A Computational Strategy for the Localization and Fracture of Laminated Composites. Part 2. Life Prediction by Mesoscale Modeling for Composite Structures
Описанный в сообщении 1 одномерный подход развит на случай двухмерного ламинатного композита Т300/914, подвергнутого статическому двухосному растяжению и сдвигу. Решение данной задачи осуществляется с помощью эволюционных моделей с эффектом задержки повреждений при ограниченной скорости их накоплен...
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| Дата: | 2011 |
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| Формат: | Стаття |
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Інститут проблем міцності ім. Г.С. Писаренко НАН України
2011
|
| Назва видання: | Проблемы прочности |
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| Онлайн доступ: | http://dspace.nbuv.gov.ua/handle/123456789/95272 |
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| Цитувати: | A Computational Strategy for the Localization and Fracture of Laminated Composites. Part 2. Life Prediction by Mesoscale Modeling for Composite Structures / A. Boutaous, M. Elchikh, M. Abdelouahab, A. Belaidi // Проблемы прочности. — 2011. — № 6. — С. 83-93. — Бібліогр.: 9 назв. — англ. |
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irk-123456789-952722016-02-21T03:02:03Z A Computational Strategy for the Localization and Fracture of Laminated Composites. Part 2. Life Prediction by Mesoscale Modeling for Composite Structures Boutaous, A. Elchikh, M. Abdelouahab, M. Belaidi, A. Научно-технический раздел Описанный в сообщении 1 одномерный подход развит на случай двухмерного ламинатного композита Т300/914, подвергнутого статическому двухосному растяжению и сдвигу. Решение данной задачи осуществляется с помощью эволюционных моделей с эффектом задержки повреждений при ограниченной скорости их накопления. Размер зоны локализации повреждения в плоскости слоев ламината зависит от временной характеристики модели, связанной с задержкой повреждения и скоростью нагружения. Показано, что совместное использование мезомодели и эволюционных моделей задержки повреждений позволяет оценить размер зоны локализации повреждения и точно оценить время разрушения. Описаний в повідомленні 1 одновимірний підхід розвинуто на випадок двовимірного ламінатного композита T300/914, що зазнає статичного двовісного розтягання і зсуву. Розв’язок даної задачі виконується за допомогою еволюційних моделей з ефектом затримки пошкоджень за обмеженої швидкості їх накопичення. Розмір зони локалізації пошкодження у площині шарів ламіната залежить від часової характеристики моделі, яка пов’язана із затримкою пошкодження і швидкістю навантаження. Показано, що спільне використання мезомоделі й еволюційних моделей затримки пошкоджень дозволяє оцінити розмір зони локалізації пошкоджень і точно оцінити час руйнування. 2011 Article A Computational Strategy for the Localization and Fracture of Laminated Composites. Part 2. Life Prediction by Mesoscale Modeling for Composite Structures / A. Boutaous, M. Elchikh, M. Abdelouahab, A. Belaidi // Проблемы прочности. — 2011. — № 6. — С. 83-93. — Бібліогр.: 9 назв. — англ. 0556-171X http://dspace.nbuv.gov.ua/handle/123456789/95272 539.4 en Проблемы прочности Інститут проблем міцності ім. Г.С. Писаренко НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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DSpace DC |
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English |
| topic |
Научно-технический раздел Научно-технический раздел |
| spellingShingle |
Научно-технический раздел Научно-технический раздел Boutaous, A. Elchikh, M. Abdelouahab, M. Belaidi, A. A Computational Strategy for the Localization and Fracture of Laminated Composites. Part 2. Life Prediction by Mesoscale Modeling for Composite Structures Проблемы прочности |
| description |
