A Computational Strategy for the Localization and Fracture of Laminated Composites. Part 1. Development of a Localization Criterion Adapted to Model Damage Evolution Time-Delay

A Computational Strategy for the Localization and Fracture of Laminated Composites. Part 1. Development of a Localization Criterion Adapted to Model Damage Evolution Time-Delay

Предложен критерий нестабильности и локализации повреждений в балке из однородного ламината T300/914 для моделирования развития повреждений с учетом эффекта задержки. Результаты, полученные для одномерного случая, свидетельствуют об одновременном появлении зоны разрушения по всей конструкции. Предло...

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Datum:2011
Hauptverfasser: Boutaous, A., Elchikh, M., Abdelouahab, M., Belaidi, A.
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Veröffentlicht: Інститут проблем міцності ім. Г.С. Писаренко НАН України 2011
Schriftenreihe:Проблемы прочности
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Zitieren:A Computational Strategy for the Localization and Fracture of Laminated Composites. Part 1. Development of a Localization Criterion Adapted to Model Damage Evolution Time-Delay / A. Boutaous, M. Elchikh, M. Abdelouahab, A. Belaidi // Проблемы прочности. — 2011. — № 5. — С. 50-58. — Бібліогр.: 18 назв. — англ.

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spelling irk-123456789-952062016-02-18T03:02:00Z A Computational Strategy for the Localization and Fracture of Laminated Composites. Part 1. Development of a Localization Criterion Adapted to Model Damage Evolution Time-Delay Boutaous, A. Elchikh, M. Abdelouahab, M. Belaidi, A. Научно-технический раздел Предложен критерий нестабильности и локализации повреждений в балке из однородного ламината T300/914 для моделирования развития повреждений с учетом эффекта задержки. Результаты, полученные для одномерного случая, свидетельствуют об одновременном появлении зоны разрушения по всей конструкции. Предложено решение, которое базируется на мезомоделировании композитов. Полученные с помощью предложенного подхода расчетные результаты позволяют выполнить точный прогноз потери устойчивости образца при ухудшении параметров его жесткости. Запропоновано критерій нестабільності і локалізації пошкоджень у балці з однорідного ламіната Т300/914 для моделювання розвитку пошкоджень з урахуванням ефекту затримки. Результати, що отримані для одновимірного випадку, свідчать про те, що зона руйнування по всій конструкції виникає одночасно. Запропоновано розв’язок, що базується на мезомоделюванні композитів. Отримані за допомогою запропонованого підходу розрахункові дані дозволяють виконати точний прогноз втрати стійкості зразка при погіршанні параметрів його жорсткості. 2011 Article A Computational Strategy for the Localization and Fracture of Laminated Composites. Part 1. Development of a Localization Criterion Adapted to Model Damage Evolution Time-Delay / A. Boutaous, M. Elchikh, M. Abdelouahab, A. Belaidi // Проблемы прочности. — 2011. — № 5. — С. 50-58. — Бібліогр.: 18 назв. — англ. 0556-171X http://dspace.nbuv.gov.ua/handle/123456789/95206 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
collection DSpace DC
language 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 1. Development of a Localization Criterion Adapted to Model Damage Evolution Time-Delay
Проблемы прочности
description Предложен критерий нестабильности и локализации повреждений в балке из однородного ламината T300/914 для моделирования развития повреждений с учетом эффекта задержки. Результаты, полученные для одномерного случая, свидетельствуют об одновременном появлении зоны разрушения по всей конструкции. Предложено решение, которое базируется на мезомоделировании композитов. Полученные с помощью предложенного подхода расчетные результаты позволяют выполнить точный прогноз потери устойчивости образца при ухудшении параметров его жесткости.
