NANOFIBER NANOCOMPOSITE FOR STRENGTHENING NOBLE
LIGHTWEIGHT SPACE STRUCTURES
N.M. Awlad Hossain
Assistant Professor
Department of Engineering and Design
EASTERN WASHINGTON UNIVERSITY
Cheney, WA – 99004
Email: [email protected]
Phone: 509-359-2871
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ABSTRACT
Structural weight reduction to minimize the launching cost is one of the desired outcomes of all
space missions. One way to achieve this objective is through the replacement of metallic
structures with advanced composites. Currently, composite materials are widely used in many
engineering applications because of their lightweight, novel multifunctional properties and low
cost. The primary focus of this research was to examine the effects of fiber geometry to improve
the strength of composites, and to pursue the possible advantages of using nanofibers instead of
conventional fibers. First, a theoretical background was established to determine that the
nanofiber composite contains significantly more surface area of fiber over the conventional
composite keeping the same fiber volume fraction. The increased surface area of fiber can help
to compensate the imperfect bonding between the fiber-matrix interphase. Then, extensive
numerical simulation was performed through analyzing three different RVE models where the
fiber volume fraction kept constant using reduced fiber diameter. Therefore, the surface area of
fiber was increased gradually in each RVE model. The effects of fiber geometry were studied by
comparing axial and shear stresses in the fiber region. As the cross sectional area of fiber
remained the same in all RVE models, the axial stress of fiber was found relatively unchanged.
On the other hand, the surface area of fiber was found to play an important role in shear stress
distribution. The shear stress of fiber was found to be reduced significantly with increasing its
surface area. Numerical simulation was then repeated with changing the boundary conditions. In
all cases, an RVE model with the highest surface area of fiber was found to offer the lowest
shear stress compared to the other RVE models. Therefore, if the composite strength depends on
surface area of fiber, then a composite consisting of nanofibers will be stronger than a
conventional composite prepared with same volume fraction, or conversely will be lighter at the
same strength.
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TABLE OF CONTENTS
Topics
I.
II.
III.
A.
B.
C.
IV.
V.
VI.
VII.
Page Number
Introduction ………………………………………………………………
Theoretical Background
………………………………………………
Analysis
………………………………………………………………
Numerical Models
………………………………………………
Boundary Conditions and Loading ………………………………………
Materials Properties ………………………………………………………
Results and Discussions
………………………………………………
Conclusions ……………………………………………………………...
Future Work ………………………………………………………………
References
………………………………………………………………
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5
7
9
9
11
12
12
24
25
26
LIST OF FIGURES
1. Simplified concept of composite materials and the Representative Volume
Element (RVE).
…………………………………………………….
6
2. Illustration of RVE of conventional and nanofiber composites
…….
8
3. Illustration of different RVE model with increased fiber surface area while
keeping the same fiber volume fraction.
……………………………….
10
4. Quarter symmetric models of conventional and nanofiber composites used
in the numerical analysis.
………………………………………………
10
5. 3D view of the conventional composite model.
……………………..
11
6. Axial stress (z-direction) distribution along the fiber axis between conventional
and nanofiber composite models.
…………………………………………..
13
7. Shear stress distribution along a nodal path (on 12 o’clock section) on the
topmost fiber layer between the conventional and nanofiber composite models. … 14
8. A closer view of shear stress (yz-direction) distribution along a nodal path
(on 12 o’clock section) on the topmost fiber layer between the conventional and
nanofiber composite models ………………………………………………..
14
9. Axial stress (z-direction) distribution along a nodal path coincident to the fiber axis
between the conventional and two other nanofiber composites. …………………. 15
10. Shear stress distribution along a nodal path (on 12 o’clock section) on the topmost
fiber layer between the conventional and two other nanofiber composites. …… 16
11. A closer view of shear stress (yz-direction) distribution along a nodal path on
the topmost fiber layer between the conventional and nanofiber composite, RVE-3. 17
12. A closer view of shear stress (yz-direction) distribution along a nodal path on the
topmost fiber layer between two nanofiber composites RVE-2 and RVE-3. …… 17
13. Contour plot of axial displacement profile of (a) conventional and (b) nanofiber
composite, RVE-2 for boundary displacement of 0.5 micron. ………………… 18
14. Contour plot of major principal stress distribution for (a) conventional and (b)
nanofiber composite, RVE-2. …………………………………………………
19
15. Contour plot of shear stress (τyz) distribution for (a) conventional and (b)
nanofibers composite models. ………………………………………………..
