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Tensile properties of natural fibers with
variation in cross-sectional areaJunji Noda
a, Yujiro Terasaki
a, Yuji Nitta
a& Koichi Goda
a
aGraduate School of Science and Engineering, Yamaguchi
University, 2-16-1 Tokiwadai, Ube 755-8611, JapanPublished online: 01 Dec 2014.
To cite this article:Junji Noda, Yujiro Terasaki, Yuji Nitta & Koichi Goda (2014): Tensile propertiesof natural fibers with variation in cross-sectional area, Advanced Composite Materials, DOI:
10.1080/09243046.2014.985421
To link to this article: http://dx.doi.org/10.1080/09243046.2014.985421
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Tensile properties of natural bers with variation in cross-sectionalarea
Junji Noda*, Yujiro Terasaki, Yuji Nitta and Koichi Goda,1
Graduate School of Science and Engineering, Yamaguchi University, 2-16-1 Tokiwadai, Ube755-8611, Japan
(Received 4 August 2014; accepted 22 September 2014)
Natural ber-reinforced green composites are increasingly being used in variousindustries. Generally speaking, the cross sections of natural bers are not circular.Moreover, they show wide variation in their cross-sectional area. Nevertheless, thetensile properties of natural bers are often evaluated on the assumption of a circularcross-sectional shape. Therefore, we proposed a new method to calculate the naturalber cross-sectional area by measuring their projection widths to estimate the propertensile properties. This method was used to investigate the effects of cross-sectionalarea variation on tensile properties. Additionally, to estimate the proper Weibull
parameters of natural bers apart from the cross-sectional area variation, we pro-posed a strength distribution function that incorporates cross-sectional area variation,based on a Weibull distribution. Finally, we formulated a strength distribution func-tion of natural bers that includes the cross-sectional area variation.
Keywords:natural bers; cross-sectional area; tensile strength; Weibull analysis
1. Introduction
Plant-based natural bers consist mainly of cellulose, which has a high elastic modulus
of 138 GPa in its crystal.[1] It is expected that natural bers will be widely substituted
for conventional glass bers. From the viewpoint of effective utilization for the bio-
mass, the application of composites that consist of the natural bers and biomass-based
biodegradable resin, designated as green composites, has also been studied.[2,3] These
natural bers mainly sustain the loads in green composites. Therefore, it is important
that basic mechanical properties such as the Youngs modulus and tensile strength of
natural bers are estimated precisely as well as the synthetic glass bers. To evaluate
the mechanical properties of natural bers, many researchers have been inclined to
assume uniform shape and size of bers, as they do with synthetic bers.[412]
Additionally, the material reliability of natural bers and synthetic bers have been
evaluated using Weibull statistics.[5,10,1216]
Plant-based natural bers consist of aggregations of many single ber cells, similar
to the kenaf bers portrayed in Figure 1.[17] A single ber cell is also an aggregation
of cellulose micro-brils (CMF), hemi-cellulose, and lignin. The tensile strength and
Youngs modulus of this single ber along the ber axis depend on the CMF angle.
*Corresponding author. Email:[email protected] member
2014 Japan Society for Composite Materials, Korean Society for Composite Materials and Taylor & Francis
Advanced Composite Materials, 2014
http://dx.doi.org/10.1080/09243046.2014.985421
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The specic gravity, strength, and Youngs modulus of these bers also depend on the
lumen, placed in the center of bers.[18,19] Figure 1 shows that the cross-sectional
shape of natural bers is complicated, differing greatly from a circle shape. For textile
engineering, the cross-sectional area is estimated indirectly by the neness as in a
denier.[20] Measurement of the direct cross-sectional area of bers is nevertheless nec-
essary to estimate the stress of bers exactly. Xu et al. precisely estimated the cross-
sectional area of sisal bers using image analysis based on many cross-sectional areaphotographs obtained using optical microscopy.[21] They reported that the conventional
estimation method of the cross-sectional area as a circle shape underestimated the area.
