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Composite Structures 109 (2014) 68–74
Contents lists available at ScienceDirect
Composite Structures
journal homepage: www.elsevier .com/locate /compstruct
Mechanical properties of a hollow-cylindrical-joint
honeycomb
0263-8223/$ - see front matter � 2013 Elsevier Ltd. All rights
reserved.http://dx.doi.org/10.1016/j.compstruct.2013.10.025
⇑ Corresponding author. Tel./fax: +86 2583792620.E-mail address:
[email protected] (Z. Li).
Qiang Chen a, Nicola Pugno b,c,d, Kai Zhao e, Zhiyong Li a,⇑a
Biomechanics Laboratory, School of Biological Science and Medical
Engineering, Southeast University, 210096 Nanjing, PR Chinab
Laboratory of Bio-Inspired & Graphene Nanomechanics, Department
of Civil, Environmental and Mechanical Engineering, University of
Trento, I-38123 Trento, Italyc Center for Materials and
Microsystems, Fondazione Bruno Kessler, I-38123 Trento, Italyd
School of Engineering and Materials Science, Queen Mary Univerisity
of London, Mile End Road, London, E1 4NS, UKe Institute of
Geotechnical Engineering, Nanjing University of Technology, 210009
Nanjing, PR China
a r t i c l e i n f o
Article history:Available online 25 October 2013
Keywords:Hollow-cylindrical-joint honeycombYoung’s
modulusPoisson’s ratioStress intensity factorStrength
a b s t r a c t
In this paper, we constructed a new honeycomb by replacing the
three-edge joint of the conventional reg-ular hexagonal honeycomb
with a hollow-cylindrical joint, and developed a corresponding
theory tostudy its mechanical properties, i.e., Young’s modulus,
Poisson’s ratio, fracture strength and stress inten-sity factor.
Interestingly, with respect to the conventional regular hexagonal
honeycomb, its Young’smodulus and fracture strength are improved by
76% and 303%, respectively; whereas, for its stress inten-sity
factor, two possibilities exist for the maximal improvements which
are dependent of its relative den-sity, and the two improvements
are 366% for low-density case and 195% for high-density
case,respectively. Moreover, a minimal Poisson’s ratio exists. The
present structure and theory could be usedto design new honeycomb
materials.
� 2013 Elsevier Ltd. All rights reserved.
1. Introduction
Honeycomb or foam structures extensively exist in
naturalmaterials, e.g. honeycomb [1], cancellous bone in animal
skeletonsand tree or grass stems [2]. From the mechanical point of
view,their mechanical properties (such as light-weight,
high-strengthand super-tough) are linked to their optimal
structures by Nature,which hints people to design different
multifunctional materials[3–6]. To this end, a considerable number
of scientists and engi-neers invented varieties of porous materials
and investigated theirmechanical properties. For example,
mechanical properties of hier-archical nanohoneycomb or nanofoam
materials, for which surfaceeffect was included, were studied
[7,8]. It is found that the elasticmodulus and strength decrease as
hierarchical level number in-creases. Also the sandwich walls with
core struts in lattice struc-tures have superior mechanical
properties to that with solidwalls [9]. Similarly, substituted
solid cell walls of the conventionalhexagonal honeycomb with
equal-mass honeycomb lattice, theYoung’s modulus of new structures
is optimized and improvedby 75% comparing to the conventional
hexagonal honeycomb[10]; also, by replacing the three-edge joint of
the regular hexago-nal honeycomb with a hollow hexagonal prism, the
Young’smodulus of the fractal-like structure is optimized [11].
Besides,replaced the solid cell walls of the conventional
hexagonalhoneycomb with a equal-mass re-entrant negative Poisson’s
ratio
honeycomb [12], the Young’s modulus of the new structure isagain
dramatically improved.
Regarding the fracture behavior of honeycomb-like structures,not
like the theory for continuum media, the common method isemploying
the stress field ahead of crack tip, then, performingstructural
analysis. There exist such models in literatures [13–15]. Maiti et
al. [13] and Choi and Lakes [14] used the conventionalsingular
stress field and the nonsingular stress field for bluntcracks to
calculate the axial force acting on the first vertical cellahead of
crack tip, respectively, and both the two models derivedfracture
strength first and then stress intensity factor. However,Choi and
Sankar [15] calculated the axial force and bending mo-ment acting
on the first vertical cell ahead of crack tip, and selectedan
effective portion of crack tip stress field considering the
exis-tence of singularity, then, the stress intensity factor was
directlyobtained.
