Content uploaded by Johannes T.B. Overvelde

Author content

All content in this area was uploaded by Johannes T.B. Overvelde on Nov 19, 2017

Content may be subject to copyright.

Tensile Instability in a Thick Elastic Body

Johannes T. B. Overvelde,1,2 David M. J. Dykstra,1Rijk de Rooij,1James Weaver,3and Katia Bertoldi1,4,*

1John A. Paulson School of Engineering and Applied Sciences, Harvard University, Cambridge, Massachusetts 02138, USA

2FOM Institute AMOLF, Science Park 104, 1098 XG Amsterdam, Netherlands

3Wyss Institute for Biologically Inspired Engineering, Harvard University, Cambridge, Massachusetts 02138, USA

4Kavli Institute, Harvard University, Cambridge, Massachusetts 02138, USA

(Received 13 April 2016; revised manuscript received 1 July 2016; published 24 August 2016)

A range of instabilities can occur in soft bodies that undergo large deformation. While most of them arise

under compressive forces, it has previously been shown analytically that a tensile instability can occur in an

elastic block subjected to equitriaxial tension. Guided by this result, we conducted centimeter-scale

experiments on thick elastomeric samples under generalized plane strain conditions and observed for the

first time this elastic tensile instability. We found that equibiaxial stretching leads to the formation of a

wavy pattern, as regions of the sample alternatively flatten and extend in the out-of-plane direction. Our

work uncovers a new type of instability that can be triggered in elastic bodies, enlarging the design space

for smart structures that harness instabilities to enhance their functionality.

DOI: 10.1103/PhysRevLett.117.094301

A variety of instabilities can be triggered when elastic

structures are subjected to mechanical loadings [1,2]. While

such instabilities have traditionally been considered as the

onset of failure, a new trend is emerging in which the

dramatic geometric changes induced by them are harnessed

to enable new functionalities [3–6]. For example, buckling

of thin beams and shells has been instrumental in the design

of stretchable electronics [7,8], complex 3D architectures

[9,10], materials with negative Poisson’s ratio [11–13], and

tunable acoustic metamaterials [14,15]. Moreover, changes

in surface curvature due to wrinkling and creases have

enabled the control of surface chemistry [16], wettability

[17], adhesion [18,19], and drag [20].

While most elastic instabilities are the result of com-

pressive forces, elastic bodies may also become unstable

under tensile loading. For example, a wrinkling instability

can be triggered in a thin elastic sheet under uniaxial

extension [21–23], and a meniscus instability can occur

when a thin layer of elastic material is confined and pulled

in the out-of-plane direction resulting in a periodic array

of fingers at its edges [24,25]. Moreover, it is well known

that a cavity can undergo a sudden expansion upon

reaching a critical internal pressure. This instability is

not only observed in the case of thin membranes

[26,27], but also persists in thick solid bodies where it is

often referred to as cavitation [28]. Since cavitation is the

only tensile instability that has been found in thick elastic

bodies, a natural question to ask is whether other elastic

tensile instabilities can occur in such systems.

About half a century ago, it was shown analytically that a

tension instability can be triggered in a block of incom-

pressible elastic material subjected to equitriaxial tension

[Fig. 1(a)][29–33]. More specifically, it has been demon-

strated that when a cube with edges of length Land made

from an incompressible Neo-Hookean material with initial

shear modulus μis subjected to uniform tractions resulting

in six tensile normal forces of magnitude F, two possible

equilibrium solutions exist (see Supplemental Material:

Analytical exploration [34]):

FIG. 1. Force-stretch bifurcation diagram for (a) a cube sub-

jected to triaxial tension, (b) a square under plane strain

conditions subjected to biaxial tension, and (c) a biaxially

stretched cross-shaped sample under plane strain conditions.

The solid and dashed lines represent stable and unstable load

paths, respectively. The contours show the maximum in-plane

principal strain. Note that (a) and (b) were obtained analytically,

while (c) was obtained using finite element analysis.

PRL 117, 094301 (2016) PHYSICAL REVIEW LETTERS week ending

26 AUGUST 2016

0031-9007=16=117(9)=094301(6) 094301-1 © 2016 American Physical Society

λ1¼λ2¼λ3¼1ð1Þ

and

F

μL2¼λ1þ1

λ2

1

;λ2¼λ1;λ3¼λ−2

1;ð2Þ

λibeing the principle stretches. As a result, when the

applied force Fis gradually increased, the block maintains

its undeformed configuration (λi¼1) until F¼2μL2.At

this point, the solution bifurcates, the initial branch

[Eq. (1)] becomes unstable, and the cube snaps to the

second branch [Eq. (2)] and, therefore, suddenly flattens. It

should be noted that this instability has only been dem-

onstrated analytically and has not been triggered exper-

imentally. In this Letter, guided by both analytical and finite

element models, we report the first experimental observa-

tion of this instability in a thick elastomeric sample that is

stretched equibiaxially under generalized plane strain

conditions.

Designing an experiment to realize the tension insta-

bility.—Although it is possible to analytically obtain the

triaxial tensile instability for a cube under triaxial tension, it

is challenging to realize the same conditions in experi-

ments. First, the instability requires the application of six

equal and orthogonal forces to a block of material. Second,

the forces need to be evenly spread across the whole

surface, and third, the boundary conditions have to adapt to

the large deformation after the instability has occurred.

