Jaehyung Ju

Joshua D. Summers1

e-mail: jsummer@clemson.edu

John Ziegert

George Fadel

Clemson Engineering Design Application and

Research Group (CEDAR),

Department of Mechanical Engineering,

Clemson University,

Clemson, SC 29634-0921

Design of Honeycombs forModulus and Yield Strainin ShearThe low in-plane modulus of honeycombs may be used for compliant structures with ahigh elastic limit while maintaining a required modulus. Numerical and finite element(FE) studies for a functional design of honeycombs having a high shear strength, (spl*)12and a high shear yield strain, (cpl*)12 are conducted with two material selections—mild-steel (MS) and polycarbonate (PC) and five honeycomb configurations, when they aredesigned to be a target shear modulus, G12* of 6.5 MPa. A numerical study of cellularmaterials theory is used to explore the elastic limit of honeycombs. FE analysis is alsoemployed to validate the numerical study. Cell wall thicknesses are found for each mate-rial to reach the target G12* for available cell heights with five honeycomb configura-tions. Both MS and PC honeycombs can be tailored to have the G12* of 6.5 MPa with0.1–0.5 mm and 0.3–2.2 mm cell wall thicknesses, respectively, depending on the numberof vertical stacks, N. The PC auxetic honeycomb with h¼ �20 deg shows high shear flexi-bility, when honeycombs are designed to be the G�

12 of 6.5 MPa; a 0.72 MPa (spl*)12 anda 13% (cpl*)12. The authors demonstrate a functional design with cellular materials witha large design space through the control of both material and geometry to generate ashear flexible property. [DOI: 10.1115/1.4004488]

Keywords: flexible design, honeycomb, auxetic, cellular solids, effective properties

1 Introduction

Two dimensional prismatic cellular materials of periodic meso-structures are called honeycombs. The hexagonal versions of suchmeso-structures are cellular materials commonly employed in var-ious applications used in the design of lightweight structures. Forexample, the in-plane moduli of hexagonal honeycombs havebeen successfully investigated with the cell wall bending model,which is called cellular material theory (CMT) summarized byGibson and Ashby [1]. There are other analytical and numericalmodels to describe in-plane effective properties of honeycombs inthe literature; a refined cell wall’s bending model by adding abeam’s stretching and hinging motion [2], an energy methodmodel [3], a refined model with round shape at cell edges [4], anda model using the homogenization method [5]. In-plane mechani-cal properties with different cell types—square, hexagonal, trian-gle, mixed squares and triangles, and diamond—were investigatedby Wang and McDowell [6]. Circular and chiral shapes of honey-combs have also been studied for a functional design [7–9]. Amultifunctional approach requiring structural stability and fastheat transfer was investigated with honeycomb structures [10].

These honeycombs have been primarily used in lightweightsandwich structures with composite sheets for which a high out-of-plane stiffness is desired [11–13]. In contrast to the highly stiffand strong properties of the out-of-plane direction (longitudinalcell axes) associated with the cell walls’ axial strength, thein-plane properties are 2 and 3 orders of magnitude weaker thanthose of the out-of-plane loading. For this reason, the mechanicalproperties for the in-plane loading have been thought be the mostlimiting for design applications. However, recently, there havebeen efforts to use the lower in-plane stiffness for designing flexi-

ble meso-structures in applications that need high deformationunder targeted loads [14–20]; a compliant microelectromechanical(MEMS) system [14], morphing airfoil structures [15–17], and aflexible tire component [18].

A high in-plane strain of a honeycomb with a required effectivemodulus can be used for high structural performance. For exam-ple, compliant mechanism based MEMS structures are requiredfor a low actuating energy while maintaining a structural stiffness[14]. A morphing airfoil having a high yield strain with a requiredaeroelastic modulus can also be used for the fuel efficient aircraftdesign while changing the shape of airfoils depending on operat-ing conditions [15–17]. A tire made of a flexible honeycomb witha core material having a capability to resist extreme thermal con-ditions may replace the conventional rubber tire for the functionalpurpose [18].

In this paper, we explore a flexible structural application ofhoneycombs in shear. While pursuing a hexagonal honeycombstructure with in-plane shear properties similar to an elastomer,we investigate the effect of various geometric parameters on thein-plane effective elastic properties (Young’s and shear moduli)of conventional and auxetic hexagonal honeycombs with mild-steel (MS) and polycarbonate (PC). Effective shear yield strains,(cpl*)12 of hexagonal honeycombs are also investigated. Under agiven volume, the effects of cell heights and wall thickness withfive cell configurations on effective properties and (cpl*)12 are dis-cussed with the two core materials.

