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https://doi.org/10.1038/s41467-019-11294-7

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Reconﬁgurable soft body trajectories using
unidirectionally stretchable composite laminae

1234567890():,;

Sang Yup Kim 1,4, Robert Baines
Rebecca Kramer-Bottiglio1

1,4, Joran Booth1, Nikolaos Vasios2,3, Katia Bertoldi2,3 &Compliant, continuum structures allow living creatures to perform complex tasks inaccessible
to artiﬁcial rigid systems. Although advancements in hyper-elastic materials have spurred thedevelopment of synthetic soft structures (i.e., artiﬁcial muscles), these structures have yet tomatch the precise control and diversity of motions witnessed in living creatures. Cephalopods
tentacles, for example, can undergo multiple trajectories using muscular hydrostat, a structure consisting of aggregated laminae of unidirectional muscle ﬁbers. Here, we present a selfadhesive composite lamina inspired by the structural morphology of the muscular hydrostat,which adheres to any volumetrically expanding soft body to govern its motion trajectory. The
composite lamina is stretchable only in one direction due to inextensible continuous ﬁbers
unidirectionally embedded within its hyper-elastic matrix. We showcase reconﬁguration of
inﬂation trajectories of two- and three-dimensional soft bodies by simply adhering laminae totheir surfaces.

1 School of Engineering and Applied Science, Yale University, New Haven, CT 06511, USA. 2 Harvard John A. Paulson School of Engineering and Applied
Sciences, Harvard University, Cambridge, MA 02138, USA. 3 Kalvi Institute of Bionano Science & Technology, Harvard University, Cambridge, MA 02138,
USA. 4These authors contributed equally: Sang Yup Kim, Robert Baines. Correspondence and requests for materials should be addressed to
R.K.-B. (email: rebecca.kramer@yale.edu)

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he introduction and critical study of hyper-elastic materials
have sparked an evolution in numerous domains of scienceand engineering, such as medicine1,2, electronics3–5, androbotics6–8. The inherent stretchability and resilience of hyperelastic materials make them apt for incorporation into devices
imitating the physiology of living creatures. In robotics, hyperelastic materials allow for the creation of compliant, continuum
soft robots with greater adaptability and ﬂexibility than conventional linkage-joint robots9–13. Many soft robots inspired by themorphology of biological organisms have performed movementsof unprecedented complexity. A typical mechanism to impart
such sophisticated movement is the incorporation of strain limiters onto a soft body. When the body inﬂates, strain limiters
exert contraction forces analogous to biological muscle ﬁbers,
governing speciﬁc shape changes14–20. McKibben and PneuNet
actuators are pneumatic artiﬁcial muscles widely used in practice,and exploit conﬁnement in woven mesh21,22 or by rigid compartments, as strain limiters23–25. However, these actuators areoften suboptimal for use in entirely-soft robots: McKibben
actuators are difﬁcult to integrate into soft bodies without rigid
ﬁxtures, and PneuNets are challenging to predict and control dueto their intricate geometry. Recent studies have established asimple yet effective way to control the deformation trajectory ofsoft actuators—by wrapping them with continuous inextensible
ﬁbers26–29. Varying the angle of wrapped ﬁbers creates uniquestrain-limiting patterns that change how an internal elastomeric
bladder expands, and allows for predictable extension, twisting,and bending. However, in this approach, programmed directionality is permanent and constrained to one ﬁxed motion,
unlike the versatility exhibited by biological organisms. Additionally, ﬁber-wrapped actuators have tedious manufacturing
processes that make them difﬁcult to integrate with twodimensional (2D) geometries or scale up.
Looking to nature for inspiration in addressing the aforementioned challenges with existing soft actuators, we noticed that
remarkably dexterous motion achieved by skeleton-free animal
parts such as elephant trunks, octopus arms, and human tongues,
is attributed to muscular hydrostats: a laminated structure composed of layers of unidirectional muscle ﬁbers30. Selective contraction of individual muscle layers in three primary axes—
transverse, longitudinal, and oblique—grants muscular hydrostats
the ability to move precisely and with inﬁnite degrees of
freedom31,32. For example, cephalopod tentacles twist by contracting oblique muscle ﬁbers and bend by simultaneously contracting transverse and longitudinal muscle ﬁbers (Fig. 1a). A
simple superposition of these unidirectional muscle contraction
primitives gives rise to sophisticated deformation.
Inspired by the ﬁber architecture of the muscular hydrostat, wedeveloped a lamina composed of a hyper-elastic matrix withunidirectionally embedded inextensible ﬁbers and an adhesive
backing (Fig. 1b). We call this composite material Stretchable
Adhesive Uni-Directional prepreg (STAUD-prepreg). STAUDprepreg is extremely stretchable in the direction perpendicular tothe ﬁbers and inextensible along the ﬁbers, boasting a 1000-folddifference in stiffness and elongation (Fig. 1c). When STAUDprepreg is adhered to volumetrically expanding soft bodies, theembedded inextensible ﬁbers induce selective contraction forces
analogous to muscle ﬁbers in the muscular hydrostat, and govern
shape change (Fig. 1d, Supplementary Movie 1). The fabricationof STAUD-prepreg is simple and scalable, leveraging a ﬁberwinding process that is already established in industry for massproduction of industrial-grade composites to create large 2D
sheets, which can be cut and applied to almost any soft actuator.
Namely, we use a bench-top ﬁber winder wherein a mixture of
two-part silicone resin is infused into polyester ﬁbers uniformly
spaced on a rotating mandrel (Fig. 1e). We then apply a silicone2