In an effort to simplify the boundary conditions, we start

by considering an incompressible elastomeric block under

plane strain conditions (i.e., λ3¼1and λ2¼λ−1

1) subjected

to four in-plane tensile forces of magnitude Fas indicated

in Fig. 1(b). When assuming a Neo-Hookean material, the

potential energy of the system Πis given by

Π¼μL2D

2λ2

1þ1

λ2

1

−2−FLλ1þ1

λ1

−2;ð3Þ

in which Dis the out-of-plane thickness of the sample. The

equilibrium solutions are then found by minimizing Π(i.e.,

∂Π=∂λ1¼0), yielding

λ1¼λ2¼λ3¼1ð4Þ

and

F

μLD ¼λ1þ1

λ1

;λ2¼λ−1

1;λ3¼1;ð5Þ

which are stable only if

∂2Π

∂λ2¼μL2D1þ3

λ4

1−FL2

λ3

1>0:ð6Þ

Interestingly, the solutions defined by Eqs. (4) and (5) are

similar to those found for the triaxial case [Eqs. (1) and (2)]

and still show a bifurcation point at F¼2μLD. Differently,

for the plane strain case, no snap-through instability is

observed since the force monotonically increases, as

indicated in Fig. 1(b). Note that during loading the out-

of-plane tensile stress that builds up in the material due to

the plane strain conditions plays an important role, since no

instability occurs if we assume plane stress conditions (see

Supplemental Material: Analytical exploration [34]).

Next, to apply uniformly distributed traction forces to the

edges of the square, we consider a cross-shaped specimen, as

typically done for biaxial experiments [37,38]. More spe-

cifically, we consider a square of edges Wwith circles of

radius R¼0.31Wcut from the corners. When assuming

plane strain conditions, and applying an outward displace-

ment to the straight boundaries of the sample, we expect its

center to undergo a triaxial state of stress. To compare the

response of the cross-shaped sample with our analytical

predictions for a square [Fig. 1(b)], we monitor the evolution

of the two diagonals with initial length L¼ﬃﬃﬃ

2

pW−2R

located at the center of the sample, as shown in Fig. 1(c),and

introduce the stretches λ1and λ2to define their deformation.

Next, to determine the response of the cross-shaped

sample upon loading, we performed 2D implicit finite

element analysis under plane strain conditions using

Abaqus (Dassault Systèmes). We captured the material

response using a nearly incompressible Neo-Hookean

model characterized by a ratio between the bulk modulus

Kand shear modulus μof K=μ¼20 [39]. The four straight

edges of the samples were loaded by a force Fin their

normal direction, while allowing movement in the orthogo-

nal direction. Moreover, to break the symmetry of the

structure, we introduced a small imperfection by increasing

the radius of two diagonally placed holes by 0.2%.

The results of our simulations are shown in Fig. 1(c),

where we report the evolution of λ1and λ2as a function of

the normalized force F=ðμLDÞ. We find that an instability is

triggered at F=ðμLDÞ≈2, resulting in a sudden flattening of

the central part of the sample similar to that predicted by the

analytical model. However, different from the analytical

results shown in Fig. 1(b), our results reveal that prior to the

instability the central domain slightly reduces in size (i.e.,

λ1¼λ2≠1). This discrepancy arises because the deforma-

tion in the numerical model is not homogeneous, as assumed

in the analytical model [see distribution of the maximum in-

plane strain ϵmax in Fig. 1(c)].

Experimental result.—Having demonstrated numerically

that an elastic instability is triggered when a cross-shaped

sample under plane strain conditions is subjected to

biaxial tension, we fabricated a thick sample from a silicon

rubber (Ecoflex 0030, Smooth-On) with a shear modulus

μ¼0.0216 MPa [Fig. 2(a) and Supplemental Material:

Experiments [34]]. To constrain the out-of-plane deforma-

tion we connected each side of the sample to a stiffer silicon

PRL 117, 094301 (2016) PHYSICAL REVIEW LETTERS week ending

26 AUGUST 2016

094301-2

elastomer (Elite Double 32, Zhermack) characterized by

μ¼0.262 MPa [40]. Moreover, to approximate the plane

strain conditions assumed in our calculations, we took

D=W ≫1(D¼132 mm and W¼14 mm). Finally, we

placed steel tubes inside the stiffer elastomer to connect it

through cables to a rigid frame that was used to stretch the

sample. Figure 2(b) shows the observed deformation at

u=W ¼2.9,u=2being the displacement applied to each

straight edge. Since boundary effects prevented us from

clearly observing the instability (Fig. S6 in Supplemental

Material [34]), we acquired x-ray transmission images at

different levels of applied deformation [Fig. 2(c) herein and

Movie 1 in Supplemental Material [34] ]. We then manually

processed the images to obtain the stretches, λ1and λ2, that

define the deformation of the diagonals of the central

domain, exactly as in our simulations. The results reported

in Fig. 2(d) indicate that at u=W ≈1an instability is

triggered that breaks the symmetry and initiates a flattening

of the center of the sample. While the experimental results

agree relatively well with the 2D plane strain simulations,

we find that the instability is triggered for smaller defor-

mations. This discrepancy is likely due to imperfections

introduced during fabrication and loading, which tend to

smoothen the sudden transition arising from the instability

and result in an earlier flattening (Fig. S7 in Supplemental

Material [34]).

Wavy pattern along the depth.—From the experiments

we find that the instability not only results in the flattening

of the center of the sample as predicted by the plane strain

simulations, but also introduces waves on the surfaces

along the depth [Fig. 2(b)]. To better understand the

formation of this wavy pattern, we conducted 3D explicit

quasistatic finite element analysis and simulated the cross-

shaped sample as used in the experiments; i.e., we modeled

both the two elastomeric materials and steel tubes to exactly

mimic the experimental conditions [41].

As shown in Fig. 3(a) herein and Movie 2 in

Supplemental Material [34], our 3D simulations confirm

the experimental observations. By monitoring the stretch of

the diagonals defining the center region of the sample

(along the depth), it becomes clear that the wavy pattern

emerges the moment the sample becomes unstable at

u=W ≈2[Fig. 3(b)]. In fact, for u=W ≲2,λ1and λ2are

constant along the depth of the sample, while for u=W ≳2,

they oscillate periodically. Moreover, we also find that after

the instability has occurred, the sample not only deforms

nonuniformly in plane, but also in the out-of-plane direc-

tion. To highlight this point, in Fig. 3(c) we report the out-

of-plane stretch λ3measured along the center line of the

sample at different levels of applied loading. The results

indicate that for u=W ≳2there is an alternation between

regions experiencing out-of-plane extension and compres-

sion along the depth of the sample.

