Advanced Composite Materials

ISSN: 0924-3046 (Print) 1568-5519 (Online) Journal homepage: http://www.tandfonline.com/loi/tacm20

Investigation about the fracture behavior and

strength in a curved section of CF/PP composite by
a thin-curved beam specimen

Tsuyoshi Matsuo, Takeshi Goto & Jun Takahashi

To cite this article: Tsuyoshi Matsuo, Takeshi Goto & Jun Takahashi (2015) Investigation about
the fracture behavior and strength in a curved section of CF/PP composite by a thin-curved beam
specimen, Advanced Composite Materials, 24:3, 249-268, DOI: 10.1080/09243046.2014.886754

To link to this article: https://doi.org/10.1080/09243046.2014.886754

Published online: 13 Feb 2014.

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Advanced Composite Materials, 2015
Vol. 24, No. 3, 249–268, http://dx.doi.org/10.1080/09243046.2014.886754

Investigation about the fracture behavior and strength in a curved

section of CF/PP composite by a thin-curved beam specimen
Tsuyoshi Matsuo*, Takeshi Goto and Jun Takahashi

Department of Systems Innovation, School of Engineering, The University of Tokyo,

7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan
(Received 5 September 2013; accepted 21 January 2014)

A curved beam specimen with a curved section bent at a right angle and straight
arms is an efficient experimental method to evaluate the interlaminar tensile strength
of a composite laminate. This paper describes a technique to apply such a test
method to determining the fracture behavior and strength in a curved section of a
thin laminate. The proposed technique is capable of estimating appropriately the
compressive or tensile strength as well as the interlaminar tensile strength at failure
in a thin-curved section, especially targeting thermoplastic composite materials. The
significant point of the proposed technique is a modified calculating method of the
stress distribution in a curved section by taking into account the large deformation
of the specimen because of thin laminate.
Keywords: curved beam specimen; thermoplastic composite; CF/PP; interlaminar
tensile strength; compressive strength

1. Introduction
With the growth of the requirement of lightweight materials and structures, laminated
composites have been developed and manufactured for a variety of applications. In par-
ticular, thermoplastic composite materials have been expected to be widely used in
mass produced transportation sectors due to its high mechanical performance and light-
weight effect. Especially, carbon fiber reinforced thermoplastic composite has a lot of
advantages to enhance mechanical properties as represented by impact energy absorp-
tion as well as to achieve high-cycle and low-cost manufacturing technologies.[1] Poly-
propylene (PP) is one of thermoplastic polymers, and its high productivity and
dramatic improvement of mechanical performance have attracted great attention on
automotive applications, providing a wide range of usage.[2] Although there were some
significant demerits about a combination of continuous carbon fiber (CF) and polypro-
pylene, taking a few examples, in relation to the inadequate impregnation of PP and
the low interfacial adhesive strength, some developed technologies have addressed
those weak points and satisfied high performance of CF/PP composite.[3–5]
In keeping with this trend, the range of application of thermoplastic composite is
spreading gradually, and a variety of shapes are being developed for designing struc-
tural member. Above all, a curved shape ranging from thin laminate to thick laminate
is an important design factor to have key roles of structural components.

*Corresponding author. Email: matsuo@sys.t.u-tokyo.ac.jp

© 2014 Japan Society for Composite Materials, Korean Society for Composite Materials and Taylor & Francis
250 T. Matsuo et al.