based adhesive to the back of the lamina after the resin fully cures
to create a self-adhesive layer. The deformation trajectories of soft
bodies are easy to program by placing STAUD-prepreg on theirsurfaces (Fig. 1f). Adhered STAUD-prepreg can be easily
detached and re-attached in various patterns and stacking
sequences to reconﬁgure the inﬂation trajectories.
Results
Inﬂuence of STAUD-prepreg on soft actuator trajectories. To
reveal the inﬂuence of STAUD-prepreg on morphing trajectories
of soft bodies, we began by constructing a freely-inﬂating
cylindrical actuator wrapped in one STAUD-prepreg. By adjusting the ﬁber orientation of this STAUD-prepreg, the cylindricalactuator exhibits a variety of motions such as contraction, elongation, and rotation (Fig. 2a). Contraction (Fig. 2a at 0°, Supplementary Movie 2) is accomplished when the STAUD-prepreg
is wrapped at θ = 0°; that is, embedded ﬁbers are aligned longitudinally to the cylinder length. The 0° ﬁbers geometricallyconstrain the pneumatic cylinder such that it is only capable ofexpanding transversely, resulting in length-wise contraction.
Elongation (Fig. 2a at 90°, Supplementary Movie 3) is achieved
with the STAUD-prepreg ﬁbers oriented at θ = 90°; that is,embedded ﬁbers are aligned perpendicular to the cylinder’slength. The 90° ﬁbers prevent the actuator from expandingtransversely, forcing it to extend axially. Rotation (Fig. 2a at 45°,
Supplementary Movie 4) occurs in a manner similar to elongation, except the angled ﬁbers (θ = 45° in this study) exert a torquewith respect to the central axis. In all cases, the deformation of acylindrical actuator is proportional to the input pressure anddirectly affected by the ﬁber spacing of the adhered prepreg.
Moreover, STAUD-prepreg are capable of programming inﬂation
trajectories of 2D objects and controlling their shape in 3D space.
For instance, applying STAUD-prepreg to a thin, planar soft body
elicits bending upon its inﬂation (Fig. 2b, Supplementary
Movie 5). Planar actuators outﬁtted with STAUD-prepreg arelight and occupy a small footprint when not actuated, making
them ideal for deployable, easily-transportable pneumatic devices.
In general, the initial shape of soft bodies (i.e., 2D vs 3D cylinder)
inﬂuences more the adherence of the STUAD-prepreg, than it
does the uniqueness of a ﬁnal shape. The 2D conﬁguration of
STAUD-prepreg is easier to adhere to 2D bodies than 3D ones
without void entrapments that can potentially degrade theadhesion quality.
Mechanical characterization of STAUD-prepreg. We conductedtensile mechanical testing of STAUD-prepreg laminae with different ﬁber orientations and spacings to ascertain its fundamental
mechanical properties. Quasi-static, unidirectional tensile tests
reveal a stark difference in stretchability (i.e., stiffness and elongation) of the material, depending on the loading direction(Fig. 2c). For example, the tensile modulus of the lamina (at 2 mm
ﬁber spacing) is 56 MPa in the longitudinal direction of the ﬁbers(E1), and 0.065 MPa in the transverse direction (E2). Moduli tend
to increase with decreasing ﬁber spacing as a result of the higher
ﬁber content in the lamina33. Theoretically calculated values frommicromechanics modeling34 conﬁrm our experimental results formoduli:E1 ¼ Vf Ef þ Vm Em

ð1Þ

E2 ¼ ðVf =Ef þ Vm =Em Þ1

ð2Þ

where Ef and Em are tensile moduli of ﬁbers and matrix, and Vf
and Vm are the volume fraction of ﬁbers and matrix, respectively(see “Supplementary method” for details).

