Composites Science and Technology 191 (2020) 108060

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Composites Science and Technology

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Origami-based deployable structures made of carbon fiber reinforcedpolymer compositesAntonio Alessandro Deleo a,1, James O’Neil a,1, Hiromi Yasuda a,b, Marco Salviato a,∗,Jinkyu Yang a,∗

a Department of Aeronautics and Astronautics, University of Washington, Seattle, WA 98195, USAb Department of Mechanical Engineering and Applied Mechanics, University of Pennsylvania, Philadelphia, PA 19104, USA

A R T I C L E I N F O

Keywords:Composite origamiA. polymer-matrix composites (PMCs)A. Flexible composites

A B S T R A C T

Deployable structures are typically made of thin membranes and slender elements, which often require foldable– yet stiff – mechanical properties. The use of carbon fiber reinforced polymer (CFRP) composites for suchdeployable structures has been limited due to the their rigid and unfoldable nature in general. Here, wedesign, fabricate, and demonstrate foldable – yet stiff – structures made of CFRP composites. To achievethis, we leverage origami design principles based on the Tachi-Miura-Polyhedron (TMP) architecture. Tomanufacture TMP structures, we devise a unique vacuum-bag-only composite fabrication method by usingcompliant urethane epoxies impregnated into woven glass fiber layers on which pre-made CFRP tiles arepositioned. We show the resulting structures feature self-deployability, high compactness, and deterministicforce–displacement characterization. Potential applications of the proposed composite origami are abundant,including deployable habitats for space exploration and disaster relief, deployable solar arrays and antennas,actively-controlled aerodynamic surfaces, and impact mitigation structures.

1. Introduction

Deployable structures are capable of modifying their footprint and/or volume in a controlled manner. They are highly demanded fornumerous engineering applications, such as solar arrays and sails inspace [1–3], disaster relief structures [4], medical devices [5], bio-inspired mechanisms, and architectures [6–8]. Often times, these de-ployable structures require features such as light weight, structural stiff-ness/strength, and multi-functional properties. Particularly, designingfoldable (during shape transition) – yet stiff (before/after deployment)– structures represents a particularly important milestone for their en-gineering applications. However, realizing these seemingly conflictingcharacteristics can be a challenging task in conventional deployablearchitectures.

To address this challenge, origami has been employed as a possi-ble design solution. Originally considered as an ancient art of paperfolding, origami has widened its engineering applications significantlyin recent decades. Examples include robotics [9,10], sun-shields fortelescopes [11], and architectures [12]. Particularly, for the purposeof designing deployable structures, origami has been adopted in solarsails [13], foldable antennas [14], and medical stents [15]. While it is

∗ Corresponding authors.E-mail addresses: salviato@aa.washington.edu (M. Salviato), jkyang@aa.washington.edu (J. Yang).

1 Equally contributing author.

theoretically possible to design simultaneously foldable and stiff struc-tures by using origami principles [16–18], their prototypes are mostlybased on non-structural materials (e.g., thin paper and plastic films),and their fabrication into a rugged form remains highly challenging todate. This is mainly due to the difficulty in implementing repeatablefolding lines, so-called crease lines, in the platform of rigid materials.Previous research has attempted to address this challenge in variousways. For example, Schenk et al. have demonstrated the fabricationof metallic Miura-Ori sheets using etched stainless steel sheets anda vacuum bag [19]. Similarly, prototypes of origami-based structureshave been fabricated using mechanical hinges [20], sandwich struc-tures [21,22], and 3D printers [23]. While each approach offers itsunique advantages and disadvantages, these methods tend to sufferfrom the lack of foldability, stiffness, lightweight, and/or compactnesswhen fabricated into deployable origami configurations.

