G Lubineau

D Violeau

P Ladevèze

ACCEPTED MANUSCRIPT

Keywords: microcracking, fatigue, damage, micromechanics

Degradations initiated near the edges of a laminate can have a significant effect on its state of degradation, even at the core. Indeed, results from the literature show that laminates which have the same stress state at the core can have completely different states of degradation, even far away from the edges. The paper discusses the influence of the edge effect on damage initiation and propagation for a specific example. A computational micromechanical approach to the degradation of laminated composites was developed recently at LMT-Cachan. This is a hybrid approach in which, depending on the scale, the mechanisms are described using continuous damage mechanics or finite fracture mechanics. Initially developed for static loading, this technique is being extended to fatigue and environmental effects.

The aim of this paper is to illustrate the capability of such an approach to take into account major observations during cyclic loading in an oxidizing atmosphere, even when edge effects are significant.

Introduction

Today, applications of laminated composites are limited mainly to parts of secondary importance subjected to relatively moderate, almost exclusively mechanical loading.

In the future, these materials will be increasingly used for vital structures subjected to high loads (not only mechanical, but also chemical or physical). Among these new applications are structural parts of the future civilian European supersonic airplane, which was studied as part of the "Supersonic Aeronautical Program" financed by the French Ministry of Equipment and Transportation. In this particular application, the material will be subjected to intensive thermal cycles (four-hour takeoff/landing cycles). In addition to classical fatigue damage, degradation due to the progressive oxidation of the material will also occur because of the relatively high temperature (approximately 130 o C at Mach 2.0) and the presence of oxygen. In fact, the degradation process is a much more complex combination of static degradation, fatigue damage and oxidation-induced degradation, resulting in the rapid development of elaborate patterns of degradation, especially microcracking ( [START_REF] Mcmanus | Prediction of thermal cycling induced matrix cracking[END_REF][START_REF] Kobayashi | Damage-mechanics analysis of matrix cracking in cross-ply cfrp laminates under thermal fatigue[END_REF][START_REF] Lafarie-Frenot | Influence of oxidative environments on damage in c/epoxy laminates subjected to thermal fatigue[END_REF]).

The safe dimensioning of these applications requires degradation to be properly described in the vicinity of the critical zones constituted by geometric discontinuities (holes, ply seams, edges in the broad sense). These zones present Email address: lubineau@lmt.ens-cachan.fr (G. Lubineau, D. Violeau and P.

Ladevèze ).

difficulties for two reasons:

The first reason is that these zones endure high out-of-plane stresses, whereas away from the edges the structure is mainly in a plane stress state. Interlaminar degradations (mainly delamination) and intralaminar degradations (e.g. microcracking) coexist and are highly coupled near the edges. Taking into account such coupling is currently a scientific challenge for which solutions were proposed only recently.

The second reason is that these zones are where high loading gradients occur.

Consequently, while the interior solution varies relatively slowly, significant stress variations take place very locally near the edges, especially in the presence of high thermal stresses. Thus, it is well-known that degradations can be initiated at the edges long before they occur in the core. Moreover, degradations initiated near the edges develop towards the core in an unstable or stable way depending on several parameters, some inherent to the ply (particularly its thickness [START_REF] Boniface | Transverse ply cracking in crossply cfrp laminates. initiation or propagation controlled[END_REF]), others related to the geometry of the sample (e.g. the stacking sequence, which influences the local stress state at the edges directly). If these degradations turn out to be unstable, they propagate towards the core in contradiction with classical laminate theory which would predict no damage. Therefore, in a laboratory sample, edge effects can have a deep impact down to the core. For a structure of industrial complexity, this effect is undoubtedly much more limited, but it is still necessary to predict accurately whether these degradations penetrate or not. In addition to these purely mechanical considerations, the edges play another important role from the physico-chemical point of view because the penetration of oxygen within the existing cracks can be very different depending on the local stress state.

Here, our objective is to show that the hybrid micromechanical model developed recently ( [START_REF] Ladevze | Multiscale computational damage modelling of laminate composites[END_REF][START_REF] Ladevze | A computational damage micromodel of laminated composites[END_REF]) is capable of reproducing this significant edge effect and to show that one must take it into account in order to analyze experimental results properly.