2 Model Development

Due to the high cost of manufacturing honeycombs with vari-ous geometric options, a numerical parametric study of an analyti-cal model is preferred to an experimental one at the initial stage ofdesign. Therefore, we employed a numerical parametric studywith a developed analytical model in this research to provide adirection for a prototype design. Hexagonal geometries easilyhandle positive to negative Poisson’s ratios by changing cellangles, which are most appropriate for our parametric study.

1Corresponding author.

Contributed by the Materials Division of ASME for publication in the JOURNAL OF

ENGINEERING MATERIALS AND TECHNOLOGY. Manuscript received September 9, 2009;

final manuscript received April 25, 2011; published online December 6, 2011. Assoc.

Editor: Mark F. Horstemeyer.

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2.1 A Brief Review of Effective Linear Elastic Propertiesand Effective Yield Strain. Unit cell geometries with conven-tional and auxetic hexagonal honeycombs are shown in Fig. 1; thecritical geometric parameters include the cell angle (h), the cellheight (h), the leg length (l), and the wall thickness (t).

Gibson and Ashby’s CMT, which has been validated experi-mentally and numerically, is used to elucidate the behavior of cel-lular materials with cell wall bending. It also describes quite thehoneycombs’ elastic behavior well including configurations withnegative cell angles [1–3]. Considering the axial and transverseshear of cell walls as well as bending, in-plane effective moduli ofhexagonal honeycombs is expressed as [1,2]

E�1 ¼ E

t

l

� �3 cos h

h=lþ sin hð Þ sin2 h

1

1þ 2:4þ 1:5�s þ cot2 hð Þ t=lð Þ2

(1)

E�2 ¼ E

t

l

� �3 h=lþ sin hð Þ

cos3 h

1

1þ 2:4þ 1:5�s þ tan2 hþ2 h=lð Þ

cos2 h

� �

t=lð Þ2� �

(2)

G�12 ¼ E

t

l

� �3 h=lþ sin hð Þ

h=lð Þ2cos h

1

F(3)

where F ¼ 1þ 2h

l

� �

þt

l

� �2 1

h=l2:4þ 1:5�sð Þ 2þ h=lþ sin hð Þþ

�

h=lþ sin h

h=lð Þ2h=lþ sin hð Þ tan2 hþ sin h

�

:

Fig. 1 Unit cell geometry for (a) conventional and (b) auxetic honeycombs

Fig. 2 2D view of honeycomb and with built PC and MS samples

Fig. 3 Five honeycomb configurations used in this study when h5 l

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The analytical plastic collapse model by Gibson and Ashbyassumes that the honeycombs begin to collapse plastically whenthe bending moment in the cell walls reaches the fully plasticmoment [1]. Several efforts have been undertaken to investigatethis in-plane crushing of honeycombs considering buckling andplastic collapse of cell walls, which covers both elastic and plasticranges of a base material [21–23].

Because the elastic ranges and the initial plastic collapse pointsof honeycombs are the only interest of this study, we used theCMT initially and a finite element (FE) subsequently for CMTvalidation. Using the standard beam theory, CMT provides a yieldpoint of honeycombs as a function of core materials’ strength overa material’s linear elastic range. In-plane yield strains at which thehoneycombs can tolerate deformation without local cell wall fail-ure when subjected to in-plane loading are given by Ref. [1]

e�pl

� �

1¼

rys

E�11

t

l

� �2 1

2 h=lþ sin hð Þ sin hj j(4)

e�pl

� �

2¼

rys

E�22

t

l

� �2 1

2 cos2 h(5)

Fig. 4 Cell heights as a function of number of unit cells for each honeycomb configuration(when a5 1 and H5 12.7 mm)

Fig. 5 Effective moduli of honeycombs with 60 deg cell angle(h5 3.33 mm, base material: MS)

Fig. 6 Effective moduli of honeycombs with 30 deg cell angle (base material: MS)

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c�pl

� �

12¼

1

4

rys

G�12

t

l

� �2 1

ðh=lÞ cos h(6)

It should be noted that the analytical expressions for the effectiveproperties and maximum effective strains are restricted to be usedin the linear elastic range.