One of the important evaluated properties of the curved laminate is the out-of-plane
property including the interlaminar tensile strength, which is closely related to the
delamination in the curved section.[6–8] In previous researches, there have been a lot
of attempts to develop test methods for measuring the out-of-plane properties.[9–14]
For example, the uniaxial out-of-plane loading test was utilized to measure through the
thickness strength of fiber reinforced plastics.[9,13] But, this method required a rela-
tively thick (over 12 mm) specimen for achieving relatively consistent evaluation, and
if using a relatively thin plate, test results showed significant scatter.[9] On the other
hand, the test methods using a curved laminate itself have been developed to achieve
validity for estimating the interlaminar tensile strength of the curved laminate,[10–
12,14] and have been also applied to verifying the strain rate dependency of the out-of-
plane property and the interface behavior between the skin and core of curved sand-
wich beam.[15,16] However, their methods made mention of only interlaminar tensile
property in curved section. But in fact, a further thin-curved laminate has some possi-
bility to generate not only the interlaminar failure but also the tensile or compressive
failure on the curved surface.
The attempt of this study is to spread the range of application of the curved beam
test method and make it possible to evaluate the tensile or compressive strength as well
as the interlaminar tensile strength, in order for more practical usage of thin-curved
laminate. A curved beam specimen with a section bent at right angle and straight arms,
which is called L-shaped specimen, was utilized due to its relative ease of manufacture
for thermoplastic composite laminate which consists of continuous carbon fiber and
polypropylene. When using the L-shaped specimen, a four-point bending test has sev-
eral advantages rather than a hinged loading tensile test because of a state of pure
bending moment in the measured section,[11,14] which means no interlaminar shear
stress. But in this study, one of valuable parameters is a tensile or compressive strength
of the curved section of thin laminate. If using the four-point-bending test, there was a
threat that some contacting stress concentration on a straight thin beam induces the first
failure at each point where the specimen is loaded by cylindrical bars and the compres-
sive failure in the curved section cannot be generated. In contrast, the tensile test does
not have such a failure mode because there is no contacting stress.
Even if the interlaminar shear stress is possibly induced in the curved section while
tensile loading, an appropriate stress analysis in the curved section can address a mean-
ingful evaluation of the curved laminate properties, taking into account a large geomet-
ric deformation of the thin-curved beam. Finite element analysis supports the validation
of such a stress analysis. This paper proposes a methodology for estimating the curved
laminate properties of thin-curved beam specimen and discusses the fracture behavior
and strength of the curved section.

2. Material-specification and experimental procedure

2.1. Materials
The developed composite material is CF/PP composed of surface-treated carbon fiber
TR50S supplied by Mitsubishi Rayon and acid-modified polypropylene supplied by
TOYOBO.[3–5] Figure 1 shows a roll of the prepreg tape with the polypropylene
impregnated into the unidirectional bundle of the carbon fibers. The in-plane fiber-
directional elastic modulus of the laminated CF/PP was determined by in-plane tensile
experiments. And, the other material properties including the out-of-plane shear
 Advanced Composite Materials 251

Figure 1. Prepreg tape of carbon fiber and polypropylene.

modulus were estimated almost the same as theoretical values calculated from the rule
of mixtures [17] or those properties of polypropylene itself.[3,5]

2.2. Specimen manufacture

The thin-curved beam specimen with the fibers aligned along the curved beam was
manufactured by a heat-and-cool pressing technology, which is one of press molding
techniques. Through the molding process, the stack of unidirectional prepreg tapes with
the width of 15 mm was heated up to almost 200 °C in a set of upper and lower
L-shaped molds as illustrated in Figure 2, after that, cooled down and compressive
pressed around 160 °C with a pressure of 10 MPa on the material by hydraulically con-
trolled pressing machine. The demolded final specimen as shown in Figure 3 has no
significant spring back effects and was formed into the shape corresponding exactly to
each inner geometry of the three sets of molds, with the same in width b = 15 mm, the

Figure 2. L-shaped molds – (left) the outer appearance (right) the inner geometry.
252 T. Matsuo et al.

