Amplifying the response of soft actuators by
harnessing snap-through instabilities
Johannes T. B. Overvelde
, Tamara Kloek
, Jonas J. A. D’haen
, and Katia Bertoldi
John A. Paulson School of Engineering and Applied Sciences, Harvard University, Cambridge, MA 02138; and
Kavli Institute, Harvard University,
Cambridge, MA 02138
Edited by John W. Hutchinson, Harvard University, Cambridge, MA, and approved July 21, 2015 (received for review March 11, 2015)
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.
The ability of elastomeric materials to undergo large de-
formation has recently enabled the design of actuators that
are inexpensive, easy to fabricate, and only require a single source of
pressure for their actuation, andstillachievecomplexmotion(1–5).
These unique characteristics have allowed for a variety of innovative
applications in areas as diverse as medical devices (6, 7), search and
rescue systems (8), and adaptive robots (9–11). However, existing
fluidic soft actuators typically show a continuous, quasi-monotonic
relation between input and output, so they rely on large amounts of
fluid to generate large deformations or exert high forces.
By contrast, it is well known that a variety of elastic instabilities
can be triggered in elastomeric films, resulting in sudden and
significant geometric changes (12, 13). Such instabilities have
traditionally been avoided as they often represent mechanical
failure. However, a new trend is emerging in which instabilities are
harnessed to enable new functionalities. For example, it has been
reported that buckling can be instrumental in the design of
stretchable soft electronics (14, 15), and tunable metamaterials
(16–18). Moreover, snap-through transitions have been shown to
result in instantaneous giant voltage-triggered deformation (19, 20).
Here, we introduce a class of soft actuators comprised of inter-
connected fluidic segments, and show that snap-through instabilities
in these systems can be harnessed to instantaneously trigger large
changes in internal pressure, extension, shape, and exerted force. By
combining experiments and numerical tools, we developed an ap-
proach that enables the design of customizable fluidic actuators for
which a small increment in supplied volume (input) is sufficient to
trigger large deformations or high forces (output).
Our work is inspired by the well-known two-balloon experiment,
in which two identical balloons, inflated to different diameters, are
connected to freely exchange air. Instead of the balloons becoming
equal in size, for most cases the smaller balloon becomes even
smaller and the balloon with the larger diameter further increases
in volume (Movie S1). This unexpected behavior originates from
the balloons’nonlinear relation between pressure and volume,
characterized by a pronounced pressure peak (21, 22). Interest-
ingly, for certain combinations of interconnected balloons, such
nonlinear response can result in snap-through instabilities at
constant volume, which lead to significant and sudden changes of
the membranes’diameters (Figs. S1 and S2). It is straightforward
to show analytically that these instabilities can be triggered only
if the pressure–volume relation of at least one of the membranes
is characterized by (i) a pronounced initial peak in pressure,
(ii) subsequent softening, and (iii) a final steep increase in pressure
(Analytical Exploration: Response of Interconnected Spherical
Membranes Upon Inflation).
Highly Nonlinear Fluidic Segments
To experimentally realize inflatable segments characterized by such
anonlinearpressure–volume relation, we initially fabricated fluidic
segments that consist of a soft latex tube of initial length Ltube, inner
radius R=6.35 mm, and thickness H=0.79 mm. We measured the
pressure–volume relation experimentally for three segments with
Ltube =22 −30 mm, and found that their response is not affected by
their length (Fig. S3). Moreover, the response does not show a final
steep increase in pressure. This is because latex has an almost
linear behavior, even at large strains.
Next, to construct fluidic segments with a final steep increase
in pressure and a response that can be easily tuned and con-
trolled, we enclosed the latex tube by longer and stiffer braids of
length Lbraid (Fig. 1A). It is important to note that the effect of
the stiff braids is twofold. First, as Lbraid >Ltube, the braids are in
a buckled state when connected to the latex tube (Fig. 1B), and
therefore apply an axial force, F, to the membrane. Second, at a
certain point during inflation when the membrane and the braids
come into contact, the overall response of the segments stiffens.
We derived a simple analytical model to predict the effect of
Lbraid and Ltube on the nonlinear response of these braided fluidic
segments (Simple Analytical Model to Predict the Response of the
Fluidic Segments). It is interesting to note that our analysis in-
dicates that for a latex tube of given length, shorter braids lower
the peak pressure due to larger axial forces (Fig. S4 Cand E).
Moreover, it also shows that Lbraid strongly affects the volume at
which stiffening occurs. In fact, the shorter the braids, the earlier
contact between the braids and the membrane occurs, reducing
the amount of supplied volume required to have a steep increase
Although instabilities have traditionally been avoided as they
often represent mechanical failure, here we embrace them to
amplify the response of fluidic soft actuators. Besides pre-
senting a robust strategy to trigger snap-through instabilities
at constant volume in soft fluidic actuators, we also show that
the energy released at the onset of the instabilities can be
harnessed to trigger instantaneous and significant changes in
internal pressure, extension, shape, and exerted force. There-
fore, in stark contrast to previously studied soft fluidic actuators,
we demonstrate that by harnessing snap-through instabilities it
is possible to design and construct systems with highly control-
lable nonlinear behavior, in which small amounts of fluid suffice
to generate large outputs.
Author contributions: J.T.B.O. and K.B. designed research; J.T.B.O., T.K., and J.J.A.D. performed
research; J.T.B.O., T.K., and K.B. analyzed data; and J.T.B.O. and K.B. wrote the paper.
The authors declare no conflict of interest.
This article is a PNAS Direct Submission.
To whom correspondence should be addressed. Email: firstname.lastname@example.org.
This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.
www.pnas.org/cgi/doi/10.1073/pnas.1504947112 PNAS Early Edition
in pressure. Conversely, if Lbraid is fixed, and the length of the
membrane is varied, both the pressure peak and the volume at
which stiffening occurs remain unaltered (Fig. S4F). However, in this
case we find that shorter tubes lower the pressure of the softening
region. Finally, the analytical model also indicates that the length of
the fluidic segments, l=λzLtube,initiallyincreasesuponinflation(Fig.