2.2 Design of Honeycombs at a Given Meso-StructuralDimension. For a shear flexure structural design of honeycombs,the height of a honeycomb sample is chosen as 12.7 mm (0.5 in.)in the x2 direction as shown in Fig. 2, which aligns with otherdesign considerations of the structure that are beyond the scope ofthis paper. Further, it allows for a refined design space in which to

Fig. 7 Effective moduli of honeycombs with 15 deg cell angle (base material: MS)

Fig. 8 Effective moduli of honeycombs with210 deg cell angle (base material: MS)

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explore the influences of the defined honeycomb parameters onthe effective honeycomb properties.

When designing honeycombs, numerous configurations areavailable with cell angle, h, cell height, h, and cell length, l. Fromthe geometric constraint for an auxetic configuration, h � 2l sin h,the maximum negative cell angle, h, is �30 deg when h¼ l

(Fig. 1). However, when realistic constraints on cell wall thick-ness are considered (0.1 mm to 20% of h), it is not possible to con-struct the auxetic honeycomb configuration of h¼�30 deg.Therefore, honeycomb geometries with a cell angle, h, higherthan �20 deg for h¼ l are considered in this study.

Specifically, five representative hexagonal configurations areconsidered and their effective properties with varying cell wallthickness are discussed (Fig. 3). The five discrete models areenough to show a continuous behavior of shear deformation overa range of the cell angles by interpolating and extrapolating theselected cell angles.

When comparing effective properties of one honeycomb withanother, the volume or the relative density is normally fixed. Inthis study, the volume is fixed; more precisely speaking, the hon-eycomb’s panel height, H, is fixed to be 12.7 mm (0.5 in.) asshown in Fig. 3

For a given honeycomb’s panel height, H, and cell angle, h, thecell height h is defined as

h ¼H

2N � 1þsin h

a

� � (7)

where N is the number of unit cells in the vertical direction (thetwo direction in Fig. 2) and a ¼ h=l. From Eq. (7), the cell

Fig. 9 Effective moduli of honeycombs with220 deg cell angle (base material: MS)

Fig. 10 Effective moduli of honeycombs with 60 deg cell angle(h5 3.4 mm, base material: PC)

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heights, h, are obtained as a function of the number of unit cells,N, in the vertical direction. In the present study, a is restricted tobe one for convenience. When a¼ 1 and H¼ 12.7 mm availablecell heights in the design space are plotted in Fig. 4.

Considering a conventional manufacturing limitation, the cellheights should be reasonably high. The lower limit of h is definedto be 2.0 mm in this study. Therefore, the honeycomb with 60 degcell angle appears to have only one cell height, 3.4 mm. Designwith the configurations of cell angles, h, 30 deg, and 15 deg showtwo cell height options; 4.23 mm and 2.12 mm for 30 deg, and5.04 mm and 2.52 mm for 15 deg. Auxetic honeycombs appear tohave more cell height options; 7.68,3.84, and 2.56 mm for �10deg cell angle, and 9.65,4.83,3.22, and 2.41 mm for �20 deg cellangle. Parametric studies of effective moduli with thicknesschanges are done when thickness varies from 0.1 mm to about25% of h, considering a manufacturing limitation.

In this study, MS and PC are used as the base materials of ourhoneycomb design, which we selected for prototyping and experi-mental validation purposes. Further, because the mechanical prop-erties of each material are outside of the target ranges for both

stiffness and strain, we can achieve honeycomb properties outsideof the realizable range for the used materials. Thus, we achievehigh shear flexure under a target stiffness using materials with lowyield strain and much higher shear stiffness. The physical valida-tion of this research will be presented in our subsequent work.

3 Design of Honeycombs With CMT

In this section, in-plane effective properties of honeycombs areinvestigated using Eqs. (1)–(7). A target shear modulus, G12* of6.5 MPa and the corresponding uniaxial moduli are found from aparametric study with the wall thickness.

3.1 Effective Properties of MS Honeycombs. Metallic hon-eycombs can be manufactured using the wire electrical dischargemachining technique, and an example part manufactured by thetechnique is shown in Fig. 2(a). A MS having 210 GPa Young’smodulus and 0.29 Poisson’s ratio is used for a parametric study ofcell wall thicknesses on effective properties of honeycombs basedon Eqs. (1)–(3). One cell height (h¼ 3.33 mm) of a MS honey-comb is available at the 60 deg cell angle as shown in Fig. 4.

