Composites: Part A 140 (2021) 106140

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

Tensile-strength-controlling factors in unidirectional carbon fiber
reinforced plastic composites
Go Yamamoto a, b, *, Keita Koizumi a, Takahiro Nakamura a, Noriyuki Hirano c,Tomonaga Okabe a, da

Department of Aerospace Engineering, Tohoku University, 6-6-01 Aramaki-Aza-Aoba, Aoba-ku, Sendai 980-8579, Japan
School of Mechanical Engineering, Sungkyunkwan University, Suwon 16419, Republic of Korea
Composite Material Research Laboratories (CMRL), Toray Industries, Inc., 1515, Tsutsui, Masaki-cho, Iyogun, Ehime 791-3193, Japand
Department of Material Science, University of Washington, Seattle, WA 98195, USAbc

A R T I C L E I N F O

A B S T R A C T

Keywords:
A. Polymer-matrix composites (PMCs)
B. Mechanical properties
B. Stress concentrations
Crack tip opening displacement (CTOD)

Factors governing the tensile strength of unidirectional carbon fiber reinforced plastic (CFRP) composites wereexperimentally investigated by focusing on the mechanical characteristics of the epoxy matrix. Correlation
analysis was conducted to reveal the mechanical characteristics of the matrix that affect the surface stressconcentration of an intact fiber caused by a fracture site in an adjacent fiber. The stress concentration factors(SCFs) were evaluated by implementing double-fiber fragmentation tests in conjunction with spring element
model simulations. From investigating six types of epoxy materials, a reasonable correlation was observed between the matrix crack tip opening displacement (CTOD) and the SCF; the SCF increased approximately linearlywith increasing CTOD. The results reported here suggest that CTOD is one of the important tensile-strengthcontrolling factors to consider for the development of stronger CFRP composites.

1. Introduction
Carbon fiber reinforced plastics (CFRPs) are applied for many engineering applications due to their high strength (stiffness)-to-weight ratio
and tailorable mechanical properties. Consequently, further enhancement of the mechanical characteristics continues to be central to CFRP
composite research. The tensile failure of CFRPs is known to be triggeredby fiber fracture sequences within 0◦ plies. The individual fiber fracturesthat occur are now known to be induced by stress concentration
generated on the intact fiber surface adjacent to the fiber break point[1]. Techniques such as Raman spectroscopy [2–3], synchrotron radiation computed tomography [4,5], and polarized light microscopy [6,7]
have been employed to investigate the origin of stress concentration in
addition to understanding fiber breaking processes. Recently, we conducted fundamental research to investigate the stress concentration
factor (SCF) by performing multi-fiber fragmentation tests, subsequently
comparing the experimental results to the corresponding spring element
model simulation, which considers the added concentrated stress on thefiber surface adjacent to a broken fiber [7]. We reported that the degreeof concentrated stress can be changed by modifying the mechanicalproperties of the epoxy matrix. Furthermore, even if the mechanicalproperties of the matrix vary, the predictive method can provide a
reasonable prediction of the tensile strength of the unidirectional CFRP
composites [7]. However, at that time, we could not provide a full
explanation regarding why the tensile strength of the unidirectional
CFRP composite varies depending on the mechanical characteristics ofthe matrix. Herein we conducted correlation analysis to reveal the mechanical characteristics of the matrix that affect the surface stress concentration of an intact fiber caused by the fracture site of an adjacentfiber. A scenario to explain the origin of stress concentration is proposed,
based on experimental evidence and conclusions drawn from correlation
analysis to investigate the relationship between possible influential
factors and SCFs.

* Corresponding author at: Department of Aerospace Engineering, Tohoku University, 6-6-01 Aramaki-Aza-Aoba, Aoba-ku, Sendai 980-8579, Japan.
E-mail addresses: gyamamoto@tohoku.ac.jp (G. Yamamoto), koizumi@plum.mech.tohoku.ac.jp (K. Koizumi), takahiro.nakamura.s1@dc.tohoku.ac.jp(T. Nakamura), noriyuki.hirano.r7@mail.toray (N. Hirano), tomonaga.okabe.a8@tohoku.ac.jp (T. Okabe).
https://doi.org/10.1016/j.compositesa.2020.106140
Received 24 March 2020; Received in revised form 23 September 2020; Accepted 27 September 2020
Available online 16 October 2020
1359-835X/© 2020 Elsevier Ltd. All rights reserved.

