Mechanics of Materials 41 (2009) 279–292
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Mechanics of Materials
journal homepage: www.elsevier.com/locate/mechmat
Curved-fiber pull-out model for nanocomposites. Part 1: Bonded
stage formulation
Xinyu Chen a, Irene J. Beyerlein c, L. Catherine Brinson a,b,*
a
b
c
Department of Mechanical Engineering, Northwestern University, 2145 Sheridan Road, Evanston, IL 60208, USA
Department of Materials Science and Engineering, Northwestern University, 2145 Sheridan Road, Evanston, IL 60208, USA
Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA
a r t i c l e
i n f o
Article history:
Received 25 January 2008
Received in revised form 27 November 2008
a b s t r a c t
This is the first part of two papers in which an analytical curved-fiber pull-out model for
nanocomposites is proposed. In nanotube-reinforced polymer composites, nanotubes are
typically curved and entangled, a reinforcement morphology that will greatly impact the
thermomechanical properties of the material. As the first step to explicitly take into
account nanotube curvature and study its effect on nanocomposite mechanical properties,
we develop a pull-out model in which the fiber has constant curvature. The model includes
the entire pull-out process, namely the bonded, debonding, and sliding stages. In this first
paper we formulate the bonded stage based on classic shear lag model assumptions and
develop a 3D finite element model to verify assumptions. The results from a parametric
study indicate that fibers with more curvature and longer embedded length need higher
debond initiation force. The finite element results and analytical results show agreement
both qualitatively and quantitatively.
Ó 2008 Elsevier Ltd. All rights reserved.
1. Introduction
1.1. Problem statement
Since the emergence of nanocomposites, intensive work
in synthesis, characterization and modeling has provided
better understanding of the material’s mechanical performance (Ajayan et al., 2003; Andrews et al., 2002; Breuer
and Sundararaj, 2004; Buryachenko et al., 2005; Coleman
et al., 2006; Fisher and Brinson, 2006; Valavala and Odegard, 2005). For instance, it has been consistently observed
that small amounts of nanotubes can increase stiffness
above that of the base polymer (Chang et al., 2005; Coleman et al., 2003; Goh et al., 2003; Liu et al., 2004; Qian
et al., 2000; Velasco-Santos et al., 2003; Zeng et al.,
2004). The effect of nanoparticles on composite toughness
has also been studied (Andrews and Weisenberger, 2004).
* Corresponding author. Address: Department of Mechanical Engineering, Northwestern University, 2145 Sheridan Road, Evanston, IL 60208,
USA. Tel.: +1 847 467 2347; fax: +1 847 510 0540.
E-mail address: cbrinson@northwestern.edu (L.C. Brinson).
0167-6636/$ - see front matter Ó 2008 Elsevier Ltd. All rights reserved.
doi:10.1016/j.mechmat.2008.12.004
Significant improvements in toughness have been observed in some spherical nanoparticle systems (Cotterell
et al., 2007; Naous et al., 2006; Ragosta et al., 2005; Xu
et al., 2008). For example Ash et al. (2002) demonstrated
78% increase in ductility of PMMA with addition of 5 wt%
of nano-alumina (39 nm diameter) particles. In contrast,
the results for nanoplate and nanotube reinforced polymers have varied a great deal. Some researchers (Moniruzzaman et al., 2006; Yasmin et al., 2006; Zheng et al.,
2004) observed substantial losses in ductility and hence
toughness, while others report minor improvements in
toughness (Gojny et al., 2004, 2005; Ma et al., 2007). In a
few cases, significant improvement on toughness with
tube-based reinforcement has been observed (Blond
et al., 2006; Chen et al., 2005; Yang et al., 2007). Fiedler
et al. (2006) measured a 45% increase in fracture toughness
of CNT/epoxy composites with 0.3% of amino-functionalised double-walled carbon nanotubes. Dondero and Gorga
(2006) reported with 0.25 wt% MWNT polypropylene
matrix’s toughness increases 32%.
Given the inherently large strain capability of nanotubes, it should be possible to consistently design a
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X. Chen et al. / Mechanics of Materials 41 (2009) 279–292
nanotube composite with significantly improved fracture
toughness. Limited success to date and the wide range
of experimentally observed results calls for a better
understanding of the underlying deformation mechanisms governing nanocomposite fracture. Such understanding is critical for design of the nanocomposite
microstructure (nanotube–polymer interface, nanotube
volume fraction, etc) for enhanced toughness.
One important toughness mechanism is nanotube pullout. As in conventional fiber pull-out, there are three
stages in nanotube pull-out. In the first stage, called the
bonded stage, the nanotube and the matrix are wellbonded. As the pull-out force increases to a certain threshold value, the debonding stage begins. During debonding,
part of the nanotube moves along the debonded interface
resisted by a friction force, while the rest of the nanotube
stays well bonded to the matrix. When debonding extends
to the entire interface, sliding occurs. In this final stage, the
entire nanotube slides through the matrix resisted by frictional forces.
The pull-out problem for nanotube-reinforced composites has been studied experimentally, analytically, and
numerically. Individual nanotube pull-out tests have been
performed using an atomic force microscopy (AFM) stages
to access the interfacial strength. The Wagner group (Barber et al., 2003. 2004, 2006; Cooper et al., 2002; Nuriel
et al., 2005) successfully traced the pull-out force and
nanotube locations to obtain the force–displacement curve
and they further were able to calculate average interfacial
shear stress and fracture energy for certain nanotube–
polymer interfaces. Analytically, continuum mechanics
models for conventional fiber/polymer interface, such as
the Kelly and Tyson model, and models based on local density approximation and classical elastic shell theory, have
been extended to describe nanotube/polymer interfaces
(Gao and Li, 2005; Lau, 2003; Wagner, 2002). Xiao and Liao
(2004) developed a nanotube pull-out model for the sliding
stage by incorporating nanotubes’ nonlinear elastic property and found the nonlinearity has a great impact on the
interfacial shear stress distribution. Other researchers
(Frankland et al., 2002; Frankland and Harik, 2003; Gou
et al., 2004; Liao and Li, 2001; Lordi and Yao, 2000; Wong
et al., 2003) considered the physical structure of nanotubes
and polymer chains at the nanoscale and applied molecular mechanics and molecular dynamics (MD) calculations
to the problem of pull-out, elucidating the stress transfer
mechanism as a function of the nanotube/polymer interface properties. An average interfacial shear stress calculated from MD simulation shows that bonded or nonbonded interactions at the interface can lead to effective
stress transfer from polymer matrix to nanotubes (Frankland et al., 2002; Gou et al., 2004; Liao and Li, 2001; Wong
et al., 2003). Although MD can describe interactions at
atomic levels through suitable potential models, it is limited by length and time scales due to the small time steps
required. The statistical nature of MD calculations requires
the MD simulation to run for a sufficiently long time to
perform enough sampling for physical properties. These
limitations make continuum mechanics approaches more
favorable for analyses at length scales in the micron
range.
