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Size-dependent and tunable elastic properties ofhierarchical honeycombs with regular square and equilateral

triangular cells

H.X. Zhu a,⇑, L.B. Yan a, R. Zhang a, X.M. Qiu b

a School of Engineering, Cardiff University, Cardiff CF24 3AA, UKb School of Aerospace, Tsinghua University, Beijing 100084, People’s Republic of China

Received 17 January 2012; received in revised form 9 May 2012; accepted 10 May 2012

Abstract

Simple closed-form results for all the independent elastic constants of macro-, micro- and nanosized first-order regular honeycombswith square and equilateral triangular cells and for the self-similar hierarchical honeycombs were obtained. It is found that, if the cell wallthickness of the first-order honeycomb is at the micrometer scale, the elastic properties of a hierarchical honeycomb are size dependent,owing to the strain gradient effects. Further, if the first-order cell wall thickness is at the nanometer scale, the elastic properties of a hier-archical honeycomb are not only size dependent owing to the effects of surface elasticity and initial stresses, but are also tunable. In addi-tion, the cell size and volume of hierarchical nanostructured cellular materials can be varied, and hierarchical nanostructured cellularmaterials could also possibly be controlled to collapse.Crown Copyright � 2012 Published by Elsevier Ltd. on behalf of Acta Materialia Inc. All rights reserved.

Keywords: Size effect; Honeycombs; Hierarchy; Elastic properties; Tunable properties

1. Introduction

In nature, living things evolve constantly to survive theirchanging environment. To support their own weight, toresist external loads and to enable different types of func-tions, their bodies should be structurally optimized andmechanically sufficiently strong. As a consequence, naturalliving materials are usually made up of hierarchical cellularstructures with basic building blocks at the micro- ornanoscale.

The elastic properties of honeycombs at the macroscalehave been extensively studied and well documented [1–7].However, the results obtained for macrohoneycombs maynot apply to their micro- and nanosized counterparts [8].It has been generally recognized that, at the micrometerscale, the strain gradient effect plays an important role in

the mechanical properties [8–14], and that, at the nanometerscale, both surface elasticity [8,15,16] and initial stresses[17,18] can greatly affect the mechanical properties of struc-tural elements. Atomistic simulations [19] suggest that, formetallic structural elements with a size of a few nanometers,the strain gradient effect is irrelevant, and surface elasticityor surface energy dominates the mechanical properties. Thesize-dependent bending rigidities have been obtained formicroplates [11] and nanoplates [18]. The size-dependenttransverse shear rigidities of micro- and nanoplates andthe size-dependent elastic properties of the first-order hon-eycombs with regular hexagonal cells were obtained inRef. [8]. Fan et al. [30] and Taylor et al. [31] have studiedthe effects of structural hierarchy on the elastic propertiesof honeycombs. However, they did not study the size-dependent effect or the tunable elastic properties for hierar-chical honeycombs. The aim of this paper is thus to obtainclosed-form results for the size-dependent and tunablemechanical properties of regular self-similar hierarchical

1359-6454/$36.00 Crown Copyright � 2012 Published by Elsevier Ltd. on behalf of Acta Materialia Inc. All rights reserved.

http://dx.doi.org/10.1016/j.actamat.2012.05.009

⇑ Corresponding author. Tel.: +44 29 20874824.E-mail address: [email protected] (H.X. Zhu).

www.elsevier.com/locate/actamat

Available online at www.sciencedirect.com

Acta Materialia 60 (2012) 4927–4939


honeycombs with square and equilateral triangular cellswhose first-order cell wall thickness is at the micron andnanometer scales.

2. Independent elastic constants of the first-order

honeycombs

The first-order honeycombs are treated as materialswhose size is much larger than the individual cells andare assumed to have uniform cell walls of length L, widthb and thickness h. The wall width b is assumed to be muchlarger than the thickness h. The focus of this paper is onsmall elastic deformation and the elastic properties.Although the cell wall can be of a metallic, biological orpolymeric material, it is always assumed to be isotropicand linear elastic, with Young’s modulus ES and Poissonratio vS, in the analysis that follows. As regular honey-combs with either square cells or equilateral triangular cellshave three orthogonal planes of symmetry, the maximumpossible number of independent elastic constants is nine[4,8,20].

2.1. Honeycombs with equilateral triangular cells

Fig. 1 shows a regular honeycomb with equilateral trian-gular cells. It is easy to show that its in-plane mechanicalproperties are isotropic. To fully determine the relationshipbetween the applied state of effective stress and theresponding state of effective strain, only five independentelastic constants – E1, m12, E3, v31 and G31 – need beobtained. The detailed derivation of these independentelastic constants is given in Appendix A.

The relative density of the first-order honeycomb withequilateral triangular cells is given by

q ¼ 2ffiffiffi3p

hL

ð1Þ

When the honeycomb is compressed by a uniform stressin the x direction, there is no junction rotation because ofthe symmetry, and the inclined cell walls undergo

combined axial compression, transverse shear deformationand plane-strain bending. Only a representative unit cellshown in Fig. 1b is needed for mechanics analysis. Byassuming that the bending stiffness of the cell walls is DB,the shear stiffness is DS and the axial compression stiffnessis DC, it is easy to obtain all five independent elastic con-stants for a perfect regular honeycomb with equilateral tri-angular cells and uniform cell walls.

The in-plane Young’s modulus of a perfect regular hon-eycomb with equilateral triangular cells can be obtained as(see Appendix A)

E1 ¼2DS � D2

C � L2 þ 24DB � D2C þ 24DB � DS � DCffiffiffi

3p

L3 � DS � DC þ 12ffiffiffi3p

L � DB � DC þ 4ffiffiffi3p

L � DB � DS

ð2ÞThe in-plane Poisson ratio is determined as

m12 ¼L2 � DS � DC þ 12DB � DC � 12DB � DS

3L2 � DS � DC þ 36DB � DC þ 12DB � DSð3Þ

and the out-of-plane Young’s modulus is given by

E3 ¼ f1ESq ð4Þwhere ES is the Young’s modulus of the solid material andf1 is a coefficient to be specified in the sections that follow,the value of which depends on the type of honeycomb celland the size scale of the cell wall thickness.

The out-of-plane shear modulus can be obtained as

G31 ¼ f2GSq ð5Þwhere GS is the shear modulus of the solid material and f2

is a coefficient to be specified.The out-of-plane Poisson ratio obviously remains the

same as that of the solid material at both the macro- andmicroscales (in this case, the strain gradient effect isabsent), thus

m31 ¼ mS ð6ÞTo simplify the analysis for honeycomb material with

cells at the nanometer scale, the surface is assumed to beisotropic and to have the same Poisson ratio mS as that of

Fig. 1. (a) A regular honeycomb with equilateral triangular cells of uniform cell wall thickness; (b) the loads applied to the cell walls of a unit cell when thehoneycomb is compressed in the x direction.

