List of Figures

Longitudinal traction: longitudinal fracture of the matrix [START_REF] Berthelot | Composite Materials: Mechanical Behavior and Structural Analysis[END_REF] ............................ Figure II. 6. Longitudinal traction: fracture of the fiber-matrix interface [START_REF] Berthelot | Composite Materials: Mechanical Behavior and Structural Analysis[END_REF] ........................ Figure II. 7. Crack growth in the case of strong adhesion fiber-matrix [START_REF] Berthelot | Composite Materials: Mechanical Behavior and Structural Analysis[END_REF] ........................... Figure II. 8. Crack growth in the case of weak adhesion fiber-matrix [START_REF] Berthelot | Composite Materials: Mechanical Behavior and Structural Analysis[END_REF] ............................. Figure II. 9. Fracture of a unidirectional composite under transverse traction [START_REF] Berthelot | Composite Materials: Mechanical Behavior and Structural Analysis[END_REF] ................ [START_REF] Szekré Nyes | Improved analysis of the modified split-cantilever beam for mode-III fracture[END_REF] Figure II. 19. Schematic illustration of the MSCB specimen [START_REF] Szekré Nyes | Improved analysis of the modified split-cantilever beam for mode-III fracture[END_REF] ......................................... Figure II. 20. Geometry of the MSCB specimen with small edge delamination [START_REF] Davidson | Mixed-mode I-II-III delamination toughness determination via a shear-torsionbending test[END_REF] ............ ECT-1 specimen and test fixture [START_REF] Marat-Mendes | Characterisation of the edge crack torsion (ECT) test for the measurement of the mode III interlaminar fracture toughness[END_REF] [START_REF] Suemasu | An experimental method to measure the mode III interlaminar fracture toughness of composite laminates[END_REF] .................................... Figure II. 25.Exploded view of the test specimen obtained by bonding of two rectangular plates (left) and cross section of the circular-shaped adhesive joint (right). [START_REF] Cricri | A novel fixture for measuring mode III toughness of bonded assemblies[END_REF] ....................... Figure II. 26.Exploded view of the shells and of the specimen being tested. [START_REF] Cricri | A novel fixture for measuring mode III toughness of bonded assemblies[END_REF] .................. Figure II. 27. Layout of the testing apparatus [START_REF] Cricri | A novel fixture for measuring mode III toughness of bonded assemblies[END_REF] Figure II. 30 STB specimen and loading [START_REF] Davidson | Mixed-mode I-II-III delamination toughness determination via a shear-torsionbending test[END_REF]

CHAPTER I. INTRODUCTION

I.1. Motivation

High performance laminated composites are widely used in aerospace industry where mass reduction is a main concern. The laminated composites have interesting mechanical properties such as their high specific strength and specific stiffness. They have good behavior under fatigue, impact loading and corrosion. In addition a structure in laminated composite can be optimized according to their mechanical loads in service, thus allow a considerable flexibility of design. On the contrary, the damage modes are quite complex and deserve special attention. The heterogeneity and anisotropy introduce difficulties of calculation, and manufacturing processes introduces defects. Especially the laminated composite structures are usually in the form of plates and shells, but the low strength of fracture between the layers presents an essential weakness. This kind of interlaminar fracture, named delamination, is one of the most dangerous damage modes. Herein, the interlaminar stresses are the key parameters; they cannot be determined by classic plaque or shell theory usually applied for calculating the laminates.

Fracture mechanics is preferred to study delamination in modern materials science because the loading field at the crack tip is singular. In fracture mechanics, three basic modes of fractures are defined: mode I (opening), mode II (in-plane shear) and mode III (out-of-plane shear). Generally, delamination in a laminated composite structure in service propagates in mixed mode I+II+III, whose prediction needs a general criterion including the participation of mode I, mode II and mode III. In literature, pure mode I and pure mode II delamination are well characterized. Then, mixed mode I+II criteria have been proposed thanks to the achievements of pure mode I, II and mixed mode I+II delamination tests. On the contrary, work regarding the participation of mode III remains a complex issue because even the pure mode III tests are very difficult to achieve. As a result, in this work we focus on characterizing the pure mode III delamination toughness, and then mixed mode delamination testing method with participation of mode III will be discussed.

I.2. Objectives and scopes

The aim of this research work is to develop testing methods for characterizing the delamination behavior of laminate composite materials under the three pure modes and mixed modes, focusing especially on the complex issue of mode III. Both experimental and numerical works were performed, validating the existing and new testing methods. Correlation between the results obtained aims, on one hand to a better understanding of the distribution of strain energy release rates (G I , G II , G III ) along the crack front and on the other hand, to propose and improve testing methods, and also to propose and validate simple approaches for the determination of delamination toughness.

Firstly, some testing methods proposed and largely used in the literature have been studied experimentally and numerically, including Double Cantilever Beam (DCB) for pure mode I; End Notched Flexure (ENF) for pure mode II; Edge Crack Torsion using tensile machine (ECT-1) and Edge Crack Torsion using torsion machine(ECT-2) for pure mode III. Secondly, novel testing methods named Edge Ring Crack Torsion (ERCT or ERC-III) tests, have been developed in order to obtain pure mode III loading condition, as well as a more uniform distribution of mode III component along the crack front. Thirdly, original testing methods for pure mode I and pure mode II delamination are proposed using the Edge Ring Crack specimens. The toughness measured by ERC tests were compared to the ones obtained by the testing methods proposed in literature, the advantages and disadvantages of ERC tests were also discussed. Finally, mixed mode I+II, mixed mode I+III and mixed mode I+II+III delamination are expected to be achieved with the Edge Ring Crack specimens. All the work above is fundamental for the establishment of a general criterion to characterize mixed mode resistance to delamination propagation in a laminated composite structure.

The scopes of the study cover:

(1) Materials:

In this research work, all tests were obtained from a woven carbon/epoxy taffeta fabric prepreg (ref: IMP503Z), whose properties are listed in Table . 1. The values with *¨ mean the measured values, which will be presented in the preliminary study in chapter III, whereas the other parameters were obtained from reference or in literature. A 13 m thick polyester film was inserted in the laminate in order to create a pre-crack, which is a classical method to create a pre-crack. In fact, there is a rich zone of resin at the crack tip, which is different from the real crack. Cyanoacrylate glue and Araldite 2012 were used to paste specimens on some devices used in our experiments.

(2) Stacking sequence

The polyester film separates the laminate into two sub-laminates. The stacking sequence of each sub-laminate was optimized in order to obtain special material parameters according to the study of VANNUCCI and GONG [1;2] . Note that 0 represents a taffeta balanced fabric ply with yarns oriented in 0° and 90° while 45 represents ±45°. For all of the stacking sequence of laminates used in this study, the fiber orientation of the adjacent 4 plies close to the crack plane is the same. The design aims at avoiding the interference caused by different fiber orientations in adjacent plies next to the crack plane when different tests are compared. In the traditional pure mode delamination tests, delamination propagates between two plies with the fiber orientation 0/0.

(3) Testing parameters and variables:

All the tests were realized at ambient temperature. Static loading was applied for all the tests using tensile/compression machine For DCB, ENF, ECT-1 ERC-I and ERC-II tests, the loading speed is 2mm/min; for ECT-2 and ERC-III tests, a torsion machine is needed and the loading speed is 0.5°/min.

(4) Finite element analysis was performed on all the tests in this study using the simulation software package LS-dyna.

I.3. Organization of thesis

This thesis is divided in 6 chapters as follows:

Chapter I mainly expresses the motivations, objectives and scopes of the work.

Chapter II gives a general literature review of all kinds of physical damages in laminated composites, especially the delamination in a laminated composite. Then the majority of criteria to characterize the damages are summarized. Delamination and criterion based on fracture mechanics are introduced particularly. Different kinds of testing methods to study delamination with the participation of mode III are discussed and compared. At last, finite element methods to evaluate strain energy release rate are summarized.

Chapiter III presents the preliminary study on the tested laminated composite. Firstly the mechanical characterization results of the composites are provided. And then the traditional tests of delamination including DCB pure mode I and ENF pure mode II have been performed and simulated.

Chapter VI focuses on the characterization of pure mode III delamination behavior of a laminated composite. At the beginning two kinds of Edge Crack Torsion tests, proposed in the literature, were carried out. The distribution of G III along the crack front was determined by finite element analysis (FEA) using virtual crack closure technique (VCCT). The performances of these tests are discussed. And then a novel mode III testing method was developed, named Edge Ring Crack Torsion test (ERCT or ERC-III later). A closed-form solution proposed by Tada is applied to determine mode III delamination toughness. Actually, a numerical study on the factors affecting the distribution of G III was carried out in this study.

In order to understand the influence of potential defaults on the performance of ERCT test, sensitivity study was performed on the relative position of the crack front, the circularity of the crack front and the specimen shape. Optimum specimen's relative pre-crack geometry is given and a method for reducing the variation of the G III along the crack front is provided.

Chapter V presents the studies of the extension of the use of Edge Ring Crack (ERC) specimens from pure mode III to pure mode I (named ERCTE or ERC-I), pure mode II (named ERCF or ERC-II) and mixed mode delamination testing. For the testing under pure mode I and pure mode II, both of experiments and FEA are carried out. Tests of delamination under mixed mode I+II, mode I+III and mixed mode I+II+III are only investigated numerically.

The thesis is concluded by Chapter VI, which gives general conclusions of this study and some recommended future work.

CHAPTER II. LITTERATURE REVIEW

II.1. Introduction

In this chapter, a general literature review is given in order to expose the background of this work. Firstly, mechanisms of damage and fracture of laminated composite are introduced, especially delamination. Secondly, most existing fracture criterions are introduced. Thirdly, mode III delamination testing methods existing in literature are discussed. Finally, finite element methods available in literature in order to calculate strain energy release rates are studied.

II.2. Mechanisms of damage and fracture of laminated composite

In the book Composite Materials: Mechanical Behavior and Structural Analysis [START_REF] Berthelot | Composite Materials: Mechanical Behavior and Structural Analysis[END_REF] , laminated composite is defined: "laminates are made of successive layers of reinforcements impregnated with resins" and mechanisms of damage and fracture of laminated composite are introduced.

Here we give a brief presentation.

II.2.1. The principal mechanisms of damage in a single ply

The fracture process of a laminated composite consists of three stages: initiation of damage, the propagation of the damaged area and the final fracture. The principal mechanisms of damage in a composite laminate are summarized as follows:

(1) The fibers fracture

In a composite material subjected to mechanical loads, fiber fracture occurs when the tensile stress σ f in a fiber reaches the fracture stress σ fu (Figure II. 1). The fiber fracture then leads to a stress concentration around the fracture. Redistribution of these constraints mainly depends on the tensile strength σ fu of fibers, the ability of the matrix to absorb the energy released, the properties of the fiber-matrix interface, etc. Figure II. 2 shows the various fracture modes of the matrix associated with the fracture of a fiber. The plastic deformation of the matrix happen if the matrix is a ductile material and the level of applied stress exceeds the elastic limit. Longitudinal traction: fracture of the fiber-matrix interface [START_REF] Berthelot | Composite Materials: Mechanical Behavior and Structural Analysis[END_REF] (7) The mixture of mechanisms of damage

In the case of a unidirectional composite subjected to a longitudinal traction, the initiation of the fracture is generally produced either by the fracture of the fibers or by transverse fracture in the matrix. After initiation, propagation of the fracture differs according to the nature of the fiber-matrix interface.

In the case of a high fiber-matrix adhesion, the fracture is initiated, either by the fracture of the fibers or by fracture of the matrix. Crack growth in the case of strong adhesion fiber-matrix [START_REF] Berthelot | Composite Materials: Mechanical Behavior and Structural Analysis[END_REF] In the case of a weak fiber-matrix adhesion, the transverse crack propagation can be schematized as Figure II. 8.

Figure II. 8. Crack growth in the case of weak adhesion fiber-matrix [START_REF] Berthelot | Composite Materials: Mechanical Behavior and Structural Analysis[END_REF] In the case of a unidirectional composite subject to transverse traction, the fracture occurs either by fracture of the matrix or by debonding of the fiber-matrix interface. The fracture of the matrix occurs when σ m reaches σ mu . This process occurs when σ mu is bellow the debonding stress in traction of the fiber-matrix interface σ d . Otherwise, the fracture of the composite occurs through the fracture of the fiber-matrix interface (Figure II. 9)

Figure II. 9. Fracture of a unidirectional composite under transverse traction [START_REF] Berthelot | Composite Materials: Mechanical Behavior and Structural Analysis[END_REF] II.2.2. Delamination

The damages of multidirectional laminated composites can be very complex. Among all kinds of damages, delamination is one of the most dangerous and common modes. It is the interlaminar fracture between two sub-laminates and the fracture plane is always tangential to the reinforcements, shown in Figure II. 10. In fact, delamination in a laminated composite is a particular type of fracture by its position and its physical mechanisms. G C can be determined using the Irwin-Kies equation [START_REF] Irwin | Critical energy rate analysis of fracture strength[END_REF] :

Eq. 1
, where P C is the critical load corresponding to crack onset; B is the width of the crack; C is the compliance and a = the initial crack length. If the specimen maintains a constant fracture mode during crack propagation, C can be calibrated as a function of a. First of all, it can be calculated analytically with certain assumption and simplification. In most cases, compliance calibration method (CC) can be realized both empirically and numerically. Empirically, experimental tests with specimens of different initial crack length are carried out aiming at evaluating the slope of the linear part from the load-displacement curve and the inverse of the slope gives the compliance for each specimen with different initial crack length. Then, the values of the compliance are interpolated as a function of the crack length a. Numerically, the evolution of C according to a can also comes from simulation results and then the same interpolation method can be used.

The steps are as below:

(1) Make a series of test specimens of the same geometry except for different initial crack length;

(2) Measure the compliance of each specimen: C i =δ i /P i (Displacement/load) on the linear domain (shown in

II.2. Fracture criteria

A large number of fracture criteria have been established in order to characterize the damage and fracture in a laminated composite. Classical mechanics and fracture mechanics are applied in order to model different kinds of damages in a laminated composite. Fracture mechanics is preferred to characterize delamination behavior dues to the singularity of stress at crack tip. The criteria can be classified as criteria without stress interaction, interactive criteria and semi-interactive criteria.

II.2.1. Criteria without stress interaction

The criteria without stress interaction are the basic criteria and contain the criteria of the maximum stress and the criteria of the maximum strain. They are very simple to apply, but they do not allow accounting the combination of experimental results. Criteria of the maximum stress and criteria of the maximum strain belong to this kind of criteria.

II.2.2. Interactive criteria

The criteria of the maximum stress and maximum strain exclude the existence of interactions between stress and strain in the axes of materials: mechanisms in longitudinal, transverse or shear fracture are supposed to occur independently. Interactive criteria were then developed by extending to orthotropic materials, Von Mises criterion. Von Mises criterion is connected to the strain energy stored per unit volume of the deformed material. However, these criteria are no longer connected exclusively to the deformation energy in the case of orthotropic materials.

One of the first interactive fracture criteria applied to anisotropic materials was introduced by R. Hill [START_REF] Hill | A theory of the yielding and plastic flow of anisotropic metals[END_REF] . It should be noted that the criterion of Hill does not take into account the difference of the behavior of materials in tension and compression, so it is suitable for composites having a very close tensile and compression strength. Moreover, this criterion does not give information on fracture modes.

The previous fracture criterion for in plane stress has been simplified by V. D. AZZI et S. W. Tsai [START_REF] Azzi | Anisotropic strength of composites[END_REF] in the case of unidirectional composites.

A generalization of the Hill's criterion was formulated by O. Hoffman [START_REF] Hoffman | The brittle strength of orthotropic materials[END_REF] . Hoffman's criterion takes into account the different behavior of the material in traction and compression.

The previous criteria are usually sufficient to describe the various experimental results. One way to improve the correlation between experimental and theoretical results is to increase the number of parameters of the theoretical equations. S. W Tsai and E. M. Wu [START_REF] Tsai | A General Theory of Strength for Anisotropic Materials[END_REF] gave a fracture criterion who allows more parameters to correlate with experimental results.

II.2.3. Semi-interactive criteria

Although the criteria discussed in the previous part take into account interaction between the applied stresses, they do not allow relating the rupture with the damage mechanisms. The semi-interactive criteria developed more recently, allow giving material properties to all identified damage mechanisms.

(1) Hashin's Criterion

The test proposed by Hashin [START_REF] Hashin | Failure criteria for unidirectional fibre composites[END_REF] clearly associates the responsible stress with a well defined damage mode, which can take into account the influence of damage identified in the calculation of the structure. This criterion is expressed in four inequalities: Fracture of fibers in traction:

Eq. 2

Fracture of fibers in compression:

Eq. 3

Fracture of matrix in traction:

Eq. 4 Fracture of matrix in compression:  ( ) Eq. 5

(2) Chang-Chang Criterion

Longitudinal compression: 15 Transverse compression:

( ) Eq.
( )
Eq. 16 Out of plane compression:

( )
Eq. 17

Delamination:

( ) ( ) ( ) Eq. 18
Where σ 11 , σ 22 , σ 33 are the normal stress along the three orthotropic axis of elemental layer. It is interesting to note that if the base material is considered as a transverse isotropic where Y i =Z i and S 12 =S 13 , as in the majority of cases for unidirectional composite, the material behavior in the delamination presented by Eq. 18 returns to the one of delamination(Eq. 12). This situation is not at all realistic in a multidirectional laminated composite.

II.2.4. Mixed mode delamination criteria

(1) Semi-empirical criterion in mixed mode I+II Theoretically, the fracture criterion under mixed mode I+II could be in a polynomial form K I et (K II ) [START_REF] Vannucci | Dé termination des stratifies quaisi-homogè nes par l'approche polaire[END_REF] . However, the form of the criterion proposed in the literature is often in a power form:

( ) ( ) Eq. 19 ( ) ( )
Eq. 20 where Ki and Gi respectively correspond to the stress intensity factors and strain energy release rate in mode i.

In the works of Gong [START_REF] Gong | Rupture interlaminaire en mode mixte I+II de composites stratifies unidirectionnels et multidirectionnels Verre/Epoxy[END_REF][START_REF] Gong | Mixed mode interlaminar fracture toughness of unidirectional Glass/Epoxy composite[END_REF] , an unexpected observation was raised by the analysis of a composite toughness measured on Glass-Epoxy: participation of mode I delamination under mixed mode I + II may be greater than tenacity under pure mode I. The ratio G I /G IC can reach 2.5 for the tested composite. This phenomenon was also observed by other authors in fragile matrix composites. In this case, the criteria given in the form of Eq. 19 and Eq. 20 are no longer valid whatever the values of the exponents m and n are.

A polynomial criterion was proposed by focusing on the participation of mode II: G TC =k 1 + k 2 G β II , where β is a constant determined empirically. The exact value of β can be easily determined by the interpolation of the experimental results. One can get the following expression applying the extreme cases under pure mode I and pure mode II:

( ) Eq. 21
This criterion has also been applied to the results published in the literature. By choosing the constant β, all the results we used are perfectly represented by this criterion. Some fronts were given by the authors: β is a constant of the material, which depends mainly on the quality of the matrix and the fiber / matrix interface; β appears to be independent of the absorbed moisture even if G IC and G IIC are highly dependent on it; β is not sensitive to the orientation of adjacent plies of fiber although G IC and G IIC are dependent heavily; β is between 2/3 and 1 for a brittle epoxy matrix composite.

(2) Empirical criterion in mixed mode I+II Despite excellent correlations between the criterion of Eq. 21 and the experimental results, it is neither reassuring participation of mode I, nor the modal report involved in this criterion. In the same studies [START_REF] Gong | Mixed mode interlaminar fracture toughness of unidirectional Glass/Epoxy composite[END_REF][START_REF] Campilho | Tensile behaviour of three-dimensional carbon-epoxy adhesively bonded single-and double-strap repairs[END_REF] , an empirical criterion has also been proposed in which the variable is the modal ratio in terms of G II /G T with G T =GI+GII ( ) Eq. 22 , where the constant k must be determined empirically.