Figure 3. Thin-curved beam specimen with a magnified image of curved section.

same in thickness t = 2 mm, and the three different kinds of inner and outer radius,
ri = 10 mm and ro = 12 mm, ri = 15 mm and ro = 17 mm, ri = 20 mm and ro = 22 mm,
respectively. Accordingly, it can be assumed that the curved beam specimen has the
orthotropic property along the tangential direction in the curved section. The material
properties as explained in Table 1 were determined as follows:
Eh ¼ 96:5 GPa Er ¼ 2:9 GPa vhr ¼ 0:26 Ghr ¼ 1:1 GPa:

2.3. Experimental setup and configuration

The test configuration as shown and schematically illustrated in Figure 4 was deter-
mined by the geometry of the fixture. The test method used a universal testing machine
Shimadzu AGS-5kNX using a 5kN load cell with a measuring accuracy of ±0.5% and
a steel-hinged loading mechanism, which was aligned and clamped on to the speci-
men’s loading arms. The dimensions and variables illustrated in Figure 5 are defined in
Table 1. According to Figure 5, the setup parameters are as follows:

initial /i ¼ 45 d ¼ 26 mm L ¼ 45 mm:

The tensile displacement was controlled at the speed of 1.0 mm/min during loading.
The testing machine recorded loads and displacements digitally until the failure was
observed on the specimen.

3. Analysis
3.1. Theoretical stress solutions
3.1.1. Previous stress solutions
Stress equations were developed by Lekhnitskii [18] for the stresses in a curved beam
section with cylindrical anisotropy. And, based on the solutions, Shivakumar [19] and
Hufenbach et al. [15] proposed a method for an analytical calculation of the radial and
tangential stress distributions respectively in the curved section. In this analytical
method, the applied tensile force F was translated to the ends of the curved section as
illustrated in Figure 6(a) and (b), and the force translation results in two separated
stress solutions for semicircular beam assuming two different loading cases with a pure
end force F and a pure bending moment M at each end.
Based on these assumptions, the radial stress distribution σr(r,θ) and the tangential
stress distribution σθ(r,θ) in the curved section are produced respectively by the end force
and the moment at the ends of the curved section as the following Equations (1)–(11).
 Advanced Composite Materials 253

Table 1. Symbolic definition.

Symbols Definition
Material property
Eθ Young’s modulus in the tangential direction
Er Young’s modulus in the radial direction
Gθr Shear modulus
νθr Poisson’s ratio
Specimen geometry
t, b Specimen thickness and width
ri, ro Specimen inner and outer radius
rm Specimen radius of the midplane
r, θ Cylindrical coordinates of any point in the curved section
Experimental geometry
d Longitudinal distance of force application
L Transversal distance of force application
ϕ Angle of the loading arm from vertical
ϕi Initial angle of ϕ
Dx Horizontal distance between points of ‘O’ and ‘B’ (refer to Figure 5)
Dy Vertical distance between points of ‘O’ and ‘A’ (refer to Figure 5)
Da Distance between points of ‘B’ and ‘C’ (refer to Figure 5)
Experimental measurement and calculation
F, δ Loading force and displacement
M Moment applied by the lever L and the offset d
M* Additional moment applied by the integration of the tangential stress σθ to the end
of the curved section
Μa Accurate moment applied to the end of the curved section
σr Radial stress at any coordinate in the curved section
σθ Tangential stress at any coordinate in the curved section
σrM Radial stress by the moment M
σθΜ Tangential stress by the moment M
σrF Radial stress by the end force F
σθF Tangential stress by the end force F
σrMa Radial stress by the accurate moment Ma
σθΜa Tangential stress by the accurate moment Ma
k, c, gM, gF Dimensionless constants
ri′, ro′ Deformed inner and outer radius
rm′ Deformed radius of the midplane
Dxi Initial horizontal distance of Dx
Dyi Initial vertical distance of Dy
ϕ′ Real angle of loading arm from vertical at point ‘D’, taking into account bending
deflection of the arm (refer to Figure 16)
φ Bending deflection angle of arm at point ‘D’ (refer to Figure 16)
σθC Compressive strength in the tangential direction
σrT Interlaminar tensile strength in the radial direction

The analysis assumed a state of plane strain. The material variables are defined in Table 1
and the geometry variables are shown in Figures 5 and 6.
The radial stress distribution σrM(r,θ) and the tangential stress distribution σθΜ(r,θ)
in the curved section under the pure bending moment M were derived as follows
[11,18]:
254 T. Matsuo et al.