4928 H.X. Zhu et al. / Acta Materialia 60 (2012) 4927–4939


the bulk material. The out-of-plane Poisson ratio m31 of ahoneycomb with nanosized equilateral triangular cells isthus mS. Therefore, in the sections that follow, we only needto obtain the results of four independent elastic constants –E1, m12, E3 and G13 – for perfect regular honeycombs withequilateral triangular cells at different size scales.

2.2. Honeycombs with square cells

The xy plane of perfectly regular honeycombs withsquare cells, as shown in Fig. 2a, is not isotropic. It is easyto verify this, and only six independent elastic constantshave to be determined: E1, m12, G12, E3, v31 and G31. Theirdetailed derivation is given in Appendix B.

The relative density of the first-order honeycomb withsquare cells and uniform cell wall thickness is given by

q ¼ 2hL

ð7Þ

The in-plane Young’s modulus is obtained as

E1 ¼rx

ex¼ DC

Lð8Þ

The in-plane Poisson ratio is determined as

m12 ¼ vSh=L ¼ 1

2mSq ð9Þ

which is the same as that given by Wang and McDowell[21] and applies to the first-order honeycomb with regularsquare cells at different size scales. The in-plane shear mod-ulus is obtained as

G12 ¼6DBDS

DSL3 þ 12DBLð10Þ

and the out-of-plane Young’s modulus is given as

E3 ¼ f3ESq ð11Þwhere f3 is a coefficient to be specified in the sections thatfollow, the value of which depends on the size scale ofthe cell wall thickness.

The out-of-plane shear modulus can be obtained as

G31 ¼ f4GSq ð12Þwhere f4 is a coefficient to be determined.

At the nanoscale, as the Poisson ratio of the surface isassumed to be mS, the out-of-plane Poisson ratio remainsthe same as that of the solid material for different-orderhierarchical honeycombs with regular square cells frommacro- and micro- down to the nanoscale,

m31 ¼ mS ð13ÞIn the sections that follow, we only need to obtain four

independent elastic constants – E1, E3, G12 and G31 – forthe first-order honeycombs with regular square cells whosecell wall thickness is uniform at different size scales.

3. Elastic constants of macrosized first-order honeycombs

For macrosized first-order honeycombs, the bending,transverse shear and axial stretching/compression rigiditiesof the cell walls are given by

DB ¼ES

1� m2S

� bh3

12ð14aÞ

DS ¼GSbh1:2

ð14bÞ

DC ¼ ESbh ð14cÞ

In Eq. (14b) a shear coefficient of 1.2 [22] is introduced forthe rectangular cross-section of the cell walls.

3.1. Honeycombs with regular equilateral triangular cells

Substituting Eqs. (1) and (14a–c) into Eqs. (2)–(5), thedimensionless results of all the four elastic constants canbe obtained as:

E1 ¼E1

13ESq¼

1þ q2

5ð1�mSÞ þq2

12ð1�m2SÞ

1þ q2

5ð1�mSÞ þq2

36ð1�m2SÞ

ð15Þ

Fig. 2. (a) A regular honeycomb with square cells of uniform cell wall thickness; (b) the loads applied to the cell walls of a unit cell when the honeycombundergoes in-plane shear deformation.

H.X. Zhu et al. / Acta Materialia 60 (2012) 4927–4939 4929


m12 ¼13þ 1

15ð1�mSÞ � q2 � 1

36ð1�m2SÞ� q2

1þ 15ð1�mS Þ � q

2 þ 136ð1�m2

SÞ� q2

ð16Þ

E3 ¼E3

Esq¼ 1 ð17Þ

G31 ¼G31

Gsq¼ 1

2ð18Þ

where the in-plane Young’s modulus is normalizedby 1

3ESq; the out-of-plane Young’s modulus is normalized

by ESq; the out-of-plane shear modulus is normalized byGSq; and the relative density q is given by Eq. (1). Theout-of-plane Poisson ratio is the same as that given inEq. (6). Wang and McDowell [21] obtained the in-plane(i.e. the xy plane) elastic properties for macrosized first-or-der honeycombs with equilateral triangular cells. Theytook cell wall bending as the sole deformation mechanism,and their result is slightly different from Eq. (15).

For the first-order honeycombs with macrosized equilat-eral triangular cells, the relationship between the in-planedimensionless Young’s modulus E1 and the relative densityq is shown in Fig. 3, and the relationship between the in-plane Poisson ratio m12 and the relative density q is pre-sented in Fig. 4. As can be seen from Figs. 3 and 4, boththe in-plane dimensionless Young’s modulus E1 and thePoisson ratio m12 vary so little with the honeycomb relativedensity q over the range from 0 to 0.35 that they can betreated as approximate constants: E1 � 1 and v12 � 0.33.These are the same as the approximate results given byWang and McDowell [21].

3.2. Honeycombs with regular square cells

Using Eqs. (8)–(12) and (14a)–(14c), the dimensionlessresults of all the independent elastic constants of the first-order honeycombs with regular square cells can be easilyobtained as:

E1 ¼E1

12qES¼ 1 ð19Þ

G12 ¼G12

18ð1�vSÞq

3GS¼ 1

1þ 35ð1�mSÞq

2ð20Þ

v12 ¼1

2vSq ð21Þ

G31 ¼G31

Gsq¼ 1

2ð22Þ

The dimensionless in-plane shear modulus G12, whichhas been normalized by 1

8ð1�vSÞGSq3, is plotted against thehoneycomb relative density q, as shown in Fig. 5. Thegreater the honeycomb relative density, the smaller thedimensionless in-plane shear modulus. This is quite differ-ent from Wang and McDowell’s dimensionless result of 1[21]. The dimensionless out-of-plane Young’s modulus(i.e. E3 ¼ E3= ðqESÞÞ is the same as that for equilateral tri-angular honeycombs. Apart from G12 and v12 ¼ 1

2vSq, all

other dimensionless elastic constants (i.e.E1 ¼ E2 ¼ E3 ¼ 1; G31 ¼ 1=2 and m31 = mS) are independentof the honeycomb relative density. Note that E1 and E3 arenormalized by different factors.

Fig. 3. Relationship between the in-plane dimensionless Young’s modulusE1 and the relative density q of regular honeycombs with macrosizedequilateral triangular cells.

