Ch El Hage

R Younès

Z Aboura

M L Benzeggagh

M Zoaeter

Ch El Hage

Analytical and numerical modeling of mechanical properties of orthogonal 3D CFRP

Keywords: 3D orthogonal; Homogenization, Modeling, Finite Elements

This study proposes an analytical and numerical model, for the prediction of mechanical properties of orthogonal 3D reinforcement composite materials, taking into account their structural parameters (mechanical properties of the components and geometrical architecture). This first step of this work requires the definition of the composite representative elementary volume (REV). Microscope studies made possible to visualize the architectural aspect of the internal structure. From these observations two types of REV, adapted to the types of modeling (analytical and by finite elements FE), were defined. The first one takes into account the whole of material in its thickness thus integrating the characteristics of the layers and the vertical reinforcements. The second one, strongly simplified in order to minimize the costs of calculations, is used in the FE approach. Moreover the analytical model is extended to the prediction of the ultimate properties by using the tonsorial criterion 3D of Tsai. The results obtaining from these modeling are compared with experimental results. This comparison highlights the interest and the limits of each approach according to the effect of the choice of the REV.

Introduction

The interest of the introduction of out off plane reinforcement into composite materials is not any more to demonstrate. In addition to increase the mechanical performances in the third

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direction, this reinforcement also improves considerably inter-laminar resistance of these materials. If the weaving technique makes it possible to propose a complex 3D preforms, the current challenge consists in developing tools able to predict the mechanical behavior of these materials starting from the structural geometrical parameters. This step is belonging works initiated in eighties which currently continued and which consist in micromechanical modeling of woven composites materials. In the first models, only the undulation in the "x" direction or stuffer yarn was taken into account [START_REF] Ishikawa | One-dimensional micromechanical analysis of woven fabric composites[END_REF]. These models evolved by taken also into account the weaving in the "y" direction or filler yarn [START_REF] Aboura | Analytical model of woven composite laminate superposition effect of two plies[END_REF] and the relative shift of the layers.

In 1994, Sankar and al [START_REF] Sankar | Analytical method for micromechanics of textile composites[END_REF] proposed an analytical model called Selective Avering method (SAM): this approach is based on a combination of average in stiffness and compliance to consider the thermo-elastic 3D properties. Scida and al (98) [START_REF] Scida | A micromechanics model for 3D elasticity and failure of woven composite materials[END_REF] propose an analytical model of homogenization by summations of average stiffness of each discretized volume of the elementary cell, to predict the threedimensional properties of 2D woven. Ping et al., 2000 [5] propose a modeling of the elastic behavior of 3D orthogonal by using similar analytical models, baptized XYZ, YXZ, ZXY and ZYX based on the conditions of iso-stress and iso-strain on micro-elements of representative elementary volume (REV) divided in series or parallel.

It acts in this paper of mechanical behavior modelling (elastic and failure) of 3D orthogonal carbon reinforcement composite material. Two REV or basic cells will be considered in this studies. An analytical model inspired of work of Scida et al. (98) [START_REF] Scida | A micromechanics model for 3D elasticity and failure of woven composite materials[END_REF] is applied to each basic cell. The results obtained are confronted with those obtained by ZYX model, with a finite elements approach based on the principle of optimization the functional calculus (Duysinx, 96) [START_REF] Duysinx | Optimisation topologique du milieu continu à la structure élastique[END_REF] and with experimental results. The effect of the choice of the basic cell is then highlighted.

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The failure, in uni-axial tension, is also predicted by the model and compared with the experimental results. It uses the 3D criterion of Tsai-Wu, while having resorts to the data base established by Khellil [START_REF] Khellil | Evaluation expérimentale d'un critère de rupture tensoriel polynomial tridimensionnel pour matériaux composites[END_REF].

Material Presentation

The material of this study is an Epoxy RTM6 matrix reinforced by 3D orthogonal T300J carbon. Its thickness is 8mm. The preform is woven according to the three orthogonal directions of a Cartesian reference mark. The reinforcements in the third direction juxtapose the longitudinal and transversal yarns along the thickness. The third yarns are vertical in the heart of the preform, and take a concave weaving form at the surfaces. The whole of the reinforcement forms a structured internal architecture. Figure 1 represents a longitudinal section of the material and a schematization of the higher surface.

The yarn proportions are 46% in warp direction, 46% in weft direction and 8% in vertical reinforcement.

Microscopic study

The observations carried out on an electronic microscope associated with images analysis, show that the material symmetry is not respected perfectly. In fact, the compaction of the preform follow-up by the resin injection creates disequilibria between geometrical dimensions of the yarns in the warp and weft direction. The analysis of the microscopic images obtained makes it possible to modeling the whole of the geometrical characteristics of the structure necessary to the next step. The sections of the wicks in the three directions are appreciably rectangular (fig. 2). Nevertheless, the horizontal section shows that the weft yarn presents, in the plane, a contracted zone due to the passage of the vertical reinforcements (fig. 2c).

Concerning the vertical reinforcement, the proportion of vertical part and the weavers part was determined. Thus the first one represents 64.14% of the total of the yarn and each curved

Analytical modeling

The selected representative elementary volume (REV) is decomposed according to its In addition, by preoccupation with a confrontation of this approa with those of the literatures, a model ZYX of Ping et al., (2000) [START_REF] Ping | Behavior of 3D orthogonal woven CFRP composites. Part II. FEA and analytical modeling approaches[END_REF] based on the principle of iso-stress and iso-strain is also applied to materials of this study.

Choice of the REV

In conventional way, the geometrical periodicity of a composite is described in term of basic cell or REV. It is defined by the condition which the entire composite can be built starting from copies of this one relocated in space. The loading response of the composite can then be calculated by analyzing the behavior of this only basic cell.

For each periodic structure, there is a multitude of possible choices. In a simplification preoccupation, symmetries in a 3D orthogonal composite impose two main roads:

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Unit cell (UC)

This REV neglects the higher and lower undulation in the vertical reinforcement. It is then possible to propose a very simple basic cell and consequently requiring a reduced computing time. This REV consists of compartments of yarn and resin with parallelepiped form, juxtaposed the ones with the others as shown on figure 3. The volumes of each component and their proportion are given in table 1.

Global cell (GC)

This REV takes into account the real form of the vertical yarn, on the totality thickness of the material (fig. 4). The 16 yarns in each warp and weft direction are modeled by parallelepiped compartments form. The vertical yarns are modeled by five types of compartment chosen according to orientations of fibers: two horizontal on the surfaces, followed by two in form of curve and then the vertical one. The curve of the yarn is modeled on the basis of four tangent circle of average radius equal to 3/2 the width of the vertical yarn (0.67mm). This value is appreciably equal to the average value of 0.63 mm determined by the microscopic analysis.

The detail of the modeling is represented in figure 5. The total volume of the REV of total cell is 21.352 mm 3 . The volume of the components and their proportion is given in table 1.

The orientations of fibers in curve compartments are taking into account in the homogenization procedure.

3D Mechanical properties

3D elastic properties: E L , E T , E H , G LT , G LH , G TH , ν LT , ν LH , ν TH given for each type of modeling by a succession of stage are represented in the flow chart (Fig. 5). Calculation is based on the geometrical properties of the basic cells and on the mechanical properties of the components.

Ultimate properties

The elastic modeling is extended to the case of the ultimate properties in particular in uniaxial tension. Indeed the determination of local stiffness allows determining the local states of stresses for a macroscopic level of loading. A criterion of failure 3D of Tsai-Wu type which takes into account the three-dimensional effect of the texture of material is then applied to each increment of loading.

The criterion is written by his tonsorial form:

1 . . . = + j i ij i i F F σ σ σ ; ( 2
)
for : i,j = 1,...... 6

The linear terms of this expression F i et F ii as well as the ultimate stress are determined by compression, tension and shearing tests on the components or, in certain cases resulting from the literature. The complexity by using the such criterion resides in fact, in the determination of the interactions' coefficients F ij ; for i ≠ j. In this study, the coefficients used result from Khellil works [START_REF] Khellil | Evaluation expérimentale d'un critère de rupture tensoriel polynomial tridimensionnel pour matériaux composites[END_REF].

Numerical modeling

Several digital techniques are proposed in the literature in order to homogenize an orthotropic material. We will quote in others:

-Theory of the effective modulus under two possible approaches in strain or stresses -Method of homogenization periodic -Energetic approach

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All these techniques require a REV checking the symmetric material on the three planes of orthotropic (fig. 6).

This study uses an energetic approach based on the principle optimization of the functional [START_REF] Duysinx | Optimisation topologique du milieu continu à la structure élastique[END_REF]

dS n a d a T . . . . ] [ . ] [ 2 1 ) ( σ σ ε - Ω = Π (3)
The first integral constitutes the deformation energy in the REV and the second represents the work applied by the load to the limits of the REV All these variables must be expressed according to displacement 3D in elementary volume. The discretization of the REV in a whole of functional calculus is done as follows: On the level of the assembly of stiffness matrix, the homogenization also requires the taking into account a boundary condition expressing equality between the axial or angular strains on the requested borders in tension or torsion. This numerical approach is realized under MatLab environment. It gives a fast convergence towards the Young modulus by a low number of trihedral meshes (592). On the other hand, convergence towards the shearing coefficient requires a high number of elements (37577).

Q K Q Q K Q d T e

Results and discussion

The mechanical properties of each type of yarn warp, weft and vertical directions, given analytically (Berthelot 92) [START_REF] Berthelot | Matériaux composites : comportement mécanique et analyse des structures[END_REF] different according to the selected REV. Indeed, the volume of each component is evolved and on the other hand the total fraction volume of the composite is maintained constant for both REV (table 3). The table 4 presents the elastic properties 3D obtained by each analytical model and finite elements. They are compared with some results of experimental tests carried out on this material. This reveals that the analytical models and finite elements give results appreciably the same ones for the properties in the longitudinal directions and transversals whatever the selected of REV. On the other hand the effect of the choice of the REV feels in the determination of the properties in the thickness. Indeed, the Global cell gives results very close to the experimentation contrary to the Unit cell which over-estimates the E 3 modulus. By comparing this model with ZXY model of Ping et al. [START_REF] Ping | Behavior of 3D orthogonal woven CFRP composites. Part II. FEA and analytical modeling approaches[END_REF] , we note a notable difference of the shear stiffness modulus. The model using the Global cell gives best satisfaction by comparison to the experimental results. The properties predicted by the model of this study are in agreement with experimentation and rather close to the results obtained by finite elements.