In this manuscript, we propose a novel solution to this challengeby fabricating deployable origami structures made of carbon fiber rein-forced polymer (CFRP) composites. These materials are stiff and strong,and are relatively lightweight compared to metals. One outstandingproblem is the realization of crease lines in the CFRP architecture. We

https://doi.org/10.1016/j.compscitech.2020.108060Received 3 September 2019; Received in revised form 20 December 2019; Accepted 6 February 2020

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Fig. 1. Geometrical parameters of the TMP. (a) 3D view of a TMP unit cell, which shows its height, 𝐻 . (b) The geometry of Miura-Ori unit cells that make up a TMP structure.(c) Fold patterns of the front and back sheets that compose the TMP in (a) as well as the geometry that defines the TMP layer. Also present are the bonding areas needed toconsolidate both sides of the TMP unit cell. (d) The geometric parameters needed to derive the force–displacement relationship.

address this problem by utilizing glass fibers infused with a urethane-based resin to act as compliant hinges, while stiff CFRP tiles serveas the facets of our origami structures. This technique offers a rel-atively simple method to manufacture composite origami, since weonly require a hot press and vacuum-bagging materials to constructour specimens. Although there has been much research on flexiblecomposites for many applications, e.g., deployable structures [24–29],what separates our work from previous flexible composite approachesis that we can fabricate foldable-yet-stiff deployable composite origamistructures in a strictly planar surface and we do not require the use ofcuts or perforations in the creases. Also, unlike conventional bendablecomposites, our fabrication method offers a sharp-cornered bendingmechanism and a consistent folding repeatability, which is ideal fororigami-based deployable architectures.

As a proof of concept for our manufacturing process, we constructthe Tachi-Miura Polyhedron (TMP) [30–33]. This structure is rigid fold-able, implying that deformation occurs only along the creases duringfolding while the origami facets remain flat. TMP is also a volumetricorigami, capable of changing its shape from one flat stage to anotherflat stage. We will show that when the origami is fabricated usingthe proposed manufacturing process, it is self-deployable by leveragingthe elastic potential energy stored in their folds. This can serve asthe physical mechanism for controllable deployment. The verificationwill include a compression test as well as a study on the geometricaldeformations and elastic responses of the structure.

The rest of the manuscript is composed as follows: the theoreticalbackground of the TMP will be summarized in Section 2. An overviewof our manufacturing process is given in Section 3. Experimental pro-cedures for force–displacement characterization and deployability aredescribed in Section 4. We discuss our experimental results in Section 5.Possible applications for our TMP specimens detailed in Section 6.Finally, we give some concluding remarks in Section 7.

2. Kinematics of Tachi-Miura polyhedron origami

Here we provide a brief overview of the kinematics of the TMP [31–33]. Fig. 1 illustrates the geometry of the TMP. Fig. 1a shows the height,𝐻 , of a TMP unit cell, defined as the distance between the top and

bottom cross-sections of the TMP. A TMP architecture can contain anynumber of layers. An example is provided in Fig. 1a which shows a TMPcomposed of two layers. Fig. 1b represents the Miura-Ori unit cell [30]– corresponding to the shaded region of the TMP in Fig. 1a – whichcomposes a quarter of the TMP structure. As the TMP is a derivationof the Miura-Ori fold pattern, it can be idealized as a rigid-foldableorigami structure. In Fig. 1b, the major crease line fold angle 𝜃𝑀 andthe minor crease line fold angle 𝜃𝑆 can be related by the angle 𝜃𝐺 andthe panel tilt angle 𝛼 by means of the following equations:

tan(𝜃𝐺) = tan(𝛼) sin(𝜃𝑀 ), (1)

sin(𝜃𝐺) = sin(𝛼) sin(𝜃𝑆 ). (2)

Fig. 1c shows the layer height, 𝑑, and the lengths 𝑚 and 𝑙. 𝑙 is definedas the distance between the center of the TMP layer to the midpointof the nearest diagonal line segment labeled as ‘‘c’’ in the figure. Thisfigure also illustrates the directionality of the mountain and valley foldsin order to generate TMP structures. The shaded region of the ‘‘Front’’side corresponds to the shaded region of Fig. 1a. The areas patternedwith horizontal lines represent the regions where adhesive needs to beapplied in order to bond two sides of the TMP unit cell together.

