Simple closed-form results for all the independent elastic constants of macro-, micro- and nanosized first-order regular honeycombs with square and equilateral triangular cells and for the self-similar hierarchical honeycombs were obtained. It is found that, if the cell wall thickness 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 hierarchical honeycomb are not only size dependent owing to the effects of surface elasticity and initial stresses, but are also tunable. In addition, the cell size and volume of hierarchical nanostructured cellular materials can be varied, and hierarchical nanostructured cellular materials could also possibly be controlled to collapse.

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Available online at www.sciencedirect.com
Acta Materialia 60 (2012) 4927–4939
www.elsevier.com/locate/actamat
Size-dependent and tunable elastic properties of
hierarchical 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, UK
b
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 honeycombs
with square and equilateral triangular cells and for the self-similar hierarchical honeycombs were obtained. It is found that, if the cell wall
thickness 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 cellular
materials could also possibly be controlled to collapse.
Crown Copyright Ó 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 the mechanical properties [8–14], and that, at the nanometer
scale, both surface elasticity [8,15,16] and initial stresses
In nature, living things evolve constantly to survive their [17,18] can greatly affect the mechanical properties of struc-
changing environment. To support their own weight, to tural elements. Atomistic simulations [19] suggest that, for
resist external loads and to enable different types of func- metallic structural elements with a size of a few nanometers,
tions, their bodies should be structurally optimized and the strain gradient effect is irrelevant, and surface elasticity
mechanically sufficiently strong. As a consequence, natural or surface energy dominates the mechanical properties. The
living materials are usually made up of hierarchical cellular size-dependent bending rigidities have been obtained for
structures with basic building blocks at the micro- or microplates [11] and nanoplates [18]. The size-dependent
nanoscale. transverse shear rigidities of micro- and nanoplates and
The elastic properties of honeycombs at the macroscale the size-dependent elastic properties of the first-order hon-
have been extensively studied and well documented [1–7]. eycombs with regular hexagonal cells were obtained in
However, the results obtained for macrohoneycombs may Ref. [8]. Fan et al. [30] and Taylor et al. [31] have studied
not apply to their micro- and nanosized counterparts [8]. the effects of structural hierarchy on the elastic properties
It has been generally recognized that, at the micrometer of honeycombs. However, they did not study the size-
scale, the strain gradient effect plays an important role in dependent effect or the tunable elastic properties for hierar-
chical honeycombs. The aim of this paper is thus to obtain
⇑ Corresponding author. Tel.: +44 29 20874824.
closed-form results for the size-dependent and tunable
E-mail address: zhuh3@cf.ac.uk (H.X. Zhu).
mechanical properties of regular self-similar hierarchical
1359-6454/$36.00 Crown Copyright Ó 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

3. Author's personal copy
4928 H.X. Zhu et al. / Acta Materialia 60 (2012) 4927–4939
honeycombs with square and equilateral triangular cells combined axial compression, transverse shear deformation
whose first-order cell wall thickness is at the micron and and plane-strain bending. Only a representative unit cell
nanometer scales. shown in Fig. 1b is needed for mechanics analysis. By
assuming that the bending stiffness of the cell walls is DB,
2. Independent elastic constants of the first-order the shear stiffness is DS and the axial compression stiffness
honeycombs is DC, it is easy to obtain all five independent elastic con-
stants for a perfect regular honeycomb with equilateral tri-
The first-order honeycombs are treated as materials angular cells and uniform cell walls.
whose size is much larger than the individual cells and The in-plane Young’s modulus of a perfect regular hon-
are assumed to have uniform cell walls of length L, width eycomb with equilateral triangular cells can be obtained as
b and thickness h. The wall width b is assumed to be much (see Appendix A)
larger than the thickness h. The focus of this paper is on
2DS Á D2 Á L2 þ 24DB Á D2 þ 24DB Á DS Á DC
C C
small elastic deformation and the elastic properties. E1 ¼ pffiffiffi 3 pffiffiffi pffiffiffi
Although the cell wall can be of a metallic, biological or 3L Á DS Á DC þ 12 3L Á DB Á DC þ 4 3L Á DB Á DS
polymeric material, it is always assumed to be isotropic ð2Þ
and linear elastic, with Young’s modulus ES and Poisson The in-plane Poisson ratio is determined as
ratio vS, in the analysis that follows. As regular honey-
combs with either square cells or equilateral triangular cells L2 Á DS Á DC þ 12DB Á DC À 12DB Á DS
m12 ¼ ð3Þ
have three orthogonal planes of symmetry, the maximum 3L2 Á DS Á DC þ 36DB Á DC þ 12DB Á DS
possible number of independent elastic constants is nine and the out-of-plane Young’s modulus is given by
[4,8,20].
E3 ¼ f1 ES q ð4Þ
2.1. Honeycombs with equilateral triangular cells where ES is the Young’s modulus of the solid material and
f1 is a coefficient to be specified in the sections that follow,
Fig. 1 shows a regular honeycomb with equilateral trian- the value of which depends on the type of honeycomb cell
gular cells. It is easy to show that its in-plane mechanical and the size scale of the cell wall thickness.
properties are isotropic. To fully determine the relationship The out-of-plane shear modulus can be obtained as
between the applied state of effective stress and the
responding state of effective strain, only five independent G31 ¼ f2 GS q ð5Þ
elastic constants – E1, m12, E3, v31 and G31 – need be where GS is the shear modulus of the solid material and f2
obtained. The detailed derivation of these independent is a coefficient to be specified.
elastic constants is given in Appendix A. The out-of-plane Poisson ratio obviously remains the
The relative density of the first-order honeycomb with same as that of the solid material at both the macro- and
equilateral triangular cells is given by microscales (in this case, the strain gradient effect is
pffiffiffi absent), thus
2 3h
q¼ ð1Þ m31 ¼ mS ð6Þ
L
When the honeycomb is compressed by a uniform stress To simplify the analysis for honeycomb material with
in the x direction, there is no junction rotation because of cells at the nanometer scale, the surface is assumed to be
the symmetry, and the inclined cell walls undergo isotropic 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 the
honeycomb is compressed in the x direction.

