

3f2

+f

5f

f 1f2

3f

6f

f 1f2

3f

1f

+f

2f

f +f

−f

f 3f

+f

4f

f +f

−f

f

3f 5

ff 1

f23f

6f

f 1f2

3f

1f

2f

3f

4f

1f2

3f

5f

6f

3s

4s

1s2

3s

6s

1f

3f

5f

6f

3s

6s (31)

We assume that compressive strains in the system can result from external stretches d1, d2, and d3 in F0, and from internal growthg1, g2, and g3 in F0

g. Fig. 3 illustrates the connection between F0 and F0g via the multiplicative decomposition of the deformation

gradient F0 into a growth part F0g, which changes the stress-free configuration, and an elastic part F0

e, which ensures compatibility(Rodriguez et al., 1994). From the total initial deformation gradient F0 and the growth tensor F0

g,

⎡

⎣⎢⎢

⎤

⎦⎥⎥

⎡

⎣⎢⎢⎢

⎤

⎦⎥⎥⎥

F Fd

dd

gg

g=

0 00 00 0

and =0 0

0 00 0

,0

1

2

3

0g

1

2

3 (32)

we obtain the elastic tensor F F F= ·0e

0 0g−1, which gives rise to stresses in the bilayered system,

⎡

⎣⎢⎢

⎤

⎦⎥⎥F F F

λλ

λ= · =

0 00 00 0

.0e

0 0g−1

1

2

3 (33)

Upon imposing the incompressibility constraint, we can express the elastic stretches as the result of both external stretches andinternal growth,

λ d g λ λ λ g g d d λ d g= / and = 1/ = [ ]/[ ] and = / .1 1 1 2 1 3 1 3 1 3 3 3 3 (34)

The deformation gradient F0 relates the reference configuration to the intermediate configuration. It defines the wavelength in theuniformly compressed configuration as l L F= 0,11, the wave number as k K F= / 0,11, and the film thickness as h HF= 0,22. Thegenerality of this formulation allows us to explore eight different loading scenarios: three modes of whole-domain compression, one-dimensional plane strain, uniaxial, and biaxial compression; two modes of film-only compression induced by one- or two-dimensional substrate prestretch; and three modes of film-only compression induced by one-, two-, and three-dimensional growth.Table 1 summarizes the eight modes of compression.

3.1. Whole-domain compression

For the case of whole-domain compression, we impose the stretches d1 and d3, and there is no growth, g g g= = = 11 2 3 . Sinceλ λ ≠ 11

23 , the roots of the characteristic equation in Eq. (21) are distinct, and the constants c2

f , c4f , and c4

s are redundant. The governingequations of Eq. (31) reduce to the following system of equations,

Fig. 3. Stress-free reference configuration,uniformly compressed intermediate configuration, and final perturbed configuration (top) and stress-free grownconfiguration (bottom).

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βd d

c c βΛ c cd d

c Λ c βΛ c c βd

c c Λ cd

c

c kh c kh d d Λ c kh d d c kh d d c Λ kh c Λ kh

dc kh d d c kh d d c c d d c c c d d c c c c c c c

0 = 2 [ + ] + [ + ] − 2 − 0 = [ − ] + 2 [ − ] + + 2 0

= 2 exp(− ) + 2 exp( ) + [ exp(− ) + exp( )]0 = − exp(− ) + exp( )

− 2 [ exp(− ) − exp( )]0 = − + − [ − ] − − 0 = − − − − + + ,

12

32 1

f3f

+d

5f

6f

12

32 3

s+d

6s

+d

1f

3f

35f

6f

+d

3s

36s

1f

f 3f

f 12

32

+d

5f

f 12

3 6f

f 12

3 1f

+d

f 3f

+d

f

35f

f 12

3 6f

f 12

3 1f

3f

12

3 5f

6f

3s

12

3 6s

1f

3f

5f

6f

3s

6s

(35)

or, equivalently, to the vanishing determinant of the following 6 × 6 matrix,

β β βΛ d d βΛ d d Λ d dβΛ d βΛ d β β Λ d

KH d d KH d d Λ d d KH Λ d d KHΛ d KH d d Λ d KH d d KH KH

d d d d d d

2 2 − 2 −− 2 − 2 2

2 exp(− / ) 2 exp( / ) exp(− ) exp( ) 0 0− exp(− / ) exp( / ) − 2 exp(− ) 2 exp( ) 0 0