This test has been very successful and is currently implemented in much FEM software, as the criterion (B-K) [START_REF] Campilho | Tensile behaviour of three-dimensional carbon-epoxy adhesively bonded single-and double-strap repairs[END_REF] , to simulate delamination behavior in laminates or that of a glued or welded joint.

(3) Predict of the delamination criteria under mixed mode I+ II+III

In the work REEDER JR (2006) [START_REF] Reeder | 3D mixed mode delamination fracture criteria-an experimentalist's perspective[END_REF] , a new framework for visualizing 3D fracture criteria is studied and new criteria are based on the following assumptions: the relationship between toughness in mode I and mode III is similar to the relationship between toughness under mode I and mode II and linear interpolation can be used between mode II and mode III. According to his study, 3D fracture criteria cannot be evaluated properly until that the data in mixed mode with a mode III component are possible to be obtained, but these assumptions appear reasonable.

This criterion is an extension of B-K criterion, which is an empirical criterion that works well with 2D data. ECT test provides mode III toughness which is generally higher than the mode II toughness. As there are no data available to describe the interactions between mode II and mode III, a reasonable assumption is that a linear interpolation governs the interaction. Combining these assumptions the proposed fracture criterion becomes Eq. 23:

( ) ( )
Eq. 23 The criterion can be written as shown in Eq. 24 in order to show the symmetry between the mode II and mode III [START_REF]Standard Tests for Toughened Resin Composites[END_REF] is the most widely used test in order to study mode I delamination behavior. This is a tensile test applied onto two arms of a symmetrical specimen including a pre-crack, which is normally realized via a non-adhesive thin insert [START_REF] Astm | ASTM standard test method for mode I interlaminar fracture toughness of unidirectional continuous fiber reinforced composite materials[END_REF] . A typical DCB test is shown in Figure II. 13. Beam theory and berry theory [START_REF] Peng | Modélisation numérique d'assemblages collés : Application à la réparation de structures en composites[END_REF] can be applied in experimental compliance calibration for data reduction.

Figure II. 13. Typical set-up for DCB test [START_REF]Determination of mode-I fracture toughness and non-uniformity for GFRP double cantilever beam specimens with an adhesive layer[END_REF] II.3.3. Pure mode II End notched flexure (ENF) test [START_REF] Russell | Factors affecting the interlaminar fracture energy of graphite/epoxy laminates[END_REF] has been widely used to study mode II delamination behavior. A typical ENF specimen is a three-point bending beam with a mid-plane pre-crack at one end of the beam as shown in Figure II. 14. The simplest analytical expression for the mode II strain energy release rate, G II , was developed by Russell and Street [START_REF] Russell | Factors affecting the interlaminar fracture energy of graphite/epoxy laminates[END_REF] and improved using Timoshenko's beam theory by Carlsson et al. [START_REF] Carlsson | On the analysis and design of end notched flexure (ENF) for mode II testing[END_REF] .

Figure II. 14. Typical set-up for ENF test [START_REF] Yoshihara | Mode II fracture mechanics properties of solid wood measured by the three-point eccentric endnotched flexure test[END_REF] II. The Crack Rail Shear test (CRS) [START_REF] Becht | Design and analysis of the crack rail shear specimen for mode III interlaminar fracture[END_REF] was proposed by Becht and Gillespie in 1988, shown in Figure II. 15. A plastic film is inserted between two sub-laminates of the specimen to create a precrack. The specimen is screwed to two rails. The configuration was identical to the ASTM standard guide D4255-83 for measuring the in plane shear properties of composite laminates. The initial delaminated portions were cut away from the inner plies after processing in order to accommodate the rail shear fixture. With the rails in place, the specimen is loaded in exactly the same manner as an ordinary two-rail shear test.

The restraint regarding this work is that mode II component at the extremities of the crack front cannot be eliminated. The test produced very low values of compliance, so the compliance calibration method cannot be applied for data reduction. In addition, the two precracks do not propagate simultaneously. Crack Rail Shear specimen geometry [START_REF] Becht | Design and analysis of the crack rail shear specimen for mode III interlaminar fracture[END_REF] II.3.4.2. Split Cantilever Beam Split Cantilever Beam test (SCB) [START_REF] Donaldson | Mode III intelaminar fracture characterization of composite materials[END_REF] was proposed by Donaldson in 1988. Each test required bonding a laminate between two aluminum bars, then loading the bars in opposite directions parallel to the crack plane as shown in Figure II. 16. As the crack extends, the aluminum bars act as cantilevers. The main drawback of these tests is the participation of mode I and mode II in the area adjacent to the ends of the crack front [START_REF] Donaldson | Mode III intelaminar fracture characterization of composite materials[END_REF][START_REF] Martin | Evaluation of the split cantilever beam for mode III interlaminar delamination testing[END_REF] . Typical spilt cantilever beam test [START_REF] Donaldson | Mode III intelaminar fracture characterization of composite materials[END_REF] The closed-form expression to determine G IIIC from SCB test is based on the beam theory and classical unidirectional specimens can be used in this test. This configuration has been modified by different authors in order to cancel mode I and mode II and to obtain a more uniform value of G III along the crack front. In general, most modified SCB tests can eliminate mode I component and reduce mode II component. Robinson and Song [START_REF] Robinson | The development of an improved mode III delamination test for composites[END_REF] tried to limit the bending moment by introducing additional constraints as shown in Figure II. 17, but they concluded that the experimental Compliance Calibration (CC) method cannot be applied in the data reduction schema.

Figure II. 17.test jig used for modified SCB test [START_REF] Robinson | The development of an improved mode III delamination test for composites[END_REF] Another modified version of SCB device (MSCB) was then realized by Sharif et al. [START_REF] Sharif | Mode III delamination using a split cantilever beam[END_REF] and this one is the most widely used SCB device [START_REF] Szekré | Interlaminar fracture analysis in the GI-GIII plane using prestressed composite beams[END_REF][START_REF] Szekré Nyes | Improved analysis of the modified split-cantilever beam for mode-III fracture[END_REF][START_REF] Khoshravan | Investigation on mode III interlaminar fracture of glass/epoxy laminates using a modified split cantilever beam test[END_REF] , whose specimen was loaded by special grips and the mode II component was significantly reduced according to their work. The data reduction method based on improved beam theory was found to be more reliable than CC method [START_REF] Szekré | Delamination fracture analysis in the GII-GIII plane using prestressed transparent composite beams[END_REF]. However, the measurement of G IIIC is disturbed because the specimen is strongly constrained at its edges near the crack front, so the variation of G III along the crack front becomes more important. As a result, it is not easy to see the crack onset on the loaddisplacement curve [27; 28] .

Figure II. 18. 3D view of the experimental equipment. Assembled state (a) and exploded view(b) [START_REF] Szekré Nyes | Improved analysis of the modified split-cantilever beam for mode-III fracture[END_REF] Figure II. 19. Schematic illustration of the MSCB specimen [START_REF] Szekré Nyes | Improved analysis of the modified split-cantilever beam for mode-III fracture[END_REF] Later, small edge delamination in the MSCB specimen was introduced by Davidson and Sediles [START_REF] Davidson | Mixed-mode I-II-III delamination toughness determination via a shear-torsionbending test[END_REF] in order to achieve more uniform evolution of G III along the crack front. The results highlighted the fact that the evolution of G III along the crack front was fairly small. Moreover the delamination onset seemed to correspond to a load drop even though a high level of non-linearity before the critical load was observed. However, the edge initial delamination penetrated a distance into the MSCB specimen from either free edge which was too small (1/16 of the width) to obtain so a number of specimens had to be rejected in their work [START_REF] Davidson | Mixed-mode I-II-III delamination toughness determination via a shear-torsionbending test[END_REF] .

Figure II. 20. Geometry of the MSCB specimen with small edge delamination [START_REF] Davidson | Mixed-mode I-II-III delamination toughness determination via a shear-torsionbending test[END_REF] II.3.4.

Edge Crack Torsion

The Edge Crack Torsion (ECT) test was proposed by Lee [START_REF] Lee | An edge crack torsion method for mode III delamination fracture testing[END_REF] . A tensile machine was used to load the specimen under torsion up to delamination growth from an edge crack. The closedform expression to determine G IIIC is based on the plate theory. Different from SCB, the specimens have to be multidirectional laminates whose twisting stiffness and strength must be strong enough to guarantee the crack propagation. Actually, the loading conditions in the first ECT tests [START_REF] Lee | An edge crack torsion method for mode III delamination fracture testing[END_REF][START_REF] Li | Simplified data reduction methods for the ECT test for mode III interlaminar fracture toughness[END_REF][START_REF] Liao | The determination of mode III fracture toughness in thick composite laminates[END_REF][START_REF] Li | Evaluation of the edge crack torsion (ECT) test for mode III interlaminar fracture toughness of laminated composites[END_REF][START_REF] Zhao | Mode III fracture behaviour of laminated composite with edge crack in torsion[END_REF][START_REF] Li | Influence of fiber volume fraction on mode III interlaminar fracture toughness of glass/epoxy composites[END_REF] cannot induce symmetrical distribution of G III along the crack front. This test is called ECT-1 thereafter.

Figure II. 21. Schematic of ECT specimen [START_REF] Marat-Mendes | Characterisation of the edge crack torsion (ECT) test for the measurement of the mode III interlaminar fracture toughness[END_REF] 35 / 125

Figure II. 22. ECT-1 specimen and test fixture [START_REF] Marat-Mendes | Characterisation of the edge crack torsion (ECT) test for the measurement of the mode III interlaminar fracture toughness[END_REF] Recent studies [START_REF] Marat-Mendes | Characterisation of the edge crack torsion (ECT) test for the measurement of the mode III interlaminar fracture toughness[END_REF][START_REF] Ratcliffe | Characterization of the edge crack torsion (ECT) test for mode III fracture toughness measurement of laminated composites[END_REF][START_REF] Pennas | The influence of strain rate on the mode III interlaminar fracture of composite materials[END_REF][START_REF] Pennas | The influence of loading rate on the mode III fracture properties of adhesively bonded composites[END_REF][START_REF] De Morais | Mode III interlaminar fracture of carbon/epoxy laminates using the edge crack torsion (ECT) test[END_REF][START_REF] Marat-Mendes | Failure criteria for mixed mode delamination in glass fibre epoxy composites[END_REF] adopted a configuration with more symmetric loading conditions, named modified Edge Ring Crack Torsion test. This test is called MECT-1 thereafter. This modification allows decreasing the likelihood of geometric nonlinearity and improving the uniformity of G III along the crack front.

Figure II. 23. MECT-1 specimen and test fixture [START_REF] Marat-Mendes | Characterisation of the edge crack torsion (ECT) test for the measurement of the mode III interlaminar fracture toughness[END_REF] Another ECT configuration using a torsion machine was proposed by H. Suemasu [START_REF] Suemasu | An experimental method to measure the mode III interlaminar fracture toughness of composite laminates[END_REF] aiming at obtaining uniform distribution of G III along the crack front. Both ends of the specimens in ECT-2 test were loosely fixed so that an axial slide was permitted during the application of the torque. This test is called ECT-2 thereafter. Photograph and schematic diagram of test fixture [START_REF] Suemasu | An experimental method to measure the mode III interlaminar fracture toughness of composite laminates[END_REF] In all ECT tests, pure mode III component is proved to be in the mid-region of the specimen, while mode II component appears near the edges. Mode II component can be reduced to a reasonable level with an appropriate choice of the specimen geometry [39-41;42] . The closedform expressions for determining G IIIC for ECT tests are not perfect since the CC method widely used in data reduction schema is not satisfactory. It was found that G IIIC increased along with the crack length [START_REF] De Morais | Mode III interlaminar fracture of carbon/epoxy laminates using the edge crack torsion (ECT) test[END_REF] , but a large number of specimens with different crack lengths had to be used. Secondly, small initial crack lengths of ECT specimens should be set for measuring G IIIC [45] , but small initial crack lengths would decrease the test accuracy because of the uncertainty in the real crack length measurement. Furthermore, it is not easy to identify the crack onset in most ECT tests because there is a significant non-linearity before the peak load in the load-displacement curve associated with a R-curve effect [39-42;46-48] . A Six-Point Edge Crack Torsion (6ECT) test was proposed by Pereira et al. [START_REF] Pereira | Design and analysis of a new sixpoint edge crack torsion (6ECT) specimen for mode III interlaminar fracture characterisation[END_REF] , whose work represents a contribution to a relatively clear identification of the critical load and evaluation of the mode III R-curve effect.

II.3.4.4. Torque Shell (TS) fixture

Recently, a novel fixture for measuring mode III toughness named Torque Shell (TS) fixture is proposed by Cricri et al. [START_REF] Cricri | A novel fixture for measuring mode III toughness of bonded assemblies[END_REF] . The plates are bonded together to form an initial angle of 90° as shown in The authors conclude that the test fixture and the approach that uses the direct identification to extract the traction-separation law are promising and it would be of interest to repeat the mode III test using a different method and also to validate results numerically. The distribution of G III along the crack front and the participation of mode I and mode II components were not presented in their work. In our opinion, there is no constraint in the z direction as mentioned in their work. So there is a possibility for the birth of mode I and mode II fracture modes.

Figure II. 25.Exploded view of the test specimen obtained by bonding of two rectangular plates (left) and cross

section of the circular-shaped adhesive joint (right) [START_REF] Cricri | A novel fixture for measuring mode III toughness of bonded assemblies[END_REF] Figure II. 26.Exploded view of the shells and of the specimen being tested [START_REF] Cricri | A novel fixture for measuring mode III toughness of bonded assemblies[END_REF] Figure II. 27. Layout of the testing apparatus [START_REF] Cricri | A novel fixture for measuring mode III toughness of bonded assemblies[END_REF] II. Unfortunately it can be seen that the participation of mode II in all these tests is never totally eliminated. The variation of G III along the crack front is not negligible at all. Generally, this evolution is slighter in the central region of the crack front, but much more significant at the edges. G III values can go to almost zero at the two edges of the specimen because of the edge effect. The longer the crack front, the better the G III distribution. In SCB test the crack front is relatively small because beam geometry must be maintained. In ECT tests the crack front length is less limited, which represents a big advantage of these tests.

Recently, an original pure mode III test has been proposed, named Edge Ring Crack Torsion (ERCT) test [START_REF] Gong | Edge Ring Crack Torsion (ERCT) test for pure mode III toughness[END_REF] . This is a torsion test on a laminated composite plate with an edge ring delamination crack between two sub-laminates. The loading mode obtained is pure mode III, and no edge effect is introduced on the G III distribution. Moreover the evolution of G III along the crack front, depending on the stacking sequence of the laminates used, can be optimized in order to be more uniform.

II.3.5. Mixed mode I+II+III

In literature, some efforts have been also taken to realize mixed mode I+II+III delamination test. However, three pure mode delamination tests are the fundamental of a mixed mode delamination test. Mixed mode tests still have large room of improvement because a perfect pure mode III delamination test cannot be achieved as mentioned above.

In the work of A. Szekré nyes [START_REF] Szekré | Application of prestressed transparent composite beams in fracture mechanics Mechanical Engineering[END_REF] The author concluded that the main advantages of the prestressed specimens are that the material can be tested at any mode ratio and the complete fracture space can be covered. The drawbacks of the test are that the mode ratio cannot be designated before the testing process, because the mode ratio depends on the external load and also on the crack length. Another mixed mode I+II+III delamination test was proposed and realized by B. Davidson and F. Sediles [START_REF] Davidson | Mixed-mode I-II-III delamination toughness determination via a shear-torsionbending test[END_REF] , whose approach is a superposition of a mixed-mode bending (MMB) [START_REF] Astm Standard | Standard test method for mixed mode I-mode II interlaminar fracture toughness of unidirectional fiber reinforced polymer matrix composites[END_REF] type arrangement for modes I and II with a modified version of the MSCB for mode III. The authors concluded that the STB is a promising method for evaluating mixed mode (including mode III component) toughness of a laminated composite.

Figure II. 30 STB specimen and loading [START_REF] Davidson | Mixed-mode I-II-III delamination toughness determination via a shear-torsionbending test[END_REF] Figure II. 31 Schematic of test set-up [START_REF] Davidson | Mixed-mode I-II-III delamination toughness determination via a shear-torsionbending test[END_REF] II.3.6. Summary

A literature review of pure mode III delamination tests and mixed mode tests has been given. Different from standardization of mode I (DCB) and mode II (ENF) tests, the investigation of delamination behavior with the participation of mode III is quite limited. Even in the case of pure mode III loading, the determination of the toughness has been proven to be a complex issue for laminated polymeric matrix composites. In most cases, the evolution of G III along the crack front is not uniform. Actually, the evolution of G III along the crack front can be observed slight in the central region of the crack front, while it becomes more significant at the extremities of the crack front due to the free edge effects. Unfortunately, mode II component can never be totally eliminated in most cases. As a result, it is difficult to propose an exact closed-form expression for mode III tests because the average value of the G III along the crack front is considered as the toughness G IIIC . And it is not so meaningful if the distribution of G III along the crack front has an important amplitude.

Determination of the critical load corresponding to a crack onset is problematic. There is always a nonlinear part before reaching to the critical load in a load/displacement curve. Some efforts have also been taken for realizing mixed mode I+II+III delamination test. Generally, it is a superposition of three basic pure modes. Therefore, the success of mixed mode delamination tests is based on standardized pure mode delamination. However, characterization of pure mode III delamination behavior remains a complex issue. That's our motivation to focus on development of pure mode III delamination tests firstly.

II.4. Finite element analysis for simulation of delamination behavior

Finite element method is used for two purposes: evaluate the stress energy release rates in a given configuration or predict the crack propagation. Several widely used finite element methods are introduced in this section.

II.4.1. VCCT method to calculate strain energy release rate

Virtual crack closure technique (VCCT) [START_REF] Rybicki | A finite element calculation of stress intensity factors by a modified crack closure integral[END_REF][START_REF] Krueger | The Virtual crack closure technique: history, approach and applications[END_REF][START_REF] Shokrieh | Simulation of mode I delamination propagation in multidirectional composites with R-curve effects using VCCT method[END_REF][START_REF] Ahna | Delamination of laminated composite plates by p-convergent partial discrete-layer elements with VCCT[END_REF][START_REF] Marjanovic | A finite element model for propagating delamination in laminated composite plates based on the Virtual Crack Closure method[END_REF] is a widely used method to determine strain energy release rate along a crack front, in which the nodal forces at the crack front and the displacements behind the crack front are used. The formulas (Eq. 25) for calculating G I , G II and G III for three dimensional eight-nodded solid components were given by Rybicki and Kanninen [START_REF] Rybicki | A finite element calculation of stress intensity factors by a modified crack closure integral[END_REF] .

, , Eq. 25 , where Δy is the sum of the one-half the element lengths on either side in the y direction and Δa is the element length in x direction as shown in Figure II. 32. Note that the crack front forces required for holding together node pair B are the same as the forces that would be required to keep node pair A and A' closed if the mesh is sufficiently refined.

Figure II. 32 VCCT for three dimensional eight-nodded solid elements [START_REF] Krueger | The Virtual crack closure technique: history, approach and applications[END_REF] II.4.2. J-integral method

The concept of the integral J was introduced by RICE [START_REF] Rice | A Path Independent Integral and the Approximate Analysis of Strain Concentration by Notches and Cracks[END_REF] . The method to calculate the stressintensity factor was an interaction J-integral method conducted with ABAQUS, and is required to separate the components of the stress-intensity factors for a crack under mixedmode loading. The method is applicable to cracks in isotropic and anisotropic materials. The interaction integrals can be expressed [START_REF] Habbit | ABAQUS user's manual, version 6.9.1[END_REF] as: 26 with Γ: an arbitrary closed contour surrounding vanishing on the faces of the crack and surrounding the crack tip; q: unit vector in the virtual crack extension direction; n: the outward normal to Γ; σ: the stress tensor; u: the displacement vector (Figure II. 33).

∫ Eq.

