Mechanical behavior and optimization of multidirectional laminate specimens under delamination by bending
Delamination in the multidirectional laminates under bending loading is often difficult to characterize because of problems involved by excessive deformations ahead of the crack tip and by non-linearities due to large displacements and transverse microcracking. In fact, these microcracks appear in t...
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irk-123456789-478772013-08-31T16:17:31Z Mechanical behavior and optimization of multidirectional laminate specimens under delamination by bending Ahmed Benyahia, A. Laksimi, A. Ouali, N. Azari, Z. Научно-технический раздел Delamination in the multidirectional laminates under bending loading is often difficult to characterize because of problems involved by excessive deformations ahead of the crack tip and by non-linearities due to large displacements and transverse microcracking. In fact, these microcracks appear in the most strained plies especially for thin materials. Consequently, the concept of linear elastic fracture mechanic cannot be used in order to characterize delamination. Thus, the objective of this work is to highlight, experimentally and analytically these problems on DCB specimens, then to carry out optimization of thickness which makes it possible to avoid the appearance of transverse cracking before starting of delamination. The material used is E-glass/epoxy multidirectional laminates. The results of optimization obtained are in good agreement with the experimental results. При навантаженні згином багатовимірних ламінатів має місце деламінація. Її опис є досить важким, оскільки у вістрі тріщини присутні великі деформації та відмічаються нелінійні ефекти, що зумовлені значними переміщеннями і утворенням поперечних мікротріщин. Останні з’являються у найбільш навантажених шарах, особливо в тонких ламінатах. Експериментально досліджено вищезгадані особливості на зразках у вигляді консольної балки та виконано оптимізацію товщини багатошарового зразка, за якої деламінація починаєтьсядо утворення поперечних мікротріщин у найбільш навантажених шарах. Об’єктом дослідження служив склоепоксидний зміцнений ламінат. Отримані результати оптимізації товщини зразка добре узгоджуються з експериментальними даними. При изгибном нагружении многомерных ламинатов имеет место деламинация, описание которой весьма затруднительно ввиду наличия больших деформаций у вершины трещины и различных нелинейных эффектов, связанных со значительными перемещениями и образованием поперечных микротрещин. Последние появляются в наиболее нагруженных слоях, особенно в тонких ламинатах. Экспериментально исследованы вышеуказанные особенности на образцах в виде консольной балки и выполнена оптимизация толщины многослойного образца, при которой деламинация начинается до образования поперечных микротрещин в наиболее нагруженных слоях. Объектом исследования служил стеклоэпоксидный упрочненный ламинат. Полученные результаты оптимизации толщины образцов хорошо согласуются с экспериментальными данными. 2006 Article Mechanical behavior and optimization of multidirectional laminate specimens under delamination by bending / A. Ahmed Benyahia, A. Laksimi, N. Ouali, Z. Azari // Проблемы прочности. — 2006. — № 6. — С. 66-78. — Бібліогр.: 9 назв. — англ. 0556-171X http://dspace.nbuv.gov.ua/handle/123456789/47877 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. Ouali, N. Azari, Z. Mechanical behavior and optimization of multidirectional laminate specimens under delamination by bending Проблемы прочности |
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Delamination in the multidirectional laminates under bending loading is often difficult to characterize because of problems involved by excessive deformations ahead of the crack tip and by non-linearities due to large displacements and transverse microcracking. In fact, these microcracks appear in the most strained plies especially for thin materials. Consequently, the concept of linear elastic fracture mechanic cannot be used in order to characterize delamination. Thus, the objective of this work is to highlight, experimentally and analytically these problems on DCB specimens, then to carry out optimization of thickness which makes it possible to avoid the appearance of transverse cracking before starting of delamination. The material used is E-glass/epoxy multidirectional laminates. The results of optimization obtained are in good agreement with the experimental results. |
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Ahmed Benyahia, A. Laksimi, A. Ouali, N. Azari, Z. |
| author_facet |
Ahmed Benyahia, A. Laksimi, A. Ouali, N. Azari, Z. |
| author_sort |
Ahmed Benyahia, A. |
| title |
Mechanical behavior and optimization of multidirectional laminate specimens under delamination by bending |
| title_short |
