14.08.origami

Origami interleaved tube cellular materials

Kenneth C Cheung1, Tomohiro Tachi2, Sam Calisch3 and Koryo Miura4

1National Aeronautics and Space Administration, Ames Research Center, Moffett Field, CA 94035, USA2Department of General Systems Studies, The University of Tokyo, Komaba 3-8-1, Meugro-Ku, 153-8902Tokyo, Japan3Massachusetts Institute of Technology, Center for Bits and Atoms, Cambridge, MA 94035, USA4 Professor Emeritus, The University of Tokyo, Japan

E-mail: [email protected], [email protected], [email protected] [email protected]

Received 1 February 2014, revised 21 April 2014Accepted for publication 24 April 2014Published 11 August 2014

AbstractA novel origami cellular material based on a deployable cellular origami structure is described.The structure is bi-directionally at-foldable in two orthogonal (x and y) directions and isrelatively stiff in the third orthogonal (z) direction. While such mechanical orthotropicity is wellknown in cellular materials with extruded two dimensional geometry, the interleaved tubegeometry presented here consists of two orthogonal axes of interleaved tubes with highinterfacial surface area and relative volume that changes with fold-state. In addition, thefoldability still allows for fabrication by a at lamination process, similar to methods used forconventional expanded two dimensional cellular materials. This article presents the geometriccharacteristics of the structure together with corresponding kinematic and mechanical modeling,explaining the orthotropic elastic behavior of the structure with classical dimensional scalinganalysis.

S Online supplementary data available from stacks.iop.org/sms/23/094012/mmedia

Keywords: rigid origami, metametrial, cellular structure, dimensional scaling analysis

1. Introduction

Origami, the art of folding a piece of paper into a three-dimensional (3D) form and/or folding it into a at state, hasreceived much attention in science and engineering, receivingcredit for many innovative designs and applications. Twomajor approaches for applied origami in mechanical engi-neering are (a) to obtain stiff and lightweight structures thatare manufactured by folding at sheets, such as folded plateshell structures [1], sandwich panels with origami cores [2, 3],and cellular meta-materials [4], and (b) to obtain exibledeployable mechanisms such as microscale devices [5],mesoscale structures such as folded stents [6] and macroscalestructures such as deployable solar panels in space [7]. Priorwork has demonstrated the ability to ll space to produce any3D shape with strings composed of very simple foldingmechanisms [8], but much of the work on the extensiblestructural properties achieved through folding has addressedtwo dimensional foldings such as tesselations. Schenk andGuest [9] study changes in the material properties by foldingegg-box and Miura-ori tessellation patterns to suggest the

ability to engineer such properties based on folding patterns.Foldable cellular materials have also been proposed [4], basedon the symmetry of Miura-ori, to produce rigid origamideployment mechanisms that are at-foldable; since thestructure deploys to a mechanically singular congurationwith at hinges, the deployed structure is not expected totransform in all of the x, y, and z directions, once deployed.Since the stiffness of the former folded-sheet types of origamistructures is obtained when the boundary is closed to formcylinders or when each part is assembled and adhered to othersheets, the straightforward expectation is that stiffness andexibility are mutually exclusive properties of origami.

However, recent studies on rigid origami, i.e., plate-and-hinge mechanisms, allow us to design deployable and yet stiffstructures through the use of geometrically compatible rigidthick materials [10] or foldable cylindrical and compositestructures [11, 12]. The proposition of the cellular structurewith a rigid-folding mechanism [12] is based on the assemblyof rigidly foldable origami tubes that tesselate to ll space byperiodic and afne transformations. Since such a cellularstructure with plates tend to produce an overconstrained or

0964-1726/14/094012+10$33.00 2014 IOP Publishing Ltd1

Smart Materials and Structures

Smart Mater. Struct. 23 (2014) 094012 (10pp) doi:10.1088/0964-1726/23/9/094012

statically indenite system, the generic result is a rigid staticstructure. However, because of the symmetry of the assembly,the overall structure becomes a one degree-of-freedom (DOF)mechanism with desirable robustness and stiffness in its fullydeployed state.