They also reported that the strength was overestimated by approximately two times.
Although their method was able to estimate the cross-sectional area precisely, the speci-
men used for tensile tests was not directly applicable because cross-sectional photo-
graphs were required.
Suzuki et al. estimated the cross-sectional area of kenaf bers using image analysis
with the specimen after tensile tests.[17] They prepared specimens by embedding them
in resin. The cross-sectional shape of the specimens was then observed at 0.2 mm inter-
vals along the ber axis. They reported that the cross-sectional area varied among indi-
vidual bers and that the cross-sectional area along the ber axis also varied within
bers. The ber strength evaluated using their method was precise because the cross-
sectional area was estimated exactly. Silva et al. reported that the precise cross-sectional
area of sisal bers was measured using micrographs obtained from scanning electron
microscopy.[15] The cross-sectional shape of natural bers was found to be polygonal,
with 57 corners.[7]
We have proposed a new method of evaluating the ber cross-sectional area pre-
cisely, called DB (date base)-based approximation in an earlier paper.[22] This DB-
based approximation method is based on the database for a distribution of actual ber
cross-sectional area obtained from cross-sectional pictorial images of ber-reinforced
composites. In this study, the DB-based approximation method was extended to variouspolygon shapes in addition to ellipses to elucidate the appropriate shape as the cross-
sectional area of natural bers and to verify the validity of DB-based approximation.
Figure 1. Laser microscope image of cross-sectional area.
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(d) Octagon with four diagonal lines equalized to four projective widths along 0,
45, 90, and 135 directions
(e) Dodecagon with six diagonal lines equalized to six projective widths along 0,
30, 60, 90, 120, and 150 directions(f ) Icositetragon with 12 diagonal lines equalized to 12 projective widths
For each image, the cross-sectional areas of the shapes above were measured. The
actual cross-sectional area painted in black was estimated using image analysis. Conse-
quently, the approximate cross-sectional areas were obtained as shown in Figure3(b)(g)
for the case of the actual cross-sectional area of 12,615 m2. Results showed that these
approximate areas were larger than the actual area.
The relations between the actual cross-sectional area and the circle assumption and
polygon assumption of the icositetragonal shape for kenaf bers are shown in
Figure 4(a) and4(b). The correlation coefcients between the actual and assumed areas
were, respectively, 0.800 and 0.909: the cross-sectional area using the polygon assump-
tion of icositetragonal shape was estimated more precisely. The other coefcients for
the case of assumed shapes described above are shown in Table 1. These results show
that the coefcients for the circle and ellipse assumption were high values of
0.800.84. The coefcients for the polygon assumption increased to about 0.9 with
increasing projective width. However, results revealed that the increase of the projective
width was not necessary to obtain high coefcients because the increase in coefcients
was saturated in the case of the hexagon shape. This table shows the average cross-
sectional areas of measurements at the various assumed shapes. The approximate
expression using the least-squares method between the actual areas and areas using the
icositetragon assumption was y= 0.606x + 852. In this study, the cross-sectional areasmeasured using the polygon assumption were transformed to their actual areas using
this equation. This method was denominated as a DB-based approximation because the
(a) Real (12615 m2) (b) Circle (14621 m
2) (c) Ellipse (18557 m
2)
(d) Hexagon
(14874 m2)
(e) Octagon
(15701 m2)
(f) Dodecagon
(17120 m2)
(g) Icositetragon
(15658 m2)
Figure 3. Assumption of various cross-sectional shapes.
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actual areas were estimated based on the database (DB) accumulated from the numer-
ous measurements that were taken.
2.2. Tensile properties of bers
Kenaf single-ber specimens having 10 mm gage length were prepared as shown in
Figure 5. For each ber of the 36 specimens, the projective width along the ber direc-
tion was measured using the laser scan micrometer (LSM-500S; Mitutoyo Corp., Japan)
during the shift to the direction of arrows. The measurement length was 8 mm, exceptfor the 1 mm from the end of the gage length to avoid interference from erroneous
measurements with the paper board of specimens. In all, 81 measurements were taken
Figure 4a. Relation between real and circle-assumed cross-sectional areas.