Mechanical behavior and optimization of multidirectional laminate specimens under delamination by bending |
| title_full |
Mechanical behavior and optimization of multidirectional laminate specimens under delamination by bending |
| title_fullStr |
Mechanical behavior and optimization of multidirectional laminate specimens under delamination by bending |
| title_full_unstemmed |
Mechanical behavior and optimization of multidirectional laminate specimens under delamination by bending |
| title_sort |
mechanical behavior and optimization of multidirectional laminate specimens under delamination by bending |
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Інститут проблем міцності ім. Г.С. Писаренко НАН України |
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2006 |
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Научно-технический раздел |
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| citation_txt |
Mechanical behavior and optimization of multidirectional laminate specimens under delamination by bending / A. Ahmed Benyahia, A. Laksimi, N. Ouali, Z. Azari // Проблемы прочности. — 2006. — № 6. — С. 66-78. — Бібліогр.: 9 назв. — англ. |
| series |
Проблемы прочности |
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2025-07-04T07:56:46Z |
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| fulltext |
UDC 539.4
M echanical Behavior and O ptim ization o f M ultid irectional Lam inate
Specim ens under D elam ination by Bending
A. Ahmed Benyahia,a A. Laksimi,b N. Ouali,a and Z. Azaric
a Laboratoire de Mécanique Avancée, Alger, Algerie
b Laboratoire Roberval FRE-CNRS-6066, Compiègne, France
c Laboratoire de Fiabilité Mécanique, Metz, France
УДК 539.4
М еханическое поведение и оптимизация образцов из многомерны х
лам инатов в условиях делам инации при изгибе
А. Ахмед Беньяхьяа, А. Лаксими6, Н. Куалиа, 3. Азарив
а Лаборатория прикладной механики, Алжир, Алжир
6 Лаборатория им. Жиля Роберваля, Компьень, Франция
в Лаборатория механической надежности, Мец, Франция
При изгибном нагружении многомерных ламинатов имеет место деламинация, описание
которой весьма затруднительно ввиду наличия больших деформаций у вершины трещины и
различных нелинейных эффектов, связанных со значительными перемещениями и образова
нием поперечных микротрещин. Последние появляются в наиболее нагруженных слоях,
особенно в тонких ламинатах. Экспериментально исследованы вышеуказанные особенности
на образцах в виде консольной балки и выполнена оптимизация толщины многослойного
образца, при которой деламинация начинается до образования поперечных микротрещин в
наиболее нагруженных слоях. Объектом исследования служил стеклоэпоксидный упроч
ненный ламинат. Полученные результаты оптимизации толщины образцов хорошо согла
суются с экспериментальными данными.
К лю ч е вы е с л о в а : многомерные ламинаты, деламинация, нелинейность, по
перечные трещины, оптимизация по толщине.
Introduction. DCB specimen is the most specimen used for the
characterization o f delamination in Mode I of composite materials [1, 2]. Its
geometry was optimized and standardized during the years 1990. However, this
optimization was carried out for unidirectional laminates where the rupture of the
plies by flexion is practically equivalent to tension load and where the delamination
behavior remains insensitive to the thickness effect of the specimen.
Due to the embedding effect of the arms at the crack tip, a correction has
been given to the analytical compliance relationship in order to take account of
the deflection angle of the arms at the crack tip during bending [3-5]. However,
the stress fields ahead of the crack tip and its effect on the mode of propagation of
delamination has not been undertaken so far. It’s obvious that the large
displacement and the excessive angle of deflexion involve a non-linear behavior
[6, 7].
© A. AHMED BENYAHIA, A. LAKSIMI, N. OUALI, Z. AZARI, 2006
66 ISSN 0556-I7IX. Проблемы прочности, 2006, N 6
M echanical Behavior and Optimization
These problems of partial embedding of the arms and non-linearity affect the
mechanical behavior of the multidirectional composites by generating additional
dissipative phenomena such as transverse cracks of the strained plies especially in
the case of large angles of the fibers orientation [8, 9]. Consequently, this behavior
makes the analysis of the experimental characterization results more difficult.
It is obvious that these problems depend on the specimen thickness, because
in previous work [8] we found that the non-linear behavior of this material
subjected to Mode II delamination by bending depends mainly on the thickness of
the used test specimen. This is why, we propose in this study to highlight the
problems caused in the case of multidirectional laminates and to find out the
relationship which expresses this phenomenon. Then we propose an optimization
of the thickness, which would ovoid the energy dissipation phenomena before
starting of delamination.