Figure 4. (a) Test assembly for experiments, (b) schematic figure of test configuration.

(
k1 )
M 1  ckþ1 r 1  ck1 kþ1 ro kþ1
r ðr; hÞ
rM ¼ 2 1   c  ; (1)
r o  b  gM 1  c2k ro 1  c2k r

(
k1 )
M 1  ckþ1 r 1  ck1  kþ1
kþ1 ro
h ðr; hÞ
rM ¼ 2 1 k þ kc ; (2)
r o  b  gM 1  c2k ro 1  c2k r

rffiffiffiffiffi
1  c2 k ð1  ckþ1 Þ2 kc2 ð1  ck1 Þ2 ri Eh
where; gM ¼   þ  ; c¼ ; k¼ :
2 kþ1 1  c2k k1 1  c2k ro Er
(3)
In Figure 6(b), the anisotropy pole is the origin of the coordinates, and the radius r is a
coordinate value of any point in the curved section and ranges from the inner radius ri
to the outer radius ro. And, the angle θ is a coordinate value of any point ranging from
the end of the curved section to the other end. And, basically the end moment of the
curved section was calculated by the addition of the moment by the loading arm length
 Advanced Composite Materials 255

Figure 5. (a) Experimental setup with parameters used for the analysis, (b) Detail of the curved
section.

Figure 6. Geometry parameters of the specimen (a) lever and (b) curved beam section.

L and the moment by the offset d of the loading fixture from the neutral axis of the
plate,

M ¼ F  ðL  sin /  d  cos /Þ: (4)

256 T. Matsuo et al.

On the other hand, the radial stress distribution σrF(r,θ) and the tangential stress dis-
tribution σθF(r,θ) only with the end force F were derived as follows [10,18]:

(
 )
F r b r b  p
o
rFr ðr; hÞ ¼ þc 
b
1  c  sin h  / þ ;
b
(5)
b  gF  r ro r 2

(
b )
F r ro b  p
rFh ðr; hÞ ¼ ð1 þ bÞ  þð1  bÞ  c  ð Þ  1  c  sin h  / þ ;
b b
b  gF  r ro r 2
(6)

2
where; gF ¼  ð1  cb Þ þ ð1 þ cb Þ  lnðcÞ; (7)
b

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Eh Eh
b ¼ 1 þ  ð1  2  mhr Þ þ : (8)
Er Ghr
Additionally, Jackson et al. [10] proposed a correction to Lekhnitskii’s end force solu-
tion to subtract out the additional moment M* applied to the end of the curved section
by the integration of the tangential stress σθ, and developed the expression for the
moment M* as the following,
Zro 
 ri þ ro  F
M ¼ rhðh¼0Þ  r   b  dr ¼  ðri þ ro Þ  cos /: (9)
2 2
ri

So, the accurate moment Ma was derived as the following equation,

ri þ ro
Ma ¼ M  M  ¼ F  ðL  sin /  d  cos /   cos /Þ: (10)
2
From Equations (1)–(3) and (10), the radial and tangential stress distributions by the pure
moment can be calculated by the replacement of M with Ma. From Equations (5)–(8), the
radial and tangential stress distributions by the pure end force reaches a maximum at θ
equals ϕ at any value of radius r. Consequently, by applying a superposition principle, the
stresses expressed from Equations (1), (2), (5), and (6) result in the stress distribution
related to the radial and tangential direction at the symmetry plane where θ equals ϕ as
follows:
rr ðrÞ ¼ rM
r ðrÞ þ rr ðrÞ;
a F
rh ðrÞ ¼ rM
h ðrÞ þ rh ðrÞ;
a F
where; h ¼ /: (11)

3.1.2. Proposed geometry modifications

In this study, the effect of loading deformation is not negligible due to very thin
specimen. The loading angle ϕ changes significantly during loading, and the specimen
radius ri, ro and rm also change with the angle between two loading arms spreading.
 Advanced Composite Materials 257