19
16. Axial stress (z-direction) distribution along a nodal path coincident to the fiber axis
between the conventional and nanofibers composite models. …………………… 20
17. Shear stress (yz-direction) distribution along a nodal path (on 12 o’clock section) on
the topmost fiber layer between conventional and nanofiber composite models. 21
18. Contour plot of major principal stress distribution for (a) conventional
and (b) nanofibers composite models. …………………………………………… 22
19. Contour plot of shear stress (τyz) distribution for (a) conventional and
(b) nanofibers composite models. ………………………………………………. 22
20. Axial stress distribution along the fiber axis between conventional and two
nanofiber composite models. ………………………………………………….
23
21. Shear stress (yz-direction) distribution along a nodal path (on 12 o’clock section)
on the topmost fiber layer between conventional and two nanofiber composite
models. …………………………………………………………………………. 23
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I.
Introduction
Structural weight reduction to minimize the launching cost is one of the targeted outcomes of
NASA’s space missions. One way to achieve this objective is through the replacement of
metallic structures with advanced composites. Currently, lightweight composite materials are
used in a variety of space applications including solar sails, sun shields, space mirrors &
telescopes, lunar habitats, scientific balloons, and inflatable antennas. The primary focus of this
research is to examine the effects of fiber geometry to improve the strength of composite
materials. The proposed research will investigate the possible advantages of using nanofibers
(diameters ≤ 10-9 meter) instead of conventional fibers (diameters ≥ 10-6 meters).
Mathematically, it can be shown that the nanofiber composite contains significantly more surface
area over the conventional composite at no cost of volume fraction. The increased surface area
can help to compensate for the imperfect bonding between the fiber-matrix interphase.
Therefore, composite materials consisting of nanofibers are expected to offer higher strength
than a conventional composite prepared with the same volume fraction.
Composite materials (or composites for short) are engineered materials made from two or
more constituent materials with significantly different physical or chemical properties, and
remain separate and distinct on a microscopic level within the finished structure. The two
constituent materials are matrix and fiber (or reinforcement). The matrix material surrounds and
supports the fiber materials by maintaining their relative positions. The fibers impart their special
mechanical and physical properties to enhance the matrix properties. A synergism produces
material properties unavailable from the individual constituent materials, while the wide variety
of matrix and fiber materials allows the designer of the product or structure to choose an
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optimum combination. Figure 1(a) represents a simplified concept of composite material with
fiber and matrix constituents. Figure 1(b) depicts the Representative Volume Element (RVE) –
the smallest cell, as shown by dotted line, to describe the individual composite constituents. The
RVE is then divided into four subcells, as shown in Figure 1(c), all of rectangular geometry.
Fig. 1: Simplified concept of composite materials and the Representative Volume Element
(RVE).
In a series of papers documented by Dr. Jacob Aboudi [1, 2], the overall behavior of
composite materials was explored through micromechanical analysis. The analysis was
performed using an RVE approach to predict the elastic, thermoelastic, viscoelastic and
viscoplastic responses of composites. The composite material was assumed homogeneously
anisotropic (properties depend on the direction) continuum without considering the interaction
effects between fibers and matrix. Goh et al. [3] developed an analytical solution describing the
average fiber stress (σf), where the fiber was surrounded by an elastic matrix. The model
proposed in [3] consists of a differential equation whose solution provided the distribution of
fiber stress (σf) and shear stress between fiber and matrix (τ) as a function of length. Expressions
for (σf) and (τ) were derived without considering the effect of fiber geometry and fiber-fiber
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interaction. According to the micromechanics theory, an increase in strength cannot be obtained
from scaling fiber size while maintaining the same fiber volume fraction. Assuming a unit depth,
volume fraction (Vf) of fiber for the RVE model is defined as follows:
Vf =
h12
Fiber Volume
=
Total Volume (h1 + h2 ) 2
(1)
It can be seen that if the volume fraction is maintained while scaling h1 down, this will directly
affect the value of h2. Consequently, the ratio h1/h2 in the micromechanics equation will remain
the same, and therefore not affect the strength at all. In conclusion, higher strength to weight
ratios cannot be obtained from scaling fiber size down at constant volume fraction.
II.