1. Experim ental Analysis.
1.1. M a teria l a n d E x p e r im e n ta l C onditions. 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 300x300 mm. A Teflon film (30 mm in
thickness) was incorporated at the mid-plane to initiate delamination (Fig. 1). The
thickness of 5 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 [+0 / - 6 / + 6 / - 6 / - 6 / + 6 / - 6 / + 6 ]s were
selected with the angles: 6 = 15,30, 45 and 60°. Each specimen was instrumented
by 10 mm strain gauges and a sensor of acoustic emission (AE) (Fig. 1) and
loaded in displacement control at a constant rate 0.5 mm/min in order to better see
the occurrence of the damage initiation. The characteristics of the acoustic
emission system are the following:
Pre-amplifier gain: 39 dB.
Threshold setting: 39 dB for this study.
Filter pre-amplification: 45 dB.
Fig. 1. Instrumentation of the DCB specimen by strain gauges and acoustic emission sensors.
The AE sensor used is a large frequency band from 200 kHz to 1 MHz. The
mechanical characteristics of the different sequences are indicated in Table 1.
1.2. S ta r tin g o f D e la m in a tio n . The instrumentation of the specimen by
strain gauges and acoustic emission sensors (AE) are necessary to detect the
microscopic delamination. This last technique (AE) enables us to identify the
damage mechanisms which occur chronologically in the material. These results
have been highlighted by microscopic observation (SEM). This experimental
technique is described in [7, 9]. The crack development for each stacking
sequence were carried out in several stages. In each one, the mechanical load was
ISSN 0556-171X. Проблемы прочности, 2006, N2 6 67
(1-10) Strain gauges AE
A. Ahm ed Benyahia, A. Laksimi, N. Ouali, and Z. Azari
followed by microscopic observations to identify the type of damage. Figure 2
and 3 show an example of load-displacement and microscopic observation of
±15° specimen. We can see that delamination starts and propagates in the median
plane of the specimen. On the other hand, for the sequence ±60°, the initial
damage is a crack transverse of one of the two median plies (Figs. 4 and 5). Thus,
for the first three sequences (±15°, ±30°, and ±45°), delamination is the first type
of occurring damage. We can then easily detect, on the curve of load
displacement, the point of delamination starting and calculate the strain energy
release rate. On the other hand, in the case of the sequence ± 60°, the starting of
delamination is preceded by transverse cracks. Consequently, calculation of the
strain energy release rate will be inaccurate because the loss of energy dissipated
by transverse cracks. These results correspond to a 5 mm thickness of the
specimen. For a reliable characterization of delamination in any type of stacking
sequence, it is necessary to avoid the occurrence of transverse cracking before the
starting of delamination which can be obtained by increasing the flexional
stiffness, i.e., the increasing thickness of the specimen.
T a b l e 1
Mechanical Characteristics of the Different Sequences
Angle 0 15° 3 o o °54 6 o o
El , GPa 44.110 39.300 27.640 18.470 16.430
Et , GPa 17.980 17.380 16.430 18.470 27.640
Glt , GPa 6.160 8.000 11.670 13.510 11.670
0.267 0.367 0.512 0.490 0.310
1 2 3 4 5 6
D is p la c e m e n t (m m )
Fig. 2. Load and AE vs. displacement for ±15° sequence.
1.3. D e fo rm a tio n s in F ro n t o f th e C rack Tip. In order to better understand
the behavior ahead of the crack tip, specimens have been instrumented by several
strain gauges. The gauges have 10 mm length and are laid out with regular
intervals o f 1mm of separation in the longitudinal direction of the specimen
(Fig. 1). The response of these gauges is illustrated in Fig. 6a and 6b for the
sequences ±15° and ±45°, respectively. The analysis of the response of these
68 ISSN 0556-171X. npoôëeMbi npounocmu, 2006, N 6
M echanical Behavior and Optimization
strain gauges shows that before macroscopic starting (line B in dotted lines), only
the gauges of the first series (J1 and J2) are sensitive to the applied loading. The
strain gauges are more and more sensitive as soon as the delamination is about
10 mm from them. Focusing on these curves, this zone seems to be under
compression which is located just in front of the crack tip, and generally appears
on the external part of the arms of a specimen loaded in opening mode. This zone
moves as a wave, forward the arm of the specimen with a speed equal to the crack
tip propagation. The length o f this wave depends on the mechanical and
geometrical characteristics of the arm. This remark enables us to confirm that the
arms of DCB specimen are not embedded perfectly at the crack tip. The strain
energy release rate may be over-estimated, because it will contain an energy
dissipated in front of the crack tip. In order to eliminate this phenomenon, it is
necessary to minimize these strains in front of the crack tip by increasing the
specimen stiffness.
