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Author's personal copy
Journal of the Mechanics and Physics of Solids 58 (2010) 1646–1660
Contents lists available at ScienceDirect
Journal of the Mechanics and Physics of Solids
journal homepage: www.elsevier.com/locate/jmps
Mechanical properties of unidirectional nanocomposites with
non-uniformly or randomly staggered platelet distribution
Z.Q. Zhang a,1, B. Liu a,n, Y. Huang b,c,n, K.C. Hwang a, H. Gao d
a
AML, Department of Engineering Mechanics, Tsinghua University, Beijing 100084, China
Department of Civil and Environmental Engineering, Northwestern University, Evanston, IL 60208, USA
c
Department of Mechanical Engineering, Northwestern University, Evanston, IL 60208, USA
d
Division of Engineering, Brown University, Providence, RI 02912, USA
b
a r t i c l e i n f o
abstract
Article history:
Received 18 October 2009
Received in revised form
25 June 2010
Accepted 2 July 2010
Unidirectional nanocomposite structures with parallel staggered platelet reinforcements are widely observed in natural biological materials. The present paper is aimed at
an investigation of the stiffness, strength, failure strain and energy storage capacity of a
unidirectional nanocomposite with non-uniformly or randomly staggered platelet
distribution. Our study indicates that, besides the volume fraction, shape, and
orientation of the platelets, their distribution also plays a significant role in the
mechanical properties of a unidirectional nanocomposite, which can be quantitatively
characterized in terms of four dimensionless parameters associated with platelet
distribution. It is found that, compared with other distributions, stairwise and regular
staggering of platelets produce overall the most balanced mechanical properties, which
might be a key reason why these structures are most widely observed in nature.
& 2010 Elsevier Ltd. All rights reserved.
Keywords:
Homogenization method
Non-uniform distribution
Biocomposites
Biomimetic composites
Mechanical properties
1. Introduction
Natural biological materials such as bone, teeth, and nacre are nanocomposites made of protein and mineral (Currey,
1977, 1984; Jackson et al., 1988; Landis, 1995; Landis et al., 1996; Rho et al., 1998; Weiner and Wagner, 1998). The protein
is compliant (Young’s modulus 50–100 MPa) with relatively high ductility and high strength-to-modulus ratio (20–50%)
(Jager and Fratzl, 2000; Ji and Gao, 2004a). The mineral, on the other hand, is stiff (Young’s modulus 50–100 GPa) with
relatively low fracture toughness ( 51 MPa m1/2) and low strength-to-modulus ratio (0.1–0.6%) (Currey, 1980; Jackson
et al., 1988; Ji and Gao, 2004a). Natural biological materials, however, are stiff and tough compared to their constituents.
Their Young’s modulus is 50 GPa for shell and 20 GPa for bone (Jackson et al., 1988; Jager and Fratzl, 2000; Ji and Gao,
2004a), comparable to that of mineral. The strength of natural biological materials is 100–300 MPa for shell and 100 MPa
for bone (Jackson et al., 1988; Jager and Fratzl, 2000; Ji and Gao, 2004a; Menig et al., 2000, 2001), comparable to the
strength of mineral. The fracture toughness of natural biological materials ranges from 2 to 7 MPa m1/2 (Jackson et al.,
1988; Ji and Gao, 2004a; Norman et al., 1995), which is several orders of magnitude higher than that of mineral (see, e.g. Ji
and Gao (2004a) for a collection of mechanical properties). It is remarkable that natural biological materials can achieve
mechanical properties that are far superior compared to their constituents.
n
Corresponding authors.
E-mail addresses: liubin@tsinghua.edu.cn (B. Liu), y-huang@northwestern.edu (Y. Huang).
1
Present address: LCS, Institute of High Performance Computing, A*STAR, Singapore 138632, Singapore.
0022-5096/$ - see front matter & 2010 Elsevier Ltd. All rights reserved.
doi:10.1016/j.jmps.2010.07.004
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Previous studies have indicated that the staggered micro-/nanostructures in natural biological materials play a crucial
role in their superior mechanical properties (Anup et al., 2007; Currey, 1977; 1984; Fratzl et al., 2004; Gao et al., 2003; Gao,
2006; Gupta et al., 2004, 2006; He and Swain, 2008; Ji and Gao, 2004a, b; Kamat et al., 2000; Laraia and Heuer, 1989; Liu
et al., 2009, 2006; Nukala and Simunovic, 2005; Weiner and Wagner, 1998). Based on prior studies on the tensile behaviors
of natural biological materials (Jackson et al., 1988; Jager and Fratzl, 2000; Kotha et al., 2001), Gao et al. (2003) and Ji and
Gao (2004a) developed a tension–shear chain model to illustrate how natural biological materials achieve high elastic
modulus and other mechanical properties. As shown by the macroscopic and TEM images of shell and bone in Fig. 1(a)–(d),
natural biological materials adopt a generic nanostructure whose load transfer path follows a one-dimensional tension–
shear chain shown in Fig. 1(e). The original tension–shear chain model of Gao et al. (2003) and Ji and Gao (2004a) is based
on the following assumptions:
(i) The elastic modulus of protein is two to three orders of magnitude smaller than that of the mineral such that the
protein cannot transfer any normal stress between the neighboring platelets; the load in the longitudinal direction (z)
can only be transferred via shear in protein.
(ii) The thickness (h) of mineral is one or two orders of magnitude smaller than its length L such that the deformation in
mineral is essentially one dimensional [i.e., depending only on z, Fig. 1(e)].
(iii) The distribution of mineral repeats itself after two mineral platelets.
(iv) The neighboring platelets overlap half of their length along the longitudinal direction.
Furthermore, the gap between mineral platelets in the longitudinal direction (z) is much smaller than the mineral
length L.