In this paper, by observing natural honeycombs, we found thatthe
cell walls of the natural honeycombs have varying cross-sections,
and the thickness reaches a maximum at both ends of cellwalls
(marked by circle in Fig. 1a). It is speculated that the mo-ments
at these ends bear the greatest bending moment, and hon-eybees
strengthen their nests by introducing more materials tothe weakest
points, exactly like the specifications of building de-sign, in
which more steel bars and concrete are placed at load-bearing
positions in beams. Meanwhile, considering the superiormechanical
efficiency of natural porous structures to solid materi-als
[8,10,11], we proposed a so-called hollow-cylindrical-joint
hon-eycomb (Fig. 1c) by replacing the three-edge joint with a
hollow
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Fig. 1. (a) Natural honeycomb; (b) conventional regular
hexagonal honeycomb; (c) hollow-cylindrical-joint honeycomb.
Q. Chen et al. / Composite Structures 109 (2014) 68–74 69
cylinder instead of the hollow hexagonal prism reported in
Ref.[11]. The related structure can be regarded as the derivative
struc-ture from the family of center-symmetrical honeycombs in
Ref.[16]. In particular, the out-of-plane properties of the
tetrachiraland hexachiral honeycomb family were well studied by
experi-ments and finite element method, and the results showed
thatbuckling strength of the honeycombs could be optimized for
appli-cations in some fields [16]. For the present structure, its
Young’smodulus and Poisson’s ratio was derived basing on
Castigliano’stheorem, and its stress intensity factor and strength
were calcu-lated by invoking the quantized fracture mechanics (QFM)
thanksto the discrete feature of the honeycomb. In the following
sections,the mechanical properties of the honeycomb with respect to
thoseof the conventional regular hexagonal honeycomb are studied
anddiscussed in detail.
2. Structural theory
Regarding the relative density of the conventional regular
hex-agonal honeycomb (Fig. 1b), it is approximately expressed
as
�qð1Þ ¼ qð1Þqs ¼ 2ffiffi3p tð1Þl� �
, where, qs is the density of constituent materi-
als, t(1) is the thickness of cell walls, the superscript (1)
denotes theconventional regular hexagonal honeycomb, and its
mechanicalproperties dependent of t(1)/l are systematically derived
by Gibsonand Ashby [17]. Different from the above expression of
relativedensity, here, it is precisely expressed by including a
quadraticterm, i.e.,
�qð1Þ ¼ 13� t
ð1Þ
l
� �þ 2
ffiffiffi3p� � tð1Þ
l
� �:
As for the relative density of the hollow-cylindrical-joint
struc-ture denoted by the superscript (2), we calculate it by a
geometri-cal analysis as,
�qð2Þ ¼ qð2Þ
qs¼ 2
3ffiffiffi3p �3 t
ð2Þ
l
� �þ 2ð2p� 3Þ r
l
� �þ 3
� �tð2Þ
l
� �ð1Þ
where t(2) is the thickness of structure’s cell walls, r is the
radius ofthe circular joint (Fig. 1c). It is noted that the
structure’s geometryrequires 0 < t(2)/r 6 2 and 0 < r/l <
0.5.
2.1. Young’s modulus
A representative unit (Fig. 2) is selected and considered as
thesum of three components, i.e., semicircle AD, beams BC and DE.
Un-der the uniaxially external tensile stress, no rotation and
horizontaldisplacement occurs at the end A, thus, the boundary
condition ofthe end A is simplified to be guided, see Fig. 2. The
force P0 actingon the end C of the beam BC is equivalent to P0
¼
ffiffiffi3p
rð2Þbl=2,where r(2) is the external stress, b is the
out-of-plane depth ofthe honeycomb, and l is the length of cell
walls.