The analytical results of failure strength by application of the criterion 3D of Tsai-Wu and the experimental results are given in table 5. Modeling proposes of ultimate stiffness, in tension in the longitudinal direction, is independently of the choice the REV. We also notes that the failure criterion 3D gives satisfaction in comparison with the experimental results. Indeed, the unit cell (UC) over-estimates the failure strength since it does not take account of the undulation of vertical yarn. The state of strength for the latter case is presented on figure 7. It

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is noticed that the checking of the criterion of Tsai-Wu is a function of the orientation's angle of the reinforcement in this zone. As an example, in longitudinal tension, the first damages appear at 180 MPa.

Conclusions

This study approached the modeling of the mechanical behavior of a composite material with reinforcement carbon orthogonal 3D. The determination of the elastic properties 3D homogenized was done by using an analytical and numerical model. This study reveals the importance of the choice of the REV on certain mechanical properties as observed in the case of out off plane elastic and failure properties. The models analytical and numerical suggested were validated following a comparison with the experimental tests. Moreover the use of the criterion of Tsai 3D gave satisfaction for the prediction of tension failure in the longitudinal direction. Tests in the three directions of space will have to be carried out in order to complete the validation of the model.

The numerical method which was interested to the elastic behavior tends to amplify the homogenized characteristics, unless refining more the mesh of the REV.

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components ( 3

 3 yarns and resin blocks). The elastic behavior of each component is then expressed by the three-dimensional stiffness matrix [C ij,k ]. If the component corresponds to the yarn, the stiffness matrix is expressed in the reference mark of the yarn (reference 123) then transformed in the basic cell reference mark (reference xyz) by taking account the orientation of yarn. The knowledge of the matrices transformed for each component k, makes it possible to deduce the global stiffness matrix from of the basic cell according to the relation (1) by taking account of the volume fraction of each component. (V t is the total volume of the REV, V k the volume of each component)

  the vector displacement on each trihedral of the REV with a meshes, K e is the matrix of corresponding stiffness, F T is the loading on the trihedral of the borders.The homogenization requires the resolution of the preceding problem in six different situations of loading: three simple tensions along the longitudinal, transverse and vertical axes and three simple shear around the same axes.

Figure 1 :Figure 2 :

 12 Figure 1: Presentation of 3D orthogonal material

Figure 3 :

 3 Figure 3: Schematic presentation of the VER modeling.

Figure 4 :

 4 Figure 4 : Modeling of the curve part of the vertical yarn in the REV global cell.

Figure 5 :

 5 Figure 5 : Flow chart for the prediction of elastic 3D properties and the failure strengths

Figure 6 :

 6 Figure 6: Finite elements modeled of REV.

Figure 7 :

 7 Figure 7 : Stress distribution on vertical yarn under σ x t

Table 1 :

 1 Volumes and proportions of the REV components

Table 2 :

 2 Elastic properties of the composite constituents

Table 3 :

 3 Stiffness results of analytical, numerical and experimental studies

Table 4 :

 4 Failure Strain and stress results

	(a) Longitudinal section	(b) Transversal section	(c) Horizontal section


Viet-Tung Nguyen

Jean-Francois Caron
email: caron@lami.enpc.fr

Finite element analysis of free-edge stresses in composite laminates under mechanical an thermal loading

Keywords: Free edge, Layer-wise, finite element, Laminates, Interfacial stresses

 is applied for general laminated and is shown to be capable of simultaneously predicting global and local responses. The analysis of free-edge stresses of composite laminates subjected to mechanical and thermal loads is performed using this C o eight-node layer-wise finite element after a classical bending validation. Laminates with finite dimensions are considered and three-dimensional out-of-plane stresses in the interior and near the free edges are evaluated. The results obtained with this finite element modelling are compared with those available in the literature. The present calculation provides accurate stresses and can be utilised as and operational tool to predict interlaminar stresses under the loads of mechanical and thermal combined.

The recent widespread use of laminated composites in various industrial fields necessitates analysis tools which are capable of predicting their mechanical behaviour more accurately an efficiently.

For laminated composites, it is well known that at the free edges interlaminar stresses arise from the mismatch of elastic properties between layers. Hence, the stress distribution in the vicinity of the free edges is in three dimensional (3D) state even though the laminates are only subjected to in-plane loading.

The interlaminar stresses are important because they have a marked effect on the failure strengths of composite laminates. Accurate determination of interlaminar stresses near the free edge is therefore crucial to correctly describe the laminate behaviour and to prevent its early failure, notably the delamination onset.

Classical single layer theories are not able to calculate these out-of-plane stresses, notably in presence of free edges. Therefore various approaches such as finite difference [START_REF] Pipes | Interlaminar stresses in composite laminates under uniform axial extension[END_REF][START_REF] Rybichi | Approximate three-dimensional solutions for symmetric laminates under in plane loading[END_REF], 3D finite element [START_REF] Wang | Some new results on the edge effect in symmetric composite laminates[END_REF][START_REF] Raju | Interlaminar stress singularities at a straight free edge in composite laminates[END_REF][START_REF] Lessard | Three-dimensional stress analysis of free-edge effects in a simple composite cross-ply laminate[END_REF][START_REF] Icardi | An evaluation of the influence of geometry and of material properties at free edges and at corners of composite laminates[END_REF], closed-form analytical approach using mixed formulation and stress based methods [START_REF] Becker | Closed-form solution for the free-edge effect in cross-ply laminates[END_REF][START_REF] Pagano | Stress fields in composite laminates[END_REF][START_REF] Kassapoglou | Closed form solutions for the interlaminar stress field in angle-ply and cross-ply laminates[END_REF], boundary layer theories [START_REF] Tang | A boundary layer theory -part II: extension of laminated finite strip[END_REF][START_REF] Wang | Boundary-layer effects in composite laminates: part 1free-edge stress singularities[END_REF][START_REF] Wang | Boundary-layer effects in composite laminates: part 2free-edge stress singularities[END_REF][START_REF] Flanagan | An efficient stress function approximation for the free-edge stresses in laminates[END_REF] and layer-wise theories [START_REF] Reddy | A generalization of two-dimensional theories of laminated composite plates[END_REF][START_REF] Tahani | Three-dimensional interlaminar stress analysis at free edges of general cross-ply composite laminates[END_REF][START_REF] Tahani | Free edge stress analysis of general cross-ply composite laminates under extension and thermal loading[END_REF] have been used to calculate interlaminar stresses near free edges. A complete discussion of the literature on the interlaminar stress analysis can be found in [START_REF] Kant | Estimation of transverse/interlaminar stresses in laminated composites -a selective review and survey of current developments[END_REF].

Finite differences, which are the oldest in the field, have been used in: [START_REF] Pipes | Interlaminar stresses in composite laminates under uniform axial extension[END_REF][START_REF] Rybichi | Approximate three-dimensional solutions for symmetric laminates under in plane loading[END_REF] to solve the two-dimensional governing elasticity equations to determine the interlaminar stresses in a long symmetric laminate under uniform axial strain.

Although 3D FE method [START_REF] Wang | Some new results on the edge effect in symmetric composite laminates[END_REF][START_REF] Raju | Interlaminar stress singularities at a straight free edge in composite laminates[END_REF][START_REF] Lessard | Three-dimensional stress analysis of free-edge effects in a simple composite cross-ply laminate[END_REF][START_REF] Icardi | An evaluation of the influence of geometry and of material properties at free edges and at corners of composite laminates[END_REF] is the most universal method that can be applied to general stacking sequence and boundary conditions, it is quite computational expensive and presents difficulties for convergence in the vicinity of free edges (see [START_REF] Carreira | Model of multilayered materials for interface stresses estimation and validation by finite element calculations[END_REF]). Consequently, some semi-analytical approaches combining the traditional FE method with certain analytical techniques have been proposed. A state space FE method that combines the conventional FE approximation and the recursive formulation of state space equation: state space formulation is introduced to solve for through-thickness stress distributions, while the traditional FE is used to approximate in-plane variations of the state variables in [START_REF] Wang | State space approach for stress decay in laminates[END_REF][START_REF] Ye | A state space finite element for laminated composites with free edges and subjected to transverse and in-plane loads[END_REF][START_REF] Zhang | Free-edge and ply cracking effect in cross-ply laminated composites under uniform extension and thermal loading[END_REF].

In order to avoid these somewhat cumbersome 3D FE methods, several approximate analytical methods have been equally proposed. Some pioneering works involve the perturbation technique by Hsu and Herakovich [START_REF] Hsu | Edge effects in angle-ply composite laminates[END_REF], the boundary layer theory by Tang and Levy [START_REF] Tang | A boundary layer theory -part II: extension of laminated finite strip[END_REF], and the approximate elasticity solution by Pipes and Pagano [START_REF] Pipes | Interlaminar stresses in composite laminates -an approximate elasticity solution[END_REF]. Mixed variational principle has been used along with stress assumption by Pagano [START_REF] Pagano | Stress fields in composite laminates[END_REF]. Wang and Choi [START_REF] Wang | Boundary-layer effects in composite laminates: part 1free-edge stress singularities[END_REF][START_REF] Wang | Boundary-layer effects in composite laminates: part 2free-edge stress singularities[END_REF] have presented a formulation of Lekhnitskii's stress function based on complex variables. Kassapoglou and Lagace [START_REF] Kassapoglou | Closed form solutions for the interlaminar stress field in angle-ply and cross-ply laminates[END_REF] studied the edge effect using the principle of minimum complementary energy and the force balance method. A large amount of development devoted to free-edge stresses analysis have been formulated based on this procedure: using stress functions and principle of complementary energy [START_REF] Flanagan | An efficient stress function approximation for the free-edge stresses in laminates[END_REF][START_REF] Yin | Free edge effects in anisotropic laminates under extension, bending and twisting. Part I: A stress-function-based variational approach[END_REF][START_REF] Yin | Free edge effects in anisotropic laminates under extension, bending and twisting. Part II: Eigenfunction analysis and the results for symmetric laminates[END_REF][START_REF] Cho | Iterative free-edge stress analysis of composite laminates under extension, bending, twisting and thermal loadings[END_REF].