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Composites Part A 140 (2021) 106140

2. Experimental and analytical procedures

broken fiber element, as determined via double-fiber fragmentation
testing. The details of model preparation, simulation procedures, andmaterial properties can be found in the Supplementary Material. Herein,
we considered the model in which the cross-section of the fiber divided
into six segments, assuming the hexagonal fiber arrangement, as illustrated in Fig. 2(a). The surface SCF of the i-th fiber segment adjacent to abroken fiber at the plane of fracture was determined by assuming that itwas αi times the average stress (σ f ) on the surface of an adjacent fiber,
which is expressed as follows:()
D
αi = 1 + γ i 1 − s ,(1)ls

2.1. Sample preparation and mechanical characterization
Epoxy specimens were prepared and their mechanical properties
were measured to reveal the mechanical properties of the epoxy materials that affect the SCF. Among the six types of epoxy specimens
investigated in this study, two specimens were synthesized using
diglycidyl ether of bisphenol A (DGEBA) as the main agent and two
curing agents (diethylenetriamine (DETA) and 4,4′ -diaminodiphenyl
sulfone (44DDS)). The other four specimens were provided by Toray
Industries, Inc., Japan; they are hereafter referred to as “A-epoxy,“ “Bepoxy,” “C-epoxy,“ and “D-epoxy,” with the order of the names indicating the magnitude of their Young’s moduli (i.e., the Young’s modulusof the A-epoxy is the lowest). The mechanical properties of the epoxy
materials were determined with uniaxial tensile loading and single-edge
notched beam (SENB) tests. Schematics of the specimen configurations
and experimental setups are shown in Fig. 1 (further details related to
specimen preparation and mechanical evaluation can be found in the
Supplementary Material).
Previously, we investigated how the SCF changes upon variation ofboth the number of fibers (2–4 fibers) and the interfiber spacing(3.5–20.0 μm) using multi-fiber fragmentation tests [7], although noapparent difference in the SCFs was observed under these conditions.
Based on the above knowledge, the double-fiber fragmentation technique is employed in the current study to assess the SCFs. A highstrength, polyacrylonitrile (PAN)-based carbon fiber (TORAYCA™
T1100G) was used to fabricate the double-fiber fragmentation composites; the detailed composite preparation procedures, SCF determination procedures, and analytical model preparation can be found in[7,8] and the Supplementary Material.

where αi is the SCF of a fiber adjacent to a broken fiber in the same plane;
γ i is the intensity attenuation factor, which is 0 for a fiber segment facing
a neighboring intact fiber and α −1 for a fiber segment facing a neighboring broken fiber; Ds is the distance from the break point; and ls is thestress recovery length. In the spring element model used in this study,
the two longitudinal spring elements in the center of the model wereassigned to the fibers and the remaining elements were assigned to thematrix. Thus, the matrix stiffness was implemented in the longitudinal
spring elements. In our previous work, the stress gradient in an intact
fiber was investigated using the three-dimensional finite element
method [7]. Hence, the additional stress concentration was added in an
ad-hoc manner to capture the experimentally measured correlations in
the fiber breaks.
The strength of the n-th fiber segment is determined by choosing arandom number ranging from 0 to 1 and solving the equation such thatthe fiber breakage probability equals a random number. The longitudinal element was removed from the model when the stress applied to afiber at the n-th fiber segment achieved the statistical distribution of thestrength of the fiber. A coordinated fracture was defined as a failure that
occurred at neighboring elements next to a broken element in the same
horizontal plane with respect to the broken fiber element, as illustrated
in Fig. 2(b). Note that we previously confirmed that employing theabove-mentioned procedures yields the predicted strengths of unidirectional CFRP composites that are reasonably consistent with theexperimental data, irrespective of the differences in the mechanicalcharacteristics of the matrix[7].

2.2. Model preparation and determination of the stress concentration
factorThe SCF (α) on the surface of an intact fiber was determined by
employing the spring element model to investigate the SCF value, withthe aim 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 theFig. 1. Schematics of the specimen configurations and experimental setups for the (a) tensile-loading test and (b) single-edge notched beam test.
2

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Composites Part A 140 (2021) 106140

Fig. 2. Illustrations to (a) define the stress concentration on the surface of the (b) i-th fiber segment.

Fig. 3. Birefringence patterns in the double-fiber fragmentation composites prepared with DGEBA/44DDS (a1) and B-epoxy (b1). The corresponding simulation
results are shown in (a2) and (b2). The symbols at the top of each fiber break point indicate the two types of fiber break; the filled symbol denotes coordinatedfracture, while the open symbol indicates other fracture. (For interpretation of the references to colour in this figure legend, the reader is referred to the web versionof this article.)3

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Composites Part A 140 (2021) 106140