The tension–shear chain model with half overlapping length, hereafter referred to as regular staggering, can be used to
interpret many underlying mechanisms in biocomposites. However, strictly speaking, there also exist other patterns of
staggering in biocomposites, such as the stairwise staggering shown in Fig. 1(b) (Fratzl et al., 2004; Meyers et al., 2006,
2008; Rho et al., 1998). Moreover, non-uniform or even random platelet alignment can occur in biomimetic materials with
‘‘mortar–brick’’ micro-/nanostructure (Bonderer et al., 2008; Tang et al., 2003). One question then arises, why does nature
choose regular and stairwise staggering? A possible answer is that these structures produce optimal mechanical properties
compared to other distribution patterns. In comparison between regular and stairwise staggering, we note that the regular
staggering has obvious weak cross-section planes as denoted by the dashed line in Fig. 1(e) such that the composite
strength is only half of that of a continuous layering composite, while the stairwise staggered structure avoids the coalignment of longitudinal gaps of platelets in concentrated weak planes and is therefore expected to have higher strength.
The tension–shear chain model of Gao et al. (2003) and Ji and Gao (2004a) does not account for non-uniform
platelet alignment and cannot address any question regarding the platelet distribution in biological and biomimetic
nanocomposites.
The purpose of this paper is to investigate the effects of non-uniform or random staggering (alignment) of platelets on
the mechanical properties of biocomposites. Since there are many possible staggered structure patterns, an interesting
question is why regular staggering and stairwise staggering are most frequently observed in nature. From a fabrication
point of view, it would seem that random staggering does not require high precision control and has a clear advantage
from the entropic point of view. Why is it not selected by nature? The present study is motivated by these questions and by
L
z
h
Fig. 1. The tension–shear chain model of biological nanocomposites. (a) Macroscopic image of bone and (b) TEM image of its generic nanostructure.
(c) Macroscopic image of shell and (d) TEM image of its microstructure. (e) Schematic illustration of the tension–shear chain model proposed by Gao et al.
(2003) and Ji and Gao (2004a).
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h
τU
L+
L
ξL
ξL
z
τL
1
2
Fig. 2. Regularly staggered nanocomposite structure with offset. (a) Overall composite, (b) its unit cell and (c) the deformed configuration subjected to a
longitudinal displacement D.
h
L
( n 1) L
n
U
U
L+
L
U
L
z
2L
n
L
n
1
2
L
3
…
U
n
Fig. 3. Stairwise staggered nanocomposite structure. (a) Nanocomposite with stairwise staggering, (b) its unit cell and (c) the deformed configuration
under a longitudinal displacement D.
the general interest to understand the basic design principles of biological systems. In particular, we will extend the
tension–shear chain model to account for non-uniform or random staggering of platelets in a unidirectional composite
structure. The paper is outlined as follows. In Section 2, the assumption (iv) above is relaxed to extend the tension–shear
chain model to the case of an arbitrary overlap length (1 x)L, as shown in Fig. 2, where 0r x o1 represents platelet
distribution, and x ¼ 12 degenerates to regular staggering considered by Gao et al. (2003) and Ji and Gao (2004a). The elastic
modulus and strength are obtained analytically in terms of x, as well as other properties of the platelet and matrix. In
Section 3, we further relax the assumption (iii) above to extend the tension–shear chain model to n 42, where n is the
number of platelets within each period, as shown in Fig. 3. In stairwise staggering, each platelet is shifted up by a distance
L/n with respect to the platelet on its left. The solutions obtained then pave the way for the study of general non-uniform
platelet distribution in Section 4, and random platelet distribution in Section 5. The implications of the present study on
natural biological materials, including the optimal platelet distribution to achieve high stiffness, strength, and resilience
(i.e., the elastic energy storage capacity) of composite materials, are discussed in Section 6.
2. Tension–shear chain model for regular staggering with offset
The tension–shear chain model is extended to a regularly staggered structure with an arbitrary overlap length between
neighboring platelets in this section. As shown in Fig. 2, all platelets are assumed to be parallel to the z direction, and are
shifted to every other neighbor by a distance of xL, which gives the overlap length (1 x)L, where L is the platelet length,
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0 r x o1; x ¼ 12 degenerates to the case considered by Gao et al. (2003) and Ji and Gao (2004a). The present configuration
will be referred to as regular staggering with offset.
Fig. 2(b) shows a unit cell of the model. The thickness of the matrix layer is related to the platelet thickness h and
volume fraction f by ðð1fÞ=fÞh. The unit cell is subjected to a displacement D, as shown in Fig. 2(c). The shear stresses in
the matrix, denoted by tU and tL, respectively, for the upper and lower parts of the matrix [Fig. 2(c)], are assumed to be
constant. The force equilibrium requires
ð1xÞtU xtL ¼ 0:
ð1Þ
The normal stress in the (first) platelet is obtained from the force equilibrium as
8
2 L
>
>
t z,
0 r z r xL,
<
h
ð2Þ
sp ¼ 2
>
>
: tU ðLzÞ, xLo z rL:
h
RL
The strain energy in the platelets in the unit cell is 2h 0 ðs2p =ð2Ep ÞÞ dz, where Ep is Young’s modulus of the platelets.
