the elastic and geometrical properties of micro- and nano-structured hierarchical random irregular...
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The elastic and geometrical properties of micro- andnano-structured hierarchical random irregular honeycombs
H. X. Zhu • H. C. Zhang • J. F. You • D. Kennedy •
Z. B. Wang • T. X. Fan • D. Zhang
Received: 7 February 2014 / Accepted: 28 April 2014 / Published online: 15 May 2014
� Springer Science+Business Media New York 2014
Abstract All the five independent elastic properties/
constants of micro- and nano-structured hierarchical and
self-similar random irregular honeycombs with different
degrees of cell regularity are obtained by analysis and
computer simulation in this paper. Cell wall bending,
stretching, and transverse shearing are the main deforma-
tion mechanisms of hierarchical honeycombs. The strain
gradient effects at the micro-meter scale, and the surface
elasticity and initial stress effects at the nano-meter scale
are incorporated into all the deformation mechanisms in the
analysis and finite element simulations. The results show
that the elastic properties of hierarchical random irregular
honeycombs strongly depend on the thickness of the first-
order cell walls if it is at the micro-meter scale, and that if
the thickness of the first-order cell walls is at the nano-
meter scale, the elastic properties of hierarchical random
irregular honeycombs are not only size-dependent, but are
also tunable and controllable over large ranges. In addition,
the geometrical properties of nano-structured hierarchical
random irregular honeycombs are also tunable and
controllable.
Introduction
The understanding of the mechanical properties of regular
and random irregular macro-sized single-order (i.e. conven-
tional) honeycombs [1–9] has well been established and
documented. These available results, however, may not apply
to their micro- and nano-sized counterparts [10] because of
the effects of the surface elasticity [11–14] and the initial
stress or strain [15, 16] at the nano-metre scale, and the strain
gradient effect at the micro-metre scale [10, 14, 17–28].
Experimental evidence [11, 29–33] has shown that the initial
surface stress can be controlled to vary by adjusting the
amplitude of an applied electric potential. For example,
although the initial surface stress of Au (111) is about 1.13 N/
m, Biener et al. [29] experimentally demonstrated that the
adsorbate-induced initial surface stress of nano-porous Au
material could be controlled to reach 17–26 N/m by adjusting
the chemical energy. Their experimental results [11, 29–33]
imply that for a single-order nano-sized honeycomb made of
a solid material with Young’s modulus ES & 100 GPa and
Poisson’s ratio vS = 0.3, the cell diameter of the honeycomb
could be controlled to increase or to reduce about 10 % if the
cell wall thickness is about 1.5 nm. This opens the possibility
to change and to control the colour, wettability, electric
capacitance, or natural frequency for nano-structured hon-
eycomb/cellular materials. Experiments by Weissmuller
et al. [31, 32] have demonstrated large recoverable defor-
mation of nano-porous materials by adjusting the amplitude
of the initial surface stress via controlling an applied electric
potential. Moreover, there is a linear correlation between the
surface stress and surface charge in anion adsorption on Au
H. X. Zhu (&) � H. C. Zhang � D. Kennedy
School of Engineering, Cardiff University, Cardiff CF24 3AA,
UK
e-mail: [email protected]
J. F. You
National Key Laboratory of Combustion, Flow and Thermo-
Structure, The 41st Institute of the Forth Academy of CASC,
Xi’an 710025, China
Z. B. Wang
CNM & IJRCNB Centers, Changchun University of Science and
Technology, Changchun, China
T. X. Fan � D. Zhang
State Key Laboratory of Composites, Department of Materials
Science and Engineering, Shanghai Jiaotong University,
Shanghai 200240, China
123
J Mater Sci (2014) 49:5690–5702
DOI 10.1007/s10853-014-8288-y
(111) [33]. These results suggest that nano-structured hon-
eycomb/cellular materials could be used as the actuating
materials or structures in practical applications. It may be
difficult to produce sufficiently large honeycomb/cellular
materials with all the cells at the nano-metre scale if their
structure is of just a single order. This difficulty could be
overcome if the honeycomb/cellular materials is nano-
structured and hierarchical, i.e. having a number of structural
levels.
Lakes [34] and Taylor et al. [35] studied the mechanical
properties of regular and conventional hierarchical cellular
materials, but did not explore the size effects at the micro- or
nano-metre scale and the possible tunable properties. The
size-dependent and tunable elastic properties have been the-
oretically studied for first-order nano-sized regular hexagonal
honeycombs [10] and for nano-structured hierarchical and
self-similar honeycombs with square, triangular [36] and
hexagonal [37] cells and open-cell foams with regular BCC
cells [37]. Living natural materials, such as bones or plant
stems, are usually cellular materials with random irregular
cells and multilevel structural hierarchy. Their basic building
blocks are usually at the micro- or nano-metre scale and their
mechanical properties may be not only size-dependent, but
also tunable and/or controllable. The degree of cell irregu-
larity can thus greatly affect the mechanical properties of the
nano-structured and hierarchical cellular materials, as has
already been observed for their macro-sized single-order
counterparts [7, 8]. The objective of this paper is to obtain
effects of cell regularity on the size-dependent and tunable
mechanical properties of micro- and nano-structured hierar-
chical and self-similar random irregular honeycombs, and
further to briefly discuss the tunable geometrical properties.
