Characterizing Carbon Fiber Polymer Composites

2013 American Transactions on Engineering & Applied Sciences.

American Transactions on
Engineering & Applied Sciences
http://TuEngr.com/ATEAS

Characterization of Mechanical, Thermal, and
Electrical Properties of Carbon Fiber Polymer
Composites by Modeling
Zhong Hu a*, Xingzhong Yan b, James Wu c, and Michelle Manzo c
a

Department of Mechanical Engineering, South Dakota State University, USA
Department of Electrical Engineering &Computer Science, South Dakota State University, USA
c
Electrochemistry Branch, NASA Glenn Research Center, USA
b

ARTICLEINFO

A B S T RA C T

Article history:
Received February 06, 2013
Received in revised form
March 06, 2013
Accepted March 08, 2013
Available online
March 11, 2013

In this paper, the mechanical, thermal and electrical
properties of carbon fiber modified thermoplastic polyimide were
numerically analyzed by finite element analysis.
A
three-dimensional model was created, in which continuous carbon
fibers are aligning and paralleling to each other and uniformly
distributing in the polymer matrix. The behaviors of the composites
in two extreme situations, i.e., parallel or perpendicular to carbon
fiber direction, were simulated. The effects of the volume fraction
of carbon fiber content on the physical properties were investigated.
It shows clearly that carbon fibers significantly improve the
mechanical strength, and thermal and electrical conductivities. The
future work includes investigation of the physical properties of the
conductive network of the composites with random carbon fiber
orientation, and different fillers, such as graphite, and carbon
nanotubes.

Keywords:
Carbon Fibers
Polymer Composites
Physical Properties
Characterization
Finite Element Analysis

2013 American Transactions on Engineering & Applied Sciences.

*Corresponding author (Z. Hu). Tel/Fax: +1-605-688-4817/+1-605-688-5878. E-mail
address: Zhong.Hu@sdstate.edu.
2013. American Transactions on Engineering &
Applied Sciences. Volume 2 No. 2 ISSN 2229-1652 eISSN 2229-1660 Online
Available at http://TuEngr.com/ATEAS/V02/133-148.pdf

133

1. Introduction
Polymers are lightweight, flexible, resistant to heat and chemicals, and non-conductive and
transparent to electromagnetic radiation. Therefore, they are used in the electronics industry for
flexible cables, as an insulating film, and for medical tubing. Thus, they are not suitable for use as
enclosures for electronic equipment because they cannot shield it from outside radiation. Also they
cannot prevent the escape of radiation from the component. (Elimat et al. 2010; Navin and Deepak
2004) Several fillers can be added to the insulating polymeric matrix in order to achieve different
conductivity ranges for a variety of industrial applications. (Delmonte 1990; Neelakanta 1995)
Polyacrylonitrile (PAN)-based carbon fibers (CFs) possess high stiffness and strength, low
expansion coefficient, and elevated thermal and electric conductivity measured along the fiber
direction. (Lei et al. 2008; Donnet et al. 1990; Park and Chou 2000; Chand 2000) Carbon fiber
(CF)/polymer composites are used in the aerospace industry on account of their high-specific
stiffness and strength, which are higher than in metallic materials. (Surendra et al. 2009)
Conducting polymers have been extensively studied because of their potential applications in
light-emitting devices, batteries, electromagnetic shield, and other functional applications.
(Anupama et al. 2010; Tsotra and Friedrich 2003; Tse et al. 1981) The ability of these composites
to serve as capacitors and other circuit element means that the structures is itself the electronics, so
that the electronics 'vanish' into the structure. (Luo and Chung 2001) In the case of continuous CF
polymer-matrix composites, CFs are the conductors (resistors) and they can be intercalated to
become electron metals or hole metals. (Chung and Wang 1999) By having the electronics vanish
into the structure, space is saved. The space saving is particularly valuable for capacitors of large
capacitance in space applications. In addition to space saving, structural electronics have the
advantage of being mechanically rugged and inexpensive, since structural materials are
necessarily rugged and inexpensive. The use of a structure as a capacitor is particularly valuable in
conjunction with structures that are powered by solar cells, as the structure (capacitor) can be used
to store the electrical energy generated by the solar cells. (Luo and Chung 2001; Chung and Wang
1999; Zhu et al. 2011; Stoller and Ruoff 2010).
In this paper, the mechanical, thermal and electrical properties of CF modified polyimide (PI)
matrix composite will be numerically analyzed by finite element analysis (FEA). (Herakovich
1998; Minus and Kumar 2007; Fei et al. 2007) A three-dimensional model will be created, in
134