4. Author's personal copy
H.X. Zhu et al. / Acta Materialia 60 (2012) 4927–4939 4929
the bulk material. The out-of-plane Poisson ratio m31 of a The out-of-plane shear modulus can be obtained as
honeycomb with nanosized equilateral triangular cells is G31 ¼ f4 GS q ð12Þ
thus mS. Therefore, in the sections that follow, we only need
to obtain the results of four independent elastic constants – where f4 is a coefficient to be determined.
E1, m12, E3 and G13 – for perfect regular honeycombs with At the nanoscale, as the Poisson ratio of the surface is
equilateral triangular cells at different size scales. assumed to be mS, the out-of-plane Poisson ratio remains
the same as that of the solid material for different-order
2.2. Honeycombs with square cells hierarchical honeycombs with regular square cells from
macro- and micro- down to the nanoscale,
The xy plane of perfectly regular honeycombs with m31 ¼ mS ð13Þ
square cells, as shown in Fig. 2a, is not isotropic. It is easy
to verify this, and only six independent elastic constants In the sections that follow, we only need to obtain four
have to be determined: E1, m12, G12, E3, v31 and G31. Their independent elastic constants – E1, E3, G12 and G31 – for
detailed derivation is given in Appendix B. the first-order honeycombs with regular square cells whose
The relative density of the first-order honeycomb with cell wall thickness is uniform at different size scales.
square cells and uniform cell wall thickness is given by
3. Elastic constants of macrosized first-order honeycombs
2h
q¼ ð7Þ
L For macrosized first-order honeycombs, the bending,
The in-plane Young’s modulus is obtained as transverse shear and axial stretching/compression rigidities
rx DC of the cell walls are given by
E1 ¼ ¼ ð8Þ
ex L ES bh3
DB ¼ Á ð14aÞ
The in-plane Poisson ratio is determined as 1 À m2 12
S
GS bh
1 DS ¼ ð14bÞ
m12 ¼ vS h=L ¼ mS q ð9Þ 1:2
2
DC ¼ ES bh ð14cÞ
which is the same as that given by Wang and McDowell
[21] and applies to the first-order honeycomb with regular In Eq. (14b) a shear coefficient of 1.2 [22] is introduced for
square cells at different size scales. The in-plane shear mod- the rectangular cross-section of the cell walls.
ulus is obtained as
3.1. Honeycombs with regular equilateral triangular cells
6DB DS
G12 ¼ ð10Þ
DS L3 þ 12DB L Substituting Eqs. (1) and (14a–c) into Eqs. (2)–(5), the
and the out-of-plane Young’s modulus is given as dimensionless results of all the four elastic constants can
E3 ¼ f3 ES q ð11Þ be obtained as:
q 2 q 2
where f3 is a coefficient to be specified in the sections that E1 1 þ 5ð1ÀmS Þ þ 12ð1Àm2 Þ
follow, the value of which depends on the size scale of E1 ¼ 1 ¼ S
ð15Þ
3
ES q 1 þ 5ð1Àm Þ þ q2 2
q2
36ð1ÀmS Þ
the cell wall thickness. S
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 honeycomb
undergoes in-plane shear deformation.

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4930 H.X. Zhu et al. / Acta Materialia 60 (2012) 4927–4939
1
3
þ 15ð1ÀmS Þ Á q2 À 36ð1Àm2 Þ Á q2
1 1
m12 ¼ 1 1
S
ð16Þ
1 þ 5ð1ÀmS Þ Á q2 þ 36ð1Àm2 Þ Á q2
S
E3
E3 ¼ ¼1 ð17Þ
Es q
G31 1
G31 ¼ ¼ ð18Þ
Gs q 2
where the in-plane Young’s modulus is normalized
by 1 ES q; the out-of-plane Young’s modulus is normalized
3
by ESq; the out-of-plane shear modulus is normalized by
GSq; and the relative density q is given by Eq. (1). The
out-of-plane Poisson ratio is the same as that given in
Eq. (6). Wang and McDowell [21] obtained the in-plane
(i.e. the xy plane) elastic properties for macrosized first-or- Fig. 4. Relationship between the in-plane Poisson ratio m12 and the
der honeycombs with equilateral triangular cells. They relative density q of regular honeycombs with macrosized equilateral
triangular cells.
took 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-plane G12 1
G12 ¼ 1
¼ 3
ð20Þ
dimensionless Young’s modulus E1 and the relative density 8ð1ÀvS Þ
q3 GS 1 þ 5ð1ÀmS Þ q2
q is shown in Fig. 3, and the relationship between the in- 1
plane Poisson ratio m12 and the relative density q is pre- v12 ¼ vS q ð21Þ
2
sented in Fig. 4. As can be seen from Figs. 3 and 4, both G31 1
the in-plane dimensionless Young’s modulus E1 and the G31 ¼ ¼ ð22Þ
Gs q 2
Poisson ratio m12 vary so little with the honeycomb relative
density q over the range from 0 to 0.35 that they can be The dimensionless in-plane shear modulus G12 , which
treated as approximate constants: E1 % 1 and v12 % 0.33. has been normalized by 8ð1ÀvS Þ GS q3 , is plotted against the
1
These are the same as the approximate results given by honeycomb relative density q, as shown in Fig. 5. The
Wang and McDowell [21]. greater the honeycomb relative density, the smaller the
dimensionless in-plane shear modulus. This is quite differ-
3.2. Honeycombs with regular square cells ent from Wang and McDowell’s dimensionless result of 1
[21]. The dimensionless out-of-plane Young’s modulus
Using Eqs. (8)–(12) and (14a)–(14c), the dimensionless (i.e. E3 ¼ E3 = ðqES ÞÞ is the same as that for equilateral tri-
results of all the independent elastic constants of the first- angular honeycombs. Apart from G12 and v12 ¼ 1 vS q, all
2
order honeycombs with regular square cells can be easily other dimensionless elastic constants (i.e.
obtained as: E1 ¼ E2 ¼ E3 ¼ 1; G31 ¼ 1=2 and m31 = mS) are independent
E1 of the honeycomb relative density. Note that E1 and E3 are
E1 ¼ 1 ¼1 ð19Þ normalized by different factors.