Figure 4b. Relation between real and icositetragon-assumed cross-sectional areas.
Table 1. Coefcient of correlation between real and assumption cross-sectional areas and aver-age areas of measurements.
Fiber Real Circle Ellipse Hexagon Octagon Dodecagon Icositet.
Coefcient of correlation 0.800 0.840 0.900 0.880 0.904 0.909Cross-sectional area (m2) 5010 7356 7045 5783 6338 6672 6865
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per ber along the ber direction at intervals of 0.1 mm. After the rst measurements,
the micrometer was rotated 15, as shown in Figure 6(b). The projective width along
the ber direction was then measured again. Such measurements were repeated from12 directions of 0 to 165. A typical distribution of the projective width from 0, 60,
and 120 directions is shown in Figure 7. Results claried that multi-angle measure-
ments along the ber direction are necessary because the projective widths are changed
at the direction and position of measurements. Using the obtained projective widths,
[mm]
Fiber
10
1010 10
5
Figure 5. Size ofber specimen.
(a) First measurement position (0) (b) Subsequent rotated position (15)
Laser
Fiber specimen
Figure 6. Measurement system using laser scan micro-meter.
Figure 7. Projection width of ber.
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the cross-sectional areas at each position based on the shape dened by chapter 2.1,
Figure 3(b)(g) were calculated. Then the average cross-sectional area of each ber
was obtained. Then, tensile tests at 0.8 mm/min were conducted using these specimens.
Table 2 shows the average cross-sectional area, Youngs modulus, and tensile strength
at each assumed shape. In this table, the approximation of cross-sectional area using
the icositetragonal shape was the transformed area of icositetragon shape based on the
DB-based approximation. The approximation of Youngs modulus and tensile strength
were the estimated modulus and strength using the transformed area. Consequently,
results suggest that these Youngs modulus and tensile strength were estimated accu-
rately because the calculated area using the DB-based approximation was close to the
actual area shown in Table 1.
3. Effect of cross-sectional area variation on tensile properties
3.1. Youngs modulus
Figure 8shows the relation between the average cross-sectional area and Youngs mod-
ulus for 36 kenafbers. Figure9shows the relation between the coefcient of variation
for the variation in within-ber cross-sectional area and Youngs modulus. From
Figure 8, the Youngs modulus of natural bers decreases concomitantly with the
Table 2. Cross-sectional area and tensile strength of each ber.
Circle Ellipse Hexagon Octagon Dodecagon IcositetragonAppro.
(Icositet.)
Cross-sectional
area (m2)
6810 6365 6192 6367 6533 6572 4835
Youngsmodulus(GPa)
11.6 12.2 12.6 12.2 11.8 11.8 14.8
Tensile strength(MPa)
235 247 256 247 241 240 310
Figure 8. Relation between Youngs modulus and average cross-sectional areas.
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increase in the cross-sectional area. This cross-sectional area dependence of the Youngs
modulus was also reported by Baley [7] and by Charlet et al. [23]. They investigated
the relation between the diameter of natural bers and Youngs modulus. Aramid bers
reportedly present a similar tendency.[24] Results presented in Figure 9 showed that
the Youngs modulus also decreases with the increase in the coefcient of variation as
well as the average cross-sectional area. The reason why the Youngs modulus
decreases according to the variation of cross-sectional area is discussed in the following
chapter in conjunction with the effects of that variation on tensile strength.
3.2. Tensile strength
Figure 10 shows the relation between the average cross-sectional area and tensile
strength. Figure11shows the relation between the coefcient of variation for the varia-
tion in within-ber cross-sectional area and tensile strength. Doan et al. reported that
the relation between the tensile strength and the cross-sectional area of bers obtained
from the convenient area measurements was investigated.[25] No dependence of the
Figure 9. Relation between Youngs modulus and coefcient of variation of cross-sectional areas.