Описанный в сообщении 1 одномерный подход развит на случай двухмерного ламинатного композита Т300/914, подвергнутого статическому двухосному растяжению и сдвигу. Решение данной задачи осуществляется с помощью эволюционных моделей с эффектом задержки
повреждений при ограниченной скорости их накопления. Размер зоны локализации повреждения в плоскости слоев ламината зависит от временной характеристики модели, связанной с задержкой повреждения и скоростью нагружения. Показано, что совместное использование мезомодели и эволюционных моделей задержки повреждений позволяет оценить размер зоны локализации повреждения и точно оценить время разрушения. |
| format |
Article |
| author |
Boutaous, A. Elchikh, M. Abdelouahab, M. Belaidi, A. |
| author_facet |
Boutaous, A. Elchikh, M. Abdelouahab, M. Belaidi, A. |
| author_sort |
Boutaous, A. |
| title |
A Computational Strategy for the Localization and Fracture of Laminated Composites. Part 2. Life Prediction by Mesoscale Modeling for Composite Structures |
| title_short |
A Computational Strategy for the Localization and Fracture of Laminated Composites. Part 2. Life Prediction by Mesoscale Modeling for Composite Structures |
| title_full |
A Computational Strategy for the Localization and Fracture of Laminated Composites. Part 2. Life Prediction by Mesoscale Modeling for Composite Structures |
| title_fullStr |
A Computational Strategy for the Localization and Fracture of Laminated Composites. Part 2. Life Prediction by Mesoscale Modeling for Composite Structures |
| title_full_unstemmed |
A Computational Strategy for the Localization and Fracture of Laminated Composites. Part 2. Life Prediction by Mesoscale Modeling for Composite Structures |
| title_sort |
computational strategy for the localization and fracture of laminated composites. part 2. life prediction by mesoscale modeling for composite structures |
| publisher |
Інститут проблем міцності ім. Г.С. Писаренко НАН України |
| publishDate |
2011 |
| topic_facet |
Научно-технический раздел |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/95272 |
| citation_txt |
A Computational Strategy for the Localization and Fracture of Laminated Composites. Part 2. Life Prediction by Mesoscale Modeling for Composite Structures / A. Boutaous, M. Elchikh, M. Abdelouahab, A. Belaidi // Проблемы прочности. — 2011. — № 6. — С. 83-93. — Бібліогр.: 9 назв. — англ. |
| series |
Проблемы прочности |
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2025-07-07T02:03:59Z |
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2025-07-07T02:03:59Z |
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| fulltext |
UDC 539.4
A Computational Strategy for the Localization and Fracture of Laminated
Composites. Part 2. Life Prediction by Mesoscale Modeling for Composite
Structures
A. Boutaous,
a
M. Elchikh,
a
M. Abdelouahab,
a
and A. Belaidi
b
a U.S.T., Oran, Algeria
b E.N.S.E.T., Oran, Algeria
ÓÄÊ 539.4
Âû÷èñëèòåëüíàÿ ìåòîäèêà ëîêàëèçàöèè ðàçðóøåíèÿ ëàìèíàòîâ.
Ñîîáùåíèå 2. Ðàñ÷åò äîëãîâå÷íîñòè êîìïîçèòíûõ êîíñòðóêöèé ñ
ïîìîùüþ ìåçîìàñøòàáíîãî ìîäåëèðîâàíèÿ
À. Áóòàó
à
, Ì. Ýëü÷èõ
à
, Ì. Àáäåëóõàá
à
, À. Áåëàéäè
á
à Óíèâåðñèòåò íàóêè è òåõíîëîãèè, Îðàí, Àëæèð
á Âûñøàÿ òåõíîëîãè÷åñêàÿ øêîëà, Îðàí, Àëæèð
Îïèñàííûé â ñîîáùåíèè 1 îäíîìåðíûé ïîäõîä ðàçâèò íà ñëó÷àé äâóõìåðíîãî ëàìèíàòíîãî
êîìïîçèòà Ò300/914, ïîäâåðãíóòîãî ñòàòè÷åñêîìó äâóõîñíîìó ðàñòÿæåíèþ è ñäâèãó. Ðåøå-
íèå äàííîé çàäà÷è îñóùåñòâëÿåòñÿ ñ ïîìîùüþ ýâîëþöèîííûõ ìîäåëåé ñ ýôôåêòîì çàäåðæêè
ïîâðåæäåíèé ïðè îãðàíè÷åííîé ñêîðîñòè èõ íàêîïëåíèÿ. Ðàçìåð çîíû ëîêàëèçàöèè ïîâðåæ-
äåíèÿ â ïëîñêîñòè ñëîåâ ëàìèíàòà çàâèñèò îò âðåìåííîé õàðàêòåðèñòèêè ìîäåëè, ñâÿçàííîé
ñ çàäåðæêîé ïîâðåæäåíèÿ è ñêîðîñòüþ íàãðóæåíèÿ. Ïîêàçàíî, ÷òî ñîâìåñòíîå èñïîëü-
çîâàíèå ìåçîìîäåëè è ýâîëþöèîííûõ ìîäåëåé çàäåðæêè ïîâðåæäåíèé ïîçâîëÿåò îöåíèòü
ðàçìåð çîíû ëîêàëèçàöèè ïîâðåæäåíèÿ è òî÷íî îöåíèòü âðåìÿ ðàçðóøåíèÿ.