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abLongitudinal
muscleContinuous fiber(Polyester)
Silicone rubber(Ecoflex 030)

Transverse
muscleObliquemuscle

c

d
X

Free
inflation

Stretch

X
0.6
0.4

0

Constrained
inflation

X

0.2

ConstrainStress,  (MPa)4.50

Adhesive layer(Silbione 4717)

0 1,max 1.0 6,max 2.0

3.0

2,maxf

STAUD-prepreg lamina

Straine
Fiber windingResin
impregnation

Retrieve prepreg,
apply adhesiveon back

Free inflating bodyContraction

Twisting

Elongation

Bending

Fig. 1 Overview and working principle of STAUD-prepreg. a Illustration of laminated layers of unidirectional muscle ﬁbers in a muscular hydrostat incephalopod tentacles and the working mechanism for achieving complex motion. b Schematic and optical micrograph of stretchable adhesive unidirectional
prepreg (STAUD-prepreg). Scale bar: 10 mm. c Quasi-static uniaxial tensile testing results of STAUD-prepreg laminae with respect to different ﬁberorientations (θ = 0°, 45°, 90° from left). d Photographs of inﬂating planar soft body. Free inﬂation (top) with uniform radial expansion and constrained
inﬂation (bottom) with one adhered lamina at θ = 90°. e Schematic of fabrication process for STAUD-prepreg using a bench-top ﬁber winder. f Illustrationof cylindrical actuators with one STAUD-prepreg lamina adhered, and the ensuing inﬂation trajectory

Analytic prediction of deformation. To predict deformation of
an inﬂating body outﬁtted with STAUD-prepreg, we used ourexperimental tensile test results to construct a stiffness matrix(Cij) of the lamina, for a relation σi ¼ Cij εij (i, j = 1, 2, 3, … 6),
where σ is stress and ε is strain. Since STAUD-prepreg is thinand its properties are symmetric about one axis, the stiffness
matrix is further simpliﬁed to ½Qx;y , for a relation½σx;y ¼ ½Qx;y ½εx;y , where ½σx;y and ½εx;y are stress and strain
matrices in the x-y plane, respectively35. This stiffness matrix
allows us to analytically predict morphing of the host body clad in
STAUD-prepreg laminae, after a slight modiﬁcation to classicallaminate theory36–38:
n Z zkn Z zkX
X½σkx;y dz ¼
½Qkx;y ½εkx;y dz
ð3Þ
k¼1

zk1

k¼1

zk1

where k is an individual lamina, n is the total number of laminae,
and z is the out-of-plane coordinate (see “Supplementary

method” for details). A prediction model from the augmentedclassical laminate theory (ACLT) shows good agreement with theexperimental strain values of the cylindrical actuators in Fig. 2a,using Eq. (3) at n = 1. Adapting ACLT to the soft robotics spacefacilitates the prediction of the morphing of soft bodies clad in
STAUD-prepreg, and enables intuitive programming of speciﬁcmotions.
Inﬂuence of multiple STAUD-prepreg on soft actuator trajectories. To impart more complex motion on a soft body, we created actuators with STAUD-prepreg patterns emulating themuscular hydrostat structure. For instance, we adhered multiple
STAUD-prepreg in a stack, or cut a single STAUD-prepreg into
various-sized patches, and distributed them across a soft body’s
surface. We used a cylindrical actuator in Fig. 2a at 90° as a basethat forced elongation, and adhered additional STAUD-prepreg
to further direct motion. When a STAUD-prepreg with θ = 0°
was adhered in a way that covered the entire base, the actuator

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a

b

0.3

ACLT
Strain,  = 90°
Pin

Free inflation

0.2

Pin

Experiment

0.1


0

0

500

1000

1500

Planar (2D)
soft bodyPressure, P (Pa)

