&RPSRVLWHV3DUW$  ²Contents lists available at ScienceDirect

Composites Part A
journal homepage: www.elsevier.com/locate/compositesa

Considering the stress concentration of ﬁber surfaces in the prediction of thetensile strength of unidirectional carbon ﬁber-reinforced plastic composites

7⁎

Go Yamamotoa, , Miho Onoderaa, Keita Koizumia, Jun Watanabeb, Haruki Okudab,
Fumihiko Tanakab, Tomonaga Okabea,caDepartment of Aerospace Engineering, Tohoku University, 6-6-01 Aramaki-Aza-Aoba, Aoba-ku, Sendai 980-8579, Japan
Composite Material Research Laboratories (CMRL), Toray Industries, Inc., 1515, Tsutsui, Masaki-cho, Iyogun, Ehime 791-3193, Japanc
Department of Materials Science and Engineering, University of Washington, Seattle, WA 98195, USAb

A R T I C LE I N FO

A B S T R A C T

Keywords:
A. Polymer-matrix composites (PMCs)
B. Fragmentation
B. Strength
B. Stress concentrations

The tensile strengths of unidirectional carbon ﬁber-reinforced plastic (CFRP) were predicted by using a springelement model that considers the surface stress concentration on ﬁbers caused by a fracture site in an adjacentﬁber. The surface stress concentration on the ﬁbers was experimentally evaluated by implementing multi-ﬁber
fragmentation tests in conjunction with a spring element model simulation. Four types of epoxy materials were
utilized to explore the eﬀects of matrix polymer properties on the surface stress concentration of the ﬁbers. The
size scaling results, coupled with the results of the spring element model simulation, designed to take into
account the surface stress concentration, were reasonably consistent with the experimental data on the tensile
strengths of the unidirectional CFRP composites, regardless of the diﬀerences in the matrix mechanical properties. Possible mechanisms by which additional stress concentration is generated on an intact ﬁber surface wereanalyzed numerically using the ﬁnite element method.

1. Introduction
Composite materials are attracting the attention of people within
the engineering sector because of their unique mechanical properties
and ease of property customization, making them highly competitive
with conventional materials. Carbon ﬁber reinforced plastic (CFRP), a
common class of composite materials, is increasingly being used as alightweight and high-stiﬀness material in various applications. The
process of determining the potential amount of weight that can besaved requires that the fracture properties of the CFRPs in the directionof the ﬁber axis be a major consideration in the design of composite
structures. Thus, improving the accuracy of tensile strength prediction
methods continues to be central to CFRP composite research.
Since the early pioneering work by Cox [1], Rosen [2], and Kelly
et al. [3], a number of models have been proposed to predict the tensile
strength of unidirectional CFRP composites. The work of Hedgepethand Van Dyke became the original report on failure process modeling of
unidirectional ﬁber reinforced composites (UD composites) [4]. They
implemented a discrete Fourier transform to analytically describe theredistribution of load resulting from ﬁber breakage in a hexagonal ﬁber
array. Subsequently, Suemasu [5,6] showed that such redistribution
can be derived by employing a method based on Green’s functions. By⁎

contrast, Rosen [2] focused on the role of the matrix in determining thestress transfer length around ﬁber break points, although without
considering the in-plane stress concentration in strength analysis.
Zweben [7] later introduced the stress concentration eﬀect caused byﬁber breaks into Rosen’s model [2]. Although the eﬀects of both inplane stress concentration and the stress recovery of broken ﬁbers are
expected to require an accurate failure process prediction, further research is needed to establish an appropriate analytical approach. Furthermore, instead of focusing on correcting this oversight, many researchers have opted to focus on simulating the failure processes ofmultiple types of UD composites comprising various matrix materials
by utilizing a procedure that implements a phantom step to solve the 2D
shear-lag equation [8–11]. Additionally, Sastry et al. [12] and Beyerlein
et al. [13] proposed a sophisticated approach, referred to as the breakinﬂuence superposition (BIS) technique and extended it to develop thequadratic inﬂuence superposition (QIS) technique within the framework of a 2D model. However, the stress concentration of a 2D model iswell known to be signiﬁcantly higher than that of the corresponding 3Dmodel. Zhou and Curtin [14] proposed a novel numerical simulation
based on the lattice Green function (LGF) approach to analyze the 3D
stress state of the composites. Building on their work, Landis et al. [15]and Okabe et al. [16] attempted to develop the 2D shear-lag model into

Corresponding author.
E-mail address: yamamoto@plum.mech.tohoku.ac.jp (G. Yamamoto).

https://doi.org/10.1016/j.compositesa.2019.04.011
Received 26 June 2018; Received in revised form 1 April 2019; Accepted 9 April 2019
$YDLODEOHRQOLQH$SULO
;(OVHYLHU/WG$OOULJKWVUHVHUYHG

&RPSRVLWHV3DUW$  ²

G. Yamamoto, et al.

simulation, which considered the added concentrated stress on the ﬁber
surface adjacent to a broken ﬁber. Moreover, they demonstrated that
applying the acquired SCF to a spring element model simulation yields
results that are consistent with experimentally obtained strengths ofunidirectional CFRP composites. The fact that fragmentation testingyields diﬀerent stress concentrations for diﬀerent types of matrix materials implies that the strengths of composites are dependent on thematrix materials. However, to our knowledge, no report has presented
the tensile strength to be predicted from the SCF obtained from fragmentation composites made with diﬀerent matrix materials. Moreover,no comparison or veriﬁcation has been oﬀered using actual composite
materials. Furthermore, the procedure for determining the surface
stress concentration remains tentative.
In this study, we considered the surface stress concentration on ﬁbers caused by a fracture site in an adjacent ﬁber into our prediction of
the ultimate tensile strengths of unidirectional CFRP composites. The
stress concentrated on the ﬁber surface was determined by performing
multi-ﬁber fragmentation tests in combination with a spring element
model simulation. We deﬁned the coordinated fracture to quantitatively evaluate the SCFs. The acquired SCFs were subsequently applied
to obtain the tensile strength prediction of unidirectional CFRP composites. The composites were fabricated with T1100G carbon ﬁber andfour types of epoxy materials with diﬀerent mechanical characteristics.
We tested these materials to validate the proposed prediction method.
The size scaling results obtained in conjunction with the results fromthe spring element model simulation were reasonably consistent withthe experimental data on the strengths of the four types of unidirectional CFRP composites used herein. A scenario on the origin of stressconcentrations generated on an intact ﬁber surface is investigatedthrough a numerical analysis based on the ﬁnite element method.