Êëþ÷åâûå ñëîâà: ïðîãíîç, ëîêàëèçàöèÿ, çàäåðæêà, ðàçðóøåíèå, ïîâðåæäåíèå,
ìåçîìîäåëèðîâàíèå, êîìïîçèò.
Introduction. We use the mesoscale modeling developed by [1–3], which has
length dimensions related to internal characteristics of composites. The model
naturally introduces the thickness of the basic folds, thereby establishing a length.
In the plane of each layer, the model with delay effect allows one to fix the missing
dimensions. The basic idea derives from the theoretical equations of the mesoscale
modeling, particularly, the properties of the evolution model with delay effect
presented on the example of a beam in tension. We show that the evolution laws
for damage delay allow one to define a rupture zone. This property essential to the
simulation phase of fracture is highlighted in the case of a beam in tension with
several defects of rigidity.
1. New Approach for the Mesoscale Modeling of Layered Structures. In
the models introduced for the study of layered structures [4, 5], the first aspect of
this approach is the use of an intermediate scale between the microscale associated
with the basic constituents of the composite (fiber, matrix) and the macroscale due
© A. BOUTAOUS, M. ELCHIKH, M. ABDELOUAHAB, A. BELAIDI, 2011
ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2011, ¹ 6 83
to the structure. At this scale, called mesoscale, a layered structure is represented
by a stack of uniform in the layer thickness and interlaminar interfaces; the layer
and the interface are the two entities called meso-constituents.
1.1. Study of the Localization of the Mesoscale Model. The mesoscale
modeling may be regarded as a localization limiter. It is developed for layered
structures and allows one to set the localization zone thickness in each ply [6], to
determine the thickness of the zone where in each ply the mesoscale model is
constructed by imposing the damage variable to be uniform by thickness. This
property of mesoscale model has an essential role in the simulation problems of
fracture [7, 8]. The localization area in the plane of the layers is determined by
using the model with delay effect.
1.2. Contribution of the Model with Delay. The structure is subjected to bulk
forces f d and surface Fd in part � 2� of the border. On the complementary part
�1� is imposed a displacement U d . The equations characterizing the mechanical
problem are written as
div( � )
�
,� � �
� �
f d 0 (1)
� �
� �
u U d� �1�, (2)
�
��
� �
n Fd� � 2�, (3)
� ( )� � ,� � �� � �E d dEe e1 (4)
� ( , ).d f de� � (5)
The above five equations represent, respectively, the equilibrium, the conditions
and the forces imposed on the structure, the relationship between stress and strain
and the model of damage evolution with delay effect. We seek a point at which
there are two different solutions that satisfy the rate problem defined by the
previous equations for a given state,
div( � ) , � �
�
0 (6)
� �
�u� 0 �1�, (7)
��
� �
n� 0 � 2�, (8)
� ( ) � .� �� �E d e1 (9)
�X represents the difference between two solutions in rate with the appropriate
equilibrium, behavior and boundary conditions.
