Composite Structures 107 (2014) 698–710

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Composite Structures

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Elastic and transport properties of the tailorable multifunctionalhierarchical honeycombs

0263-8223/$ - see front matter � 2013 Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.compstruct.2013.07.012

⇑ Corresponding author. Address: Laboratory of Bio-Inspired & GrapheneNanomechanics, Department of Civil, Environmental and Mechanical Engineering,University of Trento, Trento I-38123, Italy. Tel.: +39 0461 282525; fax: +39 0461282599.

E-mail address: nicola.pugno@unitn.it (N. Pugno).

Yongtao Sun a, Qiang Chen b, Nicola Pugno c,d,⇑a School of Mechanical Engineering, Tianjin University, Tianjin 300072, PR Chinab Laboratory of Biomechanics, School of Biological Science and Medical Engineering, Southeast University, Nanjing 210096, PR Chinac Laboratory of Bio-Inspired & Graphene Nanomechanics, Department of Civil, Environmental and Mechanical Engineering, University of Trento, Trento I-38123, Italyd Center for Materials and Microsystems, Fondazione Bruno Kessler, Via Sommarive 18, 38123 Povo (Trento), Italy

a r t i c l e i n f o

Article history:Available online 15 July 2013

Keywords:In-plane effective moduliIn-plane effective conductivityOriginal regular hexagonal honeycomb(ORHH)Multifunctional hierarchical honeycomb(MHH)

a b s t r a c t

In this paper, we analytically studied the in-plane elastic and transport properties of a peculiar hexagonalhoneycomb, i.e., the multifunctional hierarchical honeycomb (MHH). The MHH structure was developedby replacing the solid cell walls of the original regular hexagonal honeycomb (ORHH) with three kinds ofequal-mass isotropic honeycomb sub-structures possessing hexagonal, triangular and Kagome lattices.Formulas to calculate the effective in-plane elastic properties and conductivities of the MHH structureat all densities were developed. Results show that the elastic properties of the MHH structure with thehexagonal sub-structure were weakly improved in contrast to those of the ORHH. However, the triangu-lar and Kagome sub-structures result in substantial improvements by one or even three orders of mag-nitude on Young’s and shear moduli of the MHH structure, depending on the cell-wall thickness-to-length ratio of the ORHH. The present theory could be used in designing new tailorable hierarchical hon-eycomb structures for multifunctional applications.

� 2013 Elsevier Ltd. All rights reserved.

1. Introduction

Low-density cellular materials widely exist in Nature and exhi-bit fascinating mechanical properties in the aspects of strength,stiffness, toughness, etc. [1–3]. As a typical low-density cellular so-lid, honeycomb, which is mainly used as cores of light-weightsandwich panel structures [4–7], has been used in many fields,such as aerospace and automotive industries. Apart from its pecu-liar low-density and mechanical properties, honeycomb also showsother attractive functionalities, e.g., heat transfer, thermal protec-tion, catalysis application and so on. In order to find optimal topol-ogies for different multifunctional applications, varieties ofprismatic honeycombs have been developed and studied in recentyears.

Regarding the multifunctionality of honeycomb materials, Lu[8] and Gu et al. [9] reported that the regular hexagonal metalliccell, compared with triangular and square cells, provides the high-est level of heat dissipation, which is comparable to that of the

open-cell metal foams. Combining experimental and numericalmethods, Wen et al. [10] revealed that the overall thermal perfor-mances of metallic honeycomb structures are superior to otherheat sink media, such as metallic foams, lattice-frame materials,3D Kagome structures and woven textile structures. Employingthe topology optimization technique, Hyun and Torquato [11]showed that only the effective conductivity of the regular hexago-nal honeycomb tends to the Hashin–Shtrikman (H–S) upperbounds; while for triangular and Kagome honeycombs, both thein-plane effective moduli and conductivity approach the H–S upperbounds [12,13]. Hayes et al. [14] studied the mechanical and ther-mal properties of linear cellular alloys with square cells, and con-cluded that the mechanical and heat transfer characteristics ofthe honeycomb materials outperformed those of the open- andclosed-cell metallic foams with comparable relative density. Vaziriand his collaborators [15–25] focused on metallic sandwich panelswith different kinds of cellular cores, such as hexagonal honey-combs [15], square honeycombs [16–20], open-cell rhombicdodecahedron cellular structures [21] and pyramidal truss cores[22–25], and explored their multifunctional applications, such asenergy absorption [24], sustaining shock loading [16–18] andunderwater impulsive loading [19,20]. Besides, Evans et al. [26],Wadley et al. [27] and Wadley [28] reviewed the multifunctional-ities and the fabrication technologies of the multifunctional peri-odic cellular metals with different topological structures.

Y. Sun et al. / Composite Structures 107 (2014) 698–710 699

Regarding the mechanical properties of honeycomb materials,Wang and McDowell [29] investigated the in-plane stiffness andyield strength of different periodic metallic honeycombs, andshowed that the diamond, triangular and Kagome cells have supe-rior in-plane mechanical properties than the hexagonal, rectangu-lar and mixed square/triangular cells. Fleck and Qiu [30] analyzedthe damage tolerance property of 2D elastic-brittle isotropic hon-eycombs and reported that Kagome cells have much higher frac-ture toughness than those of hexagonal and triangular cells.

