Abstract and Figures

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.
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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 [36]. 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 Poissons ratio [1113], 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 [2123], 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)][2933]. 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.
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0031-9007=16=117(9)=094301(6) 094301-1 © 2016 American Physical Society
λ1¼λ2¼λ3¼1ð1Þ
and
F
μL2¼λ1þ1
λ2
1
;λ2¼λ1;λ3¼λ2
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 λ2¼λ1
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
2FLλ1þ1
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
;λ2¼λ1
1;λ3¼1;ð5Þ
which are stable only if
2Π
λ2¼μL2D1þ3
λ4
1FL2
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 L¼ffiffi
2
pW2R
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=ðμLDÞ2, 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.,
λ1¼λ21). 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
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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).
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¯
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;33
þμL2D1¯
h
2λ2
b;1þð1¯
hÞ2
λ2
b;1ð1¯
hλa;3Þ2
þð1¯
hλa;3Þ2
ð1¯
hÞ23FaLλa;1þ1
λa;1λa;3
2
FbLλb;1þ1¯
h
λb;1ð1¯
hλa;3Þ2;ð8Þ
where D¼DaþDb. Minimizing the energy results again
in two possible solutions:
λa;1¼1with 0¯
h1;ð9Þ
F
μLD ¼λa;1þ1
λ2
a;1
with ¯
h¼λ2
a;11
λ3
a;11;ð10Þ
in which λa;3¼λb;1¼λb;2¼λa;1and λa;2¼λb;3¼λ2
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 .
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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 [4446], 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
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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
33,(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
33,(S2)
and the work done by the external forces, V, is given by
V=F L (λ11) + F L (λ21) + F L (λ31)
=F L (λ1+λ2+λ33) .(S3)
It follows that the potential energy for the system, Π = UV, is given by
Π =µL3
2λ2
1+λ2
2+1
λ2
1λ2
2
3F 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
=µL3λ11
λ3
1λ2
2F L 11
λ2
1λ2= 0,(S5)
Π
∂λ2
=µL3λ21
λ2
1λ3
2F L 11
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=λ2
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
∂λ1λ2
2Π
∂λ2λ1
>0,(S9)
2
in which
2Π
∂λ2
1
=µL31 + 3
λ4
1λ2
2F L 2
λ3
1λ2(S10)
2Π
∂λ1λ2
=2Π
∂λ2λ1
=2µL3
λ3
1λ3
2
F L
λ2
1λ2
2
(S11)
2Π
∂λ2
2
=µL31 + 3
λ2
1λ4
2F 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+λ22) ,(S13)
so that the potential energy for the system becomes
Π = µL2D
2λ2
1+λ2
2+1
λ2
1λ2
2
3F L (λ1+λ22) ,(S14)
where Dis the depth of the sample, and we have used the incompressibility condition (i.e. λ1λ2λ3= 1). Equilibrium
requires that
Π
∂λ1
=µL2Dλ11
λ3
1λ2
2F L = 0,(S15)
Π
∂λ2
=µL2Dλ21
λ2
1λ3
2F L = 0,(S16)
that are only satisfied for
F
µLD =λ11
λ5
1, λ2=λ1, λ3=λ2
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=λ1
2. It follows that the potential energy for the system is given by
Π = µL2D
2λ2
1+1
λ2
1
2F L λ1+1
λ1
2,(S18)
so that the equilibrium configurations are found by setting
dΠ
1
=µL2Dλ11
λ3
1F L 11
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=λ1
1, λ3= 1.(S21)
Furthermore, stability requires
2Π
∂λ2
1
=µL2D1 + 3
λ4
1F 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
¯
a,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,33,(S27)
Ub=µL2Db
2λ2
b,1+λ2
b,2+λ2
b,33,(S28)
4
while the work done by the applied forces equals
Va=FaL(λa,1+λa,22) ,(S29)
Vb=FbL(λb,1+λb,22) .(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,33!,(S31)
Ub=µ1¯
hDL2
2 λ2
b,1+1¯
h2
λ2
b,11¯
a,32+1¯
a,32
1¯
h23!,(S32)
Va=F¯
hL λa,1+1
λa,1λa,3
2,(S33)
Vb=F1¯
hL λb,1+1¯
h
λb,11¯
a,32!,(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+UbVaVb.(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,12
λ3
a,1λ2
a,3!¯
hLF 11
λ2
a,1λa,3!,(S37)
Π
∂λa,3
=1
2D¯
hL2µ 2λa,32
λ2
a,1λ3
a,3!+1
2D(1 ¯
h)L2µ 2(1 ¯
h)2¯
h
λ2
b,1(1 ¯
a,3)32¯
h(1 ¯
a,3)
(1 ¯
h)2!