Fig. 4. Relationship between the in-plane Poisson ratio m12 and therelative density q of regular honeycombs with macrosized equilateraltriangular cells.

Fig. 5. Relationship between the in-plane dimensionless shear modulusG12 and the relative density q of regular honeycombs with macrosizedsquare cells.



4. Elastic constants of microsized first-order honeycombs

For uniform plates with thickness at the micrometerscale, the size-dependent bending stiffness is given by Yanget al. [12] as

DB ¼ES

1� m2S

� bh3

12� ½1þ 6ð1� mSÞðl=hÞ2� ð23Þ

and the transverse shear rigidity is given by Zhu [8] as

DS ¼GSbh1:2� ½1þ 6ð1� mSÞðl=hÞ2�2

1þ 2:5ð1þ mSÞðl=hÞ2ð24Þ

In Eqs. (23) and (24), h is the thickness of the cell walls andl is the material intrinsic length for the strain gradient ef-fect. If the first-order honeycomb is made of a metallicmaterial, l is at the submicron to micron scale. The axialstretching/compression stiffness of the cell walls with a uni-form thickness at the micrometer scale is the same as thatgiven in Eq. (14c) because the strain gradient effect is ab-sent. In this section, we consider the elastic properties ofthe first-order honeycombs made of uniform cell wallswhose thickness h is at the micrometer or submicron scale.


Substituting DB, DS and DC given in Eqs. (23), (24) and(14c) into Eqs. (2) and (3), the in-plane Young’s modulus(which is normalized by 1

3ESqÞ of the first-order honey-

combs with microsized equilateral triangular cells can beobtained as

E1¼E1

13�ESq

¼1þ 1þ2:5ð1þvS Þ2ðl=hÞ2

5ð1�vS Þ½1þ6ð1�vS Þðl=hÞ2 �q2þ 1

12ð1�v2S Þ� ½1þ6ð1� vSÞðl=hÞ2� �q2

1þ 1þ2:5ð1þvS Þ2ðl=hÞ2



ð25Þ

and the in-plane Poisson ratio is derived as

m12¼13þ 1þ2:5ð1þvS Þ2ðl=hÞ2

15ð1�vS Þ½1þ6ð1�vS Þðl=hÞ2 �q2� 1


1þ 1þ2:5ð1þvS Þ2ðl=hÞ2



ð26Þ

when a microsized first-order honeycomb undergoes uniax-ial compression or tension in the z direction (i.e. the out-of-plane direction), the strain gradient effect is absent. Thedimensionless out-of-plane Young’s modulus E3 in the z

direction is the same as that of the macrosized first-orderhoneycomb, and the out-of-plane Poisson ratio m31 is thesame as vs.

As the size of a honeycomb material is much larger thanits cells, when the honeycomb material with microsized cellsundergoes a globally uniform out-of-plane shear deforma-tion in the zx or zy plane (see Fig. 1), the shear strain in eachcell wall is always in the cell wall plane and uniform inamplitude, and any strain gradient effect can thus be

assumed to be absent. Therefore the out-of-plane dimen-sionless shear modulus is the same as that given by Eq.(18) for macrosized first-order honeycombs.

For the first-order honeycombs with regular microsizedequilateral triangular cells and different values of l/h, thedimensionless in-plane Young’s modulus E1, given by Eq.(25), and the in-plane Poisson ratio v12, given by Eq.(26), are plotted against the relative density in Figs. 6and 7, respectively. The Poisson ratio, vS, of the solid mate-rial is chosen as 0.3 in the calculation of the figures. As canbe seen, the thinner the cell walls (i.e. the larger the value ofl/h), the larger the dimensionless in-plane Young’s modulusE1 and the smaller the in-plane Poisson ratio v12.

Eq. (1) can be rewritten as

lh¼ 2

ffiffiffi3p

lqL

orlL¼ ql

2ffiffiffi3p

hð27Þ

where L is the length of the cell walls, which can be definedas the size of the cells. Figs. 6 and 7 demonstrate that, forthe first-order honeycombs with equilateral triangular cellsat the micrometer size scale and with a fixed relative densityq, the smaller the cell size, the larger the dimensionless in-plane Young’s modulus and the smaller the in-plane Pois-son ratio.


For the first-order honeycombs with microsized regularsquare cells, the strain gradient effect is clearly absent whenit is uniaxially deformed in either the x, y or z direction.Therefore, its dimensionless elastic constants: E1; E3; G31

and the Poisson ratios: v12 and m31 are exactly the sameas those of the first-order honeycombs with macrosizedregular square cells.

Substituting Eqs. (23) and (24) into Eq. (10), the in-plane shear modulus can be obtained as

G12 ¼G12

18ð1�vSÞq

3GS¼ 1þ 6ð1� vSÞðl=hÞ2

1þ 35ð1�vSÞ �

1þ2:5ð1þvSÞðl=hÞ2

1þ6ð1�vS Þðl=hÞ2 q2ð28Þ

which is normalized by 18ð1�vSÞGSq3.

Fig. 6. Relationship between the in-plane dimensionless Young’s modulusE1 and the relative density q of regular honeycombs with microsizedequilateral triangular cells.



The relationship between the in-plane dimensionlessShear modulus G12 and the relative density q of the first-order honeycombs with microsized regular square cells isplotted in Fig. 8. The thinner the cell walls, the larger thedimensionless in-plane shear modulus. As the size of thecells can be defined as L, for microsized honeycombs witha fixed relative density, the smaller the cell size, the largerthe dimensionless in-plane shear modulus.

It is easy to verify that, when the strain gradient effect isabsent (i.e. h/l tends to 0), all the elastic constants of thefirst-order honeycombs with either microsized equilateraltriangular cells or square cells reduce to those of their mac-rosized counterparts.

5. Elastic constants of nanosized first-order honeycombs

At the nanometer scale, both the surface elasticity [15]and the initial stresses [17,18] can greatly affect the mechan-ical properties of structural elements. Atomistic simula-tions [19] show that, for metallic structural elements withthe size of a few nanometers, no strain gradient effect exists,

and surface elasticity or surface energy dominates the influ-ence on the mechanical properties.

