World Journal of Mechanics, 2013, 3, 13-21
http://dx.doi.org/10.4236/wjm.2013.35A003 Published Online August 2013 (http://www.scirp.org/journal/wjm)
A Finite Element Study of Crack Behavior for Carbon
Nanotube Reinforced Bone Cement
Kaveh PourAkbar Saffar1*, Ahmad Raeisi Najafi2*, Manssour H. Moeinzadeh3, Leszek J. Sudak1
1Department of Mechanical and Man ufac turing Engineering, University of Calgary, Calgary, Canada
2Department of Mechanical Science and Engineering, University of Illinois at Urbana-Champaign, Urbana, USA
3Department of Industrial and Enterprise Systems Engineering, University of Illinois at Urbana-Champaign, Urbana, USA
Received May 15, 2013; revised June 15, 2013; accepted June 23, 2013
Copyright © 2013 Kaveh PourAkbar Saffar et al. This is an open access article distributed under the Creative Commons Attribution
License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
ABSTRACT
Polymethylmethacrylate (PMMA) bone cement is a polymeric material that is widely used as a structural orthopedic
material. However, it is not an ideal material fo r bone grafting due to its fragility. Carbon nanotubes (CNTs) have been
introduced in order to reinforce PMMA resulting in a composite material which exhibits improved tensile properties,
increased fatigue resistance and fracture toughness. This improvement is potentially due to bridging and arresting
cracks as well as absorption of energy. In this study, a two-dimensional finite element model is presented for the frac-
ture analysis of PMMA-CNT composite material. Instead of the classical single fiber model, the present work considers
an ensemble of CNTs interacting with a pre-existing crack. Casca is used to produce a two dimensional mesh and the
fracture analysis is performed using Franc 2D. The model is subjected to uni-axial loading in the transverse plane and
the interaction between the crack and CNTs is evaluated by determining the stress intensity factor in the vicinity of the
crack tips. The effects of geometric parameters of the CNTs and the material structural heterogeneity on crack propaga-
tion trajectory are investigated. Furthermore, the effects of CNT diameter, wall thickness and elastic mismatch between
the matrix and the nanotubes on crack growth are studied . The results illustrate that the CNTs repel cracks during load-
ing as they act as barriers to crack growth. As a result, the incorporation of CNTs into PMMA reduces crack growth but
more importantly increases the fracture resistance of bone cement.
Keywords: Bone Cement; Carbon Nanotube; Finite Element; Crack Propagation; Stress Intensity Factor
1. Introduction
Polymethylmethacrylate (PMMA) bone cement is a poly-
meric material that shows good co mpatibility with living
tissues. It is commonly used as bone filling cement dur-
ing orthopedic surgeries, such as total joint anthroplasty
[1,2]. Despite its excellent performance, several failure
mechanisms of cemented prosthetics have been well
documented [3-5].
It is well-known that both biological and mechanical
factors (such as mechanical loading, debonding at the
cement-prosthesis interface, fatigue failure and cracks in
the cement mantle) contribute to the failure of total hip
arthroplasty; howev er, it is the mechanical processes that
are considered to be the dominant feature leading to
aseptic loosening [6,7]. In particular, it is well accepted
that cracks exist in the cement mantle. These cracks
typically are generated from polymerization of the PMMA
from natural pores formed in the mantle during the
curing process or they propagate into the cement mantle
from the failure of the cement/implant interface [8]. For
example, McCormack & Prendergast [9] experimentally
investigated the amount of damage accumulation that
occurs in the cement mantle under flexural loading.
Other studies [6,10] have demonstrated that failure of the
cement mantle, due to the presence of radial cracks,
initiated loosening of the prosthetic. Thus, the fracture of
PMMA bone cement is a critical factor affecting implant
failure.
There have been numerous studies carried out to im-
prove the mechanical and physical behavior of bone ce-
ment by incorporating additive materials as reinforcing
phases to the bone cement [5,11]. However, limited suc-
cess has been achieved mainly due to filler-damage scale
mismatch, as the size of the reinforcing agents is larger
than the scale of fatigue damage. This makes the rein-
*Authors with equal contribution.
C
opyright © 2013 SciRes. WJM

K. POURAKBAR SAFFAR ET AL. 14
forcement ineffective against crack propagation and
damage accumulation [5].