Zhong Hu, Xinghong Yan, James Wu, and Michelle Manzo

which continuous CFs will be aligning and paralleling to each other and uniformly distributing in
the matrix.

The behaviors of the composite in two extreme situations, i.e., parallel or

perpendicular to the CF direction, will be simulated. The effects of CF volumetric fraction on the
physical (mechanical, thermal, and electrical) properties will be investigated.

2. Modeling by Finite Element Analysis
Although the material properties for the composite are anisotropic, for each component (CF
filler or polymer matrix) they could be treated as isotropic. Assuming that each set of properties
(mechanical, thermal, or electrical properties) is independent.

Therefore, each set of the

properties can be evaluated separately. For mechanical properties, a linear and elastic stress-strain
relationship is assumed. Therefore, for each element of one kind material (CF filler or polymer
matrix), the structural governing equation is (ANSYS Inc. 2012)
(1)
where

is the stress vector,

is the elastic strain vector, and

is the elastic stiffness

matrix. For thermal properties, the heat transfer governing equation of conduction at steady-state is
0
where

is the heat flux vector,

(2)

is the vector operator, T is temperature, and

is the

thermal conductivity matrix. For electrical properties, the electromagnetic field governing
equations are matrix. For electrical properties, the electromagnetic field governing equations are
;
where

is curl operator,

;

· is divergence operator,

is the total current density vector,
electric field intensity vector,

·

0;

·

(3)

is the magnetic field intensity vector,

is the electric flux density vector, t is time,

is the

is the magnetic flux density vector, and ρe is the electric charge

density.
Therefore, each element formulations can be developed and the equations for each element can

135

be assembled into global matrix for the entire domain of the composite (including both CF filler
and polymer matrix). In this work, a commercial FEA software ANSYS was used. A
three-dimensional element SOLID223, i.e., a 3-D 20-node high-order accuracy coupled–field
(structural, thermal, and electrical) solid element was adopted. The isotropic properties of each
material component were assumed. PAN-based carbon fibers with filament size of 5μm in
diameter are chosen as polymeric matrix filler and polyimides as polymeric matrix. The physical
properties of both materials are listed in Table 1. (Herakovich 1998; Minus and Kumar 2007; Fei at
al. 2007; Ayish and Zihlif 2010; Dupont Kapton 2010).
Table 1: Physical properties of polyimide film and carbon fibers.

Modulus
Thermal
Thermal Specific Electrical
of
Poisson Density Expansion
Dielectric
Conductivity Heat Resistivity
Materials
Elasticity Ratio (kg/m3) Coefficient
Constant
(W/m⋅K)
(J/kg⋅K)
(Ω⋅m)
-6
(GPa)
(10 /K)
Polyimide
2.5
0.34
1,420
17
0.10
1,090 1.5×1015
3.4
Film
Carbon
2000
310
0.2
1,800
0.5
12
965
0.15×10-4
Fibers
The three-dimensional solid model is shown in Figure 1, in which the cylindrical components
represent CFs and the cubic volume is the polymeric matrix, and z-axis represents the CF
orientation and x-y plane is perpendicular to the CF orientation. The dimensions of the solid model
are represented by L (length in x-axis) × W (width in y-axis) × H (height in z-axis). The FEA
meshes are shown in Figure 2. The dimensions of the model in the simulations, in terms of CF
volumetric fraction, are listed in Table 2.

Figure 1: 3-D solid model of the composite.
136

Figure 2: 3-D FEA meshes of the composite.