Fig. 3. Starting of damage by delamination in ±15° sequence.
50
0 1 2 3 4 * 5 6 7
D is p la c e m e n t (m m )
Fig. 4. Load and AE vs. displacement for ±60° sequence.
2. Thickness Optim ization of DCB Specimen. It has been noticed that the
strain energy release rate calculated experimentally is reliable if the specimen has
a sufficient stiffness permitting the delamination starting before any other type of
damage with a few possible strain in front of the crack tip. In the continuation of
this work, stiffness optimization were carried out by using an analytical approach.
ISSN 0556-171X. npodxeMbi npounocmu, 2006, N 6 69
A. Ahm ed Benyahia, A. Laksimi, N. Ouali, and Z. Azari
Fig. 5. Starting of damage by transverse cracking in ±60° sequence.
Displacement (mm)
a
Displacement (mm)
b
Fig. 6. Load and strains vs. displacement curves: ±15° (a) and ±45° (b).
2.1. D efo rm a tio n o f M a te r ia l in F ro n t o f th e C rack Tip. Several studies
were carried out in order to estimate the angle of deflection at the crack tip of the
arms of anisotropic material DCB [3-5]. They are based upon the beam theory of
the elastic Timoshenko concept. The half of DCB specimen can be schematized
by a semi-infinite beam, of which the part located in front of the crack tip is
sustained on an elastic foundation and is loaded at its free end (see Fig. 7).
70 ISSN 0556-171X. npodxeMbi npounocmu, 2006, N 6
M echanical Behavior and Optimization
an , Upper half of the DCB
'/ / / / / / / / / / / / / / / / A
p 0 > Elastic foundation
Fig. 7. Schematization of an arm of the DCB.
The deflection of the part, which is in contact with the foundation, can be
calculated by the following relationship:
For small displacements, the equation of the bent axis of a beam is given by
d 2 y
E J x x — T = ~ M z ,dz
(1)
by deriving the Eq. (1) twice, we obtain the distributed load
i4d y
e l i x^ ~ 4 = q ,
dz
(2)
q represents the reaction of the foundation; q = ky, where k is the modulus of
foundation and represents the reaction per unit of length when the strain is equal
to the unit. It is given by
E T b
O i l ' (3)
k =
Therefore
d 4 y
e t I x x — 4 = ky •
dz
(4)
Putting ß 4 =
4E l I xx
, the general solution takes the form
y = e ßz (A cos ß z + B sin ß z ) + e ßz (C cos ß z + D sin ß z )• (5)
k
The constants A, B , C , and D are given from the boundary conditions. It is
obvious to admit that the bending deformation and the moment of flexion equal
zero infinitely far away from the force P. This can be done by
y (z ) | ^ = e ^ (A cos $ z + B sin @z),
thus A = B = 0.
The expression of the deformation of the right part of the beam becomes then
y (z ) = e~@z (C cos j iz + D sin f i z ). (6)
ISSN 0556-171X. npoôëeMbi npounocmu, 2006, N9 6 71
A. Ahm ed Benyahia, A. Laksimi, N. Ouali, and Z. Azari
or
and
Constant C and D are determined by the origin conditions at z = 0
d 2 y
E l I xx- 7 t ( z = 0) = - M z = —Pa 0 ,
dz
d 3 y
E l I xx~7t ( z = 0) = —Fz = — P , dz3
d 2
— 2y = 2(5 2 e ~ ( (C sin ( z — D cos ( z ), dz2
d 3
— 3y = 2 ( 3 e —(z (C sin ( z — D cos ( z + C cos ( z + D sin ( z ).
d z 3
Thus
„ Pa 0 , ^ P ( f ia 0 - 1) D = — r-------- and C = ---- ;--------- .