Nano-structured materials have generated considerable
interest in the literature mainly because of their potential
for large improvements in the mechanical behavior as
compared to traditional structural materials. In particular,
extensive research has shown that carbon nanotubes
(CNTs) exhibit exceptional mechanical and chemical
properties as well as thermal stability. For example, in
composite science research, the use of CNTs as a rein-
forcing agent makes them ideal candidates for producing
a new generation of nano-composites [12-14]. More re-
cently, in the area of bioengineering, CNTs have been
used as biocompatible and supportive substrates [15,16].
In particular, CNTs have been considered for effective
employment in bone tissue engineering either as a scaf-
fold for bone material regeneration and growth [17,18],
as reinforcement phase for ceramic artificial bone such as
hydroxyapatite [19-21] and polymeric bone fillers, such
as PMMA [3-5,22]. More importantly, the dimensional
incompatibility between the reinforcing fibers and the
size of fatigue damage is no longer an issue. In other
words, unlike the large size of conventional fibers (o n the
order of microns), the small size of CNTs (order of na-
nometers) is comparable to the scale of fatigue damage
found in the bone cement. Consequently, the use of
CNTs as a reinforcement phase in PMMA can have a
significant benefit in preventing crack growth with the
overall goal of reducing implant motion and revision
surgery. The enhancement of the fatigue performance of
bone cement is a result of the CNT’s high surface area to
volume ratio which increases the physical interface and
improves the quality of stress transfer between the con-
stituent phases.
The use of experimentation, simulation and modeling
in the study of nano- scale materials such as CNTs on the
structural response (such as fracture characterization) is
still in its infancy. The small size of the components of
nano-composites makes it difficult to clearly observe
every factor incorporated with the real failure mecha-
nisms in experiments. So, controlled experiments at the
nano-scale are difficult. The use of molecular dynamics
simulations remains expen sive and formidable especially
in the case of large scale systems. For example, the size
of atomistic systems cannot exceed billions of particles;
however, realistic systems contain 1023 atoms and more.
Continuum mechanics approaches based on the finite
element method have been used extensively to evaluate
the elastic properties of CNT-based composites. For ex-
ample, Gawandi and colleagues addressed the problem of
a single nano-fiber interacting with a crack [23,24].
However, these works are limited because the atomistic
structure of the nanotube has been neglected in the for-
mulation of the finite element model. It should be noted
that additional, supplementary works on crack-fiber in-
teraction can be found in the following works: [25-35].
However, the study of the interaction problem between a
crack and an ensemble of multiple CNTs remains absent
from the literature. Thus, in the present work we develop
a two dimensional finite element model incorporating an
array of CNTs as the reinforcing phase in PMMA bone
cement where the atomistic structure of CNTs are taken
into account. The results clearly illustrate that CNTs re-
pel cracks during loading as they act as barriers to crack
growth. Moreover, it is observed that the incorporation of
CNTs into PMMA reduces crack growth but more im-
portantly increases the fracture resistance of bone ce-
ment.
2. Problem Statement
We consider a two dimensional setting in which single
walled carbon nanotubes (SWCNTs) are assumed to be
well dispersed and aligned parallel to one another and
embedded within the PMMA bone cement, (referred to
as the matrix). A crack is situated in the matrix and the
loading on the configuration is uni-axial tension applied
perpendicular to the X axis, as shown (see Figure 1).
Crack extension takes place in the plane of the nanotube
cross section and plane strain conditions are assumed. It
is reasonable to assume that the matrix and CNT is iso-
tropic and linear elastic subject to small deformations
[36]. In addition, all boundaries between materials are
assumed to have continuous displacements and tractions.
The SWCNT, from the structural point of view, can be
thought of as a graphene sheet rolled up into a cylind rical
Figure 1. 2D model of CNT reinforced bone cement.

K. POURAKBAR SAFFAR ET AL. 15
shape with a diameter on the order of nanometers. In
addition, the efficiency of nanotube reinforcement de-
pends on the morphology and distribution of CNTs. This
requires that the atomistic structure of CNTs be taken
into account when formulating the finite element model.
In view of this, the development of our model requires
that the structurally discrete CNTs be replaced by an
equivalent effective solid fiber continuum phase whose
length and diameter are kept the same as those of the
CNT so as to preserve the small-scale features of the
nanotube (see Figure 2). Assuming isostrain conditions,
the modulus of elasticity and Poisson’s ratio for the ef-
fective solid fiber are readily determined from the CNT
dimens i ons a nd given by Equation (1),

2
2
2
f
CNT
rrt
E
r

E (1)
where r, t, and ECNT are the radius, wall thickness and
elastic stiffness of the CNT, respectively.