For modeling the modulus of elasticity in z-axial direction (CF direction), the uniaxial tensile
testing along z-axial direction is conducted, in which the bottom face (z=0) is fixed and the top face
(z=H) is pulled 0.1% (within elastic deformation region) in +z-axial direction (i.e., z-axial strain
εz=0.001). For symmetric purpose, the displacement in x-axial direction on the face of x=0 is
constrained (ux=0 on the face of x=0), and the displacement in y-axial direction on the face of y=0 is
also constrained (uy=0 on the face of y=0).

The z-axial forces on the top face (z=H) are

accumulated as Fz, so that the average stress in z-axial direction, σz, can be calculated. The modulus
of elasticity in z-axial (CF) direction, E1, is calculated.
Table 2: The dimensions of the model in terms of CF volumetric fraction.

CF Volume Fraction

CF Diameter

Length (x-axis)

Width (y-axis)

Height (z-axis)

(%)

(μm)

(μm)

(μm)

(μm)

10
20
30
40
50

5
5
5
5
5

84.07
59.45
48.54
42.04
37.60

56.05
39.63
32.36
38.02
25.07

28.02
18.82
16.18
14.01
12.53

Figure 3: Displacement ux under z-axial tension

Figure 4: Displacement uy under z-axial

(εz=0.001).

tension (εz=0.001).
(4)

Poisson ratio in x-axis (ν12) or y-axis (ν13) under this uniaxial tensile test can also be calculated
;

(5)

137

where the average strain in x-axial and y-axial directions, εx and εy, are calculated as (as shown
in Figures 3 and 4)
(on

face)

(6)

(on

face)

(7)

For modeling the modulus of elasticity in x-axial direction, E2, (perpendicular to CF direction),
the uniaxial tensile testing along x-axial direction is conducted. The corresponding boundary
conditions are applied, i.e., the left-front face (x=0) is fixed and the right-back face (x=L) is pulled
0.1% (within elastic deformation region) in +x-axial direction (i.e., x-axial strain εx=0.001). For
symmetric purpose, the displacement in y-axial direction on the face of y=0 is constrained (uy=0 on
the face of y=0), and the displacement in z-axial direction on the face of z=0 is also constrained
(uz=0 on the face of z=0). The similar calculation steps can be adopted for calculating the average
stress and strain, σx and εy, Modulus of elasticity E2, and Poisson ratio ν23, as shown in Figures 5
and 6,
;

(8)

Figure 5: Displacement ux under x-axial tension
(εx=0.001).

Figure 6: Displacement uy under x-axial tension
(εz=0.001).

For modeling the thermal conductivity in z-axial direction (CF direction), the bottom face
(z=0) is assigned with temperature of 20°C and the top face (z=H) is assigned with temperature of
21°C (1°C difference between two faces). The equivalent thermal conductivity in z-axial direction,
138


k1, is calculated as

(

)

(on

face)

(9)

where qz is the heat flux in z-axial direction, and (Tz=H - Tz=0) is the temperature difference between
bottom face and top face. The similar steps can be adopted for calculating the equivalent thermal
conductivity in x-axial direction (perpendicular to CF direction), k2, as shown in Figures 7 and 8.

Figure 7: Heat flux (W/m2) vector distribution under
temperature difference (1°C) in z-direction.

Figure 8: Heat flux (W/m2) vector distribution
under temperature difference (1°C) in x-direction.

Similar to the thermal conductivity calculation, for modeling the electrical conductivity in
z-axial direction (CF direction), the bottom face (z=0) is assigned with voltage of 0 volt and the top
face (z=H) is assigned with voltage of 0.1 volt. The equivalent electrical conductivity in z-axial
direction, κ1, is calculated as

(

)

(on

face)

(10)

where iz is the current density in z-axial direction, and (Vz=H–Vz=0) is the voltage difference between
bottom face and top face. The similar steps can be adopted for calculating the equivalent electrical
conductivity in x-axial direction (perpendicular to CF direction), κ2.


139

Figure 9: Displacement uz under z-axial tension
(εz=0.001).

Figure 10: Stress σz (MPa) under z-axial tension
(εz=0.001).

Figure 11: Strain εx under x-axial tension (εx=0.001,
10%CF).