a 3D model. As mentioned by Xia et al. [17], the shear-lag model yields
an accurate prediction of the stress distribution around ﬁber breakpoints; however, its computational cost is known to be more expensive
than that of the above-mentioned LGF approach. Therefore, to improve
computational eﬃciency, Okabe et al. [18] proposed a novel approachreferred to as the spring element model, which couples the shear-lag
model with analytical stress recovery and enables the analysis of thestress state of the composites, with the exception of the stress recovery
region. Tavares et al. [19] recently developed the spring element model
to consider a random ﬁber packing and hybrid ﬁber structure. Moreover, improved computer techniques have enriched the understandingof the failure processes of UD composites [20–22].
According to previous reports, the failure processes of UD composites can be explained as follows: the load perturbation resulting from aﬁber failure is not uniformly distributed among the surviving ﬁbers
because it is more heavily applied to the adjacent ﬁbers. Thus, when
one ﬁber breaks, the load that it is carrying is principally transferred tothe surviving neighbors, increasing the amount of stress concentrated
on these ﬁbers relative to more distant ﬁbers and increasing the probability of failure at this position. This consequently leads to the formation of clusters of broken ﬁbers, and subsequent failure of thecomposites. Considering this, although most studies to date have only
addressed the load redistribution caused by ﬁber breakage, several
studies that have implemented fragmentation tests have reported that amatrix crack or damage to the matrix originates around a ﬁber breakpoint. Moreover, although the local stress on ﬁbers adjacent to a broken
ﬁber has been widely recognized as critical to the strength of unidirectional CFRP composites, an adequate method for predicting thestrength of these composites — one that considers the stress concentration on the ﬁber surface resulting from a ﬁber break in a neighboring ﬁber — is yet to be developed.
In the last few decades, techniques such as Raman spectroscopy[23–28], polarized-light microscopy [29,30], acoustic emission [31],and synchrotron radiation computed tomography [22,32], have been
employed to experimentally investigate the stress concentration incomposites with in-situ observations of ﬁber breakage. As an example,
Jones et al. [29] directly measured the stress concentration on ﬁbers
adjacent to a broken ﬁber by using a multi-ﬁber fragmentation technique in which nine individual ﬁbers were arranged parallel to the load
axis. The authors reported that the stress concentration tended to increase with the decreased interﬁber spacing, and that the ﬁber-fracturephase transition from the original statistical distribution of strength topossessing areas of concentrated stress occurred at a ﬁber spacing ofeight ﬁber diameters. Similarly, Van den Heuvel et al. [23] also observed the ﬁber fragmentation behavior in multi-ﬁber composites byimplementing a Raman spectroscopy technique. They found that thetransition occurred at a ﬁber spacing of nine ﬁber diameters. These
results [23,29] suggest that, for cases in which commercial composites
with a small interﬁber spacing are implemented, ﬁber breakage primarily occurs as a consequence of stress concentration, which typicallyleads to the breakage of the ﬁber directly adjacent to a fractured ﬁber;
this implies that, the fragmentation patterns in embedded ﬁbers are
similar. Furthermore, Van den Heuvel et al. also investigated the strain
distribution along the longitudinal axis of ﬁbers [26]; however, themeasured stress concentration factor (SCF) was not suﬃciently high in
inducing successive ﬁber failure in the neighboring ﬁbers. Based onthese results, we have inferred that the area excited by the Raman incident light is inclined at an angle of 90° with respect to the area ofstress concentration, as illustrated in Fig. 1. Thus, even if the measured
strain is not suﬃciently high, the signiﬁcant (localized) stress concentration is expected to generate on the surface of an intact ﬁber neara ﬁber break point. This has already been recognized and investigatedby Xia et al. [17] and Swolfs et al. [33]. Watanabe et al. [34] recently
conducted fundamental research to determine the surface SCF by performing double-ﬁber fragmentation tests, and subsequently compared
the results to the corresponding results of a spring element model

2. Experimental method
2.1. Sample preparation and mechanical characterization
High-strength, polyacrylonitrile (PAN)-based carbon ﬁber
(TORAYCATM T1100G) and four types of bisphenol-A epoxy resin materials were used to prepare multi-ﬁber and UD composites. Table 1
presents a summary of the physical and mechanical properties of the
ﬁber and the matrix. Note that the tensile-loading experiments performed herein revealed that the four types of epoxy materials exhibited
diﬀerent mechanical characteristics (details on the sample preparationand mechanical evaluation are provided in the Supplementary
Material). Thus, the epoxy materials are hereafter referred to as “Aepoxy,” “B-epoxy,” “C-epoxy,” and “D-epoxy,” with the order of thenames indicating the magnitude of the elastic modulus (lowest to
highest).
Most related studies have implemented various statistical models to
describe the statistical distribution of ﬁber strength, such as the unimodal Weibull distribution, Weibull of Weibull distribution, bimodal
Weibull distribution, lognormal distribution, and Gaussian distribution.
We chose to apply the bimodal Weibull distribution as the statistical
distribution of ﬁber strength, based on the report by Watanabe et al.
[34]. Table 2 summarizes the parameters used to construct the bimodal
Weibull distribution for the T1100G carbon ﬁber used herein (the detailed procedures for determining the bimodal Weibull parameters will
be discussed in Section 2.2). Two diﬀerent types of PAN-based carbon
ﬁbers reported by Watanabe et al. [30,34] are also indicated in Table 2.
Note that diﬀerent types of PAN-based carbon ﬁbers possess a diﬀerent
statistical distribution of strength.
Multi-ﬁber fragmentation specimens were prepared by positioning
two to four ﬁbers parallel to the loading direction, implementing aninterﬁber spacing of approximately 3.5–20.0 μm (i.e., approximately
one-half to four ﬁber diameters). After adjusting the interﬁber spacing,the ﬁbers were translated and glued onto two pieces of plastic tape withan interspatial distance of approximately 80 mm, then positioned in a&RPSRVLWHV3DUW$  ²

G. Yamamoto, et al.

Fig. 1. Schematic showing the relationship between the stress concentration area and the area excited by Raman incident light. (For interpretation of the references
to colour in this ﬁgure legend, the reader is referred to the web version of this article.)counted using the polarized-light microscope. The strain applied to theﬁber εf was calculated as follows:

Table 1
Physical and mechanical properties of the T1100G carbon ﬁber and the fourtypes of epoxy materials: Young’s modulus (E), tensile strength (σ), Poisson’s
ratio (v), failure strain (ε), and diameter (D). The average value is given, andthe range is indicated in parentheses.

Carbon ﬁber
A-epoxy
B-epoxy
C-epoxy
D-epoxy

E (GPa)σ (MPa)

v (–)

ε (%)

D (μm)

324
3.10(2.82–3.47)
3.26(3.18–3.45)
3.33(3.14–3.48)
3.80(3.72–3.93)

7000
94.0
(78.3–103.4)
97.0(95.5–98.4)
101.0(96.3–103.5)
102.9(101.3–105.2)–
0.34(0.32–0.38)
0.34(0.33–0.34)
0.34(0.39–0.36)
0.34(0.32–0.35)–
6.2(4.1–7.8)
8.1(7.8–8.4)
6.6(5.6–7.3)
7.5(7.3–7.7)

5.4n/aεf = εc ×n/an/aTable 2
Bimodal Weibull parameters of the T1100G carbon ﬁber and two diﬀerent typesof PAN-based carbon ﬁbers (i.e., T800S and T700S).