The first part of the paper focuses on some recent results from the literature ( [START_REF] Lafarie-Frenot | Influence of oxidative environments on damage in c/epoxy laminates subjected to thermal fatigue[END_REF][START_REF] Lafarie-Frnot | Influence of free edge intralaminar stresses on damage process in cfrp laminates under thermal cycling conditions[END_REF]) in which the experimental observations can be explained only by the existence of edge effects. These observations involved comparisons of the states of degradation among samples with identical inner stress states, but with very different edge effects (e.g. identical constitutive plies, but different stacking sequences). The influence of the edges is illustrated in the case of thermomechanical cycling under mechanico-chemical coupling. These results are first discussed through traditional fracture mechanics in order to illustrate clearly the influence of the edge effects on observable degradation. Therefore, it is necessary to develop a model capable of taking edge effects and their influence on the stable or unstable propagation of degradations towards the core of the sample into account correctly. A hybrid micromechanical model was proposed recently at LMT-Cachan. Initially introduced for static loading ( [START_REF] Lafarie-Frenot | Influence of oxidative environments on damage in c/epoxy laminates subjected to thermal fatigue[END_REF][START_REF] Lafarie-Frnot | Influence of free edge intralaminar stresses on damage process in cfrp laminates under thermal cycling conditions[END_REF]), this model was recently extended to include fatigue damage and oxidation effects [START_REF] Lubineau | Durability of cfrp laminates under thermomechanical loading: A micro-meso damage model[END_REF]. A general outline of this model is given in the second part.

Finally, the third part discusses the capability of this model to reproduce the experiments considered. Only preliminary qualitative results are presented here because of the current lack of data for identification. Nevertheless, we show that our model is capable of capturing the key features of the experimental observations for this difficult problem. ysis Here, we discuss the experimental results of [START_REF] Lafarie-Frnot | Influence of free edge intralaminar stresses on damage process in cfrp laminates under thermal cycling conditions[END_REF], which are summarized in Fig- The only reasonable explanation for these major differences lies in the edge effect, which is the only difference between the two sequences under this type of loading. Therefore, it is necessary to carry out a more precise analysis of the coupling among the edge effect, the microcracking state and oxidation,

ACCEPTED MANUSCRIPT

even within the core.

Influence of the edge effect on the energy release rate

First, for both sequences, let us consider the evolution of the energy release rate of a transverse microcrack during its propagation from the edge towards the core. In both cases, the crack's growth was simulated by finite elements (see Figure 2) under the assumption of a periodic microcracking pattern in the direction of the edge. The reference loading was a global ∆T = -100C temperature variation. In addition, the problem was assumed to be linear elastic.

[Fig. 2

about here.]

The energy release rates corresponding to the growth of the microcrack from edge to core were calculated by virtual crack extension. The results are shown in Figure 3.

[Fig. 3

Influence of the edge effect on the crack's opening

The edge effect also modifies the opening of the potential oxidation path. The local opening of the cracks for the two sequences is shown in Figure 4.

[Fig. 4

about here.]

In the steady-state solutions (long cracks far from the edges), the opening tends to be the same for the two sequences. But the edge effect limits the penetration of oxygen in the [-45 3 / + 45 3 ] s compared to the [0 3 /90 3 ] s case. This is especially true for very short cracks (see Case A / Λ H = 0.2, Figure 4). Consequently, not only is crack initiation delayed from a mechanical point of view (Figure 3), but weakening of the material due to oxidation is also limited.

Bases of the micromechanical model for laminates

The computational damage micromodel for laminates was introduced in [START_REF] Ladevze | Multiscale computational damage modelling of laminate composites[END_REF].

Its material foundations were detailed in [START_REF] Ladevze | A computational damage micromodel of laminated composites[END_REF]: therefore, we will merely review its main concepts, and more particularly its first extensions to fatigue and oxidizing degradation. The objective of this theory is to propose a micromechanical model of the classical elementary mechanisms which could be used as a reference material under general loading. In that sense, it differs greatly from classical micromechanical approaches, which would describe only some

ACCEPTED MANUSCRIPT

specific mechanisms under some specific loading type.

Let us first recall that we assume that any complex state of degradation of a laminated composite can be described as the result of the development of four basic mechanisms: two diffuse mechanisms (one for the ply and one for the interface), and two discrete mechanisms on the ply's scale (transverse microcracking and induced local delamination at the interface). A more detailed description of these mechanisms can be found in [START_REF] Ladevze | A computational damage micromodel of laminated composites[END_REF].

The hybrid micromechanical model under static loading

The first key step consists in grouping the mechanisms according to their characteristic scales. On the scale of the elementary ply, diffuse intra-and interlaminar damage can be considered locally homogeneous, whereas transverse microcracks and diffuse delamination can be associated with clear fracture surfaces. This leads to the following choice: the degradation of the fiber-matrix material is described using the classical continuum mechanics framework, while transverse microcracking and local delamination are described using a discrete approach based on a generalization of finite fracture mechanics [START_REF] Hashin | Analysis of cracked laminates: a variational approach[END_REF].

Continuous damage mechanics for the progressive mechanisms

For the description of damage associated with fiber/matrix debonding and small cracks within the matrix connecting close fibers, the classical form of the ACCEPTED MANUSCRIPT damage mesomodel ( [START_REF] Ladevèze | Damage modeling of the elementary ply for laminated composites[END_REF], [START_REF] Allix | Interlaminar interface modelling for the prediction of laminate delamination[END_REF]) was retained. Let 1 denote the fiber's direction, 2 the in-plane transverse direction and 3 the out-of-plane transverse direction.