Figure 12: Strain εx under x-axial tension
(εx=0.001, 50%CF).

Figure 13: Stress σx (MPa) under x-axial tension
(εx=0.001, 10%CF).

Figure 14: Stress σx (MPa) under x-axial tension
(εx=0.001, 50%CF).

3. Results and Discussion
The effects of CF volume fraction on the physical properties of the composite are investigated.
Figure 9 shows z-axial displacement under z-axial tension, which gives a uniform distribution of
the displacement. Therefore, the strain in z-axial is constant throughout the entire domain. Figure
140


10 shows z-axial stress under z-axial tension. It shows clearly that the major load applied is taken
by CFs due to their much higher stiffness. These two figures are useful for analyzing modulus of
elasticity, E1, in CF direction. Figures 11-14 show x-axial strain and stress distribution under
x-axial tension for 10%CF and 50%CF, respectively. It can be seen that the more the CF%, the
worse the uniformity of the distributions and the more the contribution from CFs. These four
figures are useful for analyzing modulus of elasticity, E2, in perpendicular to CF direction. Figures
15 and 16 show the moduli of elasticity, E1 (in CF direction) and E2 (perpendicular to CF
direction). It clearly shows that the modulus of elasticity measured in CF direction is linearly
increasing as CF% increasing, and the magnitude is primarily dominated by CF part due to its
much higher stiffness. It also shows a nonlinear increasing of the modulus of elasticity in
perpendicular to CF direction as CF% increasing due to the cross-section interaction increasing as
mentioned in Figures 11-14. However, the magnitude is primarily dominated by polymer matrix
due to the discontinuity of CFs in the plane, therefore, the load has been transferred primarily
within polymer matrix, the softer component. In CF direction, the modulus of elasticity is much

1.6E+5

Modulus of Elasticity E2(MPa)

Modulus of Elasticity E1 (MPa)

higher than that in perpendicular to the CF direction.
1.4E+5
E1 in CF direction

1.2E+5
1.0E+5
8.0E+4
6.0E+4
4.0E+4
2.0E+4
0.0E+0
0

0.1

0.2

0.3

0.4

CF Volume Fraction

Figure 15: Modulus of elasticity, E1, in CF direction.

0.5

8E+3
7E+3

E2 perpendicular to CF direction

6E+3
5E+3
4E+3
3E+3
2E+3
0

0.1

0.2

0.3

CF Volume Fraction

0.4

Figure 16: Modulus of elasticity, E2,
perpendicular to CF direction.

Figures 17-20 show the strain distributions in the cross-section perpendicular to CF direction
under z-axial tension for evaluating Poisson ratios of υ12 and υ13. It clearly shows that the more the
CF%, the worse the uniformity of the strain distribution is and the more the contribution by CF
component. Furthermore, the strain distribution in x-axial is symmetric to that in y-axial due to the
model symmetry in x-axis and y-axis.

141

0.5

Figures 21-22 show y-axial strain distribution in the cross-section perpendicular to CF
direction under x-axial tension for evaluating Poisson ratio of υ23. It clearly shows again that the
more the CF%, the worse the uniformity of the strain distribution is the more the contribution from
CF component to the deformation.

Figure 17: Strain εx under z-axial tension
(εz=0.001, 10%CF).

Figure 18: Strain εx under z-axial tension (εz=0.001,
50%CF).

Figure 19: Strain εy under z-axial tension
(εz=0.001, 10%CF).

Figure 20: Strain εy under z-axial tension (εz=0.001,
50%CF).

Figure 21: Strain εy under x-axial tension
(εx=0.001, 10%CF).

Figure 22: Strain εy under x-axial tension (εx=0.001,
50%CF).

142


Figure 23 shows the Poisson ratios of the composites. υ12 and υ13 are the same due to their
symmetry in model. Furthermore, υ12 and υ13 are almost linearly decreasing as CF% increasing,
starting from 0.34 of the polymer matrix’s Poisson ratio (0% CF). υ23 is nonlinearly decreasing and
decreasing from slower to faster as CF% increasing, starting from the same value.