All the continuum mechanics-based and molecular
mechanics-based models above only consider nanotubes
which are straight and aligned. However, in nanoreinforced polymers, nanotubes are typically curved and
entangled in-situ as shown in Fig. 1. The fine, white, hairlike filaments in Fig. 1 are the nanotubes.
The curved fiber morphology will greatly impact thermomechanical and fracture properties of the composite
systems. While the effects of nanotube curvature on stiffness have been addressed (Bradshaw et al., 2003; Fisher
et al., 2002, 2003), its influence on ductility and fracture
toughness has yet to be examined at any length scale.
For traditional (larger scale) fiber composites, the effect
of reinforcement morphology has been explored in depth.
While a weak interface can enhance toughness, it also reduces strength. A change in the morphology of the fiber
coupled with the weak interface can, however, lead to both
high toughness and high strength. One example is the so
called bone-shaped-short-fiber composites (Beyerlein
et al., 2001; Shuster et al., 1996; Zhu et al., 1999, 2001).
Composites reinforced by bone-shaped-short fibers are
able to transfer stress effectively through the enlarged fiber
ends while still providing toughness enhancements
through the weak interface. Similarly, we propose that
nanocomposites with appropriately designed interfaces
and morphologies may ultimately lead to composites with
improved stiffness, strength and toughness.
For predictive capability and design, it will be important
to account for and understand the effects of nanotube
curvature and entanglement on the critical properties of
nanocomposites, such as toughness and strength.
As a first step, in this two-part series, the curvature effect is added to a shear-lag-based model (Lawrence, 1972)
to study nanotube pull-out. Shear lag modeling is a popular and successful scheme to address fiber/matrix interface problems in conventional composites. This article is
the first part of the series which presents the formulation
for the bonded stage. It is structured as follows. First a
brief review of conventional straight fiber pull-out modeling is given. Then, a 2D analytical model for single
curved-fiber pull-out is derived. A 3D finite element simulation model is built to check some of the simplifying
assumptions made in the formulation. With the analytical
model, we examine the influence of fiber curvature on the
initial portion of the force–displacement curve when the
fiber and matrix are still bonded. Finite element simulation results are then compared with those from the
analytical formulation.
1.2. Review: different straight fiber pull-out models
Since straight fibers are prevalent in conventional fiberreinforced composites, research in modeling single straight
fiber pull-out has been extensively carried out. Fig. 2
shows a concentric cylinder model commonly used as a
representative volume element of fiber composite models
or in single fiber pull-out analyses. Cox (1952) proposed
the original shear lag model based on linear elasticity,
which involves three inherent assumptions, namely (1)
shear stress is a function of axial displacement; (2) the
fiber and matrix stresses and displacements in the axial
X. Chen et al. / Mechanics of Materials 41 (2009) 279–292
281
Fig. 1. SEM photograph of curved single walled carbon nanotubes in PMMA (functionalized tubes) (Ramanathan et al., 2005). The fine, white, hair-like
filaments are the nanotubes.
Fig. 2. Commonly used concentric cylinder geometry for fiber composites
and single fiber pull-out problems with cylindrical coordinate system.
The fiber is the inner cylinder and the matrix is the outer cylinder.
direction are independent of radial coordinates; (3) stress
in the axial direction is dominant over stress in the other
two directions. These assumptions are explained in detail
in Gao and Li (2005), Nairn (1997). Since then, the shear
lag idea has been widely applied to straight fiber pull-out
analysis for different composite systems and has been further developed with various degrees of approximation
(Cox 1952, 1990; Gao and Li, 2005; Gao et al., 1988; Hsueh,
1992a; Kerans and Parthasarathy, 1991; Kharrat et al.,
2006; Kim et al., 2004; Kim and Mai, 1998; Nairn, 1997;
Nairn and Wagner, 1996; Rosen, 1964; Tsai and Kim,
1996; Wu and Davies, 2005; Wu and Yu, 1994). For example, some researchers assumed zero radial displacement
and uniform matrix deformation confined in the cylinder
geometry (Hsueh, 1988, 1990; Takaku and Arridge 1973).
Gao and Li (2005) developed a shear lag model for carbon
nanotube/polymer composites by modeling a capped
nanotube as an effective fiber based on molecular structure
mechanics. They also modified the ‘free ends’ boundary
conditions in the original Cox model to represent a fully
embedded nanotube. They found that the large aspect ratio
of a nanotube can increase interfacial stress transfer, and
thus improve the reinforcing effects of nanotubes. Their
paper is the only effort to apply the shear lag model to
nanocomposites to date.
In addition to shear lag assumptions, other theoretical
models based on linear elasticity have also been developed
to study stress transfer in straight-fiber-reinforced composites under various assumptions (Hutchinson and Jensen, 1990; Marshall, 1992; McCartney, 1989; Mumm and
Faber, 1995; Wu et al., 2000). Finite element analyses have
also been conducted for stress distributions along the fiber
axis (Faber et al., 1986; Grande et al., 1988). Some
researchers further modeled stick-slip sliding features in
a dynamic fiber pull-out process (Sridhar et al., 2003; Tsai
and Kim, 1996).
The shear lag model has proven to provide good estimates for interfacial stress transfer. Due to its mathematical simplicity, it is widely used in straight fiber reinforced
composites. Following this background, our pull-out analysis for curved fiber reinforced composites will be built
upon a shear lag model. As in the straight fiber model,
which is axisymmetric and essentially 2D, our analytical
model for curved-fiber pull-out is also 2D. Although the
motivation of this work lies in the observed curvature of
nanotubes embedded in polymer matrix, as shown in
Fig. 1, our analytical model can also be applied at conventional length scales.