In this paper, we propose an anisotropic meta-materialbased on cellular rigid origami [12]. The material is exibleand stiff at the same time, along orthogonal axes which wename for reference: exible in the x and y directions and stiffagainst compression in the z-direction (gure 1). The inter-leaved cellular structures are constructed from material withnon-negligible thickness, and can be considered to be com-posed of facets, much like traditional closed-cell materials.While they are strictly not closed-cell structures, because thecells are connected into rows of tubes, rigid congurations ofclosed faces can contribute signicant stiffness through theirelemental bending and shearing stiffness. We establish theconstitutive volume relationships between the two axes oftubes, and predict the elastic behavior of interleaved tubecellular materials through classical dimensional scaling ana-lysis; specically, we focus on the materials elastic modulusscaling with relative density. Further, a fabrication techniquebased on the rigid-foldable deployment mechanism of thestructure is proposed, whereby distinct repeating patterns ofthin sheets are laminated in a at state, then deployed toobtain the target 3D volumetric structure.

2. Geometry

2.1. The unit cell

The overall geometry and motion of the interleaved tubecellular structure can be represented by its fundamentalrhombic dodecahedron unit cell, which is composed of

rhombi with sector angles of = 2 arctan 70.530 12 . Wecall this module a ipop module, since this is an anisotropicscaled version of the kinematic origami known as the ip-op, with sector angles of 60 , by origami artist Thoki Yenn[13]. A ipop module is comprised of two tubular com-partments and exhibits continuous at foldability in the x andy directions. Interestingly, height in the z-direction is mini-mized when it is symmetric in x and y directions, which isalso when the volume is maximized (gure 2). This modulecan tessellate a 3D space even in folded or partially foldedstates, because the shearing open rhombic faces are alwaysplanar and also parallel to xz and yz planes. Specically, theconguration can be parameterized by the angles x and y ofthese open rhombic faces (parallel to xz and yz planes,respectively). These parameters are constrained because thesurrounding rhombic faces are lled and maintain rigidity.Normalized direction vectors of the incident edges of a

Smart Mater. Struct. 23 (2014) 094012 K C Cheung et al

2

Figure 1. Interleaved tube cellular structure manufactured by 3D printing, showing foldability in XY direction (top) and rigidity in z direction(bottom). Each of the bottom images shows two copies of the same structure, with the right-hand sample oriented so that the z axis is parallelto the basal plane, and the left-hand sample oriented so that the z axis is perpendicular to the basal plane. Initial conditions are shown in theleft-hand set of images, and identical loading conditions are shown in the right-hand set of images, displaying the difference in response forthe different orientations.

rhombus can be given as

= =

l l

cos2

0

sin2

and

0

cos2

sin2

. (1)x

x

x

y

y

y

Because of the rigidity of the facet, the dot product of thesevectors are constant

= =sin2

sin2

cos1

3, (2)x

y0

and the change in the dimensions are given as

= x 2 cos2, (3)x

= y 2 cos

2, (4)

y

= + z 2 sin2

2 sin2. (5)x

y

2.2. Volumetric properties

The volumes of cells Vx and Vy along x and y axes, respec-

tively, are

= = V x sin 4 cos2

cos2

sin2, (6)x y

x y y2 3

= = V y sin 4 cos

2cos

2sin

2. (7)y x

x y x2 3

The total volume of the dodecahedron is

+ = = +

V V xyz

1

24 cos

2cos

2sin

2sin

2. (8)x y

x y x y3

Therefore, the following congurations maximize the volume

+V Vx y is maximized to 3.079163 33 3 when

= ( ) ( )sin , sin ,2 2 13 13x y , thus = = 70.53x y 0 ,and = = x y 1.6332 2

3.

Vx is maximized to 1.7781693 3 when

= ( ) ( )sin , sin ,2 2 15 53x y thus 53.13x and 96.38y , and = x 1.78945 and

= y 1.33343

, when the other volume

= V 1.067y 16153 3.