Thanks to the foregoing equations and definitions, we only requirethe four parameters (𝛼, 𝑑, 𝑚, and 𝑙) to describe the geometrical config-uration of the TMP layer. Likewise, we only need 𝜃𝑀 in order to fullydefine the posture of the TMP. Accordingly, the TMP origami can betreated essentially as a one degree of freedom structure with the heightof the TMP (𝐻) and the major crease angle (𝜃𝑀 ) being related by thefollowing equation:

cos(𝜃𝑀 ) = 𝐻𝑁𝑑

, (3)

where 𝑁 is the number of layers that make up the TMP structure(Fig. 1d).

Now let us consider the folding behavior of the TMP architecture.From Fig. 1d, we define 𝐻0 to be the initial height of the structure(i.e., 𝐻0 is the zero-energy height when the TMP structure is underno compression or tension). Upon the application of force 𝐹 and thecorresponding compressive displacement 𝑢, the height of the TMP can

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Fig. 2. (a) Variation of volume of the TMP as a function of the major crease-line fold angle. (b) Normalized force–displacement ratio for a TMP with 𝐾𝑆∕𝐾𝑀 = 1 and 𝜃𝑀0 = 0.01◦.The inset of (b) provides more information on the peak of the normalized force–displacement curve. Both plots correspond to a TMP with the following geometry: 𝑁 = 3, 𝛼 = 45◦,and 𝑑 = 𝑙 = 𝑚 = 40 mm.

be expressed as:

𝐻 = 𝐻0 − 𝑢. (4)

Given 𝜃𝑀 , the cross-sectional area of the TMP does not change alongthe 𝑍-axis shown in Fig. 1. Thus, the volume of a TMP configurationcan be calculated as follows:

𝑉 = 𝐴𝐻, (5)

where 𝐴 is the cross-sectional area of the TMP.Fig. 2a is a plot of the volume of a TMP structure (with configuration

𝑁 = 3, 𝛼 = 45◦, and 𝑑 = 𝑙 = 𝑚 = 40 mm) as the major fold angleof the TMP (𝜃𝑀 ) is varied. The volume is normalized by maximumvolume of the structure. In the case of the configuration in Fig. 2a,the maximum volume is achieved at a major fold angle of 30.36◦. It isworth mentioning that this plot is purely theoretical and is based ona zero-thickness assumption. Due to this assumption, the TMP has novolume when it is flat-folded. It is interesting to note that the TMP canexhibit a near-linear change of its volume from either flat configuration.This smooth volume change with the major fold angle makes the TMPorigami architecture particularly attractive for applications requiringan accurate and highly-controllable deployment.

The use of TMP origami as deployable structures requires the un-derstanding of the relation between the force (𝐹 ) required to fold thestructure and the displacement (𝑢) of the structure. This expressioncan be obtained in closed-form assuming that (i) the thickness of thecomposite facets can be neglected, (ii) the facets contribute to thestructure kinematics by rigid motions only (negligible deformation),and (iii) the behavior of the creases is elastic and can be captured byequivalent torsional springs. Then, using the principle of virtual work,the following expression can be derived [32]:

𝐹 = 32𝑑 sin(𝜃𝑀 )

[

𝐾𝑀𝑁 − 1𝑁

(𝜃𝑀 − 𝜃𝑀0) +𝐾𝑆cos3(𝜃𝐺) cos(𝜃𝑀 )cos(𝛼) cos(𝜃𝑆 )

(𝜃𝑆 − 𝜃𝑆0)]

,

(6)

where 𝜃𝑀0 and 𝜃𝑆0 are the major and minor crease line fold angles inthe initial configuration of the TMP, 𝜃𝑀 and 𝜃𝑆 are the major and minorcrease line fold angles in the deformed configuration, and 𝐾𝑀 and 𝐾𝑆are the major and minor crease torsional spring constants respectively.Please revisit Fig. 1b for the distinction between major and minorcrease lines that correspond to horizontal and slanted creases in Fig. 1c.

Fig. 2b shows the evolution of the normalized folding force, 𝐹𝑑∕𝐾𝑀 ,as a function of the normalized displacement, 𝑢∕𝐻0. It is interestingto note that, for this particular configuration, the normalized forceincreases steeply but smoothly with the normalized displacement until

it reaches a local maximum for 𝑢∕𝐻0 ≈ 0.0011. This is illustrated in theinset of Fig. 2b. Then, the structure exhibits a relatively mild softeningup to 𝑢∕𝐻0 ≈ 0.55 followed by hardening until the TMP is completelyfolded.