− 1 1 − − 1 −− 1 − 1 − 1 − 1 1 1

≐ 0

+d

12

32

+d

12

32

+d

12

32

+d

3 +d

3 +d

3

12

3 12

3 +d

12

32

+d

12

32

+d

3 12

3 +d

3 12

3

12

3 12

3 12

3

(36)

where Λ d d d= + 1/+d

12

12

32 for convenience. In the deformed configuration, the film thickness is h HF H d d= = /f 0,22 1 3 and the wave

number is k K F K d= / = /0,11 1. The main loading parameter is d λ=1 , while d3 is defined in terms of d1. For the one-dimensional planestrain compression, d = 13 ; for uniaxial compression, d d λ= 1/ = 1/3 1 ; and for biaxial compression d d λ= =3 1 . The first two caseshave previously been studied with a finite substrate and a traction-free boundary condition at the bottom surface (Jin et al., 2015);and for the homogeneous case β = 1 (Cao and Hutchinson, 2011).

3.2. Film-only compression

For the case of film-only compression induced by film growth, we impose growth g1, g2 and g3, and there is no deformationd d d= = = 11 2 3 . Compression affects the film only, λ g= 1/f , while the substrate is neutrally loaded, λ = 1s . Because λ λ = 11

s23s in the

substrate, the roots of the characteristic equation (Eq. (21)) are not distinct, and the constant c6s is redundant, in addition to the film

constants c2f and c4

f similar to the case of whole-domain compression. The governing equations in Eq. (31) now become

βg g c c βΛ c c c c βΛ c c βg c c c

g g c kh g g c kh Λ c kh g g c kh g g

c Λ kh c Λ kh g c kh g g c kh g g

c c c g g c g g c c c c c c c

0 = 2 [ + ] + [ + ] − 2[ + ]0 = [ − ] + 2 [ − ] + 2 0

= 2 exp(− ) + 2 exp( ) + [ exp(− /[ ]) + exp( /[ ])]0

= − exp(− ) + exp( ) − 2 [ exp(− /[ ]) − exp( /[ ])]0

= − + − /[ ] + /[ ] − − 0 = − − − − + .

12

3 1f

3f

+g

5f

6f

3s

4s

+g

1f

3f

3 5f

6f

3s

12

3 1f

f 12

3 3f

f +g

5f

f 12

3 6f

f 12

3

1f

+g

f 3f

+g

f 3 5f

f 12

3 6f

f 12

3

1f

3f

5f

12

3 6f

12

3 3s

4s

1f

3f

5f

6f

3s

(37)

We can easily eliminate the last two equations, using c c c c c= + + +3s

1f

3f

5f

6f and c c c g g c g g= −2 − [1 + 1/[ ]] − [1 − 1/[ ]]4

s1f

5f

12

3 6f

12

3 , andrephrase the system of equations as the vanishing determinant of the following 4×4 matrix,

βg g βg g βΛ g g βΛ g gβΛ βΛ βg βg

g g KHg g g g g KHg g g Λ KHg g Λ KHg gΛ KHg g g Λ KHg g g g KHg g g KHg g

2[ + 1] 2[ − 1] + 2/ − 2/2 + 2 − 2[1 + ] 2[1 − ]

2 exp(− ) 2 exp( ) exp(− / ) exp( / )− exp(− ) exp( ) − 2 exp(− / ) 2 exp( / )

≐ 0,

12

32

12

32

+g

12

3 +g

12

3

+g

+g

3 3

12

32

1 2 3 12

32

1 2 3 +g

2 1 +g

2 1

+g

1 2 3 +g

1 2 3 3 2 1 3 2 1 (38)

where Λ g g g= 1/ ++g

12

12

32. In the deformed configuration, the film thickness is h HF Hg g g= =f 0,22 1 2 3 and the wave number is equal in

the reference and deformed configurations, k KF K= =0,11 . The main loading parameter is g λ= 1/1 , while g2 and g3 are defined in

Table 1Eight modes of compression. Rows distinguish between whole-domain compression and film-only compression introduced either by substrate prestretch or by filmgrowth. Columns distinguish between the number of stretch components that are specified, denoted as one-dimensional, two-dimensional, and three-dimensional.The eight colors represent the eight cases throughout all figures of this manuscript.