The model considers that the CNTs have a diameter of
1.5 nm, a wall thickness of 0.34 nm and elastic modulus
of 0.97 TPa [37]. The PMMA is defined by two elastic
constants: elastic stiffness Em = 2.5 GPa and Poisson’s
ratio υm= 0.35. Using (1) the properties for the effective
solid fiber continuum phase are found to be Ef = 680 GPa
and υf = 0.28.
3. Finite Element Model
The 2D finite element mesh is built using Casca [38] and
Franc 2D [39] is used for the fracture analysis of the
model. Triangular elements with six nodes comprising 12
degrees of freedom are used to produce the finite element
mesh structure. In order to avoid boundary effects in the
results, the model dimensions are taken sufficiently large
compared to the CNT diameter and crack length (see
Figure 3). It should be noted that our simulations have
been tested and verified against the analytical solution
obtained from the model illustrated in Figure 4(a).
Figure 2. Nanotube as an effective fiber.
Figure 3. Finite element model of crack-CNTs interaction.
In an effort to understand how the number, location,
configuration, and material characteristics of CNTs in-
fluences’ the variation of the stress intensity factor (SIF)
at the crack tips as well as crack trajectory, finite element
simulations for two problems is investigated. In the first
problem we will consider three different configurations
of crack-CNT interaction. The first configuration consists
of a single CNT with 1.5 nm diameter placed in a suffi-
ciently large matrix subjected to a uni-axial stress, σ,
while having an internal crack of length 2L = 1.5 nm
placed in a horizontal orientation to the external load as
shown in Figure 4(a). A second configuratio n consisting
of three CNTs placed on a cross arrangement with an
internal crack placed in-between the CNTs in a horizon-
tal orientation to the externally applied load (see Figure
4(b)).
The third configuration considers the mixed mode
(modes I and II) problem for a variable inclined crack in
the PMMA matrix located in the middle of a square ar-
rangement of CNT as shown in Figure 5 (H = V = 3 nm,
2L = 1.84 nm). This illustrates the effect of changing the
crack orientation on the variation of the SIF at crack tips.
Now, with respect to the second problem, let us consider
the effects of crack propagation. The model includes dif-

K. POURAKBAR SAFFAR ET AL. 16
(a)
(b)
Figure 4. Two configurations for mode-I SIF analysis.
Figure 5. A crack with angle α oriented in the middle of a
square arrangement of CNTs in the matrix.
ferent arrangement of CNTs embedded into the PMMA
matrix (see Figures 6-8, respectively). Here, the matrix
dimensions are much larger than the interested area. So
(a)
(b) (c)
Figure 6. Finite Element Model of Crack-CNTs interaction.
The model includes fibers with 2r =1.5 nm, Ef = 0.68 TPa,
Em = 2.5 GPa. (a) H = V = 2.5 nm and primary crack length
1.42 nm; (b) H = V = 3 nm and primary crack length 1.84
nm; (c) H = V = 2.5 nm and primary crack length 1.84 nm.
(a) (b)
Figure 7. Crack propagation is severely affected by the
separation of the fibers.
there are no boundary effects in the results. Uniaxial
loading normal to the boundary is taken and the crack
propagation trajectory is obtained. Using Franc 2D, the
direction of crack propagation is determined by the ap-
plication of the maximum hoop stress. The hoop stress
(σθ) is determined around the crack tip on the circumfer-
ence of a constant radius circle. Thus, crack propagation
is in the direction of the maximum hoop stress. The cor-
responding mathematical description is provided by ex-
pressions (2) and (3).
0




 (2)
2
20




 (3)

K. POURAKBAR SAFFAR ET AL. 17
(a) (b)
(c) (d)
(e)
Figure 8. Crack propagation trajectory is affected by fiber
material properties. Stiff fiber repels the crack where soft
fiber or hole leads fiber attracting the crack. (a) Ef = Ef7 =
0.68 TPa; (b) Ef = 0.68 TPa, and a hole instead of fiber
number 7; (c) Ef = Ef7 = 0.68 TPa; (d) Ef = 0.68 TPa, Ef7 =
0.2 TPa; (e) Ef = 0.68 TPa, and a hole instead of fiber 7.
4. Results and Discussion
Figures 9(a) and (b) show the normalized SIF versus
crack location with respect to the arrangement of CNTs.