Figure II. 33 Contour for evaluation of J-integral

The subscript α represents three pure mode I, mode II, and mode III loading fields when α equals to I, II, and III, respectively. The domain form of the interaction J-integral is:

∫ Eq. 27
, where λ(s) is the virtual crack advance and dA the surface element. In the interaction Jintegral method [START_REF] Habbit | ABAQUS user's manual, version 6.9.1[END_REF] the two-dimensional auxiliary fields are introduced and superposed on the actual fields. By judicious choice of the auxiliary fields, the interaction J-integral can be directly related to the stress-intensity factors as:

Eq. 28 where B is called the pre-logarithmic energy factor matrix,

J int = [ J I int , J II int , J II int ] and K = [ K I , K II , K III ].
In linear elastic fracture mechanics, the J-integral coincides with total energy release rate, J=G T =G I +G II +G III where G I , G II and G III are associated with K I , K II and K III , respectively [START_REF] Hamed | Separation of delamination modes in composite beams with symmetric delaminations[END_REF] .B is diagonal for homogeneous and isotropic materials and the above equation is simplified to:

Eq. 29 Eq. 30 Eq. 31 with E:Young's modulus; µ :Shear modulus; for plane stress, ⁄ for plane strain conditions This integral has interesting characteristics: firstly, it is independent of the selected contour, so it is possible to choose a distant contour from the crack front, thus avoiding the singularity zone; Then, this integral is not sensitive to elements size at the crack front; finally this integral is equal to the strain energy release rate if the material behavior is linear elastic.

II.4.3. Perturbation method

This technique allows calculating G I , G II and G III quickly. It assumes that a virtual advance Δa of a crack causes only a variation in the stiffness matrix without any modification of the displacement field and the external forces. Note that the advance of crack Δa disrupts elements containing the nodes of the crack front and the variation of the stiffness matrix is carried out locally. The values determined by this method are not very sensitive to the size of elements in the vicinity of the crack if it is reasonably defined.

II.4.4. Cohesive Zone Model

The notion of cohesive force at crack front was introduced by Barenblatt [START_REF] Barenblatt | Mathematical theory of equilibrium cracks in brittle fracture[END_REF] . Cohesive Zone Model (CZM) provides an effective methodology to study and simulate fracture in solids especially delamination in composite materials. Thus, it is widely used to characterize crack propagation behavior [START_REF] Morais | Mode I cohesive zone model for delamination in composite beams[END_REF][START_REF] Morais | Cohesive zone beam modelling of mixed-mode I-II delamination[END_REF][START_REF] Xie | Closed-form solutions for cohesive zone modeling of delamination toughness tests[END_REF][START_REF] Farmand-Ashtiani | Delamination in cross-ply laminates: Identification of tractioneseparation relations and cohesive zone modeling[END_REF] . These finite elements are used to model the interface between surfaces which would be separated when the crack propagates.

In most formulations, the behavior consists of a reversible linear increase in the stress of "traction" with the relative displacement of "separation" of the nodes firstly. As the cohesive surfaces separate, traction first increases until a maximum is reached, and then subsequently reduces to zero which simulates a softening of damaged material and results in complete separation. The variation of traction in relation to displacement is plotted on a curve and is called Traction-Separation-Load-Curve (TSLC), shown in Figure II. 34, which plays a vital role in the quality of the results obtained by numerical simulation. The TSLC relates the traction between originally coincident nodes located on either side of the crack and their relative displacement (opening/sliding). The area under this curve represents the energy needed to break the element. CZM maintains continuity conditions mathematically despite physical separation. Thus, it eliminates singularity of stress and limits it to the cohesive strength of the material.

Figure II. 34. Cohesive Zone Model

There are many forms of TSLC laws in the literature (Figure II. 35): bilinear [START_REF] Alfano | Finite element interface models for the delamination analysis of laminated composites: mechanical and computational issues[END_REF] , multilinear, exponential [START_REF] Xu | Numerical simulations of fast crack growth in brittle solids[END_REF] , trapezoidal [START_REF] Dugdale | Yielding of steel sheets containing slits[END_REF] , polynomial [START_REF] Needleman | A continuum model for void nucleation by inclusion debonding[END_REF] . Herein, we introduce a widely used bilinear form in details. The parameters describing that bi-linear form are:  The area under the curve corresponding to the critical strain energy release rate G C ;  The slope of the first linear part of the curve K representing the stiffness of the cohesive area; The tensile strength σ max corresponding to the element's damage initiation;  The ultimate displacement f corresponding to the displacement of the nodes at fracture of the cohesive zone. Finite element methods to characterize delamination behavior are introduced. VCCT and CZM are discussed especially and chosen for our coming study. VCCT allows calculating strain energy release rates for each element of the crack front. It is widely used to predict crack onset and distribution of strain energy release rate along a crack front. CZM allows characterizing crack propagation by avoiding singularity of stress and limiting it to the cohesive strength of the material. However, the determination of the characteristic parameters of TSLC is still a complex issue, which are neither clearly related to the mechanical properties of the substrates nor to those of the adhesive.

II.5. Summary

In this chapter, mechanisms of damage and fracture of laminated composite, most existing fracture criteria, most delamination testing methods with the participation of mode III, finite element methods to characterize and predict delamination behavior are reviewed.

First of all, most damages modes in a laminated composite are discussed with a particular focus on delamination which is one of the most common and dangerous damage modes in a laminated composite under impact and fatigue loadings.

Then criteria for the prediction of composite damage and fracture are introduced. Fracture mechanics is preferred to characterize delamination behavior dues to the singularity at crack tip. Generally, delamination occurs in mixed mode I + II + III. In order to establish and identify a general criterion, one must firstly know how to measure the delamination toughness in each pure mode. Then mixed mode I+II and I+III delamination behaviors should be characterized. Determination of the delamination toughness of laminated composites with the participation of mode III components remains a difficult task. Currently, delamination tests in pure mode I, pure mode II, as well as mixed mode I+II are widely studied. On the contrary, work regarding the participation of mode III is quite limited.

Most pure mode III and mixed mode delamination tests involving mode III in literature are reviewed. Pure mode III delamination test is not simple to achieve with uniform distribution of G III along the crack front, which is required by an exact and meaningful closed-form expression and the achievement of pure mode III test is also the fundamental of a successful mixed mode delamination test.

In order to study delamination phenomenon in a laminated composite, finite element analysis is shown necessary. Some kinds of numerical simulation methods are summarized, among which VCCT and CZM are preferred in this study. VCCT can predict the crack onset and calculate strain energy release rates along a crack front while CZM allows characterizing crack propagation.

In the study, we focus on the delamination in pure mode III first of all, and then mixed mode delamination with participation of mode III. A dialogue between FEM and experiments is the basement of our approach. The specimens were placed under axial tensile load until complete fracture. The longitudinal and transverse deformations were observed by an extensometer and a strain gauge during the tests. The maximum tensile strengths were also obtained for all the specimens 0° and 90°, the average values of which are considered as X + and Y + . The inplane shear strength, S

CHAPTER III. PRELIMINARY STUDY

III.2.3. Results and discussions

III.3. Establishment of the finite element models

Finite element analysis was carried out using LS-dyna in order to characterize the delamination behaviors of laminated composites. The fixtures and specimens were modeled with 3D solid elements. Spring elements were arranged along the crack front in order to obtain loads at the crack front. Then VCCT was performed to determine the evolution of G I , G II and G III along the crack front. Longitudinal strain (×10 -4 )

Every ply was set up according to the stacking sequence independently and then merged together. In the pre-crack plane, only the non-delaminated area is merged. A surface contact, named Contact_automatic_surface_to surface, was applied in most finite element models on the two delamination surfaces to prevent interpenetration during the analysis. Relative sliding between points within the delamination plane was assumed to be frictionless.

(1) material model for laminate MAT 22 was chosen to simulate the composite laminates due to its better performance, where geometry nonlinear shear behavior and Changchang fracture criterion for orthotropic materials have been adopted. In fact, both of MAT22 and MAT59 were employed in same models firstly. In our context, finite element models with these two kinds of material models give similar results. The variation curve of strain energy release rate obtained is smoother from models with MAT59. However, the mechanical criterion for MAT59 is Maximum Stress Failure Criterion, which cannot fit for our situation very well. The principal mechanical properties of material were set according to Table . 1.

( As mentioned in chapter II, Irwin-Kies equation Eq. 1 is applied to evaluate delamination toughness with different compliance calibration method. Finite element analysis by VCCT was also carried out to determine the mode I delamination toughness. Eq. 32 , where a is the crack length; E is the longitudinal Young modulus; δ is the opening between the two sub-laminates; J the moment of inertia of a sub-laminate given by: Eq. 33 So the G IC is expressed as Eq. 34 , where δ C is the opening between the two sub-laminates corresponding to the critical load P C . And b is the width of the specimen.

) 2
The compliance calibration (CC) method with two compliance laws was also used in the DCB test. The compliance law proposed by Berry (Berry's Law), which is widely used in literature, is expressed by Eq. 35.

Eq. 35 where α and u are parameters intrinsic to material, determined by interpolation of the experimental compliance as a function of the initial crack length: C/a.

By applying Berry's law, the Irwin-Kies' formula is expressed as Eq. 36.

Eq. 36 Another compliance law inspired by the beam theory (Beam Law) is also common in literature as shown in Eq. 37

Eq. 37 53 / 125

where A and B are determined by interpolation of the experimental compliance C versus a [START_REF] Berthelot | Composite Materials: Mechanical Behavior and Structural Analysis[END_REF] .

By applying Beam law, the formula Irwin-Kies is expressed as Eq. 38.

Eq. 38 The dimensions of specimens are listed in And G IIC is expressed as Eq. 39.

Eq. 39 , where δ C is the opening between the two sub-laminates corresponding to the critical load P C . And a, b and L are the crack length, the width and length of the specimen.

Irwin-Kies' formula (Eq. 39) with beam theory compliance law (Eq. The mesh close to the crack front was locally refined according to literature [START_REF] Davidson | Effect of stacking sequence on energy release rate distributions in multidirectional DCB and ENF specimens[END_REF][START_REF] Davidson | Three-dimensional analysis of centerdelaminated unidirectional and multidirectional single-leg bending specimens[END_REF][START_REF] Samborski | Numerical analysis of the DCB test configuration applicability to mechanically coupled Fiber Reinforced Laminated Composite beams[END_REF][START_REF] Theotokoglou | A finite element analysis of angle-ply laminate end-notched flexure specimens[END_REF] to guarantee a good accuracy of the results. In DCB test, P C is determined as the loading value at the end of the linear part of the load/displacement experimental curve. The results of G IC from different methods are listed in Table . 10. It is shown that the value from the beam theory is 21.1% bigger than the one from compliance calibration using the Beam law, and 12.5% bigger than that of from compliance calibration using Berry's law. curve goes down progressively. The onset of the crack growth has been observed after the linear part, so the critical load corresponding to the end of the linear part is defined. In order to apply the compliance calibration method, a series of compliances have been measured on the linear part of each tested specimen having different initial crack lengths. Figure III. 14 shows the interpolation of C/a 3 with the Beam Law (Eq. 37). The interpolation is precise with a correlation coefficient larger than 0.96. In ENF test, the critical load P C is determined at the end of the linear part of the experimental load/displacement curve mentioned above. The results of G IIC from two methods are listed in Table . 11. The test reproducibility is good with a standard deviation around 5%. The theoretical value of G IIC is 11.1% larger than that the one from the compliance calibration method. As in literature, the value from the beam law is usually used as the mode II delamination toughness. 12, where G CC represents the G IC or G IIC obtained by compliance calibration (CC) methods using Beam law. In our opinion, the results from the FE method are more reliable if all constants of the material used are correct. However, the average value doesn't really have a physical sense if the variation of the strain energy release rate along the crack front is too big. If the relative variation parameter defined above as following: Δ=(|G i-max -G i-av |)/G i-av with i=I or II, Δ is not small enough in our DCB tests. As a result, it is better to consider the maximum value of the strain energy release rate as the fracture toughness in the DCB test. The toughness obtained by experimental CC method is considered as a practical approach. In general, the characterization of mode I and mode II delamination behavior has been well studied, whereas only DCB on unidirectional As mentioned above, in the characterization of the delamination behavior, the participation of mode III remains a complex issue because even the measurement of the toughness under pure mode III is problematic. Before the establishment of a general criterion with the participation of the three modes, this scientific obstacle must be overcome. This is why in this study firstly we have paid our attention on the testing method to evaluate the delamination toughness under pure mode III. At the beginning, some testing methods proposed in the literature have been studied. Eq. 40 [START_REF] Marat-Mendes | Characterisation of the edge crack torsion (ECT) test for the measurement of the mode III interlaminar fracture toughness[END_REF] was applied to calculate G IIIC , where the critical load P C is corresponding to the load at the end of the linear domain in the load-displacement curve.

( ( ))

Eq. 40 , where a, B and L are the specimen dimensions shown in Figure IV. 1. C is the compliance corresponding to a. m and A is obtained by the interpolation method Eq. 41. [START_REF] Marat-Mendes | Characterisation of the edge crack torsion (ECT) test for the measurement of the mode III interlaminar fracture toughness[END_REF] * ( )+ Eq. 41 The dimensions of ECT-1 specimen are listed in The boundary conditions were set according to the real experimental conditions. As shown in Figure IV. 3, the displacement loading curve 4.5mm was imposed at the corner using a ball. The only degree of freedom allowed is in thickness direction. The other balls were all fixed. A surface to surface contact was set between the specimen surface and the balls, as well as between the two sub-laminates of the specimen. There was no other restriction on the specimen plate.

In the model, a =20mm, B =40mm and L =90mm. The mesh close to the crack front was locally refined according to literature [38; 44] to guarantee a good accuracy of the results. The mesh for ECT-1 test is shown in In order to apply Eq. 40 for the determination of mode III toughness, the compliance calibration (CC) method has to be realized to determine the constant m. A series of tests on the specimens with different initial crack length (Table . 13) were performed to measure their compliances. According to the compliance law (Eq. 42) the interpolation between 1/C and a/B is shown in Figure IV. 5, so the constant m=0.5392 is obtained.

Figure IV. 5. Interpolation between 1/C and a/B

The mode III toughness for each specimen determined by Eq. 40 is shown in 14 is also presented in the same figure, named as G IIIC-CC .

Firstly, G IIIC-CC is about 38.6% higher than the average numerical value G IIIC-FEM . Secondly, the evolution of G III is not uniform at all, and the value drops to zero at the extremities (x/l=0 and 1), Δ of G III is 38.4%. Herein, the average value is actually not meaningful. However, the maximum value of G III from the finite element model is very close to the one from compliance calibration method. Thirdly, at the extremities of the crack front (x/l=0 and ECT-1 test proposed in literature was performed on a multidirectional composite obtained from a taffetas balanced fabric prepreg. The finite element analysis using VCCT was also carried out in order to investigate the distribution of G I , G II and G III along the crack front.

For ECT-1 test in this study, the critical strain energy release rate evaluated is probably G IIC instead of G IIIC . That shows the concern regarding ECT-1 test that may identify G IIC instead of G IIIC if the material and the geometry of specimen are not properly chosen. Some care must be taken to avoid errors in mode III delamination tests. Moreover the distribution of G III along the crack front is far from being uniform, which decrease the precision of determination of the delamination toughness if the average value is considered.

MECT-1 was also analyzed by finite element method. This test has a better performance than ECT-1 test in improving the uniformity of G III along the crack front. The evolution of G III is fairly uniform in the middle domain of the crack front while the values of G III approach 0 at the extremities. Moreover, mode II becomes also the principal mode at two extremities, where the maximum of G II is very close to the one of G III in the middle domain. Therefore, this test presents a big advantage: we can not only compare the value of G IIC and G IIIC , but also measure one of them by identifying the area where the delamination initiates.

IV An estimation method of the delamination toughness was provided in the work of H. Suemasu [44] , which doesn't perform well for multidirectional laminates. Inspired by the work of R Marat-Mendes and M. de Freitas [START_REF] Marat-Mendes | Characterisation of the edge crack torsion (ECT) test for the measurement of the mode III interlaminar fracture toughness[END_REF] , we proposed a method with compliance calibration (CC). The critical strain energy release rate was evaluated according to Eq. 42, which is a modified form of Eq. 40 [START_REF] Marat-Mendes | Characterisation of the edge crack torsion (ECT) test for the measurement of the mode III interlaminar fracture toughness[END_REF] . The compliance law remained the same as Eq. 41. The interpolation of the experimental data between 1/C and a/B gives the constant m.

( ( ))

Eq. 42 , where a, B and L are the specimen dimensions shown in In order to impose the torsion, a rotation curve 0.44rad was employed on the outermost face of one jaw while the other one was fixed. A surface to surface contact was set between the specimen surface and the jaws, as well as between the two sub-laminates of the specimen. There was no prestressing force between the jaws and the specimen. Moreover, the two points at the edge of the specimen along the loading axis were restricted. At the loading side, an axial displacement is permitted. In the fixed jaw, the point is fixed as well.

The mesh close to the crack front was locally refined according to literature [38; 44] to guarantee a good accuracy of the results. Not only the propagation at the pre-crack was observed, but also a little propagation at the opposite edge, where the expression proposed for the determination of G IIIC is no longer available. It seems that the onset of crack happens before reaching T C-3 . The critical torque is believed to be between T C-1 and T C-2 . Similar conclusions were gained in the work of Marat-Mendes R. [START_REF] Marat-Mendes | Characterisation of the edge crack torsion (ECT) test for the measurement of the mode III interlaminar fracture toughness[END_REF] . The results calculated by Eq. It is shown firstly that the uniformity of G III is better in ECT-2 test than in ECT-1 test, but the values of G III also drop to almost zero at the extremities of the crack front while the variation of G III in the middle domain along the crack front is much less significant. Δ is approximately 14.0%. Secondly, G I is negligible and G II is much smaller than in ECT-1 and MECT-1 tests.

Thirdly, the G III-CC calculated according to Eq. 42 is 16.2% smaller than G IIIC-FEM . Recall that in the ECT-1 tests, the G III-CC calculated according to Eq. 42 is always higher than G IIIC-FEM , no explanation can be given for this inverted result.

IV.1.2.4. Conclusions

In ECT-2 test, there is much less participation of G I and G II components than in ECT-1 test. Δ is big when the whole crack front is taken into consideration because the values of G III drop dramatically at the edges. However, the critical load corresponding to the onset of crack growth is not easy to define. The values of G III drop to almost zero at the extremities of the crack front. In addition, the G IIIC-FEM becomes 16.2% higher than the G IIIC-CC . If the maximum of G III is considered, the difference between G IIIC-FEM-max and G IIIC-CC is more important, the former is about 26.5% higher than the later. Besides, the loading condition has to be optimized to ensure crack propagation in the desired area.

IV.

G III G II G I G IIIC-FEM G IIIC-CC
above is therefore eliminated. Moreover the variation of G III along the crack front can be reduced by optimizing different parameters of the specimen and the testing device.

IV.2.2. Experimental

. 17 shows the schema of ERCT specimen. A pre-crack with a central circular front is created between two sub-laminates by inserting a 13 µm thick polyester film. A central disc is cut out of the inserted polyester film with the help of a circle cutter. So the circular hole is non-delaminated area while the blue film presents an Edge Notched Crack. In fracture mechanics, K III is related to strain energy release rate G III by Eq. 44 for isotropic materials if the crack growth is in its initial plane (self-similar manner), where G is the shear modulus of the isotropic material.

Eq. 44 Even though the crack in ERCT specimens is embedded into two plates in laminated composite instead of an isotropic cylinder, we want to know whether this closed-form formula can be applied for determining G III in the case of ERCT test and whether the function needs to be adapted to the ERCT test.

IV.2.3. Finite element analysis 32 plies were set up for ERCT test according to the stacking sequence highlighted in Chapter I and they were separated symmetrically into two sub-laminates by the crack plane. The steel device and bondline were also simulated. Then two kinds of model were established. For the first model, spring elements were arranged along the crack front in order to obtain loads at the crack front and then to determine the evolution of G I , G II and G III along the crack front using VCCT.