Fig. 11 Effective moduli of honeycombs with 30 deg cell angle (base material: PC)

Fig. 12 Effective moduli of honeycombs with 15 deg cell angle (base material: PC)

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A shear modulus of 6.5 MPa is obtained at a wall thickness of0.1 mm (Fig. 5). The modulus in the x2 direction is approximately40 times higher than that in the x1 direction at the wall thicknessof 0.1 mm; E1* of 1.9 MPa and E2* of 77.9 MPa.

Transverse isotropic properties are obtained with a 30 deg cellangle design. The target shear modulus G12* of 6.5 MPa isobtained at a wall thickness 0.16 mm for h¼ 4.23 mm as shownin Fig. 6(a). Its corresponding E1* and E2* are 26 MPa. Thereseems to be no solution in the 30 deg cell angle configuration withthe double stack design to reach the target shear modulus as canbe shown in Fig. 6(b): The target shear modulus is obtained with acell wall thickness of 0.08 mm, which is lower than 0.1 mm.

Two cell heights are available for honeycombs with a 15 degcell angle. The target shear modulus is reached at cell wall thick-nesses of 0.209 mm and 0.105 mm for a single unit cell row(h¼ 5.04 mm) and a double unit cell row (h¼ 2.52 mm) design,respectively, as shown in Fig. 7. Although the single unit cell rowdesign with 15 deg cell angle can reach the target shear moduluswith higher thickness than that of the 30 deg cell angle design, thecorresponding Young’s modulus, E�

11 is 4 orders of magnitudehigher than the shear modulus.

Using the available three cell heights are available with MShoneycombs of a �10 deg cell angle in Fig. 4, we can determinethe target shear modulus of 6.5 MPa at wall thicknesses of 0.37,0.18, and 0.12 mm for cell heights of 7.68,3.84, and 2.56 mm,respectively, (Fig. 8). The auxetic design produces an extremely

high stiffness up to approximately 860 MPa in the x1 directionwhile maintaining low moduli in E�

22 and G12*, which are 20 MPaand 6.5 MPa, respectively. As the number of stacks, N, increases,the honeycombs show a high sensitivity of wall thicknesses onmoduli; e.g., G12* for N¼ 1, 2, and 3 are 7.0, 7.4, and 7.9 MPa,respectively, with a wall thickness change of 0.01 mm from thereference wall thicknesses to make a G12* of 6.5 MPa. This sensi-tivity resulted from a low wall thickness of a honeycomb with anincreased N to design the same G12*.

Using the four cell heights available with the auxetic configura-tion of �20 deg cell angle in Fig. 4, the target shear modulus isobtained from all cell heights (Fig. 9). The single unit cell rowconfiguration reaches the 6.5 MPa shear modulus with a cell wallthickness of 0.49 mm and a cell height of 9.65 mm. The other con-figurations show the possibility of reaching the target shear modu-lus with cell wall thicknesses between 0.16 mm and 0.25 mms. Asthe number of stacks, N, increases, the honeycombs show a highsensitivity of wall thicknesses on moduli; e.g., G12* for N¼ 1,2,3,and 4 are 6.9,7.2,7.6, and 7.9 MPa, respectively, with a wall thick-ness change of 0.01 mm from the reference wall thickness tomake the G12* of 6.5 MPa.

Equations (1)–(3) were developed based on a unit cell withequivalent periodic boundary condition. The numerical parametricstudies in this section were conducted considering a honeycomb’sunit cell with a periodic boundary condition. The multistack con-figurations do not necessarily mean the multi-unit-cell, however.

Fig. 13 Effective moduli of honeycombs with210 deg cell angle (base material: PC)

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Fig. 14 Effective moduli of honeycombs with220 deg cell angle (base material: PC)

Fig. 15 Effective shear yield strain, (cpl*)12 as a function of density when honeycombs are designed to have a G12* of 6.5 MPa

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The effective properties of all honeycombs were obtained fromthe unit cell approach, which are the same regardless of the num-ber of vertical stack N, if the wall thicknesses are controlled(Figs. 6–9).

3.2 Effective Properties of PC Honeycombs. Polymers canalso be used as base materials for honeycombs meso-structures.There are several techniques available for manufacturing honey-combs with polymers, some of which are extrusion, resin transfermolding, or rapid prototyping (RP). One such example shown inFig. 2(b) was prepared with PC using RP. Similar parametric stud-ies of cell wall thickness on effective properties of honeycombsare carried out using PC with a 2.7 GPa Young’s modulus and0.42 Poisson’s ratio.