Another alternative method to study the edge effect involves models based on layer-wise theory [START_REF] Reddy | A generalization of two-dimensional theories of laminated composite plates[END_REF][START_REF] Tahani | Three-dimensional interlaminar stress analysis at free edges of general cross-ply composite laminates[END_REF][START_REF] Tahani | Free edge stress analysis of general cross-ply composite laminates under extension and thermal loading[END_REF]. Unlike the single layer theories, the layer-wise theories assume separate field expansions within each layer and thereby pro-vide a more kinematically correct representation of the strain and stress field in each discrete layer. However relatively less attention has been addressed to the development of layer-wise based FE for interlaminar free-edge stresses analysis. Most of layer-wise FEs developed for free edge analysis [START_REF] Spilker | Edge effects in symmetric composite laminates: importance of satisfying the traction free-degree condition[END_REF][START_REF] Liou | A three-dimensional hybrid stress isoparametric element for the analysis of lamianted composite plates[END_REF][START_REF] Tian | Straight free-edge effects in laminated composites[END_REF] are based on the hybrid stress concept using the variational principle of Pian [START_REF] Pian | Derivation of stiffness matrices by assumed stress distributions[END_REF].

Recently, Kimpara and coauthor have developed a layer-wise FE based on assumed higher order displacement in each ply [START_REF] Kimpara | Finite element stress analysis of interlayer based on selective layerwise higher-order theory[END_REF]. Imaginary springs attached at the interlayers have been used to ensure interfacial continuity.

Based on a typical layer-wise model, called the M4-5n (M ulti-particle M odel of M ultilayered M aterials with 5 kinematic fields per layer for an n-layer laminate [START_REF] Carreira | Model of multilayered materials for interface stresses estimation and validation by finite element calculations[END_REF][START_REF] Caron | Modelling the kinetics of transverse cracking in composite laminates[END_REF][START_REF] Naciri | Interlaminar stress analysis with a new multiparticle modelization of multilayered materials (M4)[END_REF][START_REF] Diaz | Software application for evaluating interfacial stresses in inelastic symmetrical laminates with free edges[END_REF]) a new layer-wise finite element has been developed in [START_REF] Nguyen | A new finite element for free edge effect analysis in laminated composites[END_REF]. The potential of this finite element for the analysis of free-edge composites under mechanical loading has been revealed. In the present work, detail investigation of free-edge stress fields under mechanical and thermal loadings is performed with this finite element. Firstly, a rectangular simply supported laminate subjected to a sinusoidal transverse load is examined using this multi-particle FE and the results will be compared with Pagano's three-dimensional elasticity solution [START_REF] Pagano | Exact solutions for rectangular bidirectional composites and sandwich plates[END_REF]. Agreement between the results will show the validity and accuracy of the proposed finite element. Then, after the reliability of the FE model would successfully be validated, straight free edge problem will be considered in comparison with existed solution in the literature. Discussion of solution's sensibility again the number of modelled sub-layers is also presented. It is assumed that the layers are perfectly bonded together so that slip between two adjacent layers may not occur. The thickness of each layer k

(k = 1, n) is denoted e k = h + k -h - k , where h - k , h + k are

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Generalised displacements

The in-plane displacement and rotation fields of layer k of the M4-5n model are defined in the following way:

U k α (x, y) = 1 e k h + k h - k U α (x, y, z)dz, φ k α (x, y) = 12 e k 2 h + k h - k z -h k e k U α (x, y, z)dz, (1)
where h k are the mid-plane z coordinate of the layer k.

The transverse displacement of layer k, U k 3 is defined such that

U k 3 (x, y) = 1 e k h + k h - k U 3 (x, y, z)dz. (2)
Thus each layer presents 5n kinematic fields. The multi-layered plate (3D object) becomes a superposition of n Reissner plates (2D object).

The generalised strains are deduced from the generalised displacements as follows (see [START_REF] Carreira | Model of multilayered materials for interface stresses estimation and validation by finite element calculations[END_REF][START_REF] Caron | Multiparticle modelling for the prediction of delamination in multi-layered materials[END_REF] for more details):

ε k αβ = 1 2 U k α,β + U k β,α , χ k αβ = 1 2 φ k α,β + φ k β,α , γ k α = φ k α + U k 3,α , D j,j+1 α = U j+1 α -U j α - e j 2 φ j α - e j+1 2 φ j+1 α , D j,j+1 ν = U j+1 3 -U j 3 . (3)
In each layer k, one can find the conventional membranar strain ε k αβ , bending curvatures χ k αβ and shear strains γ k α . At each interface Γ j,j+1 , the interlaminar strains D j,j+1 α , D j,j+1 ν are defined associated to interlaminar transverse shear stresses σ α3 (x, y, h + j ) and normal stress σ 33 (x, y, h + j ), respectively.

Generalised stresses and equilibrium equations

In layer k (1 ≤ k ≤ n), the generalised in-plane stress resultants N k , the generalised in-plane moment resultants M k and the generalised out-of-plane

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shear stress resultants Q k are defined as follows:

N k αβ (x, y) = h + k h - k σ αβ (x, y, z) dz, M k αβ (x, y) = h + k h - k (z -h k ) σ αβ (x, y, z) dz, Q k α (x, y) = h + k h - k σ α3 (x, y, z) dz. (4)
The interlaminar shear and normal stresses at interface j, j +1 (1

≤ j ≤ n-1)
are defined as

τ j,j+1 α (x, y) = σ α3 (x, y, h + j ), ν j,j+1 (x, y) = σ 33 (x, y, h + j ).
(

) 5
The equilibrium equations of the M4-5n model can be written as

N k αβ,β + τ k,k+1 α -τ k-1,k α = 0, Q k β,β + ν k,k+1 -ν k-1,k = 0, M k αβ,β + e k 2 τ k,k+1 α + τ k-1,k α -Q k α = 0. ( 6
)

Generalised constitutive equations

The M4-5n generalised constitutive equations are obtained in compliance terms, related to:

• The membranar stress resultants in layer k (1 ≤ k ≤ n)               ε k 11 ε k 22 2ε k 12               = 1 e k S k               N k 11 N k 22 N k 12               , (7)
where the second-order tensor S k represents the in-plane compliances of layer k defined by

S k αβ = S ααββ (z), S k 16 = S k 61 = 2S 1112 (z), S k 26 = S k 62 = 2S 2212 (z), S k 66 = 4S 1212 (z); • the in-plane moment resultants in layer k (1 ≤ k ≤ n)               χ k 11 χ k 22 2χ k 12               = 12 e k 3 S k               M k 11 M k 22 M k 12               , (8)
• the out-of-plane shear stress resultants in layer k

(1 ≤ k ≤ n)         γ k 1 γ k 2         = 6 5e k S k Q         Q k 1 Q k 2         - 1 10 S k Q         τ k-1,k 1 + τ k,k+1 1 τ k-1,k 2 + τ k,k+1 2         (9)
where the second-order tensor S k Q represents the shearing compliances of layer k defined by S k Qαβ = 4S α3β3 (z);

• the transverse shear stresses at interface Γ j,j+1 (1

≤ j ≤ n -1)         D j,j+1 1 D j,j+1 2         = - 1 10 S j Q         Q j 1 Q j 2         - 1 10 S j+1 Q         Q j+1 1 Q j+1 2         + 2 15 e j S j Q + e j+1 S j+1 Q         τ j,j+1 1 τ j,j+1 2         - e j 30 S j Q         τ j-1,j 1 τ j-1,j 2         - e j+1 30 S j+1 Q         τ j+1,j+2 1 τ j+1,j+2 2         (10)
• the normal stresses at interface Γ j,j+1 (1 ≤ j ≤ n -1)

D j,j+1
where the scalar S j ν denotes the normal compliance of layer j defined by

S j ν = S 33 (z) = S 3333 (z).
3 Finite element modelling of M4-5n equations

Finite element discretisation

The standard finite element formulation with C o continuous plate element has been used in this investigation. In this approach, the entire domain is

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subdivided in NE number of sub-domains or elements such that

Π(δ) = N E el=1 Π e (δ) (12)
where Π and Π e are the total potential of the system and the element, respectively. For a n-layer plate, the 5n vector of unknown nodal displacement variables are defined as

δ = {U 1 V 1 W 1 φ x 1 φ y 1 , • • • , U n V n W n φ x n φ y n } T , (13)
where the conventional engineering notations

U=U 1 , V =U 2 , W =U 3 , φ x =φ 1
and φ y =φ 2 are used. Note that to study the through-the-thickness stresses distribution, each physical layer can be subdivided in to a large number of numerical (or mathematical) layers by introducing more interfaces and therefore more unknown displacement variables.

In this study, eight-node two-dimensional shape functions N i (i=1, 8) are adopted for interpolating both the geometry and generalised displacement such that:

(x, y) = 8 i=1 N i (x i , y i ), ( 14
) δ = 8 i=1 N i δ i , (15)
where x i , y i and δ i are the nodal values. The interpolation functions matrix for the node i are N i = N i I 5n . The explicit expressions of N i can be found in [START_REF] Hinton | Finite element programming[END_REF].