V Vx y is maximized to 0.889893 3 when

= ++( )( )sin , sin ,2 2 4 73 14 7x y thus 45.65x and

118.5y , and x 1.843 and y 1.023 .

Rigid foldability of these interleaved tube structurescombined with this volume relationship theoretically allowsfor control of fold-state by differential pressurization of thetube volumes, or vice versa. By differentially pressurizingeach set of tubes, extension along each of the axes can beprescribed (gure 3) and by varying the total pressurizationthe overall stiffness can be tuned. To derive the generalgoverning equations we consider pressurizing each of the twosets of cylinders independently and consider the force exertedby the structure in the x direction. For each fundamentalipop module, conservation of energy gives:

+

=

PV

PV

Fx,x

x

xy

y

xx

x

where Px (resp. Py) is the pressure in the cells of the ipop

module in the x (resp. y) direction relative to external pres-sure, and Vx (resp. Vy) is the volume of the cells of the module

in the x (resp. y) direction. Fx is the force exerted by the entire

module in the x direction. Given that = x 2 cos2x ,

=

sinx

2x

x . To calculate the volume of one side of the

ipop module, we use the triple product to compute thevolume of a parallelepiped. Dening ly to be the image of ly


3

Figure 2. The folding motion of ipop. The module at-folds into x and y directions, while the height z is minimized when the volume ismaximized.

reected through the xy-plane, we have:

= ( )V l l l2 .x y x yGiven that

=

=

=

l

l

l

cos2, 0, sin

2,

0, cos2, sin

2,

0, cos2, sin

2,

xx x

yy y

yy y

for the full chamber (two parallelpipeds),

=V 4 sin2

cos2

cos2.x

y y x

Using the rhombus rigidity criterion,

= V 2 cos cos2,x

x

y0

we can similarly derive

=

=VV

4 sin2

cos2

cos2, 2 cos cos

2,y

x x y y

xx

y

which gives

=

=

( )

( )

F P P

P P

2 cos2

sin2

cos cos

cos cos cos cos .

xy x

y x x

y x y x

1

0

0 0

If we wish to prescribe a static unloaded conguration,we have =F 0x . Therefore, to extend to the congurationgiven to a particular value of x, we have the following

relation between the pressures:

=P P coscos

. (9)x yx

0

This gives a relationship between the two pressures (notethat to achieve the regular dodecahedron conguration where = =x y 0, we have the intuitive result: =P Px y), but if wewant to derive actual values relative to the external pressure,we must specify the desired stiffness of the actuator. Thisstiffness is given by the derivative:

=

=

+

)

( )(

( )

P

P P

sin2

cos cos sin

cos cos .

F

x

F x

xy y x

x y x

1

0

cos

0

x x

x x

y

x

In the case of maintaining a static conguration (where

= )F 0 ,x this stiffness simplies to

= Fx

P sin2

cos sin . (10)x yy

y x

Equations (9) and (10) dictate the relationship between thepressures, fold-state or conguration, and pressure inducedstiffness.

3. Mechanical model

In the deployed interleaved tube rhombic dodecahedralstructure, the fundamental repeating unit cell may be con-sidered to be the unit rhombic prism with opposing open facesand all faces conforming to the rhombus geometry of thecanonical rhombic dodecahedron, which is space-lling byafne transformation. An alternative repeating unit cell for the


4

Figure 3. Cellular structure actuated by differentially pressurizing two sets of tubes in x and y directions (colored red and green respectively).Left state maximizes V Vx y. Middle state maximizes +V Vx y. Right state maximizes V Vy x.