0.34

Poisson Ratio

0.32
0.30
υ12

0.28

υ13

0.26

υ23

0.24
0.22
0.20
0

0.1

0.2

0.3

0.4

0.5

CF Volume Fraction

Figure 23: Poisson ratios of the composites.
Thermal and electrical properties have the similar trends as the moduli of elasticity. Figures
24-25 show the thermal and electrical conductivities, k1 and κ1, in CF direction, respectively.

7

3.5E+4

Equivalent Electrical Conductivity
κ1 (S/m)

Equivalent Thermal Conductivity
k1(W/mK)

Again, CFs play the major role in these properties due to their much higher conductivities.

6

3.0E+4

k1 in CF direction

5

κ1 in CF direction

2.5E+4

4

2.0E+4

3

1.5E+4

2

1.0E+4

1

5.0E+3

0

0.0E+0

0

0.1

0.2

0.3

0.4

CF Volume Fraction

Figure 24: Thermal conductivity, k1, in CF
direction.

0.5

0

0.1

0.2

0.3

0.4

0.5

CF Volume Fraction

Figure 25: Electric conductivity, κ1, in CF direction.


143

Figures 26 and 27 show the heat flux vector distributions under temperature difference in
x-axial direction for 10% CF and 50%CF, respectively. Similarly, Figures 28 and 29 show the
electrical field vector distributions under voltage difference in x-axial direction for 10%CF and
50%CF, respectively. Again, the more the CF%, the worse the uniformity of the distributions is.

under temperature difference (1°C) in x-direction
(10%CF).

under temperature difference (1°C) in x-direction
(50%CF).

Figure 28: Electric field (V/μm) vector distribution
under voltage difference (0.1V) in x-direction
(10%CF).

Figure 29: Electric field (V/μm) vector distribution
under voltage difference (0.1V) in x-direction
(50%CF).

Figure 30 shows the thermal conductivities, k2, in perpendicular to CF direction. The modeling
results are compared with the experimental data from reference (Fei et al. 2007). Both results are
showing the nonlinearity, and the conductivities are both nonlinearly increasing as CF%
increasing. The experimental data are lower than that by modeling, because the uniformity and
bonding condition (cavities existing) in the reality are getting worse as %CF increasing which
lowered the heat transfer rate. In contrast, in the modeling a perfect CF distribution and bonding
144


between the matrix and the filler are always assumed. Comparing Figure 30 with Figure 24, it can
be seen that the thermal conductivity in CF direction is about one order higher than that in
perpendicular directions due to the difference of the thermal conductivities between these two
components.
Figure 31 shows the electric conductivities, κ2, in perpendicular to CF direction, showing the
nonlinearity again. Comparing Figure 31 with Figure 25, it can be seen that the electric
conductivity in CF direction is about 19 orders higher than that in perpendicular to CF direction.
Because CFs are perfect conductor material, so even though 10% volume of CFs could perform
very well conduction if CFs could be well aligned and keep continuity in the conduction direction.
In contrast, the electric conductivity in perpendicular to CF direction is almost zero, due to the
perfect insulator behavior of the polymer and discontinuity of CFs, which could be treated as an
open circuit.

2.5

Equivalent Electrical
Conductivity κ2 ×10-15 (S/m)

0.40

Equivalent Thermal
Conductivity
k2 (W/mK)

k2 perpendicular to CF
direction
k2 from experimental data
(Lei et al. 2007)

0.35
0.30
0.25
0.20
0.15
0.10

κ2 perpendicular to CF direction
2.0

1.5

1.0

0.5
0

0.1

0.2

0.3

0.4

0.5

CF Volume Fraction

Figure 30: Thermal conductivity, k2, in
perpendicular to CF direction.

0.6

0

0.1

0.2

0.3

0.4

0.5

CF Volume Fraction

Figure 31: Electric conductivity, κ2, perpendicular
to CF direction.

