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Extreme Mechanics Letters 1 (2014) 42–46
Contents lists available at ScienceDirect
Extreme Mechanics Letters
journal homepage: www.elsevier.com/locate/eml
Mechanical and electrical numerical analysis of soft
liquid-embedded deformation sensors analysis
Johannes T.B. Overvelde a, Yiˇ
git Mengüç a,b,c, Panagiotis Polygerinos a,b,
Yunjie Wang a, Zheng Wang b,d, Conor J. Walsh a,b, Robert J. Wood a,b,
Katia Bertoldi a,∗
aSchool of Engineering and Applied Sciences (SEAS), Harvard University, Cambridge, MA, USA
bWyss Institute for Biologically Inspired Engineering, Harvard University, Boston, MA, USA
cSchool of Mechanical, Industrial and Manufacturing Engineering (MIME), Oregon State University, Corvallis, OR, USA
dDepartment of Mechanical Engineering, The University of Hong Kong, Hong Kong
a r t i c l e i n f o
Article history:
Received 23 October 2014
Received in revised form 14 November
2014
Accepted 15 November 2014
Available online 4 December 2014
Keywords:
Finite element
Soft sensor
Liquid-embedded sensor
Large deformation
a b s t r a c t
Soft sensors comprising a flexible matrix with embedded circuit elements can undergo
large deformations while maintaining adequate performance. These devices have attracted
considerable interest for their ability to be integrated with the human body and have
enabled the design of skin-like health monitoring devices, sensing suits, and soft active
orthotics. Numerical tools are needed to facilitate the development and optimization of
these systems. In this letter, we introduce a 3D finite element-based numerical tool to
simultaneously characterize the mechanical and electrical response of fluid-embedded
soft sensors of arbitrary shape, subjected to any loading. First, we quantitatively verified
the numerical approach by comparing simulation and experimental results of a dog-bone
shaped sensor subjected to uniaxial stretch and local compression. Then, we demonstrate
the power of the numerical tool by examining a number of different loading conditions. We
expect this work will open the door for further design of complex and optimal soft sensors.
©2014 Elsevier Ltd. All rights reserved.
While engineering applications often use stiff and
rigid materials, soft materials like elastomers enable the
design of a new class of electronic devices that are flexi-
ble, stretchable, adaptive and can therefore be easily inte-
grated with the human body [1–6]. These include highly
conformable and extensible deformation sensors made
from flexible substrates with embedded circuit elements,
such as graphene sheets [7], nanotubes [8], interlocking
nanofibres [9], serpentine patterned nanomembranes of Si
[3,10,11], and conductive liquid micro channels [12–14].
A key feature of these sensors is their ability to re-
versibly stretch, bend, compress and twist to a great ex-
tent. Such deformations result in changes of the electrical
∗Corresponding author.
E-mail address: bertoldi@seas.harvard.edu (K. Bertoldi).
resistance of the sensors, which are then used to deter-
mine the applied loading conditions. Therefore, the design
of the next generation of soft sensors requires the develop-
ment of numerical tools capable of predicting not only their
mechanical performances [15,16], but also their electrical
response. Such tools will enable the design of optimized
sensors that are sensitive to desired loading conditions and
also provide crucial insights into the working principles of
these soft devices.
In this letter, we propose a 3D finite element-based nu-
merical tool that predicts both the mechanical and electri-
cal response of arbitrary shaped soft sensors subjected to
any loading condition. In particular, we focus on sensors
comprising an elastomeric matrix embedded with a net-
work of channels filled with a conductive liquid [12–14],
but the numerical approach can be easily extended to other
types of soft sensors. The analyses are performed using the
commercial finite element (FE) code Abaqus, which is an
http://dx.doi.org/10.1016/j.eml.2014.11.003
2352-4316/©2014 Elsevier Ltd. All rights reserved.
J.T.B. Overvelde et al. / Extreme Mechanics Letters 1 (2014) 42–46 43
Fig. 1. The proposed FE-based numerical tool consists of three steps. (A) The model is created using CAD software and then meshed. (B) The deformation
of the sensor is determined by using non-linear FE analysis, in which the contours show the normalized Von Mises stress σvm. (C) The resistance at different
levels of deformation is obtained by performing a steady-state linear electrical conductivity analysis. The contours show the potential across the channel.
Fig. 2. Uniaxial extension of a dog-bone shaped soft sensor. (A) Experimental images of the undeformed (ux/L=0) and deformed (ux/L=0.5) sensor. (B)
Numerical images of the undeformed (ux/L=0) and deformed (ux/L=0.5) sensor. The contours in the snapshot show the distribution of the electrical
potential. (C) Cross-sectional profile measured with a laser interferometer (green) and cross-sections used in simulations. (D) Reaction force obtained in
experiments and simulations as a function of the applied strain. (E) Cross-sections of the undeformed and deformed (ux/L=0.5) channels as predicted by
the FE analysis. (F) Electrical resistance measured in experiments and simulations as a function of the applied strain.
attractive platform because it is well-known, widely avail-
able and particularly suitable for analyses involving large
deformations. By making our code available online, we ex-
pect the proposed tool to be widely used and expanded to
design more complex soft sensors with new and improved
functions.
