Fracture Process in ± θ Laminates Subjected to Mode II Loading
This work studies the behavior of a multidirectional laminate under Mode II loading. We describe the process of delamination in a reinforced composite of glass/epoxy. The stacking sequence (plies orientation [±θ]) was selected to minimize the coupling effects. The Mode II interlaminar test under thr...
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Інститут проблем міцності ім. Г.С. Писаренко НАН України
2002
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| Цитувати: | Fracture Process in ± θ Laminates Subjected to Mode II Loading / A. Ahmed Benyahia, A. Laksimi, S. Benmedakhene, and X.L. Gong // Проблемы прочности. — 2002. — № 4. — С. 103-118. — Бібліогр.: 9 назв. — англ. |
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irk-123456789-468732013-08-31T13:03:57Z Fracture Process in ± θ Laminates Subjected to Mode II Loading Ahmed Benyahia, A. Laksimi, A. Benmedakhene, S. Gong, X. L. Научно-технический раздел This work studies the behavior of a multidirectional laminate under Mode II loading. We describe the process of delamination in a reinforced composite of glass/epoxy. The stacking sequence (plies orientation [±θ]) was selected to minimize the coupling effects. The Mode II interlaminar test under three-point bending and cantilever flexure using ENF (End Notch Flexure) and ELS (End Load Split) specimens, respectively, was performed and analyzed. The test procedures and the results of strain-energy-release rate study for crack initiation are presented. The fracture process and the mechanical behavior of the two types of specimens are analyzed. The analysis clearly shows a close link between the angle θ, the ratio a/L, and the thickness h of the specimen. Fracture by delamination can be obtained only with an optimal choice of these parameters. The analysis of the states of stresses at the tip of crack allows us to explain the phenomenon of bifurcation between plies and is confirmed by the experimental results. Исследуется поведение многомерного ламината при его нагружении по типу KII . Описан процесс деламинирования упрочненного стеклоэпоксидного композита. Для минимизации эффектов трения выбрана ориентация ± θ. Рассмотрена методика межламинарных испытаний по типу KII с использованием схем трехточечного и консольного изгиба соответственно образцов типа ENF (торцевое защемление с консольным изгибом нагрузкой, равномерно распределенной по ширине свободного торца) и ELS (шарнирно закрепленная балка с центральной нагрузкой, равномерно распределенной по ширине балки). Представлены экспериментальные методики и результаты исследования скоростей высвобождения энергии деформации при инициировании трещин. Для двух вышеуказанных типов образцов проанализирован процесс разрушения и механического поведения материала. Установлена тесная корреляционная связь между величинами угла θ, соотношением a/L и толщиной образцов h. Показано, что разрушение вследствие деламинации возможно только при условии оптимального выбора этих параметров. Анализ напряженного состояния в вершине трещины позволяет объяснить явление межслойной бифуркации, его результаты хорошо согласуются с экспериментальными данными. Досліджується поведінка багатовимірного ламіната при його навантаженні за типом KII . Описано процес деламінування зміцненого склоепоксидного композита. Для мінімізації ефектів тертя вибрано орієнтацію ± θ. Розглянуто методику міжламінарних випробувань за типом KII із використанням схем триточкового і консольного згину відповідно зразків типу ENF (торцеве затиснення з консольним згином навантаженням, що рівномірно розподілене по ширині вільного торця) і ELS (шарнірно закріплена балка з центральним навантаженням, що рівномірно розподілене по ширині балкі). Представлено експериментальні методики і результати щодо швидкостей звільнення енергії деформації при ініціюванні тріщин. Для двох вищевказаних типів зразків проаналізовано процес руйнування і механічної проведінки матеріалу. Установлено кореляційний зв’язок між величинами кута θ, відношенням a /L і товщиною зразків h. Показано, що руйнування внаслідок деламінації можливе лише за умови оптимального вибору цих параметрів. Аналіз напруженого стану у вістрі тріщини дозволяє пояснити явище міжшарової біфуркації і добре узгоджується із експериментальними даними. 2002 Article Fracture Process in ± θ Laminates Subjected to Mode II Loading / A. Ahmed Benyahia, A. Laksimi, S. Benmedakhene, and X.L. Gong // Проблемы прочности. — 2002. — № 4. — С. 103-118. — Бібліогр.: 9 назв. — англ. 0556-171X http://dspace.nbuv.gov.ua/handle/123456789/46873 539.4 en Проблемы прочности Інститут проблем міцності ім. Г.С. Писаренко НАН України |
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Научно-технический раздел Научно-технический раздел Ahmed Benyahia, A. Laksimi, A. Benmedakhene, S. Gong, X. L. Fracture Process in ± θ Laminates Subjected to Mode II Loading Проблемы прочности |