Theoretical Background
One of the focal points of the proposed research is to determine the strength of composites by
creating a more effective RVE model that will incorporate the interaction effects of fibers and
matrix. It has been suggested through other studies that the surface area (As) of fibers plays an
important role in strength of the composites. An advantage of using nanofibers is to increase the
surface area, which can help to compensate for imperfect bonding. It can be mathematically
shown that the nanofibers composite contains significantly more surface area over the
conventional composite at no cost to volume fraction. Figure 2 illustrates the concept of
maintaining the same fiber cross-sectional area while increasing fiber surface area.
Fig. 2: (a) Illustration of RVE of conventional composite with single unidirectional fiber.
(b) Illustration of RVE of modified nanocomposite with many unidirectional fibers.
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Assuming the same depth dimension, the two fiber scenarios, as shown in Figure 2,
represent the following definitions for cross sectional area:
Ax ,c =
Ax ,n =
π
4
π
4
* h12,c ,
(2)
* h12,n ,
(3)
Where Ax denotes cross-sectional area, subscripts with n and c refer to the nanofiber composite
and the conventional composite respectively, and h1 is the fiber diameter. The cross-sectional
area of the conventional composite (Ax,c) can be considered as some multiple, N, of the individual
nanofiber cross-sectional area as such:
Ax ,c = N * Ax ,n .
(4)
Substituting the definition for Ax,c and Ax,n simplifies the results in the following relationship:
h1,n =
h1,c
N
.
(5)
Next, the following two relations assess the total surface areas for each unit cell:
As ,n ~ N * π * h1,n ,
(6)
As ,c ~ π * h1,c ,
Where As is shown as proportional as the depth dimension is omitted. Considering the surface
area of nanofiber composite and substitute the relationship as found in Equation 5, the following
proportionality is obtained
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As ,n ~ N * π * h1,n
⇒ As ,n ~ N * π *
h1,c
N
⇒ As ,n ~
N * (π * h1,c )
⇒ As ,n ~
N * As ,c .
(7)
So, mathematically it can be proved that the surface area of fiber in nanofiber composite
will be higher than that of conventional composite by a factor of N , where N represents the
total number of nano fibers. Therefore, if the composite strength depends on fiber surface area,
then a composite consisting of nanofibers will be stronger than a conventional composite
prepared with same volume fraction. However, this strength advantage is not apparent in
Aboudi’s micromechanics theory. The primary purpose of this research will be to examine the
effects of fiber geometry, and pursuing the possible advantages of utilizing nanofibers within
composite materials using the RVE approach.
III. Analysis
A.
Numerical Models
Numerical models of conventional and nanofiber composites were developed using nonlinear
finite element code ABAQUS and ANSYS. First, three different RVE models were developed as
shown in Figure 3. For simplicity in the following sections, these numerical models are called
RVE-1, RVE-2, and RVE-3. Note that all dimensions are in micrometer (μm).
RVE-1, as shown in Figure 3(a), represents a conventional composite with one complete
fiber connected with matrix. RVE-2, as shown in Figure 3(b) represents a nanofiber composite,
with two nano fibers (one complete + four quarters) connected with matrix. Similarly, RVE-3, as
shown in Figure 3(c), also represents a nanofiber composite with four nano fibers (one complete
+ 4 halves+ 4 quarters). In RVE-2 and RVE-3, the fiber size was scaled down from radial
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dimension 5 micron to 3.5 micron and 2.5 micron, respectively. All RVE models have the same
volume fraction of fiber (20%). The surface area of fiber in RVE-2 and RVE-3 was increased by
40% and 100%, respectively compared to the RVE-1. For analysis simplicity and to minimize
the computational time, quarter symmetric models of the conventional and nanofiber composites,
as shown in Figure 4, were used.
Fig. 3: Illustration of different RVE model with increased fiber surface area while keeping
the same fiber volume fraction.
Fig. 4: Quarter symmetric models of conventional and nanofiber composites used in the
numerical analysis.
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B.
Boundary Conditions and Loading
Numerical analyses were performed with combining different types of boundary conditions and
loadings. For better understanding, a 3D view of the quarter symmetric conventional composite
model is shown in Figure 5. Two different boundary conditions were considered, as mentioned
below.
a. The entire back surface (z = 0), which contains both fiber and matrix nodes, was
constrained in its normal direction.
b. Only the fiber nodes located at the back surface (z = 0) were constrained in the normal
direction.