The strain energy in the matrix is
"
#
ð1fÞh
ðtL Þ2
ðtU Þ2
xL
2
þð1xÞL
f
2Gm
2Gm
where Gm is the shear modulus of the matrix. The complementary energy of the unit cell is
"
#
Z L 2
sp
ð1fÞh
ðtL Þ2
ðtU Þ2
PC ¼ 2h
dzþ 2
þð1xÞL
F D
xL
f
2Gm
2Gm
0 2Ep
"
#
2
4x L3
ð1fÞxhL
¼
þ
ðtL Þ2 2xLtL D,
3hEp
fð1xÞGm
ð3Þ
where F is the axial force on the platelet given by
F ¼ 2tL xL:
ð4Þ
Minimization of the complementary energy by imposing dPC =dtL ¼ 0 gives the shear stress tL as
tL ¼
D
4xL2
ð1fÞh
þ
3hEp fð1xÞGm
:
ð5Þ
The average stress s in the composite is
s¼
F
fF
¼
2h þ2ð1fÞh=f
2h
which can be rewritten via Eqs. (4) and (5) as
s¼
1
D
,
4
1f
L
þ 2
3fEp f xð1xÞr2 Gm
ð6Þ
where r = L/h is the platelet aspect ratio. Since D/L represents the average strain e, Young’s modulus is obtained from the
ratio of average stress to average strain,
E¼
1
1
¼ fEp
,
4
1
4
1f
þ
þ 2
3 3xð1xÞa
3fEp f xð1xÞr2 Gm
ð7Þ
where
a¼
fr2 Gm
3ð1fÞEp
ð8Þ
is a parameter combining the effects of platelet volume fraction f and aspect ratio r, as well as the matrix and platelet
elastic moduli Gm and Ep. Young’s modulus, normalized by fEp, is a universal function of a and x, which represents platelet
distribution. For x ¼ 12, Eq. (7) degenerates to Gao et al. (2003) and Ji and Gao (2004a) except for the factor 43. This is because
Eq. (7) is derived based on the principle of minimum complementary energy, which is known to give rise to a lower bound
for Young’s modulus. Eq. (7) gives an improved estimate of Young’s modulus compared to the finite element calculations
by Ji and Gao (2004a). This improvement can be partly understood by considering the limit of f approaching 1, which
corresponds to a material with vanishing organic matrix, i.e. with only platelets left. In this limit, the platelets are
not continuous. The gaps between the platelets tend to soften the material similar to internal cracks, which would
reduce the effective Young’s modulus below Ep. We should point out that our analysis is intended only for cases when the
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1.0
0.8
83
E/
Ep
0.6
3.3
0.4
0.83
0.2
0.13
0.0
0.0
= 0.021
0.1
0.2
0.3
0.4
0.5
Fig. 4. Plots of Young’s modulus over offset for different values of a ¼ fr2 Gm =3ð1fÞEp for a regularly staggered nanocomposite.
tension–shear chain model is approximately valid, which requires that the aspect ratio fall below a critical value (Chen
et al., 2009). For very large aspect ratios, the applied load will be carried by the stiff platelets alone and that the shear stress
in the soft matrix vanishes except near the ends of the mineral platelets. This stress field obviously violates the basic
assumption of the tension–shear chain model that the shear stress in the soft matrix remains uniform.
Fig. 4 shows Young’s modulus E, normalized by the platelet volume fraction and modulus fEp, versus x for a large range
of a from 0.021 to 83. Due to the symmetry with respect to x ¼ 12, only the curves for 0 r x r 12 are plotted. Young’s modulus
achieves a maximum value of
E¼
1
4
4ð1fÞ
þ
3fEp f2 r2 Gm
at x ¼ 12, which corresponds to the case of regularly staggered platelets (without offset) as studied by Gao et al. (2003) and Ji
and Gao (2004a). The limit x =0 corresponds to 100% overlap of platelets, which leads to weak planes, and therefore
vanishing Young’s modulus. Moreover, it is worth noting that the moduli associated with regularly staggering of long
platelets are insensitive to the non-dimensional overlap length x when it is close to 12.
The composite strength is defined as the average stress in the composite at which the maximum shear stress in the
matrix reaches the matrix shear strength tm
critical , or the maximum normal stress in the platelet reaches the platelet strength
spcritical ,
ð9Þ
max tU , tL ¼ tm
critical
or
2tL
xL
h
¼ spcritical :
ð10Þ
These give the composite strength as
8
r
p
>
,
r r rucritical ,
>
< fscritical 2ru
critical
scritical ¼
1
>
>
:
fspcritical ,
r 4 rucritical ,
2
ð11Þ
where
p
rucritical ¼
scritical
1
1912x9 tm
critical
ð12Þ
is the critical aspect ratio that separates matrix failure (r r r0 critical) from platelet failure (r 4 r0 critical). The composite
0
strength, normalized by fspcritical , is a universal function of the platelet aspect ratio r and rcritical, and the latter depends on
the platelet-to-matrix strength ratio and x (representing platelet distribution).
Fig. 5 shows the composite strength scritical normalized by the platelet volume fraction and strength fspcritical , versus x
for the platelet-to-matrix strength ratio spcritical =tm
critical ¼ 10 and platelet aspect ratio r ranging from 8 to 500. The solid line
represents platelet failure, and is given by a universal relation scritical ¼ fspcritical =2. The dotted lines represent matrix failure,
and they depend on the platelet aspect ratio r. The regular staggering ðx ¼ 12Þ generally gives the maximum composite
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0.8
500
Matrix failure
10
Platelet failure
100
critical /(
p
crittical )
0.6
p
critical
m
critical
0.4
50
20
0.2
=8
0.0
0.0
0.1
0.2
0.3
0.4
0.5
Fig. 5. Plots of strength over offset for different aspect ratios of platelets for a regularly staggered nanocomposite.
strength, which is scritical ¼ fspcritical =2 if the aspect ratio r exceeds spcritical =tm
critical . The limit x = 0 gives vanishing strength due
to the co-alignment of platelet gaps in weak planes. However, if r 4 spcritical =tm
critical then all
" p
#
p
x2
scritical
2rtm
critical
,1
scritical
2rtm
critical
could achieve the maximum composite strength scritical ¼ fspcritical =2. Similar to the moduli, the strengths of the regularly
staggered composites with long platelets (aspect ratio r 4 spcritical =tm
critical ) are insensitive to the non-dimensional overlap
length x when it is close to 12, which implies some robustness of the biocomposites.