In the elementary ply, for instance, the damaged material is assumed to remain transverse isotropic and its state is classically described by two damage variables: one in the transverse direction (denoted d ), and one in the shear direction (denoted d). These variables are assumed to be constant throughout the thickness of the mesoconstituent. Following that assumption, the evolution of damage is expressed as a function of the damage forces Ỹ d and Ỹ d :

d = sup τ ≤t √ Ỹ d+b 2 Ỹ d - √ Y 0 √ Yc d = b 3 d (1)
where b 2 , b 3 , Y c and Y 0 are material coefficients identified experimentally. Since the damage variables are assumed to be constant throughout the thickness of the single layer, the constitutive relation involves mean values of the damage forces throughout the thickness, so that:

Ỹ d = σ 22 2 + 2E 0 2 (1-d ) 2 Ỹ d = σ 2 12 2G 0 12 (1-d) 2 (2)
where • and • + denote the mean value throughout the thickness and the positive part respectively, in order to distinguish the behavior in traction (where cracks are active) from the behavior in compression (where cracks are assumed to be closed).

Finite fracture mechanics for cracking and delamination

The surfaces of the cracks are described using a discrete approach. Thus, the complex cracking pattern is assumed to occur within the structure through the accumulation of complete breakage of elementary surfaces, whose areas are determined from micromechanical considerations [START_REF] Ladevze | A computational damage micromodel of laminated composites[END_REF]. Consequently, the area of each type of surface defines the minimum amount of degradation considered in our approach. Thus, the mechanisms of degradation are quantified, which guarantees that our approach behaves well in terms of localization.

Two types of surfaces must be considered The damage forces for intralaminar cracks are defined as follows:

• at initiation (failure of an isolated surface):

{G i } = min[h • {∆G} h ; h • {∆G} h ] (3)
• during propagation (failure of a surface adjacent to a previously cracked surface):

{G p } = {∆G} (4)
where ∆G is the discrete energy release rate associated with complete breakage of the surface. Thus, ∆G is a mean value of the classical continuous energy release rate:

∆G = a+∆a a G(a)δa ∆a ( 5
)
where ∆a is the prescribed minimum cracking surface. The discrete values of ∆G for the crack growth shown in Figures 2 and3 are plotted in Figure 5.

One can note that the edge effect is taken into account in an average sense.

[Fig. 5

about here.]

h is a critical thickness characteristic of the material (typically of the order of twice the thickness of the elementary material ply) and h is the thickness of the unidirectional layer.

Extension to fatigue damage and environmental effects

In our approach, critical energy release rates are introduced explicitly in the analysis of the structure. A relatively easy means of introducing fatigue damage and environmental effects consists in assuming that they affect these critical quantities alone. At the same time, transverse microcracking and local delamination are assumed to remain governed by the static degradation law defined in 3.1.2. Thus, discrete mechanisms develop naturally within the structure due to the local reduction of tenacities and the verification of the static degradation law. This approach was described in [START_REF] Lubineau | Durability of cfrp laminates under thermomechanical loading: A micro-meso damage model[END_REF], and is briefly reviewed below.

Two internal variables are introduced: η f which represents the fatigue damage effect, and η ox which, in our case, represents the oxidation effect. Thus, the current critical energy release rate of the material is:

{G c } = { Ḡc } • (1 -η f ) • (1 -η ox ) (6)
An intrinsic constitutive relation for the calculation of η f is defined as a more or less complex function F of the local damage forces:

∂η f ∂N = F( Ỹ d, Ỹ d ) (7)
(A first tentative identification of F under transverse loading can be found in [START_REF] Lubineau | Durability of cfrp laminates under thermomechanical loading: A micro-meso damage model[END_REF].)

Concerning oxidation, only active surfaces are assumed to be affected. A surface is considered to be active if it is in direct contact with the atmosphere (through the skin, the edges or the cracks). If a surface becomes active with respect to oxidation, its critical energy release rate decreases immediately depending on the temperature and the partial oxygen pressure.

Let us consider the numerical example of an octagonal [0 3 /90 6 /0 3 ] sample studied in [START_REF] Lafarie-Frnot | Influence of free edge intralaminar stresses on damage process in cfrp laminates under thermal cycling conditions[END_REF]. Since this structure includes fibers at both 90 and 45 degrees with respect to the edges, it combines the results of the rectangular [0 3 /90 6 /0 3 ] and [45 3 /-45 3 /45 3 ] samples already studied in 2. The same thermomechanical cycling in an oxidizing atmosphere applied to this sample results in a complex degradation pattern (Figure 6 [a]) which clearly presents the edge effects discussed in Parts 2.1 and 2.2.

[Fig. 6 about here.]