T1100G
T800S
T700S

m1

σ02

m2

Source

7.7
6.9
5.2

4.5
4.1
4.8

9.1
8.3
6.1

13.0
13.0
12.0

This study
Refs. [30,34]
Refs. [30,34](1)

where εc is the measured composite strain, κ is the strain gauge factor(=2.13), t is the thickness of the multi-ﬁber composites (=∼2 mm), d
is the embedded ﬁber depth of the ﬁbers (=∼60 μm), and εfr is theindirectly applied strain originating from the included residual compressive strain on the ﬁber caused by the thermal expansion mismatch
between the ﬁber and the matrix, as determined via Raman spectrum
analysis. Both t and d were measured prior to fragmentation testing.
The tensile strengths of the prepared unidirectional CFRP composites were measured to validate the accuracy of the tensile strength
predicted by a spring element model derived based on the multi-ﬁber
fragmentation test results. The composites were prepared via conventional vacuum bagging and autoclave laminating technique to produce
the laminate structure of [06]. The maximum applied pressure during
processing was approximately 0.6 MPa. The respective curing and postcuring temperatures and durations were identical to those employed in
the multi-fragmentation specimen preparation. The ﬁber volume fraction and the bulk density of the resultant composites were 57% and1.79 Mg/m3, respectively. The tensile strength of each of the composites, in the form of a 12.7 mm (width) × 1.1 mm (thickness) × 230 mm (length) test specimen, was measured via a tensileloading experiment under ambient conditions. The gauge length andthe crosshead speed for the tensile tests were 127 mm and 21.2 μm/s
(1.27 mm/min), respectively. A strain gauge was used to measure thelongitudinal strain of one side of the specimen. Five samples for the fourtypes of unidirectional CFRP composites were measured. The ranges ofthe measured properties, in addition to the averaged values, are reported in subsequent sections.

n/a

σ01

2.0 ⎛ t − 2d ⎞×
− εfr,
κ⎝ t ⎠glass mold, such that the embedded depth of the ﬁbers was approximately 60 μm from the surface of the specimen. Subsequently, a preheated and degassed epoxy resin was poured into the preheated glass
mold in which the ﬁbers were bonded. Finally, the specimens were
cured in an air oven at 160 °C for 5 h prior to post-curing at 180 °C for
2 h. The specimens were cooled in the oven to room temperature before
being cut into 74 mm (length) × 26 mm (width) samples using a wheelsaw (Metkon, METACUT 251).
Multi-ﬁber fragmentation tests were performed to facilitate thederivation of a quantitative description of ﬁber failure processes byusing a polarized-light microscope (OLYMPUS, BX60) equipped with acustom-made four-point bending machine. The inner and outer span
lengths were 18 mm and 50 mm, respectively. A strain gauge (KYOWA,KFG-2-120-C1-11) was attached on the surface of the multi-ﬁber composites subjected to tensile loading to monitor the tensile strain applied
to the ﬁbers. The strain was increased in 0.1% steps until a maximum of
5.0% tensile strain was achieved and held constant during the observation of both the number and positions of the broken ﬁbers. The
number of broken ﬁbers observed under the pure bending conditions(i.e., at the central part of the specimen (=10 mm in length)), was

2.2. Model preparation
Monte-Carlo methods were implemented in the spring element
model simulation to determine the stress concentrated on the surface of
the intact ﬁbers surrounding the ﬁber break points. The spring element
model comprises longitudinal and transverse spring elements in a threedimensional hexagonal arrangement [18,30]. The longitudinal element
represents ﬁbers that exclusively carry the tensile load, whereas thetransverse element represents the matrix that only sustains the shear
load. The stiﬀnesses of the longitudinal spring element KLe and transverse spring element KTe are respectively calculated as follows:&RPSRVLWHV3DUW$  ²

G. Yamamoto, et al.

KLe = πR2

KTe =

πRl
3

∫0 BLeT EBLe dz,l∫0 BTeT GBTe dr ,

(2)

l

(3)

1
1
BLe = ⎡ − ⎤,
l⎦
⎣l

(4)

1
1
B Te = ⎡ − ⎤,
d⎦
⎣d

(5)

where L and T represent the longitudinal and transverse directions,
respectively, E is the Young’s modulus of a ﬁber, G is the shear modulus
of a matrix, R is the radius of a ﬁber, and l and d are the spring lengths
along the longitudinal and transverse directions, respectively. The
length of the transverse spring d can be obtained as follows:

⎧

d=

⎨R ⎛
⎩ ⎝

0.01
2π
3 Vf

(f ≤ 4)− 2⎞ (f > 4)⎠,

Fig. 2. Fiber breaking behavior obtained from single-ﬁber fragmentation tests.
The behaviors calculated using both the unimodal Weibull parameters and thebimodal Weibull parameters are indicated. (For interpretation of the references
to colour in this ﬁgure legend, the reader is referred to the web version of thisarticle.)(6)

where Vf and f are the ﬁber volume fraction and the number of ﬁbers in
the model, respectively. The experimentally observed average interﬁber
spacing in the multi-ﬁber fragmentation specimens was approximately
0.01 mm (10 μm). Therefore, d was set to 0.01 to analyze the results ofthe multi-ﬁber fragmentation tests performed under the condition thatthe maximum number of ﬁbers in the model is four. Conversely, ahexagonal close-packed structure was assumed for the analysis of theunidirectional CFRP composites. In spring element models with two tofour ﬁbers, the longitudinal spring elements other than ﬁbers are assigned as a matrix; thus, the stiﬀness of a matrix is implemented in thelongitudinal spring elements.
The stress proﬁle of ﬁbers generally varies because of plastic deformation and damage, such as debonding and matrix cracking. The
matrix is indirectly modeled in this study; hence, implementing debonding and matrix cracking would be diﬃcult. Therefore, for simplicity, this model considered only the eﬀect of the plastic deformation of
the matrix. We considered the situation in which the ﬁber axial stresswithin the stress recovery region σs can be expressed as a linear functionof the distance Ds from the ﬁber break point, as is described by thefollowing equation [18]:σs =2τs Ds,R(L f,0 ) of 10 mm and gauge lengths of 10, 25, and 50 mm (the detailed
procedures for determining σ01 and m1 are explained in the
Supplementary Material). Fig. S2 in the Supplementary Material shows
Weibull plots of the T1100G carbon ﬁber determined using Eq. (S1).
The intercept (σ01 = 7.7) and the slope (m1 = 4.5) were determined byapplying a least-squares technique to the acquired Weibull plots. Fiberbreaking behavior obtained from single-ﬁber fragmentation tests isshown in Fig. 2. It is observed that ﬁber break behavior calculated by
implementing unimodal Weibull parameters (indicated by the red dashed line) is not consistent with the experimentally obtained data.
Therefore, the second term parameters, σ02 and m2 , were determined byusing the elastoplastic shear-lag model to apply a curve-ﬁtting scheme
to the experimental data [35]. The ﬁtting results and resultant parameters are presented in Fig. 2 and Table 2, respectively.
Fibers are hexagonally arranged in the model; thus, the ﬁber surface
can be divided into six segments, as illustrated in Fig. 3. The surface
SCF of the i -th ﬁber segment adjacent to a broken ﬁber at the plane offracture was considered by assuming that it was α i times the average
stress on the surface of an adjacent ﬁber. This is given as follows:(7)D
α i = 1 + γi ⎛1 − s ⎞.
ls ⎠⎝where τs is the interfacial shear stress that is assumed to be constant,which means that the matrix is modeled as a perfectly elastoplastic
body. Thus, the equilibrium equation to represent the entire system isexpressed as follows:

⎡
⎢
⎣

∑

Nf − N b − Np
e=1⎤
KLe − ∑ KTe ⎥ u + ∑ πR2
e=1
e=1
⎦
Np

Nm

∫0 BLe σs dz = f ,
l

⎜

(8)

where α is the SCF of a ﬁber adjacent to a broken ﬁber in the sameplane, Ds is the distance from the break point, and l s is the stress recovery length. Thus, considering the contribution of stress concentration on an intact ﬁber surface, the ﬁber breakage probability Pf,i (σ ) canbe rewritten as follows:

where Nf and Nm are the number of ﬁber and matrix elements, respectively, Nb is the number of broken ﬁbers, and Np is the number ofﬁber elements in the plastic deformation regions.
We considered herein the situation in which the ﬁber failure is assumed to occur at the ﬁber surface, based on the fact that almost all
ﬁbers were broken by surface ﬂaws [34]. The ﬁber breakage probability
Pf (σ ) under the condition that the ﬁber is subjected to stress σ at thesurface area Sf (=2π RL f ) is expressed as follows:m⎜⎟

Sf,i ⎛ α i σ ⎞m1
Sf,i ⎛ α i σ ⎞m2 ⎫Pf,i (σ ) = 1 − exp ⎧−−,⎨Sf,0 ⎝ σ02 ⎠ ⎬⎩ Sf,0 ⎝ σ01 ⎠
⎭
⎜

m

⎜

⎟

(10)

0
(a fiber segment facing a neighboring intact fiber)
γi = ⎧⎨
α−1(afiber segment facing a neighboring broken fiber)
⎩
(11)

T

1
2
S
σ
S
σ
Pf (σ ) = 1 − exp ⎧− f ⎛ ⎞ − f ⎛ ⎞ ⎫⎨Sf,0 ⎝ σ02 ⎠ ⎬⎩ Sf,0 ⎝ σ01 ⎠
⎭

⎟

(9)

where L f is the gauge length, Sf,0 (=2πRL f,0 ) is the representative surface
area of the ﬁber (L f,0 = 10 mm), σ01 and σ02 are the Weibull scaleparameters, and m1 and m2 are the Weibull shape parameters. The
parameters in the ﬁrst term within the brackets, σ01 and m1, were obtained through single-ﬁber tensile tests under the representative length


⎟

⎜

⎟

(12)

where Sf,i is the i -th ﬁber segment of surface area (Sf,i = πRL f,o /3). As
mentioned earlier, the spring element model does not directly represent
the matrix aside from the shear load. Therefore, the additional stressconcentration is added in an ad-hoc manner to capture the experimentally measured correlations in ﬁber breaks.
The strength of the n -th ﬁber segment is determined by choosing arandom number Rn ranging from 0 to 1 and solving equation
Rn = Pf,n (σn ) . The longitudinal element was removed from the model

&RPSRVLWHV3DUW$  ²

G. Yamamoto, et al.

Fig. 3. Deﬁnition of the stress concentration on the surface of the i-th ﬁber segment.

when the stress applied to a ﬁber at the n -th ﬁber segment achieved thestatistical distribution of the strength of the ﬁber σn .

3. Results and discussion

2.3. Determination of the stress concentration factor

First, we prepared unidirectional CFRP composites consisting of thefour types of matrix polymers, and subsequently employed tensileloading tests to investigate their mechanical properties along the direction of the ﬁber axis. Fig. 4 and Table 3 summarize the representative stress-strain behavior and summary of the measured mechanical properties, respectively. Note that all composites tested in thisstudy experienced catastrophic failure after reaching a maximum load,
exhibiting a stress-strain relationship that is typically observed inconventional unidirectional CFRP composites. As presented in Table 3,
no signiﬁcant diﬀerence was observed between the Young’s modulus
values, whereas for example the composite fabricated with the D-epoxy
demonstrated strength enhanced by a factor of approximately 1.2
compared to the composite made with the A-epoxy.

3.1. Mechanical evaluation of unidirectional CFRP composites

The SCFs α on the surface of an intact ﬁber were determined by
employing the spring element model to investigate the α value, with theaim of ensuring that it was equivalent to the percentage of the coordinated fracture, which is deﬁned as a failure occurring at the elements neighboring a broken element in the horizontal plane of thebroken ﬁber element, that was determined via multi-ﬁber ﬁber fragmentation testing. Thus, the two to four longitudinal spring elements inthe center of the spring element model were assigned to the ﬁbers, andthe remaining elements were assigned to the matrix. The ﬁber elements
measuring 10 mm in length were divided into 2000 segments (i.e., theunit length of the ﬁber element was 5 μm). A coordinated fracture wasdeﬁned as a failure that occurred at neighboring elements next to abroken element in the same horizontal plane with respect to the broken
ﬁber element. Note that we previously analyzed how the ﬁber breakpoint is changed by varying α from 1.0 to 2.0 using a double-ﬁber
composite model [30]. Under the condition that α is 1.0, the ﬁber breakpositions of two ﬁbers were observed to randomly distribute along theﬁber axis direction. By contrast, the simulation results obtained considering an additional stress concentration of α = 2.0 agreed well withthe ﬁber break behavior obtained from the experiments.

