Mechanical Engineering Journal
Numerical study for tensile strength prediction ofunidirectional carbon fiber-reinforced composite consideringfiber surface stress concentration
Go YAMAMOTO* and Tomonaga OKABE*, ***Department of Aerospace Engineering, Tohoku University,
6-6-01 Aramaki-Aza-Aoba, Aoba-ku, Sendai 980-8579, Japan
E-mail: yamamoto@plum.mech.tohoku.ac.jp**Department of Materials Science and Engineering, University of Washington,
Seattle, WA 98195, U.S.A.

Received: 8 January 2019; Revised: 21 March 2019; Accepted: 16 April 2019
Abstract
The accurate tensile strength prediction of unidirectional carbon fiber-reinforced plastic composites (UDcomposites) requires approximate determination of the stress concentration on surviving fibers around a fiber
break point. Here the stress concentrated on the intact fiber surface was determined by implementing
double-fiber fragmentation tests in combination with a spring element model (SEM) simulation. The
double-fiber fragmentation composites and the UD composites were elaborated with a T1100G-type carbon
fiber and epoxy material, and tested to validate the proposed prediction method. The size scaling results,
implementing a bimodal Weibull distribution for the statistical distribution of fiber strength, coupled with theresults of the SEM simulation, designed to take into account the surface stress concentration, were reasonably
consistent with the experimental data on the tensile strengths of the UD composites. Then, the proposed
strength prediction procedure was applied to investigate the effects of the bimodal Weibull scale and shapeparameters on the tensile strength of the UD composites. It was revealed that the degree of stress concentrated
on the surface of fibers can be changed by modifying the bimodal Weibull shape and scale parameters.
However, the carbon fiber with an improved scale parameters of 20% displayed enhancement to the composite
strength by factor of ~1.07, and with an improved shape parameters of 20% showed enhancement by factor of~1.04, indicating that the degree of enhancement in the tensile strength of the UD composite was limited.
Keywords : Carbon fiber, Polymer-matrix composite (PMC), Fragmentation, Strength, Stress concentration

1. Introduction
Fiber-reinforced polymer-matrix composites (PMC) have been widely used across industries including aerospace,wind energy, pressure vessel, and automotive applications. Carbon fiber-reinforced plastic (CFRP), a typical class ofcomposite materials, is increasingly being used as lightweight and high-stiffness materials in the above-mentioned
applications. The processes of determining the potential amount of weight that can be saved requires that the fracture
properties of the CFRPs in the fiber axis direction be a major consideration for designing composite structures. The
mechanical modeling of failure of unidirectional carbon fiber-reinforced composites (UD composites) is a useful firststep for understanding of real composites including multi-ply laminates and woven composites. Thus, improving theaccuracy of tensile strength prediction methods continues to be central to CFRP composite research.
A number of extent researches aimed at finding ways to accurately predict the tensile strength of the UDcomposites (Scott et al., 2011; Swolfs et al., 2015; Watanabe et al., 2018). A local load sharing (LLS) model developedby Curtin for the numerical simulation of fracture behavior in ﬁber-reinforced composites has been widely applied to
predict the ultimate tensile strength of UD composites (Curtin et al., 2000). The author deﬁned the force redistribution
concept of the LLS as follows; the perturbation of load caused by a fiber failure is not distributed uniformly among thePaper No.19-00020

remaining fibers, but rather is carried more largely by adjacent surviving fibers. Thus, when one fiber breaks the loadthat it was carrying is transferred principally to the surviving neighbors, increasing the stress concentration of thesefibers relative to the others. Consequently, increased probability of failure of such fibers leads to the formation ofclusters of broken fibres, leading to final failure of the composites. Although the stress concentration on an intact fiber
surface around a fiber break point has been widely recognized as an important factor controlling the tensile strength of
UD composites, the strength prediction of such composite taking into account the stress concentration due to the fiber
failure is so far from satisfactory.
In this study, we considered the stress concentration on an intact fiber surface caused by a fracture site in aneighboring broken fiber into our prediction of the ultimate tensile strengths of the UD composites. The stressconcentrated on the intact fiber surface was determined by implementing double-fiber fragmentation tests incombination with a spring element model (SEM) simulation. We chose herein to apply a bimodal Weibull distribution
as the statistical distribution of fiber strength, based on the knowledge by Watanabe et al. (Watanabe et al., 2018). The
acquired stress concentration factors were subsequently implemented to obtain the tensile strength prediction of the UDcomposites. The double-fiber fragmentation composites and the UD composites were elaborated with a T1100G-type
carbon fiber and epoxy material, and tested to validate the proposed prediction method. The size scaling results
obtained in conjunction with the results obtained from the SEM simulation were reasonably consistent with theexperimental data on the tensile strengths of the UD composites. We have also systematically investigated the effects ofthe bimodal Weibull scale and shape parameters on the tensile strength of the UD composites.