Another concept related to honeycomb materials is the introduc-tion of hierarchy. Compared with their single length scale micro-structure counterparts, structural hierarchy in natural materialscan result in significantly higher stiffness or strength efficiencies(i.e. stiffness- or strength-to-weight ratios), and maintain theirflaw-tolerance or energy-absorbing property [31–36]. Therefore,many researchers [37–44] have focused on the mechanical proper-ties of the hierarchical cellular structures. Burgueno et al. [45] stud-ied the hierarchical cellular designs for load-bearing bio-compositebeams and plates. Kooistra et al. [46] investigated hierarchical cor-rugated core sandwich panels and revealed that second-ordertrusses can have much higher compressive and shear strengths thantheir equal-mass first-order counterparts when relative densitiesare less than 5%. Fan et al. [47] studied two-dimensional cellularmaterials made up of sandwich struts and showed that the relevantmechanical properties of the materials are improved substantiallyby incorporating structural hierarchy. Inspired by diatom algaewhich contains nanoporous hierarchical silicified shells, Garciaet al. [48] revealed the toughening mechanism in the superductilewavy silica nanostructures by performing a series of moleculardynamics simulations. Taylor et al. [49] introduced the functionallygraded hierarchical honeycombs by performing a set of finite ele-ment analyses, and their results suggested that the Young’s modulusof the functionally graded hierarchical honeycomb can be 1.75 timesthat of its equal-mass first-order hexagonal honeycomb if the struc-ture is designed properly. Different from the topology of the com-mon hierarchical honeycomb structures [37,38,43,49], Vaziri’sgroup recently [50] developed a new hierarchical honeycomb struc-ture by replacing every three-edge joint of a regular hexagonal lat-tice with a smaller hexagon, and showed that the Young’smodulus of the hierarchical honeycombs with one level and two lev-els can be 2.0 and 3.5 times stiffer than their equal-mass regular hex-agonal honeycomb, respectively. And more, Chen and Pugno [42–44] explored the buckling behavior of 2D hierarchical honeycombsand Young’s modulus and strength of 3D hierarchical foams by con-sidering the surface effect at the bottom level of these structures.

In this paper, along the line, we analytically studied the in-planeelastic moduli and thermal conductivity of a multifunctional hierar-chical honeycomb (MHH). The MHH structure is formed by replac-ing the solid cell walls of an original regular hexagonal honeycomb(ORHH) with three different isotropic honeycomb sub-structurespossessing hexagonal, triangular or Kagome lattices. First, we derivethe theoretical formulas of the effective elastic moduli of the regularhexagonal honeycombs for all densities. Then, the in-plane Young’s,shear and bulk moduli of the three kinds of MHH structures are cal-culated. Besides, the effective in-plane conductivities of the MHHstructures are also formulated through the H–S upper bounds.

2. Effective in-plane elastic moduli of the regular hexagonalhoneycombs for all densities

Hyun and Torquato [51] analytically studied the effective in-plane properties of the regular hexagonal honeycomb for all densi-ties via three-point approximations and expressed the effectiveYoung’s modulus Ee (Fig. 1b) as

Ee

Es¼ /ð2f� 1Þðfþ g� 1Þf3� 2/� 2ð2� /Þð1� fÞ þ ð2� f� gÞ½2/ð1� fÞ � 1�g ð1Þ

in which / is the relative density of the hexagonal honeycomb, Es isthe Young’s modulus of its constituent solid, f and g are the three-point parameters (Fig. 1a). The simulation data of the effectiveYoung’s modulus Ee [51] are also provided in Fig. 1b. It is apparentthat for the high density case (/ P 0.5), the prediction by the three-point approximations method matches very well the simulationdata, but for the lower density case, it overestimates the results,and in the super low density case, the overestimation is sogreat that the three-point approximations method is not valid anymore.

It is well-known that for the low-density regular hexagonalhoneycombs, the Euler beam theory and the Timoshenko beamtheory can effectively predict their materials constants. Here, weextend the Euler beam theory and the Timoshenko beam theoryto the all relative densities cases and compare the results withthe three-point approximations. Under the Euler beam theory, Tor-quato et al. [52] expressed the effective Young’s modulus Ee as:

Ee

Es¼ 3

2/3 ð2Þ

And employing the Timoshenko beam theory Gibson and Ashby [3]studied the elastic properties of the low density honeycombs. Forthe regular hexagonal honeycomb, the effective Young’s modulusis given by:

Ee

Es¼ 4ffiffiffi

3p t

l

� �3 1

1þ ð5:4þ 1:5msÞðt=lÞ2ð3Þ

in which ms is the Poisson’s ratio of its constituent solid andt=l ¼

ffiffiffi3p

1�ffiffiffiffiffiffiffiffiffiffiffiffi1� /

p� �is the cell-wall thickness-to-length ratio.

For honeycombs at all densities, the comparison between theEuler beam theory, Timoshenko beam theory, three-point approx-imations method and the simulation data is plotted in Fig. 1b. Wecan see that when / 6 0.5 the result calculated by the Euler beamtheory matches very well the simulation data, while the result iswell predicted by the three-point approximations method when/ > 0.5. Therefore, the effective Young’s modulus of the regularhexagonal honeycombs for all densities can be piecewisely ex-pressed as:

Ee

Es¼

32 /3 / 6 0:5

/ð2f�1Þðfþg�1Þf3�2/�2ð2�/Þð1�fÞþð2�f�gÞ½2/ð1�fÞ�1�g / > 0:5

(ð4Þ

Besides, through the three-point approximations method, Hyunand Torquato [51] also derived the effective in-plane bulk moduluske of the regular hexagonal honeycomb at all densities:

ke

ks¼ Gs/ð2f� 1Þ

ksð1� /Þ þ Gs½1þ 2/ðf� 1Þ�

¼ Gs=ks/ð2f� 1Þð1� /Þ þ Gs=ks½1þ 2/ðf� 1Þ� ð5Þ

where ks and Gs are the bulk and shear moduli of the constituent so-lid, respectively.

Due to the in-plane isotropic properties, ks, Es, ke, Ge and Ee sat-isfy the following relationships:

ks ¼Es

2ð1� msÞð6Þ

ke ¼Ee

2ð1� meÞð7Þ

Ge ¼Ee

2ð1þ meÞð8Þ

Fig. 1. (a) Three-point parameters f and g for the regular hexagonal honeycomb [51] vs the relative density /. (b) Effective Young’s modulus Ee/Es of the regular hexagonalhoneycomb vs the relative density / predicted by different methods.