3. Results and discussion

materials that influence the stress concentration generated on an intact
fiber surface, the correlation between the SCF and mechanical propertiesof the epoxy materials was analyzed. The relationships of SCF with thetensile strength, Young’s modulus, fracture toughness, crack tip opening
displacement (CTOD), and plastic zone size of the matrix materials areshown in Fig. 4(a)–(e), respectively. The dashed lines in Fig. 4 indicate
the regression lines calculated by the least-squares method, assuming alinear relationship between the SCF and the five properties of the matrix
materials. Although some variation occurs, the epoxy materials withboth higher tensile strength and Young’s modulus generally exhibit
lower SCFs, while those with higher fracture toughness lead to higher
SCFs. This implies that epoxy materials with brittle-like characteristics
are effective for decreasing the SCF, whereas those with ductile-likecharacteristics lead to an increase in the SCF. The highest correlation
relationship was observed for the tensile strength with the coefficient ofdetermination (R2 value) of 0.85. The R2 values for the Young’s
modulus, fracture toughness, CTOD, and plastic zone size were 0.76,
0.55, 0.80, and 0.70, respectively. Even though the highest correlation
was observed for the tensile strength, it represented the macroscopic
mechanical characteristics of the materials. Thus, we expect that CTOD
(and plastic zone size) seems to make sense as a measure that represents
the fracture phenomenon of fibers under the stress field around the crack
tip. The high SCF observed for the ductile-like epoxy materials can be
attributed to the large crack opening around the crack tip.
CTOD, defined as the displacement transverse to the crack tip, appears to impact the SCFs. The generalized equation describing CTOD (δ)
is expressed as follows [11]:All epoxy materials tested in the tensile loading experiments
exhibited elasto-plastic behavior that is typically observed in conventional epoxy materials. The SENB specimens experienced catastrophic
failure after reaching a maximum load and no nonlinear behavior wasobserved for all epoxy materials (Fig. S2). The mechanical properties of
all the six epoxy materials are listed in Table S2 in the Supplementary
Material.
The fiber fracture behavior of the double-fiber fragmentation composites was investigated to quantitatively determine the surface SCFs.
This investigation revealed that matrix cracking and the coordination of
fractures in adjacent fibers occur, regardless of the types of epoxy material. For example, Fig. 3 shows the birefringence pattern images at thefiber break points in the double-fiber fragmentation composites prepared with DGEBA/44DDS and B-epoxy. The corresponding two simulation results are shown in Fig. 3 (note that the images of the simulation
results were processed to make the positional dependency of the fiber
break points easier to understand, and they do not represent the actual
stress distribution obtained from the simulations). The images wereacquired at 3.26% fiber strain for DGEBA/44DDS and at 3.10% fiber
strain for B-epoxy. The symbols at the top of each fiber break point
indicate the two types of fiber break. The filled symbols denote the coordinated fracture defined based on the nature of elasto-plastic polymer
material fracture phenomena [9], while the open symbols indicate theother phenomena. As a fiber–fiber interaction criterion, fiber fracturesthat occur at an angle between 0◦ and 45◦ are defined as coordinatedfractures. It can be observed from Fig. 3(a1) and 3(b1) that the matrix
cracks propagated from the fiber break points, and a large number offiber failures occured at similar positions, indicating that the stressconcentration caused by fiber fracture was sufficiently high to cause theadjacent fiber to fracture. A similar observation was made from simulation results shown in Fig. 3(a2) and (b2). The percentages of coordinated fracture (and standard deviation) obtained by the experimental
and simulated results for DGEBA/44DDS were 75.7% (±10.3%) and
76.3% (±12.8%), respectively. The corresponding values for B-epoxy
were 48.3% (±14.1%) and 46.5% (±15.9%). The experimentally obtained percentages of coordinated fracture (and standard deviation) for
the DGEBA/DETA, A-epoxy, C-epoxy, and D-epoxy composites were
89.3% (±9.9%), 43.2% (±14.5%), 54.4% (±15.5%), and 9.1% (±4.3%),
respectively. Note that in our previous study, we prepared fiber fragmentation composites in which two T1100G carbon fibers wereembedded in B-epoxy material, and then carried out fragmentation tests
under the same conditions as those used in this study. The interfiber
spacing in the composites was approximately 200 μm. Thus interfiber
spacing is equivalent to the diameter of approximately 40 fibers and isunaffected by the fracture of the counterpart fiber[10]. The percentagesof coordinated fracture in the composites was measured using the processed images so that the interfiber spacing was 10 μm (the averageinterfiber spacing in the study). Consequently, the percentage of coordinated fracture measured to be 7.7%. This value is close to thatobserved for the composite prepared with D-epoxy, suggesting that thefiber failure process in D-epoxy matrix composites is predominantly
governed by the statistical strength distribution of the fibers.
Quantitative determination of the SCFs on the surface of intact fibers
was achieved by implementing the spring element model (Supplementary Figs. S3a–S3f). For example, for DGEBA/44DDS, the SCF on thesurface of an intact fiber was determined to be 2.16 upon comparing thesimulated percentages of coordinated fractures to the corresponding
experimental observations, indicating that for composites fabricatedwith DGEBA/44DDS, the concentrated stress acting on the fiber surface
is approximately twice that of the fiber stress with no additional surface
stress concentration. The percentages of coordinated fracture, applied
fiber strain, and SCFs for the six composites are listed in Table S3 in the
Supplementary Material.
In order to reveal the mechanical characteristics of the epoxy