The following procedure is proposed to calculate ϕ, ri′, ro′, and rm′ at a certain displace-
ment δ. The vertical distance Dy is obtained from the constant length Da and d, and the
variable angle ϕ of the deformed specimen as the following,

Dy ¼ Da  cos / þ d  sin /: (12)

Figure 5(a) shows this geometry relationship, and then the angle ϕ can be derived from
a trigonometric function for values Dy, Da, and d as the following:

0 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1
Dy  d þ Da D2a  D2y þ d 2
/ ¼ sin1 @ A: (13)
D2a þ d 2

In this equation, the vertical distance Dy is calculated by adding the half of the vertical
displacement δ to the initial distance Dyi, and the distance Da between points B and C
is calculated by the addition of the loading arm length L and the initial radius of mid-
plane rm which is the average of ri and ro,
d ri þ ro
Dy ¼ Dyi þ ; Da ¼ L þ : (14)
2 2
Taking into consideration such a large deformation, the deformed radii at failure can be
derived from a measured displacement δ as follows:
Dy  L  cos /  d  sin /
deformed radius of the midplane : rm0 ¼ ; (15)
sin /

t
deformed inner radius : ri0 ¼ rm0  ; (16)
2
t
deformed outer radius : ro0 ¼ rm0 þ : (17)
2
For the geometry modifications, the accurate moment Ma can be calculated by substitut-
ing the values, ϕ, ri′ and ro′, obtained respectively from Equation (13), (16), and (17)
into the Equation (10). In this analytical procedure, the displacement δ measured by the
universal loading machine is a significant value in order to calculate the stress distribu-
tion appropriately.

3.2. Finite element analysis

Figure 7 illustrates a 2D finite element model conducted on the curved beam specimen
to verify the relationship between load and displacement as well as the stress distribu-
tion in a curved section. Because of the symmetric geometry, only half of the model
was enough for analysis, and the symmetric plane was constrained for y direction. The
force applied to the loading fixture was straight parallel to the y direction. And, the
specimen was fixed to the loading fixture at the contact face on both sides. Abaqus
6.11 supported the analysis, in which plane strain was assumed and material directions
1, 2, and 3 were defined as presented in Figure 7 and the material properties respec-
tively along those directions were determined as indicated in Table 2. The direction 1
was parallel to the fiber direction in the specimen, the direction 2 was parallel to the
258 T. Matsuo et al.

Figure 7. Finite element model.

Table 2. Material properties used in FEA.

Material properties Unidirectional CF/PP

Elastic modulus E1 96.5
(GPa) E2 2.9
E3 2.9
G12 1.1
G13 1.1
G23 0.91
Poisson’s ratio ν12 0.26
ν13 0.26
ν23 0.59

thickness of the curved beam, in other words perpendicular to the fiber direction, and
the direction 3 was perpendicular to the plane 1-2. The effect of nonlinearity was con-
sidered due to large deformation and all of elements were four-noded quadrilateral with
20 elements in the thickness direction on the specimen model.

4. Results and discussion

4.1. Load–displacement relationship
Figure 8 presents the deformed traces of the specimen profile at some loading steps
obtained by the finite element analysis. As shown in this figure, the specimen is largely
deformed apparently as the curved section moves to the left side and the radius of the
curvature increases gradually with the load and displacement increasing.
 Advanced Composite Materials 259

Figure 8. Deformed traces by the finite element analysis.

The load-displacement relationships obtained by the experimental and the FEA

results are indicated in Figure 9, where the curves present the experimental results and
the plots present the FEA results respectively for three curved geometries. In all of
curved geometries, the widths of the specimen are equal to 15 mm. Those curves
demonstrate a slight increase in stiffness with the loading force increased. This trend is
considered as caused by the decrease of the moment on the curved beam due to the
displacement in the x direction as illustrated in Figure 8. Consequently, the finite ele-
ment analysis shows a satisfying agreement with the experimental result in case of
every curved geometry.