 = 0°

Reconfigure
trajectories

0.4

ACLT

Strain, 

0.3

Pin


Lamina on
top

0.2

 = 0°

0.1

ExperimentPin

0

Bending
0

150

300

450

600

Pressure, P (Pa)60

 = 45°

Axial strain
Transverse strain
Angle

0.5
Strain, 

Pin



0.4

40

Experiment

0.3

20

0.2

Lamina onbottomAngle,  (deg)

0.6

Planar (2D)
actuator

Pin

0.1
0

0

500

Bending

0

1000

Pressure, P (Pa)

Decreasing
fiber content
8.0

0

0

0.06

0.12
Strain, 1

0.18

0.24

Shear stress, 6 (MPa)

16

0.6

0.3

0

140

 = 45°

0.6

Elastic modulus, E1 (MPa) = 90°

0.4

0.2

0
0

1.0

2.0

3.0

Strain, 2

0

0.5

1.0

1.5

Shear strain, 6

2.0

100

E1

E2

120

E2

100

1 mm
2 mm
6 mm

80

E2

60

80
E1

60
40

E1

20
0

40

1

2
6
Inter-fiber spacing, (mm)20

Elastic modulus, E2 (kPa)0.9
 = 0°
Stress, 2 (MPa)Stress, 1 (MPa)

c 24

0

Fig. 2 Basic programmed inﬂation trajectory of 2D/3D soft bodies and mechanical characterization of STAUD-prepreg. a Inﬂating cylindrical actuatorwrapped with one STAUD-prepreg lamina at different ﬁber orientations of θ = 90°, 0° and 45°. Experimental results are juxtaposed with analytical resultsfrom augmented classical laminate theory (ACLT). Legend color codes apply to all plots. Scale bar: 25 mm. b Planar soft body consisting of two silicone
rubber ﬁlms and reconﬁguration of its inﬂation trajectory by adhering one STAUD-prepreg. Scale bar: 25 mm. c Quasi-static uniaxial tensile testing resultsfor STAUD-prepreg. E1 and E2 denote the Young’s modulus of the STAUD-prepreg along the ﬁbers (θ = 0°) and perpendicular to the ﬁbers (θ = 90°),
respectively. Higher ﬁber content in the prepreg results in lager values of E1 and E2. E1 is ~1000-times greater than E2

underwent virtually no motion due to the presence of strainlimiting ﬁbers preventing extension in both principal directions(Fig. 3a). Replacing the adhered STAUD-prepreg (θ = 0°) with
one at θ = 45° gave rise to rotational motion with a negligible
expansion in volume, unlike the actuator with one lamina
adhered at θ = 45° (see Fig. 2a at 45°). The ACLT model at n = 2
in Eq. (3) shows good agreement with experimental results andconﬁrms the utility of ACLT in predicting the deformation of
actuators clad in STAUD-prepreg. When a bulk STAUD-prepreg
is cut into patches and adhered to the elongating base actuator,
we coerce deformations that cannot be attained with actuators
whose surfaces are entirely covered by STAUD-prepreg (seeFigs. 2a, 3a), and unlock a larger workspace without any signiﬁcant jumps in mechanical complexity. For example, bending,the underlying motion for grasping and locomotion in cephalopods, is achieved in our system when one STAUD-prepreg patch4

is adhered to locally conﬁne elongation (Fig. 3b, Supplementary
Movie 6). Adjusting the ﬁber orientation of the patch results in acompound bending and twisting motion (Fig. 3c). Further complex output motion arises when multiple STAUD-prepreg patches
are attached to the actuator in different locations. Each adheredpatch creates contraction forces on the actuator based on its
respective dimensions (i.e., w and h) and elicits unique trajectories (Fig. 3d).
Numerical modeling of deformation. We performed ﬁnite element (FE) analysis on cylindrical actuators’ motion using ABAQUS commercial software (see “Supplementary method” fordetails). FE simulation results conﬁrm our experimental results
and reveal detailed strain and stress ﬁelds on the inﬂating
actuator (Fig. 3e as an example for the bending motion in

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 = 90° + 0°

0.15
Base ( = 90°)  = 90° + 45°

Strain, ACLT = 90° + 45°

t

g

30

Experiments = 45°90° + 45°

0.10

20

0.05

10
90° + 0°

0

b

0



Pin
Lamina(at  = 0°)