3.2. Tensile strength prediction of B-epoxy matrix composites
As previously mentioned, the ﬁbers surrounding a broken ﬁber were
subjected to increased stress concentration, which increased the probability of failure. Therefore, the understanding of the failure phenomena of such ﬁber is a prerequisite for the tensile strength predictionof unidirectional CFRP composites. First, we investigated the inﬂuenceof interﬁber spacing on the failure phenomena of the ﬁbers by performing a double-ﬁber fragmentation test. For this test, the B-epoxy
material was used in the specimen preparation. Fig. 5 illustrates thebirefringence patterns at the ﬁber break points in the double-ﬁber
fragmentation composites that varied by interﬁber spacing. The images

2.4. Tensile strength determination of unidirectional CFRP composites
The tensile strengths of the unidirectional CFRP composites werepredicted via the spring element model. The bimodal Weibull distribution was applied as the statistical distribution of ﬁber strength. The
model comprised 1024 ﬁbers measuring 3 mm in length that were divided into 300 segments. The stress concentration was applied to thesurface of the intact ﬁbers according to Eq. (10). The tensile strength isdeﬁned as the applied maximum composite stress, and the ﬁnal failure
is assumed to be in progress once the average ﬁber stress decreases to
90% of the maximum ﬁber stress (as shown in Supplementary Fig. S3,
maximum composite stress does not vary when the criterion of the ﬁnal
failure is changed to 70%, 80%, or 90%). Prior to comparing the simulated results to the experimental data, the simulated strengths were
subjected to size scaling, as follows [36]:nL = −n s Ls /ln(1 − Fs (∼σn )),(13)

where n s is the number of the ﬁbers (n s = 1024), Ls is the length of theσn ) is the cumulative probability of failure atcomposite (Ls = 3 mm), Fs (∼a given strength as determined by 100 spring element model simulation
runs, and nL is the size at which the characteristic strength is achieved.

Fig. 4. Representative stress-strain curves for the four types of unidirectional
CFRP composites.
&RPSRVLWHV3DUW$  ²

G. Yamamoto, et al.

appeared to be higher than those observed for ﬁber failure that weregoverned by the statistical strength distribution of ﬁbers [23]. This
result indicates that for an interﬁber spacing of one-half to four ﬁberdiameters, the failure process of the ﬁber was predominantly governedby the ﬁber–ﬁber interactions. Note that debonding was not apparent
around the ﬁber break points in any specimen, suggesting that thecomposites prepared herein exhibited good interfacial connectivity.
Next, the ﬁber fracture behavior of the multi-ﬁber fragmentation
composites consisting of up to four ﬁbers was investigated to quantitatively determine the surface SCFs. As shown in Fig. 6, the polarizedlight microscopy investigation revealed that matrix cracks and the coordination of fractures in adjacent ﬁbers were observed in multi-ﬁber
fragmentation composites, regardless of the number of ﬁbers. Fig. 7
shows the dependence of the number of ﬁber breaks in a primary ﬁberon the percentage of coordinated fractures, with the primary ﬁber being
a ﬁber with a larger number of breaks. The percentage of coordinatedfracture in the double-ﬁber composites tended to increase as thenumber of ﬁbers was increased. Figs. 5 and 6 depict that being positioned along the same plane of a fracture does not necessarily mean thatthe ﬁber will fracture; thus, the reduction in the percentage of coordinated fractures may be attributable to the fact that coordinatedfractures are less likely to occur when the number of ﬁbers is high. At
3.2% ﬁber strain (i.e., εf calculated via Eq. (1) for a composite strain εcof 5.0%), the percentages of coordinated fractures (and standard deviation) in the double-, triple-, and quadruple-ﬁber composites were48.3% ( ± 14.1%), 33.3% ( ± 17.8%), and 15.8% ( ± 5.8%), respectively. Additionally, the number of ﬁbers and interﬁber spacing were
not observed to yield any eﬀect on the fracture onset strain (i.e., thestrain at which the ﬁrst ﬁber failure was generated).
A quantitative determination of the SCFs α on the surface of an
intact ﬁber near a ﬁber break point was achieved by implementing thespring element model to investigate the value of α . Fig. 8 illustrates the

Table 3
Measured properties for the four types of unidirectional CFRP composites.
Shown are the Young’s modulus, tensile strength, and failure strain along thedirection of the ﬁber axis.

A-epoxy matrix
composite
B-epoxy matrix
composite
C-epoxy matrix
composite
D-epoxy matrix
composite

Young’smodulus (GPa)Tensile strength(GPa)

Failure strain(%)

179(178–180)
175(169–179)
171(166–175)174(172–177)

3.12(2.95–3.22)
3.17(3.05–3.32)
3.26(3.08–3.35)
3.85(3.74–3.97)

1.60(1.51–1.65)
1.65(1.59–1.74)
1.73(1.67–1.83)
1.98(1.93–2.01)

were acquired following an approximately 5.0% alleviation of thecomposite strain. The illustrations above each of the birefringence
patterns demonstrate the positional dependency of neighboring ﬁbers
with respect to the ﬁber break point. 0° indicates a perfect alignment of
the fracture site. Irrespective of the interﬁber spacing, ﬁber breakage
resulted in matrix crack initiation. Moreover, a large number of ﬁber
failures occurred at similar positions (i.e., at small angles), indicatingthat the stress concentration caused by ﬁber fracture was suﬃciently
high to cause the adjacent ﬁber to fracture, which nulliﬁes the inﬂuenceof random distributed ﬂaws along the ﬁber on the ﬁber strength. As aﬁber-ﬁber interaction criterion, ﬁber fractures that occurred at an angle
within 0° to 45° are deﬁned as coordinated fractures based on thenature of elastoplastic polymer material fracture phenomena [37]. The
percentages of coordinate fractures occurring in double-ﬁber composites with interﬁber spacings of 3.6 μm, 9.9 μm, and 20.0 μm were 73%,
57%, and 60%, respectively. Even though some variation was observed
for the percentages of coordinate fractures, the measured percentages

Fig. 5. Birefringence patterns in the double-ﬁber fragmentation composites with an interﬁber spacing of (a) 3.6 μm, (b) 9.9 μm, and (c) 20.0 μm. (For interpretationof the references to colour in this ﬁgure legend, the reader is referred to the web version of this article.)&RPSRVLWHV3DUW$  ²

G. Yamamoto, et al.

Fig. 6. Birefringence patterns in the (a)triple-ﬁber fragmentation composite and (b)quadruple-ﬁber fragmentation composite.
The matrix cracks and coordination of fractures are observed for the multi-ﬁber fragmentation composites with interﬁber spacing ranging from one-half to four ﬁberdiameters. (For interpretation of the references to colour in this ﬁgure legend, thereader is referred to the web version of thisarticle.)Fig. 7. Eﬀects of the number of ﬁber breaks in primary ﬁbers on the percentage
of coordinated fractures.