S4 Eand F). However, when the tube and braids come into contact,
further elongation is restrained by the braids and the segments
shorten as a function of the supplied volume.
Having demonstrated analytically that fluidic segments with
the desired nonlinear response can be constructed by enclosing
a latex tube by longer and stiffer braids, and that their response
can be controlled by changing Lbraid and Ltube, we now proceed to
fabricate such actuators. The stiffer braids are made from poly-
ethylene-lined ethyl vinyl acetate tubing, with an inner radius of
7.94 mm and a thickness of 1.59 mm. Eight braids are formed by
partly cutting this outer tube along its length guided by a 3D
printed socket. Finally, Nylon Luer lock couplings (one socket and
one plug) are glued to both ends of the fluidic segments to enable
easy connection (Fig. 1A). We then measure their response ex-
perimentally by inflating them with water at a rate of 60 mL/min,
ensuring quasi-static conditions (Fig. 1Band Movie S2).
We fabricated 36 segments with Lbraid =40 −50 mm and
Ltube =20 −30 mm. As shown in Fig. 1C,allfluidicsegments
are characterized by the desired nonlinear pressure—volume relation
and follow the trends predicted by the analytical model (Fig. S4 E
and F). In particular, we find that for the 36 tested segments the
initial peak in pressure ranges between 65 and 85 kPa (Fig. 1C). We
also monitored the length of the segments during inflation (Fig. 1D).
As predicted by the analytical model, we find that initially the seg-
ments elongate, but then shorten when the tube and braids come into
contact. It is important to note that no instabilities are triggered upon
inflation of the individual segments, because the supplied volume is
controlled, not the pressure.
Combined Soft Actuator
Next, we created a new, combined soft actuator by interconnecting
the two segments whose individual response is shown in Fig. 2A.
Upon inflation of this combined actuator, very rich behavior
emerges (Fig. 2Cand Movie S3). In fact, the pressure–volume
response of the combined actuator is not only characterized by two
peaks, but the second peak is also accompanied by a significant and
instantaneous elongation. This suggests that an instability at con-
stant volume has been triggered.
Numerical Algorithm. To better understand the behavior of such
combined actuators, we developed a numerical algorithm that
accurately predicts the response of systems containing nsegments,
based solely on the experimental pressure–volume curves of the
individual segments. By using the 36 segments from experiments
as building blocks, we can construct 36!=½ð36 −nÞ!n!$combined
actuators comprising nsegments (i.e., 630 different combined
actuators for n=2; 7,140 for n=3; and 58,905 for n=4), where we
assume that the order in which we arrange the segments does not
matter. It is therefore crucial to implement a robust algorithm to
efficiently scan the range of responses that can be achieved.
We start by noting that, upon inflation, the state of the ith
segment is defined by its pressure piand volume vi, and its stored
elastic energy can be calculated as
in which we neglect dynamic effects. Moreover, Videnotes the
volume of the ith segment in the unpressurized state. When the
Fig. 1. (A)Outerandstifferbraidsareaddedtothelatextubetocreatefluidic
segments with highly nonlinear response. (B)Snapshotsofasegmentcharac-
terized by ðLbraid,Ltube Þ=ð46, 20Þduring inflation at v=0, 10, 20 mL. Evolu-
tion of (C) pressure (p)and(D) length (l) as a function of the supplied volume
(v) for 36 fluidic segments characterized by Lbraid =40 −50 mm and Ltube =
20 −30 mm. (Scale bars: 10 mm.)
Fig. 2. (A)Evolutionofpressure(p)andlength(l)asafunctionofthesupplied
volume (v)fortwofluidicsegmentscharacterizedbyðLbraid ,Ltube Þ=ð46, 20Þand
ð46, 22Þmm. Snapshots of the fluidic segments at v=0, 10, 20 and v=
0, 12, 24 mL are shown as Insets,respectively.(B)Thetwofluidicsegmentsare
connected to form a new, combined soft actuator. (C)Evolutionofpressure(p)and
Snapshots of the combined actuator at v=0, 9, 18,27, 36, 45 mLare shown as Insets.
www.pnas.org/cgi/doi/10.1073/pnas.1504947112 Overvelde et al.
total volume of the system, v=Pn
i=1vi, is controlled (as in all our
experiments), the response of the system is characterized by n−1
variables v1,...,vn−1and the constraint
To determine the equilibrium configurations, we first define the
elastic energy, E, stored in the system, which is given by the sum
of the elastic energy of the individual segments
and use Eq. 2to express the energy in terms of n−1 variables
Next, we implement a numerical algorithm that finds the equi-
librium path followed by the actuator upon inflation (i.e., increasing
v). Starting from the initial configuration (i.e., vi=Vi), we in-
crementally increase the total volume of the system (v)andlocally
minimize the elastic energy (~
E). Because Eq. 4already takes into
account the volume constraint (Eq. 2), we use an unconstrained
optimization algorithm such as the Nelder–Mead simplex algorithm
implemented in Matlab (23). Note that this algorithm looks only
locally for an energy minimum, similar to what happens in the ex-
periments, and therefore it does not identify additional minima at
the same volume that may appear during inflation.
Using the aforementioned algorithm, we find that for many ac-
tuators the energy can suddenly decrease upon inflation, indicating
that a snap-through instability at constant volume has been trig-
gered. To fully unravel the response of the actuators, we also detect
all equilibrium configurations and evaluate their stability. The
equilibrium states for the system can be found by imposing
=0, ∀i∈f1, ...,n−1
Substitution of Eq. 4into Eq. 5, yields
which, when substituting Eq. 2, can be rewritten as
As expected, Eq. 7ensures that the pressure is the same in all n
segments connected in series.
Operationally, to determine all of the equilibrium configura-
tions of a combined soft actuator comprising nfluidic segments,
we first define 1,000 equispaced pressure points between 0 and
100 kPa. Then, for each of the nsegments we find all volumes
that result in those values of pressure (Fig. S5A). Finally, for
each value of pressure, we determine the equilibrium states by
making all possible combinations of those volumes (Fig. S5B).