For the second model, cohesive elements were arranged at the non-delaminated area and there was no spring element along the crack front. The model aims at verifying the onset of crack growth. A bilinear Traction-Separation-Load-Curve (TSLC) was applied according to literature [START_REF] Peng | Modélisation numérique d'assemblages collés : Application à la réparation de structures en composites[END_REF] . The parameters of the cohesive element material model are shown in Table . 16. The mesh close to the crack front was locally refined (Figure IV. 24) to guarantee a good accuracy of the results. Mesh refinement has been studied. It demonstrates that the difference on G i (i=I, II, III) calculated by VCCT between the model with the current mesh and that with twice elements in the plate plane (r, Ɵ) near the crack front is about 1%. Concerning the thickness direction z, a model with two elements in the thickness of each ply of the plate has been tested on a simplified model loaded in torsion and leads to 2.7% of variation. But the quality of the finite element is degraded by refining the mesh in the z direction, because its dimension in z is too small compared to those in r and Ɵ directions.

Actually, the finite element model was also validated by comparison of the calculated G III using VCCT and Tada formula, which are obtained from an isotropic cylinder model. The difference is less than 2%. Herein, the isotropic cylinder is in steel with length of 2x50 mm and diameter of 120 mm. An edge ring crack is located at the mid-length, whose non cracked area has diameter of 30 mm. It is interesting to note that the cylinder simulated here is not really an infinite cylinder, but Tada's formula seems to be a good approach for an isotropic material.

The where the white zone presents non-delaminated area. It demonstrates that the crack propagated indeed. Moreover, it seems that the crack propagated much more along an angle Ɵ C close to (45°). One explanation may be that the crack resistance varies as a function of the adjacent fiber orientations, and the resistance of the crack tips at Ɵ C is smaller under shear mode. We can also imagine that the penetration of air or/and water into the delaminated areas after the test depends on their fracture modes and damage level. In any case, the interpretation of the images obtained from Ultrasonic C-scan requires further study. The results of G IIIC according to Eq. 43 and Eq. 44 are listed in Table . 17. Note that G in the Table . 17 should be G zƟ for a laminate, which generally vary as a function of Ɵ. In order to make it clearer, a schematic of fiber orientation and crack front is shown in Figure IV. 28. However, the tested laminates are quasi-isotropic in our case. So G zƟ is constant and G zƟ = G 13 according to the analysis by J.M. Berthelot [START_REF] Berthelot | Maté riaux composites: comportement mé canique et analyse des structures[END_REF] . Actually it depends not only on the geometry of the device used for applying the torque, which is not strictly axisymmetrical, but also on the composite specimens, especially on the adjacent fiber orientation relative to local crack front. The ratio of d/D ring is also a parameter that determines the importance of the effects mentioned above. Δ is about 7.5% for the specimens with d=30 mm and 17.4% for those with d=50 mm. Moreover, the maximum G III is localized at Ɵ=0° and 180°, so the propagation of the crack should initiate at these angles if the toughness in pure mode III was independent of adjacent fiber orientation.

It is shown also that for the specimen with d = 30 mm, the G IIIC-Tada value agrees well with G IIIC-FEM . The difference between G IIIC-Tada and G IIIC-FEM measured on the specimens with d=30 mm is only approximately 2.2%; this difference is more significant measured on those with d=50 mm, which is approximately 20.5%. It seems that if the variation of G III along the crack front is large, the closed-form proposed by Tada would no longer be applicable.

GIII GII GI GIII-num_av GIII-exp_av G (kN/m) Ɵ(°) G III G II G I G IIIC-FEM G IIIC-Tada 82 / 125
千 GIII GII GI GIII-num_average GIII-exp_average Ɵ (°) G (KN/m) G III G II G I G IIIC-FEM G IIIC-Tada
prepreg. These tests were also simulated by the finite element method to investigate the evolution of G I , G II and G III along the crack front.

The most significant advantages of ERCT test are:  the fracture mode is pure mode III without the participation of mode I or mode II components;  Δ along the crack front is much smaller than ECT tests; it can be less than 7.6% by optimizing the geometry of the specimen. So it's acceptable to consider the average value as the material toughness;  the critical load corresponding to the crack onset is easy to define;  the agreement between G IIIC-Tada and G IIIC-FEM is good for a specimen with a small diameter d.

Some inconveniences should be discussed:  the coincidence of axes and the relative position between the crack front and the axis are significant. The gluing process and installation of the devices may introduce errors.  Preparing the specimen and curing the glue may take about 1 hour.  The rigid plates are recycled in this test and the bondlines have to be cleaned by acetone every time.

It is interesting to note that the angle Ɵ between the crack front and the fiber orientation, as shown in Figure IV. 28, varies regularly, which is different from all the traditional pure mode tests. The delamination resistance may also vary along with Ɵ. In this case, the delamination toughness evaluated by Tada formula presents the smallest value, which is also meaningful.

Moreover, it is better to propose new testing methods by using Edge Ring Crack specimen in order to compare three pure mode delamination tests without interference caused by Ɵ.

In conclusion, ERCT test is a promising pure mode III testing method for measuring G IIIC . It maybe presents better results than most tests in the literature in terms of two aspects: the elimination of mode I and II components and the reduction of G III variation along the crack front. However, further improvement of the ERCT test is still possible based on: the study of the factors affecting the evolution of G III along the crack front; the sensitivity study to experimental defaults; the development of a closed-form expression more adequate for determining G IIIC and the improvement of the testing device.

IV.3. Factors affecting the evolution of G III along the crack front of ERCT test

In this section, the factors affecting the evolution of strain energy release rate is analyzed and discussed. A polar coordinate system (r, Ɵ) is employed in order to show evolution of G III normalized by average value. Mode I and mode II components are always shown to be negligible with all the models so that they are not presented. Three influence factors are discussed in terms of device geometry, laminate geometry and relative ratio d/D ring . Moreover, an ERCT model with steel specimen is established by replacing composite material model by steel material model and keeping all the other parameters.

IV.3.1. Influence factor from the device

First of all, the original ERCT (device A) and a modified version device C are established (shown in Figure IV. 32) in order to study the influence of device geometry on the evolution of G III , which is called as device factor for short. For the device C, the loading head is replaced by a cylinder of the same height and the square steel plate is replaced by a round one with a diameter 120mm, equal to the side length of the square plate. Note that the pasted ring still remains the same in the device C with inner and outer diameters 80mm and 120mm respectively. Device A has two planes of symmetry: XOZ and YOZ while device C is totally axisymmetrical. The specimens for A and C are set as same isotropic steel in order to avoid disturbing from the material of specimen. All the influence from the composite material is named laminate factor for short. Firstly, a model of device C with composite specimen is established. In this specimen, the laminate stiffness matrix A, B, D maintain the same in all directions because the specimen is quasiisotropic; the laminate tested is also quasi-homogeneous, which means the same elastic stiffness properties in tension and in bending in all direction.

The evolution of G III along the circular crack front with steel and composite specimens using device C is shown in Figure IV. 35. The laminate factor has a small influence on the evolution, with Δ less than 3%. The values of G III increase approaching 0° and 90° while they decrease approaching ±45°. It is believed that the tiny variation results from the influence of the fiber orientations adjacent to the precrack plane. Firstly, two specimens in steel using device A were analyzed numerically which are distinguished only by changing d. "A-steel-d30" and "A-steel-d50" denote the results for the specimen with d=30mm and for that d=50mm, respectively. As shown in Figure IV. 36, the value of G III from the results A-steel-d50 increases significantly approaching 0° and decrease approaching 90°. The same tendency on the variation of G III is also observed on the results from A-steel-d30, but the amplitude of this variation is smaller than that from A-steel-d50. It demonstrates that ratio d/D ring has an important influence on the distribution of G III along the crack front. Moreover, the ratio d/D ring has an influence on the device factor. A positive correlation is found between the device factor and ratio d/D ring in a certain range. The original ERCT test on composite specimens is in the case of the test with device A, whose geometry was chosen to adapt the jaws of the testing machine. The distribution of G III is influenced by all the three factors discussed above. If there is a parameter f responsible for the evolution of G III , it should depend on the superposition of the influence of the device factor, of the ratio d/D ring and of the laminate factor. Actually, the effect of the device factor on the variation level Δ becomes smaller with a decrease of the ratio d/D ring , In this case, f should be dominated by the laminate factor; in the other case, where device factor is big when d/D ring is big, f should be controlled by the device factor. The idea is that the parameter f is maybe a useful tool for optimizing ERCT tests.

In order to understand the role of the superposition of different factors in the optimization of ERCT tests, we have studied an example. If we consider the curve of the results "C-compos-d50" resulting from laminate factor and that of "A-steel-d50" resulting from device factor, the curve of the results "A-compos-d50" should be the result of the superposition of above two factors: the laminate factor and that of device factor. The average of the curve of the results "C-compos-d50" and that of "A-steel-d50" looks similar to the curve of the results "Acompos-d50" shown also in Figure IV. 39. The definition of a parameter f seems useful to predict the evolution of G III along the crack front. However, the results presented here are not rich enough to conclude on the definition and on the use of the parameter f. It is valuable to study this parameter with much more cases in the future. , it is named as B45. B45 has a better performance in reducing the variation of G III , namely with 2.9% of improvement. Even though the improvement is negligible for our composite, the idea of the study is meaningful, which aims at optimizing the influence of the laminate factor combined with the device factor on the distribution of G III . The former may become significant when some architectures of composite are used. In these cases, the laminate factor can be neutralized by the device factor in applying this kind of installation of the specimen relative to the device. Finite element analysis was performed in order to analyze the factors affecting the evolution of G III along the circular crack front in ERCT test. It demonstrates that the distribution of G III depends on the superposition of all influence factors, in particular the device factor and the laminate factor. Moreover, the device factor is in a positive correlation with the ratio d/D ring while laminate factor is in a negative correlation with d/D ring . For ERCT test the main influent factor is the device factor. Optimization of the device is effective but it is only meaningful when the laminate factor is big. The device C can realize a pure mode III delamination test and gives a uniform distribution of G III along the crack front. The advice about the crack dimension is given. The optimal ratio d/D ring equal to approximately 0.5 is recommended because the values of G III are nearly uniform and the non-delaminated area is big enough for observation of the crack propagation.

IV.4. Modified Edge Ring Crack Torsion tests

Updated ERCT tests were proposed based on the finite element analysis above, aiming at achieving a pure mode III delamination test with uniform distribution of G III along the circular crack front. The sensitivity study of the loading position, the shape of the crack front and the shape of the specimen to modified ERCT test were also carried out.

IV.4.1. Experimental

Firstly, two kinds of Modified ERCT devices were designed and fabricated. The tests were realized under the same conditions as those in the original ERCT test. In fact, the loading transfer conditions of the two tests are different. For ERC-III-1 tests, the applied load is

.2. Circularity of the crack front

A shape default of the crack front may be introduced during the cutting of the polymer film. Thus, a finite element analysis about the shape of the crack front was carried out. In the test with device C, an oval crack front (2b=48mm at 90°, 2a=50mm at 0°) is set instead of the circular one with d=50m. The result presented in Figure IV. 51 shows that little increase of mode I and II components are found so that it's still in pure mode III delamination. Moreover it is shown in Figure IV. 52 that the variation of G III along the oval crack front is more important than that of a circular one. Note that Δ = 9% for the former while Δ=2.7% for the later. The loading axis is perpendicular to the rigid plate so that the two rigid plates are always parallel when the loading axes are blocked in the jaws of a torsional machine. However, the specimen can be inclined if the thickness of the bondline is not uniform. In this situation, the torsinal loading axis is not perpendicular to the crack front. In order to study the error caused by this situation, a FE model is established where the thickness of bondline varies from 0.2 mm to approximately 2.2 mm. An angle 1° is formed between the specimen surface and the rigid plate. The results presented in Figure IV. 53 show that little increase of mode I and II components is found so that it's still a pure mode III delamination. Figure IV. 54 shows that there is practically no effect of the default of parallelism, because the distribution of G III normalized by its average value from two cases coincide. In this section, two novel pure mode III testing methods were developed and evaluated including modified ERCT-1 (ERC-III-1) test and modified ERCT-2 (ERC-III-2). The modification on the original ERCT test concerns in particular the geometry of the parts for transmitting the applied torque to the specimens. These modified parts in both of two versions are cylindrical so as to assure axisymmetrical loading. But the loading transfer for the two tests is realized in a different manner. For ERC-III-1 tests, the load is applied on the surfaces of the specimen through a ring adhesive joint; while in ERC-III-2 tests it is applied at the circular edge of the specimen through the contact between the screws and nicks machined in the specimen.

The tests developed here were performed on a multi-directional quasi-isotropic and quasihomogeneous laminate. Finite element analysis (FEA) was used to evaluate the quality and the robustness of the tests.

The most significant advantages of MERCT tests are shown including:  the fracture mode is pure mode III without the participation of mode I or mode II components indeed;  good distribution of G III along the crack front. Δ is much smaller than ECT tests and original ERCT test. In fact, a nearly constant value of G III along the crack front is obtained in the MERCT tests. So it's more acceptable to consider the average value, G IIIC-FEM, as the material toughness;  the agreement between G IIIC-Tada and G IIIC-FEM is good with a difference approximately 8% for both MERCT tests. G IIIC-Tada is smaller than G IIIC-FEM , which guarantees the security to consider G IIIC-Tada as the delamination toughness.

Some drawbacks of MERCT tests exist in the manufacture process and installation of the testing devices. The parallelism of the rigid plates and the alignment between the crack center of the specimen and the axis of loading shafts should be checked carefully. For ERC-III-1 tests, it is time consuming to clean the glue on the surface of the rigid plates before sticking the specimen for each test. For ERC-III-2 tests, some care must be taken in order to reduce the assembly stress because a lot of screws are employed.

In conclusion, both of the two MERCT tests are able to achieve a pure mode III delamination and give nearly uniform distribution of G III along the circular crack front. It maybe presents the best result so far in terms of two aspects: the elimination of mode I and II components and the reduction of G III variation along the crack front. The robustness of MERCT tests is well confirmed. Some defaults come from the manufacturing process and the testing process are generally inevitable. Unwanted fracture mode introduced by different defaults seems no significant. But the variation of G III along the crack front is more sensitive to certain defaults, such as the circular shape of the crack front and the alignment between the crack center and the axis of loading shafts. For ECT-1 test, the critical strain energy release rate measured is probably under mixed mode instead of pure mode III. Moreover, the use of an average value of G III determined by finite element analysis, G IIIC-FEM, as the toughness is not really meaningful, because the variation of G III along the crack front is too important. As a result, the results coming from this test are not reliable, they should be out of the comparison of G IIIC . Optimization on specimen geometry and the stacking sequence may reduce errors in tests. However, we didn't optimize them in order to compare with the following tests with the specimens of the same stacking sequence.

Modified ECT-1 (MECT-1) test provides a better performance than ECT-1 test by improving the uniformity of G III along the crack front. The evolution of G III is fairly uniform in the middle domain along the crack front while the values of G III approach 0 at the extremities, where mode II component increases and becomes the principal mode. Even thought the delamination mode around the edges of the crack front should be under mixed mode II+III, the maximum of G III at the middle area of the crack front is practically the same than the one of G II at the two extremities of the crack front. The interest of this test can be described as following:

If G IIC < G IIIC, the onset of delamination should be at the two extremities of the crack front; If G IIC > G IIIC, the onset of delamination should be at the center area of the crack front; If G IIC = G IIIC, the onset of delamination should be simultaneously at all crack front.

In future work, it is valuable to realize MECT-1 tests to characterize the delamination behavior of composite.

In ECT-2 test, there is much less participation of G I and G II components. The distribution of G III is uniform in the middle domain of the crack front while G III drop to almost zero at the extremities. As ECT-1 test, the variation of G III along the crack front is too important to use its average value as material toughness. Actually, G IIIC-FEM for ECT-2 listed in Table 18 represents the average value of those calculated by finite element analysis, which becomes 20% higher than G IIIC-CC . Besides, the critical load corresponding to the onset of crack growth is not easy to define in this test because strong nonlinearity is observed before the maximum load is attained.

Concerning ERCT test, according to the results from finite element analysis the delamination mode in this test is pure mode III without the presence of mode I or mode II; the variation of G III along the crack front decreased substantially compared to ECT tests, and it can be even less than 7.6% by optimization of the geometry of the specimen. So it's acceptable to consider its average value as the material toughness. Note that G IIIC-FEM for ERCT test with specimens d=30 mm and d=50 mm are 26.6% and 3.3% smaller than the one from ECT-2 test respectively. However the variation of G III along the crack front can still not be eliminated. The bigger the diameter d, the more important the variation of G III and then the more different the average value of G III obtained from EFM and that determined by Tada's formula.

Then the factors affecting the evolution of G III along the crack front were studied. It demonstrates that the distribution of G III depends on the superposition of all influence factors, in particular that of device factor and laminate factor. Moreover, the device factor can mitigate the variation of G III due to the ratio d/D ring so in a positive manner, but the laminate factor can accentuate the variation of G III due to the ratio d/D ring , so in a negative manner. The main influence factor is the device factor, especially for a specimen with a big d/D ring . Tada's formula is found a simple and practical approach to determiner G IIIC , which gives the results good enough only when the variation of G III is small. An optimal ratio d/D ring equal to approximately 0.5 is recommended because the values of G III are nearly uniform and the nondelaminated area is big enough for observing the crack propagation.

Finally, modified ERCT (MERCT) tests were developed and evaluated in two versions based on the finite element analysis above, aiming at achieving a pure mode III delamination test with uniform distribution of G III along the circular crack front. The modification on the original ERCT test concerns in particular the geometry of the device parts for transmitting the load to the specimens in an axisymmetrical manner. For ERC-III-1 tests, the load is applied on the surfaces of the specimen through a ring adhesive joint; while in ERC-III-2 tests it is applied at the circular edge of specimen through the contact between the screws and nicks machined in the specimen. Finite element analysis (FEA) was used to evaluate the quality and the robustness of the tests, as well. With the help of the test devices developed in our study, different loading modes can be introduced so that the behavior of delamination in laminated composites can be investigated under every pure mode load, even under some mixed modes ones.

In this section pure mode I and pure mode II delamination tests were developed using ERC specimen. In this way, the three pure modes toughness measured should be more representatives with no additional effects on the results dues to geometry change.

V. Herein, the specimen was put on a rigid supporting ring of diameter D=110±2mm, and then loaded under compression in the centre of the specimen. Note that ERC specimen under pure mode I load has to be pasted to the testing device, where the adhesive joints must be strong enough to guarantee the crack propagation in the mid-plane of the composite specimen, but not in the adhesive joint between the rigid plates of the test device and the surfaces of the specimen. A structural adhesive ARALDITE 2012 has been used for this purpose. For all of these tests, great care was paid to keep the coincidence between the center of the crack of the specimens and the loading axis of the testing machine in order to ensure the wanted pure loading mode. In ERC-I model, a displacement curve 8mm was employed on both of the loading axes in order to impose a traction load. All the rest parameters were set as in ERC-III model. In ERC-II model, a displacement curve 3mm was employed on the loading ball while the supporting ring was fixed. A surface to surface contact was set between the specimen surface and the balls, between the supporting ring and specimen surface and between the two sub-laminates of the specimen. The other parameters were set the same as in ERC-III model. In conclusion, ERC-I test is a pure mode I delamination test with uniform distribution of G I along the crack front. Therefore, with the average critical load measured, we can determine the pure mode I toughness G IC . In this work, if the peak tensile load is defined as the critical load, the value of G IC determined as the average value from FEM is given as 319N/m, which is quite close to the value 348N/m measured by DCB test on the same composite. However, the definition of critical load corresponding to the crack onset needs further experimental observations. However, evident nonlinearity appears before reaching the peak tensile load. In this case, the end of the linear part should be considered as the critical load as well. By applying the force at the linear end, G IC is equal to 184N/m, which is 42.3% smaller than that calculated by peak load. By applying the load at the intersection of the curve and a line at 95% of initial slope of the linear domain, G IC is given as 210 N/m, which is 34.2% smaller than the one calculated with the peak load value. In our test, the loading axis of our testing device is not strictly corresponding to the jaws of the tensile machine. As a result, the loading axis may be not perpendicular to the crack plane. The small tilt angle may induce errors. Anyway, more ERC-I tests should be repeated in order to give a typical experimental force/displacement curve of ERC-I test and the crack onset should be verified in the future. After a small non linear part, an unstable propagation of the crack is observed, which is corresponding to a sudden drop in the compression load. Actually, most of the advantages of ERC-III are kept in the ERC-II test. Figure V. 9 shows the evolution along the crack front of G I , G II , G III normalized by G T in ERC-II test. It is seen that the relative mode I and mode III components are less than 1.5%, so they can be considered as negligible. Moreover, the distribution of G II along the crack front (Figure

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In conclusion, a pure mode II delamination test is achieved by using ERC-II test with uniform distribution of G II along the crack front. By introducing the average peak load obtained by series of ERC-II tests, the pure mode II toughness G IIC of the tested composite is measured at 1109 (N/m), which is determined as the average value from FEM. However, the value of G IIC measured by ENF test on the same composite is much lower, valued at 776 (N/m) from the same numerical method. The difference can be explained by the facts including the bigger Δ, the participation of mode III and the geometry effect in ENF test. The nonlinearity appears before force reaching the peak value but the nonlinear part is much smaller than in ERC-I test. By introducing the load at the end of the linear domain, the pure mode II toughness G IIC of the tested composite is measured at 829N/m. Firstly, by introducing different loadings, the pure mode I, the pure mode II and the pure mode III delamination tests can be performed on the ERC specimens of the same geometry. So the influence of the geometry of the specimens on the toughness measured can be avoided.