A. Boutaous, M. Elchikh, M. Abdelouahab, and A. Belaidi
84 ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2011, ¹ 6
The principle of virtual power leads to
�
� � ,� ��
0 (10)
where �� and �� represent the difference between two solutions in stress rate
and strain rate. The difference between two solutions in stresses shows no
discontinuity of the velocity of damage because the rate is a function of state
� ( ) �.� �� �E d1 (11)
Just as � ( ) � � ,� � �� � �E d1 0 with ���0 for two distinct solutions exist
( ) � � ,1 0� �
d
�
� � ���0 � �d 1. (12)
The limit point corresponds to the loss of uniqueness, it occurs at the breaking
point (d �1).
2. Illustration of Uniqueness in Two-Dimensional Case. The degradation of
the transverse and shear moduli is taken into account. To illustrate the uniqueness in
the two-dimensional case, we assume a structure whose variables d ,
d , � , and � e
are set at time t and we seek at t dt� a point from which there is a discontinuity
of the solution velocity to find out if patterns of localization exist.
Model in the transverse direction to fiber
�d k F dd
n
� � � ��2
2 if d �1 and Y Yd s
� otherwise d �1, (13)
with: F Y
Y Y
Y
d d
d
c
( )�
�
�
0
1 otherwise F Yd d
( ) .�1 (14)
Model along the shear plane
�
� � �
�
�d k F dd
n
4
4 if
�d 1 and Y Yd s
� otherwise
�d 1, (15)
with: F Y
Y Y
Y
d d
d
c
�
�
�( )
0
1 otherwise F Yd d
�( ) .1 (16)
For this type of loading layer, the relations between stress and strain rates are
written explicitly because there is no nonlinearity in the direction of fibers in
tension. The velocity relation is obtained from the strain energy.
The velocity relations between strains ans stresses can be written as
�
( )
�
( )
(
�
� �
�
�
� �11
1
0
12 21
11
12 2
0
12 211 1
1
1 1
�
� �
�
�
�
E
d
E de
�
�
� �
�
d d
de
e
)
�
( ( ))
��
� � �
� �22
12 21 11
12 211 1
A Computational Strategy for the Localization and Fracture ...
ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2011, ¹ 6 85
�
� �
�
�
�
� �
� �
� �
�
12 2
0
22
12 21
12 2
0
22
1
1 1
1
1
E
d
d
E de e
( )
�
( )
( 2 21
21� ( ))
� ,
�
d
d (17)
�
( )
( )
�
( )
�
�
� �
�
�22
12 2
0
12 21
11
2
0
1
1
1 1
1
1
�
�
� �
�
�
�
E d
d
E de
2 21
22
12 2
0
11
12 21
21
1
1 1�
�
� �
� �( )
�
( )
( ( ))
�
�
�
�
� �
d
E d
d
e
e
�d
�
� �
�
�
�
� �
� �
� �
�
12 2
0
11
12 21
12 2
0
22
1
1 1
1
1
E
d
d
E de e
( )
�
( )
( 2 21
21� ( ))
� ,
�
d
d (18)
� ( )� � ,� � �12 12
0
12 12
0
122 1 2� � �G d G de e
(19)
�
( )
�
( )
�
� �
�
�
� �11
1
0
12 21
11
12 2
0
12 211 1
1
1
�
� �
�
�
�
E
d
E de
( )
� ,
1 22�
d
e � (20)
�
( )
( )
�
( )
�
�
� �
�22
12 2
0
12 21
11
2
01
1 1
1
1
�
�
� �
�
�
�
E d
d
E de
� �
�
12 21
221( )
� ,
�
d
e (21)
� ( ) � ,� �12 12
0
122 1� �G d e
(22)
�
�
�
[ ]
�
�
�
�
�
�
�
�
�
��
��
��
���
�
�
�
�
�
�
�
�
�
�
�
�
�
K
e
e
e
22
12
�
�
�
�
�
�
�
.