2. Analytical and experimental procedures
2.1 Analytical model
Monte-Carlo methods were implemented in the SEM simulation to determine the stress concentrated on the intact
fiber surface around the fiber break points. The details of model preparation and simulation procedures can be found in
the reference (Okabe et al., 2005; Watanabe et al., 2018). In brief, the SEM comprises longitudinal and transverse
spring elements in a three-dimensional hexagonal arrangement. The longitudinal elements represent carbon fibers that
exclusively carry the tensile load, while the transverse elements indicate the matrix that only sustains the shearing load.
The stiffness matrices of the fiber element 𝑲eL and matrix element 𝑲eT are separately calculated as follows:l

K Le

SR 2 ³ BLeT Ef BLe dz,

(1)

K Te

SRl l eT
BT GBTe dr,
3 ³0

(2)

0

ª1 1 ºBLe = «  » ,
l¼
¬l

(3)

1º
ª1
BTe = «
 »,
d¼
¬d

(4)

where L and T indicate the longitudinal and transverse directions of the elements, respectively, R is the radius of the
carbon fiber (= 2.7 μm), Ef is the Young’s modulus of the carbon fiber (= 324 GPa), Gm is the shear modulus of theepoxy material (= 1.4 GPa), and l and d are the spring element lengths along the longitudinal and transverse directions,
respectively. The experimentally observed average interfiber spacings in the double-fiber fragmentation composites
was approximately 0.01 mm. Therefore, d was set to 0.01 to analyze the results of the double-fiber fragmentation tests.
Conversely, a hexagonal close-packed arrangement was assumed for the analysis of the full composite simulation and dcan be calculated as follows:

d

R

2S

3Vf  2 ,(5)

where Vf is the fiber volume fraction in the full composite model (= 57%). In this study, we considered the situation
where the fiber axial stress within the stress recovery region can be described as a linear function of the distance fromthe fiber breaking point, as is given by the following equation (Okabe et al., 2015):

Vs

2W s Ds R ,(6)where Vs is the fiber axial stress within the stress recovery region, Ws is the interfacial shear stress, and Ds is the distancefrom the fiber breaking point. Therefore, the force balance to describe the overall system is given as follows:ª ¦ Nf  Nb  Np K Le  ¦ Nm K Te º u  ¦ Np SR 2 BLeTV s dz
³0
e 1
e 1
¬ e1
¼
l

f,

(7)

where Nf and Nm are the number of fiber and matrix elements, respectively, Nb is the number of broken fiber, and Np is
the number of fiber elements in the plastic deformation region.
The statistical distribution of fiber strength was determined using both the single-fiber tensile testing andsingle-fiber fragmentation testing (Okabe et al., 2002; Watanabe et al, 2014). Table 1 shows the Weibull parameters
implemented to construct a bimodal Weibull distribution of the T1100G carbon fiber. The fiber breakage probability Pf(V) under the conditions that the fiber is subjected to stress V at the surface area Sf (= 2SRLf) is given as follows:

Pf V

S
V
°­ S V°½
1  exp ® f ( ) m1  f ( ) m2 ¾
Sf ,0 V 02 °¿°¯ Sf ,0 V 01

(8)

where Lf is the gauge length, Sf,0 (= 2SRLf,0) is the representative surface area of the fiber (Lf,0 = 10 mm), V01 and V02
are the Weibull scale parameters, and m1 and m2 are the Weibull shape parameters.
The fibers are hexagonally arranged in the full composite model based on the report by Tavares et al. (Tavares etal., 2017); thus, the fiber surface can be divided in the six segments. The surface stress concentration factor of the i-th
fiber segment next to a broken fiber at the fracture plane was considered by assuming that it was Di times the fiber
stress with no additional surface stress concentration Vf. The stress concentration factor Di is given as follows:Di

1  J i (1 

Ds
),
ls

(9)

where Ji is the additional stress concentration term, and ls is the stress recovery length; thus considering of the stressconcentration on an intact fiber surface, fiber breakage probability Pf (V) can be rewritten as follows:

Pf ,i V­° S D V
S D V ½°
1  exp ® f ,i ( i ) m1  f ,i ( i )m2 ¾
Sf ,0 V 02¯° Sf ,0 V 01
¿°

(10)

where Sf,i is the i-th fiber segment of surface area (Sf,i = SR2Lf,0/3). The strength of the n-th fiber segment is determinedby choosing a random number Rn ranging from 0 to 1 and solving equation Rn = Pf,n (Vn). The longitudinal element wasremoved from the model when the stress applied to a fiber at the n-th fiber segment reached the statistical distributionof the fiber strength Vn.
The stress concentration factor on the intact fiber surface were determined by implementing the SEM, with the aimof ensuring that it was equivalent to the percentage of the coordinated fracture, which is defined as a failure occurring

Table 1 Bimodal Weibull parameter of the T1100G carbon fiber and two different types of PAN-based carbon fibers(i.e., T800S and T700S).
Types

V01

m1

V02

m2

V01/V02

m1/m2

SourceT1100G

7.7

4.5

9.1

13.0

0.85

0.35

Yamamoto et al., 2019

T800S

6.9

4.1

8.3

13.0

0.83

0.32

Watanabe et al., 2018

T700S

5.2

4.8

6.1

12.0

0.85

0.40

Watanabe et al., 2018

at the elements neighboring a broken fiber element in the horizontal plane of the broken fiber element, that wasdetermined through the double-fiber fragmentation testing. Therefore, two longitudinal spring elements in the center of
the SEM were assigned to the fibers, while the remaining longitudinal element were assigned to the matrix. The fiber
elements with 10 mm in length were divided into 2000 segments (i.e., the unit length of the longitudinal springelements was 5 μm). A coordination of fracture was defined as a failure that occurred at neighboring elements next to abroken fiber element in the same horizontal plane with respect to the broken fiber element.

2.2 Experimental proceduresHigh-strength, polyacrylonitrile (PAN)-based carbon fiber (TORAYCATM T1100G) and bisphenol-A epoxy resinmaterials were used to elaborate both the double-fiber fragmentation composites and the UD composites. The Young’s
modulus (Ef), tensile strength (Vf), and diameter (D) of the T1100G carbon fiber are 324 GPa, 7.0 GPa, and 5.4 μm,
respectively. Figure 1 and Table 2 present a summary of the mechanical properties of the epoxy materials. Note that all
epoxy materials examined herein showed elastic‒plastic transition behavior, exhibiting a stress‒strain relationship thatis typically observed in conventional epoxy materials.
Double-fiber fragmentation composites were elaborated by positioning two fibers parallel to the loading direction,
implementing an interfiber spacing of approximately one-half to four fiber diameters. Figure 2(a) shows the schematic
illustration of the fragmentation composite. The average interfiber spacing of seven fragmentation composites wasapproximately 10 μm. Next, a pre-heated and degassed epoxy resin was poured into a pre-heated glass model where thefibers were bonded. Subsequently, the fragmentation composites were cured in an air oven at temperature of 160˚C for
5 hours prior to post-curing at temperature of 180˚C for 2 hours. Finally, the composites were cooled in the oven to
room temperature before being cut into rectangles with thickness of approximately 2.0 mm using a laboratory cut-offsaw (metlon, METACUT 251).
The fragmentation composites were deformed in a four-point bending rig while the strain was monitored using
strain gauge technique. The inner and outer span lengths for four-point bending tests were 18 mm and 50 mm,
respectively. The tensile strain was applied in tension until a maximum of 5.0% strain, and held constant at every 0.1%
steps. As a fiber‒fiber interaction criterion, fiber fractures that occurred at an angle within 0° to 45° are defined ascoordinated fractures based on the nature of elastoplastic polymer material fracture phenomena (Huang et al., 2006).
The number and position of broken fibers were recorded using the polarized-light microscope (OLYMPUS, BX60).

Fig. 1 Representative stress‒strain curve for the epoxy materials.
Table 2 Material properties of the epoxy material.
emYoung’s modulus, E(GPa)

Tangent modulus, Emp
(MPa)

3.8

80

Yield stress, V yShear modulus, Gm

(MPa)

(GPa)
103

1.4

Fig. 2

Schematics of (a) the double-fiber fragmentation composite and (b) the UD composite.