Our FE-based numerical tool consists of three steps, as
indicated in Fig. 1 (the Abaqus script files and Matlab files
used for our analysis are available online as Supporting
Information):
Step A: Creating the model. A 3D model comprising both
the flexible matrix and the circuit elements is first created
using CAD software and then meshed. In particular, for
the case of liquid-embedded soft sensors considered in
this letter, the elastomeric matrix is meshed using linear
tetrahedral elements (Abaqus element type C3D4), while
a solid mesh of the channels is not created. In fact, only
the surface mesh of the channel will be used to apply the
pressure exerted by the fluid to the elastomer.
Step B: Determining the deformation. To determine the de-
formation of the sensor under specific loading conditions,
a non-linear FE analysis is performed using the commer-
cial package Abaqus/Explicit (v6.12). In the simulations, we
fully account for contact between all faces of the model
and ensure quasi-static conditions by monitoring the ki-
netic energy and introducing a small damping factor. For
the case of liquid-embedded sensors investigated here, the
response of the flexible matrix is captured using a nearly
incompressible neo-Hookean material characterized by an
initial shear modulus µ. Moreover, we consider the chan-
nels to be completely filled with a nearly incompressible
fluid with bulk modulus K=100µand use the surface-
based fluid cavity capability in Abaqus, so that the pressure
applied by the fluid to the surface of the channel is deter-
mined from the cavity volume.
Step C: Analyzing the electrical resistance. To determine the
electrical resistance of the deformed sensor, an isothermal
steady-state linear electrical conductivity analysis (Abaqus
step Coupled thermal–electrical) is performed on the de-
formed solid mesh of the circuit elements. Assuming the
circuit elements are made of a material with electrical re-
sistivity ρ, an electrical potential difference ∆Uis applied
between the two ends of the deformed circuit mesh and
the dissipated work (W) over a time period of ∆tis cal-
culated. The electrical resistance of the channel (R) is then
obtained as
R=
∆U2∆t
W.
44 J.T.B. Overvelde et al. / Extreme Mechanics Letters 1 (2014) 42–46
Fig. 3. Compression of a dog-bone shaped soft sensor using a circular punch. (A) Experimental set-up. (B) Cross-sections of the undeformed and deformed
(uz/L= −0.11) channels as predicted by the FE analysis. (C) Numerical images of the undeformed (uz/L=0) and deformed (uz/L= −0.3) sensor. The
contours in the snapshot show the distribution of the electrical potential. (D) Reaction force obtained in experiments and simulations as a function of the
applied compressive strain. (E) Electrical resistance measured in experiments and simulations as a function of the applied compressive strain.
For the sake of convenience, in all our simulations we use
∆U=1V and ∆t=1s, so that R=1/W. Note that for
the specific case of liquid-embedded soft sensors, an ad-
ditional meshing step is required, in which the deformed
solid tetrahedral mesh of the channel is created starting
from the deformed surface mesh obtained in Step B.
To validate our simulations, we focus on a dog-bone
shaped soft sensor with a serpentine channel aligned along
its length (see Fig. 2A), and compare the numerical pre-
dictions to experimental results for two different loading
conditions: uniaxial tension and local compression. The
sample is fabricated using silicone rubber (EcoFlex 0030,
Smooth-On, Easton, PA, USA) and the channel is filled with
the conductive liquid eutectic Gallium Indium (eGaIn) (Alfa
Aesar, MA, USA) with resistivity ρ=29.4·10−8m [17].
Details on the fabrication of the sample are reported in pre-
vious papers [12,18,19]. All experiments were conducted
on a uniaxial materials testing machine (model 5544A, In-
stron Inc., MA, USA). To determine the electrical resistance
Rof the sensor, the sample was connected in series with
a resistor with resistance Rref . A potential difference ∆Uref
was applied to the entire system by a power supply, so that
R=Rref
∆U
∆Uref −∆U,
where ∆Uis the potential difference between the two ends
of the sensor’s channel measured using a data acquisition
card (BNC-2111, National Instruments Corp., TX, USA).
The dog-bone sample considered in this study has a
central slender section of length L=50 mm and a rect-
angular cross-section of 8 mm by 2 mm. Despite the fact
that the sensor is designed to have an embedded channel
with an overall length of 360 mm and a rectangular cross-
section of 0.25 mm by 0.15 mm, we find that the actual
cross-sectional shape of the channel is closer to that of a
circular segment (see Fig. 2C). Therefore, in our simulations
we consider two different cross-sectional shapes for the
channel: (i) a rectangle of 0.3 mm by 0.172 mm and (ii) a
circular segment with an angle of 111 degrees and a radius
of 232 mm. Note that in both cases the area of the channel is
equal to the experimentally measured value of 0.052 mm2.