Effective moduli are obtained for each configuration with anincreasing cell wall thickness. Only one cell height is available in 60deg honeycomb from Fig. 4. A target G12* of 6.5 MPa, for example,is obtained at 0.44 mm cell wall thickness for h¼ 3.4 mm. This con-figuration may be good for a functional design requiring weakstiffness in the x1 direction and higher stiffness in the x2 direction(Fig. 10).

Honeycombs with a 30 deg cell angle appear to be able tomimic elastic moduli of polyurethane elastomer. The target G12*

Fig. 16 Cell wall thicknesses in Fig 15

Fig. 17 Normalized (cpl*)12 as a function of cell angle, h for adesigned G12* of 6.5 MPa

Table 1 Honeycomb geometric parameters as a function of h for the same G12* of 6.5 MPa (h5l, N51 )

t (mm) (c�plÞ12

h (deg) h (mm) L (mm) MS honeycomb PC honeycomb G12* (MPa) MS honeycomb PC honeycomb

�25 11.00 11.00 0.580 2.581 6.5 0.0236 0.1823�20 9.65 9.65 0.493 2.192 6.5 0.0214 0.1647�15 8.57 8.57 0.425 1.885 6.5 0.0196 0.1503�10 7.68 7.68 0.370 1.639 6.5 0.0181 0.1386�5 6.96 6.96 0.325 1.440 6.5 0.0168 0.12900 6.35 6.35 0.288 1.276 6.5 0.0158 0.12105 5.84 5.84 0.257 1.139 6.5 0.0150 0.114510 5.41 5.41 0.232 1.024 6.5 0.0143 0.109115 5.04 5.04 0.210 0.926 6.5 0.0137 0.104620 4.73 4.73 0.191 0.842 6.5 0.0133 0.101125 4.46 4.46 0.174 0.769 6.5 0.0129 0.098330 4.23 4.23 0.160 0.706 6.5 0.0127 0.096335 4.04 4.04 0.147 0.650 6.5 0.0125 0.095040 3.87 3.87 0.136 0.600 6.5 0.0124 0.094545 3.72 3.72 0.126 0.556 6.5 0.0124 0.094750 3.60 3.60 0.117 0.515 6.5 0.0126 0.095755 3.49 3.49 0.108 0.478 6.5 0.0128 0.097860 3.40 3.40 0.100 0.442 6.5 0.0132 0.101365 3.33 3.33 0.092 0.409 6.5 0.0137 0.1067

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of 6.5 MPa is obtained at cell wall thicknesses of 0.71 mm and0.35 mm for h¼ 4.23 mm and h¼ 2.12 mm, respectively(Fig. 11). Moreover, the corresponding Young’s moduli of bothconfigurations are close to that of polyurethane (�25 MPa), mak-ing it possible to design elastomer-like honeycombs. Note that thecell angle of 30 deg provides in-plane transverse isotropy, whichmeans E11* is equal to E22*.

Honeycombs with a 15 deg cell angle result in 6.5 MPa shearmodulus at cell wall thicknesses of 0.926 mm and 0.463 mm forh¼ 5.04 mm and h¼ 2.52 mm, respectively (Fig. 12). The corre-sponding Young’s moduli are 180 MPa in the x1 direction and21.9 MPa in the x2 direction. A design with a 15 deg cell angleshows interesting results in that this honeycomb already has elas-tomerlike properties in G12* and E22*. This configuration mayprovide an adequate solution for designing elastomerlike honey-combs having more stiffness in one direction (Fig. 12).

Auxetic honeycombs with a cell angle of �10 deg have threecell height choices in the present design space: h¼ 7.68 mm,3.84 mm, and 2.56 mm (Fig. 13). The target G12* of 6.5 MPa canbe obtained with cell wall thicknesses of 1.64, 0.82, and 0.55 mmwhen cell heights are 7.68,3.84, and 2.56 mm, respectively. Thecorresponding E11* and E22* are 398.3 MPa and 18.4 MPa,

respectively. The auxetic design produces an extreme stiffness upto approximately 400 MPa in the x1 direction while maintaininglow moduli in E22* and G12*.

Again using our example in Fig. 4, four different cell heightsfrom 2.41 mm to 9.65 mm are available with �20 deg cell angle(Fig. 14). The target G12* of 6.5 MPa can be obtained with cellwall thicknesses of 2.19,1.1,0.73, and 0.55 mm when cell heightsare 9.65,4.83,3.22, and 2.41 mm, respectively. The correspondingE11* and E22* are 249.8 and 19.6 MPa, respectively.