Secondly, the evolution of the strain energy release rate along the crack front is fairly uniform, where the average value is more meaningful. It is important to develop a closed-form expression to determine easily the toughness for each ERC test.

Thirdly, if the effect of adjacent fiber orientation on the delamination resistance is important so that the delamination toughness varies along the crack front of an ERC specimen, the toughness measured in ERC tests should be the smallest one, which is also meaningful. In this case, we can imagine that the points at the crack front with the lowest resistance to delamination could be detected by ERC tests.

Finally, it is important to develop a closed-form expression to determine easily the toughness for each of ERC-I and ERC-II tests.

V.2. Mixed mode I+II V.2.1. Introduction

Actually, the ERC specimens can also be tested under some mixed mode delamination loadings. For example, a mixed mode I+II delamination test can be realized under tensile load if the two sub-laminates are not symmetrical relative to the crack plane, and a mixed mode I+III test can be realized under a combined torsion and tensile load when the two sub- laminates are symmetrical relative to the crack plane. In the next parts, only finite element analysis (FEA) will be described, it was carried out in order to study the feasibility of any mixed mode loading on ERC specimens.

Based on the achievement of pure mode I, II and III delamination test with ERC specimen, mixed mode I+II delamination test was proposed in this work, named ERC-I+II for short. FEA was carried out in order to validate the ability of the ERC-I+II test.

This test is modified from ERC-I test, the participation of mode II can be introduced under traction when the specimen is not symmetrical relative to the crack plane. As the result of Poisson's ratio, the elastic strain in the (r, Ɵ) plane for two sub-laminates created by the crack are different under the tensile loading in z direction. That results in a mode II delamination component.

V Moreover, the variation of mode I and mode II along the crack front is similar which increases approaching ±45° and decreases approaching 0° and 90°. Δ of G I is approximately 12.5% and Δ of G II is 9.3%, which is relatively small compared to traditional mixed mode I+II delamination tests generally. A mixed mode I+III delamination test was proposed making use of ERC specimen, named ERC-I+III. The mixed mode I+III can be obtained by a combination of traction and torsion using the machine MTS809, when the stacking sequence of the specimen is symmetrical relative to the crack plan, the same as pure mode ERC test.

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V.3.2. Finite element analysis

In the FE model of mixed mode I+III delamination test, all the parameters are same as those used in pure mode ERC tests except for boundary conditions. A displacement curve 0.5mm and a rotation curve 0.0025rad were set in order to apply the mixed loading mode. The different mixed mode ratio in terms of G II /G III , can be obtained by combining the tensile and torsion rate. In this chapter, pure mode I and pure mode II delamination tests were achieved using ERC specimens. Their delamination behaviors were investigated by experiments and finite element analysis. Thus pure mode I, II and III delamination tests can be performed on the same ERC specimens. The influence of the geometry of the specimens on the toughness measured can be avoided.

In addition, the evolution of the strain energy release rate along the crack front is fairly uniform, where the average value is more meaningful. It is important to develop a closedform expression to determine easily the toughness for each ERC test.

Moreover, a mixed mode I+II delamination test using the ERC specimen is investigated by finite element analysis. There is no participation of mode III component in this test. The evolution of mode I and mode II is relatively small compared to traditional mode I+II delamination tests generally. The mixed mode ratio in term of G I /G II is controlled by the stacking sequence of each sub-laminate.

Based on the success of pure mode I and III tests, a mixed mode I+III delamination test using ERC specimen was investigated by finite element analysis, which is realized under a combination load of traction and torsion. The proportion of mode I and mode III component can be controlled by modifying the proportion of traction over torsion. The distribution of mode I component along the crack front is uniform while mode III component has a small evolution.

Finally, a mixed mode I+II+III delamination test is proposed and evaluated by finite element analysis. This test is simple to achieve and easy to control the proportion of three The object of the study presented in the thesis is to develop the testing methods for characterization of the delamination behavior under pure modes and mixed modes. Especially most of our attention has been paid to the testing methods for pure mode III delamination. We focus on the onset of crack growth and the distribution of strain energy release rate along the crack front. These are fundamental work for the purpose to establish a general mixed mode delamination criterion with the participation of mode I, mode II and mode III.

Therefore, this work contains not only experimental observations, but also finite element analysis (FEA). Correlation between the results obtained aims, on the one hand to better understand the crack growth especially the crack onset; on the other hand, to propose and improve testing methods, and to propose and validate simple approaches to calculate delamination toughness.

Pure mode III testing methods are studied. Firstly, two kinds of Edge Crack Torsion tests widely used in literature were carried out. The achievements of these tests are compromised by the drawbacks. The main disadvantages are on three aspects: (1) A participation of mode II component cannot be completely eliminated; (2) The distribution of G III along the crack front is not uniform especially near their edges; (3) Nonlinearity is always observed before reaching peak load in a load/displacement curve so that it's not easy to define the critical load. The disadvantages result in difficulty of determination of delamination toughness under pure mode III.

After the study of the two existing tests, a novel original mode III testing methods, named Edge Ring Crack Torsion tests (ERCT), were evaluated by experiment and FEA. The Edge Ring Crack (ERC) specimen was employed in these tests. The circular crack front in the specimen has no extremities so that it can avoid edge effects.

Some factors affecting the evolution of G III along the crack front in ERCT test were studied. It demonstrates that the distribution of G III depends on the superposition of all influence factors, in particular that of device factor and laminate factor. Moreover, the device factor is in a positive correlation with the ratio d/D ring while laminate factor is in a negative correlation with d/D ring . In ERCT test, the main influence factor is the device factor, especially for a specimen with a big d/D ring . Tada's formula is found to be a simple and practical approach to determiner G IIIC , which gives the results good enough only when the variation of G III is small. The optimal ratio d/D ring approximately equal to 0.5 is recommended because the values of G III are nearly uniform and the non-delaminated area is big enough for observation of crack propagation.

Finally, modified ERCT (MERCT) tests in two versions were developed and evaluated based on the FEA, aiming at achieving a pure mode III delamination test with uniform distribution of G III along the circular crack front. The modification on the original ERCT test is concerning especially the geometry of the device parts for transmitting the applying load to the specimens in an axisymmetrical manner. For ERC-III-1 tests, the load is applied on the surfaces of the specimen through a ring adhesive joint while it is applied at the circular edge of the specimen through the contact between the screws and nicks in the specimen in ERC-III-2 tests. FEA was used to evaluate the quality and the robustness of the tests as well.

The most significant advantages of MERCT tests are shown including:  The fracture mode does be pure mode III without the participation of mode I or mode II components;  Uniform distribution of G III along the crack front is achieved. Δ is much smaller than ECT tests and original ERCT test. In fact, a nearly constant value of G III along the crack front is obtained in MERCT tests. Then it's more acceptable to consider the average value, G IIIC-FEM, as the delamination toughness;  The agreement between G IIIC-Tada and G IIIC-FEM is good with a difference of only approximately 8% for both MERCT tests. G IIIC-Tada is smaller than G IIIC-FEM , which guarantees the security when considering G IIIC-Tada as the delamination toughness;  The robustness of MERCT tests is well confirmed. Some defaults introduced by the manufacturing process and test process are generally inevitable. Unwanted fracture mode introduced by different defaults seems no significant. But the variation of G III along the crack front is more sensitive to certain defaults.

In conclusion, MERCT tests are promising testing methods to characterize mode III delamination behavior. In ERC-III-1 test, an unstable crack growth was observed. And in ERC-III-2 test, at onset of delamination the crack propagated a step suddenly, following by a stable crack growth. This phenomenon makes it possible to stop the test before the final fracture of the specimen to observe the crack growth onset.

The application of ERC specimen was also spread into other pure modes delamination tests. Actually, by introducing different loadings, the pure mode I (ERC-I), the pure mode II (ERC-II) and the pure mode III (ERC-III) delamination tests can be performed on the ERC specimens of the same geometry. The influence of the geometry of the specimens on the toughness measured can then be avoided. The tests ERC-I and ERC-II keep most advantages of ERC-III test. The evolution of the strain energy releases rate along the crack front is fairly, where the average value is more meaningful.

Finally, it is important to develop a closed-form expression to determine easily the toughness for both ERC-I and ERC-II tests.

Moreover, it is possible to realize delamination tests of mixed mode I+II, mixed mode I+III and mixed mode I+II+III using ERC specimens. The numerical simulation has shown that no unwanted mode is produced for mixed mode I+II and mixed mode I+III; the distribution of the strain energy release rate is not completely uniform but the variation is limited. As a result the determination of delamination toughness based on average value is still acceptable. In each mixed mode I+II+III test, the proportion of each mode is easy to control so that it is possible to study any mixed mode delamination behavior by using ERC specimens.

It is interesting to note that if the effect of adjacent fiber orientation on the delamination resistance is important and that the delamination toughness varies along the crack front of an ERC specimen, the toughness measured in ERC tests should be the smallest one, which is also meaningful. In this case, we can imagine that the points at the crack front with the lowest resistance to delamination could be detected by ERC tests.

Figure I.

  Figure I. 1. Young's modulus and shear modulus for the laminate in membrane ............... Figure I. 2. Young's modulus and shear modulus for the laminate in flexion .....................

  Figure I. 1. Young's modulus and shear modulus for the laminate in membrane ............... Figure I. 2. Young's modulus and shear modulus for the laminate in flexion .....................

Figure II. 1 .

 1 Figure II. 1. Longitudinal traction: fiber fracture [3] ............................................................. Figure II. 2. (a) Transverse fracture of matrix (b) shear fracture of matrix (c) debonding of the fiber-matrix interface (d) longitudinal fracture of the matrix[START_REF] Berthelot | Composite Materials: Mechanical Behavior and Structural Analysis[END_REF] ........................................ Figure II. 3. Local buckling of a fiber .................................................................................. Figure II. 4. Longitudinal traction: transverse fracture of the matrix [3] ............................... Figure II. 5. Longitudinal traction: longitudinal fracture of the matrix [3] ............................ Figure II. 6. Longitudinal traction: fracture of the fiber-matrix interface [3] ........................ Figure II. 7. Crack growth in the case of strong adhesion fiber-matrix [3] ........................... Figure II. 8. Crack growth in the case of weak adhesion fiber-matrix [3] ............................. Figure II. 9. Fracture of a unidirectional composite under transverse traction [3] ................ Figure II. 10. Mechanism of fracture observed in a laminate composite ............................. Figure II. 11. Three basic fracture modes ............................................................................. Figure II. 12.Experimental compliance calibration .............................................................. Figure II. 13. Typical set-up for DCB test ............................................................................ Figure II. 14. Typical set-up for ENF test ............................................................................ Figure II. 15. Crack Rail Shear specimen geometry [22] ....................................................... Figure II. 16. Typical spilt cantilever beam test [23] .............................................................. Figure II. 17.test jig used for modified SCB test [25] ............................................................ Figure II. 18. 3D view of the experimental equipment. Assembled state (a) and exploded view(b)[START_REF] Szekré Nyes | Improved analysis of the modified split-cantilever beam for mode-III fracture[END_REF] ................................................................................................................................ Figure II. 19. Schematic illustration of the MSCB specimen [28] ......................................... Figure II. 20.Geometry of the MSCB specimen with small edge delamination[START_REF] Davidson | Mixed-mode I-II-III delamination toughness determination via a shear-torsionbending test[END_REF] ............ Figure II. 21. Schematic of ECT specimen .......................................................................... Figure II. 22. ECT-1 specimen and test fixture [38] ............................................................... Figure II. 23. MECT-1 specimen and test fixture [38] ........................................................... Figure II. 24. Photograph and schematic diagram of test fixture [44] .................................... Figure II. 25.Exploded view of the test specimen obtained by bonding of two rectangular plates (left) and cross section of the circular-shaped adhesive joint (right).[START_REF] Cricri | A novel fixture for measuring mode III toughness of bonded assemblies[END_REF] ....................... Figure II. 26.Exploded view of the shells and of the specimen being tested. [50] .................. Figure II. 27. Layout of the testing apparatus [50] ................................................................. Figure II. 28. The mixed-mode I/II/III PENF specimen (d) as the superposition of the DCB (a), ENF (b) and MSCB (c) specimens .................................................................................... Figure II. 29. The mixed-mode I/II/III Double-Prestressed End-notched Flexure specimen .................................................................................................................................................. Figure II. 30 STB specimen and loading [31] ........................................................................ Figure II. 31 Schematic of test set-up [31] ............................................................................. Figure II. 32 VCCT for three dimensional eight-nodded solid elements. [55] ........................ Figure II. 33 Contour for evaluation of J-integral. ............................................................... Figure II. 34. Cohesive Zone Model ..................................................................................... Figure II. 35. Various forms of TSLC laws ..........................................................................

Figure II. 10 .

 10 Figure II. 1. Longitudinal traction: fiber fracture [3] ............................................................. Figure II. 2. (a) Transverse fracture of matrix (b) shear fracture of matrix (c) debonding of the fiber-matrix interface (d) longitudinal fracture of the matrix[START_REF] Berthelot | Composite Materials: Mechanical Behavior and Structural Analysis[END_REF] ........................................ Figure II. 3. Local buckling of a fiber .................................................................................. Figure II. 4. Longitudinal traction: transverse fracture of the matrix [3] ............................... Figure II. 5. Longitudinal traction: longitudinal fracture of the matrix [3] ............................ Figure II. 6. Longitudinal traction: fracture of the fiber-matrix interface [3] ........................ Figure II. 7. Crack growth in the case of strong adhesion fiber-matrix [3] ........................... Figure II. 8. Crack growth in the case of weak adhesion fiber-matrix [3] ............................. Figure II. 9. Fracture of a unidirectional composite under transverse traction [3] ................ Figure II. 10. Mechanism of fracture observed in a laminate composite ............................. Figure II. 11. Three basic fracture modes ............................................................................. Figure II. 12.Experimental compliance calibration .............................................................. Figure II. 13. Typical set-up for DCB test ............................................................................ Figure II. 14. Typical set-up for ENF test ............................................................................ Figure II. 15. Crack Rail Shear specimen geometry [22] ....................................................... Figure II. 16. Typical spilt cantilever beam test [23] .............................................................. Figure II. 17.test jig used for modified SCB test [25] ............................................................ Figure II. 18. 3D view of the experimental equipment. Assembled state (a) and exploded view(b)[START_REF] Szekré Nyes | Improved analysis of the modified split-cantilever beam for mode-III fracture[END_REF] ................................................................................................................................ Figure II. 19. Schematic illustration of the MSCB specimen [28] ......................................... Figure II. 20.Geometry of the MSCB specimen with small edge delamination[START_REF] Davidson | Mixed-mode I-II-III delamination toughness determination via a shear-torsionbending test[END_REF] ............ Figure II. 21. Schematic of ECT specimen .......................................................................... Figure II. 22. ECT-1 specimen and test fixture [38] ............................................................... Figure II. 23. MECT-1 specimen and test fixture [38] ........................................................... Figure II. 24. Photograph and schematic diagram of test fixture [44] .................................... Figure II. 25.Exploded view of the test specimen obtained by bonding of two rectangular plates (left) and cross section of the circular-shaped adhesive joint (right).[START_REF] Cricri | A novel fixture for measuring mode III toughness of bonded assemblies[END_REF] ....................... Figure II. 26.Exploded view of the shells and of the specimen being tested. [50] .................. Figure II. 27. Layout of the testing apparatus [50] ................................................................. Figure II. 28. The mixed-mode I/II/III PENF specimen (d) as the superposition of the DCB (a), ENF (b) and MSCB (c) specimens .................................................................................... Figure II. 29. The mixed-mode I/II/III Double-Prestressed End-notched Flexure specimen .................................................................................................................................................. Figure II. 30 STB specimen and loading [31] ........................................................................ Figure II. 31 Schematic of test set-up [31] ............................................................................. Figure II. 32 VCCT for three dimensional eight-nodded solid elements. [55] ........................ Figure II. 33 Contour for evaluation of J-integral. ............................................................... Figure II. 34. Cohesive Zone Model ..................................................................................... Figure II. 35. Various forms of TSLC laws ..........................................................................

  Figure II. 1. Longitudinal traction: fiber fracture [3] ............................................................. Figure II. 2. (a) Transverse fracture of matrix (b) shear fracture of matrix (c) debonding of the fiber-matrix interface (d) longitudinal fracture of the matrix[START_REF] Berthelot | Composite Materials: Mechanical Behavior and Structural Analysis[END_REF] ........................................ Figure II. 3. Local buckling of a fiber .................................................................................. Figure II. 4. Longitudinal traction: transverse fracture of the matrix [3] ............................... Figure II. 5. Longitudinal traction: longitudinal fracture of the matrix [3] ............................ Figure II. 6. Longitudinal traction: fracture of the fiber-matrix interface [3] ........................ Figure II. 7. Crack growth in the case of strong adhesion fiber-matrix [3] ........................... Figure II. 8. Crack growth in the case of weak adhesion fiber-matrix [3] ............................. Figure II. 9. Fracture of a unidirectional composite under transverse traction [3] ................ Figure II. 10. Mechanism of fracture observed in a laminate composite ............................. Figure II. 11. Three basic fracture modes ............................................................................. Figure II. 12.Experimental compliance calibration .............................................................. Figure II. 13. Typical set-up for DCB test ............................................................................ Figure II. 14. Typical set-up for ENF test ............................................................................ Figure II. 15. Crack Rail Shear specimen geometry [22] ....................................................... Figure II. 16. Typical spilt cantilever beam test [23] .............................................................. Figure II. 17.test jig used for modified SCB test [25] ............................................................ Figure II. 18. 3D view of the experimental equipment. Assembled state (a) and exploded view(b)[START_REF] Szekré Nyes | Improved analysis of the modified split-cantilever beam for mode-III fracture[END_REF] ................................................................................................................................ Figure II. 19. Schematic illustration of the MSCB specimen [28] ......................................... Figure II. 20.Geometry of the MSCB specimen with small edge delamination[START_REF] Davidson | Mixed-mode I-II-III delamination toughness determination via a shear-torsionbending test[END_REF] ............ Figure II. 21. Schematic of ECT specimen .......................................................................... Figure II. 22. ECT-1 specimen and test fixture [38] ............................................................... Figure II. 23. MECT-1 specimen and test fixture [38] ........................................................... Figure II. 24. Photograph and schematic diagram of test fixture [44] .................................... Figure II. 25.Exploded view of the test specimen obtained by bonding of two rectangular plates (left) and cross section of the circular-shaped adhesive joint (right).[START_REF] Cricri | A novel fixture for measuring mode III toughness of bonded assemblies[END_REF] ....................... Figure II. 26.Exploded view of the shells and of the specimen being tested. [50] .................. Figure II. 27. Layout of the testing apparatus [50] ................................................................. Figure II. 28. The mixed-mode I/II/III PENF specimen (d) as the superposition of the DCB (a), ENF (b) and MSCB (c) specimens .................................................................................... Figure II. 29. The mixed-mode I/II/III Double-Prestressed End-notched Flexure specimen .................................................................................................................................................. Figure II. 30 STB specimen and loading [31] ........................................................................ Figure II. 31 Schematic of test set-up [31] ............................................................................. Figure II. 32 VCCT for three dimensional eight-nodded solid elements. [55] ........................ Figure II. 33 Contour for evaluation of J-integral. ............................................................... Figure II. 34. Cohesive Zone Model ..................................................................................... Figure II. 35. Various forms of TSLC laws ..........................................................................