First, we load the sample uniaxially, as shown in
Fig. 2A and B and Movies S1 and S2. From the mechanical
response of the sensor, we find that the experimental mea-
sured stress–strain behavior can be fully captured mod-
eling the elastomeric matrix as a neo-Hookean material
with µ=0.0221 MPa (see Fig. 2D). As expected, we find
that both the rectangular and circular segment channels
deform in a similar manner and contract isotropically as
the deformation increases (see Fig. 2E). The decrease in
cross-sectional area of the channels results in an increase of
the electrical resistance, which is captured in both experi-
ments and simulations, as indicated in Fig. 2F. Remarkably,
the numerical simulations, which do not require any fitting
parameter, capture not only qualitatively but also quan-
titatively the evolution of Ras a function of the applied
deformation. Note that for this specific loading case our
simulations indicate that both the electrical and mechani-
cal response of the sensor are not affected by the shape of
the channels, but only by their area (see Fig. 2F).
Next, we locally compress the sample in its center with
a circular flat punch of radius 5 mm (see Fig. 3A and C
and Movies S3, S4 and S5). For this loading condition the
electrical response of the sensor is characterized by two
distinct regimes (see Fig. 3E) [13,14,19]. For low values of
J.T.B. Overvelde et al. / Extreme Mechanics Letters 1 (2014) 42–46 45
Fig. 4. Sensitivity of a dog-bone shaped soft sensor to various load cases. (A) Undeformed configuration. (B) Twist. (C) Roll. (D) Uniaxial compression in
longitudinal direction. (E) Shear. The contours in the snapshot show the distribution of the electrical potential.
applied deformation (uz/H<−0.15), the resistance R
of the sensor is constant and not affected by the applied
deformation. However, when the applied deformation is
large enough to locally close the channel (see Fig. 3B), R
increases rapidly. Remarkably, by using the same mate-
rial parameters determined in our previous analysis, the
numerical simulations exactly capture the experimentally
measured mechanical response (see Fig. 3D). Furthermore,
as shown in Fig. 3E, the simulations also correctly capture
the electrical response of the sensors and reveal that the
level of applied deformation at which Rstarts to rapidly
increase is highly sensitive to the cross-sectional shape of
the channel. Such sensitivity has been previously shown
experimentally [12,20] and is dictated by the fact that the
load required to locally close the channel is highly affected
by its shape. In particular, we find that the circular seg-
ment cross-section completely closes for smaller values of
applied strain (Fig. 3B), resulting in a sensor that is more
sensitive to local compression.
Now that we have verified the robustness of our nu-
merical tool, we use it to characterize the sensitivity of
our sensor to different loading conditions. In particular, we
consider four different loading cases: (i) we twist the sen-
sor by rotating one side around the x-axis by a 6πangle
as shown in Fig. 4B and Movies S6; (ii) we roll the sensor
by rotating one of its ends around the y-axis by a 4πan-
gle (see Fig. 4C and Movie S7); (iii) we compress the sen-
sor uniaxially along the x-axis by ux/L= −0.5 (Fig. 4D and
Movie S8); (iv) we shear the sensor by displacing one of the
ends along the y-axis by uy/L=0.5 (Fig. 4E and Movie S9).
Note that for the loading cases (i), (ii) and (iv) the sensor is
free to move in the z-direction, so that it does not stretch
uniaxially. Surprisingly, although in all these simulations
the channels are drastically deformed, the electrical resis-
tance of the sensor remains nearly unaltered, i.e. R/R0∼1.
Therefore, our simulations indicate that this dog-bone
sensor is extremely well suited for applications in which
uniaxial tensile strain applied along the longitudinal direc-
tion needs to be monitored. In fact, much higher forces are
required to make the sensor sensitive to local compression
in the z-direction, i.e. 0.4N and 2N need to be applied to in-
crease the electrical resistance by 100% under uniaxial ex-
tension and local compression, respectively. Moreover, all
other simulated loading conditions are found to leave the
electrical resistance unchanged even for extreme values of
applied deformation.
In summary, we introduced an effective finite element
procedure to characterize simultaneously the mechanical
and electrical response of soft sensors of any shape sub-
jected to arbitrarily loading conditions. The accuracy and
robustness of the proposed numerical method was verified
by comparing the numerical and experimental results for
the case of a dog-bone shaped soft sensor with a serpentine
channel aligned along its length. The simulations not only
46 J.T.B. Overvelde et al. / Extreme Mechanics Letters 1 (2014) 42–46
quantitatively capture both the mechanical and electrical
response of the sensor, but also enable us to determine the
role played by the shape of the cross-section and different
loading conditions, facilitating the design of sensors which
are only sensitive to specific loads.
This work opens the door for further simulation-based
studies of soft sensors with complex microstructures,
which would otherwise be intractable to address analyt-
ically or experimentally. In particular, the proposed nu-
merical tool may serve as a platform to accelerate the
design of devices such as embedded 3D printed circuits
[21], liquid metal pumps [22], and health-monitoring de-
vices [3,8,19,23].
Acknowledgments
This work was supported by the Materials Research
Science and Engineering Center under NSF Award No.
DMR-0820484. K.B. also acknowledges support from the
National Science Foundation (CMMI-1149456-CAREER)
and the Wyss institute through the Seed Grant Program.
Appendix A. Supplementary data
Supplementary material related to this article can be
found online at http://dx.doi.org/10.1016/j.eml.2014.11.
003.
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