Because of symmetry, only one-quarter of the sample was retained for the calculation (Figure 6 [b]). In order to comply with our micromechanical model, each ply was divided a priori into homogeneous elementary volumes (in which diffuse damage can develop) and interfaces (which, depending on their orientation, represent transverse microcracking or local delamination). This choice of introducing potentially cracked areas a priori was made for the sake of simplicity: improved strategies involving the dynamic occurrence of crack surfaces could certainly be derived, but this would require some programming which has not yet been done. Each elementary volume was meshed in three dimensions, leading to a large number of degrees of freedom: in other words, the approach proposed is relatively simple from the "modeling" point of view, but requires powerful computation capabilities. Therefore, multiscale computational strategies must be developed and used for such problems. The resolution performed here, which will be detailed in further publications, was based on the approach proposed at LMT-Cachan ( [START_REF] Ladevze | A micro-macro and parrallel computational strategy for highly heterogeneous structures[END_REF][START_REF] Ladevze | On a multiscale computational strategy with time and space homogenization for structural mechanics[END_REF]).

ACCEPTED MANUSCRIPT

At this point, due to the lack of experimental results which would allow the complete identification of the model, the results are purely qualitative. The identification of the model will be the subject of forthcoming works. Therefore, our comparisons will not be made in terms of quantitative results (e.g. evolution of the density of cracks as a function of the number of cycles), but only in terms of the morphology of the degradations observed.

[Fig. 7 about here.]

Figure 7 shows the progressive evolution of the cracking pattern within the central ply of the sample. It appears clearly that the main characteristics of the experimental results are correctly reproduced:

• as observed classically in fatigue tests ( [START_REF] Boniface | Application of the paris equation to the fatigue growth of transverse ply cracks[END_REF][START_REF] Yokozeki | Fatigue growth of matrix cracks in the transverse direction of cfrp laminates[END_REF][START_REF] Boniface | Observations on tranverse ply crack growth in a [0/90 2 ] s CFRP laminate under monotonic and cyclic loading[END_REF][START_REF] Berthelot | Transverse cracking and delamination in cross-ply glass-fiber and carbon-fiber reinforced plastic laminates: static and fatigue loading[END_REF][START_REF] Berthelot | Transverse cracking and delamination in cross-ply laminates under static and fatigue loadings[END_REF]), microcracks develop first near the edge of the sample until saturation of the microcracking rate. Then, cracks propagate from the edge toward the core. Let us note that in our approach the initiation and propagation of cracks are governed by the reduction of the critical energy release rate. This reduction is due to the development of both η f (for the fatigue part) and η ox (for the oxidation part).

Contrary to classical micromechanical modeling, no specific mechanism is introduced in the model to describe the multiplication near the edges and the growth towards the core, which seems satisfactory because, then, there is only one underlying physics.

• the influence of the edge effect is correctly accounted for. As observed experimentally, there is no crack starting from the 45-degree edge. The profile of the fracture criterion for Illustration 7-7 is shown in Figure 8. Even though the criterion is approximately homogeneous within the core, cracks which are not initiated along the 45-degree edge cannot appear in the core, whereas cracks initiated along the 90-degree edge propagate extensively.

[Fig. 8 about here.]

Conclusions and prospects

We studied the well-known and major role of edge effects in the degradation of laminates using fracture mechanics. Very different states of degradation can be observed away from the edges, even among samples having the same interior stress state. We showed that this can be explained as the result of the propagation of degradations initiated near the edges, which can be stable or unstable.

The computational micromechanical model proposed in previous works is able to predict this edge influence qualitatively, even in complex situations such as fatigue loading in an oxidizing atmosphere. Additional work is necessary in order to improve the identification part of this approach and end up with a predictive tool. Fig. 1. X-ray pictures of rectangular specimens after 1,000 thermal cycles in nitrogen and oxygen (from [START_REF] Lafarie-Frnot | Influence of free edge intralaminar stresses on damage process in cfrp laminates under thermal cycling conditions[END_REF])

.

  ure 1 and concern pure thermal cycles [-50C/150C] (triangular cycles 4/min) on 35 * 25 mm 2 rectangular samples of IM7/977-2 composite (cf. [7] for details of the experimental procedure). The stacking sequences were [-45 3 /+45 3 ] s and [0 3 /90 3 ] s , and the samples were cycled under different environments: neutral atmosphere (pure dry nitrogen) and oxidizing atmosphere (pure dry oxygen). [Fig. 1 about here.] Under such loading, the stresses in the core of each sample are due only to the differential expansion between the plies and are, of course, the same for the two stacking sequences. Therefore, a simple analysis based on the local stress state within the core would lead to similar interior degradations. However, the observations of Figure 1 clearly show that microcracks occur within the core of the central ply for the [0 3 /90 3 ] s while the central ply of the [-45 3 / + 45 3 ] s remains completely healthy. In a neutral atmosphere, this difference must be explained only by mechanical considerations. The results also show that this difference between the two sequences is amplified by oxidation.

  about here.] For long cracks, the curves tend towards a common value because the stress state of the interior solution is the same in both cases and corresponds to the steady-state value classically obtained by 2D analysis under the assumption of an infinite microcrack [9-11] . . . However, one can observe completely different types of behavior at the edge of the samples. In the [-45 3 / + 45 3 ] s sample, initiation is greatly delayed. Consequently, even if the core energy release rates of the two samples are identical, microcracking cannot develop in the [-45 3 / + 45 3 ] s because initiation does not occur, whereas it can propagate extensively in the [0 3 /90 3 ] s .