Figure 10. Relation between tensile strength and average cross-sectional areas.
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cross-sectional area on the tensile strength was found. However, because the tensilestrength depends strongly on the area, as shown in Figure 10, the size effect on
strength explained by the increase in the defect probability was expressed in this study.
Figure11results showed that the tensile strength also decreases along with the increase
in the coefcient of variation as well as the Youngs modulus.
Figure 1shows that kenaf bers have a multi-cell structure grouped by some single
cells. The effect of cross-sectional area variation on tensile properties for kenaf bers
arises from the micro-structure of each single cell. The change in the micro-bril angle
of cellulose and the lumen size depends on the growth of natural bers.[18,19] Espe-
cially, in the case of bers that have large variation of the within-ber cross-sectional
area, because a part with large local deformation attributable to the small area exists,the obtained Youngs modulus and tensile strength are expected to decrease.
4. Tensile strength distribution of natural bers
In this chapter, the strength distribution based on the Weibull statistics was proposed
considering the variation of cross-sectional area within bers and between bers. The
effect of these variations on tensile properties was investigated.
4.1. True Weibull parameter estimation except for the effect of cross-sectional area
variation
4.1.1. Conventional Weibull model
A Weibull model used for general bers to show the strength distribution is designated
as a straight bar model that considers only the variation of cross-sectional area between
bers. The cumulative probability ofber failure F is given by the following equation.
F 1exp VV0
r
r0
m 1exp
A
A0 LL0
r
r0
m (1)
Therein, m and 0, respectively, stand for the Weibull shape and scale parameters,whereas the gauge volume, area, and length are denoted, respectively, as V0, A0, and
L0. A is the average cross-sectional area of a specimen. r is the average failure stress,
Figure 11. Relation between tensile strength and coefcient of variation of cross-sectional areas.
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dened as the failure load divided by A. Transformation of each side of Equation (1)
into logarithms gives the following.
ln ln 1
1Fi lnA
A0 ln
L
L0 mln rimln r0 (2)
Therein, i1; 2;. . .;N, where N is the number ofbers.
4.1.2. Weibull model considering the cross-sectional area variation within a ber
Because natural bers have the variation of cross-sectional area within bers along the
ber direction, a stepped bar model has been proposed in an earlier paper.[22] This
model consists of n cylindrical elements that have random average area and constant
length ofx. It was assumed that the weakest ber element (the lowest stress element)
is independently broken on the size of cross-sectional area for neighborhood bers and
that the whole failure results from this element. Based on these assumptions, a Weibullmodel considering the variation of the area within bers was proposed as presented
below.
The cumulative probability Fjof the ber failure stress for the jth element in a ber
is given as the following equation.
Fj1expVjV0
rj
r0
m 1expAj Dx
A0L0rj
r0
m (3)
In that equation, Vj= Ajx and VjAj represent the volume and average area for
the jth element. Based on the weakest link model, the ber failure probability is givenas follows.
F1Ynj1
1Fj 1exp Dx
L0Xnj1
Aj
A0
rj
r0
m" # (4)
To compare Equation (4) with Equation (1), the stress j is rewritten as
rjPi
Aj A
Aj r (5)where Pi denotes the failure load for the ith ber. In the case ofx 0 considering
the continuum cross-sectional area variation, Equation (4) is expressed as presented
below.
F 1expA
A0 LL0
rr0
mR
; R 1
L
ZL0
A
A x m1
dx (6)
Therein, A(x) denotes a stochastic process that expresses the randomness of thewithin-ber cross-sectional area along its axial direction. In the case of a uniform
cross-sectional area, i.e. Ax A, Equation (6) coincides with Equation (1). However,
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the ber failure probability is rewritten as Equation (7) when the change in cross-sec-
tional area is measured discretely along the ber axial direction.