Informed by the numerical results of Figs. 3(a)–3(c),

we next extend our analytical model and assume that the

elastic block consists of two layers, aand b, which can

deform separately. We then impose generalized plane strain

conditions

FIG. 2. (a) Experimental setup to subject our thick cross-shaped sample to biaxial tension. (b) Top and bottom views of the deformed

sample at u=W ¼2.90. (c) Cross-sectional views of the sample at u=W ¼0, 0.72, 1.45, 2.17, and 2.90 obtained using a micro-CT

(computed tomography) x-ray imaging machine. (d) Relation between the normalized displacement u=W applied to the sample and the

stretches λ1and λ2of the diagonals located at the center of the sample. The results were obtained by manually processing the images and

averaging five individual measurements (the error bars indicate the standard deviation) (scale bars 10 mm).

PRL 117, 094301 (2016) PHYSICAL REVIEW LETTERS week ending

26 AUGUST 2016

094301-3

¯

hλa;3þð1−¯

hÞλb;3¼1;ð7Þ

where the stretches λa;3and λb;3are indicated in Fig. 3(d),

and ¯

h¼Da=ðDaþDbÞsets the ratio between the depth of

layer aand bin the undeformed configuration. If we further

assume that the four in-plane forces applied to each layer

depend on the initial size of the layer [i.e., Fa¼F¯

hand

Fb¼Fð1−¯

hÞ,Fbeing the total force applied to the two

blocks], the potential energy takes the form (Supplemental

Material: Analytical exploration [34])

Π¼

¯

hμL2D

2λ2

a;1þ1

λ2

a;1λ2

a;3þλ2

a;3−3

þμL2D1−¯

h

2λ2

b;1þð1−¯

hÞ2

λ2

b;1ð1−¯

hλa;3Þ2

þð1−¯

hλa;3Þ2

ð1−¯

hÞ2−3−FaLλa;1þ1

λa;1λa;3

−2

−FbLλb;1þ1−¯

h

λb;1ð1−¯

hλa;3Þ−2;ð8Þ

where D¼DaþDb. Minimizing the energy results again

in two possible solutions:

λa;1¼1with 0≤¯

h≤1;ð9Þ

F

μLD ¼λa;1þ1

λ2

a;1

with ¯

h¼λ2

a;1−1

λ3

a;1−1;ð10Þ

in which λa;3¼λb;1¼λb;2¼λa;1and λa;2¼λb;3¼λ−2

a;1.

We find that the solutions defined by Eqs. (9) and (10) are

identical to those found for a cube subjected to equitriaxial

tension [Eqs. (1) and (2)], and that a bifurcation occurs at

F=ðμLDÞ¼2.ForF=ðμLDÞ<2the system does not

deform [as illustrated in Fig. 3(e) for F=ðμLDÞ¼3=2],

while for F=ðμLDÞ>2one of the layers extends and the

other flattens in the out-of-plane direction [as illustrated

in Fig. 3(e) for F=ðμLDÞ¼5=2], resulting in a wavy

pattern that resembles the deformation shown in Figs. 2(b)

and 3(a).

Outlook.—In this work, we experimentally showed that

an instability can be triggered in a thick elastic body

subjected to in-plane tensile forces and generalized plane

strain conditions. While an instability was already analyti-

cally predicted in 1948 for a cube subjected to triaxial

tension [29], here we extended the analysis to a configu-

ration that can be tested experimentally, and found that the

modified conditions result in a wavy pattern, as portions of

the sample alternatively extend and flatten in the out-of-

plane direction.

It should be noted that tensile loading conditions can also

lead to cavitation in solids [28,42]. More specifically, for an

incompressible Neo-Hookean material subjected to equi-

triaxial tension, it can be analytically derived that cavitation

initiates at a pressure of pcav ¼5μ=2[43]. As a result, for

the elastomeric cube of Fig. 1(a) subjected to triaxial

tension, cavitation initiates at Fcav ¼5μL2=2. Although

this critical value is 25% higher than that needed to flatten

FIG. 3. (a) Numerical snapshots of the sample, (b) stretches that

define the deformation of the diagonals of the central domain (λ1

and λ2) along the depth of the sample, and (c) out-of-plane stretch

of the center line (λ3) along the depth of the sample, at u=W ¼0,

0.96, 1.91, 2.90. (d) Schematic of the bilayer under generalized

plane strain conditions used in our analytical model. (e) Deformed

states of the analytical model at F=ðμLDÞ¼3=2and 5=2for

D¼2L.

FIG. 4. Deformation of a mechanical metamaterial comprising

a square array of circular pores subjected to equibiaxial tension.

Similar to the case of the cross-shaped sample shown in Fig. 1(c),

an instability is triggered at u=W ≈2resulting in a checkerboard

pattern of pores with two different sizes. The contours represent

the maximum in-plane strain ϵmax .

PRL 117, 094301 (2016) PHYSICAL REVIEW LETTERS week ending

26 AUGUST 2016

094301-4

the cube, we expect our sample to experience such a value

of stress in the postbuckling regime. In fact, upon increas-

ing the stretch applied to the sample to u=W ¼3.26,we

immediately see some cavities forming, which slowly

increase in size when the applied deformation is maintained

for a few hours (Movie 3 in Supplemental Material [34]).

Finally, the cross-shaped samples used in our experi-

ments can be used to build a mechanical metamaterial by

arranging them on a square lattice as shown in Fig. 4.By

stretching the metamaterial biaxially (under plane strain

conditions), an instability is triggered at u=W ≈2resulting

in a checkerboard pattern of pores with two different sizes,

as indicated by the evolution of the characteristic pore sizes

l1and l2shown in Fig. 4. While the formation of this

pattern has previously been observed in simulations of

similar periodic porous structures [44–46], with the current

work we have deciphered the underlying mechanism that

leads to the instability. As such, we expect our study to

open new avenues for the design of soft structures that

harness instabilities for improved functionality.

This work was supported by the Materials Research

Science and Engineering Center under NSF Grant

No. DMR-1420570. K. B. also acknowledges support from

the National Science Foundation (CMMI-1149456-

CAREER). We thank Samuel Shian for his initial help

with the experimental setup and Sahab Babaee and John

Hutchinson for fruitful discussions.