  . The first type refers to the intralaminar surfaces, which are assumed to be parallel to the fibers and represent transverse cracking. The second type refers to interlaminar surfaces, which represent local delamination. Whatever the nature of a surface, its failure is governed by a discrete energy criterion. Consequently, the basic material quantities involved in the model are the critical energy release rates { Ḡc }, designated for each mode by { ḠI,II,III c-p } for the ply and { ḠI,II,III c-i } for the interface. A surface breaks when its damage force reaches its critical energy release rate. When this happens, friction is introduced.

Fig. 2 .Fig. 3 .Fig. 5 .Fig. 7 .Fig. 8 .

 23578 Fig.2. Simulation of the periodic propagation from edge to core. A microcracking rate ρ = 0.5 was used for the example (ρ = H L , H being the thickness of the ply and L the mean distance between cracks)

  Evolution of the energy release rate for [0 3 /90 3 ] s and [-45 3 / + 45 3 ] s laminates under pure thermal loading 21 4 Local crack opening for short and long cracks in [0 3 /90 3 ] s and [-45 3 / + 45 3 ] s (pure reference thermal loading ∆T = -100C, periodic cracking pattern ρ = 0.5) 22 5 Evolution of the discrete energy release rate for [0 3 /90 3 ] s and [-45 3 / + 45 3 ] s laminates under pure thermal loading. The length of the crack was discretized with a step equal to the thickness of the elementary layer.

	1	X-ray pictures of rectangular specimens after 1,000 thermal
		cycles [-50/150C] in nitrogen and oxygen (from [7])	19
	2	Simulation of the periodic propagation from edge to core. A
		microcracking rate ρ = 0.5 was used for the example (ρ = H L ,
		H being the thickness of the ply and L the mean distance
		between cracks)	20
	3	(pure oxygen)
			23
	6	X-ray pictures of the octagonal specimen after 1,000
		(-50 o C/150 o C) cycles in oxygen (from [7])	24
	7	Evolution of the simulated pattern of microcracks under
		thermomechanical cycling with oxidizing atmosphere (central
		ply of the octagonal sample)	25
	8	Illustration of the profile of the fracture criterion (a surface
		fails when the criterion is equal to 1)	26


N Carrère

Y Rollet

V Retel

L Boubakar

J.-F Maire

Composites structural modelling with uncertain data

Keywords: Structural calculations, Uncertain data, Variability

The aim of this paper is to propose a methodology in order to take into account the influence of uncertain data in structural calculations. A specific method, based on the approximation of the responses as a function of the uncertain data, is proposed. Classical methods from the literature are also considered. The different methods are compared on a simple example in terms of response (average value and standard deviation) and in terms of computational cost. The method proposed in the present work permits to estimate correctly the whole response and is very simple to use (pre and post processing). This method is applied to a structural calculation on a "hatshapes" part called, manufactured in the quasi-isotropic IM7/977-2 composite.

Introduction

The use of Polymer Matrix Composites (PMC) seems to be an essential evolution in terms of mass saving and innovative solutions. However, the available simulation tools, used during design of new structures, do not take into account their specific aspects (local heterogeneities, complexity of damage/rupture mechanisms, influence of the manufacturing process …) that leads † Corresponding author : N. Carrere (carrere@onera.fr) tel. +33 1 46 73 46 47, fax. +33 1 46 73 46 91 to an under-using of these materials. A number of sources of lack in confidence could be evidenced (i) numerical aspects (mesh refinement, algorithms ...), (ii) numerical models (choice of the material behaviour, of the rupture criterion …) and (iii) material uncertainties (materials parameters, boundary conditions). In order to obtain a sufficient degree of confidence, generous safety coefficients or important experimental campaigns are necessary, thus leading to maintain classical metallic solutions. The first two sources of uncertainties can be quantified and reduced, but the last one is partially "irreducible". So, it is of absolute necessity to understand the role and the influence of uncertainties on the structure behaviour. The aim of the present work is thus to propose a methodology that permits to perform structural modelling taking into account material uncertainties in order to predict their effects on the behaviour of the structure.

Data used in structural modellings are known only within a range of confidence (due for instance to the imperfections of the geometry or uncertainties on the loading conditions during structure tests, the natural variability of the mechanical properties …). A number of methods exist in the literature to estimate the effects of uncertainties on the response of a structure :

• Calculation with the extreme data (called B-values) which requires only one evaluation of the response but provides only one pessimistic value without any statistical information on the obtained result.

• Sampling procedures (Monte-Carlo). The advantage of these methods is their simplicity of implementation, giving a good estimate of the interval of the responses but at a very high cost as soon as the number of variables increases.