Geometrical model and treatment of finite element
simulations
Geometrical model of micro- and nano-structured
hierarchical and self-similar random irregular Voronoi
honeycombs
The micro- and nano-structured random irregular hierar-
chical honeycombs are assumed to be self-similar at dif-
ferent hierarchy levels. Zhu et al. [7, 8] developed a
computer code to construct representative unit periodic
random irregular Voronoi honeycombs, in which the
degree of the cell regularity a is defined as
a ¼ dd0
; ð1Þ
where d is the minimum distance between the centres of
any two neighbouring cells in a random irregular honey-
comb and d0 is the distance between two neighbouring
cells in a perfect regular hexagonal honeycomb with the
same number of complete cells, and given as [7, 8]
d0 ¼ffiffiffiffiffiffiffiffiffiffi
2A0
Nffiffiffi
3p
s
ð2Þ
In Eq. 2, A0 is the area of a representative unit volume ele-
ment (RVE) model/honeycomb and N is the total number of
complete cells within the unit RVE models. Figure 1a and b
shows the RVEs of hierarchical and self-similar honeycombs
with 300 complete cells and different degrees of regularity
a = 0 and a = 0.7 at different hierarchy levels.
It is assumed that for first-order random irregular hon-
eycombs, all the cell walls have the same uniform initial
thickness h0 and a unit initial width which is much larger
than h0. When the effects of the initial surface stresses/
strains are absent, the initial relative density of the first-
order random irregular micro- or nano-sized honeycomb
(i.e. the RVE model) is given as [5, 7, 8]
q0 ¼ h0
X
M
i¼1
l0i
!
=L20; ð3Þ
where l0i is the cell wall initial lengths and L0 is the initial
side length of the periodic unit honeycomb RVE models
(as shown in Fig. 1) and M is the total number of cell walls.
Size-dependent rigidities of first-order micro- or nano-
sized cell walls
In mechanical analyses of cellular materials, the main
deformation mechanisms [1, 2, 4–10] include cell wall/strut
bending, transverse shearing and axial stretching or com-
pression. When a first-order micro- or nano-sized honeycomb
is subjected to in-plane deformation, plane strain bending of
the cell walls is the dominant deformation mechanism [9, 10].
For micro-sized honeycombs, if the cell wall thickness is
between submicron and micron size, strain gradient [17–28]
may affect the bending and transverse shear rigidities. For a
flat and wide (compared to the thickness) plate with a uniform
thickness at the micro-metre scale, the size-dependent
bending, transverse shear and axial stretching/compression
rigidities have been obtained and given as [10]
Db ¼ESh3
12ð1� v2SÞ½1þ 6ð1� vSÞðlm=hÞ2�; ð4Þ
Ds ¼hEs
2:4ð1þ vSÞ� ½1þ 6ð1þ vSÞðlm=hÞ2�2
1þ 2:5ð1þ vSÞ2ðlm=hÞ2ð5Þ
Dc ¼ ESh; ð6Þ
where ES and vS are the Young’s modulus and Poisson’s
ratio of the solid material, h is the cell wall thickness and lmis the length scale parameter for strain gradient effect (i.e. a
J Mater Sci (2014) 49:5690–5702 5691
123
material constant which can be experimentally measured).
For metal materials, lm is usually between submicron and
micron size. In Eqs. 4 and 5, the cell wall width is assumed
to be a unity and much larger than the thickness h. It is easy
to check that when the plate thickness h is much larger than
lm (i.e. lm/h approaches 0), the bending and transverse shear
rigidities given in Eqs. 4 and 5 reduce to those of con-
ventional mechanics. The axial stretching/compression
stiffness always matches the conventional results because
there is no strain gradient effect when a cell wall undergoes
uniaxial tension or compression.
For nano-sized honeycombs, as the cell walls are very thin
and the surface to volume ratio is large, both the surface
elasticity and the initial stress or strain can greatly affect the
bending, transverse shearing and axial stretching/compression
rigidities. For simplicity in the analyses and simulations that
are to follow, both the surface and the bulk materials of the
first-order cell walls are assumed to be isotropic and to have
the same Poisson’s ratio vS. When a surface stress s0 is present,
the amplitudes of the initial stresses in the bulk material in the
length and the width directions of the first-order cell walls are
equal [38] and are obtained asr0L = r0
W = -2s0/h, where h is
the current thickness of the cell walls. At the nano-metre scale,
the yield strength of the first-order cell wall material, ry, could
reach 0.1ES or even a larger value [29, 39]. For recoverable
elastic deformation, the initial von Mises stress should not
exceed the yield strength ry of the bulk material, thus
re ¼ jrL0 j ¼ jrW
0 j ¼ j2s0
hj � ry ¼ 0:1ES. The amplitudes of
the initial elastic residual strains in the length and width
directions of the first-order cell walls are equal, and are related
to the initial stress in the bulk material by [38]
eL0 ¼ eW
0 ¼rL
0
ES
ð1� vSÞ ð7Þ
The initial elastic residual strain in the thickness direction
of the first-order cell walls is obtained as
et0 ¼
Dh
h0
� 4vSs0
ESh¼ � 2vS
ES
rL0 ð8Þ
When the effects of the initial stresses are present, the
bending, transverse shearing and axial stretching/com-
pression rigidities of a nano-sized first-order cell wall
have been obtained in Refs. [17], [10] and [12], respec-
tively, as
Db ¼ESbh3
12ð1� v2SÞ
1þ 6ln
hþ mSð1þ vSÞ
1� vS
eL0
� �
ð9Þ
Ds ¼Gsbh
1:2�½1þ 6ln
hþ vS
1þvS
1�vSeL
0 �2
1þ 10lnhþ 30ðln=hÞ2
ð10Þ
Fig. 1 Unit periodic RVE
models of the micro- or nano-
structured hierarchical and self-
similar random irregular
honeycombs with different
degrees of regularity. a a = 0.0;
b a = 0.7
5692 J Mater Sci (2014) 49:5690–5702
123
Dc ¼ Esbhð1þ 2ln=hÞ; ð11Þ
where the width of the cell walls, b, is initially unity and is
assumed to be much larger than the thickness h. In Eqs. 9–
11, ln = S/ES is the intrinsic length of the material at the
nano-metre scale and S is the surface elasticity modulus. It
should be noted that the bulk material from which the
hierarchical honeycombs are made could be any type of
metallic, or polymeric or biological materials. The material
nano-scale intrinsic length ln is typically in the range from
0.01 to 1 nm. When the effects of the cell wall initial
stresses are present, the current width, thickness and length
of the first-order cell walls can be obtained as
b ¼ b0ð1þ eW0 Þ ¼ 1þ eL
0 ð12Þ
h ¼ h0ð1þ et0Þ ¼ h0 1� 2vS
ES
rL0
� �
ð13Þ
l ¼ l0ð1þ eL0 Þ ð14Þ
The bending, axial stretching/compression and trans-
verse shearing rigidities can all be controlled to vary [10,
12, 16, 38] over large ranges because the cell wall width,
thickness and length can be controlled to reduce or increase
by about 10 % at the nano-metre scale by application of an
electric potential.