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 1. Development of a Localization Criterion Adapted to Model Damage Evolution Time-Delay
title_short A Computational Strategy for the Localization and Fracture of Laminated Composites. Part 1. Development of a Localization Criterion Adapted to Model Damage Evolution Time-Delay
title_full A Computational Strategy for the Localization and Fracture of Laminated Composites. Part 1. Development of a Localization Criterion Adapted to Model Damage Evolution Time-Delay
title_fullStr A Computational Strategy for the Localization and Fracture of Laminated Composites. Part 1. Development of a Localization Criterion Adapted to Model Damage Evolution Time-Delay
title_full_unstemmed A Computational Strategy for the Localization and Fracture of Laminated Composites. Part 1. Development of a Localization Criterion Adapted to Model Damage Evolution Time-Delay
title_sort computational strategy for the localization and fracture of laminated composites. part 1. development of a localization criterion adapted to model damage evolution time-delay
publisher Інститут проблем міцності ім. Г.С. Писаренко НАН України
publishDate 2011
topic_facet Научно-технический раздел
url http://dspace.nbuv.gov.ua/handle/123456789/95206
citation_txt A Computational Strategy for the Localization and Fracture of Laminated Composites. Part 1. Development of a Localization Criterion Adapted to Model Damage Evolution Time-Delay / A. Boutaous, M. Elchikh, M. Abdelouahab, A. Belaidi // Проблемы прочности. — 2011. — № 5. — С. 50-58. — Бібліогр.: 18 назв. — англ.
series Проблемы прочности
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fulltext UDC 539.4 A Computational Strategy for the Localization and Fracture of Laminated Composites. Part 1. Development of a Localization Criterion Adapted to Model Damage Evolution Time-Delay 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 1. Ðàçðàáîòêà êðèòåðèÿ ëîêàëüíîãî ïîâðåæäåíèÿ äëÿ ìîäåëèðîâàíèÿ ðàçâèòèÿ ïîâðåæäåíèÿ ñ ó÷åòîì ýôôåêòà çàäåðæêè 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 Ïðåäëîæåí êðèòåðèé íåñòàáèëüíîñòè è ëîêàëèçàöèè ïîâðåæäåíèé â áàëêå èç îäíîðîäíîãî ëàìèíàòà T300/914 äëÿ ìîäåëèðîâàíèÿ ðàçâèòèÿ ïîâðåæäåíèé ñ ó÷åòîì ýôôåêòà çàäåðæêè. Ðåçóëüòàòû, ïîëó÷åííûå äëÿ îäíîìåðíîãî ñëó÷àÿ, ñâèäåòåëüñòâóþò îá îäíîâðåìåííîì ïîÿâ- ëåíèè çîíû ðàçðóøåíèÿ ïî âñåé êîíñòðóêöèè. Ïðåäëîæåíî ðåøåíèå, êîòîðîå áàçèðóåòñÿ íà ìåçîìîäåëèðîâàíèè êîìïîçèòîâ. Ïîëó÷åííûå ñ ïîìîùüþ ïðåäëîæåííîãî ïîäõîäà ðàñ÷åòíûå ðåçóëüòàòû ïîçâîëÿþò âûïîëíèòü òî÷íûé ïðîãíîç ïîòåðè óñòîé÷èâîñòè îáðàçöà ïðè óõóä- øåíèè ïàðàìåòðîâ åãî æåñòêîñòè. Êëþ÷åâûå ñëîâà: ïðîãíîçèðîâàíèå, ëîêàëèçàöèÿ, ýôôåêò çàäåðæêè, ïîâðåæ- äåíèå, ìåçîìàñøòàáíîå ìîäåëèðîâàíèå, ÷èñëåííûé ðàñ÷åò, êîìïîçèòû. Introduction. From a theoretical point of view, simulation of localization and fracture of layered structures can be described in numerical specific criteria [1–3]. The study of stability and uniqueness of the solution for structures governed by the laws of elasto-plastic behavior has been studied by [4]. Following these works and those of [5], a first possibility has been to establish criteria of uniqueness and localization in order to define a field of use of behavior laws. They have been developed for the non-associated elasto-plastic laws [6], aimed to obtain a criteria of a possible onset of microcracks. For composite laminates, the ultimate point of rupture is usually close to the point determined using the criteria of uniqueness, this criterion corresponds to the point of instability and can be the endpoint from which there is a potential bifurcation of the solution [7]. The localization criterion [8] allows one to determine, for materials governed by the time independent behavior laws, the point at which localized solutions appear. We note the absence of internal length in these © A. BOUTAOUS, M. ELCHIKH, M. ABDELOUAHAB, A. BELAIDI, 2011 50 ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2011, ¹ 5 models from the classical mechanics of damage and can not properly simulate the phase fracture [1]. We show in this first part, that evolution law for damage delay allows one to define a rupture zone, this property is essential to the simulation phase of fracture and is highlighted in the case of a beam in tension. 