Neville, R. M., Pirrera, A., & Scarpa, F. (2014). Open shape morphinghoneycombs through kirigami. In ASME 2014 Conference on SmartMaterials, Adaptive Structures and Intelligent Systems, SMASIS 2014 (Vol.1). [SMASIS2014-7489] Web Portal ASME (American Society ofMechanical Engineers).

Publisher's PDF, also known as Version of record

Link to publication record in Explore Bristol ResearchPDF-document

University of Bristol - Explore Bristol ResearchGeneral rights

This document is made available in accordance with publisher policies. Please cite only the publishedversion using the reference above. Full terms of use are available:http://www.bristol.ac.uk/pure/about/ebr-terms

OPEN SHAPE MORPHING HONEYCOMBS THROUGH KIRIGAMI

Robin M. NevilleUniversity of Bristol

Bristol, England

Alberto PirreraUniversity of Bristol

Bristol, England

Fabrizio ScarpaUniversity of Bristol

Bristol, England

ABSTRACTThis work presents an “open” and deployable honeycomb

configuration created using kirigami-inspired cutting andfolding techniques. The open honeycomb differs fromtraditional “closed” honeycomb by its reduced density and itsincreased flexibility. The exploitation of these characteristicsfor multifunctional applications is the focus of this work.Potential fields in which the open honeycomb could findapplication include sandwich panel manufacturing, morphing,and deployable structures.

NOMENCLATUREh h-wall width

l l-wall width

b wall length

t sheet material thickness

θ internal cell wall angle

α hinge opening angle

H unit cell width

L unit cell length

T unit cell thickness

Amat area of sheet material in unit cell

ρmat density of sheet material

tmat thickness of sheet material

INTRODUCTIONKirigami is the ancient Japanese art of folding and cutting

paper. Nojima and Saito developed a method of creatinghoneycombs using this process [1]. Slitting, corrugation, andfolding operations are used to form a 2D sheet material into a3D cellular structure. Mating faces can be bonded together tocreate a closed honeycomb or left unbonded to create an openhoneycomb. The geometry of the honeycomb can be varied byaltering the pattern of slits, giving a design flexibility

otherwise not available with other techniques. By varying thespacing of slits on the sheet it is possible to produce ahoneycomb with a varying thickness profile [2,3]. It is alsopossible to produce different cell shapes by changing the shapeof the moulds used and adapting the slitting patternaccordingly [4] Including circles in the slitting pattern givesholes in the cell walls. Such holes have previously been usedfor ventilation [5] but could also be used for (wire) access oractuation. Such features would be difficult to machine out oftraditional honeycomb without causing damage [6]. Wedescribe how the mechanical and actuation properties of theopen honeycomb change with the Kirigami pattern and thefold angle. Finite Element Analysis (FEA) has been used toinvestigate the effect of fold angle and fold stiffness on themechanical properties of the open honeycomb. A mechanicaltesting programme is planned to validate the FEA models.Subsequent work will focus on the multifunctionality of thestructure. Of particular interest is the use of smart hinges madewith Shape Memory Polymer (SMP), and the inclusion ofventilation or access holes with tuneable wires to create classesof deployable smart honeycombs using the Kirigami process.

KIRIGAMI MANUFACTURINGThe kirigami process for creating honeycombs [1],

converts a 2D sheet of material into a 3D cellular structureusing four processing steps, as shown in Figure 1. An openhoneycomb is produced by simply omitting step 4 and leavingthe honeycomb unbonded. The kirigami process is flexible interms of input materials and processing techniques; all that isrequired is that the sheet material can be cut, folded, andbonded to itself. This affords the user some choice in terms ofmaterials selection. In this case, Victrex PEEK film was usedfor its good formability and mechanical properties[7].

The slits were made using a Blackman & White ply cutter.The corrugations were created by thermoforming the PEEKfilm between semi-hexagonal moulds using a hot press at200ºC. The slits in the sheet line up with the edges of themoulds as shown in Figure 2; this alignment is importantbecause it allows the corrugated sheet to be folded back onitself to produce a hexagonal geometry.

1 Copyright © 2014 by ASME

Proceedings of the ASME 2014 Conference on Smart Materials, Adaptive Structures and Intelligent Systems SMASIS2014

September 8-10, 2014, Newport, Rhode Island, USA

SMASIS2014-7489

After thermoforming, the corrugated sheet is folded backon itself repeatedly such that the slits (which are now semi-hexagonal) open up into hexagonal holes, resulting in the openhoneycomb geometry shown in Figure 3. This is done at roomtemperature, by hand, and simply creates plastic deformationin the material where folded.