3. The tension–shear chain model with stairwise staggering
In this section, the tension–shear chain model is extended to the case of a periodic distribution of platelets with
stairwise staggering as observed in bone [see Fig. 1(b)]. Fig. 3(a) shows the unit cell of a periodic distribution with n
platelets in each period. Each platelet is shifted by L/n with respect to the adjacent platelets, and therefore gives an overlap
length of ððn1Þ=nÞL. The model degenerates to that of Gao et al. (2003) and Ji and Gao (2004a) when n =2.
Fig. 3(b) shows the unit cell of the model consisting of n platelets and matrix. It is subjected to a displacement D, as
shown in Fig. 3(c). The shear stresses in the matrix, denoted by tL and tU, satisfy the force equilibrium
n1 U 1 L
t t ¼ 0:
n
n
ð13Þ
The normal stress obtained from force equilibrium is given separately over three regions in each platelet:
(1) the bottom region ½0, ðL=nÞ of length L/n, of which the top (z =L/n) corresponds to the gap of the platelets next to the
left side wall of the unit cell;
(2) the middle region ðL=n, ððn1Þ=nÞ=LÞ of length ððn2Þ=nÞL, of which the top (z =((n 1)/n)/L) corresponds to the gap of
the platelets next to the right side wall of the unit cell; and
(3) the top region ½ððn1Þ=nÞL, L of length L/n:
8
z
L
>
>
0 rz r ,
ðtL þ tU Þ ,
>
>
h
n
>
>
<
L
L
ðn1ÞL
oz o
,
ð14Þ
sp ¼ ðtL þ tU Þ ,
>
nh
n
n
>
>
>
Lz
ðn1ÞL
>
>
: ðtL þ tU Þ
,
r z oL:
h
n
The strain energies in the platelets and matrix within each unit cell are
"
#
Z L 2
sp
ð1fÞh L ðtL Þ2 ðn1ÞL ðtU Þ2
nh
dz and n
þ
,
n 2Gm
n
f
2Gm
0 2Ep
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respectively. The complementary energy in the unit cell is
" Z
#
2
L s2
ð1fÞhL tL
ðn1Þð1fÞhL ðtU Þ2
p
PC ¼ n h
dz þ
þ
ðn1ÞF D
nf
nf
2Gm
2Gm
0 2Ep
"
#
ð3n4ÞL3
nð1fÞhL
ðtL Þ2 LtL D,
¼
þ
6ðn1Þ2 hEp 2ðn1ÞfGm
ð15Þ
where
F ¼ ðtL þ tU Þ
L
L
¼ tL
n
n1
ð16Þ
is the axial force on each platelet. Minimization of the complementary energy with dPC =dtL ¼ 0 gives the shear stress
as
tL ¼
D
ð3n4ÞL2
3ðn1Þ2 hEp
nð1fÞh
ðn1ÞfGm
þ
:
ð17Þ
The average stress in the composite is
s¼
ðn1ÞF
n½h þ ð1fÞh=f
which can be rewritten via Eqs. (16) and (17) as
s¼
1
nð3n4Þ
3ðn1Þ2 fEp
þ
D
n2 ð1fÞ
L
,
ð18Þ
2
ðn1Þf r2 Gm
where r =L/h is the platelet aspect ratio. Its ratio to the average strain D/L gives Young’s modulus of the composite as
E¼
1
nð3n4Þ
3ðn1Þ2 fEp
þ
¼ fEp
n2 ð1fÞ
2
ðn1Þf r2 Gm
1
nð3n4Þ
3ðn1Þ2
þ
n2
3ðn1Þa
,
ð19Þ
where a, as defined in Eq. (8), serves as a unique parameter combining the effects of platelet volume fraction f and aspect
ratio r, and matrix and platelet elastic moduli Gm and Ep. Young’s modulus, normalized by fEp, is a universal function of a
and the number of platelets n within each period. For n = 2, it degenerates to Gao et al. (2003) and Ji and Gao (2004a)
(except for a factor 43 associated with the term 1/fEp resulting from the minimization of complementary energy). For the
limit of platelet volume fraction f-1 or platelet aspect ratio r-N, Young’s modulus approaches ð3ðn1Þ2 =nð3n4ÞÞfEp ,
which increases from 34fEp for n = 2 to fEp for n-N, where fEp is the Voigt (upper) limit.
Fig. 6 shows Young’s modulus E, normalized by the platelet volume fraction and modulus fEp, versus n for a large range
of a from 0.021 to 83, which corresponds to the platelet aspect ratio ranging from 8 to 500 for a fixed platelet volume
fraction f = 50% and platelet-to-matrix modulus ratio Ep/Gm = 1000. For relatively small a =0.021, 0.13, and 0.83 (i.e., platelet
1.0
83
0.8
E/ Ep
0.6
0.4
3.3
0.2
0.83
0.13
0.0
= 0.021
2
10
18
26
n
34
42
50
Fig. 6. Plots of Young’s modulus over period n for different values of a ¼ fr2 Gm =3ð1fÞEp for a stairwise staggered nanocomposite.
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aspect ratio r = 8, 20, and 50), Young’s modulus reaches the maximum
E¼
1
4
4ð1fÞ
þ
3fEp f2 r2 Gm
at n = 2, which corresponds to the case considered by Gao et al. (2003) and Ji and Gao (2004a). However, for large a =3.3 and
83 (i.e., platelet aspect ratio r = 100 and 500), Young’s modulus reaches the maximum away from n= 2. (For a finite r, the
limit n-N corresponds to weak planes, and therefore vanishing Young’s modulus.)