3.3 Design of Shear Flexible Honeycombs. When designingmaterials, the properties are often plotted as a function of densityas in the Ashby charts [24]. Here, we use the density of the hexag-onal honeycombs that is given by Ref. [1]

q� ¼ q

t

l

h

lþ 2

� �

2 cos hh

lþ sin h

� � (8)

When the yield stresses, rys (and the corresponding yield strains) ofPC and MS are 78 MPa (2.59%) and 200 MPa (0.1%), respectively,effective shear yield strains, (cpl*)12 of MS and PC honeycombs as afunction of density can be obtained from Eqs. (3) and (6). Figure 15is a (cpl*)12- density diagram of MS and PC honeycombs based onEqs. (3) and (6) when the honeycombs are designed to be a G12* of6.5 MPa. The corresponding cell wall thicknesses of MS and PChoneycombs are shown in Fig. 16. Here, the PC honeycombs have a9–16% (cpl*)12, depending upon cell configurations when the shearmodulus of the honeycombs are 6.5 MPa. Compared to these PChoneycombs, MS honeycombs have a lower (cpl*)12, due the loweryield strain of MS, eys (�0.1%).

Both honeycombs can be designed to be lighter than polyurethane,whose density is approximately 1200 kg=m3. PC honeycombs aredesigned with densities between 230 and 660 kg=m3, and MS honey-combs are designed with densities between 340 and 970 kg=m3.

PC auxetic honeycombs appear to have a high (cpl*)12, whendesigned to be the G12* of 6.5 MPa. MS honeycombs, however,exhibit only a 2% variation of (cpl*)12 with meso-structural geo-metric change, which implies that effective shear yield strains arenot quite affected by configurations of MS honeycombs, due tothe low eys and high stiffness of MS. Polymers having high

Fig. 18 Effects of 1=cosh and t2=(hl) on (cpl*)12 (normalizedvalues)

Fig. 19 Material models for FE analysis

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modulus and high eys and metals having low modulus and high eysappear to be quite appropriate for the current design targets.

Figure 17 shows interpolated and extrapolated normalized val-ues of (cpl*)12 as a function of h using Eqs. (3) and (6), when aG12* of honeycombs are designed to be 6.5 MPa. These corre-sponding geometric dimensions for MS and PC honeycombs areshown in Table 1 with the auxetic honeycombs showing a highshear flexible property as can be seen in Fig. 17. As the hincreases from a negative value, the (cpl*)12 of honeycombsdecreases until the h reaches 40 deg. A (cpl*)12 shows a slightincrease with a regular increase in h.

Due to the shear compliant characteristics of the auxetic honey-combs, honeycombs with a negative h require a higher cell wallthickness than honeycombs with a positive h in order to have thesame G12* (Fig. 16 and Table 1). For example, the cell wall thick-

ness of the PC honeycomb with a h of �20 deg is 2.219 mm tomaintain the target G12* of 6.5 MPa. On the other hand, a cellwall thickness of the PC honeycomb with a h of 60 deg is0.442 mm (Table 1).

According to Eq. (6), the (cpl*)12 is a function of t2=hl and1=cosh, when the G12* is designed as a fixed value. Figure 18shows the normalized values of t2=hl and 1=cosh as a function ofh. At a negative h, the t2=hl effect is dominant on the shear flexi-bility of honeycombs. However, the increase in a h causes anincrease in both the t2=hl decreases and the 1=cosh. This slightincrease in (cpl*)12 with a h of 60 deg can be explained with anincreased 1=cosh value.

The number of layers, N, does not affect the shear flexibility.For example, the required cell wall thicknesses to make a G12* be6.5 MPa for h¼�20 deg are 2.192 mm, 1.096 mm, 0.731 mm,

Fig. 20 Shear stress-strain curves of MS honeycombs

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and 0.548 mm with the corresponding cell heights, h (¼l), of 9.65mm, 4.83 mm, 3.22 mm, and 2.41 mm for N¼ 1,2,3, and 4,respectively. Those geometric values provide the same (cpl*)12,which is 0.1646. The same value of (cpl*)12 is caused by the samet2=hl in Eq. (6) for a fixed h. However, a higher N causes a highersensitivity of the cell wall thickness on the overall effective prop-erties as checked in Sec. 3. A proper N can be chosen based upona selection of a manufacturing method of honeycombs associatedwith a limitation of manufacturability.