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X. Chen et al. / Mechanics of Materials 41 (2009) 279–292
2. Single curved-fiber pull-out model for bonded stage
In this section, the analytical derivation for single
curved-fiber pull-out analysis at the bonded stage is first
presented. Then a 3D finite element model is constructed
to test several assumptions made in our analysis.
As mentioned, fiber pull-out includes three stages. We
are interested in connecting the three stages to obtain a
force–displacement curve, which later can be used directly
or indirectly as a bridging law to predict composite toughness and fracture behavior. Although there are many analytical models based on shear lag approach, most of them focus
only on one stage, either the bonded stage or the debonding
stage. As one of the limited numbers of paper dealing with
more than one stage, Lawrence (1972) modeled the bonded
stage based on a shear lag approach and connected it with
the debonding stage and further identified the existence of
progressive debonding and catastrophic debonding. Hsueh
also has a series of papers applying shear lag to both bonded
and debonding stages and the two stages were connected via
a debonding criterion (Hsueh 1988, 1990, 1992a, 1992b). In
this work, we choose to build upon the Lawrence model with
the following major modifications: (1) As all shear lag models to date, the Lawrence model is for a straight fiber. Therefore, in our analysis it is modified to account for fiber
curvature. (2) Lawrence assumed a ‘free end’ at the fiber
embedded end, which would not necessarily be accurate
for nanotubes entangled together as in Fig. 1. Therefore
our model denotes a parameter to account for stress due to
this entanglement. (3) In Lawrence’s work, in the debonding
stage the debonded part of the fiber was resisted by a constant friction stress. In our work, we consider two friction
models, a constant friction model and a Coulomb friction
model. The last modification is developed in the second part
of this series, which focuses on the debonding and sliding
stages and the pull-out curve of the entire pull-out process.
2.1. Analytical derivation of single curved-fiber model for
bonded stage
Based on the simple shear lag model, Lawrence (1972)
analyzed bonded and debonding stages during straight fiber pull-out and identified the possibility of progressive
debonding. From his model, we have newly derived the
force–displacement relationship for a straight fiber in the
bonded stage subjected to the modification (2) above, and
present it in Appendix I. It is important to have this solution
in hand as a basis for measuring the curvature effect.
Fig. 3(a) shows the model geometry of a curved fiber
embedded in a matrix material. Small strain conditions in
the fiber and matrix are assumed so that this problem falls
into the scope of linear elasticity and hence the shear lag
model can be applied. This small-strain assumption should
be sufficiently accurate throughout the pull-out analysis,
as the fiber and matrix in our model will eventually debond
from one another and most of the strain will be accommodated by the interface. Both fiber and matrix are isotropic
and linearly elastic. The Poisson’s effect is neglected to simplify the calculation. As a first step to account for fibers with
general curvature geometries, our current fiber is assumed
to have a constant radius of curvature R. The fiber has a cir-
cular cross-section with radius rf. It is noted that the analysis
would in general allow for noncircular cross-sections, but
we consider circular only here because of our focus on nanotube reinforcement. Also as shown in Fig. 3(a), here the fiber
is assumed to exist normal to the composite surface. Fiber
inclination effects will be considered in another paper.
Fig. 3 shows the three stages for curved-fiber pull-out. This
paper derives equations for stage I. As mentioned earlier,
stages II and III are derived in the companion paper.
The 2D curvilinear coordinate system used in our current analysis is shown in Fig. 3(a). s is the direction along
fiber (tangential direction) and r is always perpendicular
to s (radial direction). The variation in the hoop direction
for both matrix and fiber is neglected. At the fiber embedded end, s = 0. The angle characterizing the fiber geometry
is a, and aR, denoted as L later, equals the original fiber
embedded length. Our current model is valid for any a value between 0o and 180o. s0 is the fiber length outside the
matrix prior to application of the load. Pf is the pull-out
force at the fiber end. In stage I, i.e., the bonded stage,
the fiber and matrix are well-bonded. In the current model,
the only interaction between fiber and matrix at this stage
is through the interfacial shear stress si. Radial compression is not considered here although it is taken into account for the debonding and sliding stages in part II of
this series, where radial compression is more significant.
All equations in the following derivation are in normalized
form to remove any unnecessary dependencies of the form
of the solution on parameters. The normalization factor for
length is fiber radius rf and for stress and moduli it is the
fiber Young’s modulus Ef. Accordingly, that for force is
p(rf)2Ef. The asterisks indicate normalized values.
Fig. 4 considers the stress equilibrium of a small differential matrix element next to the embedded fiber in the 2D
s–r coordinates. sm
rs is the shear stress at r in the s-direction,
si is the shear stress at rf, i.e., interfacial shear stress, in the
s-direction. According to equilibrium in the fiber direction,
we have for the matrix,
m
si ¼ sm
rs r ; ð1 r r Þ;
ð1Þ
where s and s are the normalized shear stresses at r*
and at interface, respectively, and the radial position in
the matrix is normalized by rf. rm* is the normalized ‘imaginary’ matrix radius. rm* is called imaginary because there
are no boundary conditions enforced at the outer boundaries of the matrix.
From the linear elastic constitutive law, matrix shear
strain is
m
rs
cmrs ¼
sm
rs
Gm
¼
i
si
Gm r ;
ð2Þ
where Gm is matrix shear modulus normalized by Ef .
The strain–displacement relation in curvilinear coordinates1 gives:
cmrs ¼
R @um
@um
um
r
þ s þ s ;
R r @s
@r
R r
ð3Þ
where u stands for displacement.
1
The detailed derivation of elasticity equations in the current 2D
curvilinear system can be found in Appendix II.
X. Chen et al. / Mechanics of Materials 41 (2009) 279–292
283
Fig. 3. Three stages in pull-out of a single curved fiber. The figure for stage I also illustrates the model geometry. The 2D curved fiber has a constant
curvature R and circular cross-section of radius rf in curvilinear coordinate system s and r. s0 is the free length of the fiber initially not embedded in the
matrix.
m
s
ð1 R Þr us ð1Þ
:
um
þ i ln s ðr Þ ¼ ðR r Þ
R 1 Gm R
ðr R Þ
ð9Þ
Rearranging, we obtain the interfacial shear stress expressed as a linear combination of axial matrix displacement at r ¼ rm and r* = 1in a similar format to the
straight fiber pull-out model as shown in Eq. (7A).
si ¼
m m
us ðr Þ um
s ð1Þ
:
m
Þr
R r m R 1
ln ð1R
m
Gm R
ðr
Fig. 4. Stress equilibrium of a representative matrix segment in the
bonded stage. sm
rs is the shear stress at r in the s-direction, si is the shear
stress at rf, i.e., interfacial shear stress, in the s-direction.