R1
nXnj1
A
Aj m1
(7)
R denotes the variation of the within-ber cross-sectional area. When m> 2, R is
greater than 1. In other words, the variation of cross-sectional area within bers
increases concomitantly with the increase in R. For a stepped bar model, it was
assumed that a step was introduced on the center between Aj and Aj+1 elements. By
transforming each side of Equation (6) into a logarithm, it is given as shown below.
lnln 1
1
Fi lnA
A0 ln L
L0 lnRimln rimln r0 (8)Results show that the effect of the variation within bers was excluded by the term
ln Ri in Equation (8), and Equation (1) was corrected by the parameterR. The average
strength rFwas derived according to the equation presented below.
rF r0 C 1 1m
A
A0 LL0
1m
R1m (9)
As described above, the discrete cross-sectional areas at 81 positions with intervals
of 0.1 mm have been already obtained for kenaf bers. The schematic of measurement
for cross-sectional area along the ber direction is shown in Figure12.In this study, the average stress of elements j was calculated using the arithmetic
average model. When the cross-sectional areas at n + 1 positions in a ber are mea-
sured as A1A2An+1, the number of elements for the Weibull model proposed
above is n. When the representative area is assumed as the average area between Ajand Aj+1, the average stress rjof the jth element is the value presented below.
rj1lZ
l
0
rdxPil Z
l
0
dx
A
x
2PiAj
Aj1
(10)
That is, rj was dened by the arithmetic average of Aj and Aj+1. Equation (7) is
rewritten as follows.
A1 A2
Aj
An
L
l
PP
Figure 12. Schematic of measurement for cross-sectional area along ber axis.
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F1expA
A0 LL0
rr0
mR
; R1
nXnj1
2A
AjAj1
m1(11)
4.1.3. True Weibull parameter estimation
For the kenaf bers for which the tensile strength was measured previously, Weibull
parameters were obtained using the proposed model. Figure 13shows Weibull plots for
the tensile strength of kenafbers. In this study, m and 0 were calculated, respectively,
as the gradient and intercept, by assuming that the distribution shown as a linear
approximation, in which all the left side and ln ri of the right side of Equations (2) and
(8) are taken, respectively, as the Y-axis and X-axis. The cumulative failure probability
Fjj= N1 is obtained using the mean rank method. The shape and scale parame-ters of kenaf bers are presented in Table 3. Results showed that the shape and scale
parameters obtained from Equation (8) are larger than those obtained from Equation
(2). The reason for that difference is the exclusion of the variation in within-ber cross-
sectional area as well as between bers, which can be achieved by setting a new
parameter of R. Consequently, the strengths were greater and scattering was smaller
than that of the straight bar model.
4.2. Estimating the cross-sectional area distribution
Because the true Weibull parameters excluded the effects of variation within bers and
between bers estimated in a previous chapter, we attempted to quantify variations
according to the formulation of the effect of these variations. Watson et al. extended a
Figure 13. Weibull distributions using Equations (2) and (8) of tensile strength.
Table 3. Shape and scale parameters.
ExperimentsEquation
(2)Equation
(8)Equation
(18)Equation
(19)Equation
(20)
m 5.61 6.90 7.45 6.68 6.86 5.880
(MPa)336 324 334 336 327 343
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Weibull model considering the density function of cross-sectional area distribution to
estimate the strength distribution for carbon bers having different cross-sectional
areas.[26] Weibull models have been applied by Steenbakkers et al. [24] for Kevlar
bers and by Zhang et al. [27] for wool bers. Moreover, Curtin applied a Weibull dis-
tribution to the density function to predict the tensile strength of the unidirectional
CFRP composites precisely.[28] However, for natural bers, the tensile strength of ax
bers is reportedly predictable using the Curtin model assuming the interval of kink
bands as the Weibull distribution.[29] All of them used the similar extended Weibull
model to estimate the strength distribution. In this chapter, a novel extended Weibull
model was proposed by considering the variation of the cross-sectional area within
bers for kenafbers.