δ=

KIc2
,
Eσy

(2)

where KIc is the mode І fracture toughness, E is the Young’s modulus,and σy is the yielding stress. Following ASTM recommendations [11],the CTOD was calculated by substituting the yielding stress (σy ) withtensile strength (σ) in Eq. (2). As shown in Fig. 4(d), a positive linear
correlation exists between the CTOD and SCF; an epoxy material with alarger CTOD tends to exhibit a higher SCF. Furthermore, the epoxy
materials with ductile-like characteristics possessed a large plastic zone,
whose size increased with an increase in the SCF (Fig. 4(e)). This trend issimilar to that observed for the CTOD.
The size of the plastic zone ahead of a crack tip under plane strain
conditions (rp ) is expressed as follows:
( )2
1
KIc
rp = √̅̅̅.
4 2π σy(3)

Eq. (3) can therefore be used to assess differences in the extent ofstress concentration generated on intact fiber surfaces. As mentioned
earlier, the matrix crack around the original broken fiber is expected to
be one cause of additional stress concentration [7]. The matrix crack
caused by fiber breakage propagates toward the rigid intact fiber,
resulting in stable crack growth. This crack remains without reaching
the fiber/matrix interface after approaching the intact fiber.
Additional stress can continue to be applied to epoxy materials withlarge CTOD until the opening reaches a critical value because of the highstress concentration in the process zone. On the other hand, epoxy
materials with small CTOD alleviate the additional stress concentration.
These are considered to be the possible mechanism by which CTOD affects the additional stress concentration.
4. Conclusions
We have reported an experimental study in which we investigated
the tensile strength-controlling factors of unidirectional CFRP composites, focusing on assessing the mechanical characteristics of the epoxy
matrix. The degree of concentrated stress acting on the intact fiber
surface, which determines the tensile strength properties of4

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Composites Part A 140 (2021) 106140

Fig. 4. Relationship of the stress concentration factor with (a) tensile strength, (b) Young’s modulus, (c) fracture toughness, (d) crack tip opening displacement(CTOD), and (e) plastic zone size. The dashed lines indicate the regression lines calculated by the least-squares method, assuming a linear relationship between the
SCF and five properties of the matrix materials.

unidirectional CFRP composites, was evaluated by implementing
double-fiber fragmentation tests in conjunction with spring element
model simulations. Correlation analysis was conducted, with the aim ofextracting the factors that characterize the tensile strength of the unidirectional CFRP composites. The analysis of six epoxy materials withdifferent mechanical characteristics revealed that the matrix crack tip
opening displacement (CTOD), which is characterized by the tensile
strength, Young’s modulus, and mode І fracture toughness, exhibited alinear correlation with the surface SCF, with the SCF increasing from1.75 to 2.35 with an increase CTOD. These results revealed that CTOD is

one of the dominant factors influencing the tensile strength characteristics of unidirectional CFRP composites. Ultimately, our results provide
a fundamental framework for guiding the creation of high-performance
CFRP composites for epoxy matrices.
CRediT authorship contribution statement
Go Yamamoto: Conceptualization, Data curation, Formal analysis,
Funding acquisition, Investigation, Project administration, Supervision,
Validation, Visualization, Writing - original draft, Writing - review &5

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Composites Part A 140 (2021) 106140

editing. Keita Koizumi: Data curation, Investigation, Visualization.
Takahiro Nakamura: Investigation. Noriyuki Hirano: Project administration, Resources. Tomonaga Okabe: Conceptualization, Formal
analysis, Funding acquisition, Methodology, Project administration,
Software, Supervision, Validation, Writing - review & editing.

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Declaration of Competing Interest
None.
Acknowledgements
The authors thank Prof. T. Hashida and Mr. T. Suzuki of the Fracture
and Reliability Research Institute, Tohoku University, for technicalassistance in the fracture toughness measurements. This work was partly
supported by Toray Industries, Inc., the Council for Science, Technology
and Innovation (CSTI), Cross-ministerial Strategic Innovation Promotion Program (SIP), and JSPS KAKENHI [grant number 18K04721]. We
would like to thank Editage for English language editing.
Appendix A. Supplementary material
Supplementary data to this article can be found online at https://doi.
org/10.1016/j.compositesa.2020.106140.

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