| description |
This work studies the behavior of a multidirectional laminate under Mode II loading. We describe the process of delamination in a reinforced composite of glass/epoxy. The stacking sequence (plies orientation [±θ]) was selected to minimize the coupling effects. The Mode II interlaminar test under three-point bending and cantilever flexure using ENF (End Notch Flexure) and ELS (End Load Split) specimens, respectively, was performed and analyzed. The test procedures and the results of strain-energy-release rate study for crack initiation are presented. The fracture process and the mechanical behavior of the two types of specimens are analyzed. The analysis clearly shows a close link between the angle θ, the ratio a/L, and the thickness h of the specimen. Fracture by delamination can be obtained only with an optimal choice of these parameters. The analysis of the states of stresses at the tip of crack allows us to explain the phenomenon of bifurcation between plies and is confirmed by the experimental results. |
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Article |
| author |
Ahmed Benyahia, A. Laksimi, A. Benmedakhene, S. Gong, X. L. |
| author_facet |
Ahmed Benyahia, A. Laksimi, A. Benmedakhene, S. Gong, X. L. |
| author_sort |
Ahmed Benyahia, A. |
| title |
Fracture Process in ± θ Laminates Subjected to Mode II Loading |
| title_short |
Fracture Process in ± θ Laminates Subjected to Mode II Loading |
| title_full |
Fracture Process in ± θ Laminates Subjected to Mode II Loading |
| title_fullStr |
Fracture Process in ± θ Laminates Subjected to Mode II Loading |
| title_full_unstemmed |
Fracture Process in ± θ Laminates Subjected to Mode II Loading |
| title_sort |
fracture process in ± θ laminates subjected to mode ii loading |
| publisher |
Інститут проблем міцності ім. Г.С. Писаренко НАН України |
| publishDate |
2002 |
| topic_facet |
Научно-технический раздел |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/46873 |
| citation_txt |
Fracture Process in ± θ Laminates Subjected to Mode II Loading / A. Ahmed Benyahia, A. Laksimi, S. Benmedakhene, and X.L. Gong // Проблемы прочности. — 2002. — № 4. — С. 103-118. — Бібліогр.: 9 назв. — англ. |
| series |
Проблемы прочности |
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| first_indexed |
2025-07-04T06:24:20Z |
| last_indexed |
2025-07-04T06:24:20Z |
| _version_ |
1836696480881049600 |
| fulltext |
UDC 539.4
Fracture Process in ± 6 Laminates Subjected to Mode II Loading
A. Ahm ed Benyahia,a A. Laksim i,a S. Benm edakhene,a and X. L. Gongb
a Laboratoire Roberval UMR-CNRS-6066 Université de Technologie de Compiègne
(UTC), Compiègne, France
b LASMIS, GSM, Université de Technologie de Troyes (UTT), Troyes, France
У Д К
Процесс разрушения в ± 6-ламинатах, подвергнутых нагружению
по типу K II
А. Ахмед Б еньяхьяа, А. Л аксим иа, С. Бенмедахена, Кс. Л. Гонг6
a Технологический университет, Компьень, Франция
6 Технологический университет, Труа, Франция
Исследуется поведение многомерного ламината при его нагружении по типу Kп. Описан
процесс деламинирования упрочненного стеклоэпоксидного композита. Для минимизации
эффектов трения выбрана ориентация ± 6. Рассмотрена методика межламинарных испы
таний по типу KII с использованием схем трехточечного и консольного изгиба соответ
ственно образцов типа ENF (торцевое защемление с консольным изгибом нагрузкой, равно
мерно распределенной по ширине свободного торца) и ELS (шарнирно закрепленная балка с
центральной нагрузкой, равномерно распределенной по ширине балки). Представлены экспе
риментальные методики и результаты исследования скоростей высвобождения энергии
деформации при инициировании трещин. Для двух вышеуказанных типов образцов про
анализирован процесс разрушения и механического поведения материала. Установлена тес
ная корреляционная связь между величинами угла 6, соотношением a/L и толщиной образ
цов h. Показано, что разрушение вследствие деламинации возможно только при условии
оптимального выбора этих параметров. Анализ напряженного состояния в вершине тре
щины позволяет объяснить явление межслойной бифуркации, его результаты хорошо согла
суются с экспериментальными данными.
К лю чевы е слова : ±6-ламинаты, нагружение по II типу, деламинирование,
бифуркация, процесс разрушения, анализ напряженного состояния.
Introduction. The last 20 years have seen the development of a very
extensive range of tests designed to quantify the toughness of continuous fiber-
reinforced composites.
These tests are generally identified by the type of loading applied as Mode I,
Mode II, Mode III, or Mixed mode.
Delamination tests under bending conditions have been usually performed on
unidirectionally reinforced fiber composites [0° ] in order to determine
interlaminar fracture resistance G Пс.