4. Conclusion
Mechanical, thermal, and electrical properties of a carbon fiber modified polymeric matrix
composite have been numerically investigated by finite element modeling. A three-dimensional
model has been created, in which continuous CFs were aligned and parallel to each other and
uniformly distributed in the polymer matrix. The behaviors of the composite in two extreme
situations, i.e., parallel or perpendicular to carbon fiber direction, have been analyzed. The effects

145

of the CF volume fraction on these physical properties have been investigated. It shows clearly that
CFs significantly improve the mechanical strength, thermal and electrical conductivity in CF
direction, these physical properties of the composites are primarily dominated by CF volume
fraction, and are linearly increasing as the CF volume fraction increasing, due to the much higher
performance of the CFs. The physical properties in perpendicular to CF direction are nonlinearly
increasing as the CF volume fraction increasing, but the absolute values are still very low due to the
low performance of the polymer matrix.
The future work should expand this modeling technique to look into the other physical
properties, the physical properties of the conductive network of the composites with random CF
orientation, different distribution of the filler, different CF length, different fillers, such as graphite,
and carbon nanotubes, and different cross-section of the fillers, the role of interfacial interaction
between the matrix and the filler, etc.

5. Acknowledgements
This work was supported by NASA EPSCoR Funds #NNX07AL04A, the State of South
Dakota, Mechanical Engineering Department and the College of Engineering at South Dakota
State University.

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Dr. Zhong Hu is an Associate Professor of Mechanical Engineering at South Dakota State University.
He received his BS and Ph.D. in Mechanical Engineering from Tsinghua University. He has worked
for railway manufacturing industry as a senior engineer, Tsinghua University as a professor, Japan
National Laboratory as a fellow, Cornell University, Penn State University and Southern Methodist
University as a research associate. He has authored over 80 peer-reviewed publications in the
journals, proceedings and book chapters in the areas of nanotechnology and nanoscale modeling by
quantum mechanical/molecular dynamics (QM/MD); development of renewable energy related
materials; mechanical strength evaluation and failure prediction by finite element analysis (FEA) and
nondestructive engineering (NDE); design and optimization of advanced materials (such as
biomaterials, carbon nanotube, polymer and composites).
Dr. Xingzhong Yan is an assistant Professor of Electrical Engineering & Computer Science at South
Dakota State University. He received his BSc in Chemistry from Hunan Normal University, MSc in
Physical Chemistry from Chinese Academy of Science, and Ph.D. in Polymer Chemistry and Physics
from Sun Yat-sen University. He has published over 60 peer-reviewed papers in journals,
proceedings, and book chapters in the areas of solar cells, optical materials, femtosecond
spectroscopy, light and thermal management, optical sensors and electrical storage.
Dr. James Jianjun Wu joined the Electrochemistry branch of NASA in 2010. He earned his Ph.D. in
Chemistry from the University of Illinois at Urbana-Champaign and his Masters degree in
Chemistry/Analytical Chemistry from Rutgers University at New Brunswick, NJ. He holds another
Masters degree in Electrochemistry/Electroanalytical Chemistry, and a BS degree in
Chemistry/Chemical Engineering. Dr. Wu possesses postdoctoral experience and more than 10
years of industrial R&D experience prior to joining NASA. Dr. Wu has a varied experience base
with the research and development of catalysts, advanced energy storage materials and
electrochemical systems,
Michelle Manzo has served as Chief of the Electrochemistry Branch since 2005. In this role she
oversees the development of electrochemical systems for future NASA missions. She has been
involved in the development of batteries and fuel cells with an emphasis on aerospace batteries. She
began with the development of alkaline batteries, specifically nickel-hydrogen, nickel-cadmium,
silver-zinc and nickel-zinc and more has recently been addressing lithium-based systems. Michelle
has been involved with interagency aerospace flight battery systems collaborations. Michelle has
received numerous awards and recognition throughout her career. She has authored or co-authored
more than 40 papers and has been recognized with various awards that include a NASA Exceptional
Achievement Medal, a NASA Exceptional Service Medal, an R&D 100 award, and NASA Group
Achievement Awards for the Li-Ion Battery Technology Team and the NASA Spacecraft Fuel Cell
Development Team.

Peer Review: This article has been internationally peer-reviewed and accepted for
publication according to the guidelines given at the journal’s website.

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Characterizing Carbon Fiber Polymer Composites

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