It can be shown analytically that, for a r1, Young’s modulus of the composite always achieves the maximum at n= 2.
For a 41, the maximum Young’s modulus becomes
192a2
E ¼ fEp pffiffiffiffiffiffiffiffiffiffiffiffiffiffi
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ð 1 þ8a þ 1Þ3 ð3 1 þ 8a1Þ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi
at n ¼ ð1 þ 1 þ 8aÞ=2. Combining these results, the maximum Young’s modulus can be expressed as
"
#
3
256a2
a
,
E ¼ fEp max pffiffiffiffiffiffiffiffiffiffiffiffiffiffi
,
p
ffiffiffiffiffiffiffiffiffiffiffiffiffi
ffi
4
ð 1þ 8a þ 1Þ3 ð3 1 þ 8a1Þ 1 þ a
ð20Þ
which increases with a, and approaches fEp as a-N, where fEp is the Voigt (upper) limit.
The composite reaches its strength at critical shear stress in the matrix tL ¼ tm
critical or at critical normal stress in the
platelet sp ¼ spcritical . Thus the composite strength can be written as
8
n1
r
p
>
>
, r r r00 critical ,
< fscritical n r00
critical
ð21Þ
scritical ¼
>
n1
>
:
,
fspcritical
r 4 r00 critical ,
n
where
r00 critical ¼ ðn1Þ
spcritical
tm
critical
ð22Þ
00
00
is the critical aspect ratio that separates matrix failure (r r rcritical) from platelet failure (r 4 rcritical). The composite
00
strength, normalized by fspcritical , is a universal function of n, platelet aspect ratio r and rcritical.
Fig. 7 shows the composite strength scritical normalized by the platelet volume fraction and strength fspcritical , versus n
for the platelet-to-matrix strength ratio spcritical =tm
critical ¼ 10 and platelet aspect ratio r ranging from 8 to 500. The solid
curve represents platelet failure, and is given by a universal relation scritical ¼ ððn1Þ=nÞfspcritical . The dotted curves
represent matrix failure, and they depend on the platelet aspect ratio r. For each platelet aspect ratio r r spcritical =tm
critical , the
dotted curve is always lower than the solid curve such that the maximum composite strength is reached at n= 2, and is
p
m
given by 12frtm
critical . For each platelet aspect ratio r 4 scritical =tcritical , the intercept of solid and dotted lines gives the
1.0
500
σcritical /(φσ pcritical )
0.8
p
critical
m
τ critical
σ
0.6
= 10
Matrix failure
Platelet failure
100
0.4
50
0.2
20
ρ=8
0.0
2
10
18
26
n
34
42
50
Fig. 7. Plots of strength over number (n) of platelets in one period for different platelet aspect ratios for a stairwise staggered nanocomposite.
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maximum composite strength
p
frtm
critical scritical
rtm
þ spcritical
critical
,
p
p
which is reached at n ¼ 1 þ rðtm
critical =scritical Þ. The maximum composite strength approaches fscritical as r-N.
4. The tension–shear chain model with arbitrary staggering
The tension–shear chain model is extended to the case of arbitrary distribution of parallel platelets in this section.
Fig. 8(a) shows a unit cell consisting of the matrix and n platelets that are arbitrarily distributed, and the end of ith platelet
is denoted by xiL (0 r xi o1, i= 1,2, y, n) in the global coordinate z. This arbitrary distribution repeats itself outside the unit
left
cell. A local coordinate z~ ¼ zx L is introduced in Fig. 8(b) such that the ends of (i 1)th, ith, and (i+ 1)th platelets are x~ L,
i
i
right
left
left
right
0, and x~ i L, respectively, where x~ i ¼ xi1 xi if xi 1 xi Z0, x~ i ¼ xi1 xi þ 1 if xi 1 xi o0; x~ i
¼ xi þ 1 xi if
right
xi + 1 xi Z0, x~ i
¼ xi þ 1 xi þ1 if xi + 1 xi o0; and the 0th and (n + 1)th platelets are the same as the nth and 1st
L
U
platelets, respectively. Let tLi1 and tU
i1 denote the shear stress between ith and (i 1)th platelets [Fig. 8(c)], ti and ti the
shear stress between ith and (i+ 1)th platelets, and D the displacement on the unit cell.
The force equilibrium of the ith platelet gives
left
left
right
tUi1 x~ i tLi1 ð1x~ i Þ þ tLi x~ i
~ right Þ ¼ 0:
tU
i ð1x i
ð23Þ
An equilibrium stress field, which satisfies the above equation and will be used to calculate the complementary energy,
can be written as
9
8
9
8
~ left >
>
L
>
>
x
>
>
>
>
i
t
>
>
> i1 >
>
>
>
>
>
>
>
>
left
>
>
>
>
=
< tU =
< 1x~
i1
i
,
ð24Þ
¼ t*
L
right
>
>
>
ti >
>
>
>
>
1x~ i
>
>
>
>
>
>
>
>
>
>
>
>
;
: tU >
>
>
i
;
: x~ right >
i
where t* is to be determined. This stress field degenerates to Eqs. (1) and (13) for the platelet distributions in Sections 2
and 3, respectively. The normal stress in the ith platelet is obtained from the force equilibrium as
left
right
left
left
right
right
t
si ¼ * ð2x~ i x~ i Þz~ x~ i /z~ x~ i LSx~ i /z~ x~ i LS ,
ð25Þ
h
where the function /US is defined by
(
0, x o0,
/xS ¼
x, x Z 0:
The strain energy in all platelets in each unit cell is
Z L 2
n
X
si ðz~ Þ
L3 2
h
dz~ ¼ n
t* fp ,
2E
2hE
p
p
0
i¼1
L
i 1
L
(1
i )L
...