relationship between the SCF and the percentage of coordinated fractures for each fragmentation composite, which was determined bysystematically sweeping α in the spring element model simulation. The
simulation results showed that the percentage of coordinated fractures
increased in response to an increased SCF, but decreased with an increase in the number of ﬁbers. Upon comparing the simulated percentages of coordinated fractures to the corresponding experimental observations, the SCF on the surface of an intact ﬁber was determined tobe approximately 2.0. Moreover, the number of ﬁbers did not signiﬁcantly aﬀect the surface SCF, indicating that for the composites
fabricated with B-epoxy, the concentrated stress acting on the ﬁber
surface was twice as much as the ﬁber stress with no additional surface
stress concentration.
As previously mentioned, stress concentration was considered in theprediction of the tensile strength of the unidirectional CFRP composites
fabricated with the B-epoxy matrix. Fig. 9 depicts the ﬁber fracture
behaviors on the ﬁnal failure planes simulated via the spring element
model. We deﬁned the plane, including the most ﬁber breaks, as theﬁnal failure plane. A stress ratio of less than 1 in Fig. 9 indicates that theﬁber failure occurred near the ﬁnal failure plane, while a stress ratio of
0 illustrates that the ﬁber failure occurred in the ﬁnal failure plane. In
the case of no additional stress concentration condition (i.e., α = 1.0 ),the ﬁber failure was clearly randomly distributed across the plane. Thisphenomenon is consistent with the statistical distribution of ﬁber
strength. In contrast, additional concentrated stress encourages theformation of broken ﬁber clusters, which results in relatively premature
fracturing. Fig. 10 illustrates a comparison of the experimental and simulated results for the unidirectional T1100G carbon ﬁber/B-epoxy
composites. In this ﬁgure, the circles represent the size scaled strength
results calculated according to Eq. (13), whereas the cross symbols
denote the experimentally obtained tensile strengths of these composites, which ranged from 3.05 to 3.32 GPa (mean: 3.17 GPa). The simulated data obtained without consideration of added concentrated

Fig. 8. Relationship between the SCFs and the percentage of coordinatedfractures for the (a) double-, (b) triple-, and (c) quadruple-ﬁber fragmentation
composites.
&RPSRVLWHV3DUW$  ²

G. Yamamoto, et al.

Fig. 9. Distribution of stress in the axial direction on the ﬁnal failure planes. (For interpretation of the references to colour in this ﬁgure legend, the reader is referredto the web version of this article.)

3.3. Application to diﬀerent types of composites
We applied the above-mentioned strength prediction method to theunidirectional CFRP composites made with the “A-epoxy,” “C-epoxy,”and “D-epoxy.” The SCF on the surface of intact ﬁbers was acquired via
double-ﬁber fragmentation testing considering the fact that for the Bepoxy matrix composites, the number of ﬁbers did not inﬂuence the
SCFs (Fig. 8). Fig. S4 in the Supplementary Material illustrates the results of implementing SCF determination obtained via double-ﬁber
fragmentation testing and the spring element model simulation. The
SCFs were calculated as approximately 2.15 for the A-epoxy, 1.93 for
the C-epoxy, and 1.75 for the D-epoxy. A comparison of the resultspresented in Fig. 8 and Fig. S5 revealed clear diﬀerences among them.
Thus, surface SCFs were implemented to predict the tensile strengthof the three types of the unidirectional CFRP composites. Fig. 11 presents a comparison of the experimental and simulated results. In one
example, the experimentally obtained tensile strengths of the D-epoxy
Fig. 10. Experimental and simulated results for the unidirectional T1100G/Bepoxy composites.

stress were incongruent with the experimental data, whereas the predictions incorporating an SCF of 2.0 were reasonably consistent withthe experimental data. Furthermore, although the prediction with
α = 2.1 seemed to yield a better tensile strength prediction than with
α = 2.0 , we conclude that the prediction method proposed herein yields
a reasonably accurate tensile strength prediction when the matrix
crack-induced surface stress concentration of ﬁbers is appropriately
considered.
Previously, Okabe et al. [36] predicted the tensile strength of unidirectional CFRP composites made with T800H carbon ﬁber, and theresult analyzed by the 3D shear-lag model provided the upper limit of
the experimental value. However, the strength of the T800H carbon
ﬁber used to predict the tensile strength of the unidirectional CFRP
composite was lower than that speciﬁed in the supplier’s catalog data.
In addition, the validity of the strength data is unclear, because thesedata have not been compared against the results of the single-ﬁber
composite tests discussed in the study; this indicates a need for re-examination of the tensile strength prediction processes reported in [36].

Fig. 11. Experimental and simulated results for the three types of unidirectional CFRP composites. The cross symbols indicate the experimentally obtained tensile strengths of these composites.
&RPSRVLWHV3DUW$  ²

G. Yamamoto, et al.

Fig. 12. (a) Schematic representation of the
one-twelfth of the composite model. (b)Contour image of normal stress in the ﬁber
axial direction at the 2.0% strain condition.
(c) Enlarged image, taken from the whitesquare area in image (b). (For interpretationof the references to colour in this ﬁgure legend, the reader is referred to the web version of this article.)matrix composites ranged from 3.74 to 3.97 GPa (mean: 3.85 GPa), andare denoted by a cross symbol in the ﬁgure. The estimated tensile
strength simulated under the condition of α=1.75 was approximately
3.9 GPa. This value is reasonably consistent with the experimentallyobtained tensile strength. Additionally, a cluster formation on the ﬁnal
fracture plane was observed in the composite system (Supplementary
Fig. S5). Consequently, the results demonstrate that, even if the mechanical properties of the matrix materials vary, the proposed method
can yield a reasonable prediction of the tensile strength of the unidirectional CFRP composites.
The possible mechanism by which the additional stress concentration occurs is not clear; moreover is it unclear why the SCF varies depending on the matrix characteristics. A numerical analysis using theﬁnite element method was conducted to access possible mechanisms bywhich higher stress is generated on the intact ﬁber surface adjacent tothe ﬁber break point. In this analysis, a hexagonal ﬁber arrangement
was used, and only one-twelfth of the structure was modeled andanalyzed owing to reasons of structural symmetry. The schematic representation of the one-twelfth of the composite model is shown in
Fig. 12(a). The ﬁber and matrix were assumed to be elasto-plastic andelastic materials, respectively. The matrix was assumed to have rehardening characteristics with a modulus of 0.38 GPa, which is 1/10 of
the initial slope in the elastic region of the D-epoxy material. Moreover,the plasticity-free layer model of Suo, Shih, and Varias [38], referred toas the SSV model, an elastic layer with a thickness of 50 nm, was imposed around the matrix crack using the same elastic properties as theD-epoxy material. The matrix crack was assumed not to reach the intactﬁber surface and there was a 30 nm-thick elastic layer between thecrack tip and the intact ﬁber surface. Due to the elastic region surrounding the matrix crack, the stress singularity is retained. Displacement control is considered in the model, and the maximum appliedcomposite strain is 2.0%. The elastic stiﬀnesses of the ﬁber and the Depoxy materials are listed in Table 4. The elasto-plastic behavior modelFig. 13. Elasto-plastic behavior model of the D-epoxy material.