• Stochastic Finite Element Modelling (SFEM) based on perturbation, spectral Neumann expansion, projection on a polynomial chaos are the most popular approaches widely developed during these last few years [START_REF] Ghanem | Stochastic Finite Elements: a spectral approach[END_REF][START_REF] Chakraborty | An efficient 3D stochastic finite element method[END_REF][START_REF] Sudret | Comparison of finite element reliability methods[END_REF]. These methods require, in order to be applied practically, significant modifications of the computer code. Moreover, they are in general used only for variations of material properties.

The previous methods are the most popular to deal with problems of variability. It should also be noted that some authors develop approaches on non-determinist arithmetic operations (fuzzy [START_REF] Noor | Uncertainty analysis of stiffened composite panels[END_REF], interval [START_REF] Dessombz | Analysis of mechanical systems using interval computations applied to finite-element methods[END_REF]).

In the present work, we propose a specific method which permits to study the effects of various kinds of material uncertainties (geometrical parameters and material, loading or boundary conditions) on the responses of a structure. The first part of this article will be devoted to the presentation of the different approaches. They will be compared, in the second part, on a simple example in terms of statistical responses (average value and standard deviation of the output data) and in terms of cost of calculation. In the third part of this article, the method proposed in this work is applied to the case of calculation of a structure known as "hat-shaped", manufactured in the quasi-isotropic IM7/977-2 composite.

Method for uncertainties transport 2.1. Monte-Carlo methods

The principle of the Monte-Carlo method is to choose at random (according to uniform or Gaussian distributions) values for each uncertain parameter and then to perform all the calculations associated with each set of variables. The major disadvantage of this type of approach is its cost. In fact, the number of calculations necessary to consider the extreme values (for example, the B-values of the response) could quickly become tremendous if the number N of variables is high (several thousands calculations if N>5). Furthermore, since the results are available for a number of sets of variables in the whole range of uncertainty, it is possible to estimate directly (using traditional statistical tools) probabilistic values (average value, standard deviation …). It should be noted that there are many alternatives to this method which are aimed at reducing the computational cost (by using an approximation of the inverse of the stiffness matrix by Neumann expansion or by using optimized random generators).

Stochastic Finite Element Method

Several types of methods exist under the denomination "Stochastic Finite Elements Methods" (SFEM). They are all based on the same principle: both the stiffness matrix (K) and the displacement vector (u) are expanded (using a perturbation method or on a polynomial base) with respect to the random variables.

For instance, using a perturbation method, the quantities under interest are expanded using Taylor series, around the mean value of the N uncertain variables α i (also called stochastic variables).

If a first order perturbation is considered, it is assumed that :

ACCEPTED MANUSCRIPT = + = N i i i u u u 1 0 ) ( α α ) 0 ( 0 u u = 0 ) ( = ∂ ∂ = α α α i i u u (1)
The mean value and the standard variation are given by :

) 0 ( u u = [ ] 1 1 1 , ) )( ( ) var( v C u u u N i N j j i t j i = = = = α α (2)
where C α i ,α j [ ] is the covariance matrix. Note that the same expansion is used for K.

The second order perturbation gives :

j N i N j i II ij N i i i O u u u u α α α α = = = + + = 1 1 1 ) ( 0 2 ) ( = ∂ ∂ ∂ = α α α α j i II ij u u (3)
The mean value and the standard variation are written as follows :

[ ] = = + = N i N j j i II ij C u u u 1 1 , ) 0 ( α α (4) [ ] [ ] [ ] [ ] ( ) + + = l k j i l j k i l i k j t II kl II ij C C C C u u v u , , , 1 , , , , ) )( ( 4
1 ) var( α α α α α α α α (5)
The cost of this approach is due to the calculation of the derivatives by a perturbation method (2N calculations for the first order derivative and 3N for the second order derivative). A method based on an automatic derivation in the FE code is under investigation at ONERA. The method called polynomial chaos expansion (PCN, for a N order polynomial expansion) [START_REF] Ghanem | Stochastic Finite Elements: a spectral approach[END_REF] seems to be very effective. It is based on the expansion of K and u on a particular polynomial base (the base defined by the monomials is orthogonal). The base is chosen as a function of the distribution of the stochastic parameters.

ACCEPTED MANUSCRIPT

= Φ = M i i i u u 1 ) ( ) ( α α (6) with ij i j i δ α α α ) ( ) ( ) ( 2 Φ = Φ Φ and = α α α α α α d w g f g f ) ( ) ( ) ( ) ( ) (
Using the properties of the polynomial (orthogonality), it is possible to show that :

) ( ) ( ) ( ) ( α α α α i i i i u u Φ Φ Φ = (7)
The choice of the inner product (<xy>) is a function of the distribution of the stochastic parameters. The practical implementation of this method is more complex and necessitates the computation of a number of integrals. However, the results shown in the literature seem relatively precise and obtainable for reasonable computing times.