...
~ left
z
i
iL
L
U
i 1
z
1
...
U
i
L+
i ...
~ right
i
i 1
i
i 1
L
L
i
i 1
i
i 1
Fig. 8. Arbitrarily staggered nanocomposite structure. (a) Schematic of arbitrary staggering and (b) the unit cell (solid box) in a global coordinate system.
Dashed box describes the local environment of platelet i, with local coordinates defined in (b) and deformed configuration under a longitudinal
displacement D in (c).
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where
8
>
9
left 2
>
~ right Þ2 ðx~ left Þ3 ðx~ right Þ3 ð2x~ left x~ right Þ >
=
1
i Þ þ ðx i
i
i
i
i
left
right 2
left right
>
3n i ¼ 1>
>
>
: ðx~ i x~ i Þ ð1maxfx~ i , x~ i gÞ
;
n >
< ½ðx~
X
fp ¼
ð26Þ
is a dimensionless
factor depending only on the distribution of platelets. The strain energy in the matrix is
n ðð1fÞhLÞ=2fGm t2* fm , where
n left
left
right
right
1 X
fm ¼
x~ i ð1x~ i Þ þ x~ i ð1x~ i Þ
ð27Þ
2n i ¼ 1
is another dimensionless factor depending only on the distribution of platelets. The total force on the composite is nfmLt*,
which can be derived from Eq. (25). The complementary energy in the unit cell is
3
L
ð1fÞhL
PC ¼ n
fp þ
fm t2* nfm Lt* D,
ð28Þ
2fGm
2hEp
where the last term represents the external work. Minimization of the complementary energy with dPC =dt* ¼ 0 gives
D
t* ¼
2
L fp ð1fÞh
þ
hEp fm
fGm
:
ð29Þ
The average stress in the composite is
s¼
1
D
,
1 fp
1f 1 L
þ
fEp fm2 f2 r2 Gm fm
ð30Þ
where r = L/h is the platelet aspect ratio. Its ratio to the average strain D/L gives Young’s modulus of the composite as
E¼
1
b1
b ð1fÞ
þ 2
fEp f2 r2 Gm
¼
fEp
b1 þ 3ba2
,
ð31Þ
where a is defined in Eq. (8) and
b1 ¼
fp
,
fm2
ð32Þ
b2 ¼
1
,
fm
ð33Þ
are two dimensionless factors representing the effect of the non-uniform distribution of platelets on Young’s modulus of
the composite. It is interesting to note that, although there are so many distribution patterns for short platelet reinforced
composites, all the information can be condensed into only two non-dimensional factors, b1 and b2, in determining the
composite stiffness. Therefore, we can refer to b1 and b2 as the stiffness distribution factors, which can be computed by
Eqs. (26), (27), (32) and (33) for any distribution (x1,x2, y, xn). For the distribution in Section 2, b1 ¼ 43 and b1 =1/(x(1 x)).
For the distribution in Section 3, b1 ¼ nð3n4Þ=3ðn1Þ2 and b2 ¼ n2 =ðn1Þ. Table 1 lists the stiffness distribution factors for
some typical distribution patterns.
Table 1
Summary of stiffness and strength distribution factors for unidirectional composites with regular staggering, regular staggering with offset, stairwise
staggering, random staggering and continuous layering.
Regular staggering
Regular staggering with offset
Stairwise staggering
Random staggering
Continuous layering
Stiffness distribution factors
Strength distribution factors
b1
b2
b3 (r 4 rcritical)
b4 (r r rcritical)
4
3
4
3
4
nð3n4Þ
3ðn1Þ2
7
5
1
Normalized critical aspect ratio
rcritical
spcritical =tm
critical
2
2
1
2
maxfx, 1xg
xð1xÞ
maxfx, 1xg
2xð1xÞ
n2 =ðn1Þ
n=ðn1Þ
n
n 1
6
3
6
2
1
1
xð1xÞ
¼ b4 =b3
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Z.Q. Zhang et al. / J. Mech. Phys. Solids 58 (2010) 1646–1660
The composite reaches its strength when the maximum shear stress in the matrix reaches the matrix shear strength
p
tm
critical , or the maximum normal stress in the platelet reaches the platelet strength scritical . Combined with Eqs.(24) and (25),
these give the composite strength as
scritical ¼
8
1
>
m
>
r,
>
< ftcritical
r r rcritical ,
1
>
p
>
>
: fscritical b ,
r 4 rcritical ,
b4
3
ð34Þ
where
rcritical ¼
b4 spcritical
b3 t m
critical
ð35Þ
is the critical aspect ratio that separates matrix failure (r r rcritical) from platelet failure (r 4 rcritical), and b3 and b4 are two
non-dimensional factors representing the effect of non-uniform distribution of platelets on the strength of the composite,
and are given by
b3 ¼
left
right
left
right
left
right
left
right
1
max ð2x~ i x~ i Þminfx~ i , x~ i g,ðx~ i þ x~ i Þð1maxfx~ i , x~ i gÞ,
fm
b4 ¼
left
right
left
right
1
max x~ i , x~ i , 1x~ i , 1x~ i ,
fm
i ¼ 1,2,. . .,n ,
i ¼ 1,2,. . .,n :
ð36Þ
ð37Þ
Similarly, all distribution information can be condensed into another two non-dimensional factors, b3 and b4, in
determining the composite strength, and we name them the strength distribution factors. For the distribution in Section 2,
b3 = 2 and b4 ¼ 2=ð1912x9Þ. For the distribution in Section 3, b3 ¼ n=ðn1Þ and b4 = n. The strength distribution factors for
some typical distribution patterns are listed in Table 1.