Considering equilibrium of a fiber segment (see Fig. 5), we
obtain,
Consistent implicitly with shear lag assumption (1), the
variation of radial displacement in matrix along s direction
is considered negligible, i.e.,
pðrfs þ drfs Þ cos
@um
r
0
@s
ds ¼ R da
Substituting Eqs. (3) and (4) into the constitutive law Eq.
(2), we obtain the governing equation for matrix.
s
@um
um
s
þ s ¼ i :
@r R r
Gm r
ð5Þ
da
da
rfs p cos
þ si 2pds ¼ 0
2
2
ð11Þ
where
ð4Þ
ð10Þ
R Þ
ð12Þ
Note that in Fig. 5, and in the above, we have adopted the
shear lag assumption that the axial stress in the fiber is
uniform across the cross-section (or that the shear modulus of the fiber is infinite relative to that for the matrix).
This assumption applies to most fiber–polymer matrix
systems.
Eq. (5) is treated as an ordinary differential equation and
its solution is
um
ðr
Þ
¼
ðR
r
Þ
gðsÞ s
si
Gm R
ln
r R
;
r
ð6Þ
where g(s) is an arbitrary function to be defined by boundary conditions.
Considering Eq. (6) at the fiber surface, r* = 1,
um
ð1Þ
¼
ðR
1Þ
gðsÞ s
si
Gm R
lnð1 R Þ :
ð7Þ
Thus,
gðsÞ ¼
s
um
s ð1Þ
þ i lnð1 R Þ:
R 1 Gm R
ð8Þ
Substituting Eq. (8) back to Eq. (6), we obtain the solution
for the matrix displacement:
Fig. 5. Stress equilibrium of a differential fiber element in the bonded
stage.
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X. Chen et al. / Mechanics of Materials 41 (2009) 279–292
As da ? 0, we obtain the governing equation for the fiber in the bonded stage
f
s
dr
¼ 2si :
ds
ð13Þ
Interestingly Eq. (13) has the same form for both straight
and curved fibers. As we shall see in the sequel, the effect
of curvature is actually introduced through si .
Combining Eq. (13) with Eq. (10) results in
f
s
dr
¼
ds
Gm R
2 ð1R
m
Þr
ln ðrm R
Þ
m m
us ðr Þ um
s ð1Þ
:
R r m R 1
To obtain the governing equation for
derivative is taken,
2
d
rfs
2
ds
¼ 2
Gm R
ln
ð1R Þr m
ðr m R Þ
rfs , first a second
m
m
1
@um
1 dus ð1Þ
s ðr Þ
;
R r m
@s
R 1 ds
ð15Þ
where um
s ð1Þ ¼
Next recall the following strain–displacement relationships in the fiber and in the matrix
m
R
@um
1
s ðr Þ
um ðr m Þ
@s
R r m r
R rm
f
R dus
1
¼ uf R 1 ds R 1 r
efs
f
s
r
!
pffiffiffiffiffi pffiffiffiffiffi
Pf
pffiffiffiffiffi r0 cothð T L Þ sinhð T s Þ
¼
sinhð T L Þ
pffiffiffiffiffi
þ r0 coshð T s Þ
ð21Þ
and likewise for the interfacial shear stress
s
i
ð14Þ
ufs .
ems ðrm Þ ¼
Therefore, we have for the fiber stress as a function of s*
"
!
pffiffiffiffiffi Pf
1
pffiffiffiffiffi r0 cothð T L Þ
¼ 2
sinhð T L Þ
pffiffiffiffiffi
pffiffiffiffiffi
pffiffiffiffiffi
pffiffiffiffiffi i
T coshð T s Þ þ r0 T sinhð T s Þ :
ð22Þ
f
s
The distribution of r and si along the fiber s* calculated
from Eqs. (21), (22) at a given pull-out force Pf is shown
in Fig. 6(a) and (b), respectively, and the corresponding
parameters are listed in Table 1. As can be seen, both stresses are largest at the pulled end and smallest at the fiber
embedded end, which implies debonding will start from
the pulled end. The interfacial shear stress at the pulled
end reduces as the curvature increases as seen from the
ð16Þ
ð17Þ
Inserting into Eq. (15) we obtain
"
#
2
m
d rfs Gm R
ems ðrm Þ um
efs
ufr r ðr Þ
¼
2
þ
:
m
2
Þr
R
R ðR r m Þ R R ðR 1Þ
ln ð1R
ds
m
ðr
R Þ
ð18Þ
m
As in the straight fiber pull-out model, em
Þ is regarded
s ðr
as a virtual matrix strain as if no fiber exists, i.e.,
f
r
m
ems ðrm Þ e1
Þ ¼ E rsm2 (Cox 1952; Lawrence 1972; Nairn
s ðr
m
m
1997). We further assume that um
Þ and ufr are small.
r ðr
Note that these assumptions are not typical shear lag
assumptions as these two terms do not appear in the
straight fiber case. Finally with these assumptions Eq.
(18) gives the following governing equation for the axial fiber stress:
d
2
rfs
2
ds
¼ T rfs ;
whereT ¼
2Gm
ð1R Þr m
ln ðrm R Þ
ð19Þ
1
1 m2 :
Em r
ð20Þ
Note that T* contains the curvature effect.
To solve Eq. (19) we apply the following two boundary
conditions:
(1)
rfs ðs ¼ 0Þ ¼ r0 , where r0 denotes the stress at the
fiber embedded end due to its entanglement with
other nanotubes.
(2) rfs ðs ¼ L Þ ¼ rpull , i.e., stress at the pulled end
required to balance the applied stress rpull, which
equals the pull-out force Pf divided by fiber crosssection area.