+¯
hLF
λa,1λ2
a,3
¯
h(1 ¯
h)2LF
λb,1(1 ¯
a,3)2,(S38)
Π
∂λb,1
=1
2D(1 ¯
h)L2µ 2λb,12(1 ¯
h)2
λ3
b,1(1 ¯
a,3)2!(1 ¯
h)LF 11¯
h
λ2
b,1(1 ¯
a,3)!,(S39)
Π
¯
¯
h=1
2DL2µ (1 ¯
h)2
λ2
b,1(1 ¯
a,3)2+(1 ¯
a,3)2
(1 ¯
h)2+λ2
b,13!
+1
2D(1 ¯
h)L2µ 2(1 ¯
h)2λa,3
λ2
b,1(1 ¯
a,3)32(1 ¯
h)
λ2
b,1(1 ¯
a,3)2+2(1 ¯
a,3)2
(1 ¯
h)32λa,3(1 ¯
a,3)
(1 ¯
h)2!
+1
2DL2µ 1
λ2
a,1λ2
a,3
+λ2
a,1+λ2
a,33!+LF 1¯
h
λb,1(1 ¯
a,3)+λb,12
(1 ¯
h)LF (1 ¯
h)λa,3
λb,1(1 ¯
a,3)21
λb,1(1 ¯
a,3)LF 1
λa,1λa,3
+λa,12.(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,11
λ5
a,1!F L¯
h 11
λ3
a,1!= 0,(S41)
Π
∂λb,1
=µL2D(1 ¯
h) λa,1(1 ¯
h)2
λ3
a,1(1 ¯
a,1)2!F L(1 ¯
h) 11¯
h
λ2
a,1(1 ¯
a,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 ¯
h1,(S43)
F
µLD =λa,1+1
λ2
a,1
with ¯
h=λ2
a,11
λ3
a,11.(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=λ2
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 ν1/2 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
33+K
2(λ1λ2λ31)2µlog(λ1λ2λ3).(S49)
The total energy for the square under biaxial tension is then given by
Π = µL2D
2λ2
1+λ2
22+KL2D
2(λ1λ21)2µL2Dlog(λ1λ2)F L(λ1+λ22),(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
=µL21+KL2D(λ1λ21)λ2µL2D
λ1
F L = 0,(S51)
Π
∂λ2
=µL22+KL2D(λ1λ21)λ1µL2D
λ2
F L = 0,(S52)
which are only satisfied when
F
µLD =λ2
11K
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=2W2Rand L2=2W2ξR. 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).
... In this experimental analysis, all caps were fabricated out of Zhermack Elite Double 32 (with green color and initial shear modulus  = 0.35 MPa), except for the outer cap of design C, where we used Zhermack Elite Double 8 (with purple color and initial shear modulus  = 0.06 MPa). Note that these values were obtained by minimizing the error between experiments and simulations for design C and are within the range previously reported in the literature (21,(23)(24)(25). In Fig. 2A, we compare the numerical (blue lines) and experimental (red lines) pressure-volume curves for the three actuators, whereas in Fig. 2B, we display snapshots that are taken during the tests. ...
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Large deformations of soft materials can give rise to the development of various elastic instabilities. The phenomenon is associated with a sudden and dramatic change in structure morphologies. The underlying mechanism is crucial for the formation of complex morphologies in biology. Moreover, the concept of instability-induced pattern transformations is promising for designing novel materials with switchable functions and properties. In this paper, we review the state of the art in elastic instability phenomena in soft materials. We start by considering the classical buckling in beam-based structure lattice designs. Then, we discuss the instability-induced microstructure transformations in soft porous materials, and heterogeneous multiphase and fiber composites. Next, the mechanisms – often involving the post-buckling consideration – leading to the wrinkling and folding, creasing, fringe, and fingering are discussed.
... Figure 8B shows the simulation results on the shrinking of the grippers (n = 3, 4, 5, and 6). In the simulations, the parameters of the passive layer are j 1 = 423 kPa, l 1 = 21.6 kPa, 27 and a 1 = 0.00028 K -1 , 31 and those of the active layer are j 2 = j e , l 2 = l e [Eq. (S4) in Supplementary Data], and a 2 = a e [Eq. ...