The (11n -3) complete generalised strain vector ǫ is defined as follows:

ǫ = {ǫ c , ǫ ν , ǫ Q } T . ( 16
)
ǫ c = {ε 1 xx ε 1 yy 2ε 1 xy χ 1 xx χ 1 yy χ 1 xy . . . ε n xx ε n yy 2ε n xy χ n xx χ n yy χ n xy } T . ( 17
)
is the 6n membranar strain vector,

ǫ ν = {D 1,2 3 D 2,3 3 . . . D n-1,n 3 } T . ( 18
)
is the (n -1) normal strain vector and

ǫ Q = {γ 1 x γ 1 y D 1,2 x D 1,2 y γ 2 x γ 2 y . . . γ n-1 x γ n-1 y D n-1,n x D n-1,n y γ n x γ n y } T . ( 19
)
is the (4n -2) shear strain vector.

The associated generalised stress vector can be written as follows:

σ = {σ c , σ ν , σ Q } T , (20)
where

σ c = {N 1 xx N 1 yy N 1 xy M 1 xx M 1 yy M 1 xy . . . N n xx N n yy N n xy M n xx M n yy M n xy } T , σ ν = {ν 1,2 ν 2,3 . . . ν n-1,n } T , ( 21
) σ Q = {Q 1 x Q 1 y τ 1,2 x τ 1,2 y Q 2 x Q 2 y . . . Q n-1 x Q n-1 y τ n-1,n x τ n-1,n y Q n x Q n y } T .
The membranar, normal ans shear strains ǫ c , ǫ ν , ǫ Q can be written in terms of the nodal displacements δ by referring to Eq. ( 3). With the generalised displacement vector δ known at all points within the element, the generalised strain vectors at any point are determined with the aid of Eq. ( 14), as follows:

ǫ = 8 i=1 B i δ i = [B 1 , . . . , B 8 ] δ. ( 22
)
The strain energy stored in the element can be computed using stress-strain relation equations ( 7) - [START_REF] Tang | A boundary layer theory -part II: extension of laminated finite strip[END_REF]:

π e = 1 2 ω δ T B T DBδdω = 1 2 δ T K e δ. (23)
Detail expression of operator matrix B and elasticity matrix D can be found in [START_REF] Nguyen | A new finite element for free edge effect analysis in laminated composites[END_REF]. By assembling the element stiffness matrices and taking into account the boundary conditions, the nodal d.o.f. can be solved.

Layer-wise temperature variation

A layer-wise uniform temperature variation can be taken in to account using the procedure proposed in [START_REF] Diaz | Software application for evaluating interfacial stresses in inelastic symmetrical laminates with free edges[END_REF]. The tensor of inelastic strain due to a uniform temperature variation T i in the k layer is identified as

ε k a (x) = T k α k L e L ⊗ e L + α k T e T ⊗ e T + α k N e z ⊗ e z (24)
where

α k L , α k T et α k N represent thermal coefficient in material direction.
In the global x, y, z coordinate this inelastic strain tensor can be written as

ε k a (x) = T k α k 11 e x ⊗
where

α k 11 = cos 2 θ k α k L + sin 2 θ k α k T α k 22 = sin 2 θ k α k L + cos 2 θ k α k T α k 12 = cosθ k sinθ k (α k L -α k T ) α k 33 = α k N .
with θ k is the orientation of the i layer.

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Finally, according to [START_REF] Diaz | Software application for evaluating interfacial stresses in inelastic symmetrical laminates with free edges[END_REF] the inelastic strain can be expressed as

ε k a αβ (x, y) = α k αβ T i χ k a αβ (x, y) = 0 γ k a α (x, y) = 0 (26) D j,j+1 a α (x, y) = 0 D j,j+1 a ν (x, y) = α j 33 T j e j 2 + α j+1 33 T j+1 e j+1 2
The inelastic strain vector according to the multi-particle finite element formulation are:

ǫ a = {ǫ c a , ǫ ν a , ǫ Q a } T (27)
with

ǫ c a = {ε 1 a xx ε 1 a yy 2ε 1 a xy 0 0 0, . . . , ε n a xx ε n a yy 2ε n a xy 0 0 0 } T (28)
6n

ǫ ν a = {D 1,2 a ν D 2,3 a ν . . . D n-1,n a ν } T (29)
n -1 ǫ Q a = {0 0 0 0 0 0, . . . , 0 0 0 0 0 0 } T

4n -2

Numerical results

In this section, the developed finite element is initially applied to study the bidirectional bending problem of laminated plates. This case of simply supported boundary conditions is evidently used to test the reliability of the FE model, by comparing its results with existing exact elasticity solution [START_REF] Pagano | Exact solutions for rectangular bidirectional composites and sandwich plates[END_REF]. Hence, after the reliability of the proposed FE model has been successfully validated, stress analysis of free-edge problem under mechanical and thermal loads are investigated. ). For the purpose of validating the multi-particle finite element, the following Pagano's choice of material [START_REF] Pagano | Exact solutions for rectangular bidirectional composites and sandwich plates[END_REF] has been used:

E 1 /E 2 = 25, E 2 = E 3 = 7 GPa G 12 = G 13 = 0.5E 2 , G 23 = 0.2E 2 ν LT = ν T N = ν LN = 0.25
where the subscripts 1, 2, 3 signify the direction parallel to the fibres, the transverse direction perpendicular to the fibres, and the out-of-plane direction, respectively. The results of the multi-particle FE calculation will be compared with Pagano's exact three-dimensional elasticity solutions for thick laminate (a = 4h).

Due to the symmetry of the problem, only one quadrant of the plate is modelled (0 < x < a/2, 0 < y < a/2) by a 5 × 5 mesh. The numerical results

shown here are presented by means of the following normalised stresses as the same form of Pagano [START_REF] Pagano | Exact solutions for rectangular bidirectional composites and sandwich plates[END_REF]:

σ z = σ z q o , (σ xz , σ yz ) = (σ xz , σ yz ) h q o a
As previously mentioned, in the multi-particle modelling, each actual physical layer in a laminate can be treated as if it is made of as many layers as it is wished with, of course, the same fibre direction as the actual layer. The present FE model provide only direct access to the interfacial stresses by constitutive equations. Hence, one must use this sublayer modelling to study through-the-thickness stresses distributions. In Figs. 2-4 out-of-plane stresses distributions by six and one sub-layers FE modelling are plotted. The result of three-dimensional elasticity solution [START_REF] Pagano | Exact solutions for rectangular bidirectional composites and sandwich plates[END_REF] is also reported for comparison.

Excellent agreement between the multi-particle FE solution and the exact three-dimensional elasticity solution is found. This close agreement verifies the accuracy of the present finite element. It is shown in this case that regardless of the used sublayer number, interfacial stresses values do not change. Increasing the numerical layer number only helps to reproduce the stress distribution shape and does not affect the result's precision.

Free-edge laminate under uniform axial extension

In what follows, the classical straight free-edge problem are considered. This problem have been studied by several investigators. Certain similarities are observed in these results, however there are also numerous discrepancies, notably in the prediction of interlaminar stresses near free edges. It is reminded here that no exact elasticity solution to this free-edge problem has been yet founded. All the results obtained are merely based on approximate analytical and numerical analyses. For most of these investigations, the longitudinal degrees of freedom in the displacement field is neglected and the stress analysis is then restricted to a generic two-dimensional cross-section of the laminates.

ACCEPTED MANUSCRIPT

In this investigation, general finite dimension laminates with the ply configuration [0 o /90 o ] s , [90 o /0 o ] s are assumed to have the length 2a, width 2b, and thickness h = 4e, with b = 8e and a = 2b (see Fig. 5). The laminate is subjected to a prescribed uniform axial strain ε x = ∆ a . The material properties are taken to be those of a high-modulus graphite/epoxy. The elasticity constants with respect to principal material axis of each ply are:

E 1 = 137, 90 GPa E 2 = E 3 = 14, 48 GPa G 12 = G 13 = G 23 = 5, 86 GPa ν 12 = ν 13 = ν 23 = 0, 21
Due to the symmetry of the problem, only one quadrant of the plate is modelled (0 < x < a, 0 < y < b). Convergence (influence of the mesh pattern) study has been performed in [START_REF] Nguyen | A new finite element for free edge effect analysis in laminated composites[END_REF] and is not presented here. As opposed to 3D FE analysis [START_REF] Wang | Some new results on the edge effect in symmetric composite laminates[END_REF][START_REF] Raju | Interlaminar stress singularities at a straight free edge in composite laminates[END_REF][START_REF] Lessard | Three-dimensional stress analysis of free-edge effects in a simple composite cross-ply laminate[END_REF][START_REF] Icardi | An evaluation of the influence of geometry and of material properties at free edges and at corners of composite laminates[END_REF] the multi-particle FE presents converge finite stresses at free edges. Even a coarse mesh (5 × 5 elements) as shown in Fig. 5 assures a very good convergence. The following results are evaluated with this 5 × 5 mesh. [START_REF] Wang | Some new results on the edge effect in symmetric composite laminates[END_REF] and by Carreira [START_REF] Carreira | Model of multilayered materials for interface stresses estimation and validation by finite element calculations[END_REF], and Pagano's local model [START_REF] Pagano | Stress fields in composite laminates[END_REF] are also plotted for comparison. It can be seen that for y/b < 0.8 the interlaminar stress σ yz obtained by all the calculations are very close, however for 0.8 ≤ y/b ≤ 1.0 as the free edge is approached, these distributions exhibit obvious differences. The results obtained by the use of a special 3D FE procedure [START_REF] Carreira | Model of multilayered materials for interface stresses estimation and validation by finite element calculations[END_REF] for better convergence than classical one are in good agreement with present method. Note that Wang & Crossman's results have been obtained using an arbitrate mesh where convergence is not assured due to the stress singularity at the edge.

The shear stress fields σ yz in Pagano's model verify the free edge conditions (vanish of shear stress at free edge). For the present FE model, this boundary condition is only energetically taken in to account. Nevertheless the evaluation of the shear stress shape is very accurate. Except at the 0 o /90 o interface, the value of σ yz is quite close to zero (see Fig. 7). The steepness of stress gradient at the ply interface, particularly near the free edge is often attributed to a singularity at the 0 o /90 o interface in conventional 3D FE investigations [START_REF] Wang | Some new results on the edge effect in symmetric composite laminates[END_REF][START_REF] Raju | Interlaminar stress singularities at a straight free edge in composite laminates[END_REF][START_REF] Lessard | Three-dimensional stress analysis of free-edge effects in a simple composite cross-ply laminate[END_REF][START_REF] Icardi | An evaluation of the influence of geometry and of material properties at free edges and at corners of composite laminates[END_REF].