However, the structure of fiber-reinforced composites commonly used in
industry is made of stratified layers of the type [±6 ]. Initial defects from the
© A. AHMED BENYAHIA, A. LAKSIMI, S. BENMEDAKHENE, X. L. GONG, 2002
ISSN 0556-I7IX. Проблемы прочности, 2002, № 4 103
A. Ahmed Benyahia, A. Laksimi, S. Benmedakhene, and X. L. Gong
manufacture process can be located anywhere in a multidirectional laminate, and
during loading, these defects can propagate and lead to complete fracture. In such
materials, the presence of an interlaminar defect divides the composite into two
sections, analogous to the two arms of a double cantilever beam (DCB) specimen,
whose mechanical behavior must be well defined. This mechanical behavior
depends essentially on the plies orientation. Analysis of the stress and strain fields
around the crack tip, as well as the fracture mechanisms become more complex
than those in unidirectional composite structures.
Moreover, comparison is made between the values of the energy release rate
for various interfaces without taking account the stacking sequences. When a
crack propagates at large angles (especially [90° ]), the strain-energy release rate is
dissipated not only during initiation and propagation of delamination but also in
the nucleation of transverse cracks in these plies. This phenomenon, which is
called bifurcation, is due to the edge effect [1-2]. Few studies concern the Mode II
cases because of the difficulties encountered when using the concept of the linear
elastic fracture mechanic (LEFM) to obtain the energy release rate G n .
The study of Mode II in the case of laminates ± 6 requires preliminary
adaptation of specimen dimensions according to the used angle 6. The difficulties
in designing a Mode II specimen is to avoid any cracking by opening without
causing an excessive friction between the two faces of the crack.
In order to characterize the Mode II, three-point bending and cantilever
flexure tests [ENF (End Notch Flexure) and ELS (End Load Split)], respectively,
are used. However, the specimen geometry must take into account the length L,
the thickness h , and the crack length a . It must be noted that certain composite
materials like glass-epoxy, which have a rather low Young’s modulus due to a
significant length L, exhibit large displacement before the beginning of
delamination. The aim of this study is to get a better understanding of the
behavior of multidirectional composites, which depends on the orientation of
plies, stacking sequences, and crack location under Mode II loading.
1. Experim ental Analysis.
1.1. M aterial and Experim ental Procedures. The material used in this
study is an E-glass/M10-epoxy composite. The fiber volume fraction is about
52%. Test specimens were cut from plates of size 300 X 300 mm. A Teflon film
(30 /im in thickness) was incorporated at the mid-plane to initiate delamination
(Fig. 1). The thickness of 4.8 mm is composed of 16 plies. The plates were cured
in an air press according to the cure cycle recommended by the manufacturer (3
bars for 1 hour at 120°C). Four configurations [(+ 6 /—6 )2s ]s were selected with
the angles: 6 = 15°, 30°, 45°, and 60°.The starter film was inserted in the interface
between 6 /6 laminate at the mid-plane of the stacking sequence: (+6/—6/+6/—6/
—6/+6/—6/+6//+6/—6/+6/—6/—6/+6/—6/+6). In order to study the shearing mode,
two types of specimens (ELS and ENF) were used (Fig. 1), whose dimensions are
shown in Table 1.
Each specimen was instrumented with an acoustic emission transducer and
strain gauges were placed on the specimen surfaces as indicated in Fig. 1. The
specimens were loaded with displacement control at a constant rate of 0.5 mm/min
in order to observe better the occurrence of damage mechanisms.
104 ISSN 0556-171X. npo6n.eubi npounocmu, 2002, N 4
Fracture Process in ± 0 Laminates
T a b l e 1
Geometrical Characteristics of ELS and ENF Specimens
Specimens Width B,
mm
Length L,
mm
Total thickness
2h, mm
Ratio a>l L
ELS 20 60 4.8 0.4< a0/L < 0.7
ENF 20 50 4.8 0.4< a0 L < 0.7
Strain gauge
Initial crack
A.E.Transducer Strain ëauëe
Initial crack
Fig. 1. ELS (a) and ENF (b) specimens.
a
b
1.2. D ata-Reduction M ethod. The mechanical approach involves
determination of the energy release rate by the compliance method:
P 2dc
GII = 2Bda '
dc
Here, P is the applied load, B is the specimen width, and — is the partial
da
derivative of the compliance with respect to the crack length. Four specimens
were tested for each laminate.
The calibration of compliance was made according to the rule c = a + fia
[3], where a and fi are material parameters determined for various ratios a 0/ L
The values obtained for four configurations are summarized in Table 2.
The results for the energy release rate G IImax corresponding to the
maximum load Pmax are summarized in Table 3.