The crack location is considered as the normalized dis-
tance of the crack from the single CNT shown in Figure
4(a) (corresponding to Figure 9(a)), and the normalized
distance of the crack from the CNT at left of the cross
arrangement as shown in Figure 4(b) (corresponding to
Figure 9(b)). The simulation results indicate that as the
crack distance from the CNT increases the SIF also in-
creases for both arrangements. In fact, at close distances,
the CNT (on the left) has a shielding effect. Figure 9(c)
shows the SIF values for both arrangements which give
an idea on the effect of the presence of two extra CNTs
(a)
(b)
(c)
Figure 9. Normalized SIF of crack tips versus the normal-
ized distance from the fiber at left (d/r) for two configura-
tions: (a) Single CNT; (b) Cross arrangement of three CNTs;
and (c) Both arrangements altogether.
in the cross arrangement in comparison with the single
CNT arrangement. According to Figure 9(c), the cross
arrangement leads to significant stress shielding at crack
tip a, while, at crack tip b we observe a dramatic stress
amplification. This implies that the presence of two extra
CNTs in a cross arrangement pattern (at right) is effect-

K. POURAKBAR SAFFAR ET AL. 18
tive as a reinforcement in regions close enough to the
other CNT which faces crack tip a while the crack is ac-
tually encouraged to grow from crack tip b due to the
presence of CNTs at right. Obviously, this effect is de-
pendent on the separation of the two CNTs at right (i.e.
the crack is encouraged to grow from crack tip b and pass
through the space between them if this separation is more
than a certain amount). Close arrangements of CNTs
would definitely have a barrier effect against crack grow-
th. This will be discussed later.
Figure 10 shows the values of modes I and II
normalized SIF for the lower-left tip of the crack in the
model shown in Figure 5, with respect to varying α. Due
to the symmetry of the problem close values with the same
trend are expected for the other crack tip.
The model which is developed for crack propagation
analysis is shown in Figure 6. The model includes an
array of CNTs and an internal crack (at an angle of 45
degrees). The crack propagation simulation shows that
crack propagation trajectory is severely affected by the
arrangement and material properties of the CNTs. The
effect of the CNT arrangement is presented in Figures
6(a)-(c), respectively. The results show that the crack is
repelled by the CNTs and the crack follows a trajectory
between them when the distance between the CNTs is
more than a certain amount. In fact, the crack deviates
and does not enter the CNTs. On the other hand, if CNTs
are situated closer together, the crack is not able to
propagate through the space between the CNTs. In this
situation, the CNTs act as a barrier to crack growth.
Normalized SIF values for modes I and II (KI and KII)
at the crack tips for the arrangements shown in Figure 6
are also presented in Table 1. At first, proximity of the
values for crack tips a and b is observed which is due to
the problem symmetry. A comparison between the sec-
ond and third columns of this table provides sufficient
Figure 10. Normalized SIF of a crack with changing angle
in the middle of square arrangement of CNTs.
Table 1. Normalized SIF values of crack tips for models
shown in Figure 8.
H = V = 2.5 nm
2L = 1.42 nm
H = V = 3 nm
2L = 1.84 nm
H = V = 2.5 nm
2L = 1.84 nm
KI—tip a0.402 0.454 0.353
KII—tip a0.223 0.272 0.253
KI—tip b0.411 0.457 0.355
KII—tip b0.220 0.272 0.254
evidence as to why the crack grows in the space between
the distanced CNTs (e.g. Figure 6(b)), while it stops
between close CNTs (e.g. Figure 6(c)).
In the case of randomly dispersed CNTs, dispersion
of the CNTs has a significant influence on the material
properties of CNT reinforced polymer matrix nano-com-
posites [40]. In view of this, crack propagation in a ran-
domly dispersed matrix of CNTs is illustrated in Figures
7(a) and (b). These figures clearly show that crack pro-
pagation trajectory is severely affected by the separation
of the CNTs. The crack appears to select the path of least
resistance between the CNTs for propagation where the
local CNT density is low, whereas crack growth slows
or even comes to a complete stop when the local CNT
density is high (i.e., the distance between the CNTs are
small).