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terms of g1. For the case of one-dimensional growth, g g= = 12 3 ; for two-dimensional growth, g g λ= = 1/3 1 and g = 12 ; and for three-dimensional growth, g g g λ= = = 1/1 2 3 .

For the case of film-only compression induced by substrate prestretch, we can adopt a similar approach as for growth. Prestretchresults in a varying strain profile across the material; for low enough stiffness ratios, it is reasonable to assume that the substrate isstrain-free while the film experiences compressive strains. This idealized prestretch can be modeled analogous to growth. For thecase of one-dimensional prestretch, g g λ= 1/ =2 1 , g = 13 ; for two-dimensional prestretch, g g λ= 1/ =2 1

2 2, g g λ= = 1/3 1 (Tallinen andBiggins, 2015).

3.3. Numerical method for solving eigenvalue problem

We find the solution to Eqs. (36) and (38) for vanishing determinants of the system matrices. For a given stiffness ratio β and anormalized wave number KH, the value of λ at which the matrix becomes singular represents the level of compression or growth thatcauses the system to become unstable. Since the system is highly nonlinear, we use Ridders' bracketing method (Ridders, 1979) tosolve the eigenvalue condition numerically for roots in the range of λ0 < < 1. Many pairs β KH{ , } result in multiple singular values ofλ. We report the largest value of λ as the critical one, since it will be reached first when λ decreases from 1 as compression increases.We iteratively solve the resulting system for all eight modes of compression, at 100 stiffness ratios between β0.1 < < 1000, for 1000wavelengths between L H0.1 < / < 1000.

4. Results

4.1. Critical strain vs. wavelength

Fig. 4 shows critical strain λϵ = 1 − plotted against the normalized wavelength L H/ for the eight different modes of compressioncalculated for a wide range of stiffnesses β0.1 < < 1000 and wavelengths L H0.1 < / < 1000. Each curve represents a single stiffnessratio β. The curves for β = 0.1, 1, 10, 100, and 1000 are highlighted by thicker curves, from top to bottom, with β = 1 represented bythe thickest curve. For each stiffness ratio β, the wavelength with the lowest critical strain indicated by the thick black curve is thecritical strain. We note though that in cases of very fast loading, in either experimental or numerical studies, the critical strain couldbe bypassed and instabilities set in at a different point with a different wavelength.

Qualitatively, the shapes differ significantly between whole-domain compression, shown in red, orange, and yellow, and film-onlycompression, shown in green and blue. In whole-domain compression, there is a critical strain at which both very small and verylarge wavelengths become unstable. In film-only compression, especially at lower stiffness ratios, the stability curves continue to riseto the right indicating that long wavelengths are stable up to very large compressive strains. Strikingly, in whole-domaincompression, the β curves display no distinct minima for β < 1, as indicated through the thick horizontal red, orange, and yellowcurves for β = 1. However, in film-only compression, the β curves possess minima even for β < 1, as indicated through the thickgreen and blue curves with distinct minima and critical wavelengths L H1 < / < 10 for β = 1 and smaller.

Remarkably, the choice of boundary conditions has a significant influence on the stability of the system. One-dimensionalsystems under plane strain conditions in the first column of Fig. 4 are more stable than two- and three-dimensional systems in thesecond and third column, which become unstable for lower stiffness ratios β. This observation agrees well with a recent study thatcompared two- and three-dimensional growth and found the three-dimensional case to be less stable (Budday et al., 2015). It is alsoin broad agreement with studies of the morphogenesis in tubular systems where certain types of boundary conditions have astabilizing effect on pattern formation (Balbi et al., 2015; Ciarletta et al., 2014).

4.2. Critical strain vs. stiffness ratio

Fig. 5 highlights the quantitative differences between the critical axial strains λϵ = 1 − , left, and the critical normalizedwavelengths L H/ , right, for varying stiffness ratios β for the eight different modes of compression. For large stiffness ratios, β > 100,the difference between eight modes of compression is negligible. As the stiffness ratio decreases, β < 10, the differences between thedifferent modes become visible. At small stiffness ratios, the system resembles a soft film on an essentially rigid substrate and followsthe behavior of a compressed half-space as described by Biot (1963). At large stiffness ratios, the system resembles a thin stiff film ona negligible substrate and follows the behavior of a thin plate as described by the Föppl–von Kármán equations (Föppl, 1907; vonKármán, 1910). Instabilities in bilayered systems are characterized by the competition between the bending energy of the film andthe stretching energy of the substrate; the film prefers larger wavelengths, while the substrate favors shorter wavelengths (Genzerand Groenewold, 2006). As the influence of the film increases, for β ≫ 1, the wavelength L H/ increases; as the influence of thesubstrate increases for β → 1, the wavelength L H/ decreases.