The G IImax values corresponding to the sequence 0 = ±60° could not be
given because of the specimen fracture by flexure without delamination. All the
configurations seem to have fracture energies (GIImax), which converge towards a
value of approximately 2 kJ/m 2.
ISSN 0556-171X. npoôëeMbi npounocmu, 2002, N2 4 105
A. Ahmed Benyahia, A. Laksimi, S. Benmedakhene, and X. L. Gong
T a b l e 2
Compliance Parameters a and 3 Depending on 6
Laminate ±15° ±30° ±45° ±60°
a ■ 10“2, mm/N 1.42 2.21 1.80 2.36
ß-10“7, (N-mm2)"1 1.55 1.85 3.56 7.10
T a b l e 3
Energy Release Rate for ELS Specimen
Sequences 0, deg aa/ L GII max, kJ/m2
0 0.4 2.63 (0.28)
0.5 2.43 (0.21)
0.6 2.45 (0.19)
±15 0.4 1.91 (0.13)
0.5 2.07 (0.14)
0.6 1.99 (0.15)
0.7 2.13 (0.12)
±30 0.4 1.51 (0.09)
0.5 1.96 (0.12)
0.6 2.10 (0.08)
0.7 1.89 (0.08)
±45 0.4 0.92 (0.04)
0.5 1.51 (0.08)
0.6 1.38 (0.08)
0.7 1.16 (0.06)
Note. Here and in Table 4 in parentheses are given standard deviations.
T a b l e 4
Energy Release Rate for ENF Specimen
Sequences 0, deg a l GIIc, kJ/m2
15 0.4 1.380 (0.25)
0.5 1.560 (0.20)
0.6 2.060 (0.21)
0.7 2.043 (0.22)
30 0.4 1.360 (0.13)
0.5 1.680 (0.14)
0.6 1.917 (0.13)
0.7 1.927 (0.12)
45 0.4 1.520 (0.09)
0.5 1.240 (0.09)
0.6 1.550 (0.07)
0.7 1.660 (0.08)
The absence of the G IImax values for the sequence ± 60° in Table 4 is due
to the fact that this sequence practically does not have delamination, except for
the ratios a o / L that is very close to unity, and the stress field at the tip of the
initial crack is completely modified. It appears that before delamination
composites [±6] subjected to Mode II loading by flexure have highly scattered
106 ISSN 0556-171X. npoÖÄeubi npounocmu, 2002, N 4
Fracture Process in ± 0 Laminates
damages especially by transverse cracks. A transverse crack is perpendicular to
the direction of propagation, it cannot be assimilated to a pseudo-length of the
crack as required by the LEFM concepts.
Transverse cracks are responsible for the significant nonlinearity especially
for large angles.
The strain energy release rate G nmax for large angles (> 17°) remains well
below 2.5 kJ/m that represents the intrinsic resistance to delamination of the
material used in this study. This means that delamination is always preceded by
transverse cracks, which generate significant shear stresses ahead of the crack tip.
From the above observations it appears that the behavior of the angle-ply
laminates is more complex than that of unidirectional materials. The phenomenon
of bleaching (damage of glass-matrix appears as a whitening aspect) and the
specimen low stiffness involve large displacements and nonlinear behavior, which
may invalidate the fracture mechanics concepts. These remarks determine
optimization of the specimen dimensions as it is discussed below.
2. Specimen Optimization. The use of the concepts of the linear elastic
fracture mechanics is valid only in the case of the linear elastic behavior with
small displacements. Their application requires a check on two behavior
conditions in order:
- to avoid nonlinearity resulting from large displacements,
- to avoid nonlinearity of the material or fracture by flexure.
On this subject, Carlsson et al. [4] presented the analysis based on the beam
theory that allows the calculation of dimensions for an ENF specimen.
The first condition results in a maximum value of the slope y 'a , at the
loading point.
The acceptable displacement (d a) corresponding to y'a can be calculated
from Eq. (2):
The critical displacement of fracture initiation is given by the following
relation:
The condition d c < d a allows us to optimize the specimen dimensions by
combining Eqs. (2) and (3). For example, if the thickness h is the parameter,
which controls the specimen behavior, we have:
where G iimax is the energy of the crack propagation initiation, L is the
half-length of the specimen, a is the initial crack length, and E 1 is the flexure
modulus in the longitudinal direction.
dy _ 3(L2 + 3a 2>3
dx 2L3 + 3a 3
(2)
(3)
(4)
ISSN 0556-171X. npoôëeMbi npounocmu, 2002, N 4 107
A. Ahmed Benyahia, A. Laksimi, S. Benmedakhene, and X. L. Gong
For the second condition, optimization will be made according to the
permissible flexure strain. For the ratio a /L < 0.5, if there is fracture by flexure, it
occurs in the median section between the supports, where the maximum strain is
given by
6Lhd
£ m ~ 2L3 + 3a3 ' (5)
In the same way as for the first condition, the optimization of the thickness
gives
l -g Ik.
h - ’ <6a £ maE 1
where £ ma is the permissible flexure strain.
