Owing to their superior mechanical and physical properties, carbon nanotubes seem to hold a great promise as an ideal reinforcing material for composites of high-strength and low-density. In most of the experimental results up to date, however, only modest improvements in the strength and stiffness have been achieved by incorporating carbon nanotubes in polymers. In the present paper, the stiffening effect of carbon nanotubes is quantitatively investigated by micromechanics methods. Especially, the effects of the extensively observed waviness and agglomeration of carbon nanotubes are examined theoretically. The Mori-Tanaka effective-field method is first employed to calculate the effective elastic moduli of composites with aligned or randomly oriented straight nanotubes. Then, a novel micromechanics model is developed to consider the waviness or curviness effect of nanotubes, which are assumed to have a helical shape. Finally, the influence of nanotube agglomeration on the effective stiffness is analyzed. Analytical expressions are derived for the effective elastic stiffness of carbon nanotube-reinforced composites with the effects of waviness and agglomeration. It is found that these two mechanisms may reduce the stiffening effect of nanotubes significantly. The present study not only provides the relationship between the effective properties and the morphology of carbon nanotube-reinforced composites, but also may be useful for improving and tailoring the mechanical properties of nanotube composites.

1.
Iijima
,
S.
,
1991
, “
Helical Microtubles of Graphitic Carbon
,”
Nature (London)
,
354
, pp.
56
58
.
2.
Qian
,
D.
,
Wagner
,
G. J.
,
Liu
,
W. K.
,
Yu
,
M. F.
, and
Ruoff
,
R. S.
,
2002
, “
Mechanics of Carbon Nanotubes
,”
Appl. Mech. Rev.
,
55
(
6
), pp.
495
533
.
3.
Saito, R., Dresselhaus, G., and Dresselhaus, M. S., 1998, Physical Properties of Carbon Nanotubes, Imperial College Press, London.
4.
Treacy
,
M. M. J.
,
Ebbesen
,
T. W.
, and
Gibson
,
J. M.
,
1996
, “
Exceptionally High Young’s Modulus Observed for Individual Carbon Nanotubes
,”
Nature (London)
,
381
, pp.
678
680
.
5.
Yakobson
,
B. I.
,
Brabec
,
C. J.
, and
Bernholc
,
J.
,
1996
, “
Nanomechanics of Carbon Tubes: Instability Beyond Linear Response
,”
Phys. Rev. Lett.
,
76
(
14
), pp.
2511
2514
.
6.
Yu
,
M. F.
,
Files
,
B. S.
,
Arepalli
,
S.
, and
Ruoff
,
R. S.
,
2000
, “
Tensile Loading of Ropes of Single Wall Carbon Nanotubes and Their Mechanical Properties
,”
Phys. Rev. Lett.
,
84
, pp.
5552
5555
.
7.
Yu
,
M. F.
,
Lourie
,
O.
,
Dyer
,
M. J.
,
Moloni
,
K.
, and
Ruoff
,
R. S.
,
2000
, “
Strength and Breaking Mechanism of Multiwalled Carbon Nanotubes Under Tensile Load
,”
Science
,
287
, pp.
637
640
.
8.
Zhang
,
P.
,
Huang
,
Y.
,
Gao
,
H.
, and
Hwang
,
K. C.
,
2002
, “
Fracture Nucleation in Single-Wall Carbon Nanotubes Under Tension: A Continuum Analysis Incorporating Interatomic Potentials
,”
ASME J. Appl. Mech.
,
69
(
3
), pp.
454
458
.
9.
Zhang
,
P.
,
Huang
,
Y.
,
Geubelle
,
P. H.
,
Klein
,
P. A.
, and
Hwang
,
K. C.
,
2002
, “
The Elastic Modulus of Single-Wall Carbon Nanotubes: A Continuum Analysis Incorporating Interatomic Potentials
,”
Int. J. Solids Struct.
,
39
, pp.
3893
3906
.
10.
Cornwell
,
C. F.
, and
Wille
,
L. T.
,
1997
, “
Elastic Properties of Single-Walled Carbon Nanotubes in Compression
,”
Solid State Commun.
,
101
(
8
), pp.
555
558
.
11.
Yao
,
Z. H.
,
Zhu
,
C. C.
,
Cheng
,
M.
, and
Liu
,
J.
,
2001
, “
Mechanical Properties of Carbon Nanotube by Molecular Dynamics Simulation
,”
Comput. Mater. Sci.
,
22
, pp.
180
184
.
12.