Figure 9. Load-displacement relationships comparing experimental and FEA results.

260 T. Matsuo et al.

4.2. Stress distribution at failure force

As each experimental load history has a drop as indicated in Figure 9, every specimen
had a significant loss of load and stiffness caused by the failure in the curved section.
At the maximum loading force, the failure in the curved section was observed and the
strength of the curved section was obtained in each experiment.
An example in case of inner radius 15 mm plotted in a solid line indicates that the
failure loading force is approximately obtained 569 N and the failure displacement is
4.05 mm. In Figure 10, the radial stress distribution σr and the tangential stress distribu-
tion σθ at the failure force of 569 N calculated by FEA are displayed as contour figure.
These figures indicate that the highest stress area in each case of the radial or tangential
direction is on the symmetry plane of curved section where θ = ϕ as mentioned in
Equation (11).
Figure 11(a) and (b) indicate respectively the radial and tangential stress distribu-
tions on the symmetry plane of the curved section at failure when the force is 570 N
and the displacement is 4.05 mm in case of ri = 15 mm. The horizontal axis shows the
normalized radial distance (r − ri)/t, ranging from the inner surface to the outer surface

Figure 10. Stress distribution by FEA (a) radial stress : σr, (b) tangential stress: σθ.

Figure 11. Comparison of analytical and FEA stress distributions on the symmetry plane at
failure in case of ri = 15 mm – (a) radial stress and (b) tangential stress.
 Advanced Composite Materials 261

of the curved section. Those graphs compare the proposed modified solution to the pre-
vious solution and the finite element results. The results from the proposed modified
solution well fit in the finite element results in relation to both of the radial and tangen-
tial stress distributions. In contrast, the results from the previous solution are overesti-
mating the stress value. Therefore, the proposed modified solution by taking into
consideration the geometry modification expressed in Equation (13)–(17) has some
validity for estimating the analytical stress distribution in thin-curved section, while the
previous solution without the geometry modification has no validity.

4.3. Fracture behavior and strength

Three specimens in each case of inner radius, 10, 15 or 20 were investigated for the
variation of the fracture behavior and strength. Some assumed fracture pattern was
observed from every test result as indicated in Figure 12. The curved beam specimen
with the inner radius 10 mm generated an interlaminar fracture in the curved section as
shown in Figure 12(a). On the other hand, the specimen with the inner radius 15 mm
or 20 mm generated a compressive fracture on the outer surface of the curved section
as shown in Figure 12(b) and (c).
From the interlaminar failure observed in every case of ri=10 mm and the stress
analysis by the proposed modified solution, taking an example from Figure 11(a), the
interlaminar tensile strength of the thin-curved laminate was obtained around 22 MPa
as almost maximum of the radial stress in the curved section calculated by the failure
load and displacement. Figure 13 shows a comparison of the calculated interlaminar
tensile strengths by three analyses, the previous and modified solutions and the FEA,
with each average strength and variation. In a similar way, the compressive strength of
the thin-curved laminate was obtained around 720 MPa in case of ri = 15 mm or around
790 MPa in case of ri = 20 mm as the minimum stress in the stress distribution by the
modified solution like Figure 11(b). Figure14 shows a comparison of the calculated
compressive strengths by three analyses. So, Table 3 presents a summary of the
experimental results and the calculated strength values at each fracture pattern for three
types of analysis.