700 Y. Sun et al. / Composite Structures 107 (2014) 698–710

in which Ge and me are the effective in-plane shear modulus andPoisson’s ratio of the hexagonal honeycombs, respectively. DefiningEe/Es = A and ke/ks = B, Eqs. (4)–(8) provide the formula for Ge:

Ge

Es¼ AB

4B� 2Að1� msÞð9Þ

Then, the effective Poisson’s ratio me of the regular hexagonal hon-eycombs can be derived through dividing Eq. (6) by Eq. (7):

me ¼ 1� ABð1� msÞ ð10Þ

To verify the expressions of Ee, ke and Ge, Eq. (10) is depicted inFig. 2 for the honeycombs at all densities with ms = 1/3. Note that inthe calculations and following sections, the three-point parametersf and g are interpolated from Fig. 1a.

Fig. 2 shows excellent agreement with the existing results [3,53],that is to say, me tends to 1 for the extreme low densities and tends toPoisson’s ratio of constituent solid ms = 1/3 for the extreme high den-sities. This implies the validations of Eqs. (4), (5) and (9), which willbe employed to study the relevant properties of the MHH. It is worthmentioning that different from the formula in Ref. [3], here Poisson’sratio is not a constant when the relative density is low.

3. MHH with isotropic hexagonal sub-structure

3.1. Basic theory

First of all, we consider the MHH with the isotropic hexagonallattice sub-structure (Fig. 3). The thicknesses of the ORHH(Fig. 3a) and MHH (Fig. 3b) are denoted by t0 and t1, respectively,and their lengths denoted by l0 are considered to be identical. Inparticular, one of the MHH cell walls in Fig. 3b is shown in

Fig. 2. The effective Poisson’s ratio me of the regular hexagonal honeycomb withms = 1/3 vs the relative density /.

Fig. 3c, where the cell-wall thickness and length of the hexagonalcells are denoted by th and lh, respectively. The out-of-plane depthis a constant and identical for both structures.

The geometry of Fig. 3c implies:

l0 ¼ nlh þ ðnþ 1Þð2lhÞ ¼ ð3nþ 2Þlh ð11Þ

where n is the number of the solid hexagonal cell walls lying on themiddle line of the MHH cell walls (e.g., in Fig. 3c, n = 8). Definingk = lh/l0 as the hierarchical length ratio, rearranging Eq. (11) provides,

k ¼ 13nþ 2

ð12Þ

Then, defining N as the number of hexagonal cells away fromthe middle line of the MHH cell walls (e.g., in Fig. 3c, N = 1), andM the total number of half-thickness hexagonal cells in a MHH cellwall (see Appendix A), then, the relationship between M and N canbe expressed as:

M ¼ 2Nð2nþ 1Þ þ n3� 4AN ð13Þ

with AN = [(2N + 1)(N � 1) + 1]/6. After that, basing on the massequivalence between cell walls of the MHH and the ORHH, we findt0l0 � 1

2ffiffi3p t2

0 ¼ 6� 12 thlh � 1

2ffiffi3p t2

h

� �M, which gives

th

lh¼

ffiffiffi3p

1�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� 2

3ffiffiffi3p

k2M

t0

l01� 1

2ffiffiffi3p t0

l0

� �s" #ð14Þ

Besides, a geometrical analysis in Fig. 3c provides Nmax, the upperbound of N, and t1, the thickness of the MHH cell walls:

Fig. 3. (a) The original regular hexagonal honeycomb (ORHH); (b) the tailorablemultifunctional hierarchical honeycomb (MHH) with hexagonal sub-structure; (c)amplification of a hexagonal lattice cell wall in (b) (the cell walls marked by bluecircle suggest n = 8 and the dash line denotes the middle line of the MHH cell wall).

Y. Sun et al. / Composite Structures 107 (2014) 698–710 701

Nmax ¼ fl12k

� �¼ fl

3nþ 22

� �ð15Þ

t1 ¼2N

ffiffiffi3p

lh� �

þ th 1 6 N 6 Nmax � 1

2�ffiffi3p

2 l0 N ¼ Nmax

8<: ð16Þ

where ‘fl[ ]’ is the floor function, which denotes the largest integernot greater than the term in the bracket. Then, rearranging Eq.(16) gives:

t1

l0¼

2ffiffiffi3p

N þ thlh

� �k 1 6 N 6 Nmax � 1ffiffiffi

3p

N ¼ Nmax

8<: ð17Þ

On the other hand, Eq. (14) requires 1� 2= 3ffiffiffi3p

k2M� �

t0=l0

1� 1= 2ffiffiffi3p� �

t0=l0

h iP 0. Considering Eqs. (12) and (13), the

inequality provides Nmin, the lower bound of N:

Nmin ¼ ce3nþ 2�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið3nþ 2Þ2 1� 2ffiffi

3p t0

l01� 1

2ffiffi3p t0

l0

� �h iþ n

r2

2664

3775 ð18Þ

where ‘ce[ ]’ is the ceiling function, which denotes the smallest inte-ger not less than the term in the bracket. Note that Eq. (18) may giveNmin = 0, in this case Nmin = 1.

Defining the in-plane Young’s, shear and bulk moduli of theORHH as EO, GO and kO, then, from Eqs. (4), (5) and (9), we find:

EO

Es¼ AO ¼

32 /3

O /O 6 0:5/Oð2fO�1ÞðfOþgO�1Þ

f3�2/O�2ð2�/OÞð1�fOÞþð2�fO�gOÞ½2/Oð1�fOÞ�1�g /O > 0:5

(ð19Þ

kO

ks¼ BO ¼

Gs=ks/Oð2fO � 1Þð1� /OÞ þ Gs=ks½1þ 2/OðfO � 1Þ� ð20Þ

GO

Es¼ AOBO

4BO � 2AOð1� msÞð21Þ

where,

/O ¼2ffiffiffi3p t0

l0� 1

3t0

l0

� �2 t0

l06

ffiffiffi3p� �

ð22Þ

is the relative density of the ORHH.Besides, Hyun and Torquato [11,51] showed that the effective

thermal conductivity of the regular hexagonal honeycomb nearlyapproaches the H–S upper bounds. Thus, defining the thermal con-ductivities of the ORHH and the constituent solid as rO and rs, weapproximately obtain:rO

rs¼ /O

2� /Oð23Þ

Note that due to the mathematical analogy, results for the effectivethermal conductivity translate immediately into the equivalent re-sults for the effective dielectric constant, electrical conductivity andmagnetic permeability.