1. Bibliographic Aspect of Localization Limiters. Extending the calculations beyond the critical points defined by conventional criteria creates difficulties caused by the loss of uniqueness and localization of strain. To follow a solution beyond the critical points, one essential step is to introduce localization limiters. The laws of behavior called special (limiting localization) can lead the calculations until failure. With these type of laws, it is possible to obtain a fracture zone in the framework of damage mechanics. One approach to limit the phenomenon localization is to use a non-local model [9], whether in damage, in velocity localization damage or in dissipation. Another approach to regularization is to use a mechanical formula taking into account the higher order terms in the calculation of strains. In the one-dimensional case [6], taking into account the second order term in the strain, allows one to regularize the original problem. The theory of second gradient is also used as a localization limiter [10]. The adjustment can also be obtained by using a model in which the characteristic lengths are used [11]. This method is used by authors [12] for behavior laws of elasto-plastic softening. All these methods of adjustments compensate for the lack of models by introducing local classical computations in terms of length. These lengths must control the phenomenon of localization and are introduced directly into the constitutive law. In this work, we used the mesomodelisation developed by [1, 13, 14], which has internal 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. 1.1. Modeling of the Elementary Layer. The model is based on an analysis of the microlayer, taking into account the micromechanisms of damage in structures. The ply is considered homogeneous with orthotropic elastoplastic behavior damaged. At the mesolevel different mechanisms of damage in laminated composite structures are well described by simple models of behavior [3, 15]. The mechanisms considered are: transverse microcracking of the matrix, the fiber debonding and inelastic deformation of the matrix. This modeling is adapted to the ply composed of a single-direction carbon-epoxy fiber type T300/914 (E1 0 150000� ÌÐà, E2 0 10800� MPa, G12 5800� MPa, �12 0 0 32� . , b �2 5. , Y0 0 0961� . MPa, Yc �13 6161. MPa, � �Ys 0 47. MPa, and � � � �15 10 4. MPa �1). 1.2. Strain Energy of the Elementary Layer. The strain energy of the layer is split into a traction power and energy of compression to take account of the unilateral aspect of model behavior. The kinematics of damage uses three scalar damage variables noted d , �d , and dF , respectively, related to the collapse of ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2011, ¹ 5 51 A Computational Strategy for the Localization and Fracture ... shear stiffness, tensile and transverse rupture of the fibers. The damage is considered constant along the thickness of the monolayer E d E E E D cp F � � � � � � 1 2 1 11 2 1 0 11 1 0 21 0 2 0 12 0 ( ) ( )� � � � � E1 0 11 22 � � � � � � � � � � � � � � � � � � � � � � � � � � � � 1 2 1 1 22 2 2 0 22 2 2 0 12 2 12 2 � � � ( ) ( )d E E d G � � , (1) where E D cp is the strain energy associated with plane stress, “0” designates the initial quantities and � X is the positive part of X , � � � � � X X X X X if if 0 0 0 , . The behavior of fiber is linear elastic brittle in tension and nonlinear elastic in compression in the direction of the fibers [16]. A stiffness loss of 30% of the longitudinal modulus was observed near the rupture for a T300/914. The consideration of this phenomenon is determined from the function �: � � � � � � � ( ) [ ln( )] .