Figure 8 shows the effect of material properties on
crack propagation trajectory. In this model, the elastic
modulus of one of the CNTs (i.e., CNT7) is different
from the others. Different values are assumed for CNT7
and the effect of this variation upon crack propagation is
shown. Here, the elastic modulus of the CNT has a value
of Ef = Ef7 =0.68 TPa (Figures 8(a) and (c)), Ef = 0.68
TPa and Ef7 = 0.2 TPa (Figure 8(d)). In Figures 8(b) and
(e) we assume a hole instead of a CNT7 (Ef7 = 0 TPa) and
the elastic modulus of the rest of CNTs is Ef = 0.68 TPa.
It appears that the stiff CNT repels the crack whereas the
soft CNT or hole leads to attracting the crack.
Table 2 presents the normalized SIF values for the
crack tips corresponding to the different cases as illus-
trated in Figure 8. These values justify the tendency of
the crack to grow along crack tips a and b, when the lo-
cation of the crack and the stiffness of CNT7 is changed.
It is interesting to note that the SIF values and crack
propagation trajectories were also evaluated for the case
when the CNT stiffness was assumed to be Ef = 0.8 TPA
instead of 0.68 TPA. No significant changes were ob-
served which might sugge st that the behavior of the crack,
instead of the separation and arrangement of the CNTs, is
affected by the stiffness mismatch between the CNT and
the matrix rather than the slight deviation of the material
property of the CNT inclusions itself.
In general, the results of this study describe the overall
improvement of the resistance of the PMMA bone ce
ment to fracture that has been observed experimentally

K. POURAKBAR SAFFAR ET AL. 19
Table 2. Normalized SIF values of crack tips for models
shown in Figure 10.
Figure # 10a 10b 10c 10d 10e
KI—tip a 0.454 0.626 0.110 0.111 0.304
KII—tip a 0.272 0.549 0.340 0.340 0.467
KI—tip b 0.457 0.529 0.252 0.252 0.332
KII—tip b 0.272 0.401 0.299 0.299 0.323
[3-5,22] and the significant clinical applications that can
be achieved. However, it must be noted that a number of
simplifying assumptions have been made. For example,
the aligned configuration of the CNTs in the PMMA ma-
trix, as assumed in this study, cannot be easily achieved.
This would require the application of various special
methods and processes which would undoubtedly make a
CNT-PMMA bone cement practically unattractive. Con-
sequently, even though randomly oriented configuration
of CNTs leads to less effective improvement of the me-
chanical properties of the resulting nano-composite [40],
the need to create more efficient cements for joint ar-
throplasty is clinically necessary. Also we have neglected
the relatively weak non- bonding interactions and the van
der Waals forces between the CNTs. However, by choos-
ing a continuum medium for the CNT and the PMMA
matrix, it is assumed that such interactions do not affect
the crack propagation process significantly. Finally, al-
though the imp erfect bonding model between matrix and
inclusion may lead to an increase in the SIF [41], the
classical assumption of perfect bonding can still be con-
sidered valid as long as the CNT aspect ratio is taken to
be large enough [19].
5. Conclusion
This work presents a finite element study of the fracture
behavior of CNT reinforced bone cement through the
adoption of a simplified model. It is found that the ma-
terial properties and the morphological parameters of
CNTs greatly influence the fracture behavior. The results
have shown that the effect of CNTs on the crack could
lead to a decrease in the SIF at the crack tips. However,
this effect is limited to the vicinity of the CNTs. The
crack growth trajectory is also influenced by the nano-
structural arrangement of the components of the com-
posite. More importantly, the results clearly illustrate th at
CNTs could either repel or attract the crack depending
not only on their arrangement and dispersion but also on
the variation of their material properties. The results of
this study indicate that since CNTs behave as a barrier to
crack growth. As a result, the benefits to clinical app lica-
tion are o f s ignificant importan c e .
REFERENCES
[1] G. Lewis and Y. Li, “Dependence of in Vitro Fatigue
Properties of PMMA Bone Cement on the Polydispersity
Index of Its Powder,” Journal of the Mechanical Be-
havior of Biomedical Materials, Vol. 3, No. 1, 2010, pp.
94-101. doi:10.1016/j.jmbbm.2009.05.003
[2] M. Nakai, M. Niinomi, T. Akahori, H. Tsutsumi, S. It-
suno, N. Haraguchi, Y. Itoh, T. Ogasawara, T. Onishi and
T. Shindoh, “Development of Biomedical Porous Tita-
nium Filled with Medical Polymer by in-Situ Polymeriza-
tion of Monomer Solution Infiltrated into Pores,” Journal
of the Mechanical Behavior of Biomedical Materials, Vol.