Figure II. 21 .

 21 Figure II. 1. Longitudinal traction: fiber fracture [3] ............................................................. Figure II. 2. (a) Transverse fracture of matrix (b) shear fracture of matrix (c) debonding of the fiber-matrix interface (d) longitudinal fracture of the matrix[START_REF] Berthelot | Composite Materials: Mechanical Behavior and Structural Analysis[END_REF] ........................................ Figure II. 3. Local buckling of a fiber .................................................................................. Figure II. 4. Longitudinal traction: transverse fracture of the matrix [3] ............................... Figure II. 5. Longitudinal traction: longitudinal fracture of the matrix [3] ............................ Figure II. 6. Longitudinal traction: fracture of the fiber-matrix interface [3] ........................ Figure II. 7. Crack growth in the case of strong adhesion fiber-matrix [3] ........................... Figure II. 8. Crack growth in the case of weak adhesion fiber-matrix [3] ............................. Figure II. 9. Fracture of a unidirectional composite under transverse traction [3] ................ Figure II. 10. Mechanism of fracture observed in a laminate composite ............................. Figure II. 11. Three basic fracture modes ............................................................................. Figure II. 12.Experimental compliance calibration .............................................................. Figure II. 13. Typical set-up for DCB test ............................................................................ Figure II. 14. Typical set-up for ENF test ............................................................................ Figure II. 15. Crack Rail Shear specimen geometry [22] ....................................................... Figure II. 16. Typical spilt cantilever beam test [23] .............................................................. Figure II. 17.test jig used for modified SCB test [25] ............................................................ Figure II. 18. 3D view of the experimental equipment. Assembled state (a) and exploded view(b)[START_REF] Szekré Nyes | Improved analysis of the modified split-cantilever beam for mode-III fracture[END_REF] ................................................................................................................................ Figure II. 19. Schematic illustration of the MSCB specimen [28] ......................................... Figure II. 20.Geometry of the MSCB specimen with small edge delamination[START_REF] Davidson | Mixed-mode I-II-III delamination toughness determination via a shear-torsionbending test[END_REF] ............ Figure II. 21. Schematic of ECT specimen .......................................................................... Figure II. 22. ECT-1 specimen and test fixture [38] ............................................................... Figure II. 23. MECT-1 specimen and test fixture [38] ........................................................... Figure II. 24. Photograph and schematic diagram of test fixture [44] .................................... Figure II. 25.Exploded view of the test specimen obtained by bonding of two rectangular plates (left) and cross section of the circular-shaped adhesive joint (right).[START_REF] Cricri | A novel fixture for measuring mode III toughness of bonded assemblies[END_REF] ....................... Figure II. 26.Exploded view of the shells and of the specimen being tested. [50] .................. Figure II. 27. Layout of the testing apparatus [50] ................................................................. Figure II. 28. The mixed-mode I/II/III PENF specimen (d) as the superposition of the DCB (a), ENF (b) and MSCB (c) specimens .................................................................................... Figure II. 29. The mixed-mode I/II/III Double-Prestressed End-notched Flexure specimen .................................................................................................................................................. Figure II. 30 STB specimen and loading [31] ........................................................................ Figure II. 31 Schematic of test set-up [31] ............................................................................. Figure II. 32 VCCT for three dimensional eight-nodded solid elements. [55] ........................ Figure II. 33 Contour for evaluation of J-integral. ............................................................... Figure II. 34. Cohesive Zone Model ..................................................................................... Figure II. 35. Various forms of TSLC laws ..........................................................................

  Figure II. 1. Longitudinal traction: fiber fracture [3] ............................................................. Figure II. 2. (a) Transverse fracture of matrix (b) shear fracture of matrix (c) debonding of the fiber-matrix interface (d) longitudinal fracture of the matrix[START_REF] Berthelot | Composite Materials: Mechanical Behavior and Structural Analysis[END_REF] ........................................ Figure II. 3. Local buckling of a fiber .................................................................................. Figure II. 4. Longitudinal traction: transverse fracture of the matrix [3] ............................... Figure II. 5. Longitudinal traction: longitudinal fracture of the matrix [3] ............................ Figure II. 6. Longitudinal traction: fracture of the fiber-matrix interface [3] ........................ Figure II. 7. Crack growth in the case of strong adhesion fiber-matrix [3] ........................... Figure II. 8. Crack growth in the case of weak adhesion fiber-matrix [3] ............................. Figure II. 9. Fracture of a unidirectional composite under transverse traction [3] ................ Figure II. 10. Mechanism of fracture observed in a laminate composite ............................. Figure II. 11. Three basic fracture modes ............................................................................. Figure II. 12.Experimental compliance calibration .............................................................. Figure II. 13. Typical set-up for DCB test ............................................................................ Figure II. 14. Typical set-up for ENF test ............................................................................ Figure II. 15. Crack Rail Shear specimen geometry [22] ....................................................... Figure II. 16. Typical spilt cantilever beam test [23] .............................................................. Figure II. 17.test jig used for modified SCB test [25] ............................................................ Figure II. 18. 3D view of the experimental equipment. Assembled state (a) and exploded view(b)[START_REF] Szekré Nyes | Improved analysis of the modified split-cantilever beam for mode-III fracture[END_REF] ................................................................................................................................ Figure II. 19. Schematic illustration of the MSCB specimen [28] ......................................... Figure II. 20.Geometry of the MSCB specimen with small edge delamination[START_REF] Davidson | Mixed-mode I-II-III delamination toughness determination via a shear-torsionbending test[END_REF] ............ Figure II. 21. Schematic of ECT specimen .......................................................................... Figure II. 22. ECT-1 specimen and test fixture [38] ............................................................... Figure II. 23. MECT-1 specimen and test fixture [38] ........................................................... Figure II. 24. Photograph and schematic diagram of test fixture [44] .................................... Figure II. 25.Exploded view of the test specimen obtained by bonding of two rectangular plates (left) and cross section of the circular-shaped adhesive joint (right).[START_REF] Cricri | A novel fixture for measuring mode III toughness of bonded assemblies[END_REF] ....................... Figure II. 26.Exploded view of the shells and of the specimen being tested. [50] .................. Figure II. 27. Layout of the testing apparatus [50] ................................................................. Figure II. 28. The mixed-mode I/II/III PENF specimen (d) as the superposition of the DCB (a), ENF (b) and MSCB (c) specimens .................................................................................... Figure II. 29. The mixed-mode I/II/III Double-Prestressed End-notched Flexure specimen .................................................................................................................................................. Figure II. 30 STB specimen and loading [31] ........................................................................ Figure II. 31 Schematic of test set-up [31] ............................................................................. Figure II. 32 VCCT for three dimensional eight-nodded solid elements. [55] ........................ Figure II. 33 Contour for evaluation of J-integral. ............................................................... Figure II. 34. Cohesive Zone Model ..................................................................................... Figure II. 35. Various forms of TSLC laws ..........................................................................

Figure III. 12 .

 12 Figure III. 1. Experimental load/displacement curve for the specimens 0° and 90° ............ Figure III. 2. Experimental load/displacement curve for the specimen 45° ......................... Figure III. 3. Schema of Double Cantilever Beam test ........................................................ Figure III. 4. A picture of DCB specimen and test ............................................................... Figure III. 5. Schema of End Notched Flexion test .............................................................. Figure III. 6. A picture of ENF test ...................................................................................... Figure III. 7. Mesh for DCB test .......................................................................................... Figure III. 8. Mesh for ENF test ........................................................................................... Figure III. 9. An experimental load-displacement curve in DCB test .................................. Figure III. 10. Interpolation of C/a 3 for Beam compliance law (Eq. 36) .............................. Figure III. 11. Interpolation of C/a for Berry compliance law (Eq. 35) ............................... Figure III. 12. Evolution of G I , G II and G III along the crack front of DCB specimen and comparison of the average value of G IC from different methods ............................................. Figure III. 13. An experimental load-displacement curve in ENF test ................................. Figure III. 14. Interpolation of C/a 3 for Beam model compliance law (Eq. 36) ................... Figure III. 15. Evolution of G I , G II and G III along the crack front of ENF specimen and comparison of the average value of G IIC from different methods ............................................

Figure IV. 1 .

 1 Figure IV. 1. Schematic of ECT-1 test ................................................................................. Figure IV. 2. A picture of ECT-1 test ................................................................................... Figure IV. 3. Finite element model of ECT-1 test ................................................................ Figure IV. 4. (a) Experimental load-displacement curves; (b) Omniscan image of ECT-1 specimen ................................................................................................................................... Figure IV. 5. Interpolation between 1/C and a/B ................................................................. Figure IV. 6. Evolution of G I , G II and G III along the crack front of ECT-1 test ................... Figure IV. 7. SEM observations in ECT-1 specimen ........................................................... Figure IV. 8. Model of MECT-1 test .................................................................................... Figure IV. 9. Evolution of G I , G II and G III normalized by G T along the crack front in the MECT-1 test ............................................................................................................................. Figure IV. 10. Schema of Edge Crack Torsion-2 test (ECT-2) ............................................ Figure IV. 11. Model of ECT-2 test ..................................................................................... Figure IV. 12. Schematic of ECT-2 test ............................................................................... Figure IV. 13. Experimental torque/rotation curves and Omniscan image of ECT-2 test ... Figure IV. 14. Interpolation of (1/C) vs (a/B)(Eq. 41) ......................................................... Figure IV. 15. Comparison of G IIIC results from ECT-2 test ................................................ Figure IV. 16. Evolution of G I , G II and G III along the crack front of ECT-2 specimen ....... Figure IV. 17. Schema of ERCT specimen .......................................................................... Figure IV. 18. Definition of d, d ring and D ring ........................................................................ Figure IV. 19. Rigid plates in the device of ERCT .............................................................. Figure IV. 20. Schema of ERCT device ............................................................................... Figure IV. 21. A picture of ERCT test ................................................................................. Figure IV. 22. Cylindrical beam with a ring crack ............................................................... Figure IV. 23. Bilinear TSLC law used in cohesive element model .................................... Figure IV. 24. Mesh of ERCT specimen .............................................................................. Figure IV. 25. Model of ERCT test ......................................................................................Figure IV. 26. Experimental torque-rotation curves of ERCT specimen with d=30mm ..... Figure IV. 27. Ultrasonic C-scan image before and after propagation on the three ERCT specimens with d=30mm .......................................................................................................... Figure IV. 28. Definition of angle Ɵ between fiber orientation and crack front ..................

Figure IV. 29 .

 29 Figure IV. 29. Evolution of G I , G II and G III along the circular crack front of ERCT specimen with d=30mm ........................................................................................................... 81 Figure IV. 30. Evolution of G I , G II and G III along the circular crack front of ERCT specimen with d=50mm ........................................................................................................... 82 Figure IV. 31: Cohesive element model for ERCT specimen .............................................. 82 Figure IV. 32. Finite element models of device A and C ..................................................... 84 Figure IV. 33. Evolution of normalized G III along the circular crack front with device A and C ............................................................................................................................................... 85 Figure IV. 34. Top view of ERCT device ............................................................................ 85 Figure IV. 35. Evolution of normalized G III along the circular crack front with device C and steel and composite specimens ................................................................................................. 86 Figure IV. 36. Evolution of normalized G III along the circular crack front with device A and different diameters of steel specimens ..................................................................................... 87 Figure IV. 37. Evolution of normalized G III along the circular crack front with device C and different diameters of composite specimens ............................................................................ 87 Figure IV. 38. Δ vs d/D ring .................................................................................................... 88 Figure IV. 39. Evolution of normalized G III along the circular crack front with device C and A ............................................................................................................................................... 89 Figure IV. 40. Models for device B and B45 ....................................................................... 89 Figure IV. 41. Evolution of normalized G III along the circular crack front with device ...... 90 Figure IV. 42. Schema of MERCT test ................................................................................ 91 Figure IV. 43. A picture of ERC-III-1 device and specimen ............................................... 91 Figure IV. 44. A picture of ERC-III-1 test ........................................................................... 92 Figure IV. 45. Dimensions of ERC-III-2 specimen .............................................................. 93 Figure IV. 46. Schematic of ERC-III-2 test .......................................................................... 93 Figure IV. 47. Specimen and device of ERC-III-2 test ........................................................ 93 Figure IV. 48. A picture of ERC-III-2 test ........................................................................... 94 Figure IV. 49. Model of MERCT test .................................................................................. 95 Figure IV. 50. Evolution along the crack front of G III normalized by the average value with round specimen and square specimen ...................................................................................... 95 Figure IV. 51. Evolution along the crack front of G I , G II and G III normalized by G T with oval crack front ......................................................................................................................... 96 Figure IV. 52. Evolution along the crack front of G III normalized by the average value with oval and circular crack front ..................................................................................................... 96 Figure IV. 53. Evolution along the crack front of G I , G II and G III normalized by G T with inclined crack front ................................................................................................................... 97 Figure IV. 54. Evolution along the crack front of G III normalized by the average value with parallel and inclined crack front ............................................................................................... 98 Figure IV. 55. Evolution along the crack front of G I , G II and G III normalized by G T with loading axis deviation ............................................................................................................... 99 Figure IV. 56. Evolution along the crack front of G III normalized by its average value with and without the default of the alignment .................................................................................. 99 Figure IV. 57. Typical experimental torque/rotation angle curve of ERC-III-1 test .......... 100 Figure IV. 58. Typical experimental torque/rotation angle curve of ERC-III-2 test .......... 101 Figure IV. 59. Evolution along the crack front of G III normalized by the average value obtained from ERCT test and from ERC-III-1 test ................................................................ 102 Figure IV. 60. Evolution along the crack front of G I , G II and G III normalized by G T obtained from ERC-III-1 test ................................................................................................................ 102

Figure I. 1 .Figure I. 2 .

 12 Figure I. 1. Young's modulus and shear modulus (GPa) for the laminate in membrane

Figure II. 1 .

 1 Figure II. 1. Longitudinal traction: fiber fracture[START_REF] Berthelot | Composite Materials: Mechanical Behavior and Structural Analysis[END_REF]

Figure II. 2 .

 2 Figure II. 2. (a) Transverse fracture of matrix (b) shear fracture of matrix (c) debonding of the fiber-matrix interface (d) longitudinal fracture of the matrix[START_REF] Berthelot | Composite Materials: Mechanical Behavior and Structural Analysis[END_REF]

( 4 )

 4 Transverse fracture of the matrix The transverse cracking of matrix (Figure II. 4) can occur when the stress in traction σ m reaches the fracture stress of the matrix σ mu .

Figure II. 4 .( 5 )

 45 FigureII. 4. Longitudinal traction: transverse fracture of the matrix[START_REF] Berthelot | Composite Materials: Mechanical Behavior and Structural Analysis[END_REF]

Figure II. 5 .( 6 )

 56 Figure II. 5. Longitudinal traction: longitudinal fracture of the matrix[START_REF] Berthelot | Composite Materials: Mechanical Behavior and Structural Analysis[END_REF]

  Figure II. 7 shows the coupling between a longitudinal fracture of the matrix and fiber-matrix debonding.

Figure II. 7 .

 7 Figure II. 7. Crack growth in the case of strong adhesion fiber-matrix[START_REF] Berthelot | Composite Materials: Mechanical Behavior and Structural Analysis[END_REF]

Figure II. 10 .

 10 Figure II. 10. Mechanism of fracture observed in a laminate composite Fracture mechanics is usually applied to describe delamination because of the singularity at the crack front, in which we define three basic modes: mode I (opening), mode II (in-plane shear) and mode III (out-of-plane shear), shown in Figure II. 11. The delamination resistance of a composite can be characterized firstly by three intrinsic material constants: G IC , G IIC and G IIIC which are the critical strain energy release rate in the 3 pure modes; secondly by R-curve which shows the variation of the resistance to the crack propagation as a function of the crack extension.

Figure II. 11 .

 11 Figure II. 11. Three basic fracture modes

  Figure II. 12(a)); (3) Interpolate these measured C n as a function of the crack length to determine the compliance law: C = f (a) (shown in Figure II. 12(b)); (4) Determine strain energy release rate by the Irwin-Kies' formula. When P reaches P C the critical loading, then G equals G C .

Figure II. 12 .

 12 Figure II. 12.Experimental compliance calibration

3 . 4 .

 34 Pure mode III II.3.4.1. Crack Rail Shear test

Figure II. 15

 15 FigureII. 15. Crack Rail Shear specimen geometry[START_REF] Becht | Design and analysis of the crack rail shear specimen for mode III interlaminar fracture[END_REF]

  FigureII. 16. Typical spilt cantilever beam test[START_REF] Donaldson | Mode III intelaminar fracture characterization of composite materials[END_REF]

Figure II. 24

 24 Figure II. 24. Photograph and schematic diagram of test fixture[START_REF] Suemasu | An experimental method to measure the mode III interlaminar fracture toughness of composite laminates[END_REF]

  Figure II. 25. A crack starter is obtained by inserting between the two adhering surfaces a Teflon film. The specimen is placed between the shells as shown in Figure II. 26. The crack propagates when the shells are placed in a tensile machine as shown in Figure II. 27.

  , a double prestressed end-notched flexure (I/II/III) (PENF I/II/III) test was proposed and realized, whose approach is a superposition of the DCB (a), ENF (b) and MSCB(c), shown in Figure II. 28. Figure II. 29 shows the experimental equipment for the mixed mode I/II/III test. The mode-I component is fixed by the steel roller (7), while the mode-III component can be controlled by using the special grips of the MSCB test. The mode-III load is transferred to the specimen through four grub screws (8), the crack tearing displacement is controlled by a prestressing screw (3).

Figure II. 28 .

 28 Figure II. 28. The mixed-mode I/II/III PENF specimen (d) as the superposition of the DCB (a), ENF (b) and MSCB (c) specimens

Figure II. 30

 30 Figure II. 30 presents a schematic of the STB specimen and loading. In detail, P I , P II and P III represent the mode I, mode II and mode III loadings respectively. T is a torque about the z axis imposed as a boundary condition. To reproduce the MSCB loading, one would choose T = P III ×a, so that M Z = 0 at the delamination front. Figure II. 31 shows the schematic of test set-up for the mixed mode I/II/III test.

Figure II. 35 .

 35 Figure II. 35. Various forms of TSLC laws

Figure III. 1 2

 12 Figure III. 1 and Figure III.2 give the linear part of tensile stress/strain curves for the specimens 0°-1~3, 90°-1~2 and 45°-1~2 respectively. In order to protect the extensometer, tests were paused before the fracture and the extensometer was taken off, and then the specimen was reloaded until the fracture to measure the strength. As a result, we don't have a complete stress/strain curve. Note that the tests with specimens 90°-3 and 45°-3 didn't succeed. The curves for the specimens 0° and 90° are found in same figure where the slopes of their linear part are shown substantially the same for the taffetas prepreg tested. So the average value was calculated on all measured modulus from the specimens 0° and 90° with the consideration of E 11 =E 22 . Poisson's ratio  12 is also determined as the ratio of the transverse strain over longitudinal strain for the specimens 0° and 90° specimens. The inplane shear modulus G 12 is calculated according to the slope of the linear part of the curves in Figure III. 2.

( 1 )

 1 Double Cantilever Beam (DCB) pure mode I test Firstly, two steel parts were pasted on the head of DCB specimen. Then the specimen was placed under pure tensile load in the way shown in Figure III. 3 and Figure III. 4. An axial displacement speed of 2mm/min was imposed until the crack onset.

Figure III. 3 .

 3 Figure III. 3. Schema of Double Cantilever Beam test

( 2 ) 125 Figure III. 5 .

 21255 Figure III. 5. Schema of End Notched Flexion test

9 .