The study by the finite elements realized by Mall and Kochhar [5] showed
that the linear elastic analysis can be used if the critical displacement does not
3
exceed the total thickness of the specimen 1.5 times [d< -<2h)], otherwise the
G II value will be overestimated.
In the same way as for the ENF specimen, it is possible to use the
nonlinearity conditions for ELS specimens used in this study. In the case of
nonlinearity due to large displacements, an acceptable displacement (d a) can be
calculated by Eq. (7) for an acceptable slope y 'a :
dy 3(3a 2 + L2 )
dx 2(3a 3 + L3 )
The starting critical displacement is given by relation (8):
<5. (7)
L + 3a I G
On combining Eqs. (7) and (8), the condition d ,
the thickness (h) with relation (9):
5 a allows us to optimize
h > 3
G iic ( L2 + 3a 2)2
4( ya )2 a 2 E
(9)
In the case of nonlinearity due to fracture by flexure, optimization is to be
done according to the acceptable strain of fracture by flexure. For the ratio
a /L < 0.5, if there is fracture by flexure, it occurs on the cantilever, where the
strain is maximal and is given by
108 ISSN 0556-171X. npoôëeubi npounocmu, 2002, N2 4
Fracture Process in ± 0 Laminates
3Lhô
£ m ~ T3 : t 3 ' (10)L + 3a
Fracture by flexure can be avoided using the optimum thickness h. This is
the same relation as for the ENF specimen:
l 2g IIc
h - I v T ' (11)a ^maE 1
Thus, the optimization by the application of the nonlinearity conditions
requires the knowledge of the following parameters: G IImax (the energy-release
rate in Mode II), £ ma (fracture strain of the material), and E 1 (the longitudinal
modulus of elasticity obtained by flexure). It should be noted that only G IImax
(2.5 kJ/m ) is intrinsic to the laminate material whatever the orientation angle of
the plies. On the other hand, £ ma and E 1 depend on the orientation of the plies
and their values are summarized in Table 5. The £ ma values are calculated by the
application of the interactive criterion of Tsai [6]. As for the E 1 values, they are
determined by the laminate theory.
T a b l e 5
Mechanical and Energy Characteristics for Various Sequences
Sequences Q, deg E , GPa £ ma ' 10 3 GIIc, kJ/m2
(average)
0 (unidirectional) 44.110 24.55
±15 38.560 9.95
±30 26.939 6.20 2.5
±45 18.354 4.00
±60 16.432 2.53
A numerical simulation for a unidirectional laminate in this study with
G IImax = 2.5 kJ/m 2, £ ma = 24.55-10“ 3, E 1 = 44,110 MPa, h = 2.4 mm, (5c = 3h,
and a 0/L = 0.5 has given the following specimen lengths:
L < 67 mm for the first condition of nonlinearity (large displacements);
L < 42 mm for the second condition of nonlinearity (fracture by flexure).
The second condition of nonlinearity proves to be more severe and should be
used for any optimization. The length L = 40 mm appears to be very practical in
the case of the ENF specimen whose total length is 80 mm. However, in the case
of the ELS specimen, this length is insufficient because of the resistance of the
assembly. It should also be noted that this length was calculated for a
unidirectional laminate. Consequently, the application of the same condition to
laminates ± d (with d > 0°) certainly requires lower lengths (L).
The application of Eq. (11) allows us to determine, for the thickness h and a
given ratio a/L, the minimum strain em that causes fracture by delamination.
Thus, as long as £ m < £ ma, fracture occurs completely by delamination. On the
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A. Ahmed Benyahia, A. Laksimi, S. Benmedakhene, and X. L. Gong
other hand, if e m > > e ma, fracture occurs completely by flexure. As for the cases
where e m is higher but very near to e ma, the process of fracture is the
combination of fractures by flexure and delamination. In fact, in this intermediate
case, the initiation of fracture is caused by flexure (transverse fracture of the first
ply, which is extremely stretched), then delamination occurs under the effect of
increasing shear stresses due to fracture of the first ply. This calculation and
checking procedure must be carried out for all the stacking sequences and for the
ratio a /L growing from 0 to 1.
However, if for a /L < 0.5, the maximum strain is at the point m (Fig. 2), for
a /L > 0.5, the maximum strain is at the tip of the initial crack (point t). The
relation between the two strains at the points m and t is treated by the
conventional theory of beams.
a b
Fig. 2. Dangerous sections in ENF (a) and ELS (b) specimens.