For an arbitrary distribution of platelets, Young’s modulus of the composite, normalized by fEp, is a universal function
of a and the stiffness distribution factors b1 and b2. The strength of the composite, normalized by the platelet volume
fraction and strength fspcritical , is a universal function of platelet aspect ratio r, platelet-to-matrix strength ratio, and the
strength distribution factors b3 and b4.
The introduction of the stiffness and strength distribution factors makes it easier to compare the corresponding
mechanical properties between the different distribution patterns, and obviously, the smaller stiffness/strength
distribution factors imply higher stiffness/strength. Here, as an example, considering the herringbone-type distribution
of n platelets in each period shown in Fig. 9, the factor b1 ¼ ð3n3 2n2 8n þ8Þ=3nðn1Þ2 is larger than (or equal to) its
counterpart in Section 3, and b2 ¼ n2 =ðn1Þ remains the same. Therefore the distribution of platelets in Fig. 9 gives a
smaller Young’s modulus of the composite. The factor b3 = 2 is larger than (or equal to) its counterpart in Section 3, and
b4 = n remains the same. This leads to a smaller strength of the composite.
L
L/6
L/6
L/6
1
2
3
4
5
6
Fig. 9. A unidirectional nanocomposite structure with the herringbone-type platelet distribution n= 6. Solid box denotes its unit cell.
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1657
5. The tension–shear chain model with random staggering
Random platelet alignment usually emerges in biomimetic materials with ‘‘mortar–brick’’ micro-/nanostructure
(Bonderer et al., 2008; Tang et al., 2003). For random distributions of platelets, xi representing the end of ith
platelet (i= 1,2, y, n) is a random variable between 0 and 1. The average values of factors fp and fm in Eqs. (26) and
(27) are:
9
8
left
right 2
left
right 3
left
right >
>
Z 1Z 1>
½ðx~ Þ2 þ ðx~
Þ ðx~ Þ3 ðx~
Þ ð2x~ x~
Þ>
= left
<
right
1
7
dx~
,
ð38Þ
fp ¼
dx~
¼
left
right 2
left right
~
~
~
~
>
3 0 0 >
180
Þ 1maxfx , x
g
>
>
;
: ðx x
fm ¼
1
2
Z
1
0
Z
1
½x~
left
0
ð1x~
left
right
right
left
right
1
Þ þ x~
ð1x~
Þ dx~
dx~
¼ :
6
ð39Þ
These give b1 ¼ 75 and b2 =6 such that Young’s modulus of the composite with random distribution of platelets is
E¼
1
5a
¼ fEp
,
7
6ð1fÞ
10 þ7a
þ 2
5fEp f r2 Gm
ð40Þ
where a is given in Eq. (8). For very large platelet aspect ratio, Young’s modulus is E ¼ 57fEp .
left right
left
right
,1x~ ,1x~
,
For random distributions of platelets with x (i= 1,2, y, n) between 0 and 1, maxfx~ , x~
i
. . .,ng may reach 1 such that Eq. (37) gives b4 =6 since
left right
gÞ,
ð1maxfx~ , x~
i
fm ¼ 16;
i
i
i
i ¼ 1,2,
left
right
left right
left
right
maxfð2x~ x~
Þminfx~ , x~
g,ðx~ þ x~
Þ
i
i
i
i
i
i
left
right
i ¼ 1,2,. . .,ng may reach
at x~ i ¼ x~ i
¼ 12 such that Eq. (36) gives b3 =3. The strength of the
i
i
composite with random distribution of platelets is
8
r
p
>
, r r r000
>
critical ,
< fscritical 3r000
critical
ð41Þ
scritical ¼
1
>
>
: fspcritical ,
r 4 r000critical ,
3
1
2
where
r000critical ¼
2spcritical
tm
critical
:
For very large platelet aspect ratio, the strength is scritical ¼ 13fspcritical .
For random distribution of platelets, Young’s modulus and strength of the composite may reach
continuous layering composite.
ð42Þ
5
7
and
1
3
of those of the
6. Discussions and conclusions
Table 1 lists the stiffness and strength distribution factors for several typical distribution patterns. For completeness,
the distribution factors for continuous layering structure are also included by directly comparing its stiffness and strength
with Eqs. (31) and (34). From these equations, it is noted that increasing the aspect ratio of platelets r can improve
both the stiffness and strength of the staggered composites. When the aspect ratio r (or a) is relatively large, according to
Eq. (31), the stiffness distribution factor b1 plays a more dominant role than b2. Similarly, when r 4 rcritical, the
strength will depend on the strength distribution factor b3 based on Eq. (34), and will be independent of b4. It is found
from Table 1 that among various staggered distributions, the stairwise staggering with relatively large n has the
smallest b1 and b3, and therefore has higher stiffness and strength when the aspect ratio r is large. To obtain a
direct comparison, we select a group of typical material parameters f = 50%, Ep/Gm = 1000, spcritical =tm
critical ¼ 10, and
draw the normalized stiffness and strength as functions of the aspect ratio r in Figs. 10(a) and (b) for different
distribution patterns. The stiffness and strength of continuous layering composites are used to normalize the
corresponding quantities.
It is noted from Fig. 10(a) that for large aspect ratio r, the stairwise staggering with (n = 5,10) has higher stiffness
than regular staggering and random staggering distributions. The prediction by the Mori–Tanaka method (Mori and
Tanaka, 1973; Weng, 1990; Zhao and Weng, 1990) is also plotted in Fig. 10(a) and it obviously underestimates the stiffness.
It should be pointed out that the Mori–Tanaka method, as a classical homogenization-based mesomechanical approach,
can only account for the influence of the volume fraction, shape and orientation of inclusions, but cannot take the
distribution of reinforcements into account. In contrast, our study indicates that the distribution has significant effect on
the stiffness.