Fig. 6. (a) Normalized fiber axial stress and (b) normalized interfacial
shear stress distribution along normalized fiber axial position in bonded
stage for different fiber curvatures. Note that in (a) at
s* = 0,rfs ¼ r0 ¼ 1E 9 from Table 1, which is essentially zero on the
scale of these results.
Table 1
Parameters for the fiber and matrix used in Fig. 6 (Em : normalized matrix
Young’s modulus; Gm : normalized matrix shear modulus; L* normalized
fiber embedded length; rm*: normalized imaginary matrix radius; r0 :
normalized fiber embedded end stress; P f : normalized pull-out force).
Em
Gm
L*
rm*
r0
P f
1E2
5E3
33.3
20
1E9
2.5E3
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X. Chen et al. / Mechanics of Materials 41 (2009) 279–292
three curves with different values of R*. This curvature effect is examined in more detail in Section 3.1.
From the pull-out Pf–d curves, where d is the fiber displacement, we can begin to see how curvature would affect
composite toughness. The Pf–d curve can serve as a bridging law in modeling crack propagation.
The fiber displacement d*(I) (where superscript (I) denotes stage (I) is composed of two parts: the elongation of
the embedded fiber and that of the original extruded part.
dðIÞ ¼
Z
L
T
0
1
pffiffiffiffiffi
efs ds þ Pf s0 ¼ pffiffiffiffiffi Pf cothð T L Þ r
0
pffiffiffiffiffi þ r0 coth
sinhð T L Þ
!
pffiffiffiffiffi þ P f s0
T L
P f
pffiffiffiffiffi
sinhð T L Þ
ð23Þ
This displacement for the bonded stage will be used in the
companion paper, where the full pull-out curve for
bonded, debonding, and sliding stages will be determined
and impact on toughness examined.
At s ¼ L , Eq. (22) yields
simax ¼ !
pffiffiffiffiffi
1 pffiffiffiffiffi r0
pffiffiffiffiffi ;
T Pf cothð T L Þ 2
sinhð T L Þ
ð24Þ
The negative sign indicates that the shear stress acts in the
opposite direction from what is illustrated in Fig. 5, which
is physically reasonable.
The peak interfacial shear stress si increases with pullout force. As it grows to a certain value denoted as ss , debonding begins. The formulations for debonding and sliding
stages are presented in the companion paper.
To further elucidate the curvature effect we compare
our results to that for a straight fiber presented below. A
detailed derivation can be found in Appendix I. In this solution, x denotes the fiber axial direction.
r
T ¼
2Gm
ln
ð1R Þr m
ðr m R Þ
1
1
ð30Þ
Em r m2
Note that when R* goes to infinity, T* becomes Q*, and
consequently Eqs. (21)–(24) converge to the results for a
straight fiber, Eqs. (25)–(28).
2.2. 3D Finite element model
In our single curved-fiber pull-out analysis for the
bonded stage, several assumptions have been made,
including the basic shear lag assumption and negligible
matrix radial displacement. To check the validity of these
assumptions, a symmetric 3D finite element model is constructed and analyzed. The commercial software, I-DEAS,
was used to construct the finite element model and the
commercial finite element package, ABAQUS, was used to
perform the finite element simulation.
The fiber and matrix are constructed as one body and
meshed with 10-node quadratic tetrahedron elements in
I-DEAS. The pull-out simulation at the bonded stage is then
performed in ABAQUS. The properties of the fiber and maTable 2
Properties of fiber and matrix used in the 3D FE model (R, radius of fiber
curvature; rf, fiber radius; L, fiber embedded length; Ef, fiber Young’s
modulus; Em, matrix Young’s modulus).
R (m)
rf (m)
L (m)
Ef (Pa)
Em (Pa)
3.03E2
1.5E3
4.99E2
1E12
1E9
!
pffiffiffiffiffiffi pffiffiffiffiffiffi
P
pffiffiffiffiffiffi r0 cothð Q L Þ sinhð Q x Þ
¼
sinhð Q L Þ
pffiffiffiffiffiffi
þ r0 coshð Q x Þ
ð25Þ
"
!
pffiffiffiffiffiffi 1
P
pffiffiffiffiffiffi r0 cothð Q L Þ
¼
2
sinhð Q L Þ
pffiffiffiffiffiffi
pffiffiffiffiffiffi
pffiffiffiffiffiffi
pffiffiffiffiffiffi i
ð26Þ
Q coshð Q x Þ þ r0 Q sinhð Q x Þ
f
x
si
pffiffiffiffiffiffi
1
P
pffiffiffiffiffiffi dðIÞ ¼ pffiffiffiffiffiffi ðP cothð Q L Þ Q
sinhð Q L Þ
!
pffiffiffiffiffiffi r0
pffiffiffiffiffiffi þ r0 cothð Q L Þ þ P l0
sinhð Q L Þ
si max ¼ ð27Þ
!
pffiffiffiffiffiffi
1 pffiffiffiffiffiffi r0
pffiffiffiffiffiffi
Q P cothð Q L Þ 2
sinhð Q L Þ
ð28Þ
where
Q ¼
2Gm
1
1
r m2 Em
ln r m
ð29Þ
These expressions are of the same form as the corresponding expressions for a curved fiber with Q* replacing T*.
Recalling our parameter T* in Eq. (20),
Fig. 7. 3D symmetric FE model in the bonded stage with applied
boundary conditions. Front face is the symmetric plane. The red outline
shows the nanotube-matrix interface on the symmetric plane. The orange
colored outline and points represent following boundary conditions: the
matrix left surface is fully fixed, the matrix top and bottom surfaces can
only move in the 1-direction, and fiber nodes at the pulled end are given a
uniform displacement in the 1-direction. The blue colored points represent the nodes on the symmetric plane are symmetrically constrained.
(For interpretation of the references to color in this figure legend, the
reader is referred to the web version of this paper.)
286
X. Chen et al. / Mechanics of Materials 41 (2009) 279–292
trix used in analysis are listed in Table 2. These are representative of a nanotube–polymer matrix composite system. Applied boundary conditions are as illustrated in
Fig. 7: the matrix left surface is fully fixed, the matrix top
and bottom surfaces can only move in the 1-direction,
nodes on symmetric plane are properly constrained, and fiber nodes at the pulled end are given a uniform displacement in the 1-direction.
3. Results and discussion
This section presents numerical results for curved-fiber
pull-out, first for analytical model and then for the finite
element model.