As deformation occurs only along the creases and the creases aretreated as torsional springs, we can calculate the potential energy, 𝑈 ,of the structure by tracking the deformation of the major and minorcreases:

𝑈 = 12𝑁𝑀𝐾𝑀 (𝜃𝑀 − 𝜃𝑀0)2 +

12𝑁𝑆𝐾𝑆 (𝜃𝑆 − 𝜃𝑆0)2. (7)

𝑁𝑀 and 𝑁𝑆 are the number of major creases and the number of minorcreases respectively. Careful bookkeeping allows us to calculate thesevalues as follows [32]:

𝑁𝑀 = 8(𝑁 − 1), (8)

𝑁𝑆 = 8𝑁. (9)

This energy consideration will be revisited to explain self-deployabilityof the TMP in a later section.

3. Manufacturing

Now we move to describing the fabrication of composite TMPprototypes by leveraging a novel manufacturing technology combiningVacuum Bag Only (VBO) and the use of flexible urethane-matrix com-posites for the creases. The VBO technique not only guarantees highmechanical performance and limited volume fraction of voids [34,35],it is also cost-effective since it avoids the need for expensive autoclaveswhich increase the installation costs and reduce the overall productiv-ity [36]. At the same time, this method is also very flexible since severaldifferent curing cycles can be obtained, and therefore various types offibers can be utilized. Depending on the resin, the specimens can becured for longer periods of time at ambient temperature, or for muchshorter ones, using a ventilated oven.

As shown in Fig. 3, which provides a schematic of the manufac-turing process, VBO involves the use of pre-impregnated plies insteadof dry fibers. In contrast to Vacuum Assisted Resin Transfer Molding(VARTM), the vacuum is used only to supply pressure and not to allowthe resin to flow through the fibers. The specimen to be cured needsto be bagged using a hard surface, generally a metal plate or a garoliteplate whose working area is delimited by a yellow bagging tape. Inorder not to let the resin stick permanently after curing to the hardsurface, either a chemical release agent, a mechanical release agentsuch as a peel-ply or Teflon sheet or a combination of the two is used.Subsequently, the TMP origami is laid down and another layer of peel-ply or Teflon is deposited on top. Finally, the vacuum fitting is added in

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Fig. 3. (Top Left) Layup schematic and close-up example of a TMP crest. The foldable hinge is obtained by having two rigid CFRP facets traditionally cured squeezing a singledry layer of fiberglass impregnated with a very flexible urethane epoxy Shore 30A. (Top Right) Vacuum Bag Only (VBO) schematic. The whole manufacturing process starts with aclean metal (or garolite) plate set up with bagging tape and vacuum ports. A layer of Teflon is added at the bottom to allow for quick and safe release of the specimen made outof CFRP and GFRP with urethane. The sandwich TMP structure is covered again with another layer of Teflon and covered with breather and vacuum rated plastic bag. (Bottom)Step-by-step procedure. (a) First layer of CFRP facets is laid down following TMP shape. (b) The bagging tape and vacuum port are prepared. (c) The middle fiberglass layer isdeposited, covered by the top CFRP facets and the urethane resin is infused. The bag is sealed and the air is pulled out of the system. (d) Final specimen after de-bagging andpost-curing. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

a region close to the TMP on top of a double layer of breather, and thewhole plate is bagged using a vacuum-rated plastic bag. A modular 4Svacuum system by Vacmobiles [37] has been used for all the prototypesmanufactured. The system also provides an absolute pressure gaugeand a VPE ultrasonic leak detector which helped to further improvethe manufacturing quality of the specimens.