Similarly, defining the in-plane Young’s, shear and bulk moduliand thermal conductivity of the hexagonal sub-structure as Eh, Gh,kh and rh, we obtain:

Eh

Es¼ Ah ¼

32 /3

h /h 6 0:5/hð2fh�1Þðfhþgh�1Þ

f3�2/h�2ð2�/hÞð1�fhÞþð2�fh�ghÞ½2/hð1�fhÞ�1�g /h > 0:5

(ð24Þ

kh

ks¼ Bh ¼

Gs=ks/hð2fh � 1Þð1� /hÞ þ Gs=ks½1þ 2/hðfh � 1Þ� ð25Þ

Gh

Es¼ AhBh

4Bh � 2Ahð1� msÞð26Þ

rh

rs¼ /h

2� /hð27Þ

where,

/h ¼2ffiffiffi3p th

lh� 1

3th

lh

� �2 th

lh6

ffiffiffi3p� �

ð28Þ

is the relative density of the hexagonal sub-structure. Denoting theeffective Poisson’s ratio of the hexagonal sub-structure by mh, then,the relation Gh = Eh/[2(1 + mh)] holds. Then, combining Eqs. (24) and(26) gives,

mh ¼ 1� Ah

Bhð1� msÞ ð29Þ

Thus,

Gh

kh¼ Eh=½2ð1þ mhÞ�

Eh=½2ð1� mhÞ�¼ 1� mh

1þ mh¼ Ahð1� msÞ

2Bh � Ahð1� msÞð30Þ

In the meantime, defining the in-plane Young’s, shear and bulkmoduli and thermal conductivity of the MHH as EM, GM, kM andrM, it is easy to obtain:

EM

Eh¼AM ¼

32/3

M /M 60:5/M ð2fM�1ÞðfMþgM�1Þ

f3�2/M�2ð2�/MÞð1�fM Þþð2�fM�gMÞ½2/Mð1�fMÞ�1�g /M >0:5

(ð31Þ

kM

kh¼BM ¼

Gh=kh/Mð2fM�1Þð1�/MÞþGh=kh½1þ2/MðfM�1Þ� ð32Þ

GM

Eh¼ AMBM

4BM�2AMð1�mhÞð33Þ

rM

rh¼ /M

2�/Mð34Þ

where,

/M ¼2ffiffiffi3p t1

l0� 1

3t1

l0

� �2 t1

l06

ffiffiffi3p� �

ð35Þ

Combining Eqs. (19), (24) and (31) yields the relative Young’smodulus EM/EO:

EM

EO¼ AMAh

AOð36Þ

Similarly, from Eqs. (20), (25) and (32), we can get the relative in-plane bulk modulus kM/kO:

kM

kO¼ BMBh

BOð37Þ

And from Eqs. (21), (24) and (33), we obtain the relative shear mod-ulus GM/GO:

GM

GO¼ AMBMAh

2BM � AMð1� mhÞ2BO � AOð1� msÞ

AOBOð38Þ

Finally, from Eqs. (23), (27) and (34), we find the relative thermalconductivity rM/rO:

rM

rO¼ /M/hð2� /OÞ

/Oð2� /MÞð2� /hÞð39Þ

3.2. Effect of N on the relative elastic moduli and thermal conductivityof the MHH with hexagonal sub-structure

To investigate the influence of N on the relative elastic moduliEM/EO, GM/GO, kM/kO and the relative thermal conductivity rM/rO,here, we discuss the following example with parameters n = 16,k = 1/(3n + 2) = 0.02 and t0/l0 = 0.01, 0.05, 0.1, 0.2 and 0.3. Then,we can find Nmax = 25 through Eq. (15), and Nmin for each t0/l0through Eq. (18). The results are reported in Figs. 4–6, respectively.

From Fig. 4a and b, we can see that almost for all t0/l0 consid-ered, the optimal EM/EO and GM/GO, which vary between 1 and 2,exist as N increases. Note that the optimal EM/EO and GM/GO maydo not correspond to the same N. The reason is that EO/Es and

Fig. 4. (a) The relative Young’s modulus EM/EO vs N for different t0/l0. (b) The relative shear modulus GM/GO vs N for different t0/l0.

Fig. 5. The relative bulk modulus GM/GO vs N for different t0/l0.

Fig. 6. The relative thermal conductivity GM/GO vs N for different t0/l0.

702 Y. Sun et al. / Composite Structures 107 (2014) 698–710

EM/Eh depend on Poisson’s ratios ms and mh but GO/Gs and GM/Gh donot [51].

Figs. 5 and 6 show that the relative bulk modulus kM/kO and therelative thermal conductivity rM/rO increase with the increase oft0/l0, but in general, they are less than one. This implies that theeffective bulk modulus and thermal conductivity of the MHH withregular hexagonal sub-structure are less than those of the ORHHstructure. Of particular interest, there is the lowest value for thethermal conductivity, and this could be used to design low heatconductivity materials with the optimal topology.

3.3. The effects of t0/l0-the cell-wall thickness-to-length ratio of theORHH

To investigate the effects of the cell-wall thickness-to-length ra-tio t0/l0 of the ORHH on the relative elastic moduli and thermalconductivity of the MHH structure, again, we use the above exam-ple given in Section 3.2. We maintain n = 16, k = 0.02 but vary t0/l0from 0.01 to 0.5 with an increment of 0.01. In fact, under the sameN, the value of GM/GO is slightly greater than EM/EO (Fig. 4), so herewe only consider the relative Young’s modulus EM/EO influenced byt0/l0. Regarding rM/rO and kM/kO, the discussions on them will notbe treated in this section, since their values are less than one(Figs. 5 and 6), which shows the inferior properties of the MHHto those of the ORHH.