The strain subjected to the carbon fiber Hf was calculated as follows:

Hf

Hc u

2.0

N

u(

t  2d)  H fr ,t

(11)where Hc is the strain of the recorded fragmentation composite, N is the strain gauge factor (= 2.13; KYOWA,KFG-2-120-C1-11), t and d are the thickness of the fragmentation composite (= ~2 mm) and the embedded depth of thefibers (= ~60 μm), respectively, 𝜀fr is strain originating from the difference coefficient of thermal expansion between
fiber and matrix (0.27%), as investigated through Raman spectrum analysis.
The tensile strength of the UD composite was investigated to validate the accuracy of the predicted tensile strength
by the SEM derived based on the double-fiber fragmentation tests. The UD composites were elaborated throughconventional bagging and autoclave laminating procedures to prepare the laminated structure with [0 6]. Figure 2(b)shows the schematic illustration of the UD composite. The fiber volume fraction and the bulk density of the prepared
UD composites were 57% and 1.79 Mg/m3, respectively. The tensile strengths were measured by the tensile-loading
experiments under ambient conditions. A crosshead speed for the tensile experiments were 1.27 mm/min (21.2 μm/s).
The strain gauges were used to measure the longitudinal and transverse strains of one side of the UD composites.

3. Results and discussion
3.1 Tensile strength prediction
First, we measured fiber fracture behavior in the double-fiber fragmentation composites to quantitatively determine
the surface stress concentration factor. The investigations have demonstrated under the experimental conditions used in
the study that matrix cracks and the coordination of fractures in adjacent fibers were observed, regardless of theinterfiber spacing. Even though some variation has been observed for the percentages of coordinate fractures, thefragmentation tests performed herein revealed that different interfiber spacing does not apparent impact on thepercentage of coordinated fractures. At 2.5% fiber strain Hf, calculated by the equation (11) for a composite strain εc of
5.0%, the percentages of coordinated fracture was 9.09%.
A quantitative determination of the stress concentration on the surface of an intact fiber around a fiber break point
was conducted by implementing the SEM to investigate the value of stress concentration. Figure 3 depicts therelationship between the stress concentration factor and the percentage of coordinated fractures, which was determinedby systematically sweeping stress concentration factor in the SEM simulation. The simulation results revealed that thepercentage of coordinated fractures increased with the increasing stress concentration factors, and upon comparing thesimulated percentages of coordinated fractures to the corresponding experimental results, the stress concentration factor
on the surface of an intact fiber was estimated to be approximately 1.75, indicating that 1.75 times as muchconcentrated stress subjected on the intact fiber surface as the fiber stress compared with no additional surface stressconcentration conditions.
Figure 4 illustrates the fiber fracture behaviors on the final failure planes simulated through the SEM simulation.
We defined the situation where the plane, including the most fiber breaks, as the final failure plane. A stress ratio of lessthan 1 shows that the fiber failure occurred near the final failure plane, on the other hand, a stress ratio of 0 depicts that

Fig. 3 Relationship between the stress concentration factors and the percentage of coordinated fractures.

the fiber failure occurred on the final failure plane. In the case of no additional stress concertation (i.e., α = 1.0), thefiber failure was clearly randomly distributed across the final failure plane, demonstrating that this is consistent withthe statistical distribution of fiber strength. In contrast, additional stress concentration leads to the formation of broken
fiber clusters, which results in relatively premature fracturing. Note that the experimentally obtained failure strain of
the composites ranged from 1.93% to 2.01% (mean: 1.98%), and the estimated failure strain simulated under thecondition of α = 1.75 was approximately 2.20% (2.18%‒2.21%); this value is roughly consistent with theexperimentally obtained failure strain.
The tensile strength of the UD composite was predicted through the SEM technique. The model comprised 1024
fibers with 3 mm in length that were divided into 300 segments. The stress concentration was implemented to the intact
fiber surface according to equation (9). Prior to comparing the simulated strengths to the experimental results, thesimulated result were subjected to size scaling. This is given as follows (Okabe et al., 2002):nL ns Ls / ln ^1  Fs (V n )` ,(12)

where ns is the number of the fibers in the model (ns = 1024), Ls is the length of the composite model (Ls = 3 mm), andFig. 4 Distribution of fiber stress in the axial direction on the final failure planes.

Fig. 5 Experimental and simulated results for the UD composites.