Treatment of the equivalent deformation rigidities
of first-order micro- or nano-sized cell walls in finite
element simulations
Micro- and nano-sized irregular honeycombs were simu-
lated using the ANSYS finite element software [40]. Each
of the cell walls was partitioned into a number of BEAM3
elements. This type of beam element has two nodes and
takes account of the bending, stretching and transverse
shearing deformation mechanisms. For a first-order micro-
or nano-sized 2D Voronoi honeycomb with an initial rel-
ative density q0, the initial cell wall thickness h0 can be
obtained from Eq. 3. In almost all commercial finite ele-
ment software, such ABAQUS or ANSYS, there is no type
of element which can directly incorporate the size-depen-
dent effects in the finite element simulations. We thus use
an equivalent solid beam with thickness he, Young’s
modulus Ee and Poisson’s ratio ve for the beam elements in
the finite element simulations. These equivalent values are
obtained from the following three simultaneous equations:
Eeh3e
12¼ Db ð15Þ
Gehe
1:2¼ Eehe
2:4ð1þ veÞ¼ Ds ð16Þ
Eehe ¼ Dc; ð17Þ
where Db, Ds and Dc are given in Eqs. 4–6 for micro-sized
first-order honeycombs, and in Eqs. 9–11 for nano-sized
first-order honeycombs. As Db, Ds and Dc are size-depen-
dent and may also be tunable, the effects of the strain
gradient, or the surface elasticity and initial stress/strain on
the mechanical properties of the first-order micro- or nano-
sized random irregular honeycombs can be incorporated
into the finite element simulations using the equivalent
values of he, Ee and ve obtained from Eqs. 15–17. The
Poisson’s ratio vS of the solid material of the first-order cell
walls is always assumed to be 0.4 in the simulations of this
paper. It is noted that the value vS of the solid material has
very little effect on the dimensionless elastic properties of
the macro- and micro-structured hierarchical honeycombs.
It is well known that if the sized effects are absent (i.e.
lm/h = 0 in Eqs. 4–6 or ln/h = 0 in Eqs. 9–11), the
obtained dimensionless Young’s modulus of a Voronoi
honeycomb would be a function of the relative density q0
and the degree of regularity a, and be independent of the
actual cell wall thickness h. When constructing each indi-
vidual Voronoi honeycomb model with a given relative
density, the actual cell wall thickness has been obtained,
which is different for each individual models. For each
model, we use the same Young’s modulus ES and Poisson’s
ratio vS for the solid material, but a different cell wall
thickness h (which is obtained when constructing the
geometrical model). The strain gradient effects or the
surface effects are incorporated into the simulations by
choosing different values of lm/h or ln/h in Eqs. 3–6 or 9–
11. Therefore, we can obtain the correct equivalent values
of he, Ee and ve for each of the individual model from
Eqs. 15–17 and 3–6 or 9–11. As there is no identical cell
wall thickness h for different random Voronoi honeycomb
models (even with the same degree of regularity or the
same relative density), we always have different values of
he, Ee and ve for each different models.
To validate the use of the equivalent values of he, Ee and ve
obtained from Eqs. 15–17, the exact theoretical deflection
and the computed deflection were compared for a single
horizontal micro- or nano-sized plate/beam cantilever
structure when a concentrated transverse load was applied to
the free end. Although the equivalent value of ve obtained
from Eqs. 15–17 could be negative or larger than 0.5 in some
cases, which is physically meaningless for conventional
solid materials, nevertheless the numerical results for the
mechanical responses obtained from the finite element
simulations were exactly the same as those of the theoretical
predictions. This suggests that the adoption of the equivalent
values of he, Ee and ve can properly incorporate the size
effects in finite element simulations and hence correctly
predict the mechanical behaviours of micro- or nano-sized
first-order random irregular honeycombs.
J Mater Sci (2014) 49:5690–5702 5693
123
Boundary conditions in finite element simulations
As the constructed random irregular Voronoi honeycomb
models shown in Fig. 1 are periodic [7], periodic boundary
conditions [5, 7] can easily be imposed in the finite element
simulations. The periodic boundary conditions are given by
ulefti � uleft
j ¼ uright
i0 � u
right
j0 ;
vlefti � vleft
j ¼ vright
i0 � v
right
j0 ;
utopi � u
topj ¼ ubottom
i0 � ubottom
j0 ;
vtopi � v
topj ¼ vbottom
i0 � vbottom
j0 ;
hlefti ¼ hright
i0 ;
htopi ¼ hbottom
i0 :
ð18Þ
where i and j are nodes on the left or top edge of the
random irregular periodic honeycomb, while i0 and j0 are
the corresponding nodes on the right or bottom edge.