Finally, the maximum EM/EO influenced by t0/l0 is reported inFig. 7. We can see that the maximum EM/EO increases before t0/l0reaches 0.07 but after that it decreases. In other words, the optimalEM/EO of the MHH with hexagonal sub-structure exists at t0/l0 = 0.07, of which the value approximately equals 2. The result iscomparable to the finite element result given by Taylor et al. [49].

4. MHH with triangular sub-structure

4.1. Basic theory

In this section, we obtain the second topology of the MHH bysubstituting the ORHH cell walls with the equal-mass isotropic tri-angular sub-structure, see Fig. 8. As defined in Section 3, the hier-archical length ratio is expressed as

Fig. 7. The maximum EM/EO vs t0/l0.

Fig. 8. Schematics of (a) the ORHH; (b) the tailorable MHH with triangular sub-structure; and (c) amplification of a trianglar lattice cell wall in (b).

Y. Sun et al. / Composite Structures 107 (2014) 698–710 703

k ¼ ltl0¼ 1

nðn P 1Þ ð40Þ

where n is the number of solid triangular lattice cell walls lying onthe middle line of the MHH cell walls. From Fig. 8c, according to theequal-mass principle, we can find t0l0 � 1

2ffiffi3p t2

0 ¼ 3� 12 ttlt �

ffiffi3p

2 t2t

� �M,

which gives

tt

lt¼ 1ffiffiffi

3p 1�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� 4

ffiffiffi3p

3k2M

t0

l01� 1

2ffiffiffi3p t0

l0

� �s24

35 ð41Þ

where M is the total number of half thickness triangular lattice cellsin a MHH cell wall and it has the following relationship with n andN (see Appendix B):

M ¼ 2Nð2n� NÞ þ 23ðn� NÞð1 6 N 6 nÞ ð42Þ

in which N is the number of triangular lattice cells away from themiddle line of the MHH cell walls. Similar to that in Section 3, a geo-metrical analysis in Fig. 8c provides Nmax, the upper bound of N, andt1, the thickness of the MHH cell walls:

Nmax ¼ n ð43Þ

t1 ¼2N

ffiffi3p

2 lt

� �þ tt 1 6 N 6 Nmax � 1

2�ffiffi3p

2 l0 N ¼ Nmax

8<: ð44Þ

Then, rearranging Eq. (44) gives,

t1

l0¼

ffiffiffi3p

N þ ttlt

� �k 1 6 N 6 Nmax � 1ffiffiffi

3p

N ¼ Nmax

8<: ð45Þ

On the other hand, Eq. (41), requires1� 4

ffiffiffi3p

=ð3k2MÞt0=l0 1� 1= 2ffiffiffi3p� �

t0=l0

h iP 0. Then, employing

Eqs. (40) and (42), the inequality gives Nmin, the lower bound of N:

Nmin ¼ ce6n� 1�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið6n� 1Þ2 � 12n 2

ffiffiffi3p

n t0l0

1� 12ffiffi3p t0

l0

� �� 1

h ir6

2664

3775ð46Þ

Note that Eq. (46) may give Nmin = 0, in this case Nmin = 1.Like the discussion in Section 3, we would like to analyze the

effective elastic moduli and thermal conductivity of the triangularlattice sub-structure. As mentioned in the introduction, Hyun andTorquato [11] showed that for triangular and Kagome honey-combs, both the in-plane effective moduli and conductivity ap-proach the H–S upper bounds [12,13]. So, we approximately usethe H–S upper bounds to calculate the effective elastic moduliand thermal conductivity of the triangular lattice sub-structure.Defining the in-plane Young’s, shear and bulk moduli and thermalconductivity of the triangular sub-structure as Et, Gt, kt and rt, weobtain:

Et

Es¼ At ¼

/t

3� 2/tð47Þ

kt

ks¼ Bt ¼

/tGs=ks

1� /t þ Gs=ksð48Þ

Gt

Es¼ Ct ¼

12ð1þ msÞ

/t

ð1� /tÞð1þ 2Gs=ksÞ þ 1ð49Þ

rt

rs¼ /t

2� /tð50Þ

where,

/t ¼ 2ffiffiffi3p tt

lt� 3

tt

lt

� �2 tt

lt6

1ffiffiffi3p

� �ð51Þ

is the relative density of the triangular sub-structure. Denoting theeffective Poisson’s ratio of the triangular sub-structure by mt, therelation Gt = Et/[2(1 + mt)] holds. Then, combining Eqs. (47) and(49) gives,

mt ¼12

At

Ct� 1 ð52Þ

Thus,

Gt

kt¼ Et=½2ð1þ mtÞ�

Et=½2ð1� mtÞ�¼ 1� mt

1þ mt¼ 4

Ct

At� 1 ð53Þ

Defining the in-plane Young’s, shear and bulk moduli and thermalconductivity of the MHH with triangular sub-structure as EM, GM,kM and rM, we have:

EM

Et¼ AM ¼

32 /3

M /M 6 0:5/M ð2fM�1ÞðfMþgM�1Þ

f3�2/M�2ð2�/MÞð1�fMÞþð2�fM�gMÞ½2/Mð1�fM Þ�1�g /M > 0:5

(ð54Þ

kM

kt¼ BM ¼

Gt=kt/Mð2fM � 1Þð1� /MÞ þ Gt=kt ½1þ 2/MðfM � 1Þ� ð55Þ

GM

Et¼ AMBM

4BM � 2AMð1� mtÞð56Þ

rM

rt¼ /M

2� /Mð57Þ

where,

Fig. 10. The relative Young’s modulus GM/GO vs N for different t0/l0: (a) t0/l0 = 0.01; (b) t0/l0 = 0.05; (c) t0/l0 = 0.1, 0.2, 0.3.