*bertoldi@seas.harvard.edu

[1] Z. Bažant and L. Cedolin, Stability of Structures: Elastic,

Inelastic, Fracture and Damage Theories (World Scientific,

Singapore, 2010).

[2] Q. Wang and X. Zhao, MRS Bull. 41, 115 (2016).

[3] S. Singamaneni and V. V. Tsukruk, Soft Matter 6, 5681

(2010).

[4] D. Chen, J. Yoon, D. Chandra, A. J. Crosby, and R. C.

Hayward, J. Polym. Sci. B 52, 1441 (2014).

[5] B. Florijn, C. Coulais, and M. van Hecke, Phys. Rev. Lett.

113, 175503 (2014).

[6] S. Cai, D. Breid, A. Crosby, Z. Suo, and J. Hutchinson,

J. Mech. Phys. Solids 59, 1094 (2011).

[7] J. A. Rogers, T. Someya, and Y. Huang, Science 327, 1603

(2010).

[8] Y. Wang, R. Yang, Z. Shi, L. Zhang, D. Shi, E. Wang, and

G. Zhang, ACS Nano 5, 3645 (2011).

[9] J. Kim, J. A. Hanna, M. Byun, C. D. Santangelo, and R. C.

Hayward, Science 335, 1201 (2012).

[10] S. Xu, Z. Yan, K.-I. Jang, W. Huang, H. Fu, J. Kim, Z. Wei,

M. Flavin, J. McCracken, R. Wang et al.,Science 347, 154

(2015).

[11] K. Bertoldi, P. M. Reis, S. Willshaw, and T. Mullin, Adv.

Mater. 22, 361 (2010).

[12] J. T. B. Overvelde, S. Shan, and K. Bertoldi, Adv. Mater. 24,

2337 (2012).

[13] S. Babaee, J. Shim, J. C. Weaver, E. R. Chen, N. Patel, and

K. Bertoldi, Adv. Mater. 25, 5044 (2013).

[14] K. Bertoldi and M. C. Boyce, Phys. Rev. B 77,052105

(2008).

[15] P. Wang, F. Casadei, S. Shan, J. C. Weaver, and K. Bertoldi,

Phys. Rev. Lett. 113, 014301 (2014).

[16] J. Kim, J. Yoon, and R. C. Hayward, Nat. Mater. 9,159

(2010).

[17] P.-C. Lin and S. Yang, Soft Matter 5, 1011 (2009).

[18] P.-C. Lin, S. Vajpayee, A. Jagota, C.-Y. Hui, and S. Yang,

Soft Matter 4, 1830 (2008).

[19] E. Chan, E. Smith, R. Hayward, and A. Crosby, Adv. Mater.

20, 711 (2008).

[20] D. Terwagne, M. Brojan, and P. M. Reis, Adv. Mater. 26,

6608 (2014).

[21] E. Cerda and L. Mahadevan, Phys. Rev. Lett. 90, 074302

(2003).

[22] B. Davidovitch, R. D. Schroll, D. Vella, M. Adda-Bedia,

and E. A. Cerda, Proc. Natl. Acad. Sci. U.S.A. 108, 18227

(2011).

[23] H. Vandeparre, S. Gabriele, F. Brau, C. Gay, K. K. Parker,

and P. Damman, Soft Matter 6, 5751 (2010).

[24] K. R. Shull, C. M. Flanigan, and A. J. Crosby, Phys. Rev.

Lett. 84, 3057 (2000).

[25] J. S. Biggins, B. Saintyves, Z. Wei, E. Bouchaud, and L.

Mahadevan, Proc. Natl. Acad. Sci. U.S.A. 110, 12545

(2013).

[26] D. R. Merritt and F. Weinhaus, Am. J. Phys. 46, 976

(1978).

[27] J. T. B. Overvelde, T. Kloek, J. J. A. D’haen, and K.

Bertoldi, Proc. Natl. Acad. Sci. U.S.A. 112, 10863

(2015).

[28] A. N. Gent and P. B. Lindley, Proc. R. Soc. A 249, 195

(1959).

[29] R. S. Rivlin, Phil. Trans. R. Soc. A 240, 491 (1948).

[30] R. Hill, J. Mech. Phys. Solids 5, 229 (1957).

[31] M. F. Beatty, Int. J. Solids Struct. 3, 23 (1967).

[32] R. S. Rivlin, Collected Papers of R.S. Rivlin: Volume I

and II (Springer, New York, 1997), pp. 398–404.

[33] J. M. Ball and D. G. Schaeffer, Math. Proc. Cambridge

Philos. Soc. 94, 315 (1983).

[34] See Supplemental Material at http://link.aps.org/

supplemental/10.1103/PhysRevLett.117.094301, which in-

cludes Refs. [29–33,35,36], for supporting movies and an

analytical exploration (describing a cube subjected to

equitriaxial tension, a square under plane strain and plane

stress conditions subjected to equibiaxial tension, and a

block under generalized plane strain conditions subjected to

equibiaxial tension), a numerical study looking at the effect

of the material compressibility, and a description of the

experiments (fabrication and testing).

[35] R. Ogden, Non-Linear Elastic Deformations (Dover,

New York, 1988).

[36] M. C. Boyce and E. M. Arruda, Rubber Chem. Technol. 73,

504 (2000).

[37] E. Shiratori and K. Ikegami, J. Mech. Phys. Solids 16, 373

(1968).

[38] Y. Lecieux and R. Bouzidi, Int. J. Solids Struct. 47, 2459

(2010).

[39] The models were discretized using approximately 1.5×104

triangular quadratic elements (Abaqus element code CPE6).

For the nearly incompressible Neo-Hookean model we

PRL 117, 094301 (2016) PHYSICAL REVIEW LETTERS week ending

26 AUGUST 2016

094301-5

chose K=μ¼20, as it provides a good balance between

simulation time and accuracy (Supplemental Material:

Effect of material compressibility [34]).