3.1. Parametric study
In this section, a parametric study is performed to
examine the effects of different factors on the curved-fiber
pull-out behavior in the bonded stage. All parameters studied are in normalized form and therefore the normalization
factors, such as the fiber radius rf and fiber Young’s modulus Ef, do not need to be considered. The following parameters are chosen to represent a typical polymer
nanocomposite:
Em ¼ 1E 2; Gm ¼ 5E 3; ss ¼ 3:5e 5;
m
r ¼ 20: The parameters of interest are fiber radius of curvature R*, fiber length L*, and fiber axial stress from entangled fibers at the embedded end r0 : The base values of
these parameters are taken as 70, 33, 1e9, respectively,
and then varied individually within a physically reasonable
range for the simulations. Note that L* is taken as a smaller
value than the typical nanotube length, which can be from
several hundreds up to several thousands, because our
analysis focuses on one curved segment of a nanotube.
When only the bonded stage is considered, the Pf–d
curve ends when the critical interfacial shear stress is
reached, and debonding starts. Fig. 8(a) and (b) show the
effects of changing R* and L*, respectively, on the pull-out
stress to initiate debonding. The displacements shown
are purely from fiber elongation because while bonded,
no relative displacement between fiber and matrix is allowed. Increasing L*, and to a lesser extent the fiber curvature, both increase the debond initiation force with a
straight fiber (R* = infinity) requires the smallest pull-out
force to initiate debonding. Due to the same value applied
for the debonding parameter ss in both straight and curved
fiber cases, we can infer that given the same pull-out force
the interfacial shear stress in the curved fibers is smaller
than that in the straight fiber. This result implies that the
interfacial shear stresses build up slower in curved fibers
than those in straight fibers. This would be a nice quality
for composites with curved fibers since it could lead to enhanced toughness. In Fig. 8(c) the fiber embedded end
stress r0 is changed from 1e9 to 1e4. Increasing this
stress leads to a higher debond initiation force because of
the larger end stress to overcome. The plot also shows a
nonzero pull-out distance under zero pull-out force, which
is more obvious in the large r0 case. This offset is due to
fiber elongation from the residual stress r0 . Ideally the
parameter r0 should start from zero when no pull-out
Fig. 8. Effects on normalized Pf–d curves by changing following parameters from their initial values: R* = 70, L* = 33, r0 ¼ 1e 9. (a) Radius of
curvature R*, (b) fiber length L*, (c) fiber embedded end stress r0 .
loads are applied and increase with pull-out rather than
the constant value assumed here. However, the effect of
r0 within the ranges examined is relatively small. In fact,
from Fig. 8, both R* and r0 have little effect in the bonded
stage, and L* has the most significant impact on the pullout curve. Longer curved segments lead to higher debond
initiation force, which implies potential toughness
improvement of the nanocomposites as desired.
r0 is treated as a material parameter in the current formulation because it describes the axial stress from both
the bonded matrix and the surrounding entangled nanotubes. As mentioned above, this value should change during pull-out rather than a constant value. For a straight
fiber, Hsueh et al. (1997) has obtained an analytical solution for the embedded end axial stress as a function of
applied load, matrix and fiber radius, Poisson’s ratio,
Young’s modulus, fiber length, and the distance from fiber
embedded end to composite surface. Hsueh’s ‘‘imaginary
X. Chen et al. / Mechanics of Materials 41 (2009) 279–292
fiber” technique could be applied to the curved-fiber model but is not considered here.
3.2. Check of analytical assumptions through FE simulation
The 3D finite element results for the deformation and
stress field for bonded stage for the pull-out distance of
100 lm are, respectively, shown in Fig. 9(a) and (b). The
stress field in the fiber is not uniform along the hoop direction. The lower surface of the fiber displays a much higher
stress than of the upper surface. After sampling the displacement at several points near the fiber/matrix interface, it is
found that the radial displacement um
r is not negligible com-
287
m
m
pared with the axial displacement um
s and the ratio ur =us
varies from 0.26 to 7.74. The nodes nearer to the pulled
end tend to have a smaller radial displacement over axial
displacement ratio than other nodes. For the straight fiber
case, it is found that the ratio remains small and ranges from
1e5 to 1e2. The curved fiber case illustrated here is an extreme case with a large 90 o curve, and for smaller curvatures
the small um
r assumption becomes more valid. Therefore
relaxing the assumption for the matrix radial displacement
will improve the analytical model.
In Eq. (4), it is assumed that the variation of matrix radial displacement along the fiber axial direction @um
r =@s is
much smaller than the variation of matrix axial displacement along the radial direction @um
s =@r: To check this, several sets of data points are extracted near the interface. The
m
ratio of @um
r =@s to @us =@r varies from 0.0331 to 0.478.
Therefore we can say this shear lag assumption is acceptable, but again the solution could be improved by relaxing
this assumption as well. In spite of these coarse approximations, Fig. 9 shows that our analytical model does accurately capture the stress distribution along fiber axis
qualitatively. The fiber stress decreases along s-direction
from the pulled end to the embedded end. Quantitatively
speaking, with same composite dimension and material
properties and under same loading condition (pull-out displacement is 100 lm), the fiber axial stresses at pull-out
end calculated from our analysis (9.9E8 Pa) and from FE
simulation (8.5E8 Pa) are quite close.
4. Conclusions
In this paper, fiber curvature has been added into a shear
lag model to analyze the bonded stage in single curved-fiber
pull-out. A parametric study of the analytical model shows
that fiber curvature and fiber embedded length have strong
effects on the force–displacement curve. Fibers with more
curvature and longer embedded lengths can help toughen
the composites. 3D finite element results show that aside
from a stress variation around hoop direction, the current
analytical model captures the interfacial shear stress distribution qualitatively. For the same pull-out distance, the fiber stress field obtained from finite element is quite close
to that from our analytical model. However, the finite element results suggest that in cases of large fiber curvature,
the matrix radial displacement should not be ignored compared with its axial displacement. Therefore, further work
on the analytical model for the bonded stage is warranted,
in particular with regard to two issues:
Fig. 9. (a) Deformed fiber and matrix (b) Von Mises stress distribution in
fiber at the bonded stage when pull-out distance is 100 lm.