𝐹s (𝜎̃n ) is the cumulative probability of failure at a given strength as determined 100 SEM simulation runs, and nL is
the size at which the characteristic strength is achieved. Figure 5 shows a comparison of the experimental andsimulated results for the UD composites. In these figures, the circle symbols indicate the size scaled strength results
calculated based to equation (12), while the crosses denote the experimentally obtained tensile strengths of thesecomposites, which ranged from 3.74 to 3.97 GPa (mean: 3.85 GPa). The simulated result obtained without
consideration of additional stress concertation (i.e., α = 1.0) was incongruent with the experimental data, whereas thepredictions incorporating the stress concentration factors of 1.7 and 1.8 were reasonably consistent with theexperimental data. The estimated tensile strength obtained under the conditions of α = 1.75 was approximately 3.9 GPa.
Furthermore, although the prediction with α = 1.8 seems to yield a better tensile strength prediction than with α = 1.75,
we conclude that the prediction method proposed herein yields a reasonably accurate tensile strength prediction whenthe matrix crack-induced surface stress concentration of fibers is appropriately considered.
Improving the tensile strength prediction accuracy is central to CFRP composite research. As for one example, incertain types of CFRP material (Okabe et al., 2002), the model based on 3D shear-lag theory estimated overly thetensile strengths compared with the experimental data, and we could not make a full explanation of its reason. The
results reported herein indicate that the fact that matrix cracks around the fiber break points increase the degree ofstress concentrated on the intact fiber surface must be considered to obtain a high-accuracy strength prediction.

3.2 Effects of bimodal Weibull scale parameters
It can be seen from Table 1 that different types of the PAN-based carbon fibers possess different bimodal Weibull
scale and shape parameters (Watanabe et al., 2018; Yamamoto et al., 2019). Interestingly, such fiber has unique V01 andV02 values, but their ratios (i.e., V01/V02) are approximately the same; that is ~0.84. On the basis of this fact that weapplied the above-mentioned strength prediction procedures to investigate effects of the bimodal Weibull scaleparameters on the tensile strength of the UD composites. Table 3 summarizes the parameters used to construct theTable 3 Bimodal Weibull scale and shape parameters of the five different types of carbon fibers.
Types

V01

Fiber A

9.2

10.9

0.84

20% improvement over reference data

Fiber B

8.6

10.2

0.84

10% improvement over reference data

T1100G

7.7

T800S

6.9

8.3

0.83

10% reduction over reference data

Fiber C

6.2

7.3

0.85

20% reduction over reference data

m1

4.5

V02

9.1

m2

13.0

V01/V02

0.85

m1/m2

0.35

Remarks

Reference data

Fig. 6 (a) Relationship between the stress concentration factors and the percentage of coordinated fractures.
(b,c) The scaled tensile strength results for the UD composites.

bimodal Weibull distribution for five different types of carbon fibers used in this section. We considered herein thesituation where the all fiber failures have the Young’s modulus of 324 GPa and the diameter of 5.4 μm.
Figure 6(a) illustrates the results obtained through the SEM simulation to determine the stress concentration factors
in each fibers. A comparison of the experimental and simulated results presented in Figure 6(a) revealed a clear
difference in the estimated stress concentration factors. The percentages of coordinated fractures occurring in the fiber
types of Fiber A, Fiber B, T1100G (reference data), T800S, and Fiber C were calculated to be 2.08, 1.88, 1.75, 1.61,
and 1.46, respectively; an enhancement in the bimodal Weibull scale parameters leads to increase in the stressconcentration factor. Thus, acquired surface stress concentration factors were implemented to predict the tensile
strength of the UD composite. Figure 6(b) and 6(c) present the effects of the bimodal Weibull scale parameters on thetensile strength of the composites. The simulated results obtained by implementing the bimodal Weibull scaleparameters (V01/V02) of 5.4/6.4 and 10.0/11.8 are also indicated in the Figure 6(c). Contradictory to the results obtainedin the section 3.1, (which revealed that higher stress concentration on an intact fiber surface results in relatively
premature fracturing of UD composites), even if a larger stress concentration was occurring on intact fiber surfaces,
increased tensile strength was observed for the UD composites with a larger stress concentration factor. Even thoughsome variation was observed for the simulated results, the tensile strength of the series of the UD composites
containing carbon fibers with V01/V02 of approximately 0.84 gently increased in response to the increase in the bimodal
Weibull scale parameters.
Further evidence of the improved tensile strength was provided by the distribution of fiber stress in the axial
direction on the final failure planes (Figure 7). It can be seen that a larger fracture strain is observed for the composite
having a higher stress concentration factors, and the number of fiber breakage is small when comparing with the givenFig. 7 Distribution of fiber stress in the axial direction on the final failure planes of the UD composite made withfibers having V01/V02 = 10.0/11.8 and V01/V02 =7.7/9.1.