� � � � � � � � 11 11 11 2 2 1 (2) This function is chosen such that the expression of the modulus compression is a function of strain in the direction of fibers: E Ec 1 1 0 111� � � � ( ),� � (3) where � is a parameter identified as a four-points bending test. The modeling layer allows to take into account the matrix degradation and deterioration of the fiber-matrix. A remarkable property is that the crack grows parallel to fibers [17]. This property may be taken into account at the microscale. A calculation by asymptotic homogenization shows that the only moduli that are affected by the transverse modulus E2 0 and shear modulus G12 0 [18]. The state laws are given by � � �� ij e D e ij d d d E F � �, , . (4) Therefore, the elastic orthotropic damageable laws can be written in the two- dimensional case as A. Boutaous, M. Elchikh, M. Abdelouahab, and A. Belaidi 52 ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2011, ¹ 5 � � � � � � � 11 11 1 0 11 1 0 11 12 0 221 1 1 e dF E E � � � � � � � � � ( ) E1 0 � � � � � � � � , (5) � � � � � 22 22 2 0 22 2 0 12 0 11 1 01 1 e FE d E E d � � � � � � � ( ) ( ) , (6) � � 12 12 122 1 e G d � �( ) . (7) 1.3. Evolution of Damage Variables. The evolution of damage, which is governed by laws dependent on thermodynamic forces Y Yd d, ,� and YF coupled respectively to internal scalar variables d d, ,� and dF , has an expression: Y d E cst G d d D e� �� � �� � � � � � | : ( ) ,12 2 12 0 22 1 (8) Y d E cst E d d D e � � �� � ��� � � � � � � | : ( ) ,22 2 2 0 22 1 (9) Y d E cstF F D e� �� � � � �| : � � �� � � � � 1 2 1 2 22 11 2 1 0 11 2 1 21 0 2 0 12 0 ( )d E E E EF c � � � � 1 0 11 22 � � � � � � � � � . (10) In general, the laws of damage evolution can be written as a function of the history of loading time t: ED e is the energy of the monolayer and �� X is the mean value of X along the thickness of the monolayer, d A Y Y t d A Y Y tt d d d t d d d| ( | ; | , ); | ( | ; | , )� � � � �� � � � � �� � � �� � . (11) The laws of evolution Ad and Ad � of quasi-static type, depending on the type of loading, are selected to meet the second law of thermodynamics. 2. Damage Model with Delay Effect. The laws of damage evolution time- delay that express the evolution of damage is not instant over loading. There exists in this type of model a characteristic time which, combined with a characteristic speed, allows one to define a characteristic length. Value of � , � ,d dF and ��d are scalar internal variables which represent the speeds of damage. These are constants in the thickness of the layer, the thermo- dynamic force that governs the damage evolution is ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2011, ¹ 5 53 A Computational Strategy for the Localization and Fracture ... Y Y bYd d d� � . (12) The models with delay effect used in this article are as follows: Model in the direction of the fibers �d k F d d dF F F n F F� � � � � 1 1 1 1if otherwise (13) with F Y Y Y Y F F F F FC ( ) � � �0 1 otherwise F YF F( ) .�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) Model in the transverse direction to fiber �� � � � � � d k F dd n 2 2 if ��d 1 and Y Yd s� �� otherwise � �d 1 (17) with F Y Y Y Y d d d c � � � � � �( ) 0 1 otherwise F Yd d� �( ) .1 (18) Here k k k n n1 2 4 1 2, , , , , and n4 are model parameters representing the material. The parameters 1 ki are homogeneous in time characteristics. It should be noted that when the rate of damage is very low, we get the expressions of the change models without delay effect as statically identified. 2.1. Concept of the Localization Criterion for the Model with Delay Effect. The idea of the method to develop a test of stability and localization is studied in the neighborhood of the solution X 0 , the set of equations governing the mechanical problem under consideration. To study stability, we add a disturbance u to the solution X 0 . Then the set of equations is linearized near the solution X 0 , the system resulting from linearization can be written as � � u t M X u� ( ) ,0 where u represents the set of variables governing the problem considered in the neighborhood of the solution X 0 , and M X( )0 is a matrix depending on the 54 ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2011, ¹ 5 A. Boutaous, M. Elchikh, M. Abdelouahab, and A. Belaidi solution of the undisturbed problem. The system admits a solution in the form u x t u x t( , ) �( )exp( ).