4 Validation of the CMT Shear Model Using FEA

Equation (6) assumes both linear elasticity of a core materialand small deformation of honeycombs, which has a limitation to

catch a nonlinear behavior of honeycombs when subjected to alarge strain. In order to validate the elastic range of honeycombsin shear, shear stress-strain curves are generated using finite ele-ment analysis (FEA) considering the nonlinearity of both materialand geometry.

Linear and nonlinear elastic material models for MS and PC,respectively, are considered to simulate the shear behaviors ofhoneycombs. The stress-strain curves of MS and PC are shown inFig. 19. For PC, we used a nonlinear material behavior availablein the literature [25], for incorporation into an ABAQUS usersubroutine.

Using the material models, the effective shear stress-strain curvesof the five representative MS and PC honeycomb unit cells for sim-ple shear loading are plotted from FE analysis using a three-node

Fig. 21 Shear stress-strain curves of PC honeycombs

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quadratic beam element, B22 in ABAQUS (Figs. 20 and 21). The B22element allows large rotation, finite membrane strain, Timoshenko-type correction for shear deformation. After mesh convergenceanalysis, we applied four B22 elements per h and the l.

All cellular structures are designed to have a G12* of 6.5 MPa;the corresponding cell wall thicknesses, t are obtained fromEq. (3) for a given h, l, and h. Under the simple shear boundarycondition, e.g., u1¼ u2¼ 0 on the bottom surface and u1¼ theshear displacement (d) and u2¼ 0 on the top surface along withthe periodic boundary condition, effective shear stresses wereobtained by dividing the summation of reaction forces on the topnodes with the top surface area. Local maximum von Misesstresses are checked for every increment of a shear displacement.For more details on the FE preprocessing, the reader is referred toour analysis using metallic honeycombs [20].

Shear yield strains, (cpl*)12 are obtained by checking local max-imum von Mises stresses while assuming that local cell yieldingoccurs when the maximum von Mises stresses exceed the materi-al’s yield point (e.g., 200.2 and 80.1 MPa in the true stress valuesof MS and PC, respectively) in this study.

Due to the linear elastic material model, MS honeycombs ex-hibit only a nonlinear geometric behavior in the shear stress-straincurves (Fig. 20). The MS honeycomb with h¼ 60 deg shows a

high stiffening effect with an increasing shear strain as shown inFig. 20(a). The stiffening effect appears to be reduced with lowercell angles as shown in Figs. 20(b) and 20(c). The stiffening effectof the MS honeycombs with h¼ 30 deg and 15 deg decreases as astrain increases. The stiffening effect keeps decreasing with adecreasing h. The FE stress-strain curves are almost placed on thesame curve of the CMT model for MS honeycombs with negativecell angles as shown in Figs. 20(d) and 20(e). The MS honeycombwith h¼�20 deg shows the highest (cpl*)12 of 0.017, when hon-eycombs are designed to be the same G12* of 6.5 MPa.

Observing the deformation of MS honeycombs, the geometricnonlinear effect of honeycombs subjected to in-plane shear pri-marily results from the large bending of the vertical cell wall, h.For a fixed H, the honeycombs with a low effective cell height,h=H, are prone to be affected by the nonlinear bending effect of hfor a shear strain, c12*; e.g., the MS honeycomb with h¼ 60 deg.On the other hand, the honeycombs with a high h=H is relativelyless affected by the nonlinear local bending of the h for an increas-ing c12*; e.g., the MS honeycomb with h¼ �20 deg. We alsoobserved these effects in our previous studies on the shear behav-ior of honeycombs with a linear material model [19,20].

Simple shear behaviors of PC honeycombs are also simulatedwith the FE model considering both material and geometry

Fig. 22 (cpl*)12 as a function of density

Fig. 23 (spl*)12 of honeycombs as a function of density for a G12* of 6.5 MPa

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nonlinearities, and their shear stress-strain curves are plotted inFig. 21. The honeycomb with h¼ 60 deg shows a stiffness reduc-tion up to a c12* of 0.01 due to the stiffness reduction effect ofPC. As a c12* increases, the nonlinear bending effect of the hbecomes dominant, resulting in the global stiffening effect startingfrom a c12* of 0.01 as shown in Fig. 21(a). The global stiffnessreduction is more obviously noticed in the honeycomb withh¼ 30 deg as can be seen in Fig. 21(b), with the stiffness reduc-tion evident over a broad range of c12* up to 0.03. A subsequentglobal stiffening effect is also observed after passing a c12* of0.03 associated with the dominant nonlinear bending effect of theh. As a h decreases, though the stiffness reduction effect of the PCbecomes more dominant over a broader range of c12*, the nonlin-ear bending effect of the h decreases due to a higher h=H, result-ing in the global shear stiffness reduction of honeycombs(Figs. 21(c)–21(e)). The PC honeycomb with h¼�20 deg showsthe most shear compliant structure as can be seen in Fig. 21(f).