Fig. 9. The relative Young’s modulus EM/EO vs N for different t0/l0: (a) t0/l0 = 0.01; (b) t0/l0 = 0.05; and (c) t0/l0 = 0.1, 0.2, 0.3.

704 Y. Sun et al. / Composite Structures 107 (2014) 698–710

/M ¼2ffiffiffi3p t1

l0� 1

3t1

l0

� �2 t1

l06

ffiffiffi3p� �

ð58Þ

and fM and gM, interpolated from Fig. 1a, are the three-point param-eters corresponding to /M.

Combining Eqs. (19), (47) and (54) gives the relative Young’smodulus EM/EO:

EM

EO¼ AMAt

AOð59Þ

Similarly, from Eqs. (20), (48) and (55), we can find the relative bulkmodulus kM/kO:

kM

kO¼ BMBt

BOð60Þ

And from Eqs. (21), (47) and (56), we obtain the relative shear mod-ulus GM/GO:

Fig. 11. The relative bulk modulus

GM

GO¼ AMBMAt

2BM � AMð1� mtÞ2BO � AOð1� msÞ

AOBOð61Þ

Finally, from Eqs. (23), (50) and (57), the relative thermal conduc-tivity rM/rO is derived:

rM

rO¼ /M/tð2� /OÞ

/Oð2� /MÞð2� /tÞð62Þ

4.2. Effects of N on the relative elastic moduli and thermal conductivityof the MHH with triangular sub-structure

As discussed in Section 3, the influence of N on the effectiveelastic moduli and thermal conductivity of the MHH with triangu-lar sub-structure are studied, here we consider the second examplewith parameters n = 20, k = 1/n = 0.05, t0/l0 = 0.01, 0.05, 0.1, 0.2 and0.3. Then, we immediately obtain Nmax = 20 by Eq. (23) and the

kM/kO vs N for different t0/l0.

Fig. 13. Schematics of (a) the ORHH; (b) the tailorable MHH with Kagome sub-structure (in this paper the red dashline is for the convenience of linear dimension);(c) amplication of a Kagome sub-structure cell wall in (b); and (d) the represen-tative cells for the Kagome honeycomb.

Y. Sun et al. / Composite Structures 107 (2014) 698–710 705

lower bound Nmin for each t0/l0 by Eq. (46). The relative elasticmoduli EM/EO, GM/GO, kM/kO and the relative thermal conductivityrM/rO vs N are reported in Figs. 9–12, respectively.

Figs. 9 and 10 show that the relative Young’s modulus EM/EO andthe relative shear modulus GM/GO increase with the increase of N,and the thickness-to-length ratio t0/l0 has a strong influence onthem. With respect to its equal-mass ORHH, the enhancementsof the relative Young’s and shear moduli of the MHH can be one or-der (Figs. 9c and 10c) or even three orders of magnitude (Figs. 9aand 10a). Although the enhancement on Young’s modulus of theMHH decreases with the increase of t0/l0, for a smaller t0/l0 (lessthan 0.3), its stiffening effect (Figs. 9 and 10) by the triangularsub-structure is much larger than that of the hexagonal counter-part (Fig. 4a and b).

The relative bulk modulus kM/kO and the relative thermal con-ductivity rM/rO shown in Figs. 11 and 12 have similar trends tothose of the MHH with the hexagonal sub-structure reported inSection 3.2. Thus, the discussion is omitted here.

5. MHH with isotropic Kagome sub-structure

5.1. Basic theory

Kagome honeycomb has been revealed to have pronouncedfracture toughness [30] and better thermal–mechanical perfor-mance than the triangular honeycomb [54]. Therefore, in this sec-tion, we will consider the third topology of the MHH, namely,substituting the ORHH cell walls with their equal-mass Kagomesub-structure (Fig. 13), and study its effective elastic moduli andthermal conductivity. In this case, the hierarchical length ratio isexpressed as:

k ¼ lk

l0¼ 1

nn ¼ 4;6;8;10 . . . ð63Þ

where lk is the side length of triangles in Kagome cells and n is thenumber of sides of the effective triangles on the middle line of theMHH cell walls. Again, the equal-mass principle providest0l0 � 1

2ffiffi3p t2

0 ¼ 3� tklk �ffiffi3p

2 t2k

� �M, and tk/lk is derived as:

tk

lk¼ 1ffiffiffi

3p 1�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� 2

ffiffiffi3p

3k2M

t0

l01� 1

2ffiffiffi3p t0

l0

� �s24

35 ð64Þ

where M is the total number of triangles in a MHH cell wall, and it isexpressed with n and N as (see Appendix C):

M ¼ 2Nðn� NÞ 1 6 N 6n2

� �ð65Þ

Fig. 12. The relative thermal conductivity rM/rO vs N for different t0/l0.

in which N is the number of the Kagome representative cells awayfrom the middle line of the MHH cell walls (e.g., in Fig. 13c, N = 1).Similar to those in Sections 3 and 4, a geometrical analysis inFig. 13c provides Nmax, the upper bound of N, and t1, the thicknessof the MHH cell walls:

Nmax ¼n2

ð66Þ

t1 ¼2N

ffiffiffi3p

lk

� �þ 2tk 1 6 N 6 Nmax � 1

2�ffiffi3p

2 l0 N ¼ Nmax

8<: ð67Þ

Then, rearranging Eq. (67) gives

t1

l0¼

2ffiffiffi3p

N þ tklk

� �k 1 6 N 6 Nmax � 1ffiffiffi

3p

N ¼ Nmax

8<: ð68Þ

Again, Eq. (64) requires 1� 2ffiffiffi3p

=ð3k2MÞt0=l0 1� 1= 2ffiffiffi3p� �

t0=l0

h iP 0. In conjunction with Eqs. (63) and (65), the inequality producesNmin, the lower bound of N:

Nmin ¼ cen� n

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� 4

ffiffi3p

3t0l0

1� 12ffiffi3p t0

l0

� �r2

2664

3775 ð69Þ

Also, when Nmin = 0, Eq. (69) provides Nmin = 1.Here, the H–S upper bounds is again employed to analyze the

effective elastic moduli and thermal conductivity of the Kagomelattice sub-structure. Defining the in-plane Young’s, shear and bulkmoduli and thermal conductivity of the Kagome sub-structure asEk, Gk, kk and rk, respectively, we obtain:

Ek

Es¼ Ak ¼

/k

3� 2/kð70Þ

kk

ks¼ Bk ¼

/kGs=ks

1� /k þ Gs=ksð71Þ

Gk

Es¼ Ck ¼

12ð1þ msÞ

/k

ð1� /kÞð1þ 2Gs=ksÞ þ 1ð72Þ

rk

rs¼ /k

2� /kð73Þ

where,

/k ¼ffiffiffi3p tk

lk� tk

lk

� �2 tk

lk6

1ffiffiffi3p

� �ð74Þ

Fig. 14. The relative Young’s modulus EM/EO vs N for different t0/l0: (a) t0/l0 = 0.01; (b) t0/l0 = 0.05; and (c) t0/l0 = 0.1, 0.2, 0.3.

Fig. 15. The relative Young’s modulus GM/GO vs N for different t0/l0: (a) t0/l0 = 0.01; (b) t0/l0 = 0.05; and (c) t0/l0 = 0.1, 0.2, 0.3.

Fig. 16. The relative bulk modulus kM/kO vs N for different t0/l0.

706 Y. Sun et al. / Composite Structures 107 (2014) 698–710

is the relative density of the Kagome sub-structure. Denoting theeffective Poisson’s ratio of the Kagome sub-structure by mk, employ-ing Gk = Ek/[2(1 + mk)] and combining Eqs. (70) and (72) give

mk ¼12

Ak

Ck� 1 ð75Þ

Thus,

Gk

kk¼ Ek=½2ð1þ mkÞ�

Ek=½2ð1� mkÞ�¼ 1� mk

1þ mk¼ 4

Ck

Ak� 1 ð76Þ

Again, defining the in-plane Young’s, shear and bulk moduli andthermal conductivity of the MHH with Kagome sub-structure asEM, GM, kM and rM, we have:

EM

Ek¼AM ¼

32/3

M /M 60:5/Mð2fM�1ÞðfMþgM�1Þ

f3�2/M�2ð2�/M Þð1�fMÞþð2�fM�gMÞ½2/Mð1�fMÞ�1�g /M >0:5

(ð77Þ

kM

kk¼BM ¼

Gk=kk/Mð2fM�1Þð1�/MÞþGk=kk½1þ2/MðfM�1Þ� ð78Þ

GM

Ek¼ AMBM

4BM�2AMð1�mkÞð79Þ

rM

rk¼ /M

2�/Mð80Þ

where,

/M ¼2ffiffiffi3p t1

l0� 1

3t1

l0

� �2 t1

l06

ffiffiffi3p� �

ð81Þ

Fig. 17. The relative thermal conductivity rM/rO vs N for different t0/l0.

Y. Sun et al. / Composite Structures 107 (2014) 698–710 707

Combining Eqs. (19), (70) and (77) gives the relative Young’smodulus EM/EO:

EM

EO¼ AMAk

AOð82Þ

Similarly, from Eqs. (20), (71) and (78) we can get the relative bulkmodulus kM/kO:

kM

kO¼ BMBk

BOð83Þ

And from Eqs. (21), (70) and (79) we obtain the relative shear mod-ulus GM/GO:

GM

GO¼ AMBMAk

2BM � AMð1� mkÞ2BO � AOð1� msÞ

AOBOð84Þ

Finally, from Eqs. (23), (50) and (57), we find the relative thermalconductivity rM/rO:

rM

rO¼ /M/kð2� /OÞ

/Oð2� /MÞð2� /kÞð85Þ

Fig. 18. The relative Young’s modulus EM/EO vs N for different sub-structures withthe same parameters t0/l0 = 0.1 and k = 1/20.

5.2. Effects of N on the relative elastic moduli and thermal conductivityof the MHH with Kagome sub-structure

In this section, we consider the third example with parametersn = 20, k = 1/n = 0.05, t0/l0 = 0.01, 0.05, 0.1, 0.2 and 0.3. Then, Eq.(66) provides Nmax = 10 and Eq. (69) the lower bound Nmin for eacht0/l0. The results of the relative elastic moduli EM/EO, GM/GO, kM/kO

and the relative thermal conductivity rM/rO vs N are shown inFigs. 14–17, respectively.

Comparing Figs. 14 and 15 with Figs. 9 and 10, we can see thatYoung’s and shear moduli of the MHH with Kagome sub-structureare similar to those of the MHH with triangular sub-structure, sothe discussion can be referred to the Section 4.

However, it is worth to say that, different from the MHH withhexagonal and triangular sub-structures, the relative bulk moduluskM/kO and the relative thermal conductivity rM/rO of the MHH withKagome sub-structure become greater than one with the increaseof t0/l0 (Figs. 16 and 17). This is to say, when t0/l0 is great enough,the effective bulk modulus and thermal conductivity of the MHHwith Kagome sub-structure could be beyond those of the ORHHstructures.

6. Comparisons of hexagonal, triangular and Kagome sub-structures

Comparing the examples discussed in Sections 3–5, it is appar-ent that for an ORHH, the in-plane stiffness enhancements of the

Fig. 19. The relative bulk modulus kM/kO vs N for different sub-s

MHH with triangular and Kagome sub-structures could be muchgreater than that with the hexagonal sub-structure. This is illus-trated by the fourth examples with the parameters t0/l0 = 0.1 andk = 1/20 and the result is plotted in Fig. 18, in which the relativeYoung’s modulus EM/EO versus N for the MHH with the above threesub-structures are reported. Interestingly, we find that the relativeYoung’s moduli of the MHH with triangular and Kagome sub-struc-tures increase as N increases in contrast to that with hexagonalsub-structure. And more, Young’s modulus of the MHH with Kag-ome sub-structure is improved most with respect to the ORHH.

tructures with the same parameters t0/l0 = 0.3 and k = 1/20.