Fig. 5: 3D view of the conventional composite model.
For each case, symmetric boundary conditions were applied. In addition, to constrain the
rotational motion, two adjacent sides (x = 10, and y = 10) were also constrained to their normal
directions. Also, one node point on the back surface was pinned to prevent translation in the
global x- and y- directions. For each boundary condition, boundary displacement loading was
assigned to the matrix nodes of the front surface.
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The fiber surface was connected by ties to all correlated matrix surface. The numerical
model did not consider any interphase zone. The other quarter symmetric models, as shown in
Figure 4, were also analyzed using the same boundary conditions and loading as discussed
above.
C.
Material Properties
The materials properties are simply elastic and include Young’s modulus and Poisson’s ratio.
Standard mechanical properties of E-glass and Epoxy were used to mimic the fiber and matrix,
respectively.
IV. Results and Discussions
The following results were obtained when the entire back surface was constrained to its normal
direction, and the boundary displacement was applied to matrix at the front surface. All three
quarter symmetric models, as shown in Figure 4, were analyzed. For each case, axial and shear
stress distribution of the fiber was studied. As the matrix region of nanofiber composite models
has different geometry compared to conventional composite, boundary displacement assigned to
nanofiber composites was tuned until they offered the same reaction force found for the
conventional composite model.
First, we compared the stress distribution between RVE-1 and RVE-2. The axial stress
distribution along a nodal path coincident with the fiber axis (x = 0 and y = 0) is shown in Figure
6. The cross sectional area of fiber remains the same between the conventional and nanofiber
composite models. Therefore, the axial stress distribution was found to be fairly unchanged. The
axial stress was found higher near the fixed end and gradually decreased near the free end where
load was applied.
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Fig. 6: Axial stress (z-direction) distribution along the fiber axis between conventional and
nanofiber composite models.
The shear stress distribution along a nodal path (at angular orientation θ = 90 degree, as
called 12 o’clock section) on the topmost fiber layer is shown in Figure 7. The surface area of
fiber was increased by 40% in RVE-2 compared to RVE-1. Therefore, the shear stress
distribution in RVE-2, representing a nanofiber composite, was found significantly reduced. A
closer observation of the shear stress distribution, up to nondimensional length (z/L) equal to 0.5,
is shown in Figure 8. Shear stress in RVE-2 was found to be reduced by approximately 50%
compared to RVE-1. Shear stress was also found to be smaller near the fixed end and increased
exponentially towards the free end.
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Fig. 7: Shear stress distribution along a nodal path (on 12 o’clock section) on the topmost
fiber layer between the conventional and nanofiber composite models.
Fig. 8: A closer view of shear stress (yz-direction) distribution along a nodal path (on 12
o’clock section) on the topmost fiber layer between the conventional and nanofiber
composite.
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The axial stress distribution along the fiber axis of all three different RVE models is
shown in Figure 9. As the fiber cross sectional area is the same in each RVE model, the axial
stress was found almost unchanged up to the nondimensional length (z/L) equal to 0.75. Some
discrepancies were observed near the free end where boundary displacement was applied.
Fig. 9: Axial stress (z-direction) distribution along a nodal path coincident to the fiber axis
between the conventional and two other nanofiber composites.
On the other hand, the shear stress distribution of the topmost fiber layer of all three
different RVE models is shown in Figure 10. The shear stress in the RVE-2 and RVE-3,
representing nanofiber composites, was found to be reduced more with increasing the fiber
surface area. The surface area of fiber in RVE-3 is 60% more compared to RVE-2. Subsequently,
the shear stress in RVE-3 was found much smaller compared to RVE-2.
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Fig. 10: Shear stress distribution along a nodal path (on 12 o’clock section) on the topmost
fiber layer between the conventional and two other nanofiber composites.
For closer observation, the shear stress distribution of RVE-3, up to nondimensional
length (z/L) equal to 0.50, is plotted separately with RVE-1 and RVE-2. These results are shown
in Figures 11 and 12, respectively. The surface area of fiber in RVE-3 is increased by 100%
compared to the RVE-1, and by 60% compared to RVE-2. Shear stress of the topmost fiber
layer, as shown in Figures 11 and 12, is expected to decrease more with the additional increment
of the surface area of fiber.