4.2.1. Tensile strength distribution based on a Weibull model
The Weibull distribution considered the variation of cross-sectional area between bers
for average area A using the straight bar model as shown in Equation (1), which is afunction with two parameters for A and r.
FF A; r (12)When A accounted for a stochastic variable between bers and when its probability
density function was dened as gA, the strength distribution Ft is given as shownbelow.
Ft Z 1
0F
A;r g
A d
A
1Z 1
0
exp A
A0 LL0
rr0
m g A dA
(13)
It was established that the following formula was derived from Equation (13) [26]
as
Ft1exp A
A0
kr
r0
m" # (14)
where is a correction index of the size scale. For short expressions, it was assumed
that L= L0 in Equation (14). As described above, some discussions using Equation
(14) were developed for the strength distribution for various bers.[2628] Here, the
Weibull distribution considering the variation of cross-sectional area within bers as
well as between bers was proposed as shown below.
FF A; r;R (15)The difference between Equations (12) and (15) is the parameter R, which signies
the variation of areas within bers. Also, A
AR is newly dened here, with A larger
than A when R> 1. Equation (15) can be rewritten as presented below.
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Fj A; r 1exp
A
A0 LL0
rr0
m (16)
From this equation, it was qualied that this strength distribution is a Weibull distri-
bution with uniform average cross-sectional area A without within-ber variation.
When the probability density function of A is dened similarly as g*(A), Ft* takesthe value presented below.
FtZ 1
0
F A; r g A dA
1Z 1
0
exp A
A0 LL0
rr0
m g A dA
(17)
According to the experimental measurements of cross-sectional area for 34 kenaf
bers, the frequency distribution of average area A considering the average area A and
the parameter of variation within bers is shown in Figure 14. The compatibility ofthese distributions was conrmed as a normal distribution according to the Chi-square
test at the signicance level of 5%. Use of the probability density function ofgA andg*(A) proceeded as a normal distribution.
4.2.2. Probability density function using identical distributions
For a straight bar model without the variation of cross-sectional area within bers,
because the distribution of average area between bers shows agreement with the nor-
mal distribution, the probability density function of the strength distribution was also
assumed as a normal distribution. Ftis given as
Ft 1Z1
0
exp A
A0 LL0
rr00
m0( ) 1ffiffiffiffiffiffiffiffiffiffi
2pS21
p exp Al1 22S21
" #dA (18)
Figure 14. Frequency distribution of average cross-sectional area.
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where 1 is the mean of the average cross-sectional area distribution Aj and S1 is its
standard deviation. The scale and shape parameters, as estimated from Equation (2), are
used as r00 and m0. For a stepped bar model considering the within-ber variation area,
Ftis given correspondingly as
Ft1Z 1
0
exp A
A0 LL0
rr000
m00( ) 1ffiffiffiffiffiffiffiffiffiffi
2pS22
p exp A l2 22S22
" #dA (19)
where 2 is the mean of the average cross-sectional area distribution A
j , and S2 is its
standard deviation. The scale and shape parameters estimated from Equation (8) are
used as r000 and m00. For this study, Equations (18) and (19) were solved mathematically.
4.2.3. Probability density function using conditional distributions
The distributions of the average cross-sectional area for kenaf
bers were clari
ed asaccording with a normal distribution. Results show that the average tensile strength also
depends on the average cross-sectional area, as shown in Figure 10. Because
Aj A r , Equation (13) was reconsidered using the following equation.
FtZ 1
0
F A; r g Aj r
dA
1Z 1
0
exp A
A0 LL0
rr0
m g Aj r
dA
(20)
When the two-dimensional normal distribution was dened using stochastic vari-
ables and Aj as X and Y, respectively, the conditional distribution of probability den-
sity function was given as
g Aj r 1ffiffiffiffiffiffi
2pp
SYffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1q2XYp exp 1
2
AjlYqXY SY=SX rlX SY
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1q2XY
p !224
35 (21)
where X and Y, respectively, denote the means of and Aj. Also, SX and SY, respec-
tively, represent the standard deviations of and Aj. Additionally, XY is the correlation
coefcient. The mean and standard deviation were calculated from experimentally
obtained results used as Xand SX. Equation (20) was also solved mathematically.