[40] J. Shim, C. Perdigou, E. R. Chen, K. Bertoldi, and P. M.

Reis, Proc. Natl. Acad. Sci. U.S.A. 109, 5978 (2012).

[41] We modeled the Elite Double 32 and Ecoflex 0030 using a

nearly incompressible Neo-Hookean material model with

K=μ¼20. Moreover, we assumed that the steel tubes do not

deform, and used approximately 5×105linear elements to

construct the model (Abaqus element code C3D6R and

C3D8R). We further assumed that the bonding region

between the two elastomers is infinitesimally thin and

can be neglected. Note that since we used explicit analysis,

we did not apply any imperfections.

[42] J. M. Ball, Phil. Trans. R. Soc. A 306, 557 (1982).

[43] A. Gent, Int. J. Nonlinear Mech. 40, 165 (2005).

[44] K. Bertoldi and M. C. Boyce, Phys. Rev. B 78, 184107

(2008).

[45] J. T. Overvelde and K. Bertoldi, J. Mech. Phys. Solids 64,

351 (2014).

[46] J. Michel, O. Lopez-Pamies, P. P. Castañeda, and N.

Triantafyllidis, J. Mech. Phys. Solids 55, 900 (2007).

PRL 117, 094301 (2016) PHYSICAL REVIEW LETTERS week ending

26 AUGUST 2016

094301-6

Supplemental Material

ANALYTICAL EXPLORATION

To identify the key components in the design of an elastic body that undergoes an instability when subjected

to tensile forces, we analytically explore the equilibrium states of (i) an elastomeric cube subjected to equitriaxial

tension, (ii) an elastomeric square under plane stress conditions subjected to equibiaxial tension, (iii) an elastomeric

square under plane strain conditions subjected to equibiaxial tension and (iv) an elastomeric block under generalized

plane strain conditions subjected to equibiaxial tension.

In all our calculations, we assume the elastic body to be made of an incompressible Neo-Hookean material, whose

strain energy density function, W, is given by [1]

W=µ

2λ2

1+λ2

2+λ2

3−3,(S1)

where µis the initial shear modulus of the material and λiare the principal stretches, which are subjected to the

incompressibility constraint λ1λ2λ3= 1.

Cube subjected to equitriaxial tension

Similar to previous studies [2–6], we consider a cube with sides of length Lsubjected to six orthogonal outward

pointing forces as indicated in Fig. 1(a). Assuming that the cube undergoes a homogeneous deformation, its internal

energy, U, equals

U=L3W=µL3

2λ2

1+λ2

2+λ2

3−3,(S2)

and the work done by the external forces, V, is given by

V=F L (λ1−1) + F L (λ2−1) + F L (λ3−1)

=F L (λ1+λ2+λ3−3) .(S3)

It follows that the potential energy for the system, Π = U−V, is given by

Π =µL3

2λ2

1+λ2

2+1

λ2

1λ2

2

−3−F L λ1+λ2+1

λ1λ2

−3,(S4)

where we have used the fact that λ3= 1/(λ1λ2) due to incompressibility of the material. The equilibrium solutions

can then be found by minimizing Π with respect to λ1and λ2,

∂Π

∂λ1

=µL3λ1−1

λ3

1λ2

2−F L 1−1

λ2

1λ2= 0,(S5)

∂Π

∂λ2

=µL3λ2−1

λ2

1λ3

2−F L 1−1

λ1λ2

2= 0.(S6)

Interestingly, there are two distinct solutions that satisfy Eqs. (S5) and (S6),

λ1=λ2=λ3= 1,(S7)

and

F

µL2=λ1+1

λ2

1

, λ2=λ1, λ3=λ−2

1.(S8)

Moreover, note that the two solutions deﬁned by Eqs. (S7)-(S8) are stable when the Hessian of the potential energy,

H, is positive deﬁnite. This requires that

∂2Π

∂λ2

1

>0,and ∂2Π

∂λ2

1

∂2Π

∂λ2

2

−∂2Π

∂λ1∂λ2

∂2Π

∂λ2∂λ1

>0,(S9)

2

in which

∂2Π

∂λ2

1

=µL31 + 3

λ4

1λ2

2−F L 2

λ3

1λ2(S10)

∂2Π

∂λ1∂λ2

=∂2Π

∂λ2∂λ1

=2µL3

λ3

1λ3

2

−F L

λ2

1λ2

2

(S11)

∂2Π

∂λ2

2

=µL31 + 3

λ2

1λ4

2−F L 1

λ1λ3

2.(S12)

As a result, when the applied force Fis gradually increased, the block maintains its undeformed conﬁguration

(λi= 1) until F= 2µL2. At this point, the solution bifurcates, the initial branch (Eq. (S7)) becomes unstable and

the cube snaps to the second branch (Eq. (S8)) and therefore suddenly ﬂattens (see Fig. 1(a)).

Square under plane stress conditions subjected to equibiaxial tension

We next show that no instability occurs when an elastomeric square under plane stress conditions is subjected to

equibiaxial tension. Since in this case no work is done in the out-of-plane direction, the work done by the applied

forces reduces to

V=F L (λ1+λ2−2) ,(S13)

so that the potential energy for the system becomes

Π = µL2D

2λ2

1+λ2

2+1

λ2

1λ2

2

−3−F L (λ1+λ2−2) ,(S14)

where Dis the depth of the sample, and we have used the incompressibility condition (i.e. λ1λ2λ3= 1). Equilibrium

requires that

∂Π

∂λ1

=µL2Dλ1−1

λ3

1λ2

2−F L = 0,(S15)

∂Π

∂λ2

=µL2Dλ2−1

λ2

1λ3

2−F L = 0,(S16)

that are only satisﬁed for

F

µLD =λ1−1

λ5

1, λ2=λ1, λ3=λ−2

1.(S17)

Note that this solution is always stable.