Due to the mesh dependency of these models, the stress values on the edge are meaningless and prevent the use of a direct strength of materials approach to predicting delamination onset. As mention above the present model provides useful finite stress on the edge which can be used directly in a limit stress criterion as shown in [START_REF] Caron | Multiparticle modelling for the prediction of delamination in multi-layered materials[END_REF]. The present finite interfacial stresses are in this sense equivalent to the averages of the 3D FE's stresses over a characteristic distance from the edge as used in several delamination onset investigations [START_REF] Brewer | Quadratic stress criterion for initiation of delamination[END_REF][START_REF] Leguillon | The onset of delamination at stress free-edges in angle-ply laminates -analysis of two criteria[END_REF].

The distribution of σ yz through thee thickness at the free edge (y = 0.98b) of the [0 o /90 o ] s laminate is presented in Fig. 7. Each physical ply is modelled by the same N sublayer. Note that the inconvenient of all the models based on layer-wise theory is that the calculation's volume depends on the number of numerical layers and becomes very large with the increasing of the former.

Moreover, in many layer-wise models [START_REF] Tahani | Three-dimensional interlaminar stress analysis at free edges of general cross-ply composite laminates[END_REF][START_REF] Tahani | Free edge stress analysis of general cross-ply composite laminates under extension and thermal loading[END_REF], free-edge stresses are noticeably dependent on the sublayer division N and one has to incriease N to assure the convergence (N must be larger than 6 in [START_REF] Tahani | Three-dimensional interlaminar stress analysis at free edges of general cross-ply composite laminates[END_REF]). It is seen in Fig. 7 that the stress fields obtained by the present model are not affected by the sublayer ACCEPTED MANUSCRIPT division number. The multi-particle model employed here is very general in that the subdivision number can be greater, equal or less than the number of material layers n. Also the number of sublayer division can vary from layer to layer. Hence one can use a large number of sublayers in the interested material layer and a small number of numerical layers elsewhere which is very useful in the investigation of a multilayer laminate with a very large number of material layers. Fig. 8 shows an illustrated case where the 0 o ply of a [0 o /90 o ] s is modelled by 9 and 17 numerical layers and one numerical layer is used for other plies. Again a very good convergence is obtained that confirms the non existence of singularity in our model. 10 in comparison with some results found in the literature. Except a similarity of overall shape, all the considered models provide a dispersion both in magnitude and in local shape gradient. Results of Flanagan [START_REF] Flanagan | An efficient stress function approximation for the free-edge stresses in laminates[END_REF] and Cho & Kim [START_REF] Cho | Iterative free-edge stress analysis of composite laminates under extension, bending, twisting and thermal loadings[END_REF] have been obtained using stress functions and principle of complementary energy. Their results shows a pronounced singularity of stress fields in the edge that explains the large values of σ z in Fig. 10. In the work of Tian et al. [START_REF] Tian | Straight free-edge effects in laminated composites[END_REF], a 3D hybrid stress element has been used to investigate free-edge stress distribution. The values of interfacial stresses of the present model and those of [START_REF] Tian | Straight free-edge effects in laminated composites[END_REF] are quite close and both models confirm no existence of stress singularity at the interface.

The through-the-thickness distribution of σ z the free edge in a [90 o /0 o ] s lam-inate is plotted in Fig. 11. Excellent agreement between the present solutions and those of Tahani & Nosier [START_REF] Tahani | Three-dimensional interlaminar stress analysis at free edges of general cross-ply composite laminates[END_REF] and Wang & Crossman [START_REF] Wang | Some new results on the edge effect in symmetric composite laminates[END_REF] is found for this case.

Free-edge laminate under thermal loading

In this section, [0 o /90 o ] s , [90 o /0 o ] s and [45 o / -45 o ] s free-edge laminates subjected to thermal loading (residual stress due to the curing process) are considered. The results are compared with 3D FE results [START_REF] Carreira | Model of multilayered materials for interface stresses estimation and validation by finite element calculations[END_REF][START_REF] Wang | Edge effects on thermally-induced stresses in composite laminates[END_REF] and those of stress function based method [START_REF] Cho | Iterative free-edge stress analysis of composite laminates under extension, bending, twisting and thermal loadings[END_REF][START_REF] Yin | The effect of temperature gradient on the free-edge interlaminar stresses in multi-layered structures[END_REF]. The following material is used for this investigation: Once again, the present model and 3D FE [START_REF] Wang | Edge effects on thermally-induced stresses in composite laminates[END_REF] provide almost the same results.

E 1 = 137, 90 GPa E 2 = E 3 = 14, 48 GPa

Conclusions

In this study, free-edge laminates subjected to uniaxial extension and uniform temperature variation are investigated using multi-particle finite element. This two-dimensional finite element approach with very coarse mesh, provides very accurate three-dimensional free-edge stress field. Moreover the present model allows a flexible choice of sublayer scheme in order to investigate through thickness local variation of stress fields. Detail investigation of the solution's sensibility again the number of sublayer division is also given. Regardless of the used sublayer's number, the results stay practically identical. Increasing the subdivisions number is useful to reproduce shape distribution and does not affect the precision. This study confirms no existence of stress singularity at the interface.

z x ∂Ω U ∂Ω T h k e k layer n layer k layer 1 z Γ k,k+1 Γ k-1,k y h + k h - k Fig. 1.

laminates 2 . 1

 21 Laminated plate geometry, preliminariesConsider a composite laminated plate (Fig.1) consisting of n linearly elastic orthotropic layers, each of them exhibiting individual mechanical properties.

  the bottom and the top z coordinates of the layer, respectively. The occupied volume is then Ω = ω × h - 1 , h + n . The interface between layer j and j + 1 is denoted Γ j,j+1 (1 ≤ j ≤ n -1). For the sake of convenience, throughout this paper the indicial notations are used. Greek indices α, β, γ and δ indicate the components on the (x, y) plane and and run over the range 1,2. Latin indices o, p, q and r indicate the components in the (x, y, z) space and run over the range 1,2,3. Superscript k attached on any quantity identifies affiliation to the k layer. Bold face characters are used to represent tensors, matrices and vectors.

  j ν ν j-1,j + 13 35 e j S j ν + e j+1 S j+1 ν + 9 70 e j+1 S j+1 ν ν j+1,j+2

4. 1

 1 Bending of [0 o /90 o /0 o ] laminate under sinusoidal transverse loading Consider a rectangular simply supported cross-ply laminate of dimension a = b an thickness h with equal thickness for each layer. subjected to a double sinusoidal transverse load q = q o sin( πx a )sin( πy b

Fig. 6

 6 Fig. 6 shows the σ yz distribution at the 0 o /90 o interface of a [0 o /90 o ] s laminate under uniform axial extension. Each physical ply is modelled by one numerical layer. Results of 3D FE by Wang & Crossman[START_REF] Wang | Some new results on the edge effect in symmetric composite laminates[END_REF] and by Carreira[START_REF] Carreira | Model of multilayered materials for interface stresses estimation and validation by finite element calculations[END_REF], and

Fig. 9

 9 Fig. 9 shows the σ z distribution at the 0 o /90 o interfaces of [0 o /90 o ] s and [90 o /0 o ] s laminates. It is noted here that the same quality of convergence has been found and an uniform sublayer division N = 5 is used to reproduce the stress shape. The results of the present model and those of Wang & Crossman [4] are in excellent agreement. The through-the-thickness distribution of σ z at the free edge in a [0 o /90 o ] s laminate is shown in Fig. 10 in comparison with

G 12 =

 12 G 13 = G 23 = 5, 86 GPa ν 12 = ν 13 = ν 23 = 0, 21 α 1 = 0.36 × 10 -6 o C -1 , α 2 = α 3 = 28.8 × 10 -6 o C -1 Cross-ply laminates: Figs. 12 and 13 show the distribution of interlaminar stresses at the 0 o /90 o interface of laminate[0 o /90 o ] s laminate under uniform temperature variation ∆T = 1 o C. The 5 × 5 mesh as in the later example that provides a very good quality of convergence has been used. A very good agreement between all the models considered has been obtained. All of these calculations predict a compression at free edge interface for a 1 o C temperature variation (see Fig. 12). Through-the-thickness distribution of σ z at the free edge of [90 o /0 o ] s laminate for a 1 o C temperature variation is plotted in Fig 14 using a sublayer division of Cho & Kim [27]. At the 0 o /0 o interface the present model provides a larger stress value. Angle-ply laminate: In this example, a rectangular [-45 o /45 o ] s of dimension b = 8e and a = 10b is investigated. The symmetry of the problem is no longer verified in this case. Hence the whole laminate is modelled by the present FE model. A typical 20 × 10 mesh is presented in Fig 15.

Figs. 16

 16 Figs.[START_REF] Tahani | Three-dimensional interlaminar stress analysis at free edges of general cross-ply composite laminates[END_REF] and 17 show interlaminar stresses distributions at different interfaces.

Fig. 3 .Fig. 4 .

 34 Fig. 3. Distribution of shear stress σ yz through the thickness of a simply supported rectangular laminate under sinusoidal transverse loading.

Fig. 5 .Fig. 8 .Fig. 9 .Fig. 10 .Fig. 11 .Fig. 12 .Fig. 13 .Fig. 14 .

 5891011121314 Fig. 5. Laminate subjected to uniform axial extension -typical mesh pattern.

Fig. 15 .Fig. 16 .

 1516 Fig. 15. typical mesh for the angle-ply laminate

  Stacking sequence of a laminated plate.