same structure is the rhombic dodecahedron with four openfaces and an internal web composed of four rhombic faces,which is space-lling by linear translation. Figure 4 showsanother valid repeating unit cell with twice the volume of asingle ip-op, bounded by a rectangular prism with facesparallel to the X, Y, and Z planes of the cellular structure. Theenclosed rhombic dodecahedron has four open faces and aninternal web composed of four rhombic faces, with fouradditional rhombic faces allowing for space-lling by lineartranslation on the three orthogonal axes. Considering thiswebbed rhombic dodecahedron as a composition of twocylinders, the additional rhombic faces that ll out therestangular prism connect the exterior faceted sides of onecylinder to that of its neighboring dodecahedral cell (along theone axis that is orthogonal to both cylinder axes), so as toconnect these edges orthogonal to the incident cylinder axis.For simplicity and clarity, we consider this version of this unitcell for the following dimensional scaling analyses. Theresults of the following analysis are insensitive to the chosenrepeating unit cell, provided that the chosen geometry satisesthe requirement of completely tiling space to form the con-tinuous interleaved tube geometry.

The continuous interleaved tube geometry consists ofnodes with edge connectivity ze of four and eight in equalproportion (the average edge connectivity ze is 6), and edgeswith face connectivity zf of two and four with a number of

two connected edges that is twice the number of four con-

nected edges (the average face connectivity zf is 2.783 ).Assuming high elemental slenderness (thickness length inboth cell edge and face), as is common in the literature, thedensity scaling relationship in classical form [14] can besimply derived from the geometry as

t6 ,

s

where is the mass density of the structure, sis the mass

density of the constituent solid material, t is cell edge or face

thickness, is cell edge length, and 6 2.4 is a geometricconstant determined by the deployed interleaved tube rhom-bic dodecahedral geometry.

Here, we may consider the geometry as if it were createdby removing the tubular volumes from a bulk solid (gure 4(middle and right)). The resulting solid is not strictly rigid-foldable [10], though elastic deformation at the connectingedges allows the structure to deform in a manner thatresembles rigid foldability.

If we denote the length of the edge as , the height vectorof the section rhombus is given by = d e2 sin 2 z0 . Thescaling factor of each rhombus section of the tube hole is

= s 1 s , since each tube hole is obtained by translatingthe deformed rhombus section along the original zigzaggedcurve. Then, the relation between face thickness and relativedensity

s, mentioned above, follows as the dot product of the

thickness along the height vector and the normal vector n of atop facet.

= =

( )t s d n1 23 1 1 . (11)

s

Conventional foams display a lower degree of edgeconnectivity and higher degree of face connectivity than thisdeployed interleaved tube rhombic dodecahedral structure.We know from the literature that high edge connectivity iscommonly associated with cell edge behavior that is domi-nated by material stretching instead of transverse beambending, and associated improvements in specic stiffnessscaling, over conventional foams [14, 15]. The decreased faceconnectivity is expected from the geometry, and may be seenas an indication of some reduced contribution from cell face/membrane stiffness, as well as absent contribution fromcompression of enclosed cell uid, due to the presence of auid path between cells, in this geometry. The small constant


5

Figure 4.A unit cell of the interleaved tube cellular structure. Left: the white nodes have four incident edges while the black nodes have eightincident edges. The thick lines indicate edges with four incident facets while thin lines indicate edges with two incident facets. Middle:tubular deletions from the theoretical solid, to obtain the structure. Right: thickened cellular structure.

geometric factor by which relative density scales (withcomponent slenderness t

l) is similar to conventional foams,

and we may consider the interleaved tube rhombic dodeca-hedral structure to be reasonably space efcient.

The classical closed cell foam specic modulus scalingrelationship is known as

+

+

( )

( )EE

pE

1 2

11 ,

s s s s

2

2

0

s

where E is the modulus of the foam, Es is the modulus of theconstituent solid material, is the fraction of solid material inthe cell faces (versus edges), and is the Poissons ratio of thefoam [14]. The three discrete governing phenomena that

contribute to this relationship are cell edge bending ( )22

s

[14], compression of enclosed cell uid

pE

0

1 2

1ss

[16], and

membrane stretching ( )1 s [17]. The cell edge deectionterm omits material stretch, since transverse bending-baseddeection will always dominate stretch-based deectionunder the same load. If there is enough constraint to limitedge behavior to material stretch, then this term is instead

proportional, as ( )2 s [14].For the case of the interleaved tube cellular structure, as

previously mentioned, the cellular chambers are not indivi-dually enclosed; however, since the possibility of pressur-ization and the differential pressure relationship to fold-statemay be important for future applications, we include the celluid compression term above. We leave future work onpressure dependency to future studies, and focus theremainder of this work on implementations of the materialthat are not pressure dependent.