We seek a point at which there are two different solutions which satisfy the
velocity problem set from a given state
div( � ) , � �
�
0 (23)
� �
�u� 0 �1�, (24)
��
� �
n� 0 � 2�, (25)
� [ ] � .� �� K e
(26)
The two solutions must satisfy the same boundary conditions, the principle of
virtual power then leads to the following form:
� �� �T
D
�
0 with � [ ] � .� �� K e
(27)
86 ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2011, ¹ 6
A. Boutaous, M. Elchikh, M. Abdelouahab, and A. Belaidi
A sufficient condition for the problem to admit two distinct solutions is
� �� �T
D
K �
0 with � ,��0 (28)
�
��
�
�
�
�
�
�
��det ,
K K T
2
0
where K is symmetrical,
� �det( ) .K 0 (29)
This problem has a unique solution for the loss of uniqueness, which is reached for
values of the following constant damage:
� � �d
E
E
1 0
1
0
12 2
0
12� �
or d �1. (30)
The first is, a priori, negative, the second coincides with the transverse fracture.
2.1. Development and Sensitivity Analysis of a Beam with a Single Defect. A
beam [ ]90 n T300/914 composed of two parts (one has the Young modulus slightly
lower in order to force is the localization) is subject to a state of pure tension
(Fig. 1). The inequality of moduli within the two parts results in the lack of rigidity
of the beam.
This example shows the influence of the defect on the rupture zone where the
damage model with delay is used. The localization of strain and fracture occurs in
the area where the stiffness is the lowest. The proposed criterion of instability and
localization can be applied for assessment of the deformation value for which the
localization occurs causing the rupture. The inelastic strains are neglected in this
problem.
The equations solved using a program in C++ can be written as:
imposed displacement:
l ati i
i
� �
�
� ,
,1 2
(31)
ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2011, ¹ 6 87
A Computational Strategy for the Localization and Fracture ...
Fig. 1. Beam in tension-imposed displacement.
equilibrium:
E d E d1 1 1 2 2 21 1( ) ( ) ,� � �� � (32)
behavior:
� ( )d k F Y di di d i
n
� �
�
i�1 2, , (33)
with
F Y
Y Y
Y
d d
d
c
i
( )�
�
�
0
1 otherwise F Yd di
( ) ,�1 (34)
where k and n are the material constants. In this study, the parameters of the
model with delay effect are: a�0 01. mm/s, k �100, and n�0 5. .
The criterion of instability and localization is written for both parts of the
beam:
d
f
f
d
i i
e
e
0 01� � �
�
��
�
�
, we set, g
f
f
d
i i
e
i
e
e
( ) ,� �
�
��
�
�
0 01� � (35)
with: f k F Y ddi d i
n
� �
�
, i�1 2, . (36)
2.1.1. Results and Discussion. The localization occurs when the damage and
strain of the bar satisfy this criterion. In the present case, the localization and
fracture zones are located in element 1 (Fig. 2). We show in this example that the
damage is localized in the weakest element of rigidity. The intersection of two
curves representing the criterion of instability and damage as a function of strain is
the point from which there is localization (Fig. 3). It shows the strain and the value
of damage for which the localization occurs. Subsequently, the example of a beam
composed of several defects highlights the existence of a fracture zone which may
consist of several elements. It is impossible to obtain such a phenomenon with a
classical model.
88 ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2011, ¹ 6
A. Boutaous, M. Elchikh, M. Abdelouahab, and A. Belaidi
Fig. 2. Damage vs. time in the beam, localization in element 1.
2.2. Development and Sensitivity Analysis of a Bar with Several Defects.
The case of beam [ ]90 n with several defects is considered in this part (Fig. 4). The
inelastic strains are neglected, and at given loading rate the localization occurs in
the area of the defect size order. For sufficiently high loading rate the localization
also occurs in the same area if it consists of several elements. In both cases, the
curves representing the load according to the displacement are identical. This
example makes it possible to show that, in the case of model with delay efect, area
and localization of fracture are determined by the defects and the effect of damage
accumulation rate model. The criterion of instability and localization confirms the
existence of the fracture zone. The equations solved can be written as:
imposed displacement:
l ati i
i
� �
�
� ,
,1 2
(37)
equilibrium:
E d E di i i j j j( ) ( )1 1� � �� � with ( , )i j� �1 2 and ( , ),i j� �2 3 (38)
behavior:
� ( ) .d k F Y di di d i
n
� �
�
(39)
A Computational Strategy for the Localization and Fracture ...
ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2011, ¹ 6 89
Fig. 3. Damage vs. deformation and instability criterion.
Fig. 4. Beam in tension-imposed displacement.
The criterion of instability and localization is written for all three beam parts:
d
f
f
d
i i
e
e
0 01� � �
�
��
�
�
, with f k F Y ddi d i
n
� �
�
, i�1 2 3, , , (40)
where k and n are the materials constants. For further calculations we take
k �1000 and n�1, the length of the bar is 1 mm.
2.2.1. Results and Discussion. For the loading rate (a�1 mm/s) and the
parameters of the chosen delay model the localization appears in part l1 (Fig. 5).
The localization criteria are satisfied in the three elements for three different strains.
Consequently, the damage evolves differently in each part after the point defined
by the first criterion of instability. The damage is localized in the weakest element
of rigidity and attains two distinct values of damage in the other two elements.
For the loading rate (a�25), the localization occurs in the first two elements
(Fig. 6). The localization length is l l1 2� . The localization criteria are verified in
the three elements for two distinct strains. Thus, the damage evolves identically in
the two elements whose rigidities are lower. A length scale l l1 2� is then associated
with the phenomenon of fracture of the beam. For the loading rate a�50 mm/s the
localization occurs across the entire beam (Fig. 7). In this case, the three localization
criteria determined in each part of the beam are similar and show the same value of
damage at the same time.
Finally, we consider a beam similar to the previous one except that its weakest
part is cut into three
�l1 0 2. ,
�l2 0 2. , and
�l3 0 2. , with the identical values of the
Young modulus (Fig. 8). This shows the independence of the size of the fracture
with respect to the beam discretization. This example shows that for a small
perturbation of the Young modulus in the entire part I, the localization occurs
throughout the zone length l1 for the rate close to the previous one (a�1 mm/s).
For lower loading rate localization occurs again in the weakest link l1 (a�01. mm/s).
90 ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2011, ¹ 6
A. Boutaous, M. Elchikh, M. Abdelouahab, and A. Belaidi
Fig. 5. Damage and criterion with vs. time, localization in the element of length l1 (a �1 mm/s).
Starting from a certain value of loading rate of (a�15. mm/s), the load–
displacement curves obtained with discretization (three fields for l1) are identical
to those obtained with the previous discretization (one domain for l1). This
illustrates that for given loading rate the zone where fracture occurs is independent
of the beam discretization (Fig. 9).
ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2011, ¹ 6 91
A Computational Strategy for the Localization and Fracture ...
Fig. 6. Damage and criterion vs. time, localization in the elements of length l1 and l2 (a � 25 mm/s).
Fig. 7. Damage for all elements and criteria vs. time (a � 50 mm/s).
Fig. 8. Beam in tension-imposed displacement.
Conclusions. The analysis shows a dependence of the size of the localization
area with respect to the loading rate. The model with delay effect envisages
formation of a fracture zone, which makes results independent of the discretization
considered. For the usual loading rate, the size of the fracture zone is small. To
emphasize the existence of the fracture zone in several elements, the mesh must be
sufficiently fine. During the phase of localization and fracture, the velocity
variations are significant and the inclusion of inertia terms in the equations of
equilibrium reduces the influence of loading rate on the fracture zone.
This study has shown the contribution of regulating the messcale model
coupled to the model with delay effect. The model with delay effect has a
characteristic time which, combined with the loading rate, implies formation of
fracture zone in each of the layers constituting the laminated structure. The
thickness of the fracture zone in each layer is determined by the thickness of the
fold. The calculation of structures using mesoscale modeling are therefore unbiased
and unique, and combined with the delay model application can provide assessment
up to fracture. The prospects of this study is to identify the size of the fracture zone
and take into account the inertia terms in the fracture phase, in order to better
represent the physical phenomena.