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Fig. 11: A closer view of shear stress (yz-direction) distribution along a nodal path on the
topmost fiber layer between the conventional and nanofiber composite, RVE-3.
Fig. 12: A closer view of shear stress (yz-direction) distribution along a nodal path on the
topmost fiber layer between two nanofiber composites RVE-2 and RVE-3.
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Numerical simulation of the above three RVE models was then repeated with changing
the boundary condition as follows. Instead of constraining the entire back surface, only the fiber
nodes on the back surface were constrained to their normal z-direction. Therefore, matrix nodes
on the back surface were released to translate along the global z-direction. Boundary
displacement was assigned to the matrix nodes of the front surface, as used before.
First, RVE-1 and RVE-2 were analyzed and the axial displacement profile is shown in
Figure 13. For simplicity, only the fiber displacement is shown. Note that the fiber diameter in
RVE-2 was reduced by 30% keeping the same fiber volume fraction, and the surface area of
fiber was increased by 41%.
Figure 13: Contour plot of axial displacement profile of (a) conventional and (b) nanofiber
composite, RVE-2 for boundary displacement of 0.5 micron.
Figure 14 shows the contour plot of major principal stress, developed in the fiber region,
of RVE-1 and RVE-2. The maximum stress was found to be 6.177 GPa and 4.82 GPa in RVE-1
and RVE-2, respectively. Therefore, the principal stress was found to be reduced by 22% in
RVE-2 compared to the RVE-1 (conventional composite).
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Figure 14: Contour plot of major principal stress distribution for (a) conventional and (b)
nanofiber composite, RVE-2.
Figure 15 shows the shear stress (τyz) distribution for RVE-1 and RVE-2 models. The
maximum shear stress was found to be 2.479 GPa and 1.795 GPa for RVE-1 and RVE-2 models,
respectively. Therefore, the maximum shear stress was found to be reduced by 28% in the
nanofiber composite RVE-2 compared to the conventional composite RVE-1.
Figure 15: Contour plot of shear stress (τyz) distribution for (a) conventional and (b)
nanofibers composite models.
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It was evident from these contour plots that both axial and shear stress were found to be
reduced by 22% and 28%, respectively when the surface area of fiber was increased by 41% in
the nanofibers composite model.
The axial stress distribution along the fiber axis (x = 0, and y = 0) of the two different
models, RVE-1 and RVE-2, is shown in Figure 16. Boundary displacement was also tuned to
RVE-2 to receive the same reaction force at the fiber nodes as found in RVE-1. The axial stress
distribution was expected to remain unchanged because of having the same fiber cross sectional
area in each RVE model. Some difference was observed near the front surface, where boundary
displacement was applied. The stress distribution pattern looks very similar to Figure 6.
Fig. 16: Axial stress (z-direction) distribution along a nodal path coincident to the fiber axis
between the conventional and nanofibers composite models.
The shear stress distribution along a nodal path on the topmost fiber layer is shown in
Figure 17. Due to change in boundary condition, the shear stress distribution observed in Figure
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17 is significantly different from Figure 7. Shear stress was also found to be smaller near the mid
section and increased exponentially towards the fixed and free ends. As the surface area of fiber
was increased in RVE-2 compared to RVE-1, the shear stress in RVE-2 was found significantly
reduced.
Fig. 17: Shear stress (yz-direction) distribution along a nodal path (on 12 o’clock section)
on the topmost fiber layer between conventional and nanofiber composite models.
The computational analysis was then advanced by replacing the conventional composite
(i.e., RVE-1) with four nanofibers (i.e., RVE-3) keeping the fiber volume fraction same. Here,
the fiber diameter was reduced by 50% and the surface area was increased by 100%. Consistency
in results was expected as the fiber diameter was further reduced in the nanofibers composite
model. The contour plots of principal stress and shear stress of RVE-3 model is shown in Figures
18 and 19, respectively. The maximum stress values were found to be 4.143 GPa and 1.671 GPa,
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respectively. Both of the stresses were found to be reduced by 33% compared to the conventional
composite model RVE-1.
Figure 18: Contour plot of major principal stress distribution for (a) conventional and (b)
nanofibers composite models.