4.2.4. Weibull parameter comparison
The Weibull parameters using the Weibull models considering the normal distribution
and the conditional two-dimensional normal distribution were estimated from Equations
(18), (19), and (20) for kenaf bers. Figure 15 portrays the obtained tensile strength
distributions as Weibull plots. The obtained scale and shape parameters are shown in
Table 3. For Equations (18) and (19) using the normal distribution, results showed that
the difference of the parameter R, which means the within-ber variation area, was not
large, the estimated shape parameter was larger and the estimated scale parameter wassmaller. However, in the case of Equation (20) using the conditional distribution con-
sidering the dependence of cross-sectional area on the tensile strength, both parameters
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showed agreement with experimentally obtained results. Results show that the tensile
strength distribution of natural bers considering the variation of the cross-sectional
area within a ber and between bers can be formulated using Equation (20).
5. Conclusions
This study of kenaf bers was the rst to reveal correlation between the actual cross-
sectional areas obtained from many area optical microscopy measurements and the
assumed areas of circles, ellipses, and polygons having from six corners to 24 corners.
Then, precise shape of the cross-sectional area of natural bers was investigated to
improve cross-sectional area estimation methods. Results claried that the correlation
between the actual cross-sectional area and the assumed polygonal shape area was
higher than that between the actual area and the conventionally assumed circle shape.
Using the estimated cross-sectional area based on the DB-based approximation of
the polygon shape, the effect of cross-sectional area variation on Youngs modulus and
the natural ber tensile strength were investigated. Results showed that the Youngs
modulus and tensile strength decrease along with the increase of the coefcient of vari-
ation for cross-sectional area as they do also for the cross-sectional area.
Finally, the parameterR was newly proposed to assess the within-ber cross-sectional
area variation. The true Weibull parameters, excluding the effects of area variation, were
estimated experimentally. Then, a novel Weibull model of natural ber tensile strengthwas formulated. It considers both within-ber and between-ber variations. The Weibull
parameters considering the area variation were estimated from this model mathematically.
Consequently, according to the consideration of the dependence of area variation on the
strength, the tensile strength distribution including the area variation was predicted
precisely.
References
[1] Fujii T, Nishino T, Goda K, Okamoto T. Developments and applications of environmentallyfriendly composites. Tokyo: CMC Publishing Co., Ltd; 2005. Japanese.
[2] Inao T. Industrial products of plant origin material-effective use of plant origin plastics forrecycling society. Trans. Jap. Soc. Mech. Eng. 2006;109:5152. Japanese.
Figure 15. Weibull distributions using Equations (18)(20) of tensile strength.
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[3] Serizawa S, Inoue K, Iji M. Kenaf-ber-reinforced poly (lactic acid) used for electronicproducts. J. Appl. Pol. Sci. 2003;100:618624.
[4] Hornsby PR, Hinrichsen E, Tarverdi K. Preparation and properties of polypropylene com-posites reinforced with wheat and ax straw bres. J. Mater. Sci. 1997;32:443449.
[5] Lodha P, Netravali AN. Characterization of interfacial and mechanical properties of greencomposites with soy protein isolate and ramie ber. J. Mater. Sci. 2002;37:36573665.
[6] Okubo K, Fujii T, Yamamoto Y. Development of bamboo-based polymer composites andtheir mechanical properties. Composites Part A. 2002;35:377383.
[7] Baley C. Analysis of the ax bres tensile behaviour and analysis of the tensile stiffnessincrease. Composites Part A. 2002;33:939948.
[8] Joffe R, Andersons JA, Wallstrm L. Strength and adhesion characteristics of elementaryax bres with different surface treatments. Composites Part A. 2003;34:603612.