For the representative unit ABCDE, according to its
geometricalcomponents, the total strain energy stored in it is
correspondinglycomposed by three components,
U ¼ UAD þ UBC þ UDE ð2Þ
where UAD, UBC and UDE are elastic strain energies stored in
thesemicircle AD, beams BC and DE, respectively. Furthermore,the
semicircle AD is divided into two subsections: one is AB andthe
other BD. Performing force analysis (Fig. 2), we obtain innerforces
in the subsection AB: NI = R cos h, VI = R sin h, MI = �MA +Rr(1 �
cos h) when 0 < h 6 p/3 and in section BD: NII = �R cosh + P0
sinh, VII = �Rsin h � P0 cos h, MII ¼ MA þM0þP0r
ffiffi3p
2 � sin h� �
� Rrð1� cos hÞ, when p/3 6 h 6 p. Thus, the elasticstrain energy
in the semicircle AD is calculated as:
UAD¼r
2EsAð2Þ
Z p3
0N2I dhþ
Z pp3
N2IIdh
" #þk E
G
Z p3
0V2I dhþ
Z pp3
V2IIdh
" #(
þAI
Z p3
0M2I dhþ
Z pp3
M2IIdh
" #)
¼ r2EsAð2Þ
tð2Þ
l
� ��2
�
12 p MAl 2þ 4p3 Ml �3P0 rl MAl �2p rl MAl Rh i
þR 45P0 rl 2�4ð4pþ3 ffiffiffi3p Þ Ml rl þ 34ð1�2kð1þmsÞÞP0
tð2Þl� �2
� �
þp2 R2 36 rl
2þð1þ2kð1þmsÞÞ tð2Þl� �2� �
þ 8p Ml 2�36P0 Ml rl þP20 4pþ 3 ffiffi3p2� � rl 2h
þP20 p3þffiffi3p
8 þ2kð1þmsÞ p3�ffiffi3p
8
� �� �tð2Þ
l
� �2� �
8>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>:
9>>>>>>>>>>>>>>>>=>>>>>>>>>>>>>>>>;ð3Þ
where M ¼ M0 þffiffi3p
2 P0r is a defined moment.According to Castigliano’s first
theorem and boundary condi-
tions of the end A, the conditions oUAD/oR = 0 and oUAD/oMA =
0hold, then, an equation system with respect to two
dimensionlessreaction forces k1 ¼ Rl=M and k2 ¼ MA=M emerges:
k2 þ C1k1 þ C2 ¼ 0k2 þ C3k1 þ C4 ¼ 0
�ð4Þ
where,
C1¼� 124 rl �1 36 rl 2þð1þ2kð1þmsÞÞ tð2Þl� �2
� �
C2¼� 124p rl �1 45 P0 lM� � rl 2�4ð4pþ3 ffiffiffi3p Þ rl
þ34ð1�2kð1þmsÞÞ P0 lM� � tð2Þl� �2
� �C3¼� rlC4¼ 23� 32p
P0 lM
� �rl
h i
8>>>>>>>>><>>>>>>>>>:
-
Fig. 2. Force analysis in a representative unit. Note that the
red curves denote the after-deformed structures. (For
interpretation of colour in this figure legend, the reader
isreferred to the web version of this article.)
70 Q. Chen et al. / Composite Structures 109 (2014) 68–74
solving the system, we obtain k1 ¼ �ðC2 � C4Þ=ðC1 � C3Þ andk2 ¼
ðC2C3 � C1C4Þ=ðC1 � C3Þ. Then, the strain energy by Eq. (3)can be
calculated with the known forces P0 and M0. For the present
structure in Fig. 2, the two forces P0 and M0 satisfy M0
¼ffiffi3p
2 P0l2� r
,
i.e., P0 lM ¼ 4ffiffi3p . Then, the strain energy in ABCDE is
expressed as:UAD ¼
P20r
2EsAð2Þtð2Þ
l
� ��2
�
3 3p4 k22� 3p2 rl
k1k2þ p�3
ffiffiffi3p
rl
� �k2
h i
þffiffi3p
4 k1 45rl
2�ð9þ4 ffiffiffi3p pÞ rl þ 34ð1�2kð1þmsÞÞ tð2Þl� �2� �
þ3p32 k21 36
rl
2þð1þ2kð1þmsÞÞ tð2Þl� �2� �
þ 3p2 �9ffiffiffi3p
rl
þ 4pþ 3
ffiffi3p
2
� �rl
2þ p3þ ffiffi3p8� ��h
þ2kð1þmsÞ p3�ffiffi3p
8
� ��tð2Þ
l
� �2�
8>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>:
9>>>>>>>>>>>>>>>>=>>>>>>>>>>>>>>>>;ð5Þ
For the oblique cantilever beam BC, the inner forces are ex-
pressed as NIII = P0/2, V III ¼ffiffiffi3p
P0=2;MIII ¼ffiffi3p
P02
l2� r
� x�
. It isnoted that the moments at the joint B are balanced,
i.e.,MIIIð0Þ ¼ MI p3
þMII p3
, which proves the correct force analysis
to some extent. Correspondingly, its elastic strain energy
isexpressed as:
UBC ¼1
2EsAð2Þ
Z ð l2�rÞ0
N2IIIdxþ kEG
Z ð l2�rÞ0
V2IIIdxþAI
Z ð l2�rÞ0
M2IIIdx
" #
¼ P20r
2EsAð2Þtð2Þ
l
� ��2l
2r� 1
� �
� 3 12� r
l
� �2þ 1
4ð1þ 6kð1þ msÞÞ
tð2Þ
l
� �2" #ð6Þ
For the vertical cantilever beam DE, only axial deforma-tion
occurs, and thus, its elastic strain energy is easily
expressedas:
UDE ¼P20l
2EsAð2Þð7Þ
Substituting Eqs. (5)–(7) into Eq. (2), the total elastic
strainenergy U is derived. Again, employing Castigliano’s first
theorem,the displacement of the beam end C in the force direction
is de-rived as,
DV ¼@U@P0¼ P0l
EsAð2Þl
tð2Þ
� �2f
rl;tð2Þ
l
� �ð8Þ
where,
frl;tð2Þ
l
� �
¼ rl
� ��
3 3p4 k22� 3p2 rl
k1k2þ p�3
ffiffiffi3p
rl
� �k2
h iþffiffi3p
4 k1 45rl
2�ð9þ4 ffiffiffi3p pÞ rl þ 34ð1�2kð1þmsÞÞ tð2Þl� �2� �
þ3p32k21 36
rl
2þð1þ2kð1þmsÞÞ tð2Þl� �2� �
þ 3p2 �9ffiffiffi3p
rl
þ 4pþ 3
ffiffi3p
2
� �rl
2þ p3þ ffiffi3p8� ��hþ2kð1þmsÞ p3�
ffiffi3p
8
� ��tð2Þ
l
� �2�
8>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>:
9>>>>>>>>>>>>>>>=>>>>>>>>>>>>>>>;
þ 12�r
l
� �3
12�r
l
� �2þ1
4ð1þ6kð1þmsÞÞ
tð2Þ
l
� �2" #þ t
ð2Þ
l
� �2
and thus, the strain e in the representative unit is
calculated:
eV ¼DV
3l=4¼ r
Es2ffiffiffi3p
3l
tð2Þ
� �3f
rl;tð2Þ
l
� �ð9Þ
Finally, the Young’s modulus is obtained:
Eð2Þ
Es¼
ffiffiffi3p
2tð2Þ
l
� �3f�1
rl;tð2Þ
l
� �ð10Þ
if r/l tends to zero, and the quadratic term (t(2)/l)2 in f(r/l,
t(2)/l) is ne-glected due to its smallness (i.e., the shear and
axial deformationsare neglected), then, f(r/l, t(2)/l) approach
3/8, and more, Eq. (10) willbe rewritten as E(2)/Es = 2.3(t(2)/l)3,
which is the result of the conven-tional regular hexagonal
honeycomb reported in Ref. [17].
2.2. Poisson’s ratio
In Eq. (3), if we exclude M0 (i.e., let M0 disappear),P0 lM ¼
2ffiffi3p rl �1
holds, and the strain energy is now defined as UAD, which
isexpressed as:
-
Q. Chen et al. / Composite Structures 109 (2014) 68–74 71
UAD¼P20r
2EsAð2Þtð2Þ
l
� ��2
�
12 rl 2 3p
4 k22þ p� 3
ffiffi3p
2
� �k2� 3p2 rl
k1k2
� �
þffiffi3p
2rl
k1 ð27�8
ffiffiffi3pÞ rl 2þ 34ð1�2kð1þmsÞÞ tð2Þl� �2
� �
þ3p8 rl 2
k21 36rl
2þð1þ2kð1þmsÞÞ tð2Þl� �2� �
þ 10p� 33ffiffi3p
2
� �rl
2þ p3þ ffiffi3p8� �þ2kð1þmsÞ p3� ffiffi3p8� �� � tð2Þl� �2
26666666666664
37777777777775
ð11Þ
then, UAD;UAD, and M0 should satisfy:
u ¼ UAD � UADM0
ð12Þ
where u is the angular displacement caused by M0, see Fig.