2P 2E l I xx 2P 3E l I xx
The general equation becomes
-pzP e ~
y ( z ) = — 3-------- ( P a 0(cos P z + sin P z ) - cos P z ) (7)
2P E LI x1 xx
with
P = 4 ----------= 4
н \ 4 E l I xx 4
6E t
ELh 4
It should be noticed that through this formulation of the deformation, only
the thickness of the specimen arm (h) has an effect over the period of the
propagation wave of the strained zone ahead of the crack tip. The initial crack
length (a 0) has influence on the strain intensity of this zone.
In the case of this study, b = 20 mm and h = 2.5 mm, the values of k and P
are summarized in Table 2.
T a b l e 2
Values of k and P for the Different Sequences
Sequence 0 ±15° ±30° ±45° ±60°
k -105, N/mm2 2.876 2.780 2.629 2.955 4.422
P, mm-1 0.500 0.510 0.550 0.626 0.713
The final relationship of deformation of various stacking sequences are
shown in Table 3. The experimental loads P taken in the calculation of these
deformations correspond to the delamination starting for each sequence of
stratification, for a length of initial crack a 0 = 25 mm.
72 ISSN 0556-171X. Проблемы прочности, 2006, N 6
Mechanical Behavior and Optimization
T a b l e 3
Equations of the Deformations of the Various Sequences
Sequence P, N
(starting)
Equations of the deformation
±15° 63 y(z) = 2.32-10-4 e-051 z [12.75(cos0.51z+ sin0.51z) - cos0.51z ]
±30° 55 y(z) = 2.29 -10-4 e-a55 z [13.75(cos0.55z+ sin0.55z) - cos0.55z ]
±45° 50 y(z) = 2.118 -10-4 e-0626 z [15.65(cos0.626z+ sin0.626z) - cos0.626z ]
±60° 37 y(z) = 1.193-10-4 e-a713 z [17.825(cos0.7136z+ sin0.713z) - cos0.713z ]
The evolution of the deformations of the two halves of the DCB, ahead of
the crack tip, o f the various sequences can be represented, as an example by the
±15° sequence illustrated by Fig. 8. This representation assumes that the DCB
specimen is separated into two parts. This figure shows the existence of a strain
field located ahead of the crack tip. This can be explained by the fact that the
perfect embedding cannot set up at the crack tip but at a distance d (~ 10 mm)
which is composed by two quite distinct zones: a tension zone ahead of crack tip
followed by a compression zone. Although the stacking sequence has an
influence on the deformation and on the flexion angle, it does not affect the length
o f the distance d (Fig. 9). In addition, the thickness of the specimen has a
significant effect as well on the intensity of the deformations on their extent. This
remark is highlighted by the curves of Fig. 10, where for the same sequence of
stratification, the distance d = 10 mm for h = 2.5 mm increase to 20 mm for
h = 5 mm and to 40 mm for h = 10 mm. In fact, starting from this figure, it
should be noted that the surface ranging between the curve of deformation and
the axis z changes delimitations but guard the same surface when one increases
the thickness o f the specimen arms. The increasing of the arm thickness permits
to diminish the intensity of the deformation until avoiding any form of energy
dissipation in this zone.
0.003
0 2 4 6 8 10 12 14
L en g th ahead o f c rack tip [m m ]
Fig. 8. D eform ation o f the tw o arms o f ± 1 5 ° D C B specim en.
ISSN 0556-171X. n p o 6MeMbi nponnocmu, 2006, № 6 73
A. A hm ed Benyahia, A. Laksimi, N. Ouali, and Z. Azari
0.003 —
£ 0 .0 0 2 '
s
"S
£ 0 .0 0 1 ' 0)Q
3
1 ___ 1___ .. . .... ......1
0 10
Length ahead of crack tip [mm]
Fig. 9. Deformation for various stacking sequences: (1) ±15°, (2) ±30°, (3) ±45°, (4) ±60°
0.0030-
0.0025-
| 0.0020
s
'I 0.0015-
“ 0 .0010 -
0.0005 ■
0
\2
j \
i i ..... i ........ -....... ■
10 20 30
Length ahead of crack tip [mm]
40
Fig. 10. Thickness effect on deformation ahead the crack tip [h = 5 (1), 2.5 (2), and 10 mm (3)].