At β = 1, the problem becomes scale-free for whole-domain compression, as a homogeneous material is compressed uniformlyand no wavelength dominates. In fact, at this stiffness ratio, the whole-domain compression problem reduces to Biot's problem of ahomogeneous substrate under compression, which is known to be a scale-free problem (Biot, 1963; Cao and Hutchinson, 2011). InFig. 4, this case is indicated through the thick horizontal red, orange, and yellow curves. For stiffness ratios smaller than unity, β < 1,the curves in Fig. 4 become concave, reaching their minimum value at wide ranges of stiffness ratios, but having a single uniquemaximum as shown through the dashed black curves in Fig. 4 and the thin curve in Fig. 5. This maximum has no real physicalsignificance, other than indicating that there is no single wavelength favored for folding.

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Fig. 5. Critical strain λϵ = 1 − vs. stiffness ratio β (left) and critical normalized wavelength L H/ vs. stiffness ratio β (right) for the eight different modes ofcompression according to Table 1. Small stiffness ratios are associated with the half-space or substrate-only solution; large stiffness ratios are associated with the thin-plate or film-only solution. For comparison: ▴ cusping and ⋆ wrinkling for numerical and experimental studies according to Table 2.

Fig. 4. Critical strain λϵ = 1 − vs. normalized wavelength L H/ for the eight different modes of compression according to Table 1. Each curve represents a singlestiffness ratio β. The curves for β = 0.1, 1, 10, 100, and 1000 are highlighted by thicker curves, from top to bottom, with β = 1 the thickest. Minima of all β curves are

highlighted through the solid black curves; maxima or neutral conditions are highlighted through dashed black curves. For comparison: ▴ cusping and ⋆ wrinkling for1d and 2d prestretch (Tallinen and Biggins, 2015). (For interpretation of the references to color in this figure caption, the reader is referred to the web version of thispaper.)

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4.3. Effective strain, stiffness ratio, and wavelength

Fig. 6 introduces three effective quantities, the effective strain ϵeff , the normalized effective stiffness ratio β β/eff , and thenormalized effective wavelength l h L H[ / ]/[ / ] and plots them as functions of the axial strain ϵ. First, to account for differences betweengeneral systems under a constant transverse strain and plane strain systems without transverse strain, we adopt an extension of theconcept of generalized plane strain (Hong et al., 2009) and introduce the effective strain λϵ = 1 −eff eff in terms of the effective stretchλ λ λ= /eff

1 2 . For incompressible materials with λ λ λ= 1/[ ]2 1 3 , the effective strain becomes

λ λ λ λ λ λϵ = 1 − = 1 − with = .eff eff1 3

eff1 3

The effective strain relates a plane strain problem under a constant transverse strain to an equivalent plane strain problem withouttransverse strain. Fig. 6, left, shows that for the one-dimensional compression modes with λ = 13 , the effective strain is identical tothe axial strain, ϵ = ϵeff ; for uniaxial compression, the effective strain is slightly smaller, ϵ ≤ ϵeff ; and for all other modes, it is slightlylarger, ϵ ≥ ϵeff .

Second, to account for effective stiffness differences between the undeformed and deformed configurations, we introduce theeffective stiffness ratio β μ μ= /eff

feff

seff as the ratio between the effective film and substrate moduli, where the effective shear modulus,

μ λ λ μ= [ + ]eff 12 1

222 , relates the incremental stress to the incremental stretch (Biot, 1963). For incompressible materials with