Figure 1: The four steps of the Kirigami manufacturingprocess for honeycombs.

Figure 2: Slitted sheet material conforming to hexagonalmould geometry.

Figure 3: The open honeycomb geometry

Holes or other such features can be included in the slittingpattern, which will later form part of the honeycomb, as shownin Figure 4.

Figure 4: Left: cutting pattern with holes and channels. Right:The honeycomb produced by this cutting pattern

If the slitting pattern is modified, a variant on the openhoneycomb can be created with some parallelogram-shapedwalls, as shown in Figure 5. To distinguish between the openconfigurations they shall henceforth be called “open rec” and“open par”, where “rec” and “par” are short for rectangularand parallelogram, respectively, and refer to the shape of thewalls.

2 Copyright © 2014 by ASME

Figure 5: The relationship between cutting patterns andhoneycomb geometries. The black shapes on the lowest cuttingpattern represent material removed. The brown material on thecutting pattern becomes the brown “cell” on the right. Thedotted box shows the bounding volume of the cell. The celledges which touch the bounding box are shown in blue. Greyreplica cells are shown to illustrate the tesselation of thestructure.

VOLUMETRIC ANALYSISThe two distinguishing features of the open honeycomb

are its increased (and directional) flexibility and its reduceddensity. In this section the unit cell and the variable density ofthe open honeycomb are examined.

Figure 6 shows the unit cell of the closed honeycomb.

Figure 6: The unit cell of theclosed configuration.

Dimensions h, l, b, and θ are determined by the moulds andcutting pattern. Dimensions H, L, T and sheet material area Aare calculated as follows:

H closed=2h+2 l sinθ (1)

Lclosed=l cosθ (2)

T closed=b (3)

A closedmat

=2bh+2bl (4)

Figure 7 shows the unit cell of the open rec honeycomb,with a side view along the 1-direction.

Figure 7: The unit cell of the open rec configuration. The viewon the right is a view along the 1-direction.

This structure is essentially the same structure, rotatedabout the 1-axis by angle α from the vertical. The dimensionsand area A are calculated as follows.

H rec=2h+2 l sinθ (5)

Lrec=l cosθcos α+b sinα (6)

3 Copyright © 2014 by ASME

T rec=l cosθsin α+b cosα (7)

A recmat

=2bh+2bl (8)

Figure 8 shows the unit cell of the open par honeycomb,with a side view along the 1-direction.

Figure 8: The unit cell of the open par configuration. The viewon the right is a view along the 1-direction.

The dimensions and area A are calculated as follows.

H par=2h+2 l sinθ (9)

Lpar=lcosθcos α

+b sinα (10)

T par=bcos α (11)

A parmat

=2bh+2bl (12)

To isolate the effect of angle α on the density of thehoneycombs, dimensions l, h, b, and θ must be fixed. Theabsolute density is of little interest; instead the density of theopen configurations will be compared relative to the density ofthe closed configuration. The density of each configuration isgiven by:

ρ=mV

=Amat

ρmat tmat

HLT(13)

It can be seen that H and Amat are the same for allconfigurations. If the same sheet material is used to make allconfigurations then ρmat and tmat are also constant. As a resultthese variables cancel out when the relative density iscalculated as follows.

ρopenρclosed

=Lclosed T closed

LopenT open

(14)

By substituting in equations (2-3,6-7,10-11) andrearranging, the following two expressions are obtained.

ρrecρclosed

=1

(l2 cos2

θ+b2

2bl cosθ)sin 2α+1

=1

C sin 2α+1 (15)

where C=(l2cos2

θ+b2

2blcos θ)

And

ρparρclosed

=1

(b2

2bl cosθ)sin2α+1

=1

D sin 2α+1 (16)

where D=(b2

2bl cosθ)

When α = 45º, sin2α = 1 and the relative density of theopen configurations is at a minimum with a value of 1/(C+1)or 1/(D+1) depending on the configuration, as shown in Figure9.

Figure 9: The variation of relative density with angle α.

The magnitudes of these minima are dependent on thedimensions l, h, b, and θ, and thus will vary between differentsize honeycombs, but the shape of the curve will not changequalitatively. For l > 0, b > 0, and 0º < θ < 90º, C > D,meaning that for a given α the open rec configuration willalways have lower relative density than the open parconfiguration. For the honeycombs produced in this work (h =l = 5mm; b = 10mm; θ = 30º) C = 1.37 and D = 1.15, givingthe following relative densities at α = 45º:

ρrecρclosed

=0.42 (17)

4 Copyright © 2014 by ASME

ρparρclosed

=0.46 (18)

FINITE ELEMENT ANALYSISFEA models were created to predict the mechanical

properties of the various honeycomb configurations. Ofparticular interest is the shear performance of the openhoneycombs when α = 45º, since shear in many cases resolvesitself as diagonal tension and compression which in this casewould be aligned with the h-walls, and also because thiscoincides with the density minimum.