� � The criterion of stability and localization is derived from the eigenvalue problem as follows: M X u u( ) � �.0 �� The solutions to this problem are likely to increase risk of instability from a given state, if the eigenvalues are positive � 0. The case of � �0 is the transition from stability to instability. 2.2. Development of Criteria Appropriate to the Model with Delay Effect. The criterion is developed in the case of a one-dimensional elastic beam damageable-delay tensile loading subjected to forced displacement (Fig. 1). This criterion gives a necessary condition of localization of strain and damage. The equations governing the behavior of the beam in tension are given by �� �x �0, (19) � ( )� � ,� � �� � �E d dEe e1 (20) � ( , ),d f de� � (21) where Eq. (19) is the equation that denotes the equilibrium, Eq. (20) is the relationship between velocity and strain elastic strain, and Eq. (21) represents the model with delay effect. These equations are linearized around an equilibrium position noted “0” �� �� � �� � �� ( ) � � � ,� � � �E d E E de e e e1 0 0 0 (22) � � �� �� � � �� | | .d f f d d e e� (23) We search from an initial state noted “0” a perturbed solution of the above equations, which can be written as follows for any variable “X ”: X M t X M t X( , ) ( , ) ,� 0 � (24) � �X M t X M t( , ) � ( )exp( ).� (25) ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2011, ¹ 5 55 A Computational Strategy for the Localization and Fracture ... Fig. 1. Beam in tension-imposed displacement. The variations are then written for all variables in this case: � �u x t u x t( , ) �( )exp( ),� (26) �� � �( , ) � ( )exp( ),x t x t� (27) � �d x t d x t( , ) �( )exp( ).� (28) By substituting these equations into the linearized equations of the beam problem yields when � tends to zero: � ( ) | | � .� � � �� � � �� � � !! " ! ! # $ !! % ! ! E d E f f d e e e1 0 0 (29) The case � �0 corresponds to the limiting case of transition to instability. This hypothesis can be reduced to the study of behavior independent of time. The instability criterion can be written as d f f d e e 0 01� � � �� � � | | . (30) The following three equations determine the plot of effort vs. displacement and the point defined by the instability criterion � �� �E d e 2 1( ) , (31) � ,d k bE Y Y Y dc e c n � � � 2 2 02 2� (32) d bE Yc e� �1 2 2 � . (33) 3. Results and Discussion. In Fig. 2, curves 1, 2, and 3 represent the stress in terms of strain respectively for n2 2� , 1, and 0.5. The criterion of instability (33) is the peak of each curve representing the stress as a function of strain. We then plot d vs. deformation � e . Value of d has been multiplied by a factor of 100 in Fig. 2. The intersection points represent the criterion values, respectively, for n2 2� , 1, and 0.5. Similar curves can be determined by varying the parameter k2 or the loading speed. 56 ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2011, ¹ 5 A. Boutaous, M. Elchikh, M. Abdelouahab, and A. Belaidi It has been observed in this study that the model can also limit the speed of damage. Furthermore, the sensitivity of parameters k2 and n2 of model with delay effect has been studied in the case of a homogeneous beam in tension and length of unit section. We note that for a strain rate of 0.1 s�1 and a weighting factor n2 0 5� . the results of the model with delay effect does coincide with the model without delay effect only for high values of k2 . Conclusions. In this work, we verified the criterion of instability and localization which implies that the beam is likely to break through localization of strain and damage. In the case of the homogeneous beam (flawless) in static tensile, the fracture