	0.4 0.4
					Elasticity
					6 sublayers
					1 sublayer
	0.2 0.2
				Elasticity
	z/h z/h	0 0		6 sublayers 1 sublayer
	-0.2 -0.2
	-0.4 -0.4
		0 0	0.05 0.05	0.1 0.1	0.15 0.15 0.2	0.25 0.2	0.3 0.25
				σ xz (0, a/2, z) σ yz (a/2, 0, z)
	Fig. 2. Distribution of shear stress σ xz through the thickness of a simply supported
	rectangular laminate under sinusoidal transverse loading.

sublayers

Carreira [START_REF] Carreira | Model of multilayered materials for interface stresses estimation and validation by finite element calculations[END_REF] Wang & Crossman [START_REF] Wang | Some new results on the edge effect in symmetric composite laminates[END_REF] Pagano [START_REF] Pagano | Stress fields in composite laminates[END_REF]


M R'mili

M Moevus

N Godin

Statistical fracture of E-glass fibres using a bundle tensile test and acoustic emission monitoring

Keywords: A. E-glass fibres, A. Bundle tensile test, B. Sub-critical crack growth, B. moisture effect, C. Statistical fracture, C. Weibull distribution, D. Acoustic emission

Mechanical strength studies have been carried out on fibre bundles used in composite manufacturing. The variability in mechanical properties of glass fibres has been studied using bundles of about 2000 filaments. The fibre strength distributions were analysed using the survival probability-applied strain ) ( ε -S curve, in relation with various experimental conditions. We also examine the effect of lubricant's viscosity on the fracture behaviour of Eglass fibre bundles. Acoustic Emission (AE) was monitored during the bundle tensile tests in order to verify that individual filament failures are statistically independent. On tensile tests with lubricated bundles of E-glass fibres, it is shown that each individual fibre break can be detected using AE. Hence, AE monitoring of a lubricated bundle of E-glass fibres provides a convenient and relatively quick method to obtain the Weibull parameters of strength distribution.

Introduction

The strength of unidirectional composites is essentially dependent on the mechanical behaviour of their reinforcement. The evaluation of this mechanical property requires knowledge of the mechanical characteristics of the fibres. It has been shown that the variability of the tensile properties of fibres can be obtained by single fibre testing [START_REF] Wilson | Statistical tensile strength of Nextel TM 610 and Nextel TM 720 fibres[END_REF][START_REF] Lissart | Statistical analysis of failure of SiC fibers in the presence of bimodal flaw populations[END_REF][START_REF] Lavaste | Statistical analysis of strength distribution of alumina based single fibres accounting for fibre diameter variation[END_REF] or by Bundle Tensile Test (BTT) technique. The fibre bundle models with equal load sharing rule were originally introduced by Daniels [START_REF] Daniels | The statistical theory of the strength of bundles of threads[END_REF] and Coleman [START_REF] Coleman | On the strength of classical fibers bundle[END_REF]. The fibre bundle models are based on parallel set of fibres each of which has Weibull distribution function [START_REF] Weibull | A statistical distribution function of wide applicability[END_REF]. Phoenix [START_REF] Phoenix | The asymptotic strength distribution of a general fiber bundle[END_REF][START_REF] Phoenix | Probabilistic strength analysis of fibre bundle structures[END_REF][START_REF] Phoenix | Probabilistic inter fiber dependence and the asymptotic strength distribution of classic fiber bundles[END_REF] gave a theoretical review of probabilistic models for fibre bundles. In its approach the classical equal load sharing model appears as a special case of generalised models where the load on each surviving fibre is not necessary the same. These generalised models extended to case of elastic fibre bundles with random fibre slack, and also to the case of hybrid bundles with several types of fibres.

Using the most common single fibre test technique, it is possible to characterise the strength distribution of the fibres if a large number of individual fibre (n ≥ 40) are tested, and thus to estimate the parameters of the associated Weibull strength distribution. However, some uncertainties appear on the determination of the parameters of the strength distribution by this method. The specimen selection and damage of the fibres during the sampling operation are some problems introducing errors on the determination of Weibull parameters. The bundle tensile test technique allows us to overcome some of these problems. Hence it becomes an alternative method to obtain the strength distribution of the fibres [START_REF] Manders | Variability of carbon and glass fibres and the strength of aligned composites[END_REF][START_REF] Chi | Determination of single fiber strength distribution from fiber bundle testing[END_REF][START_REF] Evans | Statistical change during the corrosion of glass fibre bundle[END_REF][START_REF] Cowking | Testing E-glass fibre bundle using acoustic emission[END_REF][START_REF] Okoroafor | Relating acoustic emission signal parameters to the strength of fibres used in manufacture of polymeric composites[END_REF][START_REF] Hill | Weibull statistics of fibre bundle failure using mechanical and acoustical emission testing: the influence of interfibre friction[END_REF][START_REF] Okoroafor | Investigation of complex failure modes in fibre bundles during dynamic mechanical testing using acoustic emission and Weibull statistics[END_REF][START_REF] R'mili | Estimation of Weibull parameters from loose bundle tests[END_REF][START_REF] R'mili | Caractérisation des fibres par amélioration de l'essai sur mèche avec mesure directe de la déformation[END_REF][START_REF] Lissart | Evaluation des propriétés des monofilaments à partir d' essais de traction sur mèches[END_REF] and has been used with successfully to study the coating effect on strength distribution changes of carbon fibres [START_REF] Helmer | Coating of carbon fibers-The strength of the fibers[END_REF][START_REF] R'mili | The effect of thermal exposure on the bundle strength distribution of B4C coated carbon fibers[END_REF]. The only difficulty is to obtain a reliable load-strain ) ( ε -P curve of the bundle under quasi-static loading. The determination of the bundle elongation during the test is also a problem because it is generally deduced from the displacement of the crosshead of the testing machine.

It must be emphasized, that in BTT the major difficulties are the specimen preparation and the strain measurement during the tensile test. Variation in fibre cross-sectional area may not be important on a bundle test, but in matters in composite with matrix that affects the local load sharing and the transfer of the load intensity between surviving fibres. In fact, the magnitude of transfer load depends on the fibre radius. Whereas, the analysis on the bundle test, conducted with a reliable measurement of the true deformation, allows to overcoming the problems linked to the variation of the cross-sectional area of the fibres like the scattering of ultimate stresses which is linked to the accurate measurement of the cross-sectional area of the fibres. The large number of broken fibres and the measurement of the true deformation during the bundle test give the accurate information about the fracture behaviour of fibres.

The failure of bundles of E-glass fibre has been monitored using load-strain measurements and acoustic emission (AE). AE technique is a non-destructive method at damage evaluation [START_REF] Beattie | Acoustic emission, principles and instrumentation[END_REF][START_REF] Eitzen | Acoustic emission: establishing the fundamentals[END_REF][START_REF] Phoenix | Statistics for the time-Dependent Failure of Kevlar-49/Epoxy Composites: Micromechanical Modeling And Data Interpretation[END_REF], that uses stress waves emitted by stressed material undergoing deformation processes as plastic deformation or crack growth. As an elastic wave or AE signal travels through a material, its characteristics are constantly changing due to damping, boundary surface interactions, etc. Hence a wave form recorded by an AE device is a transformed representation of the wave created by the AE source. In conventional acoustic emission testing, the elastic wave produced by a source is converted to a voltage signal by a resonant piezo-electric sensor.

The AE technique has been used in this study to detect fibre breaks on a one -to -one basis and to determine the variability of the fibre breaks in relation with the ) ( ε -P analysis. Also Phoenix et al. [START_REF] Phoenix | Statistics for the time-Dependent Failure of Kevlar-49/Epoxy Composites: Micromechanical Modeling And Data Interpretation[END_REF] and Hamstad et al. [START_REF] Hamstad | Acoustic emission from single and multiple Kevlar 49 filament breaks[END_REF] have used silicone-oil lubricant to reduce the friction between the fibres on Kevlar bundles. They found that all the breaks could be detected and counted individually. Same results were reported in the literature, but in these cases it was be difficult to use BTT in order to obtain the fracture stress distributions of fibres. Only the first part of the load-strain curve below maximum was considered in order to obtain fibres properties. Some works [START_REF] Calard | Failure of fibre bundles[END_REF][START_REF] Creasy | A method of extracting Weibull survival model parameters from filament bundle load/strain data[END_REF][START_REF] Hidalgo | Bursts in a fiber bundle model with continuous damage[END_REF][START_REF] Kun | Damage in fiber bundle models[END_REF] investigates also the discrepancy between experimental data and theory on the tensile behaviour and strength of fibres bundle.

In the present study, experiments were conducted on lubricated fibres bundles. Due to the lubrication, multiple fractures at fibres are expected to be reduced by lowering the interactions between the individual filaments. In addition, the lubricant is used as coupling to strain data. Moreover, the effects of the test conditions on the BTT are investigated: effect of the lubricants viscosity and effect at moisture conditions on the fracture behaviour of the Eglass fibre. For the last point, Brown [START_REF] Brown | Effect of surface treatement on the hydrolytic stability of E-glass fiber bundle tensile strength[END_REF] showed that the strength, the number of fibre breaks at the peak load and Weibull parameters are strongly dependent on the humidity level of the Eglass fibre. Phoenix et al. [START_REF] Phoenix | Statistics for the time-Dependent Failure of Kevlar-49/Epoxy Composites: Micromechanical Modeling And Data Interpretation[END_REF] showed that the Kevlar fibres are more sensitive to the UV light exposure.

Theoretical background

1. Bundle model

The theoretical model of dry bundle fibre consists in a set of N o parallel fibres with statistically distributed strength. The sample is loaded parallel to fibres direction and the fibres fail if the load on them exceeds their threshold value. In stress controlled experiments, after each fibre failure the load carried by the broken fibre failure is redistributed among the unbroken ones following an equal load-sharing rule. This means that after each fibre break the stress is equally distributed on the surviving fibres assuming that no interaction acts between the broken fibres and their neighbours. The high variability in strength found in brittle fibres is well modelled by Weibull function. This variability is due to randomly distributed flaws on the fibres. The assumptions generally used in this analysis, so-called Coleman's conditions are: 1) filament length is constant within the bundle, 2) the stress-strain relationship for a single fibre follows the Hooke' s law up to failure, 3) load released with the fracture of one fibre is uniformly distributed among the surviving fibres, 4) absence of phenomena which will be able to a premature failure of fibres.