2.2. N o n -L in e a r ity E ffe c t. According to Carlsson’s studies [6], it seems
interesting to study the non-linearities of the DCB specimen behavior. These
non-linearities are due primarily to large displacements and micro-cracking by
flexion. They are shown by the maximum slope of deflection y 'm at the point of
the load application and by the maximum deformation e m corresponding to the
rupture by flexion. The non-linearities are satisfied for
y'm = y'a and e m ^ e a >
where y 'a and e a are the acceptable values.
Carlsson [6] proposed y'a = 1%, e a can be determined by the theory of the
first ply failure developed by TSAI. The calculation of these two parameters is
performed by considering the static diagram of a DCB arm (Fig. 11).
74 ISSN 0556-171X. n p o 6n e u u npouHocmu, 2006, № 6
M echanical Behavior and Optimization
Fig. 11. Static diagram of a DCB arm.
2.2.1. N on-L inearity D ue to Large D isp lacem ent. 8 m and y 'm are given
according to the beam theory by
5 m =
8P a :
E b h '
Thus
and y m
6P a ‘
Ebh 3 • (8)
= 3 aym ,
where E is the longitudinal modulus of elasticity of material, b is the width of
the arm, and h is its thickness. This yields:
(9)
Using Irwin-Kies approach to determine the energy release rate:
P 2 dCG = ______
Ic 2b da ’
where P is the applied load, b is the specimen width, and d C f da is the partial
derivative of the compliance with respect to the crack length. Knowing that
5 8a 3 dC 24a2
and — = (beam theory) for the two arms of the DCB,
P E b h 3 da E b h
provides the following relationship between the maximum displacement of the
arm and the energy release rate:
Thus, we can write
5 = 2 ^ _ G ic
m J s i E h 3 .
2a G Ic < 2 ,
ayc
E h 3 3
(10)
(11)
This last relationship allows us to determine the minimal thickness of the
specimen arm which makes the initiation of delamination without having a large
displacement state
ISSN 0556-171X. npoöneMU npouHocmu, 2006, № 6 75
A. Ahm ed Benyahia, A. Laksimi, N. Ouali, and Z. Azari
3a 2G Ic
h - j h y r • (12)
Noteworthy is that all the parameters, in this relationship, are intrinsic with
material, except the initial crack length a which remains an unspecified
geometrical characteristic and does not affect the value of h since it will be taken
constant in the case of a possible comparison between various stacking sequences.
2.2.2. N on-L inearity D ue to C racking by F lexion . The maximum strain by
flexion is given at the embedding of the specimen (crack tip) by
6Pa
E bh 2
it is related to maximum displacement by
3 h&.
(13)
^m 9 2 '
2 a
By associating the two equations (10) and (15), we obtain
(14)
3G Ic
~E hi . (15)
Thus, and in the same way as for the first condition of non-linearity, the
minimal thickness of the specimen arm ensuring initiation of delamination
without cracking by flexion can be given by
3G Ic
h - j T . (16)Ee a
For this second condition of non-linearity, all the parameters are intrinsic
with material.
2.2.3. A pplica tion o f This S tudy to Lam inates. The optimal values of the
thickness h of the specimen arm calculated according to the first and second
condition o f non-linearity are summarized in Table 4. They were calculated for
25 mm length of initial crack. The values of G ic and e a , has been already given
in [7].
Figure 12 shows the thickness evolution of the arms of the DCB specimen
necessary to avoid non-linearities before initiation of delamination according to
the angle ply. The analysis of these curves shows that for lower angles of plies
orientation (inferior to approximately 38°), the non-linearity for low thickness of
the specimen arm (approximately 1 mm) is given by large displacement. On the
other hand, for the angles higher than 38°, the non-linearity by transverse cracking
can affect the mechanical behavior. The analysis of these curves also shows that
e m
76 ISSN 0556-171X. npodxeMbi npounocmu, 2006, N 6
M echanical Behavior and Optimization
in this study (h = 2.5 mm), transverse cracking before the initiation of delamination
appears for plies orientation angle more than 47 °. This is in accordance with the
experimental results given above.