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Liquid-vapor phase change materials (PCMs), capable of significant volume change, are emerging as attractive actuating components in forming advanced soft composites for robotic applications. However, the novel and functional design of these PCM composites is significantly limited due to the lacking of the fundamental understanding of the mechanical properties, which further inhibits the broad applications of PCM based materials in the engineering structures requiring large deformation and high loading capacity. In this study we fabricate PCM-elastomer composites exhibiting large deformation and high output stress. Thermomechanical properties of these composites are experimentally and theoretically investigated, demonstrating enhanced deformation and loading capacity due to the induced vapor pressure. By controlling the distribution and content of the PCM inclusions, structures with tunable deformability under a relatively small strain in comparison with traditional soft materials are fabricated. Accompanying with the asymmetrical friction and deformation, complex locomotion and adaptable grabbing function are achieved with excellent performance.
... The equations (21) are equivalent to previously obtained relationships (16). The solutions are stable when the Hessian of the potential energy 2 2 ...
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The purpose of this work is to analyse the stability of the solution of the biaxial stretching test, in the case of Ishihara-Zahorski’s constitutive model of rubber-like material. Two solutions were obtained, the symmetric and asymmetric one. It was shown that the asymmetric solution is possible for significant forces and this solution is always stable. However, the symmetric solution is stable only for relatively small principal stretches. It is worth emphasizing that the stored energy function of the considered model is polyconvex.
... Soft adhesive layers, which could also be found in living systems [1,2], are widely used in engineering applications such as sealants, mechanical insulators, and soft robotics [3][4][5]. When a soft adhesive layer bonded between two rigid substrates is subjected to through thickness tension, various instabilities such as cavitation, fingering, and fringe can occur depending on the geometric constraints by the substrates and the material properties of the layer [6][7][8][9][10][11][12][13][14]. These instabilities can significantly affect the mechanical responses of the adhesive layer [3]. ...
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A soft adhesive layer bonded between two rigid substrates, which are being pulled apart, may exhibit diverse instability phenomena before failure, such as cavitation, fingering, and fringe instability. In this study, by subdividing the soft layers into different numbers of disconnected smaller parts, we achieve desired instability modes and mechanical responses of the layer. The partition process not only retains the monotonicity on the tensile curve but also tunes the modulus and stretchability of the adhesive layer. Meanwhile, cavitation in layers of large aspect ratios is suppressed, and the hysteresis during cyclic loading is reduced. This study provides a guideline for the structural design of soft joints and adhesive layers.
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Application of an external tensile displacement at the end of an elastic film induces its peeled from the free (fixed) end of the cantilever flexible substrate to the fixed (free) end of it. This work proposes a theoretical model for studying this simple elastic system. The analysis results for the film peeled from free end to fixed end (FRFI) demonstrate a number of unknown and unexpected behaviors: (i) There are five basic types of peeling form depending on the flexural rigidity of the substrate, the elastic modulus and thickness of film and adhesive energy; (ii) Some peeling forms have two peeling states (the substrate has two deformed shapes and the pulling force has two values when the film is peeling off) for certain peeling front; (iii) When the film is peeled to a certain place, unstable dynamic peeling will occur. In comparison to FRFI, the results of the film peeled from the fixed to free end (FIFR) appear natural. The effect of flexural rigidity and adhesive energy on the peeling behavior of FIFR is larger.
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Nature frequently employs the buckling phenomenon to facilitate the formation of complicated patterns across length-scales. Current knowledge, however, is limited to a small set of buckling-induced microstructure transformations in soft composites; and the pattern formation phenomenon remains largely unknown for a vast pool of material morphologies. Here, we investigate the unexplored rich domain of soft heterogeneous composites. We experimentally observe the formation of instability-driven domains in stratified composites with a non-dilute stiff phase. We illustrate that the discovered domain patterns are energetically favorable over wrinkling. Moreover, we introduce a closed-form analytical expression allowing us to predict the evolution of the patterns in the post-buckling regime. Finally, we show that various patterns can be pre-designed via altering material compositions. These findings can help advance our understanding of the mechanisms governing pattern formations in soft biological tissues, and potentially enable the platform for mechanical metamaterials.
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Soft, inflatable segments are the active elements responsible for the actuation of soft machines and robots. Although current designs of fluidic actuators achieve motion with large amplitudes, they require large amounts of supplied volume, limiting their speed and compactness. To circumvent these limitations, here we embrace instabilities and show that they can be exploited to amplify the response of the system. By combining experimental and numerical tools we design and construct fluidic actuators in which snap-through instabilities are harnessed to generate large motion, high forces, and fast actuation at constant volume. Our study opens avenues for the design of the next generation of soft actuators and robots in which small amounts of volume are sufficient to achieve significant ranges of motion.