[9] Gomes A, Goda K, Ohgi J. Effects of alkali treatment to reinforcement on tensile propertiesof curaua ber green composites. JSME Inter. J., Series A. 2004;47:541546.
[10] Goda K, Sreekala MS, Gomes A, Kaji T, Ohgi J. Improvement of plant based natural bersfor toughening green composites-effect of load application during mercerization of ramiebers. Composites Part A. 2006;37:22132220.
[11] Pickering KL, Beckermann GW, Alam SN, Foreman NJ. Optimising industrial hemp bre
for composites. Composites Part A. 2007;38:461468.[12] Defoirdt N, Biswas S, Vriese LD, Tran LQN, Acker JV, Ahsan Q, Gorbatikh L, Vuure AV,
Verpoest I. Assessment of the tensile properties of coir, bamboo and jute bre. CompositesPart A. 2010;41:588595.
[13] Kompella MK, Lambros J. Micromechanical characterization of cellulose bers. Polym.Test. 2002;21:523530.
[14] Zafeiropoulos NE, Baillie CA. A study of the effect of surface treatments on the tensilestrength of ax bres: Part II. Application of weibull statistics. Composites Part A.2007;38:629638.
[15] Silva FA, Chawla N, Filho RDT. Tensile behaviour of high performance natural (sisal)bers. Compos. Sci. Tech. 2008;68:34383443.
[16] Virk AS, Hall W, Summerscales J. Multiple Data Set (MDS) weak-link scaling analysis of
jute
bres. Composites Part A. 2009;40:1764
1771.[17] Suzuki K, Kimpara I, Saito H, Funami K. Cross-sectional area measurement and monola-ment strength test of kenaf bastbers. J. Jpn. Soc. Mat. Sci. 2005;54:887894. Japanese.
[18] Gassan J, Bledzki AK. Modication methods on nature bers and their inuence on theproperties of the composites. J. Eng. Appl. Sci. 1996;2:25522557.
[19] Madsen B, Thygesen A, Lilholt H. Plantbre composites porosity and volumetric interac-tion. Compos. Sci. Tech. 2007;67:15841600.
[20] The Society of Fiber Science and Technology. Fiber physical science. Tokyo: Maruzen Co.,Ltd; 1962.
[21] Xu XW, Jayaraman K. An image-processing system for the measurement of the dimensionsof natural bre cross-section. J. Compu. Appl. Tech. 2009;34:115121.
[22] Tanabe K, Matsuo T, Gomes A, Goda K, Ohgi J. Strength evaluation of curaua bers withvariation in cross-sectional area. J. Jpn. Soc. Mat. Sci. 2008;57:454460. Japanese.
[23] Charlet K, Eve S, Jernot JP, Gomina M, Breard J. Tensile deformation of a ax ber.Procedia Eng. 2009;1:233236.
[24] Steenbakkers LW, Wagner HD. Elasticity and mechanical breakdown of kevlar 149 aramidbres by a probabilistic approach. J. Mater. Sci. Lett. 1988;7:12091212.
[25] Doan TTL, Gao SL, Mder E. Jute/polypropylene composites I. Effect of matrix modica-tion. Compos. Sci. Tech. 2006;66:952963.
[26] Watson AS, Smith RL. An examination of statistical theories for brous materials in thelight of experimental data. J. Mater. Sci. 1985;20:32603270.
[27] Zhang Y, Wang X, Pan N, Postle R. Weibull analysis of the tensile behaviour of bers withgeometrical irregularities. J. Mater. Sci. 2002;37:14011406.
[28] Curtin WA. Tensile strength of ber-reinforced composites: III. Beyond the traditionalweibull model forber strength. J. Compos. Mat. 2000;34:13011332.
[29] Andersons J, Sparnins E, Porike E. Strength and damage of elementary ax bers extractedfrom tow and long line ax. J. Compos. Mat. 2009;43:26532664.
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