Mechanical properties of first-order micro- and nano-
sized random irregular honeycombs
The first-order micro- or nano-sized random irregular
honeycombs are assumed to be made of a solid material
with Young’s modulus ES, shear modulus GS and Poisson’s
ratio vS. The elastic properties of random irregular hon-
eycombs are obtained by finite element simulations from
periodic models with 300 complete cells [7, 8]. It has been
found that although the obtained Young’s moduli could be
quite different for different random irregular honeycomb
models, they are almost identical in the x and y directions
for each of the same models [7]. Each data point in the
figures in this paper represents the mean value of 20 dif-
ferent random irregular honeycomb models with the same
degree of regularity and the same set of other parameters
(e.g. degree of regularity or cell wall thickness). It is noted
that the Poisson’s ratio of the solid material of the first-
order honeycombs is assumed to be 0.4 throughout this
paper. For both micro- and nano-sized first-order random
irregular honeycombs, the in-plane Young’s modulus is
normalised by 1.5ESq03/(1 - vS
2), i.e.
ðE1Þ1 ¼ðE1Þ1ð1� v2
SÞ1:5ESq3
0
ð19Þ
The out-of-plane Young’s modulus is normalised by ESq0,
i.e.
ðE3Þ1 ¼ðE3Þ1ESq0
ð20Þ
The bulk modulus is normalised by ESq0, i.e.
ðKÞ1 ¼ðKÞ1ESq0
ð21Þ
and the out-of-plane shear modulus is normalised by GSq0,
i.e.
ðG31Þ1 ¼ðG31Þ1GSq0
ð22Þ
Size-dependent elastic properties of micro-sized first-order
random irregular honeycombs
For the first-order random irregular micro-sized honeycombs
with degrees of regularity a = 0.0 and 0.7, the size-depen-
dent relationships between the dimensionless in-plane
Young’s modulus and the honeycomb relative density are
shown in Fig. 2a and b. As can be seen, the thinner the cell
walls, the larger the dimensionless Young’s modulus; and
the larger the relative density, the smaller the dimensionless
Young’s modulus. If the relative density remains unchanged,
Eq. 3 and Fig. 2a and b suggest that the smaller the cell size,
the larger would be the dimensionless Young’s modulus. If
the cell wall thickness is much larger than the material length
parameter lm of the strain gradient effect (i.e. lm/h0 = 0), the
results in Fig. 2a and b reduce to those of the macro-sized
honeycombs as given in Ref. [7]. Figure 3 shows the effects
of cell regularity on the dimensionless Young’s modulus of
irregular honeycombs with lm/h0 = 0.5 and constant initial
relative densities q0 = 0.01 and q0 = 0.2, each data point
represents the mean value of 20 similar models and the error
bar shows the standard deviation. As can be seen from Fig. 3,
when the honeycomb relative density is small (e.g.
q0 = 0.01), the dimensionless Young’s modulus generally
reduces with the increase of the cell regularity; in contrast,
when the honeycomb relative density is large (e.g. q0 = 0.2),
the dimensionless Young’s modulus generally increases
with the increase of the cell regularity. The trend change
takes place when q0 & 0.15 for random irregular honey-
combs with lm/h0 = 0.5, and occurs due to the increasing
honeycomb relative density, as shown in Fig. 3. This
observation is consistent with the results given in Fig. 7 of
Ref. [7] for the macro-sized counterparts (i.e. lm/h0 = 0),
where the dimensionless Young’s modulus reduces with the
increasing cell regularity when q0 = 0.01, increases with the
increment of the cell regularity when q0 = 0.25 or larger,
and the trend change occurs when q0 & 0.22. It can thus
logically be conjectured that the thinner the cell walls (i.e. the
larger the value of lm/h0), the smaller is the relative density at
which the trend change takes place.
Figure 4 shows the relationships between the dimen-
sionless bulk modulus and (a) the honeycomb relative
density, (b) the degree of regularity. For first-order random
irregular honeycombs with cell wall thickness at the micro-
5694 J Mater Sci (2014) 49:5690–5702
123
metre scale, generally, the thinner the cell walls, the larger
is the dimensionless bulk modulus because at the micro-
metre scale, thinner cell walls have a larger dimensionless
bending stiffness. As can be seen from Fig. 4a, the
dimensionless bulk modulus increases very slightly with an
increase of the relative density and with a reduction of the
cell wall thickness (i.e. an increase of lm/h0); however, this
change of the dimensionless bulk modulus is so small that
it could be simply neglected. On the other hand, cell reg-
ularity has a much larger effect on the dimensionless bulk
modulus, i.e. the larger the degree of the cell regularity, the
larger is the dimensionless bulk modulus as shown in
Fig. 4b. For perfectly regular micro-sized honeycombs, the
degree of regularity is a = 1.0 and the dimensionless bulk
modulus is 0.25, which is the same as that of their macro-
sized counterparts [7] because the strain gradient effect is
completely absent in this case.
Figure 5 shows the size-dependent relationship between
Poisson’s ratio and the relative density for micro-sized
first-order random irregular honeycombs with degree of
0.00 0.05 0.10 0.15 0.20 0.25 0.300
1
2
3
4
5
6N
on-d
imen
sion
al Y
oung
's M
odul
us
Relative Density
0.00 0.05 0.10 0.15 0.20 0.25 0.30
Relative Density
lm/h=0.0
lm/h=0.2
lm/h=0.5
lm/h=1.0
(a)
0
1
2
3
4
5
6
lm/h=0.0
lm/h=0.2
lm/h=0.5
lm/h=1.0
Non
-dim
ensi
onal
You
ng's
Mod
ulus
(b)
Fig. 2 Size-dependent effect on the relationship between the dimen-
sionless Young’s modulus and the relative density of micro-sized
first-order random irregular honeycombs with degrees of regularity.