In this case, the stresses at the points m and t are given by the following
relations:
3PL 3Pa
o m = ----- 2 and o t = -----2 for an ENF specimen,
4bh 2bh
3PL 3Pa
o m = ----- r and o t = — r- for an ELS specimen.
m 2 b h 2 b h 2
2a
Therefore, o t = — o m for the two specim en types m d „ A t o y M .o a /L
2a
Consequently, for the two specimen types e t = — e m .
L
Thus, for any ratio a /L < 0.5 and e t < e m , a check for fracture by flexure
must be carried out at the point m, where the section is more dangerous. On the
other hand, for any ratio a /L > 0.5 and e t > e m, the check for fracture by flexure
is done at the point t.
The minimum values of e m (for a /L < 0.5) and e t (for a /L > 0.5), which
cause fracture by delamination, for the thickness h = 2.4 mm are presented in
Table 6. It is necessary to note that for the ratio a /L = 0, the strain e m is infinite.
That means that for this ratio (a/L = 0) the specimen does not contain any initial
crack and the initiation of fracture occurs by flexure at the point m without the
risk of delamination in the median plane of the specimen end.
The value of e m decreases with increasing ratio a /L and increases with the
orientation angle of the plies. However, when a /L > 0.5, although the strain e m
continues to decrease with an increase in the ratio a/L, the strain e t remains
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Fracture Process in ± 0 Laminates
constant. Its values correspond to the ratio a /L = 0.5 and remain constant at
a /L > 0.5.
A comparison between the minimum strains ensuring fracture by
delamination and the fracture strains in flexure is made in Table 6. It is clear that
fracture initiation is always caused by flexure in the following cases (see Fig. 3):
- for the ratio a /L < 0.2 and any orientation angle of the plies (even for a
unidirectional composite);
- for all 9 > 15° and any ratio a/L.
T a b l e 6
Minimum Strains Ensuring Fracture by Delamination
Sequences
0, deg
Cm -10“3 Ct -10'3
L II 0 <NÖII a l = 0.3 L II O 4̂ .5ÖIIla ajL > 0.5
0 <x 24.55 16.20 12.10 9.72 9.72
±15 <x 25.60 17.32 12.99 10.40 10.40
±30 <x 31.10 20.73 15.54 12.00 12.00
±45 <x 37.66 25.11 18.83 15.06 15.06
±60 <x 39.81 26.54 19.90 15.90 15.90
Angle 9 (deg)
Fig. 3. A graph giving fracture strains in flexure according to the orientation angle 9 of the plies
and the a/L ratio.
As for fracture by delamination, it occurs when the conditions of the angle 9
and the ratio a /L are limited by the contour of the shaded zone. In fact, when 9 is
lower than 15°, any couple (9, and a/L), which is below the curve of failure strain
by flexure for undamaged material, causes delamination without transverse
cracking. For example, for 9 = 10°, all the ratios a /L > 0.4 cause fracture by
delamination alone (without transverse cracking).
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The same remark can be made for the optimization of the specimen
thickness. To this effect, the application of Eq. (6) allows us to determine, for a
given fracture strain by flexure and a given ratio a/L, the necessary minimum
thickness to ensure fracture by delamination. Table 7 summarizes the minimum
values of the thickness h necessary for all the sequences of stratification and the
ratio a /L from 0 to 1. A comparison between these results and the real thickness
of the material in this study is illustrated in Fig. 4. The analysis of various curves
in this figure makes it possible to explain the fracture process in ± 0 laminates
according to the specimen thickness and the ratio a/L.
T a b l e 7
The Minimum Thickness Values Ensuring Fracture by Delamination
Sequences
9, deg
Half-thickness of the specimens
L II 0 <N©II L II p 3 L II .5OII
0l <x 2.30 1.04 0.58 0.38
±15 <x 16.37 7.27 4.10 2.62
±30 <x 60.00 26.82 15.09 9.65
±45 <x 212.00 94.60 53.20 34.05
Sequence angle 0, deg
Fig. 4. The minimum thickness ensuring fracture by delamination.
We can clearly see a close relation between the angle 0, the ratio a/L, and
the thickness h of the material. Fracture by delamination can be obtained only
with an optimum choice among these three parameters. For example, for 0 = 20°
associated with the ratio a /L = 0.4, the adequate thickness h to cause fracture by
delamination is lower or equal to 0.6 mm, whereas it can almost double for the
ratio a /L = 0.3.