2
qES
Fig. 3. Relationship between the in-plane dimensionless Young’s modulus Fig. 5. Relationship between the in-plane dimensionless shear modulus
E1 and the relative density q of regular honeycombs with macrosized G12 and the relative density q of regular honeycombs with macrosized
equilateral triangular cells. square cells.

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H.X. Zhu et al. / Acta Materialia 60 (2012) 4927–4939 4931
4. Elastic constants of microsized first-order honeycombs assumed to be absent. Therefore the out-of-plane dimen-
sionless shear modulus is the same as that given by Eq.
For uniform plates with thickness at the micrometer (18) for macrosized first-order honeycombs.
scale, the size-dependent bending stiffness is given by Yang For the first-order honeycombs with regular microsized
et al. [12] as equilateral triangular cells and different values of l/h, the
dimensionless in-plane Young’s modulus E1 , given by Eq.
ES bh3 2
DB ¼ Á Á ½1 þ 6ð1 À mS Þðl=hÞ Š ð23Þ (25), and the in-plane Poisson ratio v12, given by Eq.
1 À m2 12
S (26), are plotted against the relative density in Figs. 6
and the transverse shear rigidity is given by Zhu [8] as and 7, respectively. The Poisson ratio, vS, of the solid mate-
2 2 rial is chosen as 0.3 in the calculation of the figures. As can
GS bh ½1 þ 6ð1 À mS Þðl=hÞ Š be seen, the thinner the cell walls (i.e. the larger the value of
DS ¼ Á ð24Þ
1:2 1 þ 2:5ð1 þ mS Þðl=hÞ2 l/h), the larger the dimensionless in-plane Young’s modulus
In Eqs. (23) and (24), h is the thickness of the cell walls and E1 and the smaller the in-plane Poisson ratio v12.
l is the material intrinsic length for the strain gradient ef- Eq. (1) can be rewritten as
pffiffiffi
fect. If the first-order honeycomb is made of a metallic l 2 3l l ql
material, l is at the submicron to micron scale. The axial ¼ or ¼ pffiffiffi ð27Þ
h qL L 2 3h
stretching/compression stiffness of the cell walls with a uni-
form thickness at the micrometer scale is the same as that where L is the length of the cell walls, which can be defined
given in Eq. (14c) because the strain gradient effect is ab- as the size of the cells. Figs. 6 and 7 demonstrate that, for
sent. In this section, we consider the elastic properties of the first-order honeycombs with equilateral triangular cells
the first-order honeycombs made of uniform cell walls at the micrometer size scale and with a fixed relative density
whose thickness h is at the micrometer or submicron scale. q, the smaller the cell size, the larger the dimensionless in-
plane Young’s modulus and the smaller the in-plane Pois-
4.1. Honeycombs with regular equilateral triangular cells son ratio.
Substituting DB, DS and DC given in Eqs. (23), (24) and 4.2. Honeycombs with regular square cells
(14c) into Eqs. (2) and (3), the in-plane Young’s modulus
(which is normalized by 1 ES qÞ of the first-order honey- For the first-order honeycombs with microsized regular
3
combs with microsized equilateral triangular cells can be square cells, the strain gradient effect is clearly absent when
obtained as it is uniaxially deformed in either the x, y or z direction.
Therefore, its dimensionless elastic constants: E1 ; E3 ; G31
E1
E1 ¼ 1 and the Poisson ratios: v12 and m31 are exactly the same
Á ES q
3 as those of the first-order honeycombs with macrosized
2 2
1 þ 5ð1Àv Þ½1þ6ð1Àvðl=hÞ 2 Š q2 þ 12ð1Àv2 Þ Á ½1 þ 6ð1 À vS Þðl=hÞ2 Š Á q2
1þ2:5ð1þvS Þ
Þðl=hÞ
1 regular square cells.
S S
¼ S
Substituting Eqs. (23) and (24) into Eq. (10), the in-
1þ2:5ð1þvS Þ2 2
1 þ 5ð1Àv Þ½1þ6ð1Àvðl=hÞ 2 Š q2 þ 36ð1Àv2 Þ Á ½1 þ 6ð1 À vS Þðl=hÞ2 Š Á q2
1
plane shear modulus can be obtained as
S S Þðl=hÞ S
ð25Þ
G12 1 þ 6ð1 À vS Þðl=hÞ2
G12 ¼ ¼ ð28Þ
and the in-plane Poisson ratio is derived as 1
q3 GS 1 þ 3 Á 1þ2:5ð1þvS Þðl=hÞ2 q2
8ð1ÀvS Þ 5ð1ÀvS Þ 1þ6ð1Àv Þðl=hÞ2
S
1þ2:5ð1þvS Þ2 ðl=hÞ2
1
þ 15ð1Àv Þ½1þ6ð1Àv Þðl=hÞ2 Š q2 À 36ð1Àv2 Þ Á ½1 þ 6ð1 À vS Þðl=hÞ2 Š Á q2
1
1
m12 ¼
3 S S S which is normalized by G q3 .
8ð1ÀvS Þ S
2 2
1 þ 5ð1Àv Þ½1þ6ð1Àvðl=hÞ 2 Š q2 þ 36ð1Àv2 Þ Á ½1 þ 6ð1 À vS Þðl=hÞ2 Š Á q2
1þ2:5ð1þvS Þ
Þðl=hÞ
1
S S S
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. The
dimensionless out-of-plane Young’s modulus E3 in the z
direction is the same as that of the macrosized first-order
honeycomb, and the out-of-plane Poisson ratio m31 is the
same as vs.
As the size of a honeycomb material is much larger than
its cells, when the honeycomb material with microsized cells
undergoes a globally uniform out-of-plane shear deforma-
tion in the zx or zy plane (see Fig. 1), the shear strain in each
Fig. 6. Relationship between the in-plane dimensionless Young’s modulus
cell wall is always in the cell wall plane and uniform in E1 and the relative density q of regular honeycombs with microsized
amplitude, and any strain gradient effect can thus be equilateral triangular cells.

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4932 H.X. Zhu et al. / Acta Materialia 60 (2012) 4927–4939
and surface elasticity or surface energy dominates the influ-
ence on the mechanical properties.
For uniform plates of thickness at the nanoscale, the
combined effects of surface elasticity and initial stress on
the bending stiffness is obtained as [18]
!