 9 The boundary conditions were carefully set in order to simulate real experimental loading conditions. In the DCB model, as shown in Figure III. 7, the tensile load was imposed by a displacement curve 7.5mm in the thickness direction on the edge of one arm while the symmetrical edge was fixed. Besides, the degree of freedom in the thickness direction at the free edge was restricted by the point A (defined in Figure III. 7). In the ENF model, as shown in Figure III. 8, the compression loaded in the middle was imposed in the manner of a displacement curve 8mm using a cylinder. Two supporting cylinder were fixed. A surface to surface contact was set between the specimen surface and the cylinders, as well as between the two arms of the specimen.

Figure III. 7 . 125 Figure III. 8 .

 71258 Figure III. 7. Mesh for DCB test

Figure III. 9 .Figure III. 10 .

 910 Figure III. 9. An experimental load-displacement curve in DCB test

Figure III. 12

 12 Figure III. 12 illustrates the evolution of G I , G II and G III along the crack front computed at P C obtained by VCCT on the finite element DCB model. It demonstrates that a pure mode I delamination is obtained. The values of G I vary along the crack front and drop to 0 at the two extremities of the crack front while the maximum value appears in the middle domain. If we define a relative variation parameter as following: Δ=(|G I-max -G I-av |)/G I-av , then Δ is 29.1% in DCB test.

Figure III. 12 .

 12 Figure III. 12. Evolution of G I , G II and G III along the crack front of DCB specimen and comparison of the average value of G IC from different methods

Figure III. 13 .

 13 Figure III. 13. An experimental load-displacement curve in ENF test

Figure III. 15 .

 15 Figure III. 15. Evolution of G I , G II and G III along the crack front of ENF specimen and comparison of the average value of G IIC from different methods

IV. 1 . 1 .

 11 Edge Crack Torsion-1 (ECT-1) test IV.1.1.1. Experimental The Edge Crack Torsion-1 (ECT-1) test proposed in literature is schematized by Figure IV. 1 and Figure IV. 2.The specimen is locked at one corner without precrack and loaded on the diagonal corner using a tensile/compression machine while the specimen is supported by other two corners; the crack front is actually loaded under torsion which generates mode III. In this study, the laminate is a multidirectional ones with the stacking sequence described in chapter I (45/0/0/45/0/45/45/0)//crack//(0/45/45/0/45/0/0/45). An axial displacement speed of 2mm/min was imposed until the crack onset.

Figure IV. 1 .

 1 Figure IV. 1. Schematic of ECT-1 test

  Figure IV. 3.

Figure IV. 3 .

 3 Figure IV. 3. Finite element model of ECT-1 test

Figure IV. 4 .

 4 Figure IV. 4. (a) Experimental load-displacement curves; (b) Omniscan image of ECT-1 specimen

Figure IV. 6

 6 Figure IV. 6 shows the evolution of G I , G II and G III along the crack front computed at P C obtained by VCCT from the finite element model. Besides, the average value of G IIIC in Table.14 is also presented in the same figure, named as G IIIC-CC .

Figure IV. 6 .

 6 Figure IV. 6. Evolution of G I , G II and G III along the crack front of ECT-1 test Observation on fracture surface of ECT-1 specimen by Scanner Electronic Microscope (SEM) is shown in Figure IV. 7. According to the definition of the special areas along the crack front in Figure IV. 4(b), it can be seen that the morphology of the fracture surface in the area a is quite different from the one observed in the area b. It is believed that the former mode must be under mode II or mixed mode II+III while the later one under mode III. That means the measurement of G IIIC with this test is problematic.

Figure IV. 8 .

 8 Figure IV. 8. Model of MECT-1 test

Figure IV. 9 .

 9 Figure IV. 9. Evolution of G I , G II and G III normalized by G T along the crack front in the MECT-1 test

  The finite element model for ECT-2 test is shown in Figure IV. 11. The simulation allows optimizing the position of the sample in the jaws in order to obtain the propagation from the pre-crack without propagation at the opposite edge. The crack was actually located close to the rotation axis as shown in Figure IV. 12.

Figure IV. 11 .

 11 Figure IV. 11. Model of ECT-2 test

Figure IV. 13 .

 13 Figure IV. 13. Experimental torque/rotation curves and Omniscan image of ECT-2 test

  Figure IV. 14. Interpolation of (1/C) vs (a/B)(Eq. 41)

Figure IV. 15 .Figure IV. 16 .

 1516 Figure IV. 15. Comparison of G IIIC results from ECT-2 test The simulation of ECT-2 test was performed on the specimen with initial crack length 20mm. The evolution of G I , G II and G III along the crack front computed at T C-2 is illustrated in Figure IV. 16. The average value of G IIIC from all the tests with specimen a=20 mm obtained by Eq. 42, named G IIIC-CC , are also presented in the Figure IV. 16.

Figure IV. 17 .

 17 Figure IV. 17. Schema of ERCT specimen

Figure IV. 18 .

 18 Figure IV. 18. Definition of d, d ring and D ring Firstly, the composite specimen was pasted to the ring protruding from two rigid plates shown in Figure IV. 19. The inner d ring and outer diameter D ring of the ring are 80mm and 120mm respectively, shown in Figure IV. 18 and Figure IV. 19. Two diameters d (30mm and 50 mm)

Figure IV. 19 .

 19 Figure IV. 19. Rigid plates in the device of ERCT Figure IV. 20 shows the ERCT device and Figure IV. 21 gives a picture of ERCT test. Great care was taken in order to locate the center of the sample on the axes of the rigid plates during the pasting process. And then the rigid plates were screwed to the torsion devise. Finally, the torsion device was submitted to an imposed rotation up to the crack propagation. Thanks to the d ring diameter holes in the rigid plates, it's also possible to observe the crack propagation by Ultrasonic C-scan.

Figure IV. 20 .

 20 Figure IV. 20. Schema of ERCT device

Figure IV. 23 .

 23 Figure IV. 23. Bilinear TSLC law used in cohesive element model

  FE model of ERCT test is shown in Figure IV. 25, Ɵ where is defined in a polar coordinate. In order to impose a torque on the specimen, a rotation curve 0.005 was employed on both of the loading heads as shown in Figure IV. 25. The central point of the specimen was fixed while the loading axis was permitted to move only in the axis direction.

Figure IV. 24 .

 24 Figure IV. 24. Mesh of ERCT specimen

Figure IV. 26 .Figure

 26 Figure IV. 26. Experimental torque-rotation curves of ERCT specimen with d=30mm

Figure IV. 28 .

 28 Figure IV. 28. Definition of angle Ɵ between fiber orientation and crack front

  The curves presented in Figure IV. 29 and Figure IV. 30 illustrate the evolution of G I , G II and G III along the crack front obtained by finite element method at the critical torque T C . It is interesting to note that the participations of mode I and mode II are almost zero, which demonstrates that the crack is loaded in pure mode III indeed. The values of G III show a periodic variation instead of being constant, which can be minimized by optimization, however. In fact, the evolution of G III along the crack front results from different influence factors.

Figure IV. 29 .

 29 Figure IV. 29. Evolution of G I , G II and G III along the circular crack front of ERCT specimen with d=30mm

Figure IV. 30 .

 30 Figure IV. 30. Evolution of G I , G II and G III along the circular crack front of ERCT specimen with d=50mm Moreover, the results of cohesive element model of ERCT test show that the onset of damage will occur for =0° or 180° (Figure IV. 31), where the cohesive element are damaged and removed. It corresponds to the fact that G III reaches the maximum value at =0° or 180° shown in Figure IV. 29 and Figure IV. 30. It confirms the conclusion that the propagation of the crack should initiate at 0° and 180° if the toughness of the composite is not influenced by the fiber orientation.

Figure IV. 31 :

 31 Figure IV. 31: Cohesive element model for ERCT specimen

Figure IV. 32 .

 32 Figure IV. 32. Finite element models of device A and C

Figure IV. 35 .

 35 Figure IV. 35. Evolution of normalized G III along the circular crack front with device C and steel and composite specimens

Figure IV. 36 .

 36 Figure IV. 36. Evolution of normalized G III along the circular crack front with device A and different diameters of steel specimens And then numerical model for five composite specimens using device C are established, each specimen has different d (30, 40, 50, 60 and 75mm). All results, named by "C-compos-d30"~"C-compos-d75", are presented in Figure IV. 37. A negative correlation is found between the laminate factor and ratio d/D ring in a certain range.

Figure IV. 37 .Figure

 37 Figure IV. 37. Evolution of normalized G III along the circular crack front with device C and different diameters of composite specimens It is seen that the level of the variation of G III in term of Δ=(|G III-max -G III-av |)/G III-av , varies from 1% to 7% (Figure IV. 38). The bigger the diameter d is, the smaller the value of Δ is. The specimen with d=50mm is chosen for the coming experimental tests, where Δ is approximately 2.7%. It is recommended that d/D ring equals to approximately 0.5 because the values of G III are nearly uniform and the non-delaminated area is big enough for observation of crack propagation.

Figure IV. 39 .

 39 Figure IV. 39. Evolution of normalized G III along the circular crack front with device C and A

Figure IV. 40 .Figure IV. 41 .

 4041 Figure IV. 40. Models for device B and B45

  specimen through the adhesive joint at the specimen's surfaces, while it is on the edge of the specimen through the screws in ERC-III-2 tests.IV.4.1.1. ERC-III-1 testThe first one is shown in Figure IV. 42 and Figure IV. 43, named ERC-III-1 thereafter. The modification is only about the rigid plates and loading axes. Firstly, the loading head is changed into a cylinder loading axis of the same height (Φ=30mm and h=72mm). Moreover, the rigid plates are changed into circular ones. Note that d ring and D ring are always 80mm and 120mm respectively. In ERC-III-1 test, the same square ERCT specimens were used, because no influence of the geometry of the composite plate has been observed numerically. ERC-III-1 test is shown in Figure IV. 44. The dimensions of the specimen are d =50 mm, the side length of the square D=120mm, the total thickness of the specimen h =7mm. 3 specimens were tested.

Figure IV. 42 .

 42 Figure IV. 42. Schema of MERCT test

Figure IV. 45 .

 45 Figure IV. 45. Dimensions of ERC-III-2 specimen

Figure IV. 49 .

 49 Figure IV. 49. Model of MERCT test

Figure IV. 50 .

 50 Figure IV. 50. Evolution along the crack front of G III normalized by the average value with round specimenand square specimen

Figure IV. 51 . 3 .

 513 Figure IV. 51. Evolution along the crack front of G I , G II and G III normalized by G T with oval crack front

Figure IV. 53 .FigureFigure

 53 Figure IV. 53. Evolution along the crack front of G I , G II and G III normalized by G T with inclined crack front

Figure IV. 59 .

 59 Figure IV. 59. Evolution along the crack front of G III normalized by the average value obtained from ERCT test and from ERC-III-1 test

Figure V. 1 .

 1 Figure V. 1. A picture of ERC-I test

Figure V. 3 .

 3 Figure V. 3. Mesh for ERC-I test

Figure V. 5 .Figure V. 6 .

 56 Figure V. 5. Experimental force/displacement curve of ERC-I test

Figure V. 8 .Figure V. 10 .

 810 Figure V. 8. Typical experimental force/displacement curve of ERC-II test

Figure V. 11

 11 Figure V. 11 shows the evolution of G I , G II and G III normalized by G T along the crack front obtained by FEA using the VCCT. It is shown that there is no participation of mode I component. A mixed mode I+II delamination is realized indeed. The proportion of mode I over mode II component is approximately 2.0, which is controlled by the stacking sequence of the specimen.

Figure V. 11 .

 11 Figure V. 11. Evolution along the crack front of G I , G II , G III normalized by G T in ERC-I+II test

V. 3 . 3 .

 33 Results and discussionsThe evolution of G I , G II and G III normalized by G T along the crack front obtained by FEA using the VCCT is shown in Figure V. 12. A mixed mode I+III delamination is demonstrated to be realized indeed because there is no participation of mode II component. The proportion

  delamination modes. The values of G I , G II and G III are not constant along the crack front. Above all, ERC-I+II+III delamination test is a promising test to characterize mixed mode delamination behavior. Some improvements are necessary in the future work.

  ........................................................................ Figure II. 31 Schematic of test set-up [31] ............................................................................. Figure II. 32 VCCT for three dimensional eight-nodded solid elements. [55] ........................Figure III. 1. Experimental load/displacement curve for the specimens 0° and 90° ............ Figure III. 2. Experimental load/displacement curve for the specimen 45° ......................... Figure III. 3. Schema of Double Cantilever Beam test ........................................................ Figure III. 4. A picture of DCB specimen and test ............................................................... Figure III. 5. Schema of End Notched Flexion test .............................................................. Figure III. 6. A picture of ENF test ...................................................................................... Figure III. 7. Mesh for DCB test .......................................................................................... Figure III. 8. Mesh for ENF test ........................................................................................... Figure III. 9. An experimental load-displacement curve in DCB test .................................. Figure III. 10. Interpolation of C/a 3 for Beam compliance law (Eq. 36) .............................. Figure III. 11. Interpolation of C/a for Berry compliance law (Eq. 35) ...............................

Figure II. 33 Contour for evaluation of J-integral. ............................................................... Figure II. 34. Cohesive Zone Model ..................................................................................... Figure II. 35. Various forms of TSLC laws ..........................................................................

  These laminates are quasi-homogeneous, i.e. the same elastic stiffness properties in tension and in bending; and they are also quasi-isotropic, i.e. the same elastic stiffness properties in tension and in bending in all direction; the elimination of all possible coupling terms: B ij =A16 =A 26 =D 16 =D 26 =0, as shown in Figure I. 1 and Figure I. 2.

	To meet the geometry requirements, the stacking sequence for Double Cantilever Beam
	(DCB), End Notched Flexure (ENF) and Edge Crack Torsion (ECT-1) (using tensile machine)
	tests is designed as:
	(45/0/0/45/0/45/45/0)//crack//(0/45/45/0/45/0/0/45)
	The stacking sequence for tests Edge Crack Torsion (ECT-2) test (using torsional machine)
	and Edge Ring Crack (ERC) tests is designed as:
	(0/45/45/0/45/0/0/45/45/0/0/45/0/45/45/0)//crack//(0/45/45/0/45/0/0/45/45/0/0/45/0/45/45/0)

  The corresponding strength are named X + , Y + , Z + in traction and X -, Y -, Z -in compression, respectively. The shear stresses in the plane 12, 23 and 13 are denoted σ 12 , σ 23 , σ 13 . Their associated strengths are defined by S 12 , S 23 , S 13 .

  All composite laminates are fabricated by hand lay-up technique and a polymer film of 13 µm thick is embedded in the midplan in order to create a pre-crack. For ECT specimens, the crack tip is straight and for ERCT the crack tip is circular. The composite laminates are cured in hot-press machine with a proper curing cycle of 30 minutes at 110°C under 1 bar of pressure and then 1 hour at 125°C under 2 bars of pressure.

	Specimens h (mm)	b (mm)
	0°-1	1.68	23.13
	0°-2	1.76	22.90
	0°-3	1.66	22.71
	90°-1	1.76	23.36
	III.1. Materials and laminate manufacturing 90°-2	1.75	23.20
	90°-3	1.75	22.84
	In this research work, all specimens are obtained from a woven carbon/epoxy prepreg (ref: 45°-1 1.71 23.26 IMP503Z), which is a taffeta fabric. 45°-2 1.73 23.08
	45°-3	1.72	23.28
	Table. 3. Dimensions of the unidirectional specimens
	III.2.2. Experimental
	An 8 prepreg plies plate with the stacking sequence [0 8 ] was fabricated where 0 represents a
	taffeta balanced fabric ply with yarns oriented in 0° and 90°. And then the plate was cut into 3
	specimens for each tests along 0°, 90° and 45° relative to the fiber orientation, which were
	named 0°-1~3, 90°-1~3 and 45°-1~3 respectively. Dimensions of the specimens are shown in
	Table. 3.

III.2. Composite materials characterization

III.2.1. Characterization tests

In order to obtain the principal material constants, referenced to the standards for composite material ASTM Standard D3039M-00 and D3518M-94, some classical tests were carried out on the specimens of unidirectional laminates.

Firstly, tensile tests were performed on the specimens at 0° (with fiber orientation 0/90) and also on those at 90° (with fiber orientation 90/0), in order to verify the influence of chain/frame weaving on this balanced fabric. Tensile tests on the specimens of 0° (with fiber orientation 0/90) allow to measure -Longitudinal and transverse modulus:

E

11 , E 22 ; -Poisson's ratio  12 ; -Tensile strength in direction 1 and 2: X + , Y + ; Tensile tests on the specimens at 45° (with fiber orientation 45) provide -Shear modulus in direction 12: G 12 ; -Shear strength in direction 12: S 12 .

  material model for steel device In most finite element models, an elastic material model is employed for simulate devices. The material parameters are shown in Table.5.

	Young's modulus E 2.1*10 5 (MPa)
	Poisson's ratio ʋ	0.3
	Translational stiffness about local r-axis	1*10 8 (N/mm)
	Translational stiffness about local s-axis	1*10 8 (N/mm)
	TKT: Translational stiffness about local t-axis 1*10 8 (N/mm)

Table. 5. Mechanic properties of steel devices

(3) material model for spring element

The material parameters of spring element are shown in Table

.

6.

Table . 6

 . . Mechanic properties of steel devices(4) material model for glue Bondline of glue is used in some finite element models. An elastic material model was employed to simulate the bondline. The parameters are set as Table.7.Table. 7. Mechanic properties of bondlineFinally, the boundary conditions should be well considered. Generally, the boundary conditions are set according to the corresponding experiments. A displacement/time curve is set in tensile or compression tests while rotation/time curve is set in torsion tests. III.4. Tests of delamination under pure mode I (DCB) and pure mode II (ENF)In order to compare with mode III delamination toughness and new proposed mode I and mode II delamination tests, widely used pure mode I test Double Cantilever Beam (DCB) and mode II test End Notched Flexion (ENF) were carried out on the same composite.III.4.1. ExperimentalThe composite laminate was manufactured by hand lay-up technique according to the stacking sequence highlighted in chapter I and a 13 µm thick polyester film was inserted in the midplane in order to create a pre-crack. Specimens with different initial crack lengths, a (defined in Figure III. 3) were employed in order to apply experimental compliance calibration method. At least 3 specimens were tested for each specimens configuration.

	Young's modulus E 1800 (MPa)
	Poisson's ratio ʋ	0.3

  Table. 8. Every dimension is measured 3 times at least and the average value is presented in the table. b is the width of DCB specimen.

	Test b (mm) 2h (mm) a (mm)
	DCB1-1 21.07	3.55	41.81
	DCB1-2 21.07	3.55	46.31
	DCB1-3 21.07	3.55	54.71
	DCB1-4 21.07	3.55	67.11
	DCB1-5 21.07	3.55	77.81
	DCB1-6 21.07	3.55	83.81
	DCB2-1 20.52	3.48	41.43
	DCB2-2 20.52	3.48	46.43
	DCB2-3 20.52	3.48	57.43
	DCB2-4 20.52	3.48	68.93
	DCB2-5 20.52	3.48	78.93
	DCB2-6 20.52	3.48	84.93
	DCB3-1 20.62	3.47	41.89
	DCB3-2 20.62	3.47	43.97
	DCB3-3 20.62	3.47	54.97
	DCB3-4 20.62	3.47	64.37
	DCB3-5 20.62	3.47	76.17
	DCB3-6 20.62	3.47	81.67

Table. 8. Dimension of DCB specimens

  37) is applied in ENF test. The dimensions of ENF specimens are listed in Table. 9. Every dimension is measured 3 times at least and the average value is presented in the table.

	Test	b	2h	L (mm)	a (mm)
	ENF1-1	20.7	3.55	50	15
	ENF1-2	20.7	3.55	50	20
	ENF1-3	20.7	3.55	50	25
	ENF1-4	20.7	3.55	50	30
	ENF1-5	20.7	3.55	50	35
	ENF1-6	20.7	3.55	50	40
	ENF2	20.8	3.58	50	40
	ENF3	20.2	3.62	50	40
	ENF4	20.4	3.70	50	40

Table. 9. Dimensions of ENF specimens III.4.2. Finite element analysis 16 plies were set up for DCB and ENF tests according to the stacking sequence highlighted in chapter I and were separated symmetrically into two sub-laminates by the crack plane. In DCB model, the crack length a=40mm while the rest specimen dimensions are the same as in Table. 8. In ENF test, a=40mm as well while the other dimensions are the same as in Table.