composite strain. These observations suggest that higher surface stress concentration does not necessarily mean that therelatively premature fracturing of the UD composite. But more than that enhanced mechanical integrity, namely thehigher V01 and V02 values, of carbon fibers leads to the improved tensile strength of the UD composite, even thoughfurther detailed simulations are needed to be carried out in order to clarify the effects of the bimodal Weibull scaleparameters on the tensile strength of the UD composite. The simulation results demonstrate that the present procedures,
which implements the double-fiber fragmentation techniques in conjunction with SEM, may provide an effective routefor development of UD composites with improved tensile strength, that it may be possible to tailor and control thebimodal Weibull scale parameters.

3.3 Effects of bimodal Weibull shape parameters
Next, we applied the strength prediction procedures to investigate the effects of the bimodal Weibull shapeparameters on the tensile strength of the UD composites. Table 4 summarizes the parameters used to construct thebimodal Weibull distribution for five different types of carbon fiber used in this section. As in the previous section, we
considered the situation where the all fiber failures possess the Young’s modulus of 324 GPa and the diameter of 5.4
μm.
Figure 8(a) shows the results obtained through the SEM simulation to determine the stress concentration factors ineach fibers; an increase in the bimodal Weibull shape parameters, i.e., m1 and m2, leads to increase in the stressconcentration factor. The percentages of coordinated fractures occurring in the fiber types of Fiber D, Fiber E, T1100G(reference data), Fiber F, and Fiber G were estimated to be 1.59, 1.66, 1.75, 1.85, and 1.96, respectively. In the samemanner with the section 3.2, the acquired surface stress concentration factors were implemented to predict the tensile
strength of the UD composite.
Figure 8(b) and 8(c) show the effects of the bimodal Weibull shape parameters on the tensile strength of the UDcomposites；the tensile strength of the series of the UD composites containing carbon fibers with m1/m2 ofapproximately 0.35 increase approximately linearly in response to the increase in the bimodal Weibull shapeparameters. This may be attributable to the fact that the larger values of the Weibull shape parameter indicate a smaller
scatter of the fiber strength, therefore reduction in the strength distribution of the carbon fibers leads to theTable 4 Bimodal Weibull scale and shape parameters of the five different types of carbon fibers.
Types
V01
m1
V02
m2
V01/V02
m1/m2
Remarks
Fiber D

5.4

15.6

0.35

20% improvement over reference data

Fiber E

5.0

14.3

0.35

10% improvement over reference data

0.35

Reference data

T1100G

7.7

4.5

9.1

13.0

0.85

Fiber F

4.1

11.7

0.35

10% reduction over reference data

Fiber G

3.6

10.4

0.35

20% reduction over reference data

Fig. 8 (a) Relationship between the stress concentration factors and the percentage of coordinated fractures.
(b,c) The scaled tensile strength results for the UD composites.

enhancement in the tensile strength of the UD composite. The carbon fiber with m1/m2 = 5.4/15.6 displays enhancement
to the composite strength by factor of ~ 1.08 compared to the carbon fiber having m1/m2 = 3.6/10.4 (i.e., 50% shapeparameter improved), indicating that the degree of enhancement in the tensile strength of the UD composite is limited.

4. Conclusions
The improvement of accuracy of prediction method for tensile strength has been a central focus of CFRP
composite research. The results reported here indicated that the important factor for strength prediction with highaccuracy will be considered an additional stress concentration caused by a fracture site in an adjacent fiber. We
confirmed that employing the measured stress concentration factors and bimodal Weibull distribution to determine how
strength is statistically distributed throughout the fiber yields the predicted strengths of the UD composite that arereasonably consistent with the experimental data, thereby demonstrating the validity of the proposed prediction method.
We also demonstrated that the degree of stress concentrated on the surface of fibers can be changed by modifying thebimodal Weibull shape and scale parameters of carbon fibers. In totality, our findings provide a possible framework for
guiding the creation of stronger CFRP composites. The design of such composites will need to account, or in some way
circumvent, the stress concentration on the surface of fibers reported here.

Acknowledgements
This work was partly supported by Toray Industries, Inc., the Council for Science, Technology andInnovation(CSTI), the Cross-ministerial Strategic Innovation Promotion Program (SIP), and JSPS KAKENHI grantnumber 18K04721. The authors would like to acknowledge the vitally important encouragement and support madethrough the University of Washington-Tohoku University: Academic Open Space (UW-TU: AOS).

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