 = 0°

Bending angle,  (°)

50
 = 90°1000 2000 3000
Pressure, Pin (Pa)FE modelAngle,  (deg)

a

Pin
0

4000

Surface areal
coverage (%)

40

16
32
48
64
80

30
20
10
0
0

1000 2000 3000
Pressure, Pin (Pa)

4000

c

Experiment = 45°

 = 0°In planew= 0.32
2πr
h1
= 0.4
L

w1

Pin

h1
L

h2
L

w2

= 0.4

h2
Pin
h2
L

Normalized radius of curvatured

Out of plane

0.8

Top Bottom
(h1)
(h2)

0.6
0.4
0.2

h2
L

0

1000

= 0.2

2000

Bending angle,  (°)

100

L

Increasing
areal coverageY00
Pin = 4600 Pa

r

31
37

50

3000

4000

Pressure, Pin (Pa)

Surface areal
coverage (%)
5
13
19
25

150

= 0.2

0.0

f

Pin = 6000 Pa

Pin = 0 Pa

h2
= 0.6
L
h2
= 0.4
L

*h1/L = 0.4e = 0°

Pin

2500 5000 7500 10000
Pressure, Pin (Pa)Fig. 3b). Moreover, FE modeling allows prediction of the inﬂuence of localized patch(es) on a soft body otherwise too complicated for ACLT to predict (Fig. 3f). We utilized the FEmodeling to obtain STAUD-prepreg patch designs that whenadhered to a slender soft body elicit dexterous movements similar = 0°

 = 45°

Tip displacement, u (mm)Side view

r

Front
view

Front
view

Front

100
ux

75

uz

50
25

uy

0
–25
0

X

Pin

1000
2000
Pressure, Pin (Pa)3000

to cephalopods tentacles. We then reproduced the simulated
results in the laboratory (Fig. 3g, Supplementary Movie 7).
Experimental results show good agreement with the simulation,
testifying to the fact that FE modeling serves as an accurate
design tool.

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Fig. 3 Complex programmed inﬂation trajectory of 3D soft bodies and ﬁnite element simulation. a Elongating cylindrical actuator (at θ = 90°) with anadditional STAUD-prepreg wrapped around at θ = 0° or 45°. The addition of a STAUD-prepreg at θ = 0° immobilizes the actuator, while one at θ = 45°
gives rise to rotational motion without transverse strain ðεt Þ. Scale bar: 25 mm. b–d Bending motion of a cylindrical actuator due to a segmented STAUDprepreg patch. b Increasing the areal coverage of the patch decreases the bending angle at a given pressure. c Adhering a patch at θ = 45° elicits a mixed
motion of bending and twisting. d Two-localized patches with increasing patch dimension along the actuator length leads to a higher bending curvature.
Scale bar: 25 mm. e, f Finite element (FE) analysis results using ABAQUS on cylindrical actuators. e Bending angle as a function of input pressure and apatch’s areal coverage, as gathered from the simulation. f Two-localized patches with different ﬁber angles (at θ = 0° and 45°). g A slender cylindricalactuator mimicking the motion of cephalopod tentacles. FE simulation of the actuator with particular patch pattern and dimensions matches with theexperiment. Scale bar: 50 mma = 0°
Pin
Twist

Bend

Out-of-plane
bend and twist

 = 45° + 0°b

 = 0°, 90°, 45°

 = 0°
Pin

Pin

Grasping
object

c

No
patch

Two
patchesOnepatch

 = 0°Pin = 0°

One
long patchd
Rotational motion

 = 0°

 = 45°

Fig. 4 Practical use cases for STAUD-prepreg controlling compliant systems. a Life-like complex grasping motion akin to cephalopod tentacles
accomplished by a cylindrical actuator. Scale bar: 100 mm. b Planar soft body with eight tentacles for use as a pop-up gripper. Adhering STAUD-prepreg
patches gives rise to speciﬁc movements of individual tentacles that previously inﬂated without directional bias. Scale bar: 50 mm. c Conventional balloon
with adhered STAUD-prepreg patches. The ﬁber orientation and dimension of the applied patches determine the inﬂation trajectories of the balloon.
Scale bar: 50 mm. d A tensegrity robot consisting of rigid struts and soft membrane actuators made from STAUD-prepreg. A combination of rigid
components and soft, planar actuators allows this hybrid system to roll from face to face by inﬂating the membrane actuators to locally destabilize itself.
Scale bar: 200 mm