3, No. 1, 2010, pp. 41-50.
doi:10.1016/j.jmbbm.2009.03.003
[3] R. Ormsby, T. McNally, C. Mitchell and N. Dunne, “In-
corporation of Multiwalled Carbon Nanotubes to Acrylic
Based Bone Cements: Effects on Mechanical and Ther-
mal Properties,” Journal of the Mechanical Behavior of
Biomedical Materials, Vol. 3, No. 2, 2010, pp. 136-145.
doi:10.1016/j.jmbbm.2009.10.002
[4] R. Ormsby, T. McNally, C. Mitchell and N. Dunne, “In-
fluence of Multiwall Carbon Nanotube Functionality and
Loading on Mechanical Properties of PMMA/MWCNT
Bone Cements,” Journal of Materials Science: Materials
in Medicine, Vol. 21, No. 8, 2010, pp. 2287-2292.
doi:10.1007/s10856-009-3960-5
[5] B. Marrs, R. Andrews and D. Pienkowski, “Multiwall
Carbon Nanotubes Enhance the Fatigue Performance of
Physiologically Maintained Methyl Methacrylate-Styrene
Copolymer,” Carbon, Vol. 45, No. 10, 2007, pp. 2098-
2104. doi:10.1016/j.carbon.2007.05.013
[6] G. Lewis, “Properties of Acrylic Bone Cement: State of
the Art Review,” Journal of Biomedical Materials Re-
search, Vol. 38, No., 1997, pp. 155-182.
doi:10.1002/(SICI)1097-4636(199722)38:2<155::AID-JB
M10>3.0.CO;2-C
[7] N. Ihaddade ne, P. Erani, M. Cotifava, F. Tomei, M. Baleani,
M. Viceconti, S. Bouzid and L. Cristofolini, “Fatigue-
Fractured Surfaces of Commercial Bone Cements,” Com-
puter Methods in Biomechanics and Biomedical Engi-
neering, Vol. 10, Suppl. 1, 2007, pp. 157-158.
doi:10.1080/10255840701479610
[8] J. Jeffers, M. Browne, A. Lennon, P. Prendergast and M.
Taylor, “Cement Mantle Fatigue Failure in Total Hip Re-
placement: Experimental and Computational Testing,”
Journal of Biomechanics, Vol. 40, No. 7, 2007, pp. 1523-
1533.
[9] B. A. O. McCormack and P. J. Prendergast, “Microdam-
age Accumulation in the Cement Layer of Hip Replace-
ments under Flexural Loading,” Journal of Biomechanics,
Vol. 32, No. 5, 1999, pp. 467-476.
doi:10.1016/S0021-9290(99)00018-4
[10] M. Jasty, W. J. Maloney, C. R. Bragdon, D. O. O’Conner,
T. Haire and W. H. Harris, “The Initiation of Failure in
Cemented Femoral Components of Hip Antroplasties,”
Journal of Bone and Joint Surgery, Vol. 73B, No. 4, 1991,
pp. 551-558.
[11] R. J. Kane, W. Yue, J. J. Mason and R. K. Roeder, “Im-
proved Fatigue Life of Acrylic Bone Cements Reinforced
with Zirconia Fibers,” Journal of the Mechanical Be-
havior of Biomedical Materials, Vol. 3, No. 7, 2010, pp.

K. POURAKBAR SAFFAR ET AL. 20
504-511. doi:10.1016/j.jmbbm.2010.05.007
[12] K.-T. Lau and D. Hui, “The Revolutionary Creation of
New Advanced Materials-Carbon Nanotube Composites,”
Composites Part B: Engineering, Vol. 33, No. 4, 2002, pp.
263-277. doi:10.1016/S1359-8368(02)00012-4
[13] K. I. Tserpes, P. Papanikos and S. A. Tsirkas, “A Pro-
gressive Fracture Model for Carbon Nanotubes,” Com-
posites: Part B: Engineering, Vol. 37, No. 7-8, 2006, pp.