Without the cell uid compression term, considering theopen interleaved tube cellular structure alone, we expect thespecic modulus scaling relationship to be

+

( )EE 1 ,s s s

2

2

if edge deection behavior is dominated by microstructuralbeam bending behavior or

+

( )EE 1 ,s s s

2

if edge deection behavior is dominated by material stretch.Regardless of this edge behavior, a minority of the volumefraction of constituent material is in the edges ( )1 , forthe folded interleaved tube cellular structure, so we mayexpect membrane stretch behavior to yield good specicmodulus performance, as

EE

.s s

4. Results

To test the above application of classical cellular solidsdimensional scaling analysis to the specic modulus scalingof the interleaved tube cellular structure, we performed nite-element simulations (ELMER software [18]) with nelymeshed rigid body models (gure 5). To simplify theimplementation of periodic boundary conditions, we chosethe repeating unit cell that we introduced in the analyticalmodel above and shown in gure 4.

The simulations predict the materials elastic modulus inaccordance with the analytical model. Fitting power law

curves to the data yields E 1.5 in the z axis for a unit cellmodel with periodic boundary conditions ( E 2.2 in the zaxis for a unit cell model without periodic boundary condi-tions), and E 2.4 in the x and y axes. Performance in theseregimes are traditionally explained as expected reductionsfrom analytically idealized scaling laws [15, 19]in thiscase, E in the z axis and E 2 in the x and y axes. Thisexplanation will be addressed further in the discussionsection, below.

To evaluate the simulations we performed a brief studyof manufactured samples of structures (by stereolithographic3D printing [20]) and tested their behavior using standardload testing equipment [21] and procedures. The solid con-stituent material is UV photocured acrylic, fabricated via astereolithography process that employed a support materialwith low melting temperature (wax). While the ability toaccess all surfaces to clear support material was important forthis particular process, it is notable that the interleaved tubecellular structure geometry has limited overhang angles for anumber of orientations, and we have found that it can there-fore be constructed using a variety of commercial layeringprocesses without support material (such as lament fusion/deposition based processes).

The test specimens were cubic blocks with a 75 milli-meter dimension per side, with unit cells approximately 12.5millimeters in dimension (across the long axis of a rhombicface), and xtured to face plates on the top and bottom loadedsurfaces with high shear strength epoxy (gure 6). It isaccepted in the literature that edge effects for cellular mate-rials are minimal beyond characteristic dimensions exceedingseveral unit cells [22]. We tested four sets of specimens, withmass densities varied by changing the thickness of the cell

faces and edges, ranging between 0.07 g cm 3 and 0.18 g cm 3, and corresponding to solid volume fractions ofsix, eight, twelve, and sixteen percent.

All specimens responded as elastic solids, as expected.Modulus data were calculated from the linear elastic regime,and it was found that the specic modulus scaling fellbetween canonical stretch and transverse beam bendingmodels ( E and E 2, respectively), with a best tpower law curve of E 1.7 in the Z direction, and E 2.8 inthe XY direction. These regimes are traditionally explained asa moderate reduction in performance from idealized Eand E 2 scaling, respectively, resulting from


6

manufacturing defects [15] or stochastic variations in geo-metry that are poorly accounted for by simple periodic models[19]. To explain this difference between the idealized modeland experimental results for this particular case, we considerlikely contributing phenomena to include microstructuraldefects and edge effects produced by the stereolithographyprocess (diffusion and mixing between modeling and supportpolymers), in addition to an adjustment to the idealizedmodelan alternate stretchbend coupled model that is pre-sented in the discussion, below.