Es bh3 6S 2s0
Db ¼ 1þ À vs ð1 þ vs Þ
1 À v2 12
s Es h Es h
!
Es bh3 6ln vs ð1 þ vs Þ
¼ 1þ þ e0s ð29Þ
1 À v2 12
s h 1 À ms
The transverse shear stiffness is derived as [8]
h i2
Fig. 7. Relationship between the in-plane Poisson ratio m12 and the GS bh 1 þ 6ln þ e0s mS ð1þmS Þ
h 1ÀmS
relative density q of regular honeycombs with microsized equilateral DS ¼ Á À ln Á 2 ð30Þ
1:2 1þ 10ln
þ 30
triangular cells. h h
and the axial stretching/compression stiffness is given by
[15]
Dc ¼ Es bhð1 þ 2ln =hÞ ð31Þ
In Eqs. (29)–(31), S is the surface elasticity modulus,
ln = S/Es is the material intrinsic length at the nanoscale
and s0 is the initial surface stress, the amplitude of which
can 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 are
2s
e0s ¼ À Es 0 ð1 À vs Þ in both the cell wall length and width
h
directions, and the initial elastic residual strain in the cell
wall thickness direction is eh ¼ 4vsss0 ¼ À 1Àvs e0s . Although
0 E h
2vs
the amplitude of e0scan be varied, the range of the recover-
able elastic residual strain is limited by the yield strain of
the bulk material. For single crystal nanomaterials or poly-
Fig. 8. Relationship between the in-plane dimensionless shear modulus
meric materials, the yield strain can be 10% or even larger.
G12 and the relative density q of regular honeycombs with microsized
square cells. Atomistic simulation [26] has shown that, if the diameter of
a gold wire is sufficiently small, it can automatically under-
go plastic deformation solely owing to the presence of the
The relationship between the in-plane dimensionless initial surface stresses. It is well known that the yield
Shear modulus G12 and the relative density q of the first- strength, ry, of some conductive polymer materials or
order honeycombs with microsized regular square cells is nanosized metallic materials can be 0.1E (E is the Young’s
plotted in Fig. 8. The thinner the cell walls, the larger the modulus) or larger [27]. Biener et al. [28] have experimen-
dimensionless in-plane shear modulus. As the size of the tally found that, for nanoporous Au material, by control-
cells can be defined as L, for microsized honeycombs with ling the chemical energy, the adsorbate-induced surface
a fixed relative density, the smaller the cell size, the larger stress s0 can reach 17–26 N mÀ1. If the diameter of the lig-
the dimensionless in-plane shear modulus. aments is 5 nm, rx would be 20 GPa. As the bulk material
0
It is easy to verify that, when the strain gradient effect is discussed in this paper can be metallic, polymeric or biolog-
absent (i.e. h/l tends to 0), all the elastic constants of the ical, without losing generality, the tunable range of
first-order honeycombs with either microsized equilateral 2s
e0s ¼ À Es 0 ð1 À vs Þ is assumed to be from À0.1 to 0.1. In
h
triangular cells or square cells reduce to those of their mac- other words, the von Mises yield strength of the solid mate-
rosized counterparts. rial is assumed to be 0.1ES/(1 À vS). If the actual yield
strength of the honeycomb solid material is larger or smal-
5. Elastic constants of nanosized first-order honeycombs ler than this, the results obtained for the nanosized first-or-
der honeycombs can still be obtained by scaling those
At the nanometer scale, both the surface elasticity [15] presented in this paper up or down.
and the initial stresses [17,18] can greatly affect the mechan- When the effect of the initial surface stress s0 is absent,
ical properties of structural elements. Atomistic simula- the initial dimensions of the nanosized cell walls are
tions [19] show that, for metallic structural elements with assumed to be length L0, width b0 and thickness h0 for both
the size of a few nanometers, no strain gradient effect exists, regular triangular and square honeycombs. When the effect

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H.X. Zhu et al. / Acta Materialia 60 (2012) 4927–4939 4933
of the initial surface stress is present, the dimensions of the first-order honeycombs with microsized cells, for nanosized
cell walls become first-order honeycombs with a fixed relative density q0, the
smaller the cell size L0, the larger the normalized in-plane
L ¼ L0 ð1 þ e0s Þ ð32aÞ
Young’s modulus. If the surface elasticity modulus is nega-
b ¼ b0 ð1 þ e0s Þ ð32bÞ tive (i.e. ln = S/ES < 0), the effects are reversed. For example,
2vs when ln/h is À0.2, the dimensionless Young’s modulus
h ¼ h0 ð1 þ eh Þ ¼ h0 1 À
0 e0s ð32cÞ
1 À vs becomes 0.6, which is smaller than that of the macrosized
honeycombs (i.e. ln/h = 0). However, ln/h must be larger
and thus the relative density of both the first-order square than À1/2, otherwise the elastic modulus E1 would be nega-
and equilateral triangular honeycombs becomes tive and the honeycomb structure would deform automati-
cally [8] to a stable configuration. On the other hand, when
hL0 2vs the effect of the surface elasticity modulus is absent and the
q¼ q0 ¼ 1 À e0s q0 =ð1 þ eos Þ ð33Þ
h0 L 1 À vs effect of the initial stress/strain is present, Eq. (34) reduces to
2s h i
where e0s ¼ À Es 0 ð1 À vs Þ and 1 2 1 mS ð1þmS Þ
pffiffiffi
h 1þ mS ð1þmS Þ Á q þ 12ð1Àm2 Þ Á 1 þ eos Á 1Àm Á q2
5ð1ÀmS Þ½1þeos Á 1Àm Š S S
q0 ¼ 2 3h0 =L0 ð33aÞ E1 ¼ S
h i
1þ h 1 i Á q2 þ 1 2 Á 1 þ eos Á mS ð1þmS Þ Á q2
36ð1Àm Þ 1ÀmS
for equilateral triangular honeycombs and 5ð1ÀmS Þ 1þeos Á
mS ð1þmS Þ S
1ÀmS
q0 ¼ 2h0 =L0 ð33bÞ 2vS
1 À 1ÀvS eos
for square honeycombs. Á
1 þ eos
ð36Þ
5.1. Honeycombs with regular equilateral triangular cells
Eq. (36) can be approximated well by E1 % ð1 À 2vS eos =
Substituting DB, DS and DC and the dimensions of the ð1 À vS ÞÞ=ð1 þ eos Þ, and the error is smaller than 1% if
nanosized cell walls given by Eqs. (29)–(33) into Eq. (2), q0 6 0.35. Thus the in-plane dimensionless Young’s modu-
the in-plane dimensionless Young’s modulus of honey- lus of a nanosized first-order regular honeycomb with equi-
combs with nanosized equilateral triangular cells can be lateral triangular cells can be varied by adjusting the
obtained as amplitude of the initial surface stress (and hence the initial
E1 strain eos), and the tunable range depends on the Poisson
E1 ¼ 1
3
E S q0 ratio vS of the solid materials. When eos is varied from
 Ã
À Á n 2
1þ10ln þ30ðlh Þ ð1þ2ln Þ 2
h i 0.1 to À0.1, the tunable range of the dimensionless in-plane
1 þ 2ln þ 5ð1ÀmS Þ Á h h
1 ih q þ 1 2 Á 1 þ 6ln þ eos Á mS ð1þmS Þ Á q2
h
1þ6ln þeos Á
h
12ð1Àm Þ
mS ð1þmS Þ
1ÀmS
h 1ÀmS S Young’s modulus will be from 0.909 to 1.111 if vS = 0,
¼ h i 0.831–1.206 if vS = 0.3, and 0.727–1.5 if vS = 0.5.