Figure 19: Contour plot of shear stress (τyz) distribution for (a) conventional and (b)
nanofibers composite models. The axial stress distribution along the fiber axis (x = 0 and y = 0) of three different RVE
models is shown in Figure 20. No significant different was observed between them as the fiber
cross sectional area was the same. The shear stress is also plotted for the topmost fiber layer at an
angular orientation θ = 90 degree for the three different RVE models as shown in Figure 21. An
RVE model with higher surface area of fiber was found to offer reduced shear stress. Therefore,
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the RVE-3 with the highest surface area of fiber was found to offer the least shear stress. The
shear stress distribution pattern in Figure 21 has similar trend as shown in Figure 17.
Fig. 20: Axial stress distribution along the fiber axis between conventional and two
nanofiber composite models.
Fig. 21: Shear stress (yz-direction) distribution along a nodal path (on 12 o’clock section)
on the topmost fiber layer between conventional and two nanofiber composite models.
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V.
Conclusions
The primary purpose of this research was to examine the effects of fiber geometry, and pursuing
the possible advantages of utilizing nanofibers within composite materials using the
representative volume element (RVE) approach. Mathematically, it was shown that nanofibers
composite contains higher surface area over the conventional composite at no cost of volume
fraction. Three different RVE models were analyzed where the surface area of fiber was
increased gradually, keeping the fiber volume fraction constant. The effects of increased surface
area were investigated by comparing the axial and shear stresses of fiber region. As the cross
sectional area of fiber remained the same in all RVE models, the axial stress along the fiber axis
was found almost unchanged. Therefore nanofiber or the fiber geometry was found to be neutral
on the axial strength. The axial stress was found to be the highest near the fixed end and
gradually decrease towards the free end. This increased surface area of fiber was found to play
an important role in shear stress distribution. The shear stress of fiber was found to be reduced
significantly with increasing its surface area. Therefore, an RVE model with the highest surface
area of fiber was found to offer the least shear stress compared to the other RVE models.
Numerical simulations were repeated with changing the boundary conditions, and consistencies
in results were observed. In all cases, an RVE model with increase surface area of fiber was
found to offer reduced shear stress. Therefore, on the basis of shear stress, developed in the fiber
region, a composite consisting of nanofibers will be stronger than a conventional composite
prepared with same volume fraction, or conversely will be lighter at the same strength. It is
anticipated that the nanofibers composite is not only beneficial to reduce the shear stress but also
helps to compensate for imperfect fiber-matrix bonding.
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VI. Future Work
Additional research work is essential to explore the detailed micromechanical behavior of
nanofibers composite. In particular, the following items deserved to be potential candidate for
further investigation:
1. Currently, the effect of nanofibers on composite strength has been studied by using the
RVE approach. To make the research outcomes robust, the nanofibers composite needs to
be analyzed using the “unit cell” approach too. A unit cell has similar concept like the
RVE has. The basic difference is that the unit cell’s dependence on geometric symmetry
that creates a repeatable structure. A unit cell must be symmetrical such that multiple cells
combined at any space dimension will eventually build the actual structure of the material.
2. In this current research, the geometry of nanofibers has been found to be neutral on axial
strength but has significant impact on shear strength of composite material. The axial stress
along the fiber axis was found almost unchanged for the nanofibers composite compared to
the conventional composite. However, it is anticipated that the nanofibers composite plays an
important role in case of surface defects. Most of the composites fail under tensile loading by
fiber breakage. After a fiber break, the fiber can no longer sustain the load it originally
endured, and so must transfer the load to the nearest fiber. The length of fiber needed to
transfer 90 percent of the original load to the next nearest fiber is considered the “ineffective
fiber length”. It is worth to investigate whether the nanofibers composite offers shorter
ineffective fiber length compared to the conventional composites. Shorter ineffective fiber
length is desired for better composite strength.
3. Additional research is also essential to simulate the details of broken fiber scenario in
composite material. It is important to know how the stress distribution, both axial and
shear, will change and vary in a particular fiber when the other nearest fiber is broken.
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References
[1]
[2]
[3]
Aboudi, J., “Micromechanical Analysis of Composites by the Method of Cells,” ASME,
1989.
Aboudi, J., “Micromechanics Prediction of Fatigue Failure of Composite Materials,”
Journal of Reinforced Plastic Composites, 1989.
Goh, K.L., Aspden, R.M., and Hukins, D.W.L., “Shear Lag Models for Stress Transfer
from an Elastic Matrix to a Fiber in a Composite Materials,” Int. J. Materials and
Structural Integrity, Vol. 1, Nos. 1/2/3, 2007.
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