Note that by using Eq. 2we can also determine the total volume
in the system at each equilibrium state, and then plot the pres-
sure–volume response for the combined actuator (Fig. S5C).
Finally, we check the stability of each equilibrium configura-
tion. Because an equilibrium state is stable when it corresponds
to a minimum of the elastic energy ~
Edefined in Eq. 4, at any
stable equilibrium solution the Hessian matrix
is positive definite. Note that the second-order partial derivatives
in Eq. 8can be evaluated as
vk!, if i=j
vk!, if i≠j,
Fig. 3. (A)Experimentallymeasuredpressure–volume relations for all 36 fab-
ricated fluidic segments. (B)Experimentallymeasuredlength–volume relations
for all 36 fabricated segments. (C)Numericallydetermined elastic energy, E,fora
combined actuator comprising the two segments whose individual behavior is
highlighted in Aand B.Theenergyisshownforincreasingvaluesofthesupplied
blue and red circular markers, respectively. (D)Equilibriumconfigurationsforthe
combined actuators. At v=19 mL an unstable (1, 1) transition is found, resulting
in a significant internal volume flow. A second instability of type (1, 2) is then
triggered at v=22 mL. (E)Numericallydeterminedpressure–volume and length–
volume relations for the combined soft actuator.
Overvelde et al. PNAS Early Edition
in which pi
diagonal terms of the Hessian matrix are identical and using Sylvest-
Numerical Results. To demonstrate the numerical algorithm, we
focus on two segments where the experimentally measured
pressure–volume and length–volume responses are highlighted
in Fig. 3 Aand B. In Fig. 3Cwe report the evolution of the total
elastic energy of the system, E, as a function of the volume of the
first segment, v1, for increasing values of the total supplied vol-
ume, v, and in Fig. 3Dwe show all equilibrium configurations in
the v1–v2plane. We find that initially (0 <v<5 mL) the volume
of both segments increases gradually. However, for 5 <v<19 mL,
v1remains almost constant and all additional volume that is
added to the system flows into the second segment. Moreover, at
v=6 mL a second local minimum for Eemerges, so that for
6<v<19 mL the system is characterized by two stable equilib-
rium configurations. Although for v>13 mL this second mini-
mum has the lowest energy, the system remains in the original
energy valley until v=19 mL. At this point the local minimum of
Ein which the system is residing disappears, so that its equilib-
rium configuration becomes unstable, forcing the actuator to snap
to the second equilibrium characterized by a lower value of E.
Interestingly, this instability triggers a significant internal volume
flow from the second to the first segment (Fig. 3D) and a sudden
increase in length (Fig. 3E). Further inflating the system to v=22
mL triggers a second instability, at which some volume suddenly
flows back from the first to the second segment. After this second
instability, increasing the system’s volume further inflates both
All transitions that take place upon inflation (i.e., at v=5, 19, and
22 mL) are highlighted by a peak in the pressure–volume curve (Fig.
3E), and correspond to instances at which one or more of the in-
dividual segments cross their own peak in pressure. These state
transitions can either be stable or unstable (Fig. 3 C–E). A stable
transition always leads to an increase of the elastic energy stored in
the system, and an instability results in a new equilibrium configu-
ration with lower energy. Each state transition can therefore be
characterized by the elastic energy release, which we define as a
normalized scalar Δ^
E=ðEpost −EpreÞ=Epre .Hereandinthefol-
lowing, the subscripts pre and post indicate the values of the quantity
immediately before and after the state transition. Moreover, to
better understand the effects of each transition on the system, we
define the associated normalized changes in internal volume
distribution, length and pressure as Δ^
l=ðlpost −lpreÞ=ðlpre Þand Δ^
In Fig. 4 we report Δ^
l, and Δ^
pversus the normalized
change in energy, Δ^
E, for all transitions that occur in the 630
combined soft actuators comprising n=2 segments. Note that
there are more than 630 data points, because all actuators show
two or more state transitions. We find that −0.1 ≤Δ^
E≤4 · 10−5,
indicating that some of the transitions are stable (i.e., Δ^
and others are unstable (Δ^
E<0). We furthermore observe that
the energy increase for stable transitions is very small, and is
therefore sensitive to the increment size used in the numerical
algorithm. By contrast, the elastic energy released during un-
stable transitions can be as high as 10% of the stored energy.
We also characterize each state transition according to the
changes induced in the individual segments, and use ðα,βÞto
identify the number of segments to the right of their pressure
peak before (α) and after (β) the state transition. For combined
soft actuators comprising n=2 segments, the numerical results
show three possible types of transitions: ð0, 1Þ, in which both
segments are initially on the left of their peak in pressure and
then one of them crosses its pressure peak during the state
transition (blue markers in Fig. 4); ð1, 2Þ, in which the second
segment also crosses its peak in pressure (green markers in Fig.
4); ð1, 1Þ, in which both segments cross their pressure peak, but
one while inflating and the other while deflating (red markers in
Fig. 4). We find that transitions of type ð0, 1Þoccur in all com-
bined actuators and are always stable. Therefore, the associated
changes in elastic energy, length, pressure, and the internal vol-
ume distribution are approximately zero. By contrast, transitions
of type ð1, 1Þare always unstable and result in both high elastic
energy release (up to 10%) and high internal volume flow (up to
80%). Unlike ð1, 1Þ, transitions of type ð1, 2Þcan be either stable
or unstable. The unstable transitions result in moderate energy
release (up to 2.5%), but can lead to significant and instantaneous
changes in length (up to 14%). Therefore, our analysis clearly in-
dicates not only that snap-through instabilities at constant volume
can be triggered in soft fluidic actuators, but also that the associated
released energy can be harnessed to trigger sudden changes in
length, drops in pressure, and internal volume flows.