This last point is very important. Then, up to the maximum of load, the dry bundle model and the experimental data are in good agreement. On the contrary, just after the peak load, the experimental load-strain curve exhibit catastrophic failure with a large magnitude of load drop that was not predicted by the theoretical model. This difference is certainly due to fibres interactions as friction between the fibres, stress concentration near the ends of the bundle.

Therefore specific cares are taken to reduce these effects and to obtain experimental curves in better accordance with the model.

Statistical distribution of fibre strength: Single Weibull distribution

Using BTT, the properties of fibres are generally described in terms of the strain ε rather than in terms of the stress σ . These two descriptions are equivalent, as for ceramic fibres, the applied stress σ and strain ε follow Hooke's law up to failure:

σ ε = E f (1)
where E f is the fibre Young's modulus.

The bundle test is then also likely to give the two Weibull parameters of perfect elastic fibres using a ) ( ε -P curve. The bundle theory has been used by Chi and al. [START_REF] Phoenix | Probabilistic strength analysis of fibre bundle structures[END_REF] to assimilate the strengths of loose bundle of carbon fibre with those of single carbon fibres. Chi and al. showed that the behaviour of bundles can be described from the simple Weibull strength distribution of fibres and from their geometrical characteristics. Indeed, for a large number of fibres, the number of survival fibres at an applied strain, ε is given by:

( )                 - = m l o N N ε ε ε exp (2)
where l ε is a scale parameter for a given gauge length l and m is the Weibull modulus, and N o is the number of initially loaded fibres. The survival probability of fibres, ( ) S ε , corresponding to the number N( ) ε of fibres that survive at a given strain ε divided by the initial number of loaded fibres N o , can be expressed as:

S N N o l m ( ) ( ) exp ε ε ε ε = = -               (3)
A commonly used approach relies on least-squares fit of a linear form of Eq. ( 3).

[ ]

l Ln mLn LnS Ln ε ε ε - = - ) ( (4)
which is a linear equation with slope m . It is easy to see from Eq. ( 3) that l ε is the strain corresponding to a survival probability of 1/e or 0.37.

Assuming that the applied load is uniformly distributed among the surviving fibres and that fibres have a linear load-strain relationship up to breakage, the load at an applied strain ε during a bundle tensile test is given by:

) ( ) ( ) ( ε ε ε ε ε σ S AE N AN E AN P f o f = = = ( 5
)
where A is the cross-sectional area of each of the fibres, E f their Young's modulus. The most common technique for estimating the Weibull parameters is to plot the observed data in the linear form of Eq. (4).

Experimental procedure

Specimen preparation and investigated material

Fibre bundle specimens were prepared according to the protocol described in a previous paper [START_REF] R'mili | Caractérisation des fibres par amélioration de l'essai sur mèche avec mesure directe de la déformation[END_REF]. In the present study, tensile tests on bundles of 30, 60 and 100 mm gauge length have been performed on E-glass fibres manufactured by Vetrotex firm. The mechanical behaviour of bundles has been studied in a tension test on bundle of about 204x10 filaments.

The mean fibre diameter is 15 ± 2 µ m and the elastic modulus is 71 ± 2 GPa.

Tensile test

All the tensile bundle tests have been carried out in room atmosphere (23 ± 2°C and 30 ± 20% of relative humidity). These tests were performed under with a pneumatic testing machine with a 250 N load cell. Two small thermo-retractable stumps of 3 mm long are threaded on the bundle to define the gauge length and to allow clamping of the mechanical extensometer with specific clamps [START_REF] R'mili | Caractérisation des fibres par amélioration de l'essai sur mèche avec mesure directe de la déformation[END_REF]. To put and optimise the position of the extensometer, the sample is previously loaded under 15% of the ultimate load. Fig. 1 shows a view of bundle testing tab, clamping system of the extensometer and Acoustic Emission sensors. Nominal strain values were measured directly with a mechanical extensometer. The tensile tests have been performed

with a crosshead speed of 2 m µ /s. To point out the effect of the humidity, several tests have been conducted after water immersion. In these cases the bundles were lubricated just prior to the loading. In order to examine the effect of lubricant oils on the fracture behaviour change, several lubricants with various viscosities were used.

Acoustic emission system

A two channel Mistras 2001 data acquisition system of Euro Physical Acoustics (EPA)

was used for the recording of AE data. Two resonant PZT transducers (Acoustic Emission type µ 80) were acoustically connected to the end of the samples with silicon grease. Table 1 summarizes the typical setting of the AE system. The amplitude distribution covers the range used. This threshold allows to minimize electronic background noise due to the grips, the tensile testing machine and others electrical machines. Preliminary measurements have been performed to optimise the acquisition parameters to the case of the fibres studied (table 1).

Simulated AE events with pencil lead breaks were used to verify that AE signals were well transmitted across the metallic-sample interface, and to evaluate the wave speed. Each simulated signals recorded was analysed by the Mistras system and six parameters are calculated from the waveforms: rise time, counts, energy, duration, amplitude, counts to peak.

These were used for the conventional analysis of AE data. Furthermore each waveform is digitised and stored. The signals detected are also subjected to a calculation procedure to determine the location of AE signal source. During the AE data analysis, only the events located between the two sensors are considered.

Methods of analysis

P -ε curve analysis

The experimental procedure to estimate the survival probability from the ) ( ε -P curve was described in detail by Chi and al. [START_REF] Chi | Determination of single fiber strength distribution from fiber bundle testing[END_REF] for a single Weibull distribution. Eq. ( 5) can be expressed as:

) ( ) ( ε ε ε ε S S AE N P o f o ℜ = = (6)
and

o f o N AE d dP = = ℜ =0 ε ε (7) where o ℜ is the initial slope of ) ( ε - P curve.
It is observed from Eq. ( 6) that the survival probability of the fibres is given by:

S o ( ) ( ) ε ε = ℜ ℜ (8)
where ℜ = ( ) / ε ε P is the slope of the straight line connecting the origin and the current point of ) ( ε -P curve. Thus the survival probability ( ) S ε , at any strain level ε , can be determined experimentally from the ) ( ε -P curve using Eq. ( 8). This method allows us to overcome the characteristics of the bundle like: Young's modulus (E f ), the initial number (N 0 ) and individual cross-section area of fibres (A). A schematic describing determination of ℜ o and ℜ( )

ε from experimental ) ( ε - P
curve is given Fig. 2.

S -ε curve analysis

The ) ( ε -P curves are dependent on bundle characteristics, essentially of initial number of fibres and cross section. The survival probability function is independent of fibres Young's modulus, of the initial number and of the individual cross-section area of fibres (Eq.

3). Therefore, the ) ( ε -S can be regarded as a master curve and can be used as an indicator of the evolution of mechanical properties of the fibres in a given experimental conditions. Thus in this study, we chose to use ) (

ε -

S

diagrams to analyse our experimental data. In the present work, for estimating Weibull parameters, a non-linear least squares method is applied using a non-linear curve fitting Eq. (3). We used the chi-square test ( ) χ 2 to evaluate the accuracy of curve fitting. The results are compared with those obtained by linear fit method Eq. (4).

Results and discussion

Validation of experimental measurements 4.1.1. Influence of the lubrication

In order to examine the effect of the lubricant oils viscosity on the fracture change of Eglass bundles, several lubricants were used. Viscosity versus shear rate for various lubricant oils is shown in Fig. 3. Typical load-strain curves obtained with a dry bundle (1), petrol oil (2), vaseline oil (3) and light-oil lubricant (4) are given in Fig. 4. This figure shows that the fracture of fibre is linked to the lubricant viscosities. The maximum of load and the corresponding strain are significantly reduced in the dry bundle. On the other hand, a significant increase in the stress and the corresponding strain is obtained with light-oil lubricant. It is clear that for the highest viscosity (vaseline oil) a decrease of the mechanical properties is observed. As the lubricant viscosity increases, the lateral interaction of the fibre in a bundle enhances local load sharing and thus contributes to the premature fracture of the neighbouring fibres. Hence, it can observe that dry bundles (1) lead to the premature and unstable fracture in comparison with the lubricated bundles. This phenomenon is due to damage induced by inter fibres friction [START_REF] Hill | Weibull statistics of fibre bundle failure using mechanical and acoustical emission testing: the influence of interfibre friction[END_REF], and by a dynamic failure during the breaking process [START_REF] Chi | Determination of single fiber strength distribution from fiber bundle testing[END_REF]. When a fibre fails, it retracts that induces a relative displacement with respect to the neighbours. Through shearing the viscous medium that will induce slight tensile overloads on the nearest neighbours. Fig. 5 shows the appearance difference of bundles between a dry and lubricated one. After the test, the dry bundle has the appearance of a cotton-wool, as observed by some authors [START_REF] Evans | Statistical change during the corrosion of glass fibre bundle[END_REF][START_REF] Hill | Weibull statistics of fibre bundle failure using mechanical and acoustical emission testing: the influence of interfibre friction[END_REF]. For the lightoil lubricant, the fracture is stable indicating that the lubricant reduces the friction damage and the premature fracture by dynamic effect. Indeed, during the fracture process of the bundle with lubricant oil, the fibre remains certainly in their initial position without interaction with neighbouring fibres. It is clear that the efficiency of the lubrication depend on the inter-fibres spacing, the fibres diameters distribution and the wet-ability of the lubricant. For the next, all the tests are performed with the light-oil lubricant.

Loading -unloading test

The necessary condition required to obtain ) (ε S with a good accuracy, is that during the tensile test the ) ( ε -P curve of the bundle exhibit no residual strain when the bundle is totally unloaded before the total fracture of the bundle. It can be verified with particular tensile tests where an unloading-reloading cycle to as carried out after the peak load. Examples of these curves obtained with bundle of glass and carbon fibres are shown in Figs. 6a and6b respectively. These figures show that the ) ( ε -P curve during the loading-unloading cycle is relatively linear, the residual displacement tends to zero and the area at the hysteresis loop is very small. This is typical of elastic damage behaviour as described in Fig. 2. For glass fibres, an important load loss (indicated by the superposed circle) is observed during the unloadingreloading sequence. This phenomenon can be attributed to sub-critical crack growth process. This is confirmed by carbon fibres that are not sensitive to stress assisted corrosion in room atmosphere. This will be discussed later.