For uniform plates of thickness at the nanoscale, thecombined effects of surface elasticity and initial stress onthe bending stiffness is obtained as [18]

Db ¼Es

1� v2s

bh3

121þ 6S

Esh� 2s0

Eshvsð1þ vsÞ

� �

¼ Es

1� v2s

bh3

121þ 6ln

hþ e0s

vsð1þ vsÞ1� ms

� �ð29Þ

The transverse shear stiffness is derived as [8]

DS ¼GSbh1:2�

1þ 6lnh þ e0smS

ð1þmS Þ1�mS

h i2

1þ 10lnh þ 30 ln

h

� �2ð30Þ

and the axial stretching/compression stiffness is given by[15]

Dc ¼ Esbhð1þ 2ln=hÞ ð31ÞIn Eqs. (29)–(31), S is the surface elasticity modulus,ln = S/Es is the material intrinsic length at the nanoscaleand s0 is the initial surface stress, the amplitude of whichcan be varied by adjusting the applied electric potential[23–25]. When the initial surface stress s0 is present, the ini-tial elastic residual strains of the bulk material aree0s ¼ � 2s0

Esh ð1� vsÞ in both the cell wall length and widthdirections, and the initial elastic residual strain in the cellwall thickness direction is eh

0 ¼4vss0

Esh ¼ �2vs

1�vse0s. Although

the amplitude of e0scan be varied, the range of the recover-able elastic residual strain is limited by the yield strain ofthe bulk material. For single crystal nanomaterials or poly-meric materials, the yield strain can be 10% or even larger.Atomistic simulation [26] has shown that, if the diameter ofa gold wire is sufficiently small, it can automatically under-go plastic deformation solely owing to the presence of theinitial surface stresses. It is well known that the yieldstrength, ry, of some conductive polymer materials ornanosized metallic materials can be 0.1E (E is the Young’smodulus) or larger [27]. Biener et al. [28] have experimen-tally found that, for nanoporous Au material, by control-ling the chemical energy, the adsorbate-induced surfacestress s0 can reach 17–26 N m�1. If the diameter of the lig-aments is 5 nm, rx

0 would be 20 GPa. As the bulk materialdiscussed in this paper can be metallic, polymeric or biolog-ical, without losing generality, the tunable range ofe0s ¼ � 2s0

Esh ð1� vsÞ is assumed to be from �0.1 to 0.1. Inother words, the von Mises yield strength of the solid mate-rial is assumed to be 0.1ES/(1 � vS). If the actual yieldstrength of the honeycomb solid material is larger or smal-ler than this, the results obtained for the nanosized first-or-der honeycombs can still be obtained by scaling thosepresented in this paper up or down.

When the effect of the initial surface stress s0 is absent,the initial dimensions of the nanosized cell walls areassumed to be length L0, width b0 and thickness h0 for bothregular triangular and square honeycombs. When the effect

Fig. 7. Relationship between the in-plane Poisson ratio m12 and therelative density q of regular honeycombs with microsized equilateraltriangular cells.

Fig. 8. Relationship between the in-plane dimensionless shear modulusG12 and the relative density q of regular honeycombs with microsizedsquare cells.



of the initial surface stress is present, the dimensions of thecell walls become

L ¼ L0ð1þ e0sÞ ð32aÞb ¼ b0ð1þ e0sÞ ð32bÞ

h ¼ h0ð1þ eh0Þ ¼ h0 1� 2vs

1� vse0s

� �ð32cÞ

and thus the relative density of both the first-order squareand equilateral triangular honeycombs becomes

q ¼ hL0

h0Lq0 ¼ 1� 2vs

1� vse0s

� �q0=ð1þ eosÞ ð33Þ

where e0s ¼ � 2s0

Esh ð1� vsÞ and

q0 ¼ 2ffiffiffi3p

h0=L0 ð33aÞfor equilateral triangular honeycombs and

q0 ¼ 2h0=L0 ð33bÞfor square honeycombs.


Substituting DB, DS and DC and the dimensions of thenanosized cell walls given by Eqs. (29)–(33) into Eq. (2),the in-plane dimensionless Young’s modulus of honey-combs with nanosized equilateral triangular cells can beobtained as

E1¼E1

13ESq0

¼

1þ 2lnh

� �þ 1

5ð1�mS Þ �1þ10ln

h þ30 lnhð Þ2

1þ2ln

hð Þ1þ6ln

h þeos �mS ð1þmS Þ

1�mS

h i q2þ 112ð1�m2

S Þ� 1þ 6ln

h þ eos � mS ð1þmS Þ1�mS

h i�q2

1þ 15ð1�mS Þ �

1þ10lnh þ30 ln

hð Þ2

1þ6lnh þeos �

mS ð1þmS Þ1�mS

h iq2þ 136ð1�m2

S Þ�

1þ6lnh þeos �

mS ð1þmS Þ1�mS

h i1þ2ln

h½ � �q2

�1� 2vS

1�vSeos

1þ eos

ð34Þwhere the in-plane Young’s modulus is normalized by13� ESq0 and h and q0 are given by Eqs. (32c) and (33a),

respectively.When the effect of the initial surface stress s0 is absent

(i.e. the initial elastic residual strain e0S of the bulk materialis 0) and the effect of the surface elasticity is present, h = h0

and q = q0, and Eq. (34) reduces to

E1¼1þ 2ln

h

� �þ 1

5ð1�mS Þ �1þ2ln

hð Þ 1þ10lnh þ30 ln

hð Þ2

1þ6lnhð Þ �q2þ 1

12ð1�m2S Þ� 1þ 6ln

h

� ��q2

1þ 15ð1�mS Þ �

1þ10lnh þ30 ln

hð Þ21þ6ln

hð Þ �q2þ 136ð1�m2

S Þ� 1þ6ln

hð Þ1þ2ln

hð Þ �q2

ð35Þ

Fig. 9 shows the effect of the surface elasticity on the rela-tionship between the normalized in-plane (i.e. xy plane)Young’s modulus (given by Eq. (35)) and the relative densityq0 of the nanosized first-order honeycombs with regularequilateral triangular cells. If the surface elasticity modulusis positive (i.e. ln = S/ES > 0), the thinner the cell walls, thelarger the normalized Young’s modulus. Similarly to the

first-order honeycombs with microsized cells, for nanosizedfirst-order honeycombs with a fixed relative density q0, thesmaller the cell size L0, the larger the normalized in-planeYoung’s modulus. If the surface elasticity modulus is nega-tive (i.e. ln = S/ES < 0), the effects are reversed. For example,when ln/h is �0.2, the dimensionless Young’s modulusbecomes 0.6, which is smaller than that of the macrosizedhoneycombs (i.e. ln/h = 0). However, ln/h must be largerthan �1/2, otherwise the elastic modulus E1 would be nega-tive and the honeycomb structure would deform automati-cally [8] to a stable configuration. On the other hand, whenthe effect of the surface elasticity modulus is absent and theeffect of the initial stress/strain is present, Eq. (34) reduces to