5. Fabrication by lamination

The interleaved tube cellular structure folds to a at state withmultiple overlapping layers, and may be manufactured bysequential lamination of two dimensional patterns withappropriate edge bonding. Reversing the process of at-folding enables us to then produce the three dimensionalcellular structure based on the one-DOF deploymentmechanism from the laminated at state. The approach issimilar to the fabrication of a conventional 2.5D honeycombstructure. Since adjacent layers are connected at the edges forthe ideal geometric model, we rst modify the geometry to

have a nite width at the gluing paths, without losing itsfundamental mechanism. Figure 7 shows the folding motionof a modied structure that has nite double covered area.Here, each zig-zagged tube structure in one direction ischamfered to produce a thin quadrangle gluing facet thatreplaces the gluing path. Then, the at state is produced bylaminating four distinct repeating layer patterns shown ingure 8. The patterns shown are comprised of cutting andfolding lines and gluing (or sewing) paths to the next layer,and the process can be performed by manual or automatedroll-to-roll presses, or CNC controlled cutting and gluingmachines. Another efcient implementation of this structureis to use stereo-lithography or 3D printing technology to printthe structure in a nearly at state, noting again that it is trivialto orient the structure so that overhang angles are minimized,allowing for unsupported 3D manufacturing processes,as well.

6. Discussion

The typical 3D closed-cell foam structure has pseudo iso-tropic elastic modulus governed by transverse cell edgebending, resulting in nearly quadratic scaling with respect to


7

Figure 5. Interleaved tube simulated specic modulus scaling for XY direction (indicated by open square ) and Z direction (indicated bysolid dot ). Exaggerated deformations shown, to illustrate deformation mode.

Figure 6. 3D printed acrylic interleaved tube cellular material specic modulus scaling for XY direction (indicated by open square ) and Zdirection (indicated by solid dot ), experimental.

its relative density [14]. It is well known that the mechanicalproperties of such a cellular structure can be improved up toproportional relative elastic modulus, based on the degree ofgeometric constraint [14]many such examples of engi-neered cellular solids exist. From the standpoint of mechan-ical performance, the interleaved tube cellular materialspresented here are shown to approach the mechanical per-formance of the best known engineered orthotropic cellularmaterials. Some 2.5D honeycombs, such as typical hexagonalhoneycomb core materials, display similarly orthotropicbehavior with nearly proportional elastic modulus scalingwith density on one axis and signicantly reduced modulusscaling in the orthogonal plane [14].

The results here present the materials elastic modulus oforder 1 1.5 with respect to its relative density in its stifforientation (z axis) and elastic modulus of order 2 in theexible orientations (x and y axes).

Prior work has suggested that a bulk behavior model maybe derived from a combination of micro-structural componentstretch and moment induced component bending, resulting inobserved behavior that is dominated by the latter, larger

phenomenon [23]. Stretchbend coupled models have alsobeen applied to biomaterials with mechanical performancethat exceeds the predictions of traditional models, based onmicro-structural topology [24].

A dimensional scaling argument that is parallel to theclassical stretch and transverse component bending models[14], with a bending moment induced component deection

yields a characteristic modulus scaling of E 1.5. Consider amicrostructural element model with a length and thicknesst, made from a constituent solid material with modulus Es, anda bulk cellular material loading condition that results in aspecic load P on the component. For a cellular geometrywith a high degree of connectivity, we consider component

deection as resulting from both stretch ( PE tstretch s

2 [25]) and

moment induced bending ( ME Imoment s

2

[25]), with the

moment arm being proportional to the element thickness, asopposed to its length as with transverse component bendingbased deection. We expect then that the linear response is


8

Figure 7. Cellular structure with gluing tabs. The modication maintains the rigid-foldability of the cellular structure.