Ð å ç þ ì å
Îïèñàíèé â ïîâ³äîìëåíí³ 1 îäíîâèì³ðíèé ï³äõ³ä ðîçâèíóòî íà âèïàäîê äâî-
âèì³ðíîãî ëàì³íàòíîãî êîìïîçèòà T300/914, ùî çàçíຠñòàòè÷íîãî äâîâ³ñíîãî
ðîçòÿãàííÿ ³ çñóâó. Ðîçâ’ÿçîê äàíî¿ çàäà÷³ âèêîíóºòüñÿ çà äîïîìîãîþ åâîëþ-
ö³éíèõ ìîäåëåé ç åôåêòîì çàòðèìêè ïîøêîäæåíü çà îáìåæåíî¿ øâèäêîñò³ ¿õ
íàêîïè÷åííÿ. Ðîçì³ð çîíè ëîêàë³çàö³¿ ïîøêîäæåííÿ ó ïëîùèí³ øàð³â ëàì³-
íàòà çàëåæèòü â³ä ÷àñîâî¿ õàðàêòåðèñòèêè ìîäåë³, ÿêà ïîâ’ÿçàíà ³ç çàòðèìêîþ
ïîøêîäæåííÿ ³ øâèäê³ñòþ íàâàíòàæåííÿ. Ïîêàçàíî, ùî ñï³ëüíå âèêîðèñòàí-
íÿ ìåçîìîäåë³ é åâîëþö³éíèõ ìîäåëåé çàòðèìêè ïîøêîäæåíü äîçâîëÿº îö³íè-
òè ðîçì³ð çîíè ëîêàë³çàö³¿ ïîøêîäæåíü ³ òî÷íî îö³íèòè ÷àñ ðóéíóâàííÿ.
92 ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2011, ¹ 6
A. Boutaous, M. Elchikh, M. Abdelouahab, and A. Belaidi
Fig. 9. Damage throughout the first three elements and criteria vs. time (a �15. mm/s).
1. L. Gornet, D. Lévêque, et L. Perret, “Modélisation, identification et simulation
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3. P. Ladevèze and G. Lubineau, “An enhanced mesomodel for laminates based
on micromechanics,” Compos. Sci. Tech., 62, 533–541 (2002).
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Received 09. 05. 2010
ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2011, ¹ 6 93
A Computational Strategy for the Localization and Fracture ...
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/SUO <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>
/SVE <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>
/ENU (Use these settings to create Adobe PDF documents for quality printing on desktop printers and proofers. Created PDF documents can be opened with Acrobat and Adobe Reader 5.0 and later.)
>>
/Namespace [
(Adobe)
(Common)
(1.0)
]
/OtherNamespaces [
<<
/AsReaderSpreads false
/CropImagesToFrames true
/ErrorControl /WarnAndContinue
/FlattenerIgnoreSpreadOverrides false
/IncludeGuidesGrids false
/IncludeNonPrinting false
/IncludeSlug false
/Namespace [
(Adobe)
(InDesign)
(4.0)
]
/OmitPlacedBitmaps false
/OmitPlacedEPS false
/OmitPlacedPDF false
/SimulateOverprint /Legacy
>>
<<
/AddBleedMarks false
/AddColorBars false
/AddCropMarks false
/AddPageInfo false
/AddRegMarks false
/ConvertColors /NoConversion
/DestinationProfileName ()
/DestinationProfileSelector /NA
/Downsample16BitImages true
/FlattenerPreset <<
/PresetSelector /MediumResolution
>>
/FormElements false
/GenerateStructure true
/IncludeBookmarks false
/IncludeHyperlinks false
/IncludeInteractive false
/IncludeLayers false
/IncludeProfiles true
/MultimediaHandling /UseObjectSettings
/Namespace [
(Adobe)
(CreativeSuite)
(2.0)
]
/PDFXOutputIntentProfileSelector /NA
/PreserveEditing true
/UntaggedCMYKHandling /LeaveUntagged
/UntaggedRGBHandling /LeaveUntagged
/UseDocumentBleed false
>>
]
>> setdistillerparams
<<
/HWResolution [2400 2400]
/PageSize [612.000 792.000]
>> setpagedevice
|