λ λ λ= 1/[ ]2 1 3 , the effective stiffness ratio becomes

⎡⎣⎢

⎤⎦⎥

⎡⎣⎢

⎤⎦⎥β

μμ

λλ

λ λλ λ

μμ

μ μ λλ λ

= = + with = 12

+ 1 .eff feff

seff

f12

s12

s12

s32

f12

f32

f

s

eff12

12

32

In the case of whole-domain compression, the film and substrate undergo identical compression, λ λ=f s, their stiffnesses increase intandem, μ μ μ μ/ = /f

effseff

fs, and preserve their stiffness ratio, β β=eff . In the case of film-only compression, the film undergoes

compression, λ < 1f , while the substrate does not, λ = 1s , and the stiffness ratio at the onset of buckling,β μ μ λ λ λ μ μ= / = [ + 1/[ ]] /eff

feff

seff 1

2 12

12

32

f s, is not identical to the initial stiffness ratio β; it increases throughout loading, β β≥eff as

the effective stiffness of the film increases, μ μ≥feff

f . Fig. 6, middle, shows that for whole-domain compression, the normalizedeffective stiffness ratio remains constant to one, β β/ = 1eff ; for film-only compression, it increases rapidly with increasingcompression, β β/ ≥ 1eff .

Third, to account for kinematic changes between the undeformed and deformed configurations, we introduce the normalizedeffective wavelength as the wavelength in the current configuration l normalized by the film thickness in the current configuration h,

lh

d dg g g

LH

khg g gd d

KH= and = .12

3

1 2 3

1 2 3

12

3

As the film increases in thickness during compression, using the deformed rather than the undeformed kinematic quantities has asignificant effect, especially in ultrasoft materials, which buckle only at large strains. Fig. 6, right, shows that for prestretchedsystems, the normalized effective wavelength remains constant to one, l h L H[ / ]/[ / ] = 1; for all other systems, it decreases rapidly withincreasing compression, l h L H[ / ]/[ / ] = 1 ≤ 1.

Fig. 7 illustrates the critical strain and wavelength for varying stiffness ratios, similar to Fig. 5, but now for the effective strain ϵeff ,the normalized effective wavelength l h L H[ / ]/[ / ], and the normalized effective stiffness ratio βeff . While Fig. 5 suggests that the eightcompression modes display marked differences for moderate stiffness ratios, at β < 10, these discrepancy disappears for the effectivequantities in Fig. 7. The effective strain and the effective wavelength essentially scale the ordinate, while the effective stiffness ratioscales the abscissa. Effective measures are convenient for comparing and unifying numerical and experimental results. The solidblack line in Fig. 7, left, indicates the effective strain beyond which cusping, a localized instability distinct from folding, occurs underplane strain conditions. When translating data from a number of other studies in Table 2 into effective strain measures, we find aremarkable alignment with cusping, indicated through the triangular markers, at ϵ = 0.35eff (Hohlfeld and Mahadevan, 2011; Honget al., 2009), for several different loading modes. The dashed black horizontal line in Fig. 7, left, indicates the effective strain

Fig. 6. Effective strain λϵ = 1 −eff eff vs. axial strain ϵ (left), normalized effective stiffness ratio β β/eff vs. axial strain ϵ (middle), and normalized effective wavelength

l h L H[ / ]/[ / ] vs. axial strain ϵ (right) for the eight different modes of compression according to Table 1.

M.A. Holland et al. J. Mech. Phys. Solids 98 (2017) 350–365

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associated with Biot's (1963) scale-free instability at ϵ = 0.456eff . This instability dominates at small stiffness ratios, regardless of themode of compression. This is intuitive, since the system reduces to a soft material on an essentially rigid substrate as β → 0.

4.4. Wrinkling and creasing thresholds

Fig. 7, left, highlights two important lines, the critical effective strain for Biot's (1963) scale-free instability at ϵ = 0.456eff and forcusping at ϵ = 0.35eff (Hohlfeld and Mahadevan, 2011; Hong et al., 2009). Each curve of the eight different modes of compressionintersects the Biot's instability at the critical wrinkling threshold βw and the cusping instability at the critical creasing threshold βc.

Table 3 summarizes the critical thresholds for wrinkling βw and creasing βc for all eight modes of compression. For β β> w, thecurves in Fig. 4 have unique minima. For β β< c, creasing is energetically favorable over wrinkling. Because the cusping instabilityoccurs at a lower strain than Biot's instability, the latter is never actually realized. Instead, we predict creasing for β β< c andwrinkling for β β> c. Notably, systems under two-dimensional prestretch and two- and three-dimensional growth with β = 0.86c canbecome unstable for moderate or inverted stiffness ratios β ≤ 1. This implies that the present theory can indeed explain corticalfolding during brain development (Goriely et al., 2015), even for experimentally measured stiffness ratios small than one (Buddayet al., 2015; Weickenmeier et al., 2016).