The aim of this FEA was to investigate the effect ofseveral variables on the mechanical performance of thehoneycomb configurations. The input variables were angle α,hinge (or fold) stiffness k, and the boundary conditions, whichwere used to simulate the presence or absence of sandwichpanel skins. For each open configuration loading case, α wasvaried from 5º to 80º with intervals of 5º. k was given thevalues 1e-10, 0.1, 1, 1e10. The crude spread of this variablewas intended to save computational time while still givingsome indication of the effect of k across the entirety of itspossible range, from k = 0 (frictionless hinge) to k = ∞ (totallyrigid joint). Any observed effects could then be investigated inmore detail later.

The output variables were the flatwise modulus E3 and thetransverse shear moduli G13 and G23. G13 and G23 were notmeasured with sandwich skins absent, since shear loading bydefinition requires some kind of skin to introduce the load.Figure 10 shows a schematic of the test matrix, with each dotrepresenting an individual model run. ABAQUS/CAE FEAsoftware was used throughout, and a Python script was used toiteratively populate the test matrix.

Figure 10: Schematic of the FEA test matrix. Each black dotrepresents one model run, including the dots on the plots.NOTE that the lines on the plots do not show real trends andare only intended to illustrate the parameter space for eachconfiguration.

A double-size unit cell was used to allow the hinge to beincluded in the model geometry. For the open configurationsthe model consisted of two separate strip instances connectedby hinge elements along the relevant edges. Figures 11, 12,and 13 show schematics of how the unit cell models wereconstructed for the various configurations.

Figure 11: Schematic of the closedconfiguration unit cell model. A simplifiedmesh is shown for convenience. The blue linesrepresent couplings between edge nodes. Thegreen lines mark the coupling between the topface nodes. The red arrow marks the loadintroduction point. The black triangles markthe nodes on the bottom face constrained byBCs. The h-walls (tinted blue) are doublethickness.

Shell elements (S4R) were used throughout the models.Following a convergence study an element size of 0.25mm wasselected (for reference, the dimensions of the unit cell aregiven by h = l = 5mm; b = 10mm; θ = 30º).

Hinge connectors were used to connect the strip instancesin the open configuration models. Connectors are anABAQUS feature which allow the user to couple multipleDegrees Of Freedom (DOF) of two or more nodes, such thatthose nodes behave as if connected by a joint. Of the availablejoint types, the hinge was the most suitable for these models.The hinge connector has the option of including a springstiffness (D44) and it is this variable that was used toimplement the hinge stiffness k.

The presence of sandwich skins was modelled as follows.On the bottom face of the honeycomb all 6 DOF wererestrained using Boundary Conditions (BCs). On the top face,All rotations were restrained with BCs, and all 3 translationswere coupled such that the top face nodes moved as one; theseconstraints were chosen to represent the cell walls beingembedded in adhesive.

5 Copyright © 2014 by ASME

Figure 12: Schematic of the open rec configuration unit cellmodel. A simplified mesh is shown for convenience. The bluelines represent couplings between edge nodes. The green linesmark the coupling between the top face nodes. The pink linesrepresent hinge connections. The red arrow marks the loadintroduction point. The black triangles mark the nodes on thebottom face constrained by BCs.

For the cases in which the skins were absent, only flatwisecompression was simulated. The bottom face of the honeycombwas restrained in U3 using BCs, leaving the other 5 DOF free.One node on the bottom surface was restrained in U1 & U2 toprevent the entire model from sliding. The top face nodes werecoupled in U3 to represent a flat loading platen compressingthe structure.

To simulate the effect of a continuous honeycomb, theoutermost nodes were coupled to their counterparts on theother side of the cell using equations to couple their DOF. Thiswas done on a node-by-node basis, using a for loop in thePython script, so that the wall edges were not made rigid. Thisis illustrated in Figures 11, 12, and 13.

Load was introduced through a single point, and the restof the nodes on the top face were coupled to this point toensure load redistribution. A load of 1N was used for allmodels. This low load was to ensure that the structure behavedin a linear fashion.