Fig. 10(b) shows that the regular staggering has the highest strength at small aspect ratios r (e.g., r r15), but stairwise
staggering with larger period n (e.g., n =5,10) would achieve higher strength provided that the aspect ratio is large enough.
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Z.Q. Zhang et al. / J. Mech. Phys. Solids 58 (2010) 1646–1660
1.2
50%
Continuous layering
1.0
Stair-wise staggering
n=5
1000
Gm
1.0
p
Regular staggering
/
Random staggering
0.2
n=5
0.8
Regular staggering
0.6
Random staggering
0.2
Mori-Tanaka Method
0.0
0.0
0
100
200
300
400
500
0
20
40
60
40
50%
32
Ep
Gm
1000
p
critical
m
critical
10
10
50%
p
wcritical/( wcritical)
24
/
critical
Regular staggering
16
Stair-wise staggering
n=5
8
0
20
40
60
6
80
Gm
1000
p
critical
m
critical
120
10
Random staggering
Stair-wise staggering
4
n=5
0
Continuous layering
0
Ep
100
n=10
2
n=10
Random
staggering
80
Regular staggering
8
p
10
0.4
0.4
critical
p
critical
m
critical
Stair-wise staggering n=10
n=10
0.6
50%
Continuous layering
critical
E/( Ep)
0.8
1.2
Ep
critical
1658
100
120
Continuous layering
0
20
40
60
80
100
120
Fig. 10. Comparison of mechanical properties of unidirectional nanocomposites with regular staggering, stairwise staggering (n= 5,10), random
staggering, and continuous layering with respect to the platelet aspect ratio: (a) Young’s modulus, (b) strength, (c) failure strain, and (d) energy storage
capacity.
Every distribution tends to reach its strength limit fspcritical U1=b3 when the aspect ratio r is beyond the corresponding
critical value rcritical. Higher strength limit is realized by stairwise staggering with larger period n, which has smaller b3 as
indicated in Table 1. It is worth noting that the strength of random staggering is significantly lower than that of controlled
distribution such as regular staggering and stairwise staggering.
From Fig. 10(a, b) and Table 1, it seems that the continuous layering composite is the best distribution with respect to
stiffness and strength, but why does nature not adopt this microstructure? An answer to this question will
require consideration of failure strain ecritical ¼ scritical =E and energy storage capacity wcritical ¼ s2critical =ð2EÞ, which also
depend on the distribution via the four distribution factors (b1, b2, b3, and b4). Fig. 10(c) shows the failure strain
as a function of the aspect ratio r for regular staggering, stairwise staggering (n= 5,10), random staggering, and continuous
layering distributions. The failure strain is normalized by the failure strain of continuous layering structure,
equal to platelet failure strain epcritical ¼ spcritical =Ep . It is found that, different from the stiffness and strength, the failure
strain ecritical of staggering distribution decreases as the aspect ratio r increases, and the continuous layering distribution
has the lowest failure strain. Hence, there should exist an optimal aspect ratio r that gives best the overall best mechanical
properties. In this sense, the energy storage capacity may serve as an index in evaluating the overall mechanical properties,
since it is essentially the product of the strength and the failure strain. Fig. 10(d) shows the energy storage capacity as a
function of the aspect ratio r, normalized by that of continuous layering distribution fwpcritical ¼ fðspcritical Þ2 =ð2Ep Þ. Fig. 10(d)
suggests that each energy storage capacity curve for staggering distribution reaches its peak at a critical aspect ratio.
The peak values for regular staggering and stairwise staggering (n =5,10) are several times and even one order larger than
that of the continuous layering case. The peak value of the energy storage capacity for random staggering is less than that
of regular staggering and stairwise staggering (n =5,10) but almost two times of that for the continuous layering
composites.
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1659
In summary, we have developed analytical models on the stiffness, strength, failure strain, and energy storage
capacity of unidirectional platelet-reinforced composites with arbitrary distribution. The following conclusions can
be drawn:
(1) It is found that stairwise staggering (including regular staggering with n= 2) could achieve overall excellent
performance, namely high stiffness and strength comparable to those of the reinforcing platelets as well as large failure
strain and energy storage capacity comparable to those of the soft matrix. This may be a key reason why regular and
stairwise staggering are most widely observed in natural biological materials, and may be regarded as the optimal
design resulting from millions of years of evolution. The performance of random staggering is obviously lower than
stairwise staggering, which indicates that precisely controlled microstructure is an efficient way to further improve
performance of biomimetic and other man-made composites.
(2) Besides the volume fraction, shape, and orientation of the reinforcements, the distribution of platelets also plays a
significant role in the mechanical properties of unidirectional composites. Four distribution factors, the stiffness
distribution factors b1, b2 and the strength distribution factors b3, b4, are identified to completely characterize the
influence of the reinforcement distribution, and they can be easily computed. It should be emphasized that classical
homogenization-based mesomechanical approaches, such as the Mori–Tanaka method, cannot take this distribution
effect into account.
(3) The analytical models and distribution factors developed in the present paper allow one to compare different designs
of unidirectional composites and may guide the development of high performance biomimetic and other man-made
composites.
Acknowledgments
BL acknowledges the support from National Natural Science Foundation of China (Grant nos. 10702034, 10732050,
90816006) and National Basic Research Program of China (973 Program 2007CB936803). YH acknowledges the support
from ONR Composites for Marine Structures Program (Grant N00014-01-1-0205, Program Manager Dr. Y.D.S. Rajapakse)
and the NSF (Grant #0800417). The work of HG was partially supported by the A*Star Visiting Investigator Program ‘‘Size
Effects in Small Scale Materials’’ hosted at the Institute of High Performance Computing in Singapore.
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