Square under plane strain conditions subjected to equibiaxial tension

Diﬀerently from the plane stress case, an instability can be observed when an elastomeric square under plane strain

conditions is subjected to equibiaxial tension. Since for plane strain λ3= 1, the incompressibility constraint reduces

to λ1=λ−1

2. It follows that the potential energy for the system is given by

Π = µL2D

2λ2

1+1

λ2

1

−2−F L λ1+1

λ1

−2,(S18)

so that the equilibrium conﬁgurations are found by setting

dΠ

dλ1

=µL2Dλ1−1

λ3

1−F L 1−1

λ2

1= 0.(S19)

Interestingly, there are two distinct solutions that satisfy Eq. (S19),

λ1=λ2=λ3= 1,(S20)

3

FIG. S1: Incremental solution obtained with the fmincon function in Matlab for the general plane strain case. (a) Relation

between the normalized force F/(µLD) and the potential energy Π/(µL2D). (b) Relation between the normalized force

F/(µLD) and the characteristic stretches λa,1,λa,3and λb,1.

and

F

µLD =λ1+1

λ1

, λ2=λ−1

1, λ3= 1.(S21)

Furthermore, stability requires

∂2Π

∂λ2

1

=µL2D1 + 3

λ4

1−F L 2

λ3

1>0.(S22)

It should be noted that the solutions deﬁned by Eqs. (S20)-(S21) are similar to those found for the triaxial case (Eqs.

(S7)-(S8)) and still show a bifurcation point at F= 2µLD. Diﬀerently, for the plane strain case no snap-through

instability is observed since the force monotonically increases, as indicated in Fig. 1(b).

Block under generalized plane strain conditions subjected to equibiaxial tension

Our experimental and 3D numerical results indicate that the instability not only results in the ﬂattening of the

center of the sample as predicted by the plane strain simulations, but also introduces waves on the surfaces along

the depth. Informed by these results, we extend our analytical model and assume that the elastic block consists of

two layers, aand b, which deform separately and homogeneously (Fig. 3(d)). The deformation of the system is then

fully described by six stretches, λa,i for layer aand λb,i for layer b(i= 1,2,3), subjected to the incompressibility

constraints

λa,1λa,2λa,3= 1,(S23)

λb,1λb,2λb,3= 1.(S24)

Next, we impose generalized plane strain conditions, so that

Daλa,3+Dbλb,3=Da+Db=D, (S25)

which can be rewritten as

¯

hλa,3+1−¯

hλb,3= 1,(S26)

where ¯

h=Da/(Da+Db) sets the ratio between the depth of layer aand bin the undeformed conﬁguration.

Furthermore, given the dimensions and forces indicated in Fig. 3(d), the potential energy for layers aand bis given

by

Ua=µL2Da

2λ2

a,1+λ2

a,2+λ2

a,3−3,(S27)

Ub=µL2Db

2λ2

b,1+λ2

b,2+λ2

b,3−3,(S28)

4

while the work done by the applied forces equals

Va=FaL(λa,1+λa,2−2) ,(S29)

Vb=FbL(λb,1+λb,2−2) .(S30)

By further assuming that the four in-plane forces applied to each layer depend on the initial size of the layer (i.e.

Fa=F¯

hand Fb=F(1 −¯

h), Fbeing the total force applied to the two blocks) and using the conditions from

Eqs. (S23)-(S26), Eqs. (S27)-(S30) can be rewritten as

Ua=µ¯

hDL2

2 λ2

a,1+1

λ2

a,1λ2

a,3

+λ2

a,3−3!,(S31)

Ub=µ1−¯

hDL2

2 λ2

b,1+1−¯

h2

λ2

b,11−¯

hλa,32+1−¯

hλa,32

1−¯

h2−3!,(S32)

Va=F¯

hL λa,1+1

λa,1λa,3

−2,(S33)

Vb=F1−¯

hL λb,1+1−¯

h

λb,11−¯

hλa,3−2!,(S34)

in which the independent variables have been reduced to λa,1,λa,3,λb,1and ¯

h. Finally, the potential energy for the

system can be obtained as

Π = Ua+Ub−Va−Vb.(S35)

and the stable equilibrium solutions can be determined by minimizing the potential energy Π,

∂Π

∂λa,1

= 0,∂Π

∂λa,3

= 0,∂Π

∂λb,1

= 0,∂Π

∂¯

h= 0,(S36)

where

∂Π

∂λa,1

=1

2D¯

hL2µ 2λa,1−2

λ3

a,1λ2

a,3!−¯

hLF 1−1

λ2

a,1λa,3!,(S37)

∂Π

∂λa,3

=1

2D¯

hL2µ 2λa,3−2

λ2

a,1λ3

a,3!+1

2D(1 −¯

h)L2µ 2(1 −¯

h)2¯

h

λ2

b,1(1 −¯

hλa,3)3−2¯

h(1 −¯

hλa,3)

(1 −¯

h)2!

+¯

hLF

λa,1λ2

a,3

−

¯

h(1 −¯

h)2LF

λb,1(1 −¯

hλa,3)2,(S38)

∂Π

∂λb,1

=1

2D(1 −¯

h)L2µ 2λb,1−2(1 −¯

h)2

λ3

b,1(1 −¯

hλa,3)2!−(1 −¯

h)LF 1−1−¯

h

λ2

b,1(1 −¯

hλa,3)!,(S39)

∂Π

∂¯

¯

h=−1

2DL2µ (1 −¯

h)2

λ2

b,1(1 −¯

hλa,3)2+(1 −¯

hλa,3)2

(1 −¯

h)2+λ2

b,1−3!

+1

2D(1 −¯

h)L2µ 2(1 −¯

h)2λa,3

λ2

b,1(1 −¯

hλa,3)3−2(1 −¯

h)

λ2

b,1(1 −¯

hλa,3)2+2(1 −¯

hλa,3)2

(1 −¯

h)3−2λa,3(1 −¯

hλa,3)

(1 −¯

h)2!