Figure 8. Four cutting-folding-gluing patterns that repeatedly appear from the bottom to top in the order: (a)(b)(c)(d)(a) . Note thatthe glue is placed on top of the sheet, so that the next layer is welded on the previous layer

dominated by the deection due to the bending moment

= +

( ) ( )Pt

E I

P

E t,

s sstretch moment

2 2

3

since I t4 [25], and given the conventional assumption thatrelative modulus is proportional to load and inversely pro-

portional to deection, length, and Es, i.e., E

E

P

lEs s[14], we

expect relative modulus to be proportional to the cube of the

component aspect ratio, i.e.,

E

E

t

s

3

3 . Taking the relative

density of lattice structures to be proportional to the square of

the component aspect ratio, i.e., t

s

2

2 [14], we nd that the

expected relative modulus of a cellular material with thismechanism will scale with the relative density by an exponent

of 32, i.e., ( )EE

1.5

s s[23].

Whether or not we accept a stretch dominated orstretchbend coupled model for the specic modulus scalingwith density, these interleaved tube cellular solids will likelynot be competitive against standard 2.5D honeycomb mate-rials, for identical applications. However, the ability to con-veniently access the interleaved enclosed tubular volumes forow or pressurization may provide interesting applicationopportunities.

Since the cylindrical chambers can be accessed from theexterior boundary of the structure, active pressurization canbe made to dominate the stiffness behavior of the bulk sys-tem, and to exceed what is possible through solid material thathas not been pre-stressed. Differential pressurization betweenthe two cylinder axes can also be used to drive actuation nearthe fully deployed state, and differential pressurizationbetween both cylinder axes and the external environment canbe used to drive overall deployment actuation. The ability toseparate the cell uid pressurization process from the struc-ture manufacturing process enables this functionality, as wellas the ability to make the cell uid compression term dom-inate the mechanical behavior of the bulk system.

Interleaved tube cellular structures may also act as apump to drive the two channels of uid. For instance, byactuating one of the spatial directions, uid could be drawninto the structure from both the x and y directions. Besidessimply pumping uids, this could be an effective way to builda heat exchanger due to the large shared surface area betweenthe sets of cylinders. The performance of a heat exchanger isusually measured by its capacity for thermal transmissionrelative to pressure required to achieve a desired ow rate.There is a precedent for the application of cellular materials tobuild heat exchangers [2628] but most of these are singleuid designs. The interleaved tube cellular structure affordscomplete employment of the constituent membrane materialas interfacial surface between the two ow axes. Further,movements of a cellular material have not been previouslyconsidered as a mechanisms for optimizing ow conditions.

In addition, a practical layer-by-layer fabrication methodof the structure is proposed. The principle is to construct thestructure in a at-folded state with multiple layers by

laminating sheets with specied shapes and gluing patterns.Then, the sheets are stretched or inated to produce a foldingmotion to deploy the 3D structure. This is potentiallyapplicable to roll-to-roll processes, stereo-lithography and 3Dprinting, or sheet lamination with CNC proling machines(cutting and gluing), so that we can apply the structure atdifferent scales, from microelectromechanical systems todeployable structures for architecture and aerospace.

Acknowledgments

The authors would like to acknowledge the support of theNASA Ames Research Center, the University of Tokyo, andthe MIT Center for Bits and Atoms. The rst authors con-tributions were supported by the NASA STMD CenterInnovation Fund. The second author is supported by the JSTPresto Program.

References

[1] Yoshimura Y 1951 On the Mechanism of Buckling of aCircular Cylindrical Shell under Axial Compression Thereports of the Institute of Science and Technology Universityof Tokyo NACA-TM-1390

[2] Miura K 1975 New Structural Form of Sandwich Core J. Aircr.12 473441

[3] Klett Y and Drechsler K 2011 Designing TechnicalTessellations ed P Wang-Iverston, R Lang and M YimOrigami5: Fifth Int. Conf. on Origami in ScienceMathematics and Education (5OSME) (Boca Raton, FL:CRC Press) pp 30522

[4] Schenk M and Guest S 2013 Geometry of Miura-foldedmetamaterials Proc. Natl Acad. Sci. 110 327681

[5] Bassik N, Stern G M and Gracias D H 2009 Microassemblybased on hands free origami with bidirectional curvatureAppl. Phys. Lett. 95 091901