4.5. Critical pressure vs. stiffness ratio

Fig. 8 illustrates the critical Biot pressure Pbiot and the hydrostatic pressure Phydr for varying stiffness ratios β. The Biot pressure,the compressive stress in the axial direction, is a valuable measure of critical compression as initially pointed out by Biot (1963),

⎡⎣⎢⎢

⎤⎦⎥⎥

⎡⎣⎢

⎤⎦⎥P σ μ

Jdg

g gd d

μJ

λλ λ

= − = − − = − − 1 ,biot 1112

12

12

32

12

32 1

2

12

32

(39)

where FJ λ λ λ= det( ) = 1 2 3. We compare the Biot pressure against the hydrostatic pressure,

⎡⎣⎢⎢

⎤⎦⎥⎥P σ σ σ μ

Jdg

dg

g gd d

μJ

λ λ λ= − 13

[ + + ] = −3

+ − 2 = −3

[ + − 2 ].hydr 11 22 3312

12

32

32

12

32

12

32 1

232

22

(40)

Fig. 8 suggests that, unlike the effective strain, the critical Biot pressure and the hydrostatic pressure exaggerate the differencesbetween the eight modes of compression.

4.6. Föppl–von Kármán strain vs. wavelength

The Föppl–von Kármán equations (Föppl, 1907; von Kármán, 1910) describe the deflections of a large thin plate, assuming thatthe wavelength is much greater than the thickness of the plate (Dervaux et al., 2009) and that the thickness does not change. TheFöppl–von Kármán theory has been criticized for a lack of clarity, notably by Truesdell, who described it as “handed down by somehigher power (a Hungarian wizard, say)” and said that “nobody can make sense out of the derivations” (Ciarlet, 1983). Major issueswith the theory include approximate geometry and a conflation of the reference and deformed configurations. Despite thesecriticisms, the Föppl–von Kármán equations have been successfully applied to physical phenomena, including the buckling of stifffilms on soft substrates. An elegant way to circumvent these issues is to model the thin film as a growing surface equipped with itsown boundary energy and prestretch or growth characteristics (Holland et al., 2013; Papastavrou et al., 2013).

Fig. 7. Critical effective strain ϵeff vs. effective stiffness ratio βeff (left) and critical normalized effective wavelength l h L H[ / ]/[ / ] vs. effective stiffness ratio βeff (right)

for the eight different modes of compression according to Table 1. For comparison: ▴ cusping and ⋆ wrinkling for numerical and experimental studies according toTable 2. Solid and dashed black lines indicate the critical effective strain for cusping at ϵ = 0.35eff (Hohlfeld and Mahadevan, 2011; Hong et al., 2009) and Biot's(1963) instability at ϵ = 0.456eff .

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Because of their simplicity, the Föppl–von Kármán equations remain widely used in the context of growth, to relate the growth-induced film pressure P and the wavenumber K (Bayly et al., 2013; Budday et al., 2014a). Assuming incompressibility, the criticalFöppl–von Kármán stress Pfvk can be equated with the compressive axial stress P in an incompressible material induced by one-dimensional compression or growth,

⎡⎣⎢

⎤⎦⎥

⎡⎣⎢⎢

⎤⎦⎥⎥

⎡⎣⎢

⎤⎦⎥P μ β KH

KHP σ β μ

gdg

gd

β μg

λλ

= 26

[ ] + 1 and = − = − − = − − 1fvk

211

1

12

12

12

12

112

12

(41)

where μ denotes the substrate stiffness. From Eq. (41)(2), we can see that this pressure is not equal for one-dimensionalcompression, with d λ=1 and g = 11 , and one-dimensional growth, with g λ=1 and d = 11 , because of the volume normalization of thestress. Equating the two pressures yields the critical condition,

KHβKH

dg

gd

13

[ ] + 2 + − ≐ 0,2 12

13

1

12

(42)

which can be solved by Ridders' (1979) method to determine the critical strain λϵ = 1 − at which the system becomes unstable. Formodest stiffness ratios, the geometrical assumptions of the Föppl–von Kármán equations – namely that the thickness remainsconstant and the wavelength is much greater than the thickness – no longer hold. For instance, when β = 1000, the system becomesunstable at about 1% strain, while for β = 5, about 20% strain is required to induce an instability. The thickness of an incompressiblematerial will change significantly in this strain regime, resulting in an incorrectly inflated normalized wavelength (Sultan and

Table 2Eight modes of compression. Analytical, numerical, and experimental studies in the literature; plotted in Fig. 5 with the indicated symbols.