+1

2DL2µ 1

λ2

a,1λ2

a,3

+λ2

a,1+λ2

a,3−3!+LF 1−¯

h

λb,1(1 −¯

hλa,3)+λb,1−2

−(1 −¯

h)LF (1 −¯

h)λa,3

λb,1(1 −¯

hλa,3)2−1

λb,1(1 −¯

hλa,3)−LF 1

λa,1λa,3

+λa,1−2.(S40)

Since we could not solve Eq. (S36) analytically, we used the fmincon function in Matlab (Mathworks) to ﬁnd

the lowest energy states. More speciﬁcally, we started from the undeformed conﬁguration for which F= 0 and

λa,i =λb,i =1(i= 1,2,3) and set ¯

h= 0.5. We then incrementally increased Fby ∆F=µLD/100 to ﬁnd the

5

FIG. S2: Eﬀect of material compressibility. (a) Relation between the applied force Fand the stretches of the diagonals λ1and

λ2as a function of K/µ for a cross-shape sample under plane strain conditions subjected to equibiaxial tension. (b) Relation

between the applied force Fand the stretches of the diagonals λ1and λ2as a function of K/µ for a square under plane strain

conditions subjected to equibiaxial tension. (c) Evolution of the critical stretch as a function of K/µ for a square under plane

strain conditions subjected to equibiaxial tension.

equilibrium path (note that we used ∆F=µLD/10000 around the instability). The minimal energy path predicted

by our numerical calculations is shown in Fig. S1(a). Upon loading, the system does not deform until F / (µLD) = 2

as shown in Fig. S1(b), and as such the internal energy Uof the block remains zero. However, at F/ (µLD)=2

a snap-through instability occurs instantly lowering the potential energy Π (see inset in Fig. S1(a)). Interestingly,

for F/ (µLD)>2 both layers ﬂatten and deform into two ﬂat plates rotated 90 degrees with respect to each other.

Moreover, we ﬁnd that for F/ (µLD)>2, λa,1=λa,3=λb,1as shown in Fig. S1(b).

Next, guided by our numerical results, we turn back to Eqs. (S35)-(S40) and assume that λa,3=λa,1, and λb,1=λa,1.

The terms from Eqs. (S37)-(S40) then simplify, such that

∂Π

∂λa,1

=µL2D¯

h λa,1−1

λ5

a,1!−F L¯

h 1−1

λ3

a,1!= 0,(S41)

∂Π

∂λb,1

=µL2D(1 −¯

h) λa,1−(1 −¯

h)2

λ3

a,1(1 −¯

hλa,1)2!−F L(1 −¯

h) 1−1−¯

h

λ2

a,1(1 −¯

hλa,1)!= 0,(S42)

while ∂Π/∂λa,3= 0 and ∂Π/∂ ¯

h= 0 are automatically satisﬁed when Eqs. (S41)-(S42) are satisﬁed. Interestingly,

there are two distinct solutions to Eqs. (S41)-(S42)

λa,1= 1 with 0 ≤¯

h≤1,(S43)

F

µLD =λa,1+1

λ2

a,1

with ¯

h=λ2

a,1−1

λ3

a,1−1.(S44)

Finally, by combining Eqs. (S23)-(S24) and (S43)-(S44), all the stretches in the two layers can be determined

λa,3=λb,1=λb,2=λa,1,(S45)

λa,2=λb,3=λ−2

a,1.(S46)

Importantly, we ﬁnd that the solutions deﬁned by Eqs. (S43)-(S44) are identical to those found for a cube subjected

to equitriaxial tension (Eqs. (S7)-(S8)), and that a bifurcation occurs at F / (µLD) = 2. For F/ (µLD)<2 the system

does not deform (as illustrated in Fig. 3(e) for F / (µLD)=3/2), while for F / (µLD)>2 one of the layer extends

and the other ﬂattens in the out-of-plane direction (as illustrated in Fig. 3(e) for F / (µLD)=5/2), resulting in a

wavy pattern that closely resembles the deformation shown in Figs. 2(b) and 3(a).

EFFECT OF MATERIAL COMPRESSIBILITY

In our simulations, the choice of K/µ = 20 (resulting in a Poisson’s ratio of 0.475) was dictated by numerical

considerations. Note that the stable time increment for our 3D explicit analyses is given by

∆t=L

c,(S47)

6

where Lis a characteristic length of the elements and cis the wave speed in a 3D-medium,

c=sE(1 −ν)

ρ(1 + ν)(1 −2ν),(S48)

νbeing the Poisson’s ratio. From Equation (S48) it becomes clear that when the materials approaches incompressibility

(i.e ν→1/2 and K/µ → ∞), ∆tbecomes very small, resulting in signiﬁcantly longer simulation time (note that

Abaqus only allows for K/µ ≤100, corresponding to a Poisson’s ratio of ν= 0.495). To ﬁnd a compromise between

simulation time and accuracy (as we wanted to preserve a nearly incompressible behavior in our analysis), we simulated

the response of a cross-shaped sample under plane strain conditions subjected to equibiaxial tension for diﬀerent values

of K/µ. First, the results reported in Fig. S2(a) indicate that the instability is triggered for all the considered values

of K/µ. However, as K/µ decreases, smaller forces are needed to trigger the bifurcation. Second, we ﬁnd that the

response of the sample characterized by K/µ = 20 and K/µ = 80 are very close to each other, so we choose K/µ = 20

as it provides a good balance between accuracy and simulation speed.

Note that we can also study analytically the eﬀect of the ratio K/µ by determining the response of a compressible

square under plane strain conditions. To this end, we use the strain energy of a nearly incompressible Neo Hookean

material [1, 7]

W=µ

2λ2

1+λ2

2+λ2

3−3+K

2(λ1λ2λ3−1)2−µlog(λ1λ2λ3).(S49)

The total energy for the square under biaxial tension is then given by

Π = µL2D

2λ2

1+λ2

2−2+KL2D

2(λ1λ2−1)2−µL2Dlog(λ1λ2)−F L(λ1+λ2−2),(S50)

where we have used the fact that λ3= 1 due to plane strain conditions. The equilibrium solutions are found by

minimizing Π with respect to λ1and λ2,

∂Π

∂λ1

=µL2Dλ1+KL2D(λ1λ2−1)λ2−µL2D

λ1

−F L = 0,(S51)