The TMP origami is manufactured by using Carbon Toray T700plain weave for the facets which are cured using the traditional hot-press method. The curing cycle is one hour ramp to 149 ◦C (300 ◦F)and two hours soak with a constant pressure of 689 kPa (100 psi).The cured facets had the following mechanical properties: 𝐸1 = 𝐸2 =85 GPa, a shear modulus of 𝐺12 = 5 GPa, and a Poisson’s ratio of 𝜈12 =0.1 [38]. The fiber volume fraction of the facets was around 0.68 as permanufacturing specification sheet. For the crests, we use a Fiberglassorca cloth plain weave provided by Fiberlay Inc. [39]. To guarantee thedesired flexibility and elasticity of the crests, a Sharkthane Flex Pro 30–20 urethane epoxy was used [40]. The system of glass fiber reinforcedpolymer (GFRP) and urethane epoxy yielded the following mechanicalproperties: 𝐸1 = 𝐸2 = 1.93 GPa, a shear modulus of 𝐺12 = 3.40 MPa,and a Poisson’s ratio of 𝜈12 = 0.21 with an observed fiber volume frac-tion between 0.5 and 0.6 [41]. The urethane resin was cured at roomtemperature and no experiments nor tests were performed at highertemperature where the glass transition temperature property mighthave played a key role in mechanical behavior. Future applicationsand studies might involve the analyses of such temperature-dependentparameters.

Fig. 3a–d shows the main steps of the manufacturing process. Ini-tially, one side of the CFRP facets is laid down using wood and acryliclaser-cut stencils to assure their proper location (Fig. 3a). As it will bediscussed in the results section, proper alignment of the facets is critical

for producing optimal specimens as a misalignment can produce non-uniform bending stiffness along creases. After this step, the urethaneepoxy is mixed using the manufacturer recommended ratio and a thinlayer is initially deposited onto the inner side of the CFRP. Once it isspread uniformly, a single layer of dry satin weave glass fiber fromFiberlay [39] is laid down and another layer of urethane epoxy ispoured to be absorbed by the fibers. Once the fibers are properly wet,the final side of the CFRP facets is laid down on top of the ones alreadydeposited as shown in Fig. 3c. The whole laminate is then properlybagged and leak-tested before curing. Once the TMP is cured, it isproperly checked for air pockets since the viscosity of the urethaneepoxy is much higher than a classic conventional epoxy. The finalcomposite origami structure consists of two (top and bottom) pre-curedCFRP facets sandwiched together with the urethane-impregnated GFRPmid-layer. This makes it a 3-ply structure. The GFRP is exposed only atthe location of the crests.

A typical example of a composite TMP laminate after curing and de-bagging is shown in Fig. 3d. Any additional glass fiber not sandwichedbetween the facets is discarded from the laminate. We can now eitherchoose to cut through the laminate along major crease-lines in order tocreate unit cells with a smaller number of layers or we can keep all ofthe layers and manufacture a new laminate with the same number oflayers. Either way, we bond two fabricated sheets (i.e., front and backparts as described in Fig. 1c) using high-strength 3M VHBTM AdhesiveTransfer Tape applied along bonding regions also seen in the figure.This tape is packaged so that there is a backing on one side. We applyhigh pressure using a hot press (heat is turned off) onto the laminatesto ensure that the tape is distributed evenly when the backing of thetape is peeled off. Finally, both sides of the TMP are lined up andconsolidated together at the bonding regions to form a unit cell.

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Fig. 4. Experimental apparatus utilized for characterizing the force–displacement response of the composite TMP specimens.

Fig. 5. Deployment apparatus for the TMP. (a) Overview of method for securing the TMP in the initial/stored configuration using nylon string. (b) A nylon string is used inconjunction with bolts to hold TMP at a fixed 𝜃𝑀 .

4. Methodology

4.1. Force–displacement characterization

In this paper, we characterize the force–displacement behavior of aTMP unit cell manufactured using our process described in the previoussection. We run compression tests on the specimen and gather force anddisplacement data. The experimental apparatus for these compressiontests is described in Fig. 4. A Kyowa LUX-B-50N load cell (max force50N) is attached to a load frame to collect the force data. Since thecompliance of the TMP structure is significantly larger than the one ofthe load-frame, the cross-head displacement of the machine is used forthe experimental characterization of the displacement of the specimen.