2.According to the structural analysis in Fig. 2, the
displacementscaused by the shear and axial forces in part BC
are:
dV ¼2ffiffiffi3p P0r
EsAð2Þtð2Þ
l
� ��2l
2r� 1
� �3
12� r
l
� �2þ 3
2kð1þ msÞ
tð2Þ
l
� �2" #
dN ¼12
P0r
EsAð2Þl
2r� 1
� �ð13Þ
Therefore, the horizontal displacement of the point C is
calcu-lated as:
DH ¼ �ul2� r
� �sin
p6� dV sin
p6þ dN cos
p6
ð14Þ
Finally, the Poisson’s ratio is obtained as:
m ¼ � eHeV¼ �DH=ð
ffiffiffi3p
l=4ÞDV=ð3l=4Þ
¼ �ffiffiffi3p
DHDV
ð15Þ
Fig. 3. (a) Preexisting crack in the honeycomb; (b) fracture in
vertical cell walls; (c) fract
2.3. Fracture strength and stress intensity factor
In this section, we consider two fracture mechanisms in
animperfect honeycomb (Fig. 3) as stated in the literature [15].
Oneis that the crack propagates due to the tensile-bending failure
ofthe vertical cell wall (Fig. 3b), and the other is due to the
bendingfailures of the curved cell wall (B0D0 in Fig. 3c) of the
circular jointand the inclined cell walls (B0C0 in Fig. 3d). For
the two mecha-nisms, their fracture strengths and stress intensity
factors are de-rived, respectively, and the competition between
them, like thatin Ref. [18], is discussed.
2.3.1. Mechanism (1): tensile-bending failure of the vertical
cell wallAccording to Choi and Sankar [15], the axial force PE
and
moment ME acting on the first cell wall ahead of crack tip
areexpressed as,
PE ¼Z a
0
KIffiffiffiffiffiffiffiffiffi2prp bdr ¼ KIb
ffiffiffiffiffiffi2ap
r
ME ¼Z a
0
KIffiffiffiffiffiffiffiffiffi2prp brdr ¼ KIb
3
ffiffiffiffiffiffiffiffi2a3
p
r ð16Þ
where a = fl is an effective length, f ¼ 0:411ð�qð2ÞÞ0:308 [15]
is adimensionless factor, KI is the mode-I stress intensity factor.
If thefailure of the vertical cell wall occurs, the maximum bending
stressrmax in the cell wall produced by the forces should equal the
tensilestrength rscr of the cell wall materials, i.e.,
rscr ¼PE
btð2Þþ 6ME
bðtð2ÞÞ2¼ Kð2ÞIC;1
ltð2Þ
� �2 ffiffiffiffiffi2fpl
rtð2Þ
lþ 2f
� �ð17Þ
where Kð2ÞIC;1 is the critical stress intensity factor, in which
the secondsubscript 1 denotes the first failure mechanism. Then,
rearrangingEq. (17), the stress intensity factor is obtained
as,
Kð2ÞIC;1rscr
ffiffiffiffiffiplp ¼ 1ffiffiffiffiffi
2fp
tð2Þl þ 2f
� � tð2Þl
� �2ð18Þ
ure at the point D0 and force analysis; (d) fracture at the
point B0 and force analysis.