Experimental Results. To validate the numerical predictions, we
measured experimentally the response of several combined ac-
tuators. In Fig. 5Awe show the results for the system whose
predicted transitions are indicated by the diamond gray markers
in Fig. 4. We compare the numerically predicted and experimen-
tally observed mechanical response, finding an excellent agree-
ment. In particular, for this combined actuator we find that the
pressure–volume curve is characterized by two peaks, indicating
Fig. 4. A–Cshow Δ^
pversus the normalized change in energy Δ^
Efor all state transitions that occur in the 630 combined soft actuators comprising n=2
fluidic segments. Blue, red, and green markers correspond to ð0, 1Þ,ð1, 1Þ,andð1, 2Þtransitions, respectively; (A)Δ^
www.pnas.org/cgi/doi/10.1073/pnas.1504947112 Overvelde et al.
that two transitions take place upon inflation. Although the ð0, 1Þ
transition is stable, the ð1, 2Þtransition is unstable and results in an
instantaneous and significant increase in length of 11% and a high
pressure drop of 23% (Fig. 5Aand Movie S4). This unstable
transition is also accompanied by a moderate internal volume re-
distribution of 22%, resulting in the sudden inflation of the top
actuator (see snapshots in Fig. 5Aand numerical result in
In Fig. 5Bwe present the results for the combined actuator
whose response is indicated by the square gray markers in Fig. 4.
Our analysis indicates that one stable ð0, 1Þtransition and two
unstable transitions are triggered during its inflation. The first
snap-through instability is a ð1, 1Þtransition and is accompanied
by a significant and sudden volume redistribution (see snapshots
in Fig. 5Band numerical result in Fig. S6B) and a large increase
in length (Movie S5), and the second instability is a ð1, 2Þtran-
sition and results in smaller values for Δ
v. Again, we
observe an excellent agreement between experimental and nu-
merical results, indicating that our modeling approach is accu-
rate and can be used to effectively design soft actuators that
harness instabilities to amplify their response.
Although the results reported in Fig. 5 Aand Bare for actuators
free to expand, these systems can also be used to exert large forces
while supplying only small volumes. To this end, in Fig. 5 Cand Dwe
show the force measured during inflation when the elongation of the
actuators is completely constrained. We find that also in this case an
instability is triggered, resulting in a sudden, large increase in the
exerted force. Note that the volume at which the instability occurs is
slightly different from that found in the case of free inflation. This
discrepancy arises from the fact that the pressure–volume relation of
each segment is affected by the conditions at its boundaries.
The proposed approach can be easily extended to study more
complex combined actuators comprising a larger number of
segments. By increasing n, new types of state transitions can be
triggered. For example, transitions of type ð2,1Þare also observed
for n=3(Fig. S7 A–C), in which two segments deflate into a
single one, causing all three segments to cross their peak in
pressure. In Fig. 6, we focus on an actuator that undergoes an
unstable ð2,1Þtransition at v=29 mL. We first inflate the actu-
ator to v=28 mL, and then decouple it from the syringe pump
and connect it to a small reservoir containing only 1 mL of water.
Remarkably, by adding only 1 mL of water to the system, we are
able to trigger a significant internal volume flow of ∼20 mL that
results in the deflation of two segments into one segment (Fig. 6
and Movie S6).These results further highlight that snap-through
instability can be harnessed to amplify the effect of small inputs.
In summary, by combining experimental and numerical tools we
have shown that snap-through instabilities at constant volume can
be triggered when multiple fluidic segments with a highly nonlinear
pressure–volume relation are interconnected, and that such un-
stable transitions can be exploited to amplify the response of the
system. In stark contrast to most of the soft fluidic actuators pre-
viously studied, we have demonstrated that by harnessing snap-
through instabilities it is possible to design and construct systems in
which small amounts of fluid suffice to trigger instantaneous and
significant changes in pressure, length, shape, and exerted force.
To simplify the analysis, in this study we have used water to
actuate the segments (due to its incompressibility). However, it is
important to note that the actuation speed of the proposed actu-
ators can be greatly increased by supplying air. In fact, we find that
water introduces significant inertia during inflation, limiting the
actuation speed. It typically takes more than 1 s for the changes in
length, pressure, and internal volume induced by the instability to
fully take place (Movie S7). However, by simply using air to actuate
the system and by adding a small reservoir to increase the energy
stored in the system, the actuation time can be significantly re-
duced (from Δt=1.4 to 0.1 s for the actuator considered in Movie
S7), highlighting the potential of these systems for applications
where speed is important. Although this actuation time is similar to
that of recently reported high-speed soft actuators (3), only a small
volume of supplied fluid is required to actuate the system because
we exploit snap-through instabilities at constant volume. As a
Fig. 5. (Aand B) Experimental (solid lines) and numerical (dashed lines)
pressure–volume curves for two soft actuators comprising n=2 fluidic seg-
ments. (A) Results for a combined actuator with ðLbraid,LtubeÞ=ð48, 30Þand
ð50, 20Þmm. The transitions for this actuator are highlighted by diamond
markers in Fig. 4. Snapshots of the combined actuators 0.5 mL before and after
each state transition (at v=4, 26 mL) are also shown. (B)Resultsforacombined
actuator with ðLbraid ,LtubeÞ=ð44 , 30Þand ð48, 26Þmm. The transitions for this
actuator are highlighted by square markers in Fig. 4. Snapshots of the combined
actuators 0.5 mL before and after each state transition (at v=5, 16, 24 mL )
are also shown. Experimentally measured exerted force as a function of the
supplied volume for a combined actuator with (C)ðLbraid,Ltube Þ=ð48, 30Þand
ð50, 20Þmm and (D)ðLbraid,Ltube Þ=ð44, 30Þand ð48, 26Þmm with con-
Fig. 6. Snapshots of a combined actuator with ðLbraid ,Ltube Þ=ð40,28Þ,ð44, 30Þ,
and ð50, 24Þmm. The numerical analysis predicts a ð2, 1Þstate transition at
v=29 mL (see gray triangle in Fig. S7 A–C). The combined actuator is inflated
to v=28 mL, and then decoupled from the syringe pump and connected to a
small reservoir containing only 1 mL of water. An additional volume of 1 mL
supplied to the system is enough to trigger a significant internal volume flow
of ∼20 mL that results in the deflation of two segments into one segment
Overvelde et al. PNAS Early Edition
result, small compressors are sufficient to inflate these actuators,
making them highly suitable for untethered applications.