Table 3Wrinkling and creasing thresholds for the eight different modes of compression. Above the wrinkling threshold βw, wrinkling becomes unstable at a single wavelengthat Biot's instability; below the creasing threshold βc, creasing occurs before wrinkling. Therefore, creasing is expected to occur first for β β≤ c; wrinkling is expected toprecede for β β> c.

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Boudaoud, 2008). In addition to the change in thickness, the wavelength will decrease as a soft material undergoes significantcompression (Jiang et al., 2007). We can quantify the thickness and wavelength in the deformed configuration using the totaldeformation gradient under either compression or growth,

h F Hgd

H kF

Kd

K= = and = 1 = 1 .f 220 1

1 110

1 (43)

Using these dimensions of the deformed configuration, equating the two pressures of Eq. (41) yields the modified critical condition,

KHgd βKH

dg

dg

gd

13

[ ] + 2 + − ≐ 02 12

14

12

1

12

13

1

12

(44)

which we can again solve using Ridders' (1979) method. Eq. (44) differs from Eq. (42) only by a scaling of the normalized wavelengthKH while the critical strain ϵ is not affected by using the dimensions of the deformed configuration. Fig. 9 shows the directcomparison of the critical strains for the original Föppl–von Kármán equations, dashed black curves, and the modified Föppl–vonKármán equations for compression, red solid curves, and growth, blue solid curves.

4.7. Föppl–von Kármán wavelength and critical strain vs. stiffness ratio

Fig. 10 illustrates identical critical strains, left, and different wavelengths, right, for the original and modified Föppl–von Kármánequations. The curves demonstrate that the original Föppl–von Kármán equations, thick dashed curves, predict results at odds withthe assumptions they are based on; namely at low stiffness ratios, the wavelength is predicted to be on the order of the film thickness,rather than being significantly larger. If the changes in thickness and wavelength were truly negligible as assumed, accounting for thechanges would not change the results noticeably. This is the case for large stiffness ratios, β > 100. Indeed, as the stiffness ratioapproaches infinity, β → ∞, the behavior of the system approaches that of a thin plate with no supporting material (Cai and Fu,1999), and the original solution of Eq. (42) and the modified solutions of Eq. (44) converge to one another. For smaller stiffnessratios, however, the predicted wavelengths differ significantly. Essentially, the fundamental assumptions of the Föppl–von Kármánequations are appropriate for stiff films on soft substrates, but not for soft films.

The modifications of Eq. (44) have relatively little effect on the critical strain ϵ for most of the range of stiffness ratios. However,in the regime of low stiffness contrast, for β1 < < 10, the difference between both conditions becomes significant. The originalFöppl–von Kármán equations (42) overestimate the stability of low-stiffness systems considerably, predicting, for instance, thatgrowing materials could remain stable while experiencing up to 90% compressive strain, when our calculations predict instabilitiesaround 40%, lower row. Experiments and numerical simulations indicate that the bilayered system would actually be susceptible tocusping at 35% strain (Hohlfeld and Mahadevan, 2011; Hong et al., 2009).

Our findings do not suggest though that incorporating realistic spatial dimensions in Eq. (44) addresses every criticism of theFöppl–von Kármán equations. Nonetheless, our results indicate that the original equations (42) are unsuitable for predictingbuckling of bilayered materials in the low-stiffness ratio regime. Interestingly, in a weakly nonlinear analysis to determine theimperfection sensitivity of the bilayered problem with modest stiffness ratios (Cai and Fu, 1999), the results were strikingly similarto those predicted by the Föppl–von Kármán equations in Fig. 10, thin dashed curve. As with the Föppl–von Kármán equations, noattempt was made to find the material dimensions that correlate to the spatial dimensions at the onset of buckling.