The elastic damage behaviour of the bundle is a result of individual fracture of fibres. So the failure of each fibre is followed by a redistribution of load among all the surviving fibres which do not induce further fibre breaks in the vicinity. These remarks imply that the compliance changes are only the results of fibres fractures and the friction between the fibres can be neglected. Thus the survival probability ) (ε S can be experimentally measured with a good accuracy.

Analysis of the acoustic emission data

Typical mechanical and AE results are presented in Fig. 7. This figure shows the tensile load-strain curve for two gauge lengths (30 and 100 mm) and linear location of AE events between the two sensors. The solid lines indicate the approximate locations of the bundle ends.

Only the events localised in the length have been taken into account during the AE data analysis. The ) ( ε -P curve presents a linear phase, which is followed by a non-linear behaviour. Moreover, the failure of bundle is stable beyond peak load. Whatever the length, the fibre break are distributed over the entire gauge length during the bundle failure. It can noticed that for the short gauge length some breaks are located, only at the beginning of the test, near the bundles ends due to the end effects. For larger gauge lengths, this fracture mechanism is reduced. The ″end-effect″ model was already discussed in details by Stoner and al. [START_REF] Stoner | An end-effect model fort he single filament tensile test[END_REF].

The friction between fibres is a source of additional AE that increases with the gauge length.

This phenomenon is well highlighted by the histograms of amplitude distributions (see Fig. 8.).

One can note the presence of two distributions. A principal distribution with amplitudes ranged from approximately 80 to 95 dB, can be attributed to the fibre breaks. The amplitudes of the signals recorded in this distribution during the tensile test increase with the applied strain. This is due to the fact that weak fibres fail under low strain and the fibres with high strength at the end of the test. A second distribution, located towards the low amplitudes, is observed principally on the bundle with large gauge lengths. This second distribution of AE signal is low and can be allotted to the friction between fibres. For the next, this distribution is not taken into account for the statistical analysis of fibre breaks.

Fig. 9 shows the evolution of the cumulative count of AE events and the load with the applied strain for a bundle of 30 and 100 mm gauge length. The number of signals recorded on the bundle with gauge length of 100 mm is around 2000. Hence all the fibre breaks have been detected. On the other hand for 30 mm gauge length, less than 2000 signals are recorded due to the end effect. To confirm that ″end effect″ mechanism was predominant for the short gauge length, the ) ( ε -S curves obtained from ) ( ε -P and AE analyses have been calculated. Fig. 10 shows that for 30 mm gauge length (Fig. 10a) a significant difference is observed between the survival probability obtained by using the ) ( ε -P and AE analysis. On the contrary for 100 mm gauge length (Fig. 10b) the two methods give similar results. Therefore, for the short gauge length one part of fibre breaks is due to the ″end effect″ discussed below. In a general way, this is a good accordance between the number of acoustics events and the number of fibres fractures determined by the analysis of the ) ( ε -P curve. Agreement of 5 % was obtained (Table 2). This shows that the two techniques give similar results for fibre breaks

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especially for the large gauge lengths. The cumulative count of AE events and amplitude distributions show that each fibre breaks individually. The part of multiple fibre breaks is also reduced and single fibre breaks are dominant during the mechanical test. This result is in good agreement with the fact that catastrophic collapse of the bundle did not occur. Because fibre breaks occur one-by-one and are randomly distributed along the gauge length, fibre breaks are statistically independent. On the location graph, no cluster is observed. The signals assigned to the fibre breaks have higher amplitude, quick rising and decaying. This AE signature can be used to train supervised classifier in composite materials [START_REF] Godin | Integration of the Kohonen' s self-organising map and kmeans algorithm for the segmentation of the AE data collected during tensile tests on cross-ply composites[END_REF][START_REF] Godin | Clustering of acoustic emission signals collected during tensile tests on unidirectionnal glass/polyester composites using supervised and unsupervised classifiers[END_REF][START_REF] Huguet | Use of acoustic emission to identify damage modes in glass fibre reinforced polyester[END_REF]. Nevertheless, in that case the AE signature of the fibre breaks in a matrix will be certainly different and new type of flaws and fracture mechanisms appears for composite materials.

Effect of relative humidity level

The ) ( ε -S curves (Fig. 11) exhibit a significant change of the bundle fracture behaviour with the relative humidity (RH). Failure stresses are measured at two humidity levels 30 % RH and after immersion in water. The results show an important decrease of the strength between the bundles loaded in room atmosphere (30% RH) and for bundles tested under water.

The strength estimate at 50% of survival probability was found equal to 1.44 GPa for dry bundle (30% RH) and only 0.92 GPa for fibres tested under water. This result points out that the moisture has a significant effect on fracture of the bundles and affects strongly the fracture behaviour and the mean strength of glass fibres. This result is attributed to the effects of stress corrosion in humid environment where flaws growth due to stress-corrosion mechanisms. It is well known that the strength of glass-fibres is sensitive to the relative humidity [START_REF] Gougeon | Evolution of strength silica optical fibers under various moisture condition[END_REF][START_REF] Brown | Effect of surface treatment on the hydrolytic stability of E-glass fiber bundle tensile strength[END_REF] because in these fibres, pre-existing flaws subjected to stress and in presence of water vapour grow subcritically up to critical size leading to fibre break. This is due to the weakening of SiOSi bonds by water molecules, and to the stress concentration around the crack tip can leading then to the fracture of these bonds.

Statistical analysis

Fig. 12 shows for different gauge lengths (30, 60 and 100 mm) the strength distribution derived from the acoustic emission analysis. In the Weibull diagram, Fig. 12 exhibits approximately a linear and small effect of gauge length changes. Usually in the Weibull plots the analysis derived from the mechanical data exhibits a pseudo bi-modal behaviour. In the present case, the apparition of non-linearity is not due to a second population of defects. But is due to the fibre breaks by the ″end effects″ and that can not be removed from the experimental data obtained with mechanical analysis. For the data obtained by acoustic emission analysis, the absence of non-linearity can be noticed. The use of the location procedure allows to separate ″end effects″ from the true fibre strength distribution. Therefore for the large gauge lengths, the fibre breaks are essentially controlled by one type of defect, certainly surface defects. The Weibull parameters were determined by the linear fit method Eq. ( 4) and the non linear fit method Eq. (3). The parameter values with various gauge length are summarized in table 2. Also the Weibull parameters extracted by using ) ( ε -P analysis are reported in brackets (). As part of 30 mm gauge length, the results from the two methods are approximately similar. In the Weibull plot no change in observed between 60 and 100 mm gauge length. Consequently, for the large lengths the fracture behaviour of E-glass fibre can be described well by a two-parameter Weibull function.

Conclusion

Bundle tensile tests on E-glass fibre have been performed to failure with AE monitoring. The identification of fibres breaks by ″end-effect″ process, lubricant oil viscosity, and controlled breaks progression after the peak load, in addition to direct measurement of true strain by extensometer are key points to obtain a reliable determination of the statistical parameters of the fibre breaks. The AE technique gives additional information on the fibre breaks process. The linear location and the fibre breaks detection allow us to overcome the problems linked with the load-strain analysis like ″end-effect″. Therefore the statistical Weibull parameters can be extracted from the survival probability function-applied strain with a good accuracy.

We have demonstrated that under adequate oil lubrication condition, premature failure of fibres is significantly reduced. It is also shown that glass fibres are strongly sensitive to relative humidity level. Finally, up to 30 mm gauge length, the fracture behaviour of E-glass fibres appears to be principally controlled by one type of defect, attributed to surface defects.

(

(3)

Fig. 1 .Fig. 2 .

 12 Fig.1. View of the experimental device to test E-glass fibre bundle with AE monitoring.

Fig. 5 .

 5 Fig.5. Effect of lubricant on bundle appearance after failure: dry bundle (a) and lubricated with petrol-oil (b).

Fig. 6 .Fig. 8 .Fig. 10 .

 6810 Fig.6. Typical load-strain curves with loading-unloading sequence for E-glass fibre (a) and carbon fibre (b), points out moisture effect on E-glass fibres.

Fig. 11 .

 11 Fig.11. Moisture effect on fracture behaviour of bundles: Experimental load-strain curves for bundles wetted with water (1) and for bundle wetted with petrol oil (2) obtained for a 60 mm gauge length.

Fig. 12 .

 12 Fig.12. Experimental survival probability in Weibull diagram for 30 (1), 60 (2) and 60 mm (3) gauge lengths.

Table 1 :

 1 Typical setting of AE acquisition system

		Sampling rate			8 MHz
	Preamplifier gain			40 dB
	Threshold of detection			32 dB
	Type of sensor			Micro-80 PAC
		Coupling			Silicon grease
	Peak definition time			50 µs
		(PDT)
	Hit definition time			100 µs
		(HDT)
	Hit lockout time			300 µs
		(HLT)
		Wave velocity			3500 m/s
			Linear fit	Non-linear fit
			Eq. (4)		Eq. (3)
	(mm)	N	m	ε L (%)	m	ε L (%)
	30	1878	3.0	2.76	3.3	2.77
		(2086)	(2.5)	(2.63)	(2.9)	(2.61)
	60	1949	3.6	2.14	4.0	2.15
		(1947)	(4.7)	(2.15)	(4.6)	(2.17)
	100	2051	2.8	2.07	3.1	2.10
		(2085) () Weibull parameters obtained by the ( P -ε ) analysis. (3.3) (2.06) (3.4)	(2.10)

Table 2 :

 2 Weibull parameters obtained by linear method and the non-linear fit. Comparison between the AE and the ( P -ε ) analysis.

( of AE events = 100 mm Fig. 9. AE cumulative counts and load variations with applied strain for bundle with 30 and 100 mm gauge length.