E1¼1þ 1

5ð1�mSÞ½1þeos�mS ð1þmS Þ

1�mS��q2þ 1

12ð1�m2SÞ� 1þ eos � mSð1þmS Þ

1�mS

h i�q2

1þ 1

5ð1�mSÞ 1þeos�mS ð1þmS Þ

1�mS

h i �q2þ 136ð1�m2

S Þ� 1þ eos � mSð1þmSÞ

1�mS

h i�q2

�1� 2vS

1�vSeos

1þ eos

ð36ÞEq. (36) can be approximated well by E1 � ð1� 2vSeos=ð1� vSÞÞ=ð1þ eosÞ, and the error is smaller than 1% ifq0 6 0.35. Thus the in-plane dimensionless Young’s modu-lus of a nanosized first-order regular honeycomb with equi-lateral triangular cells can be varied by adjusting theamplitude of the initial surface stress (and hence the initialstrain eos), and the tunable range depends on the Poissonratio vS of the solid materials. When eos is varied from0.1 to �0.1, the tunable range of the dimensionless in-planeYoung’s modulus will be from 0.909 to 1.111 if vS = 0,0.831–1.206 if vS = 0.3, and 0.727–1.5 if vS = 0.5.

The in-plane Poisson ratio can be obtained as

m12¼

13þ 1

15ð1�mSÞ �1þ10ln

h þ30 lnhð Þ2

1þ6lnh þeos�

mS ð1þmS Þ1�mS

h iq2� 136ð1�m2

SÞ�

1þ6lnh þeos�

mS ð1þmS Þ1�mS

h i1þ2ln

hð Þ �q2

1þ 15ð1�mSÞ �

1þ10lnh þ30 ln

hð Þ2

1þ6lnh þeos�

mS ð1þmS Þ1�mS

h iq2þ 136ð1�m2

SÞ�

1þ6lnh þeos�

mS ð1þmS Þ1�mS

h i1þ2ln

hð Þ �q2

ð37ÞWhen the effect of the initial surface stress s0 is absent, Eq.(37) reduces to

m12 ¼13þ 1

15ð1�mS Þ1þ10ln

h þ30 lnhð Þ2

1þ6lnh½ � � q2 � 1

36ð1�m2S Þ� 1þ6ln

h½ �1þ2ln

hð Þ � q2

1þ 15ð1�mSÞ

1þ10lnh þ30 ln

hð Þ21þ6ln

h½ � � q2 þ 136ð1�m2

S Þ� 1þ6ln

h½ �1þ2ln

hð Þ � q2

ð38Þ

Fig. 10 shows that the effect of the surface elasticity onthe in-plane Poisson ratio (given by Eq. (38)) of nanosizedfirst-order honeycombs depends upon the value of ln/h.The thinner the cell walls, the smaller the Poisson ratio.On the other hand, the effect of the initial stress or strainon the Poisson ratio is so small that the result is notpresented.



From the stretching stiffness of nanoplate given by Eq.(31), the normalized out-of-plane Young’s modulus ofthe first-order honeycombs with nanosized equilateral tri-angular cells can be easily obtained as

E3 ¼E3

Esq0

¼ 1þ 2ln

h

� �� 1� 2vS

1� vSeos

� �=ð1þ eosÞ ð39Þ

The normalized out-of-plane shear modulus can be easilyderived as

G31 ¼G31

Gsq0

¼ 1

21þ 2ln

h

� �1� 2vS

1� vSeos


For the nanosized first-order honeycombs with a fixed rel-ative density, the thinner the cell walls, the larger will be thenormalized Young’s modulus and the out-of-plane shearmodulus if ln = S/ES is positive. If the surface elasticitymodulus is negative, the trend of the effects is reversed.As the Poisson ratio of the surface is assumed to be thesame as that of the bulk material, the out-of-plane Poissonratio of a nanohoneycomb, m13, is thus equal to vS. Whenthe effect of the surface elasticity is present and the effectof the initial stress/strain is absent, both the out-of-plane

Young’s modulus and shear modulus are proportional to(1 + 2ln/h). On the other hand, when the effect of surfaceelasticity is absent (or fixed) and the effect of the initialstress/strain is present, both the out-of-plane dimensionlessYoung’s modulus and shear modulus depend not only onthe amplitude of the initial strain eos, but also on the Pois-son ratio vS of the solid material.


Substituting Eqs. (29)–(33) into Eq. (8), the in-planedimensionless Young’s modulus of the first-order honey-combs with nanosized square cells can be obtained as

E1 ¼E1

Esq0

¼ 1

21þ 2ln

h

� �1� 2vS

1� vSeos


which is normalized by ESq0.The in-plane dimensionless shear modulus of nanosized

first-order honeycombs, which is normalized by 18ð1�vSÞ q

30GS,

can be obtained as

G12¼G12

18ð1�vSÞq

30GS¼

1þ 6lnh þ e0S

mSð1þmSÞð1�mSÞ

1þ 35ð1�vSÞ �

1þ10lnh þ30 ln

hð Þ21þ6ln

h þe0SmS ð1þmS Þð1�mS Þ

q2

�1� 2vS

1�vSeos

1þ eos

!3

ð42ÞWhen the effect of the initial surface stress is absent (i.e. e0S

is 0) and the effect of the surface elasticity is present, Eq.(42) reduces to

G12 ¼G12

18ð1�vSÞGSq3

0

¼1þ 6ln

h

1þ 35ð1�mSÞ �

1þ10lnh þ30 ln

hð Þ21þ6ln

hq2

ð43Þ

The relationship between the dimensionless in-plane shearmodulus G12 and the relative density q of the first-orderhoneycombs with nanosized square cells is plotted inFig. 11. If the surface elasticity modulus is positive (i.e.ln = S/ES > 0), the thinner the cell walls, the larger thedimensionless shear modulus. If the relative density q ofnanosized honeycombs is fixed, the smaller the cell wall

Fig. 10. Size-dependent effect on the relationship between the in-planePoisson ratio m12 and the relative density q of regular nanosized equilateraltriangular honeycombs when the effect of the cell wall initial elasticresidual strain is absent.

Fig. 9. Relationship between the in-plane dimensionless Young’s modulusE1 and the relative density q of regular honeycombs with nanosizedequilateral triangular cells.