of the D-epoxy material is shown in Fig. 13. Fig. 12(b) and (c) show
contour images of normal stress in the ﬁber axial direction at the 2.0%
strain condition.
It was revealed that an SCF of α = ∼1.7, as observed for the Depoxy matrix composite, was indeed generated on the intact ﬁber surface by deﬁning the SCF as the ratio of the stress of an outermost surface element of the intact ﬁber to the stress in an element suﬃciently
away from the ﬁber break point. In addition, as shown in Fig. 14, thestress recovery behavior of the broken ﬁber was reasonably consistent
with the behavior obtained from the spring element model simulation
under the condition α = 1. Fig. 15 shows the positional dependence of
the SCF in the circumferential direction of the intact ﬁber; the SCF
decreases rapidly with the increase in the angle, and the SCF decreases
almost 1 at about 60°. Note that we previously investigated the SCFs
using analytical models that included/excluded SSV and rehardening
treatments; the SCFs obtained were less than 1.4 in all the analyses,
indicating that no signiﬁcant increase in the SCFs was observed whenexcluding the two treatments. We consider that the above-mentioned
mechanism is one of the scenarios in which a higher stress is generated
on the intact ﬁber surface. It remains unclear why the SCF varies withthe matrix characteristics, and further research is needed to clarify thisreason; however, because the degree of SCF depends on the gap between the crack tip and the intact ﬁber surface, we speculate that theTable 4
Elastic stiﬀness Cij (GPa) and Young’s moduli Eij (GPa) of the T1100G carbon
ﬁber [39] and D-epoxy material. The x3-axis is parallel to the ﬁber axis direction.

Fiber
D-epoxy

C11

C33

C13

C44

C66

E11

E33

24.8
5.8

314.7

22.7
3.0

20.5
1.4

5.6

17.0
3.8

287.9



&RPSRVLWHV3DUW$  ²

G. Yamamoto, et al.

the eﬀects of matrix polymer properties on the stress concentrated on
the ﬁber surface. Consequently, the multi-ﬁber fragmentation test results demonstrated that for an interﬁber spacing of one-half to four ﬁberdiameters, the failure process of the ﬁber was predominantly governed
by ﬁber-ﬁber interactions, irrespective of the matrix polymer properties. We also demonstrated that the degree of stress concentrated on thesurface of ﬁbers can be changed by modifying the mechanical properties of the matrix polymer. Additionally, utilizing an epoxy matrix witha higher Young’s modulus and increased tensile strength in the composite preparation reduced the SCF from approximately 2.15 to 1.75.
We have also shown a numerical scenario on the origin of the stressconcentrations that are generated on the intact ﬁber surface by implementing the SSV model and employing the rehardening characteristics of epoxy materials. Finally, we conﬁrmed that employing themeasured SCFs and bimodal Weibull distribution to determine how
strength is statistically distributed throughout the ﬁber yields the predicted strengths of the four types of unidirectional CFRP compositesthat are reasonably consistent with the experimental data, thereby
demonstrating the validity of the proposed prediction method.

Fig. 14. Comparison of the stress recovery behavior calculated from the ﬁnite
element simulation and spring element model simulation.

Declarations of interest
None.
Acknowledgements
The authors thank Mr. W. Shoichiro of the Department of Aerospace
Engineering, Tohoku University, and Dr. R. Higuchi of the Department
of Aeronautics and Astronautics, The University of Tokyo, for technicalassistance in the FEM analysis. This work was partly supported by
Toray Industries, Inc., Japan, the Council for Science, Technology andInnovation (CSTI), Japan, the Cross-ministerial Strategic Innovation
Promotion Program (SIP), Japan, and JSPS KAKENHI grant number
18K04721. The authors would like to thank the reviewers for theiruseful comments and recommendations. 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). The authors would like to thank Editage(www.editage.jp) for the English language editing.

Fig. 15. Positional dependence of the SCF in the circumferential direction of
the intact ﬁber. (For interpretation of the references to colour in this ﬁgurelegend, the reader is referred to the web version of this article.)

Appendix A. Supplementary material
diﬀerence in the gap distance was caused by the diﬀerence in the modeI stress intensity factor of the epoxy matrix.
As previously mentioned, improving the accuracy of the tensile
strength prediction is central to CFRP composite research. For instance,in certain types of CFRP material [36], our previous model overestimated the ultimate tensile strength compared with the experimentally obtained data, and we could not make a full explanation of its
reason. The results indicate that the fact that matrix cracks near theﬁber break points increase the amount of stress concentrated on theﬁber surface must be considered to obtain a high-accuracy strength
prediction. Moreover, the surface SCF must be properly estimated to
improve prediction accuracy. Therefore, further work is currently underway to investigate the relationship between the surface SCF andmatrix polymer properties, and this will provide useful information tocomposite researchers. Our ﬁndings provide a potential framework thatcan be implemented to facilitate the development of stronger CFRP
composites.

Supplementary data to this article can be found online at https://doi.org/10.1016/j.compositesa.2019.04.011.
References[1] Cox HL. The elasticity and strength of paper and other ﬁbrous materials. Br J Appl
Phys 1952;3(3):72–9.
[2] Rosen BW. Tensile failure of ﬁbrous composites. AIAA J 1964;2(11):1985–91.
[3] Kelly A, Tyson W. Tensile properties of ﬁbre-reinforced metals: copper/tungsten
and copper/molybdenum. J Mech Phys Solids 1965;13(6):329–50.
[4] Hedgepeth JM, Van Dyke P. Local stress concentrations in imperfect ﬁlamentarycomposite materials. J Compos Mater 1967;1(3):294–309.
[5] Suemasu H. An analytical study of probabilistic aspect of strength of unidirectionalﬁber reinforced composite under tensile loads. Trans Japan Soc Compos Mater
1982;8(1/2):29–36.
[6] Suemasu H. Probabilistic aspects of strength of unidirectional ﬁbre-reinforced
composites with matrix failure. J Mater Sci 1984;19(2):574–84.
[7] Zweben C. Tensile failure of ﬁbers composites. AIAA J 1968;6(12):2325–31.
[8] Ochiai S, Hojo M. Stress disturbances arising from cut ﬁbre and matrix in unidirectional metal matrix composites calculated by means of a modiﬁed shear lag
analysis. J Mater Sci 1996;31(14):3861–9.
[9] Ochiai S, Hojo M, Inoue T. Shear-lag simulation of the progress of interfacial debonding in unidirectional composites. Compos Sci Technol 1999;59(1):77–88.
[10] Landis CM, McMeeking RM. A shear-lag model for a broken ﬁber embedded in acomposite with a ductile matrix. Compos Sci Technol 1999;59(3):447–57.
[11] Tanaka M, Ochiai S, Hojo M, Ishikawa T, Kajii S, Matsunaga K, et al. Fracture behavior of unidirectional Si-Ti-C-O ﬁber-bonded ceramic composite materials. Key
Eng Mater 1999;164–165:141–4.
[12] Sastry AM, Phoenix SL. Load redistribution near non-aligned ﬁbre breaks in a two-