Our results indicate that by combining fluidic segments with
designed nonlinear responses and by embracing their nonlinearities,
we can construct actuators capable of large motion, high forces, and
fast actuation at constant volume. Although here we have focused
specifically on controlling the nonlinear response of fluidic actua-
tors, we believe that our analysis can also be used to enhance the
response of other types of actuators (e.g., thermal, electrical and
mechanical) by rationally introducing strong nonlinearities. Our
approach therefore enables the design of a class of nonlinear sys-
tems that is waiting to be explored.
Materials and Methods
All individual soft fluidic segments and combined actuators investigated in
this study are tested using a syringe pump (Standard Infuse/Withdraw PHD
Ultra; Harvard Apparatus) equipped with two 50-mL syringes that have an
accuracy of ±0.1% (1000 series, Hamilton Company). The segments and the
combined actuators are inflated at a rate of 60 and 20 mL/min, respectively,
ensuring quasi-static conditions. Moreover, during inflation the pressure
is measured using a silicon pressure sensor (MPX5100; Freescale Semiconductor)
with a range of 0–100 kPa and an accuracy of ±2.5%, which is connected to a
data acquisition system (NI USB-6009, National Instruments). The elongation of
the actuators is monitored by putting two markers on both ends of each actu-
ator, and recording their position every two seconds with a high-resolution
camera (D90 SLR, Nikon). The length of the actuator is then calculated from the
pictures using a digital image processing code in Matlab. Each experiment is
repeated 5 times, and the final response of the actuator as shown in the paper is
determined by averaging the results of the last four tests. Finally, we measured
the force exerted by the actuators during inflation when their elongation is
completely constrained. In this case we use a uniaxial materials testing machine
(model 5544A; Instron, Inc.) with a 100-N load cell to measure the reaction force
ACKNOWLEDGMENTS. This work was supported by the Materials Research
Science and Engineering Center under National Science Foundation Award
DMR-1420570. K.B. also acknowledges support from the National Science
Foundation (CMMI-1149456-CAREER) and the Wyss institute through the
Seed Grant Program.
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www.pnas.org/cgi/doi/10.1073/pnas.1504947112 Overvelde et al.
Overvelde et al. 10.1073/pnas.1504947112
Analytical Exploration: Response of Interconnected
Spherical Membranes Upon Inflation
To identify the key components in the design of an inflatable
system that undergoes a snap-through instability upon inflation,
we analytically explore the equilibrium states of two spherical
membranes, connected in series. We start by describing the equa-
tions governing the response of a single hyperelastic spherical
membrane uponinflation (25, 26), andthen discuss the be havior of
two interconnected spherical membranes (27–30).
Response of a Hyperelastic Spherical Membrane upon Inflation. We
consider a spherical membrane with initial radius R, and initial
thickness H, and assume that the membrane is thin (i.e., H!R),
so that the contribution from bending and shear forces can be
neglected. When the membrane is subjected to a uniform pres-
sure p, a biaxial state of stress is achieved and the principal
stretches are given by
rand hdenoting respectively the radius and thickness of the
membrane in the deformed configuration. Moreover, equilib-
and the assumption of a plane stress condition yields
where σθ,σϕ, and σrare the principal Cauchy stresses.
We consider the membrane to be made of a hyperelastic, in-
compressible material, the response of which is captured by the
strain energy W. Because of incompressibility we have
and therefore the stress in the material is given by
Here, we model the material response using the incompressible
Gent model (31), for which the strain energy is defined by
where μis the initial shear modulus and Jmis a constant related
to the strain saturation of the material (as the stresses become
infinite when Jm−λ2
ϕ+3 approaches zero). Note
that the nonlinear pressure–volume response of an inflated
spherical membrane depends greatly on the constitutive mate-
rial model. Although in this study we have focused on a Gent
model to achieve a final steep increase in pressure, when using
a Varga, neo-Hookean, or three-term Ogden model no strain
stiffening is observed upon inflation (32, 33). Although the
material parameters entering in the Ogden model can be tuned
to account for strain stiffening or to even suppress the pressure
peak encounter during inflation, here we preferred to use the
Gent model because only μand Jmneed to be changed to
control the nonlinear response (26).
Substitution of Eqs. S1,S2, and S6 into Eq. S5 yields
which can be rewritten as
where we use Eq. S4 to express has a function of r(i.e.,
h=HR2=r2), and in which we set λ=λθfor convenience. Eq. S8
clearly shows that the initial radius R, initial thickness H, and
shear modulus μonly scale the value of the pressure and that the
only parameter that changes the shape of the p−λcurve is the
stretching limit Jm. Note that the volume inside the membrane
can be easily determined as
As an example, in Fig. S1 we show the p−vcurves for three
spherical membranes characterized by the same radius Rand
shear modulus μand (a)Haand Jm,a,(b)Hb=1.1Haand Jm,b=
It is important to note that in our analysis we assume that the
deformation is homogeneous throughout the membrane, al-
though it has been shown that this assumption can be violated by
spherical membranes as they can exhibit asymmetric bifurcation
modes (32, 34). However, these asymmetric modes do not affect
the main features of the nonlinear pressure–volume relation
typical for such membranes, and as such our analysis provides
enough detail to qualitatively study the response of systems of
Response of Two Interconnected Spherical Membranes upon Inflation.