Fig. 11. Size-dependent effect on the relationship between the in-planedimensionless shear modulus G12 and the relative density q of regularhoneycombs with nanosized square cells when the effect of the cell wallinitial elastic residual strain is absent.



length or the cell size L, the larger the dimensionless elasticmodulus. If the surface elasticity modulus is negative (i.e.ln = S/ES < 0), the effects are reversed. For example, ifln=h ¼ �0:1;G12 becomes 0.4, which is smaller than thatfor macrosized honeycombs (i.e. ln/h = 0). Eq. (43) impliesthat ln/h must be larger than �1/6, otherwise the in-planeshear modulus G12 becomes negative and the honeycombstructure will automatically deform until the structural sta-bility is regained [8].

When the effect of the surface elasticity modulus isabsent and only the effect of the initial surface stress is pres-ent, Eq. (42) reduces to

G12 ¼G12

18ð1�vSÞGSq3

0

¼1þ e0S

mSð1þmSÞð1�mSÞ

1þ 35ð1�mSÞ �

1

1þe0SmS ð1þmS Þð1�mS Þ

q2

1� 2vS1�vS

eos

1þ eos

!3

ð44ÞFig. 12 shows the effect of the initial residual elastic

strain e0S of the bulk material (or the initial stress) on thein-plane dimensionless shear modulus of the first-orderhoneycombs with nanosized perfect regular square cells.When vS = 0.3 and the initial strain eos is varied from 0.1to �0.1, the dimensionless in-plane shear modulus G12

can change from 0.6 to 1.5.For the first-order nanosized honeycombs with regular

square cells, it is easy to obtain the same dimensionlessout-of plane Young’s modulus E3 and shear modulus G31

as those given by Eqs. (39) and (40) for the nanosized first-order honeycombs with regular equilateral triangular cells.

It is easy to verify that, for regular nanosized honey-combs with either square cells or equilateral triangularcells, all the elastic constants will reduce to those of theirmacrosized counterparts when the effects of both the sur-face elasticity and the initial stress/strain are absent.

6. Size-dependent and tunable elastic properties of

hierarchical honeycombs

The hierarchical honeycombs are assumed to be self-similar [29]. At all different hierarchy levels, they are

treated as materials whose size is much larger than the indi-vidual cells at the same hierarchy level. The relative densityof the nth-order self-similar hierarchical honeycombs witheither equilateral triangular cells or square cells can be eas-ily obtained as

qn ¼ 1� 2vS

1� vSeos

� �ðq0Þ

n=ð1þ eosÞ ð45Þ

where q0 is given by Eq. (33a) for the first-order honey-combs with equilateral triangular cells and by Eq. (33b)for the first-order honeycombs with square cells. Whenthe initial strain eos is 0, qn reduces to (q0)n.

For the first-order honeycombs with regular equilateraltriangular cells of wall thickness at the macro-, micro- ornanoscale, the five independent elastic constants areobtained as given in Sections 3–5, depending upon the sizescale of their cell wall thickness. For the nth-order self-sim-ilar hierarchical honeycombs with equilateral triangularcells, the five independent dimensionless elastic constantscan be obtained as

ðE1Þn ¼ðE1Þn13ESq0

� q0

3

� �n�1

� ðE1Þ1 ð46Þ

ðv12Þn � 1=3 ð47Þ

ðE3Þn ¼ðE3ÞnEsq0

¼ ðq0Þn�1 � ðE3Þ1 ð48Þ

ðG31Þn ¼ðG31ÞnGsq0

¼ q0

2

� �n�1

� ðG31Þ1 ð49Þ

and (v31)n = vS, where n is the hierarchy level of the self-similar hierarchical honeycombs. In Eqs. (46), (48) and(49), ðE1Þ1; ðE3Þ1 and ðG31Þ1 are the dimensionless Young’smoduli in the x and z directions and the dimensionlessshear modulus in the xz plane of the first-order honey-comb, respectively. It is easy to check that, when n P 2,the errors of Eqs. (46) and (47) are smaller than 0.2% ifq0 6 0.35.

For the nth-order self-similar hierarchical honeycombswith square cells, the six independent dimensionless elasticconstants can be obtained as

ðE1Þn ¼ðE1Þn12q0ES

¼ q0

2

� �n�1

� ðE1Þ1 ð50Þ

ðE3Þn ¼ðE3Þnq0ES

¼ ðq0Þn�1 � ðE3Þ1 ð51Þ

ðG12Þn ¼ðG12Þn

18ð1�vS Þ ðq0Þ

3GS

� q0

8

� �3ðn�1Þ� ðG12Þ1 ð52Þ

ðv12Þn ¼q0

2

� �n�1

� ðv12Þ1 ð53Þ

ðG31Þn ¼ðG31ÞnGsq0

¼ q0

2

� �n�1

� ðG31Þ1 ð54Þ

and (m31)n = mS. In Eqs. (50)–(54), the elastic constants ofthe first-order honeycombs with regular square cells of wallthickness at the macro-, micro- or nanoscale, ðE1Þ1;ðE3Þ1; ðG12Þ1; ðm12Þ1 and ðG31Þ1, are given in Sections 3–5,

Fig. 12. Effect of cell wall initial elastic residual strain on the relationshipbetween the in-plane dimensionless shear modulus G12 and the relativedensity q of regular honeycombs with nanosized square cells when theeffect of the surface elasticity modulus is absent.



depending upon the size scale of their cell wall thickness.Therefore, the elastic constants of the nth-order self-similarhierarchical honeycombs are functions of those of the first-order honeycombs. It is easy to check that, when n P 2, theerror of Eq. (53) is smaller than 0.4% if q0 6 0.35. Fan et al.[30] and Taylor et al. [31] studied the effects of structuralhierarchy on the elastic properties of honeycombs. How-ever, they did not study the size-dependent effect or the tun-able elastic properties of hierarchical honeycombs. Zhu [8]studied the size-dependent and tunable mechanical proper-ties for single-order regular hexagonal honeycombs, butnot for hierarchical honeycombs.