mS ð1þmS Þ
1þ10ln þ30ðlh Þ
n 2 1þ6ln þeos Á
1 þ 5ð1ÀmS Þ Á h
1 h i q2 þ 1
Á
h 1ÀmS
Á q2 The in-plane Poisson ratio can be obtained as
mS ð1þmS Þ 36ð1Àm2 Þ ½1þ2lhn Š
1þ6ln þeos Á S
h 1ÀmS
h i
2vS m ð1þm Þ
1 À 1ÀvS eos 10ln ln 2 1þ6ln þeos Á S1Àm S
h 1þ h þ30ð h Þ i q2 À 1 2 Á
h
Á 1 1
þ 15ð1ÀmS Þ Á
S
Á q2
1 þ eos 3 mS ð1þmS Þ 36ð1ÀmS Þ ð1þ2lhn Þ
6ln
1þ h þeos Á 1Àm
ð34Þ S
h i
m12 ¼
mS ð1þmS Þ
where the in-plane Young’s modulus is normalized by n 2 1þ6ln þeos Á
1þ10ln þ30 lh ð Þ i 2 h 1ÀmS
1
Á ES q0 and h and q0 are given by Eqs. (32c) and (33a), 1 þ 5ð1ÀmS Þ Á h
1 h
q þ 1
36ð1Àm2 Þ
Á Á q2
3 m ð1þm Þ
1þ6ln þeos Á S1Àm S S ð1þ2lh
n
Þ
h
respectively. S
When the effect of the initial surface stress s0 is absent ð37Þ
(i.e. the initial elastic residual strain e0S of the bulk material When the effect of the initial surface stress s0 is absent, Eq.
is 0) and the effect of the surface elasticity is present, h = h0 (37) reduces to
and q = q0, and Eq. (34) reduces to 2
1þ10ln þ30ðlh Þ
n
½1þ6lhn Š 2
 à 1 1
þ 15ð1ÀmS Þ h
Á q2 À 36ð1Àm2 Þ Á
1
Áq
À Á ð1þ2lhn Þ 1þ10ln þ30ðlh Þ
n 2
À Á 3 ½ 1þ6lnŠ ð1þ2lhn Þ
1 þ 2ln þ 5ð1ÀmS Þ Á
1 h
Á q2 þ 12ð1Àm2 Þ Á 1 þ 6ln Á q2
1
m12 ¼ h S
ð38Þ
h ð1þ6lhn Þ h
n 2
E1 ¼ 2
S
ð35Þ 1 1þ10ln þ30ðlh Þ
h 1 ½1þ6ln Š
1
1 þ 5ð1ÀmS Þ Á
1þ10ln þ30ðlh Þ
h
n
1
Á q2 þ 36ð1Àm2 Þ Á
ð1þ6lhn Þ
Á q2 1þ Á q2 þ 36ð1Àm2 Þ Á 1þ2hn Á q2
ð1þ6lhn Þ S ð1þ2lhn Þ
5ð1ÀmS Þ ½1þ h Š
6ln
S ð lh Þ
Fig. 9 shows the effect of the surface elasticity on the rela- Fig. 10 shows that the effect of the surface elasticity on
tionship between the normalized in-plane (i.e. xy plane) the in-plane Poisson ratio (given by Eq. (38)) of nanosized
Young’s modulus (given by Eq. (35)) and the relative density first-order honeycombs depends upon the value of ln/h.
q0 of the nanosized first-order honeycombs with regular The thinner the cell walls, the smaller the Poisson ratio.
equilateral triangular cells. If the surface elasticity modulus On the other hand, the effect of the initial stress or strain
is positive (i.e. ln = S/ES 0), the thinner the cell walls, the on the Poisson ratio is so small that the result is not
larger the normalized Young’s modulus. Similarly to the presented.

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4934 H.X. Zhu et al. / Acta Materialia 60 (2012) 4927–4939
Young’s modulus and shear modulus are proportional to
(1 + 2ln/h). On the other hand, when the effect of surface
elasticity is absent (or fixed) and the effect of the initial
stress/strain is present, both the out-of-plane dimensionless
Young’s modulus and shear modulus depend not only on
the amplitude of the initial strain eos, but also on the Pois-
son ratio vS of the solid material.
5.2. Honeycombs with regular square cells
Substituting Eqs. (29)–(33) into Eq. (8), the in-plane
dimensionless Young’s modulus of the first-order honey-
combs with nanosized square cells can be obtained as
Fig. 9. Relationship between the in-plane dimensionless Young’s modulus
E1 1 2ln 2vS
E1 and the relative density q of regular honeycombs with nanosized E1 ¼ ¼ 1þ 1À eos =ð1 þ eos Þ ð41Þ
equilateral triangular cells. Es q0 2 h 1 À vS
which is normalized by ESq0.