Figure 13: Schematic of the open par configuration unit cellmodel. A simplified mesh is shown for convenience. The bluelines represent couplings between edge nodes. The green linesmark the coupling between the top face nodes. The pink linesrepresent hinge connections. The red arrow marks the loadintroduction point. The black triangles mark the nodes on thebottom face constrained by BCs.

RESULTSThe results of the FEA modelling are shown in Figures

14-19. The moduli of the open and closed configurations arecompared in two ways. The absolute modulus and the specificmodulus are compared between configurations. Thecomparison of specific moduli takes into account the change indensity due to variations in α.

6 Copyright © 2014 by ASME

Figure 14: Absolute modulus E3 compared between the openand closed configurations.

Figure 15: Specific modulus E3/ρ compared between the openand closed configurations.

Figure 16: Absolute modulus G13 compared between the openand closed configurations.

Figure 17: Specific modulus G13/ρ compared between the openand closed configurations.

Figure 18: Absolute modulus G23 compared between the openand closed configurations.

Figure 19: Specific modulus G23/ρ compared between the openand closed configurations.

DISCUSSIONIt can be seen that the effect of the hinge stiffness is

negligible. The plots in Figures 14-19 show data series for thefour different hinge stiffnesses, but it is mostly impossible todistinguish the individual lines because they are so closetogether. The hinges' lack of influence on the moduli is notunexpected; in these loading cases the hinge is either totally

7 Copyright © 2014 by ASME

restrained by the sandwich skins, or the loading is such thatthe hinge behaviour doesn't play a big part in the deformationmechanism of the structure. It is anticipated that the hingeswill have the greatest effect on the bending behaviour of thestructure.

Figure 14 And Figure 15 show that the presence orabsence of sandwich panel skins has a significant effect on thecompressive moduli. However, the effect of the absence ofsandwich skins is not as great as expected. For an openconfiguration without skins, it should be possible for the stripsto simply splay outwards under a compressive load, with onlythe stiffness of the hinge to resist this motion. This is one ofthe cases where the hinges were expected to have a significanteffect on the structure's behaviour. This was not observed inthe models, and this may be due to the side constraints whichsimulate a continuous honeycomb; these may have preventedthe cell from expanding as it normally would.

As expected, the open honeycombs' absolute moduli arelower than that of the closed honeycomb. This is to be expectedbecause for the open configurations the same amount ofmaterial has been spread over a greater volume. However, it isalso important to note that there is overall less connectivitybetween the cell walls in the open configurations. In the closedconfiguration, all cell walls are supported by two other walls atthe joints, and this, along with the the back-to-back h-walls,provides good buckling support. This is not true for the openconfigurations; especially the open rec configuration in whichevery cell wall has at least one free edge even when sandwichskins are present. The open honeycombs' specific moduli arealso mostly lower than that of the closed honeycomb, but insome cases the specific moduli of the open par configurationapproach that of the closed configuration. For both openconfigurations, the variation of alpha produces stronglynonlinear changes in moduli. This presents the possibility oftailoring the moduli and density for a specific purpose, bycontrolling the dimensions of the honeycomb. As shown inequations (17-18) the open configurations can havesignificantly lower density compared to their closedcounterpart. This suggests that open honeycombs could be usedfor applications which are weight-critical but not stiffness-critical.

It can be seen that the open par configuration consistentlyoutperforms the open rec configuration. This is due to thegreater contact area between the open par honeycomb and theload introduction surface, which allows better load distributionin the structure and fewer stress concentrations. Indeed, thepoor load distribution in the open rec configuration is whatinspired the open par configuration. Figure 20 shows how theopen rec configuration develops large stress concentrations inthe l-walls in reponse to a compressive load. The h-walls, bycontrast, contribute little due to their free edge, which isvirtually stress-free.

A particularly interesting result is the specific G23 of theopen par configuration, because it surpasses that of the closedhoneycomb. A possible explanation for this can be thefollowing. In Gibson & Ashby's classical analysis of the shearproperties of a honeycomb the effect of the h-wallsperpendicular to the shear direction is neglected, because thesewalls will deform in pure bending, the stiffness of which isnegligible compared to one of the other walls in shear.However, in the case of the open par honeycomb, the h-walls

are angled such that they are not loaded under pure bendingbecause there is also present a component of axial loading.This loading behaviour allows the h-walls to contribute to G23

in a way that is not possible in the closed honeycomb. Also,this G23 is at a maximum for α = 45º, which confirms thehypothesis that the cell walls provide optimal shear support atα = 45º. Interestingly, the G23 curves for the open recconfiguration are totally different. This is due to the fact thatthe h-walls only ever touch one sandwich skin and thus do notprovide a continuous path for the shear force to be transmitted.Instead the shear force is resisted almost entirely by the l-walls, which touch each sandwich skin with one corner anddevelop large stress concentrations.