a a = 0.0; b a = 0.7
0.0 0.2 0.4 0.6 0.8 1.01.0
1.5
2.0
2.5
3.0ρ0=0.01
ρ0=0.20
Non
-dim
ensi
onal
You
ng's
Mod
ulus
Degree of Regularity
Fig. 3 Effects of cell regularity on the dimensionless Young’s
modulus of micro-sized random irregular honeycombs with lm/
h = 0.5
0.00 0.05 0.10 0.15 0.20 0.25 0.300.239
0.240
0.241
0.242
0.243
lm/h=0.0
lm/h=0.2
lm/h=0.5
lm/h=1.0
Non
-dim
ensi
onal
Bul
k M
odul
us
Relative Density
(a)
0.0 0.2 0.4 0.6 0.8 1.00.18
0.20
0.22
0.24
0.26
ρ0=0.01
ρ0=0.20
Non
-dim
ensi
onal
Bul
k M
odul
us
Degree of Regularity
(b)
Fig. 4 Relationships of the dimensionless bulk modulus of micro-
sized first-order random irregular honeycombs against a relative
density when a = 0.7; b degree of regularity
J Mater Sci (2014) 49:5690–5702 5695
123
regularity a = 0.7. Generally, the thinner the cell walls or
the larger the honeycomb relative density, the smaller is
Poisson’s ratio; and the degree of cell regularity has very
little effect on Poisson’s ratio. When the relative density
tends to zero, Poisson’s ratio always approaches unity. As
it has already been demonstrated in Ref. [7] that random
irregular honeycombs are isotropic in-plane, the size-
dependent dimensionless in-plane shear modulus of the
first-order micro-sized honeycombs can thus be obtained
from the in-plane dimensionless Young’s modulus and
Poisson’s ratio, and so the results for the in-plane shear
modulus are not presented separately.
As there is no strain gradient effect in the out-of-plane
deformation, the out-of-plane dimensionless Young’s
modulus of first-order micro-sized random irregular hon-
eycombs is obviously unity, the out-of-plane Poisson’s
ratio v31 = vS, and the out-of-plane dimensionless shear
modulus are always approximately 0.5. Thus, all the out-
of-plane elastic properties of micro-sized first-order ran-
dom irregular honeycombs are exactly the same as those of
their macro-sized counterparts.
Size-dependent and tunable elastic properties of nano-
sized first-order random irregular honeycombs
When the cell wall thickness is at the nano-metre scale,
both the surface elasticity [12] and the initial stresses/
strains [10, 15, 16] can affect the elastic properties of first-
order random irregular honeycombs. When the effects of
the initial stresses/strains are absent, the size-dependent
relationships between the dimensionless Young’s modulus
and the relative density are given in Fig. 6a and b for first-
order nano-sized random irregular honeycombs with
degrees of regularity a = 0.0 and a = 0.7. As can been
seen, the thinner the cell walls, the larger is the dimen-
sionless Young’s modulus; and the larger the relative
density, the smaller the dimensionless Young’s modulus. If
the relative density of the first-order nano-sized irregular
honeycombs remains the same, Eq. 3 and Fig. 6a and b
suggest that the smaller the cell size, the larger would be
the dimensionless Young’s modulus. These results are
consistent with previous findings for nano-sized first-order
honeycombs with perfectly regular hexagonal cells (i.e.
a = 1.0) [10]. When the cell wall thickness h0 is much
larger than the material intrinsic length ln at the nano-metre
scale (i.e. ln/h0 = 0), the results reduce to those of the
macro-sized counterparts given in Ref. [7].
When the effects of the surface elasticity are absent (i.e.
ln/h0 = 0), the effects of the initial stresses/strains on the
relationships between the dimensionless Young’s modulus
and the relative density are shown in Fig. 7a and b for the
first-order nano-sized random irregular honeycombs with
0.00 0.05 0.10 0.15 0.20 0.25 0.300.4
0.5
0.6
0.7
0.8
0.9
1.0
1.1
lm/h=0.0
lm/h=0.2
lm/h=0.5
lm/h=1.0
Pois
son'
s R
atio
Relative Density
Fig. 5 Size-dependent effect on the relationship between Poisson’s
ratio and the relative density of micro-sized first-order random
irregular honeycombs with a = 0.7
0.00 0.05 0.10 0.15 0.20 0.25 0.300
2
4
6
8
10
ln/h=0.0
ln/h=0.2
ln/h=0.5
ln/h=1.0
Relative Density
Non
-dim
ensi
onal
You
ng's
Mod
ulus
(a)
0.00 0.05 0.10 0.15 0.20 0.25 0.300
2
4
6
8
10
ln/h=0.0
ln/h=0.2
ln/h=0.5
ln/h=1.0
Relative Density
Non
-dim
ensi
onal
You
ng's
Mod
ulus
(b)
Fig. 6 Size-dependent effect on the relationship between the dimen-
sionless Young’s modulus and the relative density of nano-sized
random irregular honeycombs when the effect of the cell wall initial
stress/strain is absent. a a = 0.0; b a = 0.7
5696 J Mater Sci (2014) 49:5690–5702
123
degrees of regularity a = 0.0 and a = 0.7. At the nano-
metre scale, the amplitude of the recoverable initial elastic
strain in the cell wall length direction can be controlled to
vary over a range from -0.1 to 0.1 by application of an
electric potential [10, 32, 33, 36–38]. Figure 7a and b
demonstrate that the dimensionless Young’s modulus of
the nano-sized first-order random irregular honeycombs
can be controlled either to increase about 60 % or to reduce
about 35 % when the initial strain in cell wall length
direction is controlled to vary from -6 to 6 %. In agree-
ment with the previous results [10, 36–38], whether the
effects of the surface elasticity are present (i.e. ln/h0 = 0)
or absent (i.e. ln/h0 = 0), the percentage of the tunable
range of the dimensionless Young’s modulus remains
almost unchanged if the applied initial strain in the cell
wall length direction is controlled to vary over the same
range. If the values of the initial strain applied in the cell
wall length direction are different from those given in
Fig. 7a and b, the dimensionless Young’s modulus of the
nano-sized first-order random irregular honeycombs can be
obtained by scaling up or scaling down the results.