T a b l e 4
Optimal Thickness of the Specimen Arms for Various Sequences
Sequence e-10“3 Gtc, J/m2 , % h, mm
first condition second condition
0 24.55 185 1 0.923 0.021
±15° 9.95 185 1 0.965 0.145
±30° 6.20 185 1 1.088 0.535
±45° 4.00 185 1 1.236 1.980
±60° 2.53 185 1 1.283 5.277
0 10 20 30 40 50 60 70
Angle of stratification [deg]
Fig. 12. Thickness optimization for the DCB specimen arm.
Conclusions. The objective of this study was the optimization thickness of
the DCB specimen, which makes it possible to avoid all non-linearity, especially
by transverse cracking, before the initiation of delamination in the multidirectional
laminates. The experimental results, according to the response of the strain
gauges showed the existence of deformations ahead of crack tip, which explains
why embedding is not as perfect as it was assumed. They also showed by
microscopic observations, that the initiation of delamination in Mode I of
sequence for 2.5 mm arm thickness is always preceded by a cracking transverse
of the more strained one of the two arms of the DCB specimen. As for the
analytical results obtained, they are in good agreement with the experimental
results, and allowed us the determination of the minimal thickness of DCB
specimen, which satisfies the application the Linear Elastic Fracture Mechanics
(LEFM) concepts.
IS S N 0556-171X. npoôëeMbi npounocmu, 2006, N 6 77
A. Ahm ed Benyahia, A. Laksimi, N. Ouali, and Z. Azari
Р е з ю м е
При навантаженні згином багатовимірних ламінатів має місце деламінація.
Її опис є досить важким, оскільки у вістрі тріщини присутні великі дефор
мації та відмічаються нелінійні ефекти, що зумовлені значними переміщен
нями і утворенням поперечних мікротріщин. Останні з ’являються у най
більш навантажених шарах, особливо в тонких ламінатах. Експерименталь
но досліджено вищезгадані особливості на зразках у вигляді консольної
балки та виконано оптимізацію товщини багатошарового зразка, за якої
деламінація починаєтьсядо утворення поперечних мікротріщин у найбільш
навантажених шарах. Об’єктом дослідження служив склоепоксидний зміц
нений ламінат. Отримані результати оптимізації товщини зразка добре
узгоджуються з експериментальними даними.
1. C. E. Harris and D. H. Morris, A C haracterization o f the F racture B ehavior
o f Thick, N otched, L am ina ted G raphite/E poxy C om posites, VPI Institute and
State University, Blacksburg, Grant No. NASA-NAG-1-264 (1983).
2. J. G. Williams, M o d e I Transverse C racking in an E poxy a n d G raphite F iber
R ein forced E poxy, M.S. Thesis Texas A&M University (1981).
3. J. G. Williams, “End corrections for orthotropic DCB specimen,” Compos.
Sci. Technol., 35, 367-376 (1989).
4. F. E. Penado, “A closed form solution for the energy release rate of the
double cantilever beam specimen with an adhesive layer,” J. Compos.
M ater., 27, No. 4, 383-407 (1993).
5. R. Olsson, “A simplified improved beam analysis of the DCB specimen,”
Compos. Sci. Technol., 43, 329-338 (1992).
6. L. A. Carlsson, J. W. Gillespie, and R. B. Pipes, “On the analysis and design
of end notched flexure (ENF) specimen for Mode II testing,” J. Compos.
M ater., 20, 594-604 (1986).
7. A. Ahmed Benyahia, E tude des M écan ism es de D élam inage sous l 'E ffe t de
C ontraintes C om plexe G énérées p a r des So llic ita tions S im ples d 'O uverture
e t de C isa illem ent dans les S tra tifies ± в , Thesis, Université de Technologie
de Compiègne, No. 1040 (1997).
8. A. Ahmed Benyahia, A. Laksimi, S. Benmedakhene and X. L. Gong,
“Fracture process in ± в laminates subjected to Mode II loading,” Strength
M ater., 34, No. 4, 390-401 (2002).
9. A. Laksimi, A. Ahmed Benyahia, M. L. Benzeggagh, X. L. Gong, “Initiation
and bifurcation mechanisms of cracks in multi-directional laminates,” Compos.
Sci. Technol., 60, 597-604 (2000).
Received 01. 06. 2006
78 ISSN 0556-171X. П роблем и прочности, 2006, № 6
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