Surface Response based Analysis of the Variability Method

The method proposed in the present paper could be called "experimental design numerical method" since it is strongly inspired by the experimental design methods. However, the traditional experimental design is defined to minimize the effects of the experimental errors and the noise whereas, in the present case, all the values are computation results. This method consists in building an analytical relation (most generally polynomial) between stochastic variables and the outputs using a number of runs of the complete modelling (usually a FE calculation). The choice of the values of the stochastic variables for the computations is made in order to obtain the most precise approximation. Moreover, some tools have been developed to estimate the quality of this approximation.

Remark : the approximation could be performed using a polynomial chaos expansion. In this case, the terms of the polynomial are calculated by eq. 7. The integrals are calculated by a Gauss integral approximation.

The proposed approach is divided into several stages (Figure 1) :

• Choice of the sets of variables to be tested,

• Realization of the calculations,

• Determination of the approximation (polynomial) function that relates the outputs to the stochastic variables,

• Calculation of the estimators of error made by the approximation,

• Calculation of the extreme values, statistical distributions of the responses, B-values... thanks to the approximation function.

One of the key points of this method is the distribution of the sets of variables to be tested. Two strategies have been developed: one based on the neural network [START_REF] Kohonen | Self Organizing Maps[END_REF] that permits to occupy as far as possible the whole hypercube defined by the range of the stochastic variables, the other one consists in occupying the tops of the hypercube as far as possible. The sets of variables being chosen, FE simulations are performed on a cluster of PC by using the natural parallelization of the problem (3x[N+1 ] structural analyses).

The outputs are extracted from all the calculations and the relations between the uncertain variables α i and the outputs are approximated using a polynomial form. The error evaluation of the polynomial approximation is performed using various techniques of cross-validations (bootstrap [START_REF] Efron | An introduction to the bootstrap[END_REF], leave-one-out [START_REF] Stone | Cross-validatory choice and assessment of statistical predictions[END_REF]). It consists in identifying X times the approximation using sets of calculations randomly chosen through the N available ones (N randomly or (N-M) choices among the N for the bootstrap and the leave-M-out methods respectively). These techniques, which allow estimating the generalization error, improve considerably the reliability of the approximation.

In practice, the approximation is first chosen as linear. If the errors are too significant (a minority of the investigated cases), it is possible to increase the degree of the polynomial interpolation. In order to avoid increasing significantly the number of monomials (384 for N=11 with a polynomial of degree 3), the Gram-Schmidt orthogonalization algorithm is used [START_REF] Stoppiglia | Ranking a random feature for variable and feature selection[END_REF]. Following this method, the monomials are iteratively ranked in order of their decreasing contribution to the output. It has been shown that in several cases it is possible to drastically reduce the errors by introducing only some additional monomials. Other approximations or regression methods are based on neural networks for highly non-linear probabilistic transfer problems.

Finally, once the approximation is identified and validated, it is very easy to obtain for each output a number of results such as extreme values (minimum and maximum of each output), the B-values (using Monte-Carlo techniques thanks to the polynomial approximation). Using the developed error approximation, it is possible to determine the confidence in these results.

Comparison of the methods on a simple example

This section is devoted to the comparison of the presented methods (in terms of average value, standard deviation and computing time). The very simple case under investigation is a 2D square element subjected to a simple tensile test (see Figure 2). The behaviour of the material is isotropic, with a stochastic Young modulus and a deterministic Poisson's ratio ( µ= 0.31). The applied load F is equal to 100N and E is defined by: ) 1 (

ξ E r E E + = (8)
where GPa E 200 = is the mean value of the Young modulus. r E (=0.1) is the coefficient of variation. ξ is a centered Gaussian random variable.

The results obtained with the different approaches are presented in Figure 2. When the number of random variables is sufficiently low (N<5), it is well known that for a great number of samples, the results obtained by the Monte-Carlo method (MC) could be considered as having converged (as compared with the exact solution). On this example, the 2 nd or 4 th order polynomial chaos methods (PC2 or PC4) lead to very close results as compared with the MC method. However, the cost of a calculation with the PC method is drastically less than that which the MC method. The 1 st order perturbation method is very efficient in terms of computational cost but underestimates the results.

The method proposed in the present work gives very close results as compared with the MC ones.

Its computational cost is low since only some deterministic FE calculations are necessary (here 7 calculations were used to identify the approximation). The average and the variance of displacement can be calculated:

• Analytically, starting from the equations ( 1) and ( 2) for which the derivatives are very simple to calculate due to the polynomial approximation of the output

• By a MC method using the polynomial approximation that allows a quasi immediate estimate of the output.

The method proposed in this paper permits to obtain very close results as compared with the other ones found in the literature, for a much lower computational cost. Moreover, the use of this method does not require any modification of the computer code since the whole analysis is performed in post-processing. Finally, it should be noted that this method permits to study material uncertainties but also geometrical ones (boundary conditions, loading …).