The in-plane dimensionless shear modulus of nanosized
1
first-order honeycombs, which is normalized by 8ð1ÀvS Þ q3 GS , 0
can be obtained as
!3
G12 1 þ 6ln þ e0S mS ð1þmS Þ
h ð1ÀmS Þ
2vS
1 À 1ÀvS eos
G12 ¼ 1 ¼ n 2
Á
q3 G
8ð1ÀvS Þ 0 S 3 1þ10ln þ30ðlh Þ 1 þ eos
1 þ 5ð1ÀvS Þ Á 6lnh mS ð1þmS Þ q2
1þ h þe0S ð1ÀmS Þ
ð42Þ
When 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 1 þ 6ln
h
G12 ¼ 1
¼ 2 ð43Þ
Fig. 10. Size-dependent effect on the relationship between the in-plane G q3
8ð1ÀvS Þ S 0 3 1þ10ln þ30ðlh Þ
h
n
Poisson ratio m12 and the relative density q of regular nanosized equilateral 1þ 5ð1ÀmS Þ
Á 1þ6ln
q2
h
triangular honeycombs when the effect of the cell wall initial elastic
residual strain is absent. The relationship between the dimensionless in-plane shear
modulus G12 and the relative density q of the first-order
From the stretching stiffness of nanoplate given by Eq. honeycombs with nanosized square cells is plotted in
(31), the normalized out-of-plane Young’s modulus of Fig. 11. If the surface elasticity modulus is positive (i.e.
the first-order honeycombs with nanosized equilateral tri- ln = S/ES 0), the thinner the cell walls, the larger the
angular cells can be easily obtained as dimensionless shear modulus. If the relative density q of
nanosized honeycombs is fixed, the smaller the cell wall
E3 2ln 2vS
E3 ¼ ¼ 1þ Á 1À eos =ð1 þ eos Þ ð39Þ
E s q0 h 1 À vS
The normalized out-of-plane shear modulus can be easily
derived as
G31 1 2ln 2vS
G31 ¼ ¼ 1þ 1À eos =ð1 þ eos Þ ð40Þ
Gs q0 2 h 1 À vS
For the nanosized first-order honeycombs with a fixed rel-
ative density, the thinner the cell walls, the larger will be the
normalized Young’s modulus and the out-of-plane shear
modulus if ln = S/ES is positive. If the surface elasticity
modulus is negative, the trend of the effects is reversed.
As the Poisson ratio of the surface is assumed to be the
same as that of the bulk material, the out-of-plane Poisson
Fig. 11. Size-dependent effect on the relationship between the in-plane
ratio of a nanohoneycomb, m13, is thus equal to vS. When
dimensionless shear modulus G12 and the relative density q of regular
the effect of the surface elasticity is present and the effect honeycombs with nanosized square cells when the effect of the cell wall
of the initial stress/strain is absent, both the out-of-plane initial elastic residual strain is absent.

10. Author's personal copy
H.X. Zhu et al. / Acta Materialia 60 (2012) 4927–4939 4935
treated as materials whose size is much larger than the indi-
vidual cells at the same hierarchy level. The relative density
of the nth-order self-similar hierarchical honeycombs with
either equilateral triangular cells or square cells can be eas-
ily obtained as
2vS
qn ¼ 1 À eos ðq0 Þn =ð1 þ eos Þ ð45Þ
1 À vS
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. When
the initial strain eos is 0, qn reduces to (q0)n.
For the first-order honeycombs with regular equilateral
Fig. 12. Effect of cell wall initial elastic residual strain on the relationship triangular cells of wall thickness at the macro-, micro- or
between the in-plane dimensionless shear modulus G12 and the relative nanoscale, the five independent elastic constants are
density q of regular honeycombs with nanosized square cells when the
effect of the surface elasticity modulus is absent.
obtained as given in Sections 3–5, depending upon the size
scale of their cell wall thickness. For the nth-order self-sim-
length or the cell size L, the larger the dimensionless elastic ilar hierarchical honeycombs with equilateral triangular
modulus. If the surface elasticity modulus is negative (i.e. cells, the five independent dimensionless elastic constants
ln = S/ES 0), the effects are reversed. For example, if can be obtained as
ln =h ¼ À0:1; G12 becomes 0.4, which is smaller than that q nÀ1
ðE1 Þn
for macrosized honeycombs (i.e. ln/h = 0). Eq. (43) implies ðE1 Þn ¼ 1 % 0 Á ðE1 Þ1 ð46Þ
that ln/h must be larger than À1/6, otherwise the in-plane E q
3 S 0
3
shear modulus G12 becomes negative and the honeycomb ðv12 Þn % 1=3 ð47Þ
structure will automatically deform until the structural sta- ðE3 Þn nÀ1
bility is regained [8]. ðE3 Þn ¼ ¼ ðq0 Þ Á ðE3 Þ1 ð48Þ
Es q0
When the effect of the surface elasticity modulus is
ðG31 Þn q0 nÀ1
absent and only the effect of the initial surface stress is pres- ðG31 Þn ¼ ¼ Á ðG31 Þ1 ð49Þ
ent, Eq. (42) reduces to Gs q0 2
!3 and (v31)n = vS, where n is the hierarchy level of the self-
G12 1 þ e0S mS ð1þmS Þ 2vS
1 À 1ÀvS eos
G12 ¼ 1 ¼
ð1ÀmS Þ similar hierarchical honeycombs. In Eqs. (46), (48) and
3 1
G q3 1 þ 5ð1ÀmS Þ Á
8ð1ÀvS Þ S 0 mS ð1þmS Þ q
2 1 þ eos (49), ðE1 Þ1 ; ðE3 Þ1 and ðG31 Þ1 are the dimensionless Young’s
1þe0S ð1ÀmS Þ
moduli in the x and z directions and the dimensionless
ð44Þ shear modulus in the xz plane of the first-order honey-
Fig. 12 shows the effect of the initial residual elastic comb, respectively. It is easy to check that, when n P 2,
strain e0S of the bulk material (or the initial stress) on the the errors of Eqs. (46) and (47) are smaller than 0.2% if
in-plane dimensionless shear modulus of the first-order q0 6 0.35.