In this study we choose the TMP configuration of 𝛼 = 45◦ and𝑑 = 𝑙 = 𝑚 = 40 mm. We construct the prototype to have three layers(𝑁 = 3). The TMP is compressed by an acrylic plate that is coatedwith Teflon tape while it rests on another Teflon-coated acrylic plate.The edges of the TMP that make contact with these plates also haveTeflon coatings. The reason for using Teflon is to reduce the effects offrictional forces in the system. In our experiments, the composite TMPspecimen is pre-compressed about 10 mm from its flat-unfolded statebefore the compression tests begin. This is because the TMP specimencould not stand by itself in the test frame when they are in the flat-unfolded state. This is also to facilitate the volumetric folding motionsof the TMP specimens without planar buckling that can happen in itsflat-folded state.

4.2. Deployment

To verify the deployability of the composite TMP, an apparatus isprepared that compresses and secures the TMP in place with nylonstrings (Fig. 5a). The TMP is secured at a major crease-line angle of𝜃𝑀 = 78◦. When the TMP is compressed, there is elastic potentialenergy stored in the creases that will allow the TMP to be self-deployed.Therefore, to keep the TMP in the folded state, we use a nylon stringthat is tied around the TMP to secure it as shown in Fig. 5a. Additionalnylon string is used in the front and back of the TMP in order to limitnon-axial movement. This includes rotation along the axial direction ofthe TMP and any translational shifting along the plate as well as anyloss of contact between the bottom of the TMP and the acrylic platedue to the elastic nature of the creases and the TMP’s relatively lowmass. To reduce the friction during deployment, the acrylic plate theTMP rests on is coated with Teflon tape.

For the controlled deployment motions, we also need to set not onlythe initial folded, but also the final unfolded posture of the TMP. Asshown in Fig. 5b, the nylon string is used again inside the TMP speci-men to hold it at a given 𝜃𝑀 . We install a constant-length nylon stringthat is held in place by bolts installed in the center of the trapezoidalfacets. For our deployment tests, we choose the final unfolded angle of𝜃𝑀 = 30.5◦. This angle is close to the one (30.36◦) that results in theTMP’s maximum volume state based on Fig. 2a.

To deploy the specimen, we cut the external nylon string manuallyusing wire cutters. Potentially, an electrical system, e.g., nichrome-wire deployment system used for CubeSats [42], could be utilized fora controllable deployment. To monitor the deployment motions of the

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Fig. 6. Force–displacement results for the composite TMP specimen in Fig. 4 and detailed in Section 4.1. The blue curve and the colored area represents the mean of the experimentsand the standard deviation, respectively, of the experimental trials. The red curve is fitted to this data based on the theoretical force–displacement relationship outlined in Section 2.The quality of the curve fit as well as the spring constants and initial fold angle are given in the figure. (For interpretation of the references to color in this figure legend, thereader is referred to the web version of this article.)

system, we use a high speed camera (Chronos 1.4 high-speed camera)and capture video at 1500 frames per second.

5. Results and discussion

5.1. Force-displacement characterization

Fig. 6 shows the force–displacement results of the composite TMPspecimen. This specimen is compressed five times, and the averagedempirical data is represented by the blue curve. The colored areaaround the blue curve represents the standard deviation of the ex-periments. The experimental data obtained shows a decrease in forceas the displacement is increased (i.e., negative stiffness). As we fur-ther increase the compressive displacement, we witness a nearly zerostiffness. This trend implies that the folding motions of the TMP canbe controlled in a smooth manner without necessitating kinematicallysingular mechanisms.