[6] You Z and Kuribayashi K 2003 A Novel Origami Stent Proc. ofthe 2003 Summer Bioengineering Conf. pp 2578

[7] Tanizawa K and Miura K 1978 Large displacementcongurations of Bi-Axially compressed innite plate Trans.Japan Soc. Aeronaut. Sci. 20 17787

[8] Cheung K C, Demaine E D, Bachrach J R and Grifth S 2011Programmable assembly with universally foldable strings(moteins) IEEE Trans. Robot. 27 71829

[9] Schenk M and Guest S 2011 Origami folding: a structuralengineering approach ed P Wang-Iverston, R Lang andM Yim Origami5: Fifth Int. Conf. on Origami in Science,Mathematics and Education (5OSME) (Boca Raton, FL:CRC Press) pp 25363

[10] Tachi T 2011 Rigid-Foldable Thick Origami edP Wang-Iverston, R Lang and M Yim Origami5: Fifth Int.Conf. on Origami in Science Mathematics and Education(5OSME) (Boca Raton, FL: CRC Press) pp 25363

[11] Tachi T 2009 One-DOF cylindrical deployable structures withrigid quadrilateral panels Proc. of the IASS Symposium 2009pp 2295306

[12] Tachi T and Miura K 2012 Rigid-Foldable cylinders and cellsJ. Int. Assoc. Shell Spat. Struct. 53 21726

[13] Yenn T Flip op (http://erikdemaine.org/thok/ipop.html)[14] Gibson L and Ashby M 1999 Cellular Solids: Structure and

Properties. Cambridge Solid State Science Series


9

(Cambridge: Cambridge University Press) Available from:http://books.google.com/books?id=IySUr5sn4N8C.

[15] Deshpande V and Fleck N 2000 Isotropic constitutive modelsfor metallic foams J. Mech. Phys. Solids 48 125383

[16] Gent A and Thomas A 1963 Mechanics of foamed elasticmaterials Rubber Chem. Technol. 36 597610

[17] Patel M and Finnie I 1970 Structural features and mechanicalproperties of rigid cellular plastics J. Mater. 5 90932

[18] (CSC - IT Center for Science: Elmer (http://csc./english/pages/elmer)

[19] Roberts A and Garboczi E J 2002 Elastic properties of modelrandom three-dimensional open-cell solids J. Mech. Phys.Solids 50 3355

[20] 3d Systems InVision Si2. 3D Systems, (Rock Hill, SC 3dSystems InVision Si2) (http://3dsystems.com)

[21] Instron 4411. Instron, (Norwood, MA: Instron 4411) (http://instron.us)

[22] Andrews E, Gioux G, Onck P and Gibson L 2001 Size effectsin ductile cellular solids. Part II: experimental results Int. J.Mech. Sci. 43 70113

[23] Cheung K C and Gershenfeld N 2013 Reversibly assembledcellular composite materials Science 341 121921

[24] Broedersz C P, Mao X, Lubensky T C and MacKintosh F C2011 Criticality and isostaticity in bre networks Nat. Phys.7 9838

[25] Timoshenko S P and Gere J M 1961 Theory of Elastic Stability(New York: McGraw-Hill)

[26] Wang B and Cheng G 2006 Design of cellular structure foroptimum efciency of heat dissipation ed M Bendse,N Olhoff and O Sigmund IUTAM Symposium onTopological Design Optimization of Structures, Machinesand Materials (Solid Mechanics and Its Applications) vol137 (Netherlands: Springer) pp 10716

[27] Wadley H N and Queheillalt D T 2007 Thermal applications ofcellular lattice structures Materials science forum vol 539(Zurich: Trans Tech Publ) pp 2427

[28] Kumar V, Manogharan G and Cormier D R 2009 Design ofperiodic cellular structures for heat exchanger applicationsSolid Freeform Fabrication Symposium Symposium Proc.(Austin, TX: University of Texas) pp 73848


10

14.08.origami

Documents