As for the material in this study (h = 2.4 mm), fracture takes place by simple
delamination for any couple (0, a/L) pertaining to the small hatched rectangle
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Fracture Process in ± 0 Laminates
(Fig. 4). Thus, for all the sequences studied, fracture cannot occur only by
delamination if the specimens are loaded in Mode II by flexure that was checked
by experiments. It is easy to see that microcracks generated by flexure in Mode II
grow by coalescence and lead to the bifurcation phenomenon, especially in the
adjacent delamination plies. This damage process becomes more important and
takes place in the plies having a large angle 6. This damage mechanism occurs
even in Mode I [6, 7]. But the question is how this bifurcation phenomenon
appears around the crack tip in this case, and what types of stresses are involved
in this mechanism. In order to have a better understanding of the appearance of
transverse cracks in this case, analysis of the fracture process at the crack tip is
presented.
3. Analysis of the F ractu re Process in ENF and ELS Specimens. The
results of optimization of ENF and ELS specimens allow us to identify 3 modes
of fracture: fracture by delamination alone, transverse fracture only by flexure,
and fracture by the combination of the last two modes.
With regard to the first mode, fracture by delamination alone is present in
two types of laminates (unidirectional and multidirectional (±6)) for which the
couple (6, a/L) is inside the small hatched area of Fig. (3). The delamination
propagation occurs at the median interface (+6/+6). This fracture mode can be
explained in two ways:
- the material toughness in Mode II (2.5 kJ/m ) is lower than the work of the
external forces necessary to cause fracture by flexure;
- the ratio of the shear stress at the crack tip (r t ) to the fracture shear stress
(r R ) is much larger than the ratio of the maximum flexure stress ( a mf ) to the
fracture stress ( a Rf ) by flexure (r t j r R > > a mf / a Rf ).
For the second mode, fracture by flexure alone occurs for the values of (6,
a/L) very far from the rectangle OABC (Fig. 4). It is necessary that the distance of
the (6, a/L) values from the rectangle OABC be such that the amplification of the
shear stress after failure of the first ply could not generate delamination. For
example, for the practical ratio a /L > 0.5 (the ratio that allows a stable
propagation), all the laminates (±6) for which 6 > 60° fracture only by flexure.
The third mode characterized by both delamination and fracture by flexure is
more complex and requires a particular study. Indeed, in this case, fracture is
initiated by transverse cracks within two extremely stressed plies (+6) causing a
transfer of delamination towards the interface (+6/—6) to a ply of the median
plane. For better understanding of this mode of delamination plane, it would be
necessary to analyze the distribution of stresses surrounding the tip of the initial
crack.
However, the majority of works treating stress distributions in fractured
laminates concern the sequences loaded under simple tension and necessarily
containing plies oriented at 90°. Having the weakest ultimate deformation, the
plies oriented at 90° are the best case for studying transverse cracking. The basic
methods of calculation developed for the stress distribution analysis are:
- shear-lag method;
- variational approach;
- theory of internal variables;
- statistical approach.
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A. Ahmed Benyahia, A. Laksimi, S. Benmedakhene, and X. L. Gong
In this study, the shear-lag method is adopted for the analysis of stress
distribution, which is studied around the tip of the initial crack. In fact, the most
dangerous section under the effect of flexure is located at the tip of the initial
crack, where a /L > 0.5 (stable propagation). In the elastic linear phase, the stress
field surrounding the tip of the initial crack is presented in Fig. 5.
Fig. 5. Flexure and shear stress fields in the vicinity of the initial crack tip at the elastic phase.
When the bending stress reaches the limiting value a j within the stretched
median ply (+0), a transverse crack is initiated on its right side at the tip of the
initial crack. At this crack, the neutral axis moves from (A.N.1) towards (A.N.2)
(Fig. 6) annulling the bending stress. The remaining part of the specimen’s arm
section has a new stress distribution. The evolution of the neutral axis and of the
new distribution of bending stresses around the transverse crack is illustrated in
Fig. 6. While moving away from the crack, the bending stress a j (x ) in the
failure ply increases exponentially until reaching the maximum whose value
remains somewhat lower than ~ g .
I ̂ Initial crack-tip
CTf \ Evolution of the neutral axis
Fig. 6. Redistribution of bending stresses and evolution of the neutral axis near a transverse crack.
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trasverse crack
A l
+6 j
- e
Fig. 7. Appearance and distribution of shear stresses near a transverse crack due to flexure.
Fig. 9. Scheme of the crack propagation process in the ± 0 laminate loaded under Mode II
conditions.
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A. Ahmed Benyahia, A. Laksimi, S. Benmedakhene, and X. L. Gong
Fig. 10. Crack bifurcation at the crack tip: (a) crack initiation, (b) crack propagation.
Fig. 11. Coalescence of microcracks.