(1) Unlike the straight fiber, curved-fiber pull-out is not
an axisymmetric problem. As seen from Fig. 9 (b),
the stress field at fiber/matrix interface varies along
the hoop direction. A 3D analytical model is required
to take into account the variation in the hoop
direction.
(2) Once the radial compressive stresses are considered
in a newly developed 3D model, matrix deformation
can be analyzed more accurately and provide alternatives to neglecting the radial displacement.
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X. Chen et al. / Mechanics of Materials 41 (2009) 279–292
In spite of these possible improvements, the results
from our current analytical model are reasonable and can
be extended to include the debonding and sliding stages
to obtain information on effect of curvature on the full
pull-out scenario applicable to nanocomposites. In our second paper of this series, debonding and sliding stages are
analyzed and the results are combined with the result for
bonded stage in this paper to generate the entire pull-out
curve. The effect of curvature on the pull-out curve is then
studied.
Fig. 2A. Stress equilibrium of a differential fiber element.
Matrix equilibrium shown in Fig. 3A generates
si ¼ sm
rx r :
Acknowledgements
Strain–displacement gives
This work is supported by the National Science Foundation under Grant No. 0404291. I.J.B. acknowledge support
by a Los Alamos Laboratory Directed Research and Development Project (No. 20030216) and an Office of Basic Energy Sciences Project FWP 06SCPE401.
m
cmrx ¼
Here a single straight fiber pull-out model based on the
shear lag by Lawrence (1972) is reviewed and the corresponding pull-out force-displacement relation is derived.
As shown in Fig. 2, straight fiber pull-out is considered
an axisymmetric problem, in which all the stress, strain
and displacement components depend only on radial and
axial coordinates. A similar model geometry is also shown
in Fig. 1A. Fiber and matrix are co-cylinders with diameters
of df and dm, respectively. Initially, the fiber has a length of
l0 extruding out of matrix and has an embedded length of L.
All equations in the following derivation are in normalized form to remove any unnecessary dependencies of the
form of the solution on some parameters. The normalization factor for length is fiber radius rf. That for stress and
moduli is the fiber Young’s modulus Ef and accordingly,
that for force is p(rf)2Ef. The asterisks indicate normalized
values.
From Fig. 2A, we have
ð1AÞ
m
m
si
dux
dur
dux
sm
rx
þ
¼
¼ ;
G
r
Gm
dr
dx
dr
m
ð3AÞ
based on basic shear lag assumption, implied by assumption (1):
m
Appendix I. Single straight fiber pull-out in the bonded
stage
drfx
¼ 2si :
dx
ð2AÞ
m
dux
dur
:
dr
dx
ð4AÞ
Integrating Eq. (3A) on both sides,
Z
m Þ
um
x ðr
m
dux ¼
f
ux
Z
r m
si dr
Gm r 1
ð5AÞ
:
We have
m
f
um
x ðr Þ ux ¼
si
Gm
ln r m :
ð6AÞ
Reorganizing, we get
si ¼ Gm
f
m
um
x ðr Þ ux
:
ln r m
ð7AÞ
Combined with the equilibrium equation for the fiber Eq.
(1A), we have
f
m m
drfx
ux ðr Þ ux
:
¼ 2Gm
m
ln r
dx
ð8AÞ
In order to get an ODE of rfx , the displacement is related to
stress through strain–displacement relation and an elastic,
isotropic constitutive law.
2
d
rfx
2
dx
¼
2Gm f
2Gm
m
ðe em
x ðr ÞÞ ¼
ln r m x
ln rm
rfx m
rm
x ðr Þ
Em
;
ð9AÞ
Fig. 1A. Model geometry of single straight fiber pull-out in the bonded
stage. Fiber and matrix both have a circular cross-section of diameter df
and dm, respectively, in axisymmetric coordinate system x and r. L is the
fiber embedded length. l0 is the free length of the fiber initially not
embedded in the matrix. P is the applied pull-out force at the fiber pulled
end.
Fig. 3A. Stress equilibrium of a representative matrix segment in the
bonded stage.
289
X. Chen et al. / Mechanics of Materials 41 (2009) 279–292
where rm
x is regarded as a virtual matrix stress generated
under pull-out stress as if there is no fiber, i.e.,
m
rm
x ðr Þ ¼
P
rfx
¼
:
r m2 rm2
ð10AÞ
Substituting Eq. (10A) into (9A),
2
d
rfx
¼
2
dx
2Gm
1
1
rf Q ðrm Þrfx :
r m2 Em x
ln r m
ð11AÞ
The general solution for Eq. (11A) is
pffiffiffiffiffiffi
pffiffiffiffiffiffi
rfx ¼ A sinhð Q x Þ þ B coshð Q x Þ:
ð12AÞ
Using the following boundary conditions:
1Þ
rfx ð0Þ ¼ r0 ) B ¼ r0 ;
ð13AÞ
where r0 denotes the stress from neighboring nanotubes
at the embedded end. Note that the classic shear lag assumes a free-end rfx ð0Þ ¼ 0 and thus has a simpler form
solution than we obtain here.
2Þ
rfx ðL Þ ¼ rpull
pffiffiffiffiffiffi
rpull r0 coshð Q L Þ
pffiffiffiffiffiffi ;
)A¼
sinhð Q L Þ
ð14AÞ
where r
is the applied stress at the pulled end and it
equals the normalized pull-out force P*.
Substituting Eqs. (13A), and (14A) into Eq. (12A), we
have the fiber axial stress
pull
rfx
!