Now we compare the empirical data with the analytical modeldiscussed in Section 2. To this end, we first need to identify the springconstants of our creases (i.e., 𝐾𝑀 and 𝐾𝑆 for major and minor creaselines) by fitting the theoretical force–displacement relationship given inEq. (6) with the measured force–displacement data. Additionally, 𝜃𝑀0is also used as a fitting parameter since the specimen is nearly flat-unfolded when no compression is applied and, depending on how onehandles the specimen, it is difficult to measure what the exact initialmajor crease-line fold angle is. We used photographs of the side ofthe TMP in order to get an estimate of the angle for our curve fittinganalysis. We must note that we cannot fit the data with 𝜃𝑀0 = 0◦

because the force reaches singularity when this occurs (see Eq. (6)).Based on the curve-fitting, the torsional spring values are identified

to be 𝐾𝑀 = 3.10 Nmmrad and 𝐾𝑆 = 2.91 Nmm

rad . The spring constant ofthe major creases being about 10% larger than the spring constant ofthe minor creases can be explained by the different fiber orientationrelative to the crease line. In fact, the major crease lines have fibersoriented perpendicular to the edges of the facets whereas the minorcreases have fibers oriented at 45◦ in both directions (see Fig. 1c wheremajor (minor) creases are shown in horizontal (slanted) orientations).The spring constants can be modified by changing the fiber directionand/or by selecting a different urethane or hyperelastic material forthe matrix of the creases. For the initial folding angle, we obtain 𝜃𝑀 =0.01◦ as the best fit value. This is plausible since the zero-energy stateof our prototype is a nearly flat state.

The analytical curve based on the aforementioned parameters isshown in Fig. 6. We observe satisfactory agreement between the analyt-ical (red curve) and empirical (blue) results (𝑅2 of 0.960). Thus, we findour rigid origami model from Section 2 captures the kinematic motionsof the TMP successfully. The minor discrepancy between the analyticaland experimental results could be due to manufacturing defects andany misalignments in the facets, and the finite-thickness effect of theTMP prototype. Also, we see that at the beginning of compression, thevariance of the force is greater than at the end of compression. Thisbehavior is a direct result of the TMP having to initially overcome thefriction and any minor instability at the beginning of the experiment.

We also point out that our experiments could not capture the criticalpeak force as predicted by the analytical curve. This is because weneeded to impose about 10 mm pre-compression to the TMP for thereasons described in Section 4.1. To capture this critical point, themanufacturing process needs to be improved to address any issues ofinstability and also to reduce friction.

Interestingly, based on the force–displacement plot from Fig. 6,although it will require about 6 N of force to begin folding the TMP unitcell from its flat-unfolded state, we do not have to apply this same forceto keep the prototype folded. The prototype needs less than 5 N in orderto stay folded. Additionally, the zero-stiffness near the initial storedconfiguration (max compression) is useful for deployment purposes asthe effort needed to compress the specimen to storage will decreasewith displacement.

5.2. Deployment

The results of deployment tests are shown in Fig. 7a. As can be seenin this figure, the TMP prototype exhibits smooth deployment from itsinitial folded configuration due to the robust and repeatedly foldablecreases. When released from compression, the TMP is relatively quickto expand to its minimum 𝜃𝑀 configuration. As can be seen, it takes thespecimen around 13 ms to deploy to this configuration. The changein height of the specimen during deployment is around 78 mm. Thismeans specimen deployed to its maximum volume at a speed of about6 m/s assuming the speed is constant. Since the initial configurationhas a 𝜃𝑀 = 78◦, based on Fig. 2a, the normalized volume is 0.233. Ifthe max volume configuration is assumed to have a normalized volumeof 1 for simplicity, this means the change in volume is around 300%compared to the initial configuration.

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Fig. 7. (a) A time-lapse of the deployment of the composite TMP specimen as it is released from its initial/stored configuration. As shown, this particular prototype can besmoothly deployed in approximately 13 ms. (b) The energy landscape of a TMP without finite creases with the same spring constants. The lines are placed such that the 𝑢∕ℎ0 ofa TMP without finite creases is the same as the manufactured TMP during deployment.

Fig. 7b depicts the potential energy of a TMP that does not havefinite creases like the manufactured prototype but has the configuration𝛼 = 45◦, 𝑑 = 𝑙 = 𝑚 = 40 mm, and 𝑁 = 3. The experimentallydetermined spring constants from Section 4 were used to generatethe plot. In the figure, the initial and deployed configurations aremarked. The normalized displacement for these configurations are thesame for both the equivalent TMP without finite creases and for themanufactured TMP during deployment. From the initial configurationto the deployed configuration, there is a significant release of energy(71% drop in energy). This deployment can be controlled if the TMP isconstrained as was ours.