occurs simultaneously throughout the structure. We will prove in the second part of this work that we can correctly predict the moment of fracture when the beam has a lack of rigidity, using a development that treats the provision regulating the use of mesomodeling. Ð å ç þ ì å Çàïðîïîíîâàíî êðèòåð³é íåñòàá³ëüíîñò³ ³ ëîêàë³çàö³¿ ïîøêîäæåíü ó áàëö³ ç îäíîð³äíîãî ëàì³íàòà Ò300/914 äëÿ ìîäåëþâàííÿ ðîçâèòêó ïîøêîäæåíü ç óðàõóâàííÿì åôåêòó çàòðèìêè. Ðåçóëüòàòè, ùî îòðèìàí³ äëÿ îäíîâèì³ðíîãî âèïàäêó, ñâ³ä÷àòü ïðî òå, ùî çîíà ðóéíóâàííÿ ïî âñ³é êîíñòðóêö³¿ âèíèêຠîäíî÷àñíî. Çàïðîïîíîâàíî ðîçâ’ÿçîê, ùî áàçóºòüñÿ íà ìåçîìîäåëþâàíí³ êîì- ïîçèò³â. Îòðèìàí³ çà äîïîìîãîþ çàïðîïîíîâàíîãî ï³äõîäó ðîçðàõóíêîâ³ äàí³ äîçâîëÿþòü âèêîíàòè òî÷íèé ïðîãíîç âòðàòè ñò³éêîñò³ çðàçêà ïðè ïîã³ðøàíí³ ïàðàìåòð³â éîãî æîðñòêîñò³. ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2011, ¹ 5 57 A Computational Strategy for the Localization and Fracture ... �e � �10 1 Fig. 2. Load vs. displacement. 1. L. Gornet, D. Lévêque, et L. Perret, “Modélisation, identification et simulation éléments finis des phénomènes de délaminage dans les structures composites stratifiées,” Mécanique & Industries, 1, No. 3, 267–276 (2000). 2. A. Boutaous, B. Peseux, L. Gornet, and A. Bélaidi, “A new modeling of plasticity coupled with the damage and identification for carbon fibre composite laminates,” Compos. Struct., 74, 1–9 (2006). 3. P. Ladevèze, “Multiscale modelling and computational strategies for composites,” in: Proc. 2nd Int. Conf. on Testing, Evaluation and Quality Control of Composites (TEQC 1987, Sept. 22–24, University of Surrey), pp. 189–193. 4. R. Hill, “A general theory of uniqueness and stability in elastic-plastic solids,” J. Mech. Phys. Solids, 6, 236–249 (1958). 5. J. W. Rudnicki and J. R. Rice, “Conditions for the localization of deformation in pressure-sensitive dilatant materials,” Ibid, 23, 371–394 (1975). 6. A. Boutaous, Modélisation du Comportement de l’Endommagement d’un Stratifié Composite au Niveau des Boucles d’Hystérésis, Thèse de Doctorat, USTO-MB, Oran (2007). 7. O. Allix, “Délamination et localisation,” Colloque National en Calcul des Structures, Hermès (1993), 2, pp. 611–620. 8. J. R. Rice and J. W. Rudnicki, “A note on some features of the theory of localization of deformation,” Int. J. Solids Struct., 16, 597–605 (1980). 9. G. Pijaudier-Cabot and A. Benallal, “Strain localization and bifurcation in a nonlocal continuum,” Ibid, 30, 1761–1775 (1993). 10. R. De Borst, L. Sluys, H.-B. Mühlhaus, and J. Pamin, “Fundamental issues in finite element analysis of localization of deformation,” Eng. Comput., 10, 99–121 (1993). 11. R. De Borst and H.-B. Mühlhaus, “Gradient-dependent plasticity: formulation and algorithmic aspects,” Int. J. Num. Meth. Eng. Comput., 35, 521–539 (1992). 12. R. De Borst, “Simulation of strain localization: a reappraisal of the cosserat continuum,” Eng. Comput., 8, 317–332 (1991). 13. P. Ladevèze, O. Allix, J.-F. Deü, and D. Lévêque, “A mesomodel for localisation and damage computation in laminates,” Comput. Meth. Appl. Mech. Eng., 183, 105–122 (2000). 14. P. Ladevèze and G. Lubineau, “An enhanced mesomodel for laminates based on micromechanics,” Compos. Sci. Tech., 62, 533–541 (2002). 15. P. Ladevèze et G. Lubineau, “Pont entre les “micro” et “meso” mécaniques des composites stratifies,” C.R. Mécanique, 331, 537–544 (2003). 16. J. F. Harper and T. O. Heumann, “The strain dependance of elastic modulus in unidirectional composites,” Int. J. Numer. Meth. Eng., 60, 233–253 (2004). 17. E. Le Dantec, Contribution à la Modélisation du Comportement Mécanique des Composites Stratifiés, Ph.D. Thèse, Université Paris 6 (1989). 18. F. Devries, H. Dumontet, G. Duvaut, and F. Lene, “Homogeneization and damage for composites structures,” Int. J. Numer. Meth. Eng., 27, 285–298 (1989). Received 09. 05. 2010 58 ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2011, ¹ 5 A. Boutaous, M. Elchikh, M. Abdelouahab, and A. 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