pffiffiffiffiffiffi pffiffiffiffiffiffi
P
pffiffiffiffiffiffi r0 cothð Q L Þ sinhð Q x Þ
¼
sinhð Q L Þ
pffiffiffiffiffiffi
þ r0 coshð Q x Þ;
ð15AÞ
and the interfacial shear stress
1
2
si ¼ "
pffiffiffiffiffiffi
P
pffiffiffiffiffiffi r0 cothð Q L Þ
sinhð Q L Þ
s
~
r ;i ; ~
g i ~
g j ¼ dij ¼
g i ¼~
1ði ¼ jÞ
0ði–jÞ
g j ¼ g ij ; ~
g i ~
g j ¼ g ij :
;~
g i ~
ð19AÞ
In detail,
@~
r
s
s
¼ sin a i cos a j;
@r
R
R
@~
r Rr
s R r
s
¼
cos a i þ
sin a j;
g~2 ¼
@s
R
R
R
R
ð20AÞ
s
s
g~1 ¼ sin a i cos a j;
R
R
R
s
R
s
~
2
cos a i þ
sin a j;
g ¼
Rr
R
Rr
R
ð21AÞ
g~1 ¼
2
Rr
;
R
¼ g~1 g~1 ¼ 1;
g 11 ¼ g~1 g~1 ¼ 1; g 22 ¼ g~2 g~2 ¼
g 12 ¼ g 21 ¼ g~1 g~2 ¼ 0; g 11
2
R
; g 12 ¼ g 21 ¼ g~1 g~2 ¼ 0:
g 22 ¼ g~2 g~2 ¼
Rr
Ckij ¼ Ckji ¼ g~k ~
g i;j ;
ð22AÞ
ð23AÞ
where
ð16AÞ
When r0 is set to zero, Eq. (16A) has the same form as given by Lawrence (1972). In our formulation we have an
analytical form for the arbitrary constants in his equation.
Note that si max is reached at x* = L*.
i max
Based on continuum mechanics (Green and Zerna,
1992; Malvern, 1969), basic elasticity equations for small
strain condition including equilibrium, stress–displacement relationship, and constitutive law are derived for
2D orthogonal curvilinear system as follows.
The 2D curvilinear system we employ is shown in
Fig. 4A. Point (r,s) in r–s curvilinear system can be represented by a vector ~
r of R ðR rÞ sinða Rs Þi þ ðR rÞ
s
cosða RÞj in x–y Cartesian coordinate system. Base vectors
~
g i and metric tensors g ij ; g ij are defined as
gi ; ~
The Christoffel symbols of the second kind
!
#
pffiffiffiffiffiffi
pffiffiffiffiffiffi
pffiffiffiffiffiffi
pffiffiffiffiffiffi
Q coshð Q x Þ þ r0 Q sinhð Q x Þ :
Appendix II. Derivation of basic elasticity equations in
2D orthogonal curvilinear system
1
s 1
s
~
g 1;2 ¼ ~
g 2;1 ¼ cos a i sin a j;
g 1;1 ¼ 0; ~
R
R
R
R
Rr
s R r
s
~
g 2;2 ¼ 2 sin a i 2 cos a j:
ð24AÞ
R
R
R
R
!
pffiffiffiffiffiffi 1 pffiffiffiffiffiffi r0
pffiffiffiffiffiffi
Q P cothð Q L Þ :
¼
2
sinhð Q L Þ
ð17AÞ
When s
¼ s debonding begins, where s is the critical
shear stress for fiber-matrix separation.
Fiber displacement is the sum of elastic elongation of
the embedded and the extruded part.
i max
dðIÞ ¼
Z
0
L
s,
s
1
pffiffiffiffiffiffi
efx dx þ P l0 ¼ pffiffiffiffiffiffi P cothð Q L Þ
Q
!
pffiffiffiffiffiffi P
r0
p
p
ffiffiffiffiffiffi
ffiffiffiffiffiffi
þ r0 cothð Q L Þ þ P l0
sinhð Q L Þ sinhð Q L Þ
ð18AÞ
Fig. 4A. 2D curvilinear system s and r and x–y Cartesian coordinate
system.
290
X. Chen et al. / Mechanics of Materials 41 (2009) 279–292
Thus,
Ck11 ¼ 0; C112 ¼ C121 ¼ 0; C212 ¼ C221
C122 ¼
Rr
R2
The constitutive law does not change with coordinate system. Therefore, for an isotropic material with no Poisson
effect, we still have
1
¼
;
rR
; C222 ¼ 0:
ð25AÞ
er ¼
rr
E
; es ¼
rs
E
;
and crs ¼
srs
G
ð36AÞ
Static equilibrium without body force is
References
r s ¼ r ðsij g~i g~j Þ ¼ sij:i g~j ¼ 0; i:e:; sij:i
¼ sij;i þ skj Ciki þ sik Cjki ¼ 0:
ð26AÞ
In r-direction j = 1, we have equilibrium:
1
2
1
2
21
11
21
11 1
s11
;1 þ s;2 þ s ðC11 þ C12 Þ þ s ðC21 þ C22 Þ þ s C11
þ s12 C121 þ s21 C112 þ s22 C122 ¼ 0:
ð27AÞ
In s-direction j = 2, we have:
1
2
1
2
22
12
22
11 2
s12
;1 þ s;2 þ s ðC11 þ C12 Þ þ s ðC21 þ C22 Þ þ s C11
þ s12 C221 þ s21 C212 þ s22 C222 ¼ 0:
ð28AÞ
We have to transform contravariant components to physical components as follows.
s11 ¼ rr ; s12 ¼ s21 ¼
R
srs ;
Rr
s22 ¼
R2
ðR rÞ2
rs :
ð29AÞ
The equilibrium equations represented by physical components are thus expressed as follows.
@ rr
R @ srs rs rr
þ
þ
¼ 0 in r-direction:
R r @s
@r
Rr
@ srs
R @ rs
2
srs ¼ 0 in s-direction:
þ
R r @s
Rr
@r
ð30AÞ
ð31AÞ
As for the strain–displacement relations under a small
strain condition, based on
1
2
cij ¼ ðvijj þ vjji Þ;
where vijj ¼ vi;j C
ð32AÞ
r
ij vr
we have
@v
c11 ¼ v1j1 ¼ 1 C111 v1 C211 v2 ;
@r
c12 ¼ c12 ¼ v1j2
1 @v1
@v2
¼
C112 v1 C212 v2 þ
C121 v1 C221 v2
2 @s
@r
@v2
1
2
c22 ¼ C22 v1 C22 v2 :
@s
ð33AÞ
Again, we have to transform covariant components to
physical components as follows.
Rr
us ; c11 ¼ er ; c12 ¼ c21
R
2
Rr
Rr
¼
ers ; c11 ¼
es :
R
R
v1 ¼ ur ; v2 ¼
ð34AÞ
Therefore, the final form of the strain–displacement relation is
@ur
;
@r
1 R
@ur R r @us us
1
¼ crs ;
ers ¼
þ
þ
2 R r @s
R @r
2
R
R @us
ur
es ¼
:
R r @s R r
er ¼
ð35AÞ
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