Having determined the pressure–volume curve for a single spher-
ical membrane, we now determine the equilibrium states for a
system of two interconnected membranes. For both mem-
branes, we use the pressure–volume relation defined by Eqs. S8
and S9. Specialization of Eqs. 9and 14 to a system comprising
two spherical membranes, yields
Moreover, Eq. 17 reduces to
so that an equilibrium configuration for a system comprising n=2
spherical membranes is stable when
Overvelde et al. www.pnas.org/cgi/content/short/1504947112 1 of 10
where λiindicates the stretch in the ith membrane. Moreover, Hi
and Ridenote the initial thickness and radius of the ith mem-
brane and μiand Jlm,iits shear modulus and strain saturation
In Fig. S2 we show the response of the systems obtained by
interconnecting membranes aand b(Fig. S2A), band c(Fig.
S2B), and aand c(Fig. S2C). Note that the pressure–volume
curves for the individual segments are shown in Fig. S1. When
connecting membranes aand b(Fig. S2A), upon inflation
membrane ainflates first. Then, when this membrane starts to
stiffen, membrane balso increases in volume. However, for this
combined system all equilibrium points are stable. More in-
teresting behavior is observed when combining membrane band
c(Fig. S2B). Initially, this system behaves similarly to the system
shown in Fig. S2, with membrane cinflating first. However, at
v=Va=42 a snap-through instability is triggered, resulting in a
sudden decrease in pressure and increase in the diameter of
membrane b. Therefore, at the instability a significant amount
of volume flows from membrane cto membrane b.Wealso
note that this system is characterized by equilibrium configu-
rations that are disconnected from the main curve, so that they
will never be reached upon inflation. Finally, even more com-
plex behavior can be achieved by connecting membranes aand
c(Fig. S2C). Here, a first instability causes all volume to flow
from membrane ato membrane c,andasecondinstability
forces some volume to flow back from membrane cto mem-
Simple Analytical Model to Predict the Response of the
The analytical predictions reported above reveal that snap-
through instabilities at constant volume can be triggered in a
system comprising interconnected spherical membranes only if
the pressure–volume relation of at least one of the membranes
is characterized by (i) a pronounced initial peak in pressure,
(ii) subsequent softening, and (iii) a final steep increase in
pressure. As for the case of the Gent model used in the pre-
vious analytical exploration, such response can be achieved by
using membranes made of elastomeric materials that exhibit
stiffening at large strains. However, it is important to note that
commercially available latex tubes are characterized by an al-
most linear behavior, so that their response can be nicely
captured using a neo-Hookean model (35), for which the strain
energy is defined as
where μ=0.49 MPa is the initial shear modulus of the latex
material and λiare the principal stretches. Note that the neo-
Hookean strain energy in Eq. S14 can be obtained from the Gent
energy density in the limit as Jm→∞.
In the following, we first derive the pressure–volume rela-
tionship for a latex tube, and then we develop a simple analytical
model to capture the response of the highly nonlinear braided
fluidic segments used in this study.
Response of a Latex Tube Upon Inflation. We start by investigating
analytically the response of a cylindrical latex membrane with
initial radius R, initial length Ltube, and initial thickness H, and
assume that the membrane is thin (i.e., H#R), so that the
contribution from bending and shear forces can be neglected. In
particular, we determine the response when the membrane is
inflated by an internal pressure p. For simplicity, we neglect end
effects and local instabilities resulting from bulging (36–38), and
assume that the cylindrical membrane deforms uniformly. The
principal stretches are then given by
r,l, and hdenoting the membrane’s radius, length, and thickness
in the deformed configuration, respectively. Moreover, as in ex-
periments, we assume that the tube has closed ends, so that it is
subjected to an axial force pπr2. For such a membrane, equilib-
and the assumption of plane stress conditions yields
where σidenote the principal Cauchy stresses.
We consider the membrane to be made of an incompressible
neo-Hookean material (Eq. S14) and because of incompressibility
The stress in the material can then be expressed as
pis a Lagrange multiplier. Substitution of Eqs. S15–S17
and S19 into Eqs. S21 and S22 yields
Combining Eqs. S19,S23, and S24 results in
which can be solved to obtain
where we used λr≥1 and λz≥1. Finally, substitution of Eq. S26
into Eq. S23 provides the evolution of the pressure as a function
of the axial stretch λz
Overvelde et al. www.pnas.org/cgi/content/short/1504947112 2 of 10
Note that the pressure–volume relationship for an inflated latex
tube can be easily obtained from Eq. S27, because the volume
enclosed by the membrane, v, is given by
In Fig. S3 we report the pressure–volume relation measured ex-
perimentally, and find excellent agreement with the analytical
model (Eq. S27). Furthermore, we immediately see that no stiff-
ening occurs, even for very high volumes. As a result, we do not
expect to trigger any instability at constant volume in systems com-
prising interconnected latex tubes. Moreover, we also note that it is
not possible to tune the pressure response of these inflatable tubes
by changing their length, as the pressure does not depend on Ltube.
Response of a Braided Fluidic Segment. To construct fluidic segments
with a final steep increase in pressure and whose response can be
easily tuned and controlled, we enclosed the latex tube of initial length
Ltube by longer and stiffer braids of initial length Lbraid (Fig. 1A). The
effect of the stiff braids is twofold: (i)asLbraid >Ltube ,thebraidshave
to be buckled first before attaching them the latex tube, resulting in
an axial force, F,appliedtothemembrane;(ii)atacriticalpoint
during inflation the braids come into contact with the latex tube,
stiffening the response of the segment upon further inflation.
Effect of the axial force. To determine the effect of the braids on the
initial response of the segments, we first estimate the axial force F
introduced by nbraids that enclose the latex tube. For the sake
of simplicity, we model each individual braid as two rigid seg-
ments of length Lbraid=2, connected by a torsional spring with
stiffness K(Fig. S4A). As the braids are longer than the latex
tube, before attaching them together, they are shortened by
u=Lbraid −l=Lbraid sin θ(Fig. S4B), causing them to buckle.
Balance of the work done by the axial force Fand the elastic
FLbraid sin θ=2nKθ,[S29]
from which we obtain
Lbraid sin θ.[S30]
For the fluidic segment considered in this study θ<40° we ap-
proximate sin θ≈θ, so that Eq. S30 reduces to
Note that this approximation yields a constant axial force F, and
introduces an error that is within 9%.