-
72 Q. Chen et al. / Composite Structures 109 (2014) 68–74
The quantized fracture mechanics (QFM) presented by
Pugno[19,20], has already been applied to carbon nanotubes and
graph-ene. Thanks to the geometric similarity between the present
struc-ture and graphene, QFM is employed here to derive
thehoneycomb’s fracture strength. If a preexisting crack with
length2c exists in the honeycomb (Fig. 3a), the effective length a
is de-fined as the fracture quanta, then, the QFM strength of the
honey-comb is expressed as rð2Þcr;1 ¼ K
ð2ÞIC;1=
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffipðc
þ a=2Þ
p. Furthermore, we
assume that the crack length satisfies 2c ¼ nffiffiffi3p
l, where n is thenumber of cracked cells, then, the strength of
the honeycomb isexpressed,
rð2Þcr;1rscr¼ 1ffiffiffi
fp
tð2Þl þ 2f
� �
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3p
nþ fq tð2Þ
l
� �2ð19Þ
2.3.2. Mechanism (2): bending failure of cell wallsBefore
discussing the bending failure mechanism, let us study
the strength of a perfect honeycomb first. Its strength is
reachedwhen a cell wall fails due to the maximum bending moment.
Wecan see that the moment at the end B (or B
0due to symmetry) of
the beam BC (or B0C0) gradually disappears if the ratio r/l
ap-proaches 0.5, and the position of the maximum moment
switchesfrom the end B (or B
0) of the oblique cantilever beam BC (or B0C0) to
the end D (or D0) of the semicircle AD. According to the force
anal-ysis with above-obtained reaction forces R and MA, and
denotingthe maximum moments at the two positions as MB (or MB0 )
andMD (or MD0 ), respectively, the dimensionless moments
areexpressed as:
MBP0 l¼
ffiffi3p
212� rl
MDP0 l¼
ffiffi3p
4 k2 � 2k1 rl
þ 1
� 8<: ð20Þ
Second, for an imperfect honeycomb, the fracture location
lo-cates at the points B0 or D0. For these two cases, the force
analysisare shown in Fig. 3c and d, respectively. It is noted that
0.5 PE inFig. 3c and d corresponds to P0 in Fig. 2, and thus,
substituted P0in Eq. (20) with 0.5PE, the bending moment at the
points B0 or D0
in the cracked honeycomb can be expressed as,
MB0PEl¼ MBþ0:5MEPEl ¼
ffiffi3p
412� rl
þ f6MD0PEl¼ MDþ0:5MEPEl ¼
ffiffi3p
8 k2 � 2k1 rl
þ 1
� þ f6
8<: ð21Þwhere MB and MD, which can be calculated by Eq. (20),
are bendingmoments at the points B0 or D0 caused only by the axial
force PEthanks to the structural symmetry. For brittle materials,
the maxi-mum moment Mmax ¼ 16 rscrbðt
ð2ÞÞ2, thus, the failure occurs whenMmax ¼maxðMB0 ;MD0 Þ, then,
we find,
Kð2ÞIC;2rscr
ffiffiffiffiffiplp ¼
1ffiffiffiffi2fp
3ffiffi3p
212�
rlð Þþf
� tð2Þl
� �2MD0 < MB0
1ffiffiffiffi2fp
3ffiffi3p
4 k2�2k1rlð Þþ1ð Þþf
� tð2Þl
� �2MD0 > MB0
8>><>>: ð22Þ
similarly, the strength is derived by QFM,
rð2Þcr;2rscr¼
1ffiffifp
3ffiffi3p
212�
rlð Þþf
� ffiffiffiffiffiffiffiffiffiffiffiffi3p
nþfp tð2Þ
l
� �2MD0 < MB0
1ffiffifp
3ffiffi3p
4 k2�2k1rlð Þþ1ð Þþf
� ffiffiffiffiffiffiffiffiffiffiffiffi3p
nþfp tð2Þ
l
� �2MD0 > MB0
8>><>>: ð23Þ
considering the limiting case that r/l tends to zero and f = 1,
then,
MD < MB holds and Eq. (22) becomes Kð2ÞIC;2 ¼ 0:307rscr
ffiffiffiffiffiplp
tð2Þl
� �2,
which is identical to the result of the stress intensity factor
of reg-ular hexagonal honeycombs in Ref. [17]; and more, if n = 0,
Eq. (23)
becomesrð2Þ
cr;2rscr¼ 0:435 tð2Þl
� �2� 49 t
ð2Þ
l
� �2, which is also the result of the
compressive strength of perfect regular hexagonal honeycombs
inRef. [17].
Overall, as the ratio r/l varies, the fracture location changes.