662-669. doi:10.1016/j.compositesb.2006.02.024
[14] V. A. Buryachenko and A. Roy, “Effective Elastic Moduli
of Nanocomposites with Prescribed Random Orientation
of Nanofibers,” Composites Part B: Engineering, Vol. 36,
No. 5, 2005, pp. 405-416.
doi:10.1016/j.compositesb.2005.01.003
[15] W. Wei, A. Sethuraman, C. Jin, N. A. Monteiro-Riviere
and R. J. Narayan, “Biological Properties of Carbon
Nanotubes,” Journal of Nanoscience and Nanotechnology,
Vol. 7, No. 4-5, 2007, pp. 1284-1297.
doi:10.1166/jnn.2007.655
[16] S. K. Smart, A. I. Cassady, G. Q. Lu and D. J. Martin,
“The Biocompatibility of Carbon Nanotubes,” Carbon,
Vol. 44, No. 6, 2006, pp. 1034-1047.
doi:10.1016/j.carbon.2005.10.011
[17] L. P. Zanello, B. Zhao, H. Hu and R. C. Haddon, “Bone
Cell Proliferation on Carbon Nanotubes,” Nano Letters,
Vol. 6, No. 3, 2006, pp. 562-567. doi:10.1021/nl051861e
[18] B. Zhao, H. Hu, S. K. Mandal and R. C. Haddon, “A
Bone Mimic Based on the Self-Assembly Of Hydroxya-
patite on Chemically Functionalized Single-Walled Car-
bon Nanotubes,” Chemical Materials, Vol. 17, No. 12,
2005, pp. 3235-3241. doi:10.1021/cm0500399
[19] K. PourAkbar Saffar, A. R. Arshi, N. Jamilpour, A.
Raeisi Najafi, G. Rouhi and L. Sudak, “A Cross-Linking
Model for Estimating Young ’s Modulus of Artificial Bone
Tissue Grown on Carbon Nanotube Scaffold,” Journal of
Biomedical Materials Research Part A, Vol. 94A, No. 2,
2010, pp. 594-602. doi:10.1002/jbm.a.32737
[20] A. A. White, S. M. Best and I. A. Kinloch, “Hydroxyapa-
tite-Carbon Nanotube Composites for Biomedical Appli-
cations: A Review,” International Journal of Applied
Ceramic Technology, Vol. 4, No. 1, 2007, pp. 1-13.
doi:10.1111/j.1744-7402.2007.02113.x
[21] Y. Chen, Y. Q. Zhang, T. H. Zhang, C. H. Gan, C. Y.
Zheng and G. Yu, “Carbon Nanotube Reinforced Hy-
droxyapatite Composite Coatings Produced through Laser
Surface Alloying,” Carbon, Vol. 44, No. 1, 2006, pp. 37-
45. doi:10.1016/j.carbon.2005.07.011
[22] B. Marrs, R. Andrews, T. Rantell and D. Pienkowski,
“Augmentation of Acrylic Bone Cement with Multiwall
Carbon Nanotubes,” Journal of Biomedical Materials
Research Part A, Vol. 77, No. 2, 2006, pp. 269-276.
doi:10.1002/jbm.a.30651
[23] A. A. Gawandi, J. M. Whitney, R. B. Brockman and G. P.
Tandon, “Interaction between a Nanofiber and an Arbi-
trarily Oriented Crack,” Journal of Composite Materials,
Vol. 42, No. 1, 2008, pp. 45-68.
[24] A. A. Gawandi, J. M. Whitney, G. P. Tandon and R. B.
Brockman, “Three-Dimensional Analysis of the Interac-
tion between a Matrix Crack and Nanofiber,” Composites
Part B: Engineering, Vol. 40, No. 8, 2009, pp. 698-704.
doi:10.1016/j.compositesb.2009.04.001
[25] C. Atkinson, “On the Stress Intensity Factors Associated
with the Cracks Interacting with an Interface between
Two Elastic Media,” International Journal of Engineer-
ing Science, Vol. 13, No. 5, 1975, pp. 487-504.
doi:10.1016/0020-7225(75)90018-X
[26] A. Romeo and R. A. Ballarini, “A Crack Very Close to a
Bimaterial Interface,” ASME Journal of Applied Me-
chanics, Vol. 62, No. 3, 1995, pp. 614-619.
doi:10.1115/1.2895990
[27] Z. Li and L. Yang, “The Near-Tip Stress Intensity Factor
for a Crack Partially Penetrating an Inclusion,” Journal of
Applied Mechanics, Vol. 71, No. 4, 2002, pp. 465-469.
doi:10.1115/1.1651539
[28] P. S. Steif, “A Semi-Infinite Crack Partially Penetrating a
Circular Inclusion,” Journal of Applied Mechanics, Vol.