Finally, we experimentally estimate the torsional stiffness K
by performing uniaxial compression tests on braids of different
length and comparing the experimentally measured critical forces
to Eq. S31. From the results reported in Fig. S4Bwe obtain that
for the braids used in this study
Next, we investigate the effect of the axial force Fon the latex
tube. To account for the axial force, Eq. S17 modifies as
so that Eq. S24 becomes
and Eqs. S16 and S23 remain unaltered. Note that Eq. S34 can
also be used to estimate the axial stretch introduced into the tube
by the braids in the unpressurized state (i.e., p=0)
By combining Eqs. S4,S23, and S34 we obtain
from which the stretch in radial direction can be determined as
where we assumed λr≥0 and λz≥λz,0. By substituting Eq. S37
into Eq. S34 the pressure–axial stretch relationship is finally
and the relation between the volume and the axial stretch is
in which we assumed that λz≥0 and
In Fig. S4Cwe show the analytically predicted pressure–vol-
ume curves for a latex tube of length Ltube =30 mm subjected to
an axial force 0 ≤F≤gN (where g=9.81 m s
is the gravi-
tational acceleration). As previously observed (37), the results
show that an increase in the axial force applied to the mem-
brane results in a lower peak pressure, indicating that the
axial force resulting from the braids can be used to control the
initial response of the fluidic segments. In Fig. S4Cthe ana-
lytical predictions (dashed lines) are also compared with ex-
perimental results (continuous lines) obtained inflating a latex
tube of length Ltube =30 mm with a weight attached to one of
its ends. The good agreement between analytical and experi-
mental results indicate that, despite the simplifications in-
troduced in the derivation, our analysis captures all of the im-
Effect of contact. At a critical point during inflation, the braids
and the membrane come into contact, stiffening the overall re-
sponse of the segments. Here, we assume that contact occurs
when vbraid =v, where vbraid is the volume enclosed by the braids,
which can be approximated as
Overvelde et al. www.pnas.org/cgi/content/short/1504947112 3 of 10
3 tan θ!R+λzLtube tan θ
3 tan θ,
Because the pressure–volume relationship provided by Eqs.
S38 and S39 is valid only for v<vbraid, we now proceed to de-
termine how these modify after contact. We assume that the
contact between the tube and the braids results in an uniform
pressure, pbraid,appliedtothebraids(Fig.S4D). Because the
beams are modeled as rigid, this pressure results in two ef-
fective forces with radial, Fr,andaxial,Fa, components (Fig.
S4D), given by
Balancing the external work and the elastic energy yields
2+F−Fa"Lbraid sin θ=2nKθ,[S44]
where Fis defined in Eq. S31. Furthermore, we assume that
after contact the response is fully dominated by the braids
in which ^
p=pðvbraid =vÞcan be calculated from Eqs. S38–S40.
Finally, combining Eqs. S42–S45 we obtain
p=8nKθ−4FLbraid sin θ
braidðcos θsin θ−sin2θÞ
In Fig. S4Ewe show the predicted response for six segments
characterized by Ltube =20 mm and Lbraid =40 −50 mm. The results
indicate that for shorter braids, the peak in pressure is lower, dem-
onstrating that the braids can be used to control the initial response
of the fluidic segments. Moreover, the predicted response also
clearly shows that Lbraid strongly affects the volume at which stiff-
ening occurs. In fact, the shorter the braids, the earlier contact be-
tween the braid and the membrane occurs, reducing the amount of
volume that needs to be supplied to have a steep increase in pressure.
However, if the braid length is kept constant at Lbraid =50 mm
and Ltube is varied (Ltube =20 −30 mm), both the pressure peak
and the volume at which stiffening occurs remain unaltered (Fig.
S4F). However, in this case we find that shorter tubes lower the
pressure of the softening region. Finally, the analytical model also
indicates that the length of the fluidic segments, l=λzLtube, initially
increases. However, when the tube and braids come into contact,
further elongation is restrained by the braids and the segments
shorten as a function of the supplied volume (Fig. S4 Eand F).
Therefore, this simple analytical model indicates that, by en-
closing inflatable tubes with stiffer and longer braids, fluidic
segments with the desired nonlinear response can be realized.
Importantly, we have also found that by changing Lbraid and Ltube
their pressure–volume response (i.e., height of the initial pres-
sure peak, softening response, and volume at which the final
steep increase in pressure occurs) can be tuned and controlled.
Therefore, we expect that by rationally interconnecting these
braided fluidic segments, we can design systems in which snap-
through instabilities at constant volume can be triggered.
Finally, it is important to note that we expect this model to
predict only qualitatively and not quantitatively the response of
the segments. This is mainly due to the effect of boundary conditions
(i.e., the deformation is not uniform throughout the membrane) and
inextensibility of the braids. Moreover, it has also been shown that
local instabilities resulting in bulges (36–38) are triggered during
the inflated of tubes. Although our model aims to provide design
guidelines, accounting for all these effects would have lead to a
significantly more complicated and less intuitive model, which
falls outside the scope of this study.
Fig. S1. Relation between the pressure and volume for three different hyperelastic spherical membranes upon inflation.
Overvelde et al. www.pnas.org/cgi/content/short/1504947112 4 of 10
Fig. S2. Response of two interconnected spherical membranes upon inflation. The response of the individual membranes is shown in Fig. S1. The equilibrium
states and their stability have been obtained numerically. The pressure–volume relation and the relation between the volume of the individual membranes are
shown for systems comprising (A) membranes aand b,(B) membranes band c, and (C) membranes aand c.
Fig. S3. Evolution of pressure (p) as a function of the supplied volume (v) for three latex tubes characterized by Ltube =22, 26, and 30 mm as measured in
experiments (solid lines) and predicted by the analytical model (dashed lines). Snapshots of a tube characterized by Ltube =22 mm at v=0, 20, 40 mL are shown
on the left. (Scale bar: 10 mm.)
Overvelde et al. www.pnas.org/cgi/content/short/1504947112 5 of 10
Fig. S4. Analytical prediction of the response of individual fluidic segments. (A) Diagram of the braids before contact with the membrane (i.e., v<vbraid ).