honeycombs with nanosized perfect regular square cells. For the nth-order self-similar hierarchical honeycombs
When vS = 0.3 and the initial strain eos is varied from 0.1 with square cells, the six independent dimensionless elastic
to À0.1, the dimensionless in-plane shear modulus G12 constants can be obtained as
can change from 0.6 to 1.5. q nÀ1
ðE1 Þn
For the first-order nanosized honeycombs with regular ðE1 Þn ¼ 1 ¼ 0 Á ðE1 Þ1 ð50Þ
square cells, it is easy to obtain the same dimensionless qE
2 0 S
2
out-of plane Young’s modulus E3 and shear modulus G31 ðE3 Þn
ðE3 Þn ¼ ¼ ðq0 ÞnÀ1 Á ðE3 Þ1 ð51Þ
as those given by Eqs. (39) and (40) for the nanosized first- q0 ES
order honeycombs with regular equilateral triangular cells. ðG12 Þn q 3ðnÀ1Þ
It is easy to verify that, for regular nanosized honey- ðG12 Þn ¼ 1 3
% 0 Á ðG12 Þ1 ð52Þ
8ð1ÀvS Þ
ðq0 Þ GS 8
combs with either square cells or equilateral triangular q nÀ1
cells, all the elastic constants will reduce to those of their ðv12 Þn ¼ 0 Á ðv12 Þ1 ð53Þ
macrosized counterparts when the effects of both the sur- 2
face elasticity and the initial stress/strain are absent. ðG31 Þn q0 nÀ1
ðG31 Þn ¼ ¼ Á ðG31 Þ1 ð54Þ
Gs q0 2
6. Size-dependent and tunable elastic properties of
hierarchical honeycombs and (m31)n = mS. In Eqs. (50)–(54), the elastic constants of
the first-order honeycombs with regular square cells of wall
The hierarchical honeycombs are assumed to be self- thickness at the macro-, micro- or nanoscale, ðE1 Þ1 ;
similar [29]. At all different hierarchy levels, they are ðE3 Þ1 ; ðG12 Þ1 ; ðm12 Þ1 and ðG31 Þ1 , are given in Sections 3–5,

11. Author's personal copy
4936 H.X. Zhu et al. / Acta Materialia 60 (2012) 4927–4939
depending upon the size scale of their cell wall thickness. cellular material collapse automatically [8]. In practical
Therefore, the elastic constants of the nth-order self-similar applications, one failing part could help protect others.
hierarchical honeycombs are functions of those of the first-
order honeycombs. It is easy to check that, when n P 2, the 7. Conclusion
error 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 structural This paper presents the detailed derivation and closed-
hierarchy on the elastic properties of honeycombs. How- form results of all the independent elastic constants of
ever, they did not study the size-dependent effect or the tun- self-similar hierarchical honeycombs with either regular
able elastic properties of hierarchical honeycombs. Zhu [8] square or equilateral triangular cells. The results imply that
studied the size-dependent and tunable mechanical proper- many interesting geometrical, mechanical and physical
ties for single-order regular hexagonal honeycombs, but properties and functions that do not exist in single-order
not for hierarchical honeycombs. macrosized cellular materials become possible in their hier-
For regular self-similar hierarchical honeycombs with archical nanostructured counterparts. If the cell wall thick-
the thickness of their first-order cell walls at the nanometer ness of the first-order honeycomb is at the micrometer
scale, the dimensionless elastic properties ðE1 Þn ; ðE3 Þn ; scale, the elastic properties of a hierarchical honeycomb
are size dependent owing to the strain gradient effects. If
2vS
ðG12 Þn and ðG31 Þn contain a common factor 1 À 1ÀvS eos =
the cell wall thickness of the first-order honeycomb is at
ð1 þ eos Þ and can thus be varied over a large range by the nanometer scale, in addition to the size dependence
adjusting the amplitude of the initial strain eos. When the owing to the effects of surface elasticity, the elastic proper-
initial strain is absent, the initial cell diameter, area and ties of a hierarchical honeycomb are tunable because of the
volume of an nth-order self-similar hierarchical honeycomb effects of the initial stresses/strains, the amplitudes of which
are assumed to be (L0)n, (A0)n and (V0)n respectively. When are controllable. More interestingly, if the cell wall/strut
the initial strain eos is present, the dimensionless cell diam- thickness of the first order cellular material is at the nano-
eter, area and volume of an nth-order self-similar hierarchi- meter scale, the cell size, surface color, wettability, material
cal honeycomb become strength, stiffness, natural frequency and many other inter-
esting physical properties of a hierarchical nanostructured
ðLÞn =ðL0 Þn ¼ 1 þ eos ð55Þ cellular material could be varied over a large range.
2
ðAÞn =ðA0 Þn ¼ ð1 þ eos Þ ð56Þ
Acknowledgement
and
This work is supported by the EC project PIRSES-GA-
3 2009-247644.
ðV Þn =ðV 0 Þn ¼ ð1 þ eos Þ ð57Þ
Appendix A. Derivation of the elastic properties of regular
If eos can be controlled to change from À0.1 to 0.1, the honeycombs with equilateral triangular cells
dimensionless cell diameter, area and volume of an nth-or-
der self-similar hierarchical honeycomb would vary over A.1. In-plane Young’s modulus and Poisson ratio
ranges 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. When a first-order regular honeycomb with equilateral
For example, by adjusting the cell size, a hierarchical cellu- triangular cells is uniaxially compressed in the x direction,
lar material could possibly be controlled to change its color as shown in Fig. 1a, there is no junction rotation because of
or wettability. Micro- or nanosized porous materials are the symmetry of the structure and the applied load. Only a
often used to select/separate materials with a specific parti- representative unit cell structure, as shown in Fig. 1b, is
cle size in medical industry. The results given in Eqs. (55)– thus needed for the analysis. Node A is assumed to have
(57) suggest that the size of the selected/separated particles no displacement and rotation. The horizontal load applied
is tunable and controllable. Although it is difficult to alter at B is assumed to be P1 and the horizontal load applied at
the cell size and the mechanical properties, such as the stiff- C is P2. Obviously, there is no load in the y direction at B
ness, the buckling force and the natural frequency, of a sin- and C.
gle-order macro- or microsized porous material, the results The inclined cell wall AB undergoes bending, transverse
obtained in this section suggest that it is possible to realize shear and axial compression. Its deformation in the x direc-
those for its hierarchical counterpart with the first-order tion is given by
cell wall/strut thickness at the nanometer scale.