∂Π

∂λ2

=µL2Dλ2+KL2D(λ1λ2−1)λ1−µL2D

λ2

−F L = 0,(S52)

which are only satisﬁed when

F

µLD =λ2

1−1K

µλ2

1+ 1

λ1

with λ2=λ1,(S53)

or

F

µLD =qK2

µ2+ 6K

µ+ 1 + 2K

µλ2

1+K

µ+ 1

2K

µλ1

with λ2=qK2

µ2+ 6K

µ+1+K

µ+ 1

2K

µλ1

.(S54)

These two equilibrium solutions are shown in Figure S2(b) for various ratios of K/µ. Similarly to the numerical results

reported in Figure S2(a) for the cross-shaped sample, the analysis indicates that the instability still occurs even if the

material is compressible. Moreover, by comparing Equations (S53) and (S54) we can determine the stretch, λcr, at

which the solutions become unstable

λcr =v

u

u

u

t

1

2+v

u

u

tµ2

4K2 2+6K

µ+K2

µ2+ 2s1+6K

µ+K2

µ2!,(S55)

In Figure S2(c) we report the evolution of λcr as a function of K/µ. The results clearly indicate that the critical

stretch for K/µ = 20 is very close to that found for the incompressible case.

7

FIG. S3: Fabrication of the sample. (a) The 3D printed mold and the steel tubes prior to assembly. (b) Assembled mold. (c)

Casting of the Ecoﬂex 0030. (d)-(e) After two hours some inner parts are removed to make room for the second casting step.

(f) Casting of the Elite Double 32. (g) After approximately one day, the sample is fully cured and can be removed from the

mold.

EXPERIMENTS

Fabrication

The samples were fabricated using a molding process (Fig. S3)). To cast a multi-material sample, and to allow

for easy removal after curing, the mold was assembled from several 3D printed parts (Stratasys Connex500) and

steel tubes (Fig. S3(a)). Casting the sample consisted of several steps. We ﬁrst assembled the mold as shown in

Fig. S3(b). Second, a silicone-based rubber Ecoﬂex 0030 (Smooth-On, Inc.) was cast in the center of the mold as

shown in Fig. S3(c). Before casting, we degassed the Ecoﬂex for 1 minute to remove any air bubbles still present after

mixing of the two components. After letting the Ecoﬂex cure for 2 hours at room temperature, part of the mold was

removed to prepare for the second casting step (Figs. S3(d)-(e)). Next, another silicone-based rubber Elite Double 32

(Zhermack) was cast around the four metal tubes already present in the mold (Fig. S3(f)). Note that at this point

the Ecoﬂex was not yet fully cured, but was able to support itself. This improved the bonding between the Ecoﬂex

0030 and the Elite Double 32 in the ﬁnal samples. We tested diﬀerent curing durations, but found that curing the

Ecoﬂex for 2 hours in the ﬁrst step provided the best bonding. Finally, the sample was removed from the mold after

approximately 1 day (Fig. S3(g)).

The cured Ecoﬂex 0030 was tested under uniaxial tension using a single-axis Instron (model 5544A; Instron, Inc.)

with a 1000-N load cell. The material behavior up to a stretch of 300% is reported in Fig. S4. We used a least squares

method to ﬁt an incompressible Neo-Hookean model (Eq. (S1)) to the measured data, and found that the material

response is best capture with an initial shear modulus µ= 0.0216 MPa.

Testing

The samples were stretched equibiaxially using a custom made setup consisting of an aluminium frame to which

the sample was connected by four steel cables (Fig. S5). Four screws were manually tightened to stretch the sample.

Since boundary eﬀects prevented us to clearly observe the instability upon stretching (Movie 1), we acquired x-ray

transmission images (HMXST225, X-Tek) after each time we turned the screws a full turn. Note that a full turn of

the screws results in an applied displacement u/W = 0.18. The results are shown in Fig. 2 and Movie 1.

8

FIG. S4: Nominal stress versus stretch for the cured Ecoﬂex 0030 obtained from a uniaxial tension test. The experiments were

ﬁtted using a Neo-Hookean material model with initial shear modulus µ= 0.0216 MPa.

FIG. S5: (a) Test-setup used to biaxially stretch the sample. (b) Close-up view of the undeformed sample suspended in the

test-setup.

9

ADDITIONAL FIGURES

FIG. S6: (a) Front view of the sample at u/W = 2.90 obtained using a digital camera (D90 SLR, Nikon). (b) Cross-sectional

view of the sample at u/W = 2.90 obtained using a micro-CT X-ray imaging machine (HMXST225, X-Tek).

FIG. S7: 2D ﬁnite element simulations highlighting the eﬀect of imperfections. Each simulation consists of a square with edges

of length Wand circles of radius R1= 0.31Wand R2=ξ0.31W=ξR1cut from its opposite corners, so that the the two

diagonals located at the center of the sample are given by L1=√2W−2Rand L2=√2W−2ξR. An outward displacement

is applied to the straight boundaries of the sample, and we monitored the evolution of the two diagonals with length λ1L1and

λ2L2in the stretched conﬁguration. The simulations are performed assuming both plane stress (a-c) and plane strain (d-f)

conditions.

10

[1] R. Ogden, Non-Linear Elastic Deformations (Dover New York, 1988).

[2] R. S. Rivlin, Philosophical Transactions of the Royal Society of London A: Mathematical, Physical and Engineering Sciences

240, 491 (1948).

[3] R. Hill, Journal of the Mechanics and Physics of Solids 5, 229 (1957).

[4] M. F. Beatty, International Journal of Solids and Structures 3, 23 (1967).

[5] R. S. Rivlin, Collected Papers of R.S. Rivlin: Volume I and II (Springer New York, New York, NY, 1997), chap. Stability

of Pure Homogeneous Deformations of an Elastic Cube under Dead Loading, pp. 398–404.

[6] J. M. Ball and D. G. Schaeﬀer, Mathematical Proceedings of the Cambridge Philosophical Society 94, 315 (1983).

[7] M. C. Boyce and E. M. Arruda, Rubber Chemistry and Technology 73, 504 (2000).