Demonstrations using STAUD-prepreg. We demonstrated
several practical use cases for STAUD-prepreg. First, linearlyscaling up the dimensions of the actuator in Fig. 3d gives rise to alife-like grasping motion; a combined movement of wrapping andbending reminiscent of cephalopod tentacles (Fig. 4a, Supplementary Movie 7). We then created a planar actuator consistingof eight tentacles that inﬂate uniformly without directional bias.
Attaching STAUD-prepreg patches of varying ﬁber orientation
enabled us to individually control the tentacles’ shapes during
inﬂation (Fig. 4b, Supplementary Movie 8). The planar actuator
outﬁtted with STAUD-prepreg is capable of grasping andtwisting an object, although it is signiﬁcantly lighter and more
6

compactable compared to conventional soft grippers. The versatility of STAUD-prepreg is further demonstrated by its application to every-day soft bodies, such as balloons (Fig. 4c,
Supplementary Movie 9). When adhered to a balloon, STAUDprepreg patches govern its inﬂation trajectory based on thequantity, dimensions, and ﬁber orientations of the patches. This
reafﬁrms how balloons—cheap and ubiquitous as they are—could
be used as programmable actuators in soft robots. Moreover, we
made planar membrane actuators to change the shape of skeletal
robotic systems. In particular, we created thin elastomer membranes outﬁtted with a layer of STAUD-prepreg to interface thecompressive members of a tensegrity robot. The membrane

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actuators are positioned on the exterior faces of the tensegrity,
providing ample internal space for the robot to deform and rollwhen the actuating-faces inﬂate (Fig. 4d, Supplementary
Movie 10).
We have shown that STAUD-prepreg is a versatile material
capable of transforming entirely-soft bodies and hybrid soft-rigid
architectures into customized, programmable, and reconﬁgurable deforming systems. The effect of adhered patches
on inﬂating soft bodies is easy to predict due to the simple strainlimiting behavior of the embedded unidirectional ﬁbers. We
veriﬁed that the shape change of actuators outﬁtted with STAUDprepreg is analytically predictable using ACLT, and utilized
numerical modeling in ABAQUS to predict compounded motion.
Ultimately, we demonstrated highly complex, predictable shapechange of 2D and 3D soft bodies reminiscent of the ﬁber-directedmotion of a muscular hydrostat. The fundamental mechanisms of
localized, reconﬁgurable strain-limiting ﬁbers presented hereintake a step closer to devising soft robot actuation ever more likereal-life ﬁber musculature that can modulate directional contractions in situ. We expect STAUD-prepreg to become an essential
component for controlling synthetic all-soft bodies, but also
foresee its adoption into sports, medicine, and rehabilitation
spaces due to its ease of implementation and manufacture.

zk  zk1 and its mid-plane is a distance zk. The input to the model is a stress term
obtained by converting pressure to stress assuming the cylindrical actuators can bemodeled as expanding pressure vessels. This model is also supplemented by geometrical boundary conditions outlined in the supplemental document. Results in
Figs. 2a and 3a show good agreement between experimentally observed and analytically predicted deformation of the various lamina-wrapped cylindricalactuators.
Finite element analysis. We performed ﬁnite element analysis (FEA) simulations
using Abaqus (SIMULIA, Providence RI) commercial software in order to fully
resolve the strain and stress distributions of each laminae and further account forboundary effects in the various cylindrically shaped actuators. FEA simulations
allowed us to predict the deformation of more complex geometrical layouts andlocalized patches of STAUD-prepreg, and to subsequently replicate these simulated
movements and designs in real-life systems (Fig. 3d; Supplementary Movie 7). All
simulated models were generated using shell elements using the composite shell
section formulation that enables numerical integration of composite shells consisting of several laminae in various orientations. Each STAUD-prepreg lamina wasmodeled as an anisotropic continuum 3D shell whose constitutive behavior isgoverned by the Holzapfel-Gasser-Ogden (HGO) anisotropic hyper-elastic
continuum model.

Data availability
The data in support of the ﬁndings and detailing the studies are available from thecorresponding author upon request.