For regular self-similar hierarchical honeycombs withthe thickness of their first-order cell walls at the nanometer

scale, the dimensionless elastic properties ðE1Þn; ðE3Þn;ðG12Þn and ðG31Þn contain a common factor 1� 2vS

1�vSeos

� �=

ð1þ eosÞ and can thus be varied over a large range byadjusting the amplitude of the initial strain eos. When theinitial strain is absent, the initial cell diameter, area andvolume of an nth-order self-similar hierarchical honeycombare assumed to be (L0)n, (A0)n and (V0)n respectively. Whenthe initial strain eos is present, the dimensionless cell diam-eter, area and volume of an nth-order self-similar hierarchi-cal honeycomb become

ðLÞn=ðL0Þn ¼ 1þ eos ð55ÞðAÞn=ðA0Þn ¼ ð1þ eosÞ2 ð56Þ

and

ðV Þn=ðV 0Þn ¼ ð1þ eosÞ3 ð57Þ

If eos can be controlled to change from �0.1 to 0.1, thedimensionless cell diameter, area and volume of an nth-or-der self-similar hierarchical honeycomb would vary overranges from 0.9 to 1.1, 0.81 to 1.21 and 0.729 to 1.331,respectively. This can be of very important applications.For example, by adjusting the cell size, a hierarchical cellu-lar material could possibly be controlled to change its coloror wettability. Micro- or nanosized porous materials areoften used to select/separate materials with a specific parti-cle size in medical industry. The results given in Eqs. (55)–(57) suggest that the size of the selected/separated particlesis tunable and controllable. Although it is difficult to alterthe cell size and the mechanical properties, such as the stiff-ness, the buckling force and the natural frequency, of a sin-gle-order macro- or microsized porous material, the resultsobtained in this section suggest that it is possible to realizethose for its hierarchical counterpart with the first-ordercell wall/strut thickness at the nanometer scale.

In addition, Eqs. (42) and (52) imply the possibility that,if ln/h is very close to �1/6 (but ln/h should be slightly lar-ger than �1/6, the structure would not be stable), control-ling eos to a negative value (say �0.1) could result in anegative ðG12Þ, and hence could make the hierarchical

cellular material collapse automatically [8]. In practicalapplications, one failing part could help protect others.

7. Conclusion

This paper presents the detailed derivation and closed-form results of all the independent elastic constants ofself-similar hierarchical honeycombs with either regularsquare or equilateral triangular cells. The results imply thatmany interesting geometrical, mechanical and physicalproperties and functions that do not exist in single-ordermacrosized cellular materials become possible in their hier-archical nanostructured counterparts. If the cell wall thick-ness of the first-order honeycomb is at the micrometerscale, the elastic properties of a hierarchical honeycombare size dependent owing to the strain gradient effects. Ifthe cell wall thickness of the first-order honeycomb is atthe nanometer scale, in addition to the size dependenceowing to the effects of surface elasticity, the elastic proper-ties of a hierarchical honeycomb are tunable because of theeffects of the initial stresses/strains, the amplitudes of whichare controllable. More interestingly, if the cell wall/strutthickness of the first order cellular material is at the nano-meter scale, the cell size, surface color, wettability, materialstrength, stiffness, natural frequency and many other inter-esting physical properties of a hierarchical nanostructuredcellular material could be varied over a large range.

Acknowledgement

This work is supported by the EC project PIRSES-GA-2009-247644.

Appendix A. Derivation of the elastic properties of regular

honeycombs with equilateral triangular cells

A.1. In-plane Young’s modulus and Poisson ratio

When a first-order regular honeycomb with equilateraltriangular cells is uniaxially compressed in the x direction,as shown in Fig. 1a, there is no junction rotation because ofthe symmetry of the structure and the applied load. Only arepresentative unit cell structure, as shown in Fig. 1b, isthus needed for the analysis. Node A is assumed to haveno displacement and rotation. The horizontal load appliedat B is assumed to be P1 and the horizontal load applied atC is P2. Obviously, there is no load in the y direction at Band C.

The inclined cell wall AB undergoes bending, transverseshear and axial compression. Its deformation in the x direc-tion is given by

Dx ¼ Dxb þ Dx

s þ Dxc ðA1Þ

where Dxb; Dx

s and Dxc are the deformations in the x direction

owing to cell wall bending, transverse shear and cell wallaxial compression, respectively. By assuming that the bend-ing stiffness of the cell walls is DB, the shear stiffness is DS,



and the axial compression stiffness is DC, Eq. (A1) can berewritten as

Dx ¼P 1 � sin 60

� � L2

� �3

3DBcos 30

� þP 1 � sin 60

� � L2

DS

� cos 30� þ

P 1 � cos 60� � L

2

DCsin 30

�

¼ P 1L3

32DBþ 3P 1L

8DSþ P 1L

8DCðA2Þ

where L is the length of the cell walls.From Fig. 1b, the dimension of the representative unit

cell in the x direction is

lx ¼L2� cos 60

� ¼ L4

ðA3Þ

The compressive strain of the inclined element AB in the x

direction is therefore

ex ¼Dx

lx¼ P 1L2

8DBþ 3P 1

2DSþ P 1

2DCðA4Þ

The stress component of the honeycomb in x direction dueto force P1 is thus

rx1 ¼P 1

L � sin 60� � b¼ 2P 1ffiffiffi

3p

LðA5Þ

where the width b of the honeycomb is much larger thanthe cell wall thickness h and is assumed to be 1 forsimplicity.

For the horizontal element AC, the deformation com-patibility condition requires that the compressive strainof AC due to force P2 should be the same as that of theinclined element AB in the x direction. Thus,

P 2 ¼ DC � ex ¼ DC �P 1L2

8DBþ 3P 1

2DSþ P 1

2DC

� �ðA6Þ

The stress component of the honeycomb in x direction dueto force P2 is derived as

rx2 ¼P 2

L � sin 60� � b ¼2P 2ffiffiffi

3p

L

¼ P 1 � L � DC

4ffiffiffi3p

DB

þffiffiffi3p

P 1 � DC

DS � Lþ P 1ffiffiffi

3p

LðA7Þ

The total compressive stress in the x direction is thus ob-tained as

rx ¼ rx1 þ rx2 ¼P 1LDC

4ffiffiffi3p

DB

þffiffiffi3p

P 1DC

DSLþ

ffiffiffi3p

P 1

LðA8Þ

The Young’s modulus of the honeycomb is therefore

Ex¼rx

ex¼

P 1�L�DC

4ffiffi3p

DBþffiffi3p

P 1�DCDS �L þ

ffiffi3p

P 1

L

P 1�L2

8DBþ 3P 1

2DSþ P 1

2DC

¼DS �DC �L2þ12DB�DCþ12DB�DS

4ffiffi3p

DB�DS �LDS �DC �L2þ12DB�DCþ4DB�DS

8DB�DS �DC

¼ 2DS �D2C �L2þ24DB �D2

Cþ24DB �DS �DCffiffiffi3p

L3 �DS �DCþ12ffiffiffi3p

L �DB �DCþ4ffiffiffi3p

L �DB �DS

ðA9Þ

The deformation of the inclined cell wall in the y directionis

Dy ¼ Dyb þ Dy

s þ Dyc ðA10Þ

Eq. (A10) can be rewritten as

Dy ¼P 1 � sin 60� � L

2

� �3

3DBsin 30� þ

P 1 � sin 60� � L2

� �DS

� sin 30� �P 1 � cos 60� � L

2

� �DC

cos 30�

¼ffiffiffi3p

P 1 � L3

96DBþ

ffiffiffi3p

P 1 � L8DS

�ffiffiffi3p

P 1 � L8DC

ðA11Þ

From Fig. 1b, the dimension of the representative unit cellin the y direction is

ly ¼L2� sin 60

� ¼ffiffiffi3p

L4

ðA12Þ

The expansion strain in the y direction can be obtained as

ey ¼Dy

ly¼ P 1L2

24DBþ P 1

2DS� P 1

2DCðA13Þ

The Poisson ratio is therefore

v12 ¼ey

ex

¼ L2DSDC þ 12DBDC � 12DBDS

3L2DSDC þ 36DBDC þ 12DBDSðA14Þ

As the honeycomb structure is isotropic, the in-plane shearmodulus G12 can be obtained from Ex and v12, and given as