6. Applications

Since we can configure the geometry of the deployable TMP struc-tures and control their deployment, these structures can be utilizedin a variety of engineering applications. One possible application forthis structure is that of a disaster relief shelter. A TMP structure canbe deployed to its maximum volume state or a state where its cross-sectional area is maximized in order to provide shelter to those indistress. In addition, previous studies on volumetric origami haveshown not only the reconfigurability but also the self-locking and/orload-bearing capabilities [43–45], which is advantageous for designingsturdy structures. The TMP structure for this type of application mightneed a covering, and for this, relatively simple solutions such as tarpscould be employed [46]. More recent work by the authors also exploredthe possibility to have active aerodynamics surfaces using as wellorigami-type structures made out of composites [47].

With a careful selection of geometric parameters, our TMP struc-tures could be designed to be slender for use as a deployable boom.

As a boom, radio antennas or other mission critical equipment such assolar panels for space missions could be mounted onto the TMP. In fact,if solar panels are to be mounted onto the TMP, it could be designedto be flatter and less slim to maximize the surface area. Should themission require adjustments in the orientation of the solar panels, theTMP could fold or unfold as needed to optimize the solar intensity andtherefore power being generated from the solar panels. Something tobe mindful of in these applications: As urethane is a sealant, any im-purities in the curing of the TMP could lead to undesirable outgassing.Special steps should be taken to ensure this will not effect the missionobjectives.

For some of these applications, we will need to consider the scalabil-ity of the proposed TMP structures using our manufacturing approach.The kinematic behavior described in Section 2 is of course scalable.Although we are confident that our manufacturing process could bealso scaled up for large TMP structures, we have to consider the effectsof the increase in mass on the system. Our current analytical modelfor the force–displacement relationship of TMP neglects gravitationaleffects. With some modification of the relationship or with numericalapproaches, we expect to predict the effect of gravity. Furthermore, wecan also tune the composite TMP structures by modifying the materialsused in their construction. If a less brittle material is required for anapplication, aramid fibers could be utilized instead of carbon fiber. Ifyou want a matrix that is not as adhesive as urethane, one can useother hyperelastic materials in our manufacturing process. It has beenshown that silicone rubber can also provide flexibility to compositelaminates [48].

7. Conclusions

The objective of this project was to verify that rigid origami struc-tures such as Tachi-Miura Polyhedron (TMP) could be manufactured

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using composite materials for a broad range of engineering applica-tions such as space, disaster-relief, medical, and even architecturaldeployable structures where a high strength-to-weight ratio is crucial.

The goal of manufacturing a complete TMP origami in compositeswas achieved by sandwiching CFRP facets and GFRP crests using avery flexible urethane epoxy. A Vacuum Bag Only (VBO) methodwas utilized due to the overall process being simpler, cheaper, andas reliable as conventional composite manufacturing methods such asVARTM and autoclave. A prototype manufactured using this methodwas shown to be deployable and the possibility to store compositeTMP structures and deploy them in different stages was confirmed. Theprototype was also shown to have a deterministic force–displacementbehavior predicted by a rigid origami model.

Future investigations in regards to TMP will focus on the tunabilityof the geometry of the structure as well as the fiber orientations alongits creases. Furthermore, other hyperelastic materials that can formthe matrix of the creases will be studied and possibly implemented toobserve any interesting behavior. The robustness of our manufacturingprocess will be verified in other foldable and deployable structures thatalso possess tunable properties.

Declaration of competing interest

The authors declare that they have no known competing finan-cial interests or personal relationships that could have appeared toinfluence the work reported in this paper.

CRediT authorship contribution statement

Antonio Alessandro Deleo: Investigation, Writing - review & edit-ing. James O’Neil: Investigation, Writing - review & editing. HiromiYasuda: Supervision, Writing - review & editing. Marco Salviato:Supervision, Writing - review & editing. Jinkyu Yang: Supervision,Writing - review & editing.

Acknowledgment

JO, HY, and JY acknowledge the financial support from the Wash-ington Research Foundation.

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