(B) Experimentally obtained force-displacement relation for braids with Lbraid =40 −50 under uniaxial compression. The Inset shows the critical force, F,
measured from experiments. (C)Evolutionofpressure(p) as a function of the supplied volume (v) for a latex tube subjected to an axial force F=0−1 g(g
denoting the gravitational acceleration). Continuous and dashed lines correspond to experimental data and analytical predictions, respectively.(D) Diagram of
the braids after contact with the membrane. (E) Predicted response for six segments characterized by Ltube =20 mm and Lbraid =40 −50 mm. (F) Predicted
response for six segments characterized by Ltube =20 −30 mm and Lbraid =50.
Overvelde et al. www.pnas.org/cgi/content/short/1504947112 6 of 10
Fig. S5. Numerical procedure to determine the equilibrium configurations for a combined actuator with n=2. (A) We identify the volumes of each segment
that correspond to a given value of pressure (in this case ppoint ). (B) We find all corresponding equilibrium configurations for the combined actuator by
combining all those volumes. (C) Because v=v1+v2, we can also identify all equilibrium points in the pressure–volume curve for the combined actuator.
Fig. S6. Numerical results for two combined soft actuators with n=2. (A) Relation between the individual volumes of the segments for a soft actuator with
ðLbraid ,LtubeÞ=ð48, 30 Þand ð50, 20Þmm. (B) Relation between the individual volumes of the segments for a soft actuator with ðLbraid,Ltube Þ=ð44, 30Þand ð48, 26Þmm.
Overvelde et al. www.pnas.org/cgi/content/short/1504947112 7 of 10
Fig. S7. (A–C)Δ^
pversus the normalized change in energy Δ
Efor all state transitions that occur in the 7,140 combined soft actuators comprising n=3
fluidic segments. Blue, red, green, purple, orange, and yellow markers correspond to ð0, 1Þ,ð1, 1Þ,ð1, 2Þ,ð2,1Þ,ð2, 2Þ,andð2, 3Þtransitions, respectively. (D)Ex-
perimental (solid line) and numerical (dashed line) evolution of pressure and length as a function of the supplied volume for a soft actuator with n=3, charac-
terized by ðLbraid ,Ltube Þ=ð40, 28Þ,ð44,22Þ,andð48, 26Þmm. (E) Numerically determined relation between the individual volumes of the three segments.
Movie S1. Two-balloon experiment. (i) Two identical balloons, inflated to different diameters, are connected to freely exchange air. Surprisingly, instead of
the balloons becoming equal in size, for most cases the smaller balloon becomes even smaller and the balloon with the larger diameter further increases in
volume. This unexpected behavior is the result of the nonlinear relation between the balloons’pressure and volume. (ii) By changing the initial volume of the
two balloons, it is also possible to realize a system in which the balloons become equal in size when connected.
Overvelde et al. www.pnas.org/cgi/content/short/1504947112 8 of 10
Movie S2. Inflation of a fluidic segment. Inflation of a single fluidic segment characterized by Lbraid =46 and Ltube =20 mm. In the experiment, we inflated the
segment with water at a rate of 60 mL/min.
Movie S3. Inflation of a combined soft actuator. Inflation of a combined soft actuator consisting of two interconnected fluidic segments, characterized by
ðLbraid,Ltube Þ=ð46, 20Þand ð46, 22Þmm. In the experiment, we inflated the segment with water at a rate of 20 mL/min. At v=23 ml an instability at constant
volume is triggered, causing a sudden increase in length, internal volume flow, and pressure drop.
Movie S4. Inflation of a combined soft actuator–sudden increase in length. Inflation of a combined soft actuator consisting of two interconnected fluidic
segments, characterized by ðLbraid ,LtubeÞ=ð48, 30Þand ð50, 20Þmm. In the experiment, we inflated the segment with water at a rate of 20 mL/min. At v=26 ml
an instability at constant volume is triggered, causing a sudden increase in length of 10 mm, which corresponds to an increase of 11%.
Overvelde et al. www.pnas.org/cgi/content/short/1504947112 9 of 10
Movie S5. Inflation of a combined soft actuator–sudden internal volume flow. Inflation of a combined soft actuator consisting of two interconnected fluidic
segments, characterized by ðLbraid ,LtubeÞ=ð44, 30Þand ð48, 26Þmm. In the experiment, we inflated the segment with water at a rate of 20 mL/min. For this
system two instabilities at constant volume are triggered. The first one occurs at v=16 mL and results in the release of ∼10% of the stored elastic energy and in
a sudden internal volume flow. The second instability is triggered at v=24 mL, causing some fluid to flow back.
Movie S6. Amplifying the response of a combined soft actuator comprising three fluidic segments. Amplification of the response of a combined soft actuator
consisting of three interconnected fluidic segments, characterized by ðLbraid ,Ltube Þ=ð40, 28Þ,ð44, 30Þ,andð50, 24Þmm. We first inflate the actuator to v=28 mL,
and then decouple it from the syringe pump and connect it to a small reservoir containing only 1 mL of water. Remarkably, by adding only 1 mL of water to the
system, we are able to trigger a significant internal volume flow of ∼20 mL, that results in the deflation of two segments into one segment.
Movie S7. Actuation time. Actuation time for a soft actuator consisting of two interconnected fluidic segments, characterized by ðLbraid ,Ltube Þ=ð44, 30Þand
ð48, 26Þmm. We first inflate the actuator to v=16 mL, and then decouple it from the syringe pump and connect it to a small reservoir containing only 1 mL of
water. When the system is inflated with water it takes more than 1 s for the changes in length, pressure, and internal volume induced by the instability to fully
take place. By replacing water with air, the time is reduced from 1.4 s to 300 ms. Moreover, by adding an additional reservoir of air to increase the energy
stored in the system, the actuation time can be further decreased to 100 ms.
Overvelde et al. www.pnas.org/cgi/content/short/1504947112 10 of 10