4. Conclusions
Four types of unidirectional CFRP composites with diﬀerent mechanical characteristics were prepared in this study. The tensile
strengths of the fabricated composites were predicted via a numericalsimulation based on the results obtained from multi-ﬁber fragmentation
experiments. Furthermore, the primary aim of this study was to explore&RPSRVLWHV3DUW$  ²

G. Yamamoto, et al.

1998;58(6):933–44.
[26] Van Den Heuvel PWJ, Wubbolts MK, Young RJ, Peijs T. Failure phenomena in twodimensional multi-ﬁbre model composites: 5. a ﬁnite element study. Compos Pt AAppl Sci Manuf 1998;29(9–10):1121–35.
[27] Van Den Heuvel PWJ, Peijs T, Young RJ. Failure phenomena in two-dimensional
multi-ﬁbre microcomposites. Part 4: A Raman spectroscopic study on the inﬂuenceof the matrix yield stress on stress concentrations. Compos Pt A-Appl Sci Manuf
2000;31(2):165–71.
[28] Van den Heuvel PWJ, Goutianos S, Young RJ, Peijs T. Failure phenomena in ﬁbrereinforced composites. Part 6: A ﬁnite element study of stress concentrations in
unidirectional carbon ﬁbre-reinforced epoxy composites. Compos Sci Technol
2004;64(5):645–56.
[29] Jones KD, DiBenedetto AT. Fiber fracture in hybrid composite systems. Compos Sci
Technol 1994;51(1):53–62.
[30] Watanabe J, Tanaka F, Higuchi R, Matsutani H, Okuda H, Okabe T. A study of stressconcentrations around ﬁber breaks in unidirectional CF/epoxy composites using
double-ﬁber fragmentation tests. Adv Compos Mater 2018;27(6):575–87.
[31] Pickering KL, Bader MG, Kimber AC. Damage accumulation during the failure ofuniaxial carbon ﬁbre composites. Compos Pt A-Appl Sci Manuf 1998;29(4):435–41.
[32] Scott AE, Mavrogordato M, Wright P, Sinclair I, Spearing SM. In situ ﬁbre fracture
measurement in carbon-epoxy laminates using high resolution computed tomography. Compos Sci Technol 2011;71(12):1471–7.
[33] Swolfs Y, McMeeking RM, Verpoest I, Gorbatikh L. Matrix cracks around ﬁbrebreaks and their eﬀect on stress redistribution and failure development in unidirectional composites. Compos Sci Technol 2015;108:16–22.
[34] Watanabe J, Tanaka F, Okuda H, Okabe T. Tensile strength distribution of carbon
ﬁbers at short gauge lengths. Adv Compos Mater 2014;23(1):535–50.
[35] Okabe T, Takeda N. Elastoplastic shear-lag analysis of single-ﬁber composites andstrength prediction of unidirectional multi-ﬁber composites. Compos Pt A-Appl Sci
Manuf 2002;33(10):1327–35.
[36] Okabe T, Takeda N. Size eﬀect on tensile strength of unidirectional CFRP composites experiment and simulation. Compos Sci Technol 2002;62(15):2053–64.
[37] Huang H, Talreja R. Numerical simulation of matrix micro-cracking in short ﬁber
reinforced polymer composites: Initiation and propagation. Compos Sci Technol
2006;66(15):2743–57.
[38] Suo Z, Shih CF, Varias AG. A theory for cleavage cracking in the presence of plastic
ﬂow. Acta Mater 1993;41(5):1551–7.
[39] Tane M, Okuda H, Tanaka F. Nanocomposite microstructures dominating anisotropic elastic modulus in carbon ﬁbers. Acta Mater 2019;166:75–84.

dimensional unidirectional composite using break-inﬂuence superposition. J Mater
Sci Lett 1993;12(20):1596–9.
[13] Beyerlein IJ, Phoenix SL. Stress concentrations around multiple ﬁber breaks in anelastic matrix with local yielding or debonding using quadratic inﬂuence superposition. J Mech Phys Solids 1996;44(12):1997–2036.
[14] Zhou SJ, Curtin WA. Failure of ﬁber composites: A lattice green function model.
Acta Metall Mater 1995;43(8):3093–104.
[15] Landis CM, Beyerlein IJ, McMeeking RM. Micromechanical simulation of the failureof ﬁber reinforced composites. J Mech Phys Solids 2000;48(3):621–48.
[16] Okabe T, Takeda N, Kamoshida Y, Shimizu M, Curtin WA. A 3D shear-lag model
considering micro-damage and statistical strength prediction of unidirectional ﬁberreinforced composites. Compos Sci Technol 2001;61(12):1773–87.
[17] Xia Z, Curtin WA, Peters PWM. Multiscale modeling of failure in metal matrix
composite. Acta Mater 2001;49(2):273–87.
[18] Okabe T, Sekine H, Ishii K, Nishikawa M, Takeda N. Numerical method for failure
simulation of unidirectional ﬁber-reinforced composites with spring element model.
Compos Sci Technol 2005;65(6):921–33.
[19] Tavares RP, Otero F, Turon A, Camanho PP. Eﬀective simulation of the mechanics
of longitudinal tensile failure of unidirectional polymer composites. Int J Fract
2017;208(1-2):269–85.
[20] Chou HY, Bunsell AR, Mair G, Thionnet A. Eﬀect of the loading rate on ultimate
strength of composites. Application: Pressure vessel slow burst test. Compos Struct2013;104:144–53.
[21] Thionnet A, Chou HY, Bunsell A. Fiber break processes in unidirectional composites. Compos Pt A-Appl Sci Manuf 2014;65:148–60.
[22] Swolfs Y, Morton H, Scott AE, Gorbatikh L, Reed PAS, Sinclair I, et al. Synchrotron
radiation computed tomography for experimental validation of a tensile strength
model for unidirectional ﬁbre-reinforced composites. Compos Pt A-Appl Sci Manuf
2015;77:106–13.
[23] Van Den Heuvel PWJ, Van Der Bruggen YJW, Peijs T. Failure phenomena in multiﬁbre model composites: Part 1. An experimental investigation into the inﬂuence ofﬁbre spacing and ﬁbre-matrix adhesion. Compos Pt A-Appl Sci Manuf
1996;27(9):855–9.
[24] Van Den Heuvel PWJ, Peijs T, Young RJ. Failure phenomena in two-dimensional
multi-ﬁbre microcomposites: 2. A Raman spectroscopic study of the inﬂuence ofinter-ﬁbre spacing on stress concentrations. Compos Sci Technol
1997;57(8):899–911.
[25] Van Den Heuvel PWJ, Peijs T, Young RJ. Failure phenomena in two-dimensional
multi-ﬁbre microcomposites - 3. A Raman spectroscopy study of the inﬂuence ofinterfacial debonding on stress concentrations. Compos Sci Technol