Forthe whole structure, its strength rð2Þcr can be determined
byrð2Þcr ¼minðrð2Þcr;1;r
ð2Þcr;2Þ according to Eqs. (19) and (23). Correspond-
ingly, its stress intensity factor Kð2ÞIC is also obtained by
minimizingEqs. (18) and (22), i.e., Kð2ÞIC ¼minðK
ð2ÞIC;1;K
ð2ÞIC;2Þ.
3. Results and discussion
Here, we study the influences of the relative density and
param-eter r/l on the Young’s modulus, Poisson’s ratio, fracture
strengthand stress intensity factor of the honeycomb normalized by
itscounterparts of the conventional regular hexagonal honeycomb.The
Poisson’s ratio ms of the constituent material is assumed tobe 0.3.
The range for r/l is from 0.1 to 0.45 and �qð2Þ is set in therange
from 0.01 to 0.21, then, according to Eq. (1) and r/l, we
findt(2)/r varying from 0.001 to 1.741, which satisfies the
conditions0 < t(2)/r < 2.
For the normalized Young’s modulus, we compare the result ofthe
present structure by the theory with those of the similar
struc-ture studied by experiments and finite element results [11],
andparametrically investigate the influences of the relative
densityand r/l. The structure in Ref. [11] is controlled by the
ratio a/l, inwhich a is the side length of the hexagon (see Fig.
4a). The resultsare reported in Fig. 4. We can see that the present
theory (the linein Fig. 4a) agrees well with the experimental (the
circle in Fig. 4a)and finite element results (the square in Fig.
4a) even though theyhave different geometries. The former has an
optimal value whenr/l � 0.31, which is less than the latter’s
optimal value whena/l � 0.33, this is because r is less than a if a
hexagon is equivalentto a circle. The parametric study shows that
the optimal value, withrespect to the conventional honeycomb,
decreases (Fig. 4b) as therelative density increases, and the
normalized Young’s modulustends to one, as r/l approaches 0.1 which
means that the Young’smodulus of the honeycomb tends to that of the
conventional hon-eycomb. Corresponding to the conventional
honeycomb, theimprovement of the Young’s modulus is up to 76%
when�qð2Þ ¼ 0:01 and r/l = 0.31.
For the structure’s Poisson’s ratio, the present prediction is
alsocompared with the finite element results reported in Ref. [11],
anda good agreement is again obtained, see Fig. 5a. If r/l ? 0.1,
thePoisson’s ratio tend to the well-known result (i.e., one), and a
low-est value m = 0.315 is reached when r/l = 0.38, which is less
than thereported value 0.37 when a/l = 0.4 under the common
relative den-sity �qð2Þ ¼ 0:06. The reason can be referred to the
correspondingdiscussion of Young’s modulus. The parametric study on
the Pois-son’s ratio is performed with respect to the relative
density and r/l,and the result is plotted in Fig. 5b, which shows
that the Poisson’sratio is between 0.313 and 0.996.
For the fracture strength, the mechanism (1) is absent and
themechanism (2) prevails, thus, the fracture location switches
fromthe point B0 to the point D0 (Fig. 6a). Interestingly, if the
bendingfailure occurs at the point B0, it reaches a minimum whenr/l
� 0.25. This is because a smaller r results in a greater
cell-wallthickness t(2), which requires a greater bending moment to
fail; agreater r results in a smaller moment arm of the beam B0C0,
whichalso needs a greater force to fail, and r � 0.25l is
in-between.Whereas, if the failure is at the point D0, an
increasing r results ina decreasing t(2), thus, the failure bending
moment becomes smal-ler and smaller. Moreover, as the number of
cracked cells (or cracklength) increases, the normalized fracture
strength increases(Fig. 6b). This is because the fracture quanta
plays a moreimportant role in the case of a shorter crack which
reduces the
-
Fig. 4. (a) Comparison between the present theory of the present
structure, experiments and finite element results from the
literature [11], when �qð2Þ ¼ �qð1Þ ¼ 0:1; (b)normalized Young’s
modulus vs. relative density �qð2Þ and r/l.
Fig. 5. (a) Comparison between the present theory of the present
structure and finite element results from the literature [11], when
�qð2Þ ¼ �qð1Þ ¼ 0:06; (b) Poisson’s ratio vs.relative density �qð2Þ
and r/l.