4.4. Effect of the material property and geometry

From the above-mentioned analysis result, the difference between the deformed angle ϕ
and the initial angle ϕi obtained from the displacement δ is significantly sensitive to the
evaluated stress. So, the more appropriately evaluated angle ϕ, which should be equal
to the actual angle of loading arm from vertical at the end of the curved section, makes
it possible to improve the accuracy of the evaluated stress and strength. For an example
in case of ri = 20 mm, Figure 15 indicates how much the accuracy of the calculated
rotation angle dϕ, which subtracts the evaluated angle ϕ from the initial angle ϕi, influ-
ences the evaluated compressive strength obtained by the modified solution. This rela-
tionship means that 1 degree error produces about 5% error of strength.
Such a influence reflects in the difference of the evaluated compressive strength
between the modified solution and the FEA in case of ri = 20 mm, which is more than
in case of ri = 15 mm as indicated in Figure 14 and Table 3, because the modified
solution does not take into account the actual bending deformation of loading
arms which increases when the radius of the curved section increases and the thickness
decreases. Figure 16 shows the geometric relationship considering the bending
262 T. Matsuo et al.

Figure 12. Damaged specimens with magnified images of damaged section (a) ri = 10 mm,
(b) ri = 15 mm, (c) ri = 20 mm.
 Advanced Composite Materials 263

Figure 13. Comparison of interlaminar tensile strengths at failure in case of ri = 10 mm.

Figure 14. Comparison of compressive strengths at failure in case of ri = 15 mm and 20 mm.

deformation of the arm. For easy understanding and easy calculation, the figure
assumes the curved section as a partial linear beam of the extended loading arm, the
point B on the extended line as fixed and the loading point as horizontal free while
loading. In this way, the deformed arm is illustrated as a bending curve connecting the
point B, D′, E′, and C′ in Figure 16. So, the real angle ϕ′ of the deformed arm from
vertical at the end of the curved section, the point D′, can be derived as follows:

/0 ¼ /i  u: (18)
Here, the angle φ is a deflection angle of the arm at the point D′ (or D as the initial
point), and from the fundamental bending beam theory is expressed as the following
equation,

6  F  sin /i
u¼ fðDa  dÞ2  ðL  dÞ2 g: (19)
Eh  b  t 3
 264

Table 3. Experimental results and calculations for interlaminar or compressive stress at failure.

Interlaminar tensile strength, Compressive strength, σθC

Experimental value σrT [MPa] [MPa]
Failure
Failure Disp.,
Test Original inner force, Ffail dfail Fracture Fracture location, normalized Previous
r  ri Modified Previous Modified
no radius, ri [mm] [N] [mm] behavior radial distance, solution solution FEA solution solution FEA
t
1 10 485 2.90 Interlaminar 0.3 36.5 23.7 25.6 unknown
2 379 2.41 Interlaminar 0.3 28.5 20.2 21.1
3 407 2.66 Interlaminar 0.3 30.6 20.8 22.3
Ave. 31.8 21.6 23.0

4 15 569 4.05 Compression 1.0 unknown 994 722 741

T. Matsuo et al.

5 556 3.99 Compression 1.0 971 711 687

6 598 4.35 Compression 1.0 1045 734 742
Ave. 1003 722 723

7 20 601 4.99 Compression 1.0 unknown 1139 823 721

8 594 4.94 Compression 1.0 1125 817 727
9 502 4.39 Compression 1.0 951 724 646
Ave. 1072 788 698
 Advanced Composite Materials 265

Figure 15. Geometric relationship of the loading arm with bending deformation.

As is clearly seen from Figure 16, the angle ϕ from the modified solution should be
less than the actual angle ϕ′ as the following,

/\/0 : (20)
From Equations (18) and (20), the evaluated angle φ should satisfy the following
relationship,

u\/i  / ¼ d/: (21)

Figure 16. Relationship between the rotation angle of ϕ and the compressive strength in case
of ri = 20 mm.
266 T. Matsuo et al.