Dx ¼ Dx þ Dx þ Dx
b s c ðA1Þ
In addition, Eqs. (42) and (52) imply the possibility that,
if ln/h is very close to À1/6 (but ln/h should be slightly lar- where Dx ; Dx and Dx are the deformations in the x direction
b s c
ger than À1/6, the structure would not be stable), control- owing to cell wall bending, transverse shear and cell wall
ling eos to a negative value (say À0.1) could result in a axial compression, respectively. By assuming that the bend-
negative ðG12 Þ, and hence could make the hierarchical ing stiffness of the cell walls is DB, the shear stiffness is DS,

12. Author's personal copy
H.X. Zhu et al. / Acta Materialia 60 (2012) 4927–4939 4937
and the axial compression stiffness is DC, Eq. (A1) can be The deformation of the inclined cell wall in the y direction
rewritten as is
À Á3
P 1 Á sin 60 Á L
2 P 1 Á sin 60 Á L
2
Dy ¼ Dy þ Dy þ Dy
b s c ðA10Þ
Dx ¼ cos 30 þ
3DB DS Eq. (A10) can be rewritten as
P 1 Á cos 60 Á L
2 À Á3 ÀÁ
 cos 30 þ sin 30 P 1 Á sin 60 Á L2 P 1 Á sin 60 Á L
2
DC Dy ¼ sin 30 þ
3DB DS
P 1 L3 3P 1 L P 1 L ÀÁ
¼ þ þ ðA2Þ P 1 Á cos 60 Á L
32DB 8DS 8DC Â sin 30 À 2
cos 30
DC
where L is the length of the cell walls. pffiffiffi pffiffiffi pffiffiffi
From Fig. 1b, the dimension of the representative unit 3P 1 Á L3 3P 1 Á L 3P 1 Á L
¼ þ À ðA11Þ
cell in the x direction is 96DB 8DS 8DC
L L From Fig. 1b, the dimension of the representative unit cell
lx ¼ Á cos 60 ¼ ðA3Þ in the y direction is
2 4
pffiffiffi
The compressive strain of the inclined element AB in the x L 3L
ly ¼ Á sin 60 ¼ ðA12Þ
direction is therefore 2 4
Dx P 1 L2 3P 1 P1 The expansion strain in the y direction can be obtained as
ex ¼ ¼ þ þ ðA4Þ
lx 8DB 2DS 2DC Dy P 1 L2 P1 P1
ey ¼ ¼ þ À ðA13Þ
The stress component of the honeycomb in x direction due ly 24DB 2DS 2DC
to force P1 is thus
The Poisson ratio is therefore
P1 2P 1
rx1 ¼ ¼ pffiffiffi ðA5Þ ey L2 DS DC þ 12DB DC À 12DB DS
v12 ¼ ¼ 2
L Á sin 60 Á b 3L e 3L D D þ 36D D þ 12D D ðA14Þ
x S C B C B S
where the width b of the honeycomb is much larger than
the cell wall thickness h and is assumed to be 1 for As the honeycomb structure is isotropic, the in-plane shear
simplicity. modulus G12 can be obtained from Ex and v12, and given as
For the horizontal element AC, the deformation com- Ex
G12 ¼
patibility condition requires that the compressive strain 2ð1 þ v12 Þ
of AC due to force P2 should be the same as that of the pffiffiffi 2 pffiffiffi pffiffiffi
3L Á DS Á DC þ 12 3DB Á DC þ 12 3DB Á DS
inclined element AB in the x direction. Thus, ¼
4L3 Á DS þ 48L Á DB
P 1 L2 3P 1 P1 ðA15Þ
P 2 ¼ D C Á ex ¼ D C Á þ þ ðA6Þ
8DB 2DS 2DC
The stress component of the honeycomb in x direction due A.2. Out-of-plane shear modulus and Young’s modulus
to force P2 is derived as
P2 2P 2 To derive the out-of-plane shear modulus, the size of the
rx2 ¼ ¼ pffiffiffi honeycomb material is assumed to be much larger than the
L Á sin 60 Á b 3L
pffiffiffi honeycomb cells and thus much larger than the cell wall
P 1 Á L Á DC 3P 1 Á DC P1 length L. When the honeycomb material is subjected to
¼ pffiffiffi þ þ pffiffiffi ðA7Þ
4 3DB DS Á L 3L an out-of-plane pure shear stress syz (which is in the yz
The total compressive stress in the x direction is thus ob- plane), the shear load T on the cell wall of the representa-
tained as tive unit cell is shown in Fig. A1.
pffiffiffi pffiffiffi The equilibrium in the y direction requires
P 1 LDC 3P 1 DC 3P 1
rx ¼ rx1 þ rx2 ¼ pffiffiffi þ þ ðA8Þ Q ¼ T Á sin 600 ðA16Þ
4 3DB DS L L
and
The Young’s modulus of the honeycomb is therefore
pffiffi pffiffi pffiffiffi
P 1 ÁLÁDC 3P 1 ÁDC DS ÁDC ÁL2 þ12DB ÁDC þ12DB ÁDS
pffiffi 3 2
rx 4pffiffiDB þ DS ÁL þ L
3P 1
3 4 3DB ÁDS ÁL Q¼ sÁyz L ðA17Þ
Ex ¼ ¼ P ÁL2 3P ¼ D ÁD ÁL2 þ12D ÁD þ4D ÁD 4
ex 1
þ 1 þ P1 S C B C B S
8DB 2DS 2DC 8DB ÁDS ÁDC
where syzpis the effective out-of-plane shear stress acting on
ffiffi
2DS Á D2 Á L2 þ 24DB Á D2 þ 24DB Á DS Á DC
C C the area 43 L2 of the representative unit cell of the honey-
¼ pffiffiffi 3 pffiffiffi pffiffiffi ðA9Þ
3L Á DS Á DC þ 12 3L Á DB Á DC þ 4 3L Á DB Á DS comb material, as shown in Fig. A1.