Fig. 6. (a) Normalized fracture strength vs. r/l when n = 7; (b)
normalized fracture strength vs. number of cracked unit cells n and
r/l when �qð2Þ ¼ �qð1Þ ¼ 0:1.
Q. Chen et al. / Composite Structures 109 (2014) 68–74 73
improvement of the fracture strength, and its influence weakens
asthe crack length increases. In particular, the maximal
improvementof the fracture strength with respect to the
conventional honey-comb is up to 300% when r/l = 0.1 and n = 25,
and for each n, themaximal improvement is always reached at r/l =
0.1. It is worthmentioning that the critical failure at both points
B0 and D0 occurssimultaneously when r/l = 0.33, and its strength is
improved by264% when r/l = 0.33 and n = 25.
Finally, the result of the stress intensity factor is reported
inFig. 7. Different from the optimal value of the Young’s
modulus
when r/l = 0.31 (Fig. 4b) and the maximal value of the
fracturestrength when r/l = 0.1 (Fig. 6b), the maximal value of the
normal-ized stress intensity factor varies from r/l = 0.1 to r/l =
0.33 as therelative density decreases. Addressing this point, we
study its crit-ical conditions depicted in Fig. 7a. It shows that
the maximal valueappears at both r/l = 0.1 and r/l = 0.33 when
�qð2Þ ¼ 0:03; while it isat r/l = 0.33 when �qð2Þ ¼ 0:02 and at r/l
= 0.1 when �qð2Þ ¼ 0:04.Therefore, combining Fig. 7b, it can be
concluded that the maximalvalue of the normalized stress intensity
factor is at r/l = 0.33 if�qð2Þ < 0:03 while at r/l = 0.1 if
�qð2Þ > 0:03. Compared to the
-
Fig. 7. (a) Normalized stress intensity factor vs. r/l, note
that the three horizontal lines are corresponding to the maximal
values of the three cases, respectively; (b)normalized fracture
strength vs. relative density �qð2Þ and r/l.
74 Q. Chen et al. / Composite Structures 109 (2014) 68–74
conventional regular honeycomb, the stress intensity factor
ismaximally improved by 366% when �qð2Þ ¼ 0:01 and r/l = 0.33 andby
195% when �qð2Þ ¼ 0:21 and r/l = 0.1. It is worth mentioning thatif
the natural honeycomb has a relative density greater than 0.03,the
result illustrates why more silk and wax are centrally locatedat
the three-edge joint, according to strength and fracture tough-ness
of which the maximal values are at smaller r/l.
4. Conclusion
In this paper, we have constructed a
hollow-cylindrical-jointhoneycomb, and developed a theory to
calculate its Young’s mod-ulus, Poisson’s ratio, fracture strength
and stress intensity factor.With respect to the conventional
honeycomb, the results showedthat its Young’s modulus can be
optimized and comparable to thatin the literature. The smallest
Poisson’s ratio is obtained whenr/l � 0.38. For the maximal
improvement of its fracture strength,it can be obtained by
decreasing the radius of the circular joint.Whereas, a critical
relative density 0.03 exists for the maximalimprovement of stress
intensity factor, namely, the maximal valueis reached when the
ratio r/l equals 0.1 and the honeycomb’s rela-tive density is
greater than 0.03; otherwise, the maximal value isobtained when r/l
equals 0.33 and the honeycomb’s relative den-sity is less than
0.03. The present structure and theory could beused as a guide to
design new honeycomb materials.
Acknowledgements
CQ is supported by the Priority Academic Program Develop-ment of
Jiangsu Higher Education Institutions (No. 1107037001)and the
National Natural Science Foundation of China (NSFC)(No. 31300780).
NP thanks the European Research Council (ERCStG Ideas 2011 BIHSNAM,
ERC Proof of Concept REPLICA2 2013)and the European Union (Graphene
Flagship) for support.
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Mechanical properties of a hollow-cylindrical-joint honeycomb1
Introduction2 Structural theory2.1 Young’s modulus2.2 Poisson’s
ratio2.3 Fracture strength and stress intensity factor2.3.1
Mechanism (1): tensile-bending failure of the vertical cell
wall2.3.2 Mechanism (2): bending failure of cell walls
3 Results and discussion4
ConclusionAcknowledgementsReferences