Ebbesen
,
T. W.
,
Lezec
,
H. J.
, and
Hiura
,
H.
,
1996
, “
Electrical Conductivity of Individual Carbon Nanotubes
,”
Nature (London)
,
382
, pp.
54
56
.
13.
Wei
,
J. H.
,
Xie
,
S. J.
,
Wang
,
S. G.
, and
Mei
,
M. L.
,
2001
, “
Dimensional Model of Carbon Nanotubes
,”
Phys. Lett. A
,
292
, pp.
207
211
.
14.
Calvert
,
P.
,
1999
, “
Nanotube Composites: A Recipe for Strength
,”
Nature (London)
,
399
, pp.
210
211
.
15.
Thostenson
,
E. T.
,
Ren
,
Z.
, and
Chou
,
T. W.
,
2001
, “
Advances in the Science and Technology of Carbon Nanotubes and Their Composites: A Review
,”
Compos. Sci. Technol.
,
61
, pp.
1899
1912
.
16.
Haggenmueller
,
R.
,
Gommans
,
H. H.
,
Rinzler
,
A. G.
,
Fischer
,
J. E.
, and
Winey
,
K. I.
,
2000
, “
Aligned Single-Wall Carbon Nanotubes in Composites by Melt Processing Methods
,”
Chem. Phys. Lett.
,
330
, pp.
219
225
.
17.
Bower
,
C.
,
Rosen
,
R.
,
Jin
,
L.
,
Han
,
J.
, and
Zhou
,
O.
,
1999
, “
Deformation of Carbon Nanotubes in Nanotube-Polymer Composites
,”
Appl. Phys. Lett.
,
74
(
22
), pp.
3317
3319
.
18.
Lourie
,
O.
,
Cox
,
D. M.
, and
Wagner
,
H. D.
,
1998
, “
Buckling and Collapse of Embedded Carbon Nanotubes
,”
Phys. Rev. Lett.
,
81
(
8
), pp.
1638
1641
.
19.
Wagner
,
H. D.
,
Lourie
,
O.
,
Feldman
,
Y.
, and
Tenne
,
R.
,
1998
, “
Stress-Induced Fragmentation of Multiwall Carbon Nanotubes in a Polymer Matrix
,”
Appl. Phys. Lett.
,
72
(
2
), pp.
188
190
.
20.
Lourie
,
O.
, and
Wagner
,
H. D.
,
1998
, “
Transmission Electron Microscopy Observations of Single-Wall Carbon Nanotubes Under Axial Tension
,”
Appl. Phys. Lett.
,
73
(
24
), pp.
3527
3529
.
21.
Jia
,
Z. J.
,
Wang
,
Z.
,
Xu
,
C.
,
Liang
,
J.
,
Wei
,
B.
,
Wu
,
D.
, and
Zhu
,
S.
,
1999
, “
Study on Poly(methyl methacrylate)/Carbon Nanotube Composites
,”
Mater. Sci. Eng., A
,
271
, pp.
395
400
.
22.
Qian
,
D.
,
Dickey
,
E. C.
,
Andrews
,
R.
, and
Rantell
,
T.
,
2000
, “
Load Transfer and Deformation Mechanisms in Carbon Nanotube-Polystyrene Composites
,”
Appl. Phys. Lett.
,
76
, pp.
2868
2870
.
23.
Po¨tschke
,
P.
,
Fornes
,
T. D.
, and
Paul
,
D. R.
,
2002
, “
Rheological Behavior of Multiwalled Carbon Nanotube/Polycarbonate Composites
,”
Polymer
,
43
(
11
), pp.
3247
3255
.
24.
Andrews
,
R.
,
Jacques
,
D.
,
Rao
,
A. M.
,
Rantell
,
T.
,
Derbyshire
,
F.
,
Chen
,
Y.
,
Chen
,
J.
, and
Haddon
,
R. C.
,
1999
, “
Nanotube Composite Carbon Fibers
,”
Appl. Phys. Lett.
,
75
(
9
), pp.
1329
1331
.
25.
Odegard
,
G. M.
,
Gates
,
T. S.
,
Wise
,
K. E.
,
Park
,
C.
, and
Siochi
,
E. J.
,
2002
, “
Constitutive Modeling of Nanotube-Reinforced Polymer Composites
,”
Compos. Sci. Technol.
,
63
(
11
), pp.
1671
1687
.
26.
Ajayan
,
P. M.
,
Schadler
,
L. S.
,
Giannaris
,
C.
, and
Rubio
,
A.