  Table. 10. Results of G IC from different methods for DCB test

	G IC (N/m)	G IC-Theory G IC-Beam law G IC-Berry's law
	Test 1	587.5	485.6	522.7
	Test 2	550.6	444.1	478.8
	Test 3	533.0	449.2	483.5
	Average	557	460	495
	Standard deviation	28	24	23
		58 / 125

  av , then Δ is 29.1% in DCB test.Finally, the values of G IC in Table.10 are also presented in Figure III. 12, where G IC-Theory , G IC-Beam law and G IC-Berry's law are bigger than G IC-FEM , the average value of G I computed at P C using VCCT, by 60%, 32.3% and 42.3% respectively. It is interesting to note that G Imax obtained by FEA agrees well with that from the bean theory, so closer to those from compliance calibration methods. Actually, when the G Imax value attains the toughness of the material, the onset of the crack propagation occurs. Herein the average value has less physical

sense. As in literature, G IC-Beam law and G IC-Berry's law are usually considered as a precise mode I delamination toughness. It is decided that only G IC-Beam law will be compared with other delamination test thereafter.

  [START_REF] Irwin | Critical energy rate analysis of fracture strength[END_REF] mm 3 ) Table. 11. Results of G IIC from different methodsFigure III. 15 shows the evolution of G I , G II and G III along the crack front computed at P C obtained by VCCT using the finite element ENF model. It demonstrates that a main mode II delamination is obtained. The values of G II are nearly uniform in the middle domain while they have a relatively big variation near the extremities of the crack front, where a small participation of mode III appears. The relative variation parameter Δ is approximately 7.2%. Finally, the values of G IIC-Theory and G IIC-Beam law in Table.11 are also presented in the same figure, which are bigger than G IIC-FEM by 37.7% and 24.0%, respectively.

	G IIC (N/m)	G IIC-Theory G IIC-Beam law
	Test 1	970	911
	Test 2	1123	982
	Test 3	934	830
	Test 4	1032	932
	Average	1015	914
	Standard deviation	83	63
	60 / 125

  In this study, the DCB and ENF specimens are obtained from a multidirectional laminate. The variation of G I in DCB test is quite big as discussed above. As shown in Table.12, if we compare the maximum value of G I obtained by the FE method with the toughness obtained by CC method using Beam compliance law, G IC-CC is 2.2% bigger than G IC-FEM-max and G IIC-CC is 15.7% bigger than G IIC-FEM-max, respectively. The difference between G C-FEM-max and G C-FEM-av is small.In general, the values evaluated by VCCT in DCB and ENF tests are smaller than those evaluated by CC method. If the G C-FEM represents the delamination toughness, CC method cannot be accepted strictly because a crack has already propagated before reaching G C-CC .

	Test CHAPTER IV. STUDY OF DELAMINATION UNDER PURE G C-FEM-max (N/m) G C-FEM-av (N/m) G C-CC (N/m) MODE III specimens is standardized. Mode IV.1. Pure mode III delamination tests proposed in literature
	I	DCB	450	348		460
	II	ENF	790	776		914
		Table. 12. Toughness of pure mode I, pure mode II
	G (kN/m)
	1.1
	1
	0.9
	0.8
	0.7
	0.6		GI
	0.1 0.5 0.4 0.3 0.2		GII GIII GIIC-FEM GIIC-Theory GIIC-Beam Law
	0
	0	0.2	0.4	0.6	0.8	1
			x/b
			61 / 125

  set up for ECT-1 test according to the stacking sequence and were separated symmetrically into two sub-laminates by the crack plane.

			mm)
	1-1	44.00 87.00	15
	1-2	44.00 87.00	20
	1-3	44.00 87.00	25
	2-1	44.00 87.00	15
	2-2	44.00 87.00	20
	2-3	44.00 87.00	25
	3-1	44.00 87.00	15
	3-2	44.00 87.00	20
	3-3	44.00 87.00	25
	Table. 13. Dimensions of ECT-1 specimen

Table. 13. Every dimension is measured 3 times at least and the average value is presented in the table. ECT-1 Test B (mm) L (mm) a (IV.1.1.2. Finite element analysis 64 / 125 16 plies were

  Table. 14. The standard deviation of the results is small.

	1.2
	1
	0.8
	0.6
	0.4
	0.2
	0
	0	2	4	6	8	10 12 14
						Test	G IIIC-CC (N/m)
						1	1218
						2	1351
						3	1362
					Average value	1310
					Standard deviation	80
					Table. 14. G IIIC measured by ECT-1 tests

  1), mode II component becomes predominant. Moreover at one edge, the mode I component becomes more than 76.7% of G IC measured by DCB test. According to the Omniscan image (Figure IV. 4(b)), the onset of crack growth occurs in the area a (x/L=1) where G II becomes important but G III very small, moreover it seems that crack onset occurs under pure mode II or under mixed mode I+II+III instead of pure mode III.

  42 are compared in the Figure IV. 15. It is seen that the values of G IIIC measured on the specimens with different initial crack lengths (10 mm, 15 mm, 20 mm, 25 mm) are very close to each other if T C-1 or T C-2 are considered as the critical load while the difference becomes too large if T C-3 is used. Actually, these G

IIIC do not correspond to the onset of crack growth, but to first instable crack propagation. In summary, it is more possible to consider T C-1 or T C-2 as critical load. In the next part, only T C-2 is used to determine G IIIC , also because it is easier to measure experimentally.

  2. Edge Ring Crack Torsion (ERCT or ERC-III) testIV.2.1. IntroductionIt is seen that the participation of mode II component in all the ECT tests can never be totally eliminated. The variation of G III along the crack front is too big to be negligible. Generally, this variation is slighter in the central domain of the crack front, but much more significant at the extremities of the crack front. Values of G III can go to almost zero at the extremities.

		G (kN/m)
	千	1.4 1.6
		1.2
		1.0
		0.8 0.6 0.4		GIII GII GI GIII-num_average
		0.2		GIII-exp_average
		0.0
		0	0.2	0.4	0.6	0.8	1
				x/L
	An original pure mode III test was proposed in previous work in our laboratory, named Edge
	Ring Crack Torsion (ERCT or ERC-III) test. This is a torsion test on a composite plate

including an edge ring delamination crack between two sub-laminates. The loading mode obtained is pure mode III, and the crack front is circular, so closed without edges (that means without the extremities of the crack front). The edge effect in ECT specimens mentioned

  Table. 17. G IIIC obtained from Tada's formula for ERCT tests

					) G IIIC-Tada (N/m)
	30	204±4	5400	3.13±0.06	910±34
	50	742±3	5400	3.17±0.02	932±11

  Tada formula expressed by Eq. 43 for ERCT test is more suitable in MERCT test because the loading condition is completely axisymmetrical and so similar to the original situation. Herein, the pure mode III toughness can be determined by Tada's formula Eq. 43 and Eq. 44, denoted G IIIC-Tada or by finite element analysis, named G IIIC-FEM that presents an average value of G III at the critical load. This average is physically meaningful because the variation of G III along the crack front is limited. III-1 test, the crack propagates towards the centre in an unstable manner. Figure IV. 57 shows a typical experimental torque/rotation angle curve, where a sudden drop in the torque corresponding to the crack propagation is observed. Note that the experimental curve is not really linear up to the peak load, but the repeatability of the test is relatively good. III-2 test, the crack propagates towards the centre in a stable manner. Figure IV. 58 shows a typical experimental torque/rotation angle curve, where a sudden drop in the torque is observed. This point should be corresponding to the crack onset, where a local unstable crack growth occurs. And then the torque kept on increasing with a stable propagation of the crack until a total fracture of the specimen. Note that the experimental curve is not really linear up to the first peak load. And the first peak torque is considered as the critical load.Figure IV. 58. Typical experimental torque/rotation angle curve of ERC-III-2 testMERCT test (ERC-III-1) has been simulated by a finite element model to obtain the distribution of the strain energy release rates. Figure IV. 59 shows the evolution of normalized G III along the crack front using the ERCT and MERCT devices. The modification of the mode III device conduces to more uniform distribution of G III in the ERC-III-1 specimen with d=50mm, where Δ is less than 2.8%. As a comparison, Δ from an original ERCT test is approximately 17.4% for a specimen with d=50mm, and Δ is approximately 14.0% from ECT-2 test. The improvement on the distribution of G III seems effective by using axisymmetrical device. Moreover, mode I and mode II components over the total strain energy release rate stay below 1.4% as shown in Figure IV. 60, so can be considered as negligible. As a comparison, mode II component from ECT-2 test becomes significant at the sides of the crack front. Therefore, the interest of this pure mode III test is evident. In conclusion, MERCT test can be considered as a pure mode III delamination testing method and with uniform distribution of G III along the crack front.

	IV.4.4. Results and discussions	1.2
		1
	0.8 IV.4.4.1. Modified Edge Ring Crack Torsion-1 (ERC-III-1) test
	0 0.2 0.4 0.6 0 0.2 0.4 0.6 0.8 1 1.2 In the ERC-Figure IV. 57. Typical experimental torque/rotation angle curve of ERC-III-1 test GIII deviation GII deviation GI deviation 0° 0° 0.2 0.4 0.6 0.8 T (kN.m) IV.4.4.2. Modified Edge Ring Crack Torsion-2 (ERC-III-2) test 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 In the ERC-0 α (°)
		GIII deviation
		GIII concentric
		100 / 125 101 / 125

  ). By introducing the average critical load obtained from ERC-III-2 tests, G IIIC-FEM =1254 (N/m) was obtained. The G IIIC-FEM from ERC-III-1 test is 16% smaller than that from ERC-III-2 test. Furthermore, taking Δ into consideration, these values are quite close to that measured by the Edge Cracktorsion test (ECT), valued at 1213 (N/m).The results of G IIIC from two MERCT tests according to Eq. 43 and Eq. 44 are listed in Table.18. It also shows that the value from ERC-III-1 test is about 17% smaller than the one from ERC-III-2 test. Additionally, values of G IIIC-Tada agree well with G IIIC-FEM . For these two tests, G IIIC-Tada is about 8% bigger than G IIIC-FEM .

			1.2
			1
			0.8
	Test	0.6 T C-av (N.m) G (MPa) K III (MPa.m 1/2 ) G IIIC-Tada (N/m)
	ERC-III-1	754±31	0.4 5400	3.22	961±81
	ERC-III-2	829±21	0.2 5400	3.54	1159±5 9
	0 Table. 18. G IIIC obtained from Tada's formula for MERCT tests 0°
	IV.4.5. Conclusions
				GIII ERCT
				GIII MERCT
			1
			0.8
			0.6
			0.4
			0.2
			0	0°
				GIII
				GII
				GI
			103 / 125

  Above all, ERC-III tests are promising testing methods to characterize mode III delamination behavior. In ERC-III-1 test, an unstable crack growth was observed. And in ERC-III-2 test, at onset of delamination the crack propagated a step suddenly, following by a stable crack growth. This phenomenon makes it possible to stop the test before the final fracture of the specimen to observe the onset of crack growth. IV.5. Conclusions IV.5.1. Comparison of pure mode III toughness G IIIC obtained by different methods The average values of G IIIC measured by different tests are compared in Table.19. Theoretically, the toughness under pure delamination mode in terms of the critical strain energy release rate is believed intrinsic to the material but the measured values are influenced not only by the testing and calculating methods but also by the geometry of the tested specimens. This means that the measurement of G IIIC is problematic.

	Type of test	G IIIC-FEM (N/m) G IIIC-CC or G IIIC-Tada (N/m) Difference
	ECT-1 (a=20mm)	945	1310±80	39%
	ECT-2 (a=20mm)	1213	1017±44	16%
	ERCT (d=30mm)	890	910±34	2%
	ERCT (d=50mm)	1173	932±11	21%
	ERC-III-1 (d=50mm)	1049	961±80	8%
	ERC-III-2 (d=50mm)	1254	1159±59	8%

  .1. Pure mode I and pure mode II In chapter III, the Edge Ring Crack Torsion test was studied. The main advantage of the Edge Ring Crack specimen, called ERC thereafter, is the total absence of the extremities of crack front and so no edge effects on the results. The application of ERC specimen can be spread into other modes of delamination.

	Mode G C-FEM (N/m) G C-CC or G C-Tada (N/m)	Test
	I	450	460±23	DCB
	II III CHAPTER V. DELAMINATION UNDER PURE MODE I, 776 914±63 ENF 1254 1159±59 ERC-III-2
	PURE MODE II AND MIXED MODE USING MODIFIED Table. 20. Toughness of pure mode I, pure mode II and pure mode III EDGE RING CRACK SPECIMEN
			108 / 125

V

  .2.2. Finite element analysisIn the FE model of mixed I+II delamination test, all of the geometric and material parameters are the same as pure mode ERC tests except for the stacking sequence (0/45/45/0/45/0/0/45/45/0/0/45/0/45/45/0)//crack//(0/45/45/0), which is highlighted in Chapter I. A displacement curve 2mm was employed on both of the loading axes in order to impose a traction load. All the other parameters were set as in ERC-I model.

V.2.3. Results and discussions

Table. 2. Comparison of the main pure mode III tests

ACKNOWLEDGEMENTS

First of all, I would like to express my sincere and heartfelt thanks to Professor Gong XiaoJing, Professor De Luycker Emmanuel and Professor Hurez Anita for their patience and guidance through my studies during the thesis. They have tried their best to help me get over the difficulties.

List of Tables

 Table. 1 [START_REF] Chang | A progressive damage model for laminated composites containing stress concentrations[END_REF]modifies the Hashin's criterion integrating the nonlinear behavior in shear, frequently observed for composites:

where α comes from the non-linear shear equation The expression of this criterion can be summarized in: Fracture of fibers in traction:

Eq.

Fracture of fibers in compression:

Eq.

Fracture of matrix in traction:

Eq.

Fracture of matrix in compression:

Eq.

The criteria in the third category are widely implemented in finite element software.

Longitudinal traction:

Transverse traction:

In plane shear:

Out of plane shear:

Figure IV. 21. A picture of ERCT test

In order to propose a closed-form expression for determining G IIIC , we started from the case of a cylinder of diameter D with an external crack of diameter d submitted to a torque T. The stress intensity factor K III is given by a semi analytical expression Eq. 43 from Tada Handbook [START_REF] Tada | The stress analysis of cracks handbook[END_REF] developed for a metal cylinder with an external circular notch, shown in The most significant advantages of MERCT tests are:  The fracture mode does be pure mode III without the participation of mode I or mode II components;  Good distribution of G III along the crack front. Δ is much smaller than ECT tests and original ERCT test. In fact, a nearly constant value of G III along the crack front is obtained in the MERCT tests. So it's more acceptable to consider the average value, G IIIC-FEM, as the material toughness;  The agreement between G IIIC-Tada and G IIIC-FEM is good with a difference only approximately 8% for both MERCT tests. G IIIC-Tada is smaller than G IIIC-FEM , which guarantees the security to consider G IIIC-Tada as the delamination toughness;  The robustness of MERCT tests is well confirmed. Some defaults introduced by the manufacture process and test process are generally inevitable. Unwanted fracture mode introduced by different defaults seems no significant. But the variation of G III along the crack front is more sensitive to certain defaults,

In conclusion, MERCT tests are promising testing methods to characterize mode III delamination behavior. In ERC-III-1 test, an unstable crack growth was observed. And in ERC-III-2 test, at onset of delamination the crack propagated a step suddenly, following by a stable crack growth. This phenomenon makes it possible to stop the test before the final fracture of the specimen to observe the crack growth at the onset.

IV.5.2. Comparison of G IC , G IIC and G IIIC

The delamination toughness under the three pure modes is listed in Table . 20. They were obtained from DCB, ENF and ERC-III-2 tests. ERC-III-2 test was preferred because it has smaller standard deviation. It is shown that in the cases of DCB and ENF tests the average value from finite element analysis can be quite different from the one determined by compliance calibration method. For DCB test, the distribution of G I along its straight crack front is not uniform with Δ=29.1%. In addition, the length of the crack front is much shorter compared to ECT and ERC tests so that the variation has a bigger influence on the result. So G IC-FEM given here should be not meaningful. Instead, the maximum value G IC-FEM-max should be considered, which is shown in Table . 20. For the rest, average values by FEM are considered. Concerning ENF test, the distribution of G II is relatively good with Δ=7.2%, so G IIC-FEM given here should be meaningful. Recall that the participation of mode III at the edges of the crack front is observed in ENF test, which could interfere with the measurement of G IIC . However in the case of ERC-III-2 where the variation of G III is small with Δ=2.7%, the difference between the value from FEM and the one from Tada's formula is only 8%.

Based on the toughness obtained from CC or Tada's methods, G IIIC is 27% higher than G IIC, the former becomes 62% than the later if we refer to the average values from FEM. The difference between G IIC and G IIIC does seem significant for the composite tested. This observation should be generalized, because the pure mode II and pure mode III are two independent fracture modes. So the corresponding behavior can be very different one from another for some materials, or similar for others materials.

of mode I over mode III component is approximately 2.6, which is controlled by the proportion of traction and torsion.

Moreover, the distribution of mode I component along the crack front is uniform while mode III component increases close to ±0° and ±90° and decreases approaching ±45°. Δ of G III is approximately 5.0%, which is relatively small compared to traditional mixed mode I+III delamination tests generally.

V.4.3. Results and discussions

The evolution of G I , G II and G III normalized by G T along the crack front obtained by FEA using the VCCT is shown in Figure V. 13. A mixed mode I+II+III delamination is demonstrated to be realized indeed. The mixed mode ratio in terms of G I /G T for the participation of mode I, G II /G T for that of mode II and G III /G T for that of mode III is 27%: 11%: 62%, respectively. The mixed ratio can be controlled by the stacking sequence of the specimen, as well as the proportion of traction and torsion.

It is interesting to note that the variation of the mode I and mode II components seems following the same tendency: they decrease close to 0°, ±90° and 180°, and increase close to ±45°. But the variation tendency of mode III component is just on the contrary, it increases while mode I and mode II components decrease. Δ of mode I component is approximately 14.8%, Δ of mode II component is approximately 27.5% and Δ of mode III component is approximately 9.0%. The level of the variation for mode II is too high to be validated. Recall that all of the data reduction methods mentioned here can be applied to determine precisely the fracture toughness only if the variation of the strain energy release rate along the crack front is small enough. So the optimization of the specimen geometry is necessary.

Next, the observation about the onset of crack growth and the process of propagation should be continued. Actually, limited by experimental conditions and the time, the observation of the crack onset and the crack growth by C-SCAN and of fracture surface by SEM were not realized on all specimens, the former provides the information about the adjacent fiber orientation effects; the later should be essential to understand the damage mechanisms in the tested composite under different delamination loadings. But these observations need to stop the test and unload the specimen in the current conditions of the laboratory. Sometimes, it is difficult to stop the test in time after the critical load is reached in certain ERC tests since delaminating growth is unstable. In these cases, others in-situ non-destructive monitoring methods have to be adopted.

Concerning the mixed mode delamination testing, the methods presented in the thesis needs to be evaluated experimentally. Series of mixed mode I+II, I+III, and I+II+III delamination tests should be carried out with different mixed mode ratios by experiment and finite element analysis.

Finally, it's now possible to propose a semi-empirical criterion to predict delamination resistance for any mixed mode ratio, especially with the participation of mode III. The delamination resistance can be expressed by a general criterion as f(G I , G II , G III , G IC , G IIC , G IIIC )=0. It should be based on the measurement of the toughness under pure mode I, pure mode II and pure mode III loading. As in Eq. 21 proposed for mixed mode I+II, the empirical constant can be determined by the interpolation of G TC as a function of the mixed mode ratio G II /G T . In the same manner, mixed mode I+III constant could be determined by the interpolation of G TC as a function of the mixed mode ratio G III /G T . Finally, the measurement of the toughness by mode I+II+III delamination testing will allow validating the criterion.