When the effects of the initial stresses/strains are absent,
the relationships between the dimensionless Young’s
modulus and the cell regularity are shown in Fig. 8 for
nano-sized random irregular honeycombs with ln/h = 0.5
and different values of initial relative density, i.e.
q0 = 0.01 and q0 = 0.2. The standard deviation is about
10 % of the dimensionless Young’s modulus or smaller. As
for the case of micro-sized random irregular honeycombs,
the dimensionless Young’s modulus reduces with an
increase in the degree of the cell regularity when the
honeycomb relative density is small, e.g. q0 = 0.01, and
increases with an increase in the cell regularity when the
honeycomb relative density is large, e.g. q0 = 0.20. For
nano-sized random irregular honeycombs with ln/h = 0.5,
the trend change occurs when the relative density is about
q0 = 0.15. The larger the value of ln/h, the smaller is the
relative density q0 at which the trend change occurs.
Figure 9a shows the effects of the cell wall thickness (or
the surface elasticity) on the relationship between the
dimensionless bulk modulus and the relative density of the
nano-sized first-order random irregular honeycombs with
a = 0.7. As can be seen, it is the cell wall thickness rather
than the relative density which affects the dimensionless
bulk modulus, and cell wall stretching/compression is the
dominant deformation mechanism when a honeycomb is
under a hydrostatic stress. Figure 9b shows that when
amplitude of the initial strain in the cell wall direction is
controlled to vary from -6 to 6 %, the dimensionless bulk
modulus of nano-sized random irregular honeycombs with
a = 0.7 can be controlled to change from 15 % increment
to 14 % reduction, while the effect of the initial relative
0.00 0.05 0.10 0.15 0.20 0.25 0.30
0.4
0.8
1.2
1.6
2.0
2.4
Relative Density
Non
-dim
ensi
onal
You
ng's
Mod
ulus
ε L0=-0.06
ε L0=-0.03
ε L0=0
ε L0=0.03
ε L0=0.06
(a)
0.00 0.05 0.10 0.15 0.20 0.25 0.30
0.4
0.8
1.2
1.6
2.0
2.4
ε L0 = -0.06
ε L0 = -0.03
ε L0 = 0
ε L0 = 0.03
ε L0 = 0.06
Non
-dim
ensi
onal
You
ng's
Mod
ulus
Relative Density
(b)
Fig. 7 Effects of the cell wall initial stress/strain on the relationship
between the dimensionless Young’s modulus and the relative density
of nano-sized random irregular honeycombs when the surface
elasticity effect is absent. a a = 0.0; b a = 0.7
0.0 0.2 0.4 0.6 0.8 1.02
3
4
5
6ρ0=0.01
ρ0=0.20
Non
-dim
ensi
onal
You
ng's
Mod
ulus
Degree of Regularity
Fig. 8 Relative density effect on the relationship between the
dimensionless Young’s modulus and the cell regularity of nano-sized
random irregular honeycombs with ln/h = 0.5 when the cell wall
initial stress effect is absent
J Mater Sci (2014) 49:5690–5702 5697
123
density on the dimensionless bulk modulus is almost neg-
ligible. The dimensionless bulk moduli of nano-sized ran-
dom irregular honeycombs with a = 0.7, shown in Fig. 9a
and b, can well be approximated by
0:24ð1þ 2lnhÞð1� 2vS
1�vSeL
0 Þ=ð1þ eL0 Þ.
Figure 10 shows the relationship between the dimen-
sionless bulk modulus and the degree of regularity of nano-
sized random irregular honeycombs with ln/h = 0.5. The
small standard deviation value indicates that the bulk
moduli obtained from each of the 20 similar models is very
close. As can be seen, the greater the degree of the cell
regularity, the larger the dimensionless bulk modulus. This
is because the greater the cell regularity, the more domi-
nant a role the cell wall stretching/compression mechanism
plays when a honeycomb is under a hydrostatic stress. The
results of Fig. 10 agree well with those for micro-sized and
the macro-sized first-order random irregular honeycombs,
indicating that the honeycomb relative density does not
affect this relationship. Comparing the results in Fig. 10 to
those in Fig. 6 of paper [7], it is easily deduced that the
effect of the surface elasticity on the dimensionless bulk
modulus of nano-sized honeycombs can be well described
by a factor ð1þ 2lnhÞð1� 2vS
1�vSeL
0 Þ=ð1þ eL0 Þ because cell wall
stretching/compression is the dominant deformation
mechanism.
When the effects of the initial stress/strain are absent,
the effects of the cell wall thickness (i.e. ln/h or the surface
elasticity effects) on the relationship between the in-plane
Poisson’s ratio and the relative density of nano-sized
honeycombs with a = 0.7 are shown in Fig. 11a. As can
be seen, the in-plane Poisson’s ratio tends to unity as the
honeycomb relative density tends to zero, reduces
approximately linearly with an increase of the relative
density, and becomes approximately 0.78 (for the case ln/
h = 0) when q0 = 0.30. Generally, the thinner the cell
wall (i.e. the larger the ln/h value), the smaller is the in-
plane Poisson’s ratio. However, the effect of the surface
elasticity (or the cell wall thickness) on the in-plane Pois-
son’s ratio is very small. Figure 11b shows that the rela-
tionship between the in-plane Poisson’s ratio and the
relative density of nano-sized honeycombs (with a = 0.7)
can be controlled to vary by adjusting the amplitude of the
initial strain. The controllable range of the Poisson’s ratio
is approximately in proportion to the range of the tunable
initial strain applied in the cell wall direction and to the
honeycomb relative density.