5. Limitations

Our study successfully examines the instabilities of bilayered systems subjected to eight different modes of compression; yet, ithas several important limitations. First, our analysis is restricted to wrinkling instabilities in the 1–2 plane, and does not fully

Fig. 8. Critical Biot pressure Pbiot vs. stiffness ratio β (left) and hydrostatic pressure Phydr vs. stiffness ratio β (right) for the eight different modes of compression

according to Table 1.

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capture three-dimensional buckling effects. Nonetheless, our model is able to provide reasonable predictions, for the critical strainsλϵ = 1 − and normalized wavelengths L H/ in the case of fully three-dimensional buckling (Tallinen and Biggins, 2015). The post-

buckling behavior of materials undergoing three-dimensional buckling is very interesting, manifesting a variety of shapes includingcheckerboards, hexagons, and herringbones (Audoly and Boudaoud, 2008a, 2008b; Cai et al., 2011; Li et al., 2009). The generalideas of energy minimization still hold, though, which most likely accounts for the relative accuracy of the two-dimensionalanalytical solution we have presented here. Second, wrinkles consist of infinitesimal strains in a finite region, while creases exhibitlarge strains in an infinitesimal region (Jin et al., 2015). As pointed out by other studies, when linearizing around the uniformlycompressed configuration, highly localized large-strain instabilities like creases are ignored. The linear perturbation analysis asperformed here can only predict the onset of infinitesimal deformations past a critical threshold. Analyses of creasing, cusping,subcritical instabilities, and post-buckling behavior require carefully designed studies that are out of the scope of our analyticalinvestigation. Third, another limitation is the nature of the isotropic, hyperelastic materials used. In reality, both the film and thesubstrate are often highly anisotropic and inelastic. Previous studies have modified the Föppl–von Kármán equations for viscousmaterials (Huang, 2005) to imitate the growth of the substrate (Bayly et al., 2013; Budday et al., 2014a), but to replicate morecomplex behaviors such as anisotropic prestretch or growth, numerical studies become necessary (Holland et al., 2015).

6. Conclusion

Here we have presented a comprehensive study of wrinkling instabilities in bilayered structures, suitable for materials with awide range of stiffness ratios, from β0.1 < < 1000, under a variety of loading conditions. Through the governing equations of theeigenvalue problem, we have illustrated whole-domain compression and film-only compression as distinct special cases of

Fig. 9. Critical strain λϵ = 1 − vs. wavelength L H/ , for eight different modes of compression compared to the original Föppl–von Kármán equations (dashed blackcurves) and the modified Föppl–von Kármán equations for compression (red solid curves) and growth (blue solid curves). Each curve represents a single stiffnessratio β. The curves for β = 0.1, 1, 10, 100, and 1000 are highlighted by thicker curves, from top to bottom, with β = 1 the thickest. (For interpretation of the references

to color in this figure caption, the reader is referred to the web version of this paper.)

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instabilities in bilayer systems. In total, we analyzed eight modes of compression, three modes of whole-domain compression andfive of film-only compression. Film-only compression includes substrate prestretch and film growth, which had not previously beenstudied via a linear perturbation analysis. While the eight compression modes are relatively indistinguishable for high stiffnesscontrasts, they display significant differences in the low stiffness contrast regime. Our study shows that these differences disappearwhen using measures of effective strain, effective stiffness, and effective wavelength. Our results suggest that future instabilitystudies in planar geometries should use these effective measure to standardize their findings. Finally, we revisited the classicalFöppl–von Kármán equations and modified them to account for geometric changes that take place when soft materials arecompressed under significant strain. Our study lays the foundation for studying instabilities in soft bilayered systems with variousapplications in nature including biology, development, and disease. For materials with stiffness ratios β < 10, the choices of stiffnessratio, boundary conditions, and mode of compression become important and significantly affect the instability analysis; these choicesmust be carefully considered in future studies.

Acknowledgments

This work was supported by the National Science Foundation Graduate Research Fellowship, by the National Science FoundationEast Asia and Pacific Summer Institutes Fellowship No. 1515340, and by the Stanford Graduate Fellowship to Maria A. Holland; bythe Stanford Bio-X Interdisciplinary Initiatives Program and by the National Institutes of Health Grant U01 HL119578 to EllenKuhl; and by the National Natural Science Foundation of China Grants 11432008 and 11542005 to Bo Li and Xi-Qiao Feng. Theauthors also gratefully acknowledge discussions with Alain Goriely, which led to the analysis of effective stiffness ratios.

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