Received: 12 March 2019 Accepted: 27 June 2019

Methods
Materials. To create an elastomer matrix for the lamina, elastomeric platinumcure resin (Ecoﬂex 00–30, Smooth-On) was mixed in a 1:1 ratio by weight. The
mixture was subjected to 2000 rpm for 1 min in a planetary mixer (Thinky ARE310) and degassed at 2200 rpm for an additional minute. The pneumatic cylinders,planar actuators, and membrane actuators used in the demonstrations were alsofabricated from this resin, to create soft base bladders sufﬁciently compatible for
the STUAD-prepreg to remain adhered. Polyester continuous ﬁbers (100% SpunThread, PRC) were used as received. The resin for the self-adhesive backing of thelamina was created by mixing a 1:1 weight ratio of silicone adhesive constituent(Silbione 4717, Elkem Silicones). Tabs for uniaxial tension tests were made frommuslin fabric (Product #: 8808K11, McMaster Carr) infused with a platinum-cureresin (Dragon Skin 10, SmoothOn).

References
1.

2.
3.
4.

Prepreg fabrication. We built a ﬁber-winding machine to fabricate STAUDprepreg with tunable properties in a scalable fashion. The ﬁber winder consistsof a spool holster, a linearly-translating ﬁber distributor, a rotating winding
mandrel, and controlled via an Arduino with an attached Adafruit V2 motor
shield. To create bulk lamina, we entered the desired ﬁber spacing into theArduino code. We then secured the polyester ﬁber spool into its holster adjacentto the ﬁber-winding machine. Next, the machine automatically wound polyesterﬁbers onto the rotating drum. Subsequently, the elastomer resin was poured onto
the ﬁbers and smoothed with a resin scraper to disperse it evenly. We allowed
the drum to keep rotating for 12 h to allow the resin to disperse evenly as it
cured. A self-adhesive layer was then created on the back of the fully-cured
prepreg by applying silicone adhesive that cures at room condition for 12 h. This
self-adhesive layer was previously reported to maintain its adhesive force up to
20 attachment/detachment cycles, depending on the preparation of target surfaces39. We note that sufﬁcient areal coverage between the adhesive layer and an
inﬂatable body is required for robust attachment. For example, in Fig. 3b theadhesion of the STUAD-prepreg to a cylindrical actuator to induce bending
motion was reliable with areal coverage >16%.
Augmented classical laminate theory. We constructed a simpliﬁed compliancematrix in the fashion that considers both geometric non-linearity and material
non-linearity of ﬁber-reinforced elastomers (Supplementary Eq. 12). Furthermore,in line with the original classical laminate theory, we account for the angle of ﬁbersof each lamina in a laminate by subjecting them to a transformation matrix(Supplementary Eq. 13). Lastly, we consider the effect of cyclic strain relaxation of
STAUD-prepreg via a ratio of regression equations gathered from cycled and uncycled STAUD-prepreg stress-strain curves. The ﬁnal formulation of ACLT is
written as:
n Z zk
X
k¼1

zk1

"
½ekx;y dz ¼ Γc

n
X
k¼1

1
½Skx;y#1n Z zk
X
k¼1

zk1

5.
6.
7.
8.

9.
10.
11.
12.

13.
14.
15.
16.
17.

½σkx;y dz

ð4ÞHere, e and σ are Eulerian terms referring to the current principal material
coordinates, Γc is a cycle-depending scaling factor to adjust for strain relaxation,
S is a simpliﬁed compliance matrix which relates in-plane stresses to in-planestrains for laminae, all for an n layered-laminate, where each layer k has thickness

18.
19.
20.

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8

Acknowledgements
We thank Dr. Michelle C. Yuen for assisting with demonstrations. This work was supported by a Small Business Technology Transfer grant (No. 80NSSC17C0030) from the
National Aeronautics and Space Administration and an Ofﬁce of Naval Research Young
Investigator award (No. N00014–17–1–2604).

Author contributions
S.Y.K. and R.K.B. conceived the idea. S.Y.K. and R.B. performed the manufacturing. R.B.
and N.V. performed the material characterization and modeling. S.Y.K., R.B. and J.B.
performed the demonstrations. K.B. and R.K.B. reviewed and commented on the
manuscript. All authors participated in writing the manuscript.

Additional information
Supplementary Information accompanies this paper at https://doi.org/10.1038/s41467019-11294-7.
Competing interests: The authors declare no competing interests.
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