G12 ¼Ex

2ð1þ v12Þ

¼ffiffiffi3p

L2 � DS � DC þ 12ffiffiffi3p

DB � DC þ 12ffiffiffi3p

DB � DS

4L3 � DS þ 48L � DB

ðA15Þ

A.2. Out-of-plane shear modulus and Young’s modulus

To derive the out-of-plane shear modulus, the size of thehoneycomb material is assumed to be much larger than thehoneycomb cells and thus much larger than the cell walllength L. When the honeycomb material is subjected toan out-of-plane pure shear stress syz (which is in the yz

plane), the shear load T on the cell wall of the representa-tive unit cell is shown in Fig. A1.

The equilibrium in the y direction requires

Q ¼ T � sin 600 ðA16Þand

Q ¼ s�yz

ffiffiffi3p

4L2 ðA17Þ

where syz is the effective out-of-plane shear stress acting onthe area

ffiffi3p

4L2 of the representative unit cell of the honey-

comb material, as shown in Fig. A1.



In this part, we derive the out-of-plane shear modulusG13 for the first-order honeycombs with nanosized equilat-eral triangular cells. Eqs. (A16) and (A17) lead to

T ¼ syz �1

2L2 ðA18Þ

As the shear stress or strain is uniform in the cell walls (be-cause the honeycomb material is assumed to be much lar-ger than the cell size L), the shear strain ci

yz in theinclined cell wall shown on the right side of Fig. A1 is re-lated to the shear force T by

ciyz ¼ T

�GShLþ 2 � SL

2ð1þ mSÞ

� �

¼ syzL2GShð1þ 2ln=hÞ ðA19Þ

where S is the surface elasticity modulus, the surface Pois-son ratio is assumed to be vS (the same as that of the solidmaterial), S

2ð1þvSÞ is the shear modulus of the surface, ln = S/ES is the material intrinsic length at nanometer scale, and h

is the thickness of the nanosized cell walls.The shear strain of the inclined cell wall is related to the

out-of plane shear strain (i.e. the shear strain in the yz

plane) of the first-order honeycomb by

cyz ¼ci

yz

cos 300¼ Lsyzffiffiffi

3p

GShð1þ 2ln=hÞðA20Þ

As the relative density of the first-order honeycombs withequilateral triangular cells is q ¼ 2

ffiffiffi3p

hL, the out-of-plane

shear modulus of the nanosized honeycomb is thus

G13 ¼ Gyz ¼syz

cyz=ðGSqÞ ¼

1

2ð1þ 2ln=hÞ ðA21Þ

which is normalized by Gsq. As can be seen in Eq. (A21), thedimensionless out-of-plane shear modulus of honeycombswith nanosized cells is size dependent. When the surface ef-fect is absent (i.e. h� ln), Eq. (A21) reduces to the dimen-sionless out-of-plane shear modulus for a honeycombswith macro- or microsized equilateral triangular cells.

The stretching stiffness of a nanoplate is given by [15]

DC ¼ ESbhð1þ 2ln=hÞ ðA22Þ

The dimensionless out-of-plane Young’s modulus of nano-sized first-order honeycombs with equilateral triangularcells can be easily obtained as

E3 ¼E3

Esq¼ 1þ 2ln

hðA23Þ

which is normalized by ESq. Similarly, when the surface ef-fect is absent, it reduces to that of honeycombs with macro-or microsized cells.

Appendix B. Derivation of the elastic properties of regular

honeycombs with square cells

B.1. In-plane Young’s modulus and Poisson ratio

When a regular first-order honeycomb with square cellsis uniaxially compressed in the x direction, there is no junc-tion rotation because of the symmetry of the structure andthe applied load. There is no stress or strain in vertical cellwalls. By taking account the expansion of the horizontalcell walls, the in-plane Poisson ratio of the first-order hon-eycomb can be obtained as v12 ¼ v21 ¼ h

L vS ¼ 12qvS. The in-

plane dimensionless Young’s modulus can be easilyobtained as E1 ¼ E1

12ESq¼ DC

ES h, where q ¼ 2hL is the honeycomb

relative density.To derive the in-plane shear modulus, a pure shear stress

sxy is applied to the honeycomb shown in Fig. 2a. Only arepresentative unit cell structure (shown in Fig. 2b) isneeded for the analysis. The force equilibrium conditionrequires

P ¼ sxyL ðA24ÞIf junction A shown in Fig. 2b has no rotation, the verticaldisplacement of the middle point D of the horizontal cellwall owing to bending and transverse shear should be

DAD ¼ DB þ DS ¼P L

2

� �3

3DBþ

P L2

DSðA25Þ

Obviously, under pure in-plane shear deformation, themiddle points of the horizontal cell walls of the deformedsquare honeycomb always remain in the same straight line,which is assumed to be in the horizontal position in order

Fig. A1. The out-of-plane shear load on the area of a representative cell unit.



to simplify the analysis. Thus, the deformed cell wall ADhas a clockwise rotation of hA, which is given as

hA ¼DAD

L=2¼ PL2

12DBþ P

DSðA26Þ

Consequently, owing to cell wall bending, transverse shearand junction rotation, the displacement of point C in thehorizontal direction is obtained as

jDCj ¼ 2hAðL=2Þ ¼ PL3

12DBþ PL

DSðA27Þ

The shear strain c is therefore

cxy ¼jDCjL=2¼ sxyL3

6DBþ 2sxyL

DSðA28Þ

The in-plane shear modulus of a regular square honey-combs is thus

G12 ¼sxy

cxy¼ 6DBDS

DSL3 þ 12DBLðA29Þ

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size dependent and tunable elastic properties of hierarchical honeycombs with regular square and...

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