Figure 20: The open rec configuration under flatwisecompression loading. The contour plot shows Von Mises stressand reveals the stress concentrations at the corners of the l-walls. Note: this is an early model and has a different meshdensity from the models used in this paper.

It is interesting to note that the closed configuration modelagrees with Zhang & Ashby's analysis of a honeycomb incompression with doubly-thick h-walls [8]. The stress in thedoubly-thick walls is double that in the single walls.

It should be noted that after alpha reaches about 60degrees, the open par configuration becomes impractical,because the unit cell becomes so stretched that it would bedifficult to manufacture.

POTENTIAL APPLICATIONS

ManufacturingWhen α is small, the open par configuration's specific

moduli are close to those of the closed configuration, as can beseen in Figures 15, 17, and 19. A honeycomb like this couldpotentially be used as a “drapeable” honeycomb – slightlyinferior mechanical properties but much easier to form intocomplex/tight curvatures due to its greatly reduced bendingstiffness. Such a honeycomb could easily form cylindricalshapes because the deformation of the hinges produces none ofthe secondary curvature normally observed in honeycombs inbending.

8 Copyright © 2014 by ASME

Morphing As previously mentioned, holes can be easily included in

the cell walls of the honeycomb. These have the potential tocreate a morphing structure when wires are fed through thechannels created. Figure 21 shows a demonstrator, whichshows how tensioning the wire can produce different,deformed shapes depending on the wire's location in thehoneycomb.

Figure 21: A morphing demonstrator created by threadingcables through the channels in a honeycomb. The knots at theends of the cables allow them to exert force on the honeycombwhen pulled.

Deployable Structures Due to the honeycomb's increased flexibility and its

reliance on the hinges for bending stiffness, this geometrycould be used to create a deployable structure using shapememory materials. Figure 22 shows a possible conceptwhereby shape memory hinges control the expansion of thehoneycomb. This could be useful for parts requiring a specificcurvature.

Figure 22: A concept for adeployable honeycomb createdusing shape memory material forthe hinges (purple) and strips(grey)

CONCLUSIONSIn this paper two open honeycombs are presented and

analysed using analytical methods and FEA. Thesehoneycombs display a significantly different behaviour fromthat of the traditional closed honeycombs. The particulardeformation mechanism shown by open honeycombs could beexploited for applications related to the manufacturing ofcellular structures, morphing, and deployable structures.

Future work will include a mechanical testing programmeto validate the results of the FEA. Subsequent work will focuson developing the multifunctional aspects of the openhoneycombs concept.

ACKNOWLEDGMENTSSpecial thanks go to Mr. Ian Chorley (ACCIS Technician)

for help and advice manufacturing and testing of the PEEKhoneycombs.

The Authors are also grateful to Dr. Alan Wood fromVictrex plc for the advice provided about the processing of thePEEK films.

The Authors would also like to thank Dr. Ian Farrow forencouraging the investigation into open honeycombs.

REFERENCES[1] Nojima T, Saito K. Development of newly designed ultra-light core structures. JSME Int J Ser A 2006;49(1):38–42.

[2] Saito K, Pellegrino S, Nojima T. Manufacture of arbitrarycross-section composite honeycomb cores based on origamitechniques. In: Proceedings of the ASME 2013 internationaldesign engineering technical conferences and computers andinformation in engineering conference; 2013.

[3] Saito K, Agnese F, Scarpa F. A cellular Kirigami morphingwingbox concept. J Intell Mater Syst Struct 2011;22(9):935–44.

[4] Hou Y, Neville R, Scarpa F, Remillat C, Gu B, Ruzzene M.Graded conventional–auxetic Kirigami sandwich structures:flatwise compression and edgewise loading. Compos Part BEng 2013(11).

[5] Neville et al. Transverse stiffness and strength of Kirigamizero-ν PEEK honeycombs. Composite Structures 2014;114(2014) 30–40

[6] Bitzer T. Honeycomb technology – materials, design,manufacturing, applications and testing. 1st ed. London:Chapman & Hall; 1997.

[7] Victrex PLC. APTIV film thermoforming guide; 2011.

[8] Zhang J, Ashby MF. The out-of-plane properties ofhoneycombs. Int J Mech Sci 1992;34(6):475–89.

9 Copyright © 2014 by ASME