The CMT based shear stress-strain curves overestimate theelastic limit of honeycombs in shear, because the effective prop-erty based model does not cover the local microrotation of cellwalls, which induces more local stresses, resulting in a lower(cpl*)12.

The FEA results of (cpl*)12 of MS and PC honeycombs as afunction of density as shown in Fig. 22 are compared with theresults from CMT, which are also presented in Fig. 15. The FEAresults also show that the PC auxetic honeycombs have a high(cpl*)12 associated with a combined effect of a high yield strain,eys of PC and geometric effect. CMT overestimates (cpl*)12 of MSand PC honeycombs by approximately 21.5–29.1% and21.7–30.2%, respectively, because CMT does not cover the non-linear behaviors of a core material and the large local cell wallbending. The honeycombs with h¼ 60 deg show the highesterrors; 29.1% and 30.2% with MS and PC, respectively due to thehigh stiffening effect as shown in Figs. 20(a) and 21(a). Consider-ing the difference in errors between MS and PC honeycombs,nonlinearity of the core material does not appear to be high com-pared with geometric nonlinearity.

Figure 23 shows the shear yield strengths, (spl*)12 of MS andPC honeycombs as a function of density. CMT also generallyoverestimates (spl*)12 of honeycombs by 9.8–20.8% and �4.53–33.4% with MS and PC honeycombs. However, CMT underesti-mates (spl*)12 of the PC honeycomb with h¼ 60 deg due to itshighly nonlinear stiffening effect as shown in Fig. 21(a). Interest-ingly, the values of (spl*)12 of PC honeycombs are higher thanthose of MS honeycombs even though rys of PC is approximately2.5 times lower than that of MS due to the low yield strain, eys ofMS.

In terms of designing a high shear strain and high shear strengthof honeycombs for a fixed G12*, a (spl*)12� (cpl*)12 diagram isplotted in Fig. 24 by combining Figs. 22 and 23. The PC auxetichoneycomb with h ¼� 20 deg shows the highest flexible prop-erty; a (spl*)12 of 0.78 MPa and a (cpl*)12 of 12.9%. Though theCMT qualitatively ranks the elastic limits of honeycombs in shear,it does not appear to be enough to explain them quantitatively(Fig. 24).

5 Concluding Remarks

Using analytical and FE studies, honeycomb dimensions wereobtained when MS and PC honeycombs were designed with thesame G12*. For a fixed G12*, (spl*)12 and (cpl*)12 of auxetic andconventional hexagonal honeycombs with MS and PC were inves-tigated with geometric and materials selections. The key findingsof the present study are that:

• While both MS and PC present the possibility of tailoringhoneycombs having a G12* of 6.5 MPa with 0.1–0.5 mm and0.3–2.2 mm cell wall thicknesses, respectively, depending onthe number of vertical stacks, N, the high MS modulusgreatly enhances the sensitivity of the effective MS honey-comb properties to the cell wall thickness;

• A single layer design is preferable to a multilayer one interms of a lower sensitivity of the cell wall thickness on theeffective properties;

• The PC auxetic honeycomb with h ¼� 20 deg shows highshear flexibility when all honeycombs are designed to be theG12* of 6.5 MPa; a 0.72 MPa (spl*)12 and a 13% (cpl*)12;

• While the CMT model can qualitatively rank the shear flexi-bility of honeycombs, the qualitative analysis is limitedregarding the elastic limits of honeycombs as it does not han-dle the local cell wall microrotation, leading a local stressconcentration.

Honeycombs have a large design space controlling both materi-als and geometry. In this study, we detailed an example of honey-comb design, where dual target properties (an effective modulusand an effective strain) are desired. These results will be experi-mentally validated in future work. We will also explore the cellconfigurations of h 6¼ l to obtain the dual target properties as acontinuation of the present study.

Acknowledgment

This work was supported by National Institute of Standards andTechnology (NIST)—Advanced Technology Program (ATP).

Fig. 24 ( spl*)122 (cpl*)12 diagram for high shear strength and shear strain design

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