However, because of the transverse crack, the bending stress is transformed
into the shear stress (r j ) and then becomes an additional stress to the remaining
part of the section. As the shear modulus is the same for the plies + 6 and — 6,
the new shear stress r j is distributed through the thickness on both sides of the
interface (+6) and (—6) (Fig. 7). The effect of the shear stress r j on the
continuation of the fracture mode is decisive. Indeed, the stress r j added to the
stress r prior to bifurcation causes redistribution of the shear stress (Fig. 8). This
successive redistribution of the shear stress is the cause of the stepwise crack
propagation. Consequently, the dangerous zone under the effect of shear is the
interface + 0/—0 toward the adjacent ply of the median plane. Thus, the
delamination expected on the median interface + 0/+0 is transferred towards the
interface + 0/—0.
In any case, according to observations of Purslow [9], the evolution of
delamination in Mode II occurs as shown in Figs. 9 and 10. Fracture initiation
takes place at the matrix-fiber interface and the coalescence occurs by forming
strips at 45° according to shearing fracture (Fig. 11).
Conclusions. The angle of orientation 6 has a significant influence on the
value of the energy release rate G IIc.
The Mode II tests on ENF and ELS specimens with angles ± 6 require a
preliminary geometrical optimization in order to avoid transverse cracking apart
from the median plane, which dissipates energy. This study shows the existence of
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Fracture Process in ± в Laminates
a close relation between the angle в, the ratio a/L, and the thickness h of the
material. For the 5-mm thickness and the sequences of stratification studied,
fracture cannot occur by delamination only without transverse cracking.
The analysis of the stress state at the crack tip allows us to explain the
phenomenon of bifurcation between plies and it is in good agreement with the
experimental results. A successive redistribution of the shear stresses is the main
cause of bifurcation and stepwise crack propagation.
Fracture initiation takes place at the matrix-fiber interface and the
coalescence occurs by the formation of strips at 45° according to fracture by shear.
Р е з ю м е
Досліджується поведінка багатовимірного ламіната при його навантаженні
за типом K п . Описано процес деламінування зміцненого склоепоксидного
композита. Для мінімізації ефектів тертя вибрано орієнтацію ± в. Розгля
нуто методику міжламінарних випробувань за типом K п із використанням
схем триточкового і консольного згину відповідно зразків типу ENF (тор
цеве затиснення з консольним згином навантаженням, що рівномірно роз
поділене по ширині вільного торця) і ELS (шарнірно закріплена балка з
центральним навантаженням, що рівномірно розподілене по ширині балкі).
Представлено експериментальні методики і результати щодо швидкостей
звільнення енергії деформації при ініціюванні тріщин. Для двох вищевказа
них типів зразків проаналізовано процес руйнування і механічної прове-
дінки матеріалу. Установлено кореляційний зв’язок між величинами кута в ,
відношенням a /L і товщиною зразків h. Показано, що руйнування внаслі
док деламінації можливе лише за умови оптимального вибору цих парамет
рів. Аналіз напруженого стану у вістрі тріщини дозволяє пояснити явище
міжшарової біфуркації і добре узгоджується із експериментальними даними.
1. D. J. Nicholls and J. P. Gallagher, “Determination of G Ic in angle ply
composites using a cantilever beam test method,” J. Reinforced Plastics and
Composites, No. 2, 2-17(1983).
2. P. Robinson and D. Q. Song, “A modified DCB specimen for Mode I of
multidirectional laminates,” J. Composite M ater., 26, No. 11, 1554-1577
(1992).
3. J. Whitney and L. M. Pinnel, “Characterization of interlaminar Mode II
fracture using beam specimens,” in: Science and Technology o f Composite
M aterials (ECCM IV), Stuttgart (1990), pp. 865-868.
4. L. A. Carlsson, J. W. Gillepsie, Jr., and R. B. Pipes, “On the analysis and
design o f End-Notched Flexure (END) specimens for Mode II testing,”
J. Composite Mater., 20, 594-604 (1986).
5. J. Mall and P. Kochhar, “Finite-element analysis of end-notch flexure
specimens,” J. Comp. Tech. Research, 8, No. 2, 54-57 (1986).
6. S. W. Tsai, Theory o f Composites Design, Think Composites (1992).
ISSN 0556-171X. Проблеми прочности, 2002, № 4 117
7. Ahm ed Benyahia, Etude des M ecanismes de Délam inage sous l'E ffet de
Contrainte Complexes Générées P ar des Sollicitation Simple d'O uverture et
de Cisaillement Dans les Stratifiés ± 0, Thesis Université de Technologie de
Compiègne, No. 1040 (1997).
8. A. Laksimi, A. Benyahia, M. Benzeggagh, and X. L. Gong, ”Study of crack
initiation and bifurcation mechanisms in a multidirectional laminate,”
Composites Science and Technology, 60, 597-604 (2000).
9. D. Purslow, “Matrix fractography of fiber-epoxy,” in: Composites, Royal
Aircraft Establishment, Technical Report 86046 (1986).
Received 19. 09. 2001
A. Ahmed Benyahia, A. Laksimi, S. Benmedakhene, and X. L. Gong
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