In case of the relationship as an example of Figure 16 (Test No. 7 in Table 3), the rota-
tion angle dϕ between ϕi and ϕ is 5.7° from Equation (13), while the bending deflection
angle φ is 7.8° from Equation (19). Their calculated values do not satisfy the relation-
ship of Equation (21), so have no validity. For that reason, if φ decreases less than dϕ
by increasing the specimen width b, the evaluated value from the modified solution
using the test result has potential to satisfy the condition expressed in Equation (21).
In this study, the relatively thin-curved laminate is investigated, so it is not so
important to clarify the out-of-plane stiffness as represented by the parameter Er.
Figure 17 shows a parameter study about the influence of the variable radial modulus
Er to the evaluated strength in case of ri = 10 mm. Clearly from this analysis, the evalu-
ated value from the modified solution does not depend so much on the out-of-plane
stiffness.
The aim of this study is to evaluate the tangential strength as well as the interlami-
nar tensile strength of the curved surface by using a thin-curved laminate as mentioned
in the introduction. Therefore, it is helpful to investigate how much thickness of the
curved laminate is useful for applying the modified solution. Figure 18 shows the rela-
tionship between the thickness and the ratio of the maximum compressive stress (the
absolute value of the minimum tangential stress) to the maximum interlaminar tensile
stress (the maximum radial stress) at a certain loading force obtained from Equation
(11), and compares the modified solution with the previous solution. In the modified
solution, the variable displacement δ is assumed to be inversely proportional to the
cube of the thickness and is calculated referring to the measured displacement from the
test result when the thickness is 2 mm.
When the thickness increases, or not to mention the loading force is low, the modi-
fied solution is approaching to the previous solution because the displacement δ is com-
ing close to zero. And it is also found that the ratio of the maximum compressive
stress to the maximum interlaminar tensile stress decreases with the increase of the
thickness. So, whether the ratio is more than σθC/σrT (the ratio of the compressive
strength divided by the interlaminar tensile strength) or not, the failure pattern changes.
Because the strength ratio σθC/σrT can be obtained as 33 from the test results explained
in Section 4.3, the thickness borderline between the compressive failure and the inter-
laminar failure is found as 1.6 mm with ri = 10 mm or 2.1 mm with ri = 15 mm. This

Figure 17. Relationship between the modulus in the radial direction Er and the interlaminar
strength in case of ri = 10 mm.
 Advanced Composite Materials 267

Figure 18. Effects of the thickness and the difference between the modified and the previous
solution.

means that if the thickness ranges from 1.6 mm to 2.1 mm, the curved laminate with
ri = 10 mm generates the interlaminar failure and the curved laminate with ri = 15 mm
generates the compressive failure. Of course, it depends on the material strength related
to σθC/σrT. On the other hand, the previous solution just using thicker curved laminate
than 2.1 mm with ri = 15 mm satisfies the condition for evaluating only the interlaminar
strength. The relationship as indicated in Figure 18 has an effective guideline for
designing a curved geometry, related to the thickness and the radius, taking into consid-
eration the elastic deformation.

5. Conclusion
Thin-curved beam specimens formed from unidirectional CF/PP laminates were investi-
gated for verifying the compressive strength as well as the interlaminar tensile strength
and the relationship between the curved geometry and the fracture behavior in the curved
section. In the process of analysing the stress distribution in the curved section, the geo-
metrical large deformation is a key factor of modifying the conventional analytical stress
solution. So, the geometric modified solution was proposed and recognized as an avail-
able solution by comparing to the finite element analysis which satisfies the agreement
with the experimental load-displacement results. Based on these results, the fracture
behavior and strength of the curved section were estimated from the geometric parame-
ters related to the stress distribution calculated by the proposed modified solution. That is
to say, the curved laminate with the thickness decreasing or the inner radius increasing
has a tendency to generate the compressive fracture on the curved surface. In contrast,
the curved laminate with the thickness increasing or the inner radius decreasing has a ten-
dency to generate the interlaminar fracture in the curved section. Those experimental pro-
cedures and evaluating methods will contribute toward a technique applied to designing
and manufacturing a variety of laminated composite structures with thin curvature.

Acknowledgments
A part of this work belongs to Japanese METI-NEDO project ‘Development of sustainable hyper
composite technology’ since 2008fy. The authors gratefully acknowledge the support of this study
by all of concerned participators in the project.
268 T. Matsuo et al.

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