For nano-sized first-order random irregular honey-
combs, the normalised out-of-plane Young’s modulus can
be easily obtained as
ðE3Þ1 ¼ðE3Þ1ESq0
¼ 1þ 2ln
h
� �
1� 2vS
1� vS
eL0
� �
= 1þ eL0
� �
ð23Þ
0.00 0.05 0.10 0.15 0.20 0.25 0.300.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4ln/h=0.0
ln/h=0.2
ln/h=0.5
ln/h=1.0
Non
-dim
ensi
onal
Bul
k M
odul
us
Relative Density
(a)
0.00 0.05 0.10 0.15 0.20 0.25 0.300.16
0.24
0.32
0.40
0.48ε L
0 =-0.06
ε L0 =-0.03
ε L0 =0
ε L0 =0.03
ε L0 =0.06
Non
-dim
ensi
onal
Bul
k M
odul
us
Relative Density
(b)
Fig. 9 Relationship between the dimensionless bulk modulus and
relative density of nano-sized random irregular honeycombs with
a = 0.7. a Surface elasticity effect; b initial stress/strain effect
0.0 0.2 0.4 0.6 0.8 1.00.35
0.40
0.45
0.50
0.55
ρ0=0.01
ρ0=0.20
Non
-dim
ensi
onal
Bul
k M
odul
us
Degree of Regularity
Fig. 10 Relative density effect on the relationship between the
dimensionless bulk modulus and the regularity of nano-sized random
irregular honeycombs with ln/h = 0.5 when the cell wall initial stress
effect is absent
5698 J Mater Sci (2014) 49:5690–5702
123
The dimensionless bulk modulus and the out-of-plane shear
modulus can well be approximated by
ðKÞ1 ¼ðKÞ1ESq0
¼ KðaÞ 1þ 2ln
h
� �
1� 2vS
1� vS
eL0
� �
= 1þ eL0
� �
ð24Þ
ðG31Þ1 ¼ðG31Þ1Gsq0
¼ 1
21þ 2ln
h
� �
1� 2vS
1� vS
eL0
� �
= 1þ eL0
� �
ð25Þ
and the out-of-plane Poisson’s ratio is (v31)1 = vS, i.e. the
same as that of the solid material. In the above equations,
eL0 ¼ � 2s0
hESð1� vSÞ and K(a) give the relationship between
the dimensionless bulk modulus of the macro-sized first-
order random irregular honeycombs and the cell regularity,
which is a function of the regularity a, obtained in Ref. [7]
and is almost the same as the relationship given in Fig. 4b.
The effect of the relative density q0 on K(a) is very small
and thus negligible.
Tunable geometrical properties of nano-sized first-
order random irregular honeycombs
For nano-sized first-order random irregular honeycombs,
when the effects of the initial stress/strain are absent, the
initial cell diameter, cross-sectional area and volume are
assumed to be D0, A0 and V0, respectively. When the
effects of the initial stress/stain are present, those geo-
metrical parameters become
ðDÞ1 ¼ðDÞ1D0
¼ 1þ eL0 ð26Þ
ðAÞ1 ¼ðAÞ1A0
¼ ð1þ eL0 Þ
2 ð27Þ
ðVÞ1 ¼ðVÞ1V0
¼ ð1þ eL0 Þ
3; ð28Þ
where the initial strain e0L in the cell wall length/width
direction can be controlled to vary over the range -0.1 to
0.1 by application of an electric potential. Therefore, the
geometrical properties of the nano-sized first-order random
irregular honeycombs are tunable and controllable.
Mechanical properties of micro- and nano-structured
hierarchical and self-similar random irregular
honeycombs
The micro- and nano-structured hierarchical random
irregular honeycombs are assumed to be geometrically
self-similar [34–37], as shown in Fig. 1a and b. At each of
the different hierarchy levels, the honeycomb is assumed to
be a material whose size is much larger than the individual
cells at that hierarchy level. The relative density of an nth
order hierarchical and self-similar random irregular hon-
eycomb is tunable and given as
qn ¼ 1� 2vS
1� vS
eL0
� �
ðq0Þn=ð1þ eL0 Þ; ð29Þ
where q0 is given in Eq. 3 for the first-order random
irregular honeycomb. When the initial strain e0L is 0, qn
reduces to (q0)n.
For nth order micro- or nano-structured hierarchical and
self-similar random irregular honeycombs, the five inde-
pendent dimensionless elastic constants can be obtained as
ðE1Þn ¼ðE1Þnð1� v2
SÞ1:5ESq0
� ½f ða; q0Þ�n�1ðE1Þ1 ð30Þ
ðv12Þn � vðq0Þ; n� 2 ð31Þ
ðE3Þn ¼ðE3ÞnESq0
¼ ðq0Þn�1 � ðE3Þ1 ð32Þ
0.00 0.05 0.10 0.15 0.20 0.25 0.300.6
0.7
0.8
0.9
1.0
1.1
ln/h=0.0
ln/h=0.2
ln/h=0.5
ln/h=1.0
Pois
son'
s R
atio
Relative Density
(a)
0.00 0.05 0.10 0.15 0.20 0.25 0.300.70
0.75
0.80
0.85
0.90
0.95
1.00
1.05
ε L0=-0.06
ε L0=-0.03
ε L0=0
ε L0=0.03
ε L0=0.06
Pois
son'
s R
atio
Relative Density
(b)
Fig. 11 Relationship between the in-plane Poisson’s ratio and the
relative density of nano-sized random irregular honeycombs with
a = 0.7. a Surface elasticity effect; b initial stress/strain effect
J Mater Sci (2014) 49:5690–5702 5699
123
ðG31Þn ¼ðG31ÞnGsq0
� ðq0
2Þn�1 � ðG31Þ1 ð33Þ
and (v31)n = vS, where n is the hierarchy level of the
micro- or nano-structured hierarchical and self-similar
honeycombs. In Eqs. 30, 32 and 33, ðE1Þ1; ðE3Þ1, and
ðG31Þ1 are the dimensionless in-plane and out-of-plane
Young’s moduli, and the dimensionless out-of-plane shear
modulus of the first-order micro- or nano-sized random
irregular honeycomb, respectively. Their results have been
obtained and given in ‘‘Size-dependent elastic properties of
micro-sized first-order random irregular honeycombs’’ and
‘‘Size-dependent and tunable elastic properties of nano-
sized first-order random irregular honeycombs’’ sections.