,
2000
, “
Single-Walled Nanotube-Polymer Composites: Strength and Weaknesses
,”
Adv. Mater. (Weinheim, Ger.)
,
12
(
10
), pp.
750
753
.
27.
Nardelli
,
M. B.
,
Fattebert
,
J. L.
,
Orlikowski
,
D.
,
Roland
,
C.
,
Zhao
,
Q.
, and
Bernholc
,
J.
,
2000
, “
Mechanical Properties, Defects and Electronic Behavior of Carbon Nanotubes
,”
Carbon
,
38
, pp.
1703
1711
.
28.
Nemat-Nasser, S., and Hori, M., 1993, Micromechanics: Overall Properties of Heterogeneous Materials, North-Holland, New York.
29.
Mori
,
T.
, and
Tanaka
,
K.
,
1973
, “
Average Stress in Matrix and Average Elastic Energy of Materials With Misfitting Inclusions
,”
Acta Metall.
,
21
, pp.
571
574
.
30.
Hill
,
R.
,
1965
, “
A Self-Consistent Mechanics of Composite Materials
,”
J. Mech. Phys. Solids
,
13
, pp.
213
222
.
31.
Mura, T., 1987, Micromechanics of Defects in Solids, Martinus Nijhoff Publishers, Dordrecht.
32.
Popov
,
V. N.
,
Doren
,
V. E.
, and
Balkanski
,
M.
,
2000
, “
Elastic Properties of Crystals of Single-Walled Carbon Nanotubes
,”
Solid State Commun.
,
114
, pp.
395
399
.
33.
Andrews
,
R.
,
Jacques
,
D.
,
Minot
,
M.
, and
Rantell
,
T.
,
2002
, “
Fabrication of Carbon Multiwall Nanotube/Polymer Composites by Shear Mixing
,”
Macromolecular Materials and Engineering
,
287
, pp.
395
403
.
34.
Shaffer
,
M. S. P.
, and
Windle
,
A. H.
,
1999
, “
Fabrication and Characterization of Carbon Nanotube/Poly(vinyl alcohol) Composites
,”
Adv. Mater. (Weinheim, Ger.)
,
11
, pp.
937
941
.
35.
Vigolo
,
B.
,
Penicaud
,
A. P.
,
Couloun
,
C.
,
Sauder
,
S.
,
Pailler
,
R.
,
Journet
,
C.
,
Bernier
,
P.
, and
Poulin
,
P.
,
2000
, “
Macroscopic Fibers and Ribbons of Oriented Carbon Nanotubes
,”
Science
,
290
, pp.
1331
1334
.
36.
Fisher
,
F. T.
,
Bradshaw
,
R. D.
, and
Brinson
,
L. C.
,
2002
, “
Effects of Nanotube Waviness on the Modulus of Nanotube-Reinforced Polymers
,”
Appl. Phys. Lett.
,
80
(
24
), pp.
4647
4649
.
37.
Fisher
,
F. T.
,
Bradshaw
,
R. D.
, and
Brinson
,
L. C.
,
2003
, “
Fiber Waviness in Nanotube-Reinforced Polymer Composites: I. Modulus Predictions Using Effective Nanotube Properties
,”
Compos. Sci. Technol.
,
63
(
11
), pp.
1689
1703
.
38.
Bradshaw
,
R. D.
,
Fisher
,
F. T.
, and
Brinson
,
L. C.
,
2003
, “
Fiber Waviness in Nanotube-Reinforced Polymer Composites: II. Modeling via Numerical Approximation of the Dilute Strain Concentration Tensor
,”
Compos. Sci. Technol.
,
63
(
11
), pp.
1705
1722
.
39.
Curtin, W. A., 2002, private communication with Y. Huang.
40.
Stephan
,
C.
,
Nguyen
,
T. P.
,
Chapelle
,
M. L.
, and
Lefrant
,
S.
,
2000
, “
Characterization of Single-Walled Carbon Nanotubes-PMMA Composite
,”
Synth. Met.
,
108
, pp.
139
149
.
41.
Jones, R. M., 1999, Mechanics of Composite Materials, Taylor & Francis, Philadelphia.
42.
Shi
,
D. L.
,
Feng
,
X. Q.
,
Huang
,
Y.
, and
Hwang
,
K. C.
,
2004
, “
Critical evaluation of the stiffening effect of carbon nanotubes in composites
,”
Key Eng. Mater.
,
261–263
, pp.
1487
1492
.
You do not currently have access to this content.