54, No. 1, 1987, pp. 87-92. doi:10.1115/1.3172999
[29] A. Raeisi Najafi, A. R. Arshi, M. R. Eslami, S. Fariborz
and M. H. Moeinzadeh, “Haversian Cortical Bone Model
with Many Radial Microcracks: An Elastic Analytic So-
lution,” Medical Engineering and Physics, Vol. 29, No. 6,
2007, pp. 708-717.
doi:10.1016/j.medengphy.2006.08.001
[30] A. Raeisi Najafi, A. R. Arshi, K. PourAkbar Saffar, M. R.
Eslami, S. Fariborz and M. H. Moeinzadeh, “A Fiber-
Ceramic Matrix Composite Material Model for Osteonal
Cortical Bone Micromechanics Fracture: General Solu-
tion of Microcracks Interaction,” Journal of the Mecha-
nical Behavior of Biomedical Materials, Vol. 2, No. 3,
2009, pp. 217-223. doi:10.1016/j.jmbbm.2008.06.003
[31] N. A. Noda, Y. Takase and T. Hamashima, “Generalized
Stress Intensity Factors in the Interaction within a Rec-
tangular Array of Rectangular Inclusions,” Archive of
Applied Mechanics, Vol. 73, No. 5-6, 2003, pp. 311-322.
doi:10.1007/s00419-002-0249-2
[32] Y. Qiao, X. Kong and E. Pan, “Fracture Toughness of
Thermoset Composites Reinforced by Perfectly Bonded
Impenetrable Short Fiber,” Engineering Fracture Me-
chanics, Vol. 71, No. 18, 2004, pp. 2621-2633.
doi:10.1016/j.engfracmech.2004.02.007
[33] M. B. Bush, “The interaction between a Crack and a Par-
ticle Cluster, ” International Journal of Fracture, Vol. 88,
No. 3, 1997, pp. 215-232.
doi:10.1023/A:1007469631883
[34] K. Kim and L. J. Sudak, “Interaction between a Radial
Matrix Crack and a Three-Phase Circular Inclusion with
Imperfect Interface in Plane Elasticity,” International
Journal of Fracture, Vol. 131, No. 2, 2005, pp. 155-172.
doi:10.1007/s10704-004-3636-6
[35] P. G. Park and L. J. Sudak, “Stress Intensity Factor for an
Interphase Crack Interacting with Two Imperfect Inter-
faces,” Mathematics and Mechanics of Solids, Vol. 15,
No. 3, 2010, pp. 353-367.
doi:10.1177/1081286508101512
[36] G. D. Seidel and D. C. Lagoudas, “Micromechanical
Analysis of the Effective Elastic Properties of Carbon

K. POURAKBAR SAFFAR ET AL. 21
Nanotube Reinforced Composites,” Mechanics of Mate-
rials, Vol. 38, No. 8-10, 2006, pp. 884-907.
doi:10.1016/j.mechmat.2005.06.029
[37] J. P. Lu, “Elastic Properties of Carbon Nanotubes and
Nanoropes,” Physical Review Letters, Vol. 79, No. 7,
1997, pp. 1297-1300. doi:10.1103/PhysRevLett.79.1297
[38] D. Swenson, M. James and B. Hardeman, “CASCA: A
Simple 2-D Mesh Generator, Version 1. 4 User’s Guide,”
Kansas State University, Manhattan, Kansas, 1997.
[39] P. Wawrzynek and A. Ingraffea, “FRANC2D: A Two-
Dimensional Crack Propagation Simulator,” Version 2. 7
User’s Guide, NASA CR 4572, 1994.
[40] X.-L. Xie, Y.-W. Mai and X.-P. Zhou, “Dispersion and
Alignment of Carbon Nanotubes in Polymer Matrix: A
Review,” Materials Science and Engineering: R: Reports,
Vol. 49, No. 4, 2005, pp. 89-112.
doi:10.1016/j.mser.2005.04.002
[41] Y. Liu, C. Q. Ru, P. Schiavone and A. Mioduchowski,
“New Phenomena Concerning the Effect of Imperfect
Bonding on Radial Matrix Cracking in Fiber Compos-
ites,” International Journal of Engineering Science, Vol.
39, No. 18, 2001, pp. 2033-2050.
doi:10.1016/S0020-7225(01)00049-0.