Fig. 20. The relative thermal conductivity rM/rO vs N for different sub-structures with the same parameters t0/l0 = 0.3 and k = 1/20.

Fig. A.1. Schematics of MHH cell walls in Fig. 1b: (a) N = 1 and (b) N = 2.

708 Y. Sun et al. / Composite Structures 107 (2014) 698–710

For the comparisons on bulk modulus and thermal conductivityof the three MHHs, the parameters t0/l0 = 0.3 and k = 1/20 are em-ployed, and the results are depicted in Figs. 19 and 20, respectively.From the two figures, we can say that the MHH with Kagome sub-structure is the optimal structure to design the bulk modulus andthermal conductivity of the multifunctional regular hexagonalhoneycomb.

7. Conclusions

In this paper, we have studied the in-plane elastic and trans-port properties of the MHH, which is formed by replacing theORHH solid cell walls with three types of equal-mass isotropichoneycomb sub-structures. The analytical results show that withthe hexagonal sub-structure it is difficult to greatly increaseYoung’s and shear moduli of the MHH. Whereas, triangular andKagome sub-structures share a similar improvement on theMHH’s Young’s and shear moduli, and the improvement is sub-stantial, from one order to three orders of magnitude dependingon the cell-wall thickness-to-length ratio t0/l0 of the ORHH.Meanwhile, if t0/l0 is great enough, the effective bulk modulusand transport ability of the MHH with Kagome sub-structurecan exceed those of the ORHH structure. These interesting find-ings show a possibility to design hierarchical honeycombs formultifunctional applications, e.g., the metallic MHH can be usedas the core of light weight sandwich panels in electronic packagesand airborne devices, where both the structural and thermal char-acteristics are desirable.

Acknowledgements

The research related to these results has received funding fromthe European Research Council under the European Union’s Sev-enth Framework Programme (FP7/2007-2013)/ERC Grant agree-ment nu [279985] (ERC Starting Grant BIHSNAM and ERCProof of Concept REPLICA, PI NMP on ‘‘Bio-inspired hierarchicalsuper nanomaterials’’). Q. Chen is supported by the Scientific Re-search Starting Fund from Southeast University. Y. Sun appreciatesthe China Scholarship Council (CSC) for the financial supports.

Appendix A. MHH cell wall with hexagonal sub-structures

Fig. A.1 shows the representative cell walls of the MHH withregular hexagonal sub-structures in Fig. 3b. The mass of the sub-structure is distributed uniformly among the half-thickness hexag-onal sub-structure cells within the blue hexagon.

From Fig. A.1 we can see that the number of the half-thicknesshexagonal sub-structure cells M can be determined by n and N asthe following form:

M ¼ 2N½nþ ðnþ 1Þ� þ 16ð2nÞ � 4AN

¼ 2Nð2nþ 1Þ þ n3� 4AN ðA:1Þ

in which A1 = 1/6 and A2 = 1.Here, AN depends on N, we find it generally expressed as:

AN ¼ð2N þ 1ÞðN � 1Þ þ 1

6ðN P 1Þ ðA:2Þ

Appendix B. MHH cell wall with triangular sub-structures

Fig. B.1 schematically shows the cell wall of the MHH with tri-angular sub-structures (Fig. 8b). The hierarchical length ratio isk = 1/n. M is the total number of the half-thickness triangular cellsin one sub-structure cell wall. It is easily to get the following rela-tion between M, N and n:

N ¼ 1 : M ¼ 2ð2n� 1� 1Þ þ 23ðn� 1Þ

N ¼ 2 : M ¼ 2ð2n� 2� 1� 1� 2Þ þ 23ðn� 2Þ

N ¼ 3 : M ¼ 2ð2n� 3� 1� 1� 2� 2� 3Þ þ 23ðn� 3Þ

ðB:1Þ

Fig. B.1. Schematics for the representative cell walls of the MHH with triangularsub-structures: (a) N = 1 and (b) N = 2.

Fig. C.1. Schematics for the representative cell walls of the MHH with Kagome sub-structures: (a) N = 1 and (b) N = 2.

Y. Sun et al. / Composite Structures 107 (2014) 698–710 709

Then, by inductive method, we find:

M ¼ 2ð2n� N � BNÞ þ23ðn� NÞ1 6 N 6 n ðB:2Þ

with

BN ¼ N2 ðB:3Þ

Substituting Eq. (B.3) into Eq. (B.2) gives:

M ¼ 2Nð2n� NÞ þ 23ðn� NÞð1 6 N 6 nÞ ðB:4Þ

Appendix C. MHH cell wall with Kagome sub-structures

Fig. C.1 schematically shows the cell wall of the MHH with Kag-ome sub-structures. The hierarchical length ratio is k = 1/n. M is thetotal number of the triangular cells included in one Kagome sub-structure cell wall. Then, the relationship between M, N and n areexpressed as:

N ¼ 1 : M ¼ 2½ðn� 1Þ � 0�N ¼ 2 : M ¼ 2½2ðn� 1Þ � 2�N ¼ 3 : M ¼ 2½3ðn� 1Þ � 2� 4�

ðC:1Þ

Likewise, we find:

M ¼ 2½Nðn� 1Þ � CN � 1 6 N 6n2

� �ðC:2Þ

with

CN ¼ NðN � 1Þ ðC:3Þ

Substituting Eq. (C.3) into Eq. (C.2) gives:

M ¼ 2Nðn� NÞ 1 6 N 6n2

� �ðC:4Þ

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