Hierarchical materials may have a number of structural
levels. For hierarchical and self-similar random irregular
honeycombs with n C 2 and fixed values q0 = 0.2 or
q0 = 0.3, the numerical results of the function
f(a, q0) = (E1)n/[(E1)n-1q03] in Eq. 30 are obtained by
computer simulation, and are plotted against the honey-
comb regularity a, as shown in Fig. 12. It is found that the
value of f(a, q0) depends mainly on q0 and is not sensitive
to the values of En-1 and (v12)n-1, and thus remains almost
unchanged for hierarchy levels n C 2 if q0 is a constant. As
can be seen from Fig. 12, when q0 is 0.2, the value of
f(a, q0) is consistently larger than unity, and when q0 is
0.3, the value of f(a, q0) is slightly smaller than but very
close to unity. The smaller the value q0 and the lower the
degree of the cell regularity a, the larger is the value of
f(a, q0). Therefore, it can be concluded that, for micro- or
nano-structured hierarchical and self-similar honeycombs
with the same value of q0 and the same first-order cell wall
thickness (i.e. ðE1Þ1 remains the same), the larger the
hierarchy level n, the larger is the dimensionless in-plane
Young’s modulus ðE1Þn if q0 B 0.28; or the smaller is the
dimensionless in-plane Young’s modulus ðE1Þn if q0 [ 0.3.
In other words, if the overall relative density qn = q0n
remains unchanged for micro- or nano-structured hierar-
chical and self-similar honeycombs and first-order cell wall
thickness remains the same (i.e. ðE1Þ1 remains the same),
the dimensionless in-plane Young’s modulus ðE1Þn of a
hierarchical random irregular honeycomb given Eq. 30 can
be designed to be larger or smaller than that of its single-
order counterpart, depending on the values q0 and a.
For micro- or nano-structured hierarchical and self-
similar regular hexagonal honeycombs (a = 1.0), when
q0 = 0.2 and q0 = 0.3, the results given in Eqs. 30–33 are
consistent with those of Ref. [37]. For hierarchical and self-
similar honeycombs with n C 2, the value of the in-plane
Poisson’s ratio (v12)n is approximately the same as that
given by the solid curve in Fig. 11a for ln/h = 0, which
depends mainly on the value of q0 and is almost inde-
pendent of the cell regularity a and the material Young’s
modulus. Therefore, the Poisson’s ratio of micro- or nano-
structured hierarchical honeycombs with n C 2 is in gen-
eral almost not tunable. Equation 32 indicates that the
dimensionless out-of-plane Young’s modulus ðE3Þn of
micro- or nano-structured hierarchical and self-similar
honeycombs depends only on the value of the overall rel-
ative density qn and the thickness of the first-order cell
walls. In contrast, Eq. 33 shows clearly that for micro- or
nano-structured hierarchical and self-similar honeycombs
with the same overall relative density qn, the larger the
number of structural hierarchy level n, the smaller is the
dimensionless out-of-plane shear modulus ðG31Þn.
Tunable geometrical properties of nano-structured
hierarchical and self-similar random irregular
honeycombs
For nth order nano-structured hierarchical and self-similar
random irregular honeycombs, when the effects of the
initial stress/strain are absent, the initial cell diameter,
cross-sectional area and volume are assumed to be (D0)n,
(A0)n and (V0)n, respectively. When the effects of the initial
stress/stain are present, those geometrical parameters vary
and are normalised to
ðDÞn ¼ðDÞnðD0Þn
¼ 1þ eL0 ð34Þ
ðAÞn ¼ðAÞnðA0Þn
¼ ð1þ eL0 Þ
2 ð35Þ
0.0 0.2 0.4 0.6 0.8 1.00.8
1.0
1.2
1.4
1.6ρ0=0.20
ρ0=0.30
f (α
, ρ0)
Degree of Regularity
Fig. 12 Relationship between the value of f(a, q0) = (E1)n/
[(E1)n-1q03] and cell regularity of micro- or nano-structured hierar-
chical and self-similar random irregular honeycombs with n C 2 and
different relative densities q0 = 0.2 and q0 = 0.3
5700 J Mater Sci (2014) 49:5690–5702
123
ðVÞn ¼ðVÞnðV0Þn
¼ ð1þ eL0 Þ
3; ð36Þ
where e0L is the initial strain of the first-order cell walls in
their length/width direction, which can be controlled to
vary over -0.1 to 0.1 by application of an electric poten-
tial. Therefore, the geometrical properties of the nano-sized
nth order hierarchical and self-similar random irregular
honeycombs are tunable and controllable. These are the
same as those of perfect regular honeycombs and inde-
pendent of the degree of the cell regularity.
Conclusion
All the five independent elastic constants: E1, v12, E3, G31 and
v31 = vS are obtained for the micro- and nano-structured first-
order and the hierarchical and self-similar random irregular
honeycombs. It is found that if the thickness of the first-order
cell walls is at the micro-metre scale, the mechanical prop-
erties of random irregular hierarchical honeycombs are size-
dependent due to the strain gradient effects; and that if the
thickness of the first-order cell walls is at the nano-metre scale,
the mechanical properties of the random irregular hierarchical
honeycombs are not only size-dependent due to the surface
elasticity effects but are also tunable and controllable due to
the effects of the initial stresses/strains. In addition, for nano-
structured hierarchical random irregular honeycombs, the cell
diameter, cross-section area and volume can all be controlled
to vary over large ranges by application of an applied electric
potential. The results obtained in this paper could serve as a
guide in the design of many different advanced functional
materials with tunable and controllable properties in strength/
stiffness, colour, wettability, selectivity, natural frequency, or
electric capacitance.
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