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We create mechanical metamaterials whose response to uniaxial compression can be programmed by lateral confinement, allowing monotonic, non-monotonic and hysteretic behavior. These functionalities arise from a broken rotational symmetry which causes highly nonlinear coupling of deformations along the two primary axes of these metamaterials. We introduce a soft mechanism model which captures the programmable mechanics, and outline a general design strategy for confined mechanical metamaterials. Finally, we show how inhomogeneous confinement can be explored to create multi stability and giant hysteresis.
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Biological surfaces display fascinating topographic patterns such as corrugated blood cells and wrinkled dog skin. These patterns have inspired an emerging technology in materials science and engineering to create self-organized surface patterns by harnessing mechanical instabilities. Compared with patterns generated by conventional lithography, surface instability patterns or so-called ruga patterns are low cost, are easy to fabricate, and can be dynamically controlled by tuning various physical stimuli—offering new opportunities in materials and device engineering across multiple length scales. This article provides a systematic review on the fundamental mechanisms and innovative functions of surface instability patterns by categorizing various modes of instabilities into a quantitatively defined thermodynamic phase diagram, and by highlighting their engineering and biological applications.
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A review of constitutive models for the finite deformation response of rubbery materials is given. Several recent and classic statistical mechanics and continuum mechanics models of incompressible rubber elasticity are discussed and compared to experimental data. A hybrid of the Flory-Erman model for low stretch deformation and the Arruda-Boyce model for large stretch deformation is shown to give an accurate, predictive description of Treloar's classical data over the entire stretch range for all deformation states. The modeling of compressibility is also address.
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Complex three-dimensional (3D) structures in biology (e.g., cytoskeletal webs, neural circuits, and vasculature networks) form naturally to provide essential functions in even the most basic forms of life. Compelling opportunities exist for analogous 3D architectures in human-made devices, but design options are constrained by existing capabilities in materials growth and assembly. We report routes to previously inaccessible classes of 3D constructs in advanced materials, including device-grade silicon. The schemes involve geometric transformation of 2D micro/nanostructures into extended 3D layouts by compressive buckling. Demonstrations include experimental and theoretical studies of more than 40 representative geometries, from single and multiple helices, toroids, and conical spirals to structures that resemble spherical baskets, cuboid cages, starbursts, flowers, scaffolds, fences, and frameworks, each with single- and/or multiple-level configurations. Copyright © 2015, American Association for the Advancement of Science.
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Thin polymer films may undergo a wide variety of elastic instabilities that include global buckling modes, wrinkling and creasing of surfaces, and snapping transitions. Traditionally, these deformations have usually been avoided as they often represent a means of mechanical failure. However, a new trend has emerged in recent years in which buckling mechanics can be harnessed to endow materials with beneficial functions. For many such applications, it is desirable that such deformations happen reversibly and in response to well-defined signals or changes in their environment. While significant progress has been made on understanding and exploiting each type of deformation in its own right, here we focus on recent advances in the control and application of stimuli-responsive mechanical instabilities. © 2014 Wiley Periodicals, Inc. J. Polym. Sci., Part B: Polym. Phys. 2014
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We report a new class of tunable and switchable acoustic metamaterials comprising resonating units dispersed into an elastic matrix. Each resonator consists of a metallic core connected to the elastomeric matrix through elastic beams, whose buckling is intentionally exploited as a novel and effective approach to control the propagation of elastic waves. We first use numerical analysis to show the evolution of the locally resonant band gap, fully accounting for the effect of nonlinear pre-deformation. Then, we experimentally measure the transmission of vibrations as a function of the applied loading in a finite-size sample and find excellent agreement with our numerical predictions. The proposed concept expands the ability of existing acoustic metamaterials by enabling tunability over a wide range of frequencies. Furthermore, we demonstrate that in our system the deformation can be exploited to turn on or off the band gap, opening avenues for the design of adaptive switches.
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Smart Morphable Surfaces enable switchable and tunable aerodynamic drag reduction of bluff bodies. Their topography, resembling the morphology of golf balls, can be custom-generated through a wrinkling instability on a curved surface. Pneumatic actuation of these patterns results in the control of the drag coefficient of spherical samples by up to a factor of two, over a range of flow conditions.