Influence of geometrical uncertainties

In the following example, geometrical (dimensional uncertainties but also uncertainties on boundary conditions and loading) are investigated. The structure under investigation (called "hatshaped" structure, see Figure 3) is defined by the following 9 parameters (θ 1 , θ 2 , θ 3 , r, l, a g , a d , r 4 and r 5 ) with uncertainty intervals of ± 10 % for each parameter. In order to validate the proposed method, comparison with reference results obtained by a MC method (1024 FE calculations with uniform distribution on each stochastic parameter) is performed. This structure is a 8-ply quasiisotropic IM7-977-2 composite. The behaviour of the ply is described by a transverse-isotropic linear elastic model (5 elastic parameters). The structure is subjected to an imposed displacement.

Several outputs are investigated (load, maximal stress, maximal strain, rupture criteria such as maximal admissible fiber deformation…).

Figure 4 presents all the load/displacement curves obtained thanks to the 1024 FE calculations.

The first remark is that, according to the Central Limit Theorem, even if the distribution of the stochastic variables is uniform, if the number of runs is not too small, the histogram presents a Gaussian shape. Three outputs are investigated: stiffness of the structure (load/déplacement ratio), maximal strain criterion for the fibers and, finally, a damage criterion (quadratic criterion which is a function of the stress field). The first output is a global one as the other ones are evaluated by seeking, for each calculation, the maximum value on the whole structure. The statistical treatment of the MC results are presented in table 1 (average value and standard deviation of 1024 calculations as well as the minimal and maximum values).

Moreover, thanks to cross-validation approaches it is possible to evaluate the quality (confidence) of the values provided in table 1. For that, the "bootstrap" [START_REF] Efron | An introduction to the bootstrap[END_REF] method used in descriptive statistics, is well adapted. The results obtained with this method with samples of 50, 200, 512 and 1024 calculations are presented in table 2.

The method proposed in this paper has been applied and identified using a set of 20, 30 or 60 calculations. In the first part of table 3, the average values and the variations are indicated when the method is identified 100 times, starting from sets of different calculations. One can notice the good robustness of the method. It should be noted that using only one set of calculations (0, 30 or 60), it is possible to obtain results equivalent to those obtained using cross-validation methods (here the method used is of type "leave-one-out"). Very similar results are obtained on the other outputs. It is also important to note that the effect of the type of distribution of the stochastic parameters on the output can be investigated using the proposed method. For instance, if a normal distribution is chosen instead of the uniform distribution, the standard deviation decreases from 14.2 to 8.6. The confidence on the values of table 3 depends obviously on the quality of the approximation (linear in this case). If the degree of confidence is low, it is possible to increase the quality of the approximation by increasing the number of monomials (chosen using the Gram-Schmidt orthogonalization method, [START_REF] Stoppiglia | Ranking a random feature for variable and feature selection[END_REF]. The reduction of the average quadratic error (estimated on all 1024 calculations) on the fiber rupture criterion is presented in figure 5.

Conclusion

Various methodologies may be found in the literature allowing to take into account the effect of uncertainties on the response of a structure. A method, based on the surface response has been proposed in the present paper. These various methods have been compared on simple examples in terms of response (average value and standard deviation of the output) and in terms of computational cost. The method proposed in this paper permits to evaluate a number of statistical results. This method is very simple to use since it does not need any modification of the FE code (procedure used in pre/post processing). It has been shown that the distribution of the output and the associated confidence could be evaluated using a few number of calculations (3x[N+1]).

Figure captions

Figure 1 :Figure 2 :

 12 Figure 1 : Principle of the proposed method

Figure 3 :Figure 4 :Figure 5 :

 345 Figure 3 : Geometry and boundary conditions of the « hat-shaped » structure

FiguresFigure 1 :Figure 2 :

 12 Figures

Figure 3 :Figure 4 :Figure 5 :

 345 Figure 3 : Geometry and boundary conditions of the « hat-shaped » structure

Table captions Table 1 :

 captions1 Results obtained by the Monte-Carlo method

Table 2 :

 2 Results obtained using a re-sampling method

Table 3 :

 3 Results obtained by the present method : comparison between 100 re-identification of the polynomial approximation and a re-sampling method.

Table 1 :

 1 Results obtained by the Monte-Carlo method

	Tables
	Monte-Carlo method		Mean value	Standard deviation Minimal value	Maximal value
	Stiffness (N/mm)		335.8		14.2	298.9	379.5
	Fibers strain (%)		1.46		0.04	1.35	1.58
	Hill criterion (MPa)		127.5		3.85	117.1	139.9
	Stiffness (N/mm)	N	Mean value	Standard deviation	Minimal value	Maximal value
	Bootstrap	50 200 512 1024	336.5 ± 6.0 335.4 ± 2.9 335.2 ± 1.9 335.7 ± 1.5	14.9 ± 6.2 14.6 ± 2.7 13.8 ± 1.8 14.1 ± 1.1	304.7 ± 9.8 300.3 ± 5.7 299.7 ± 2.8 299.5 ± 2.0	375.6 ± 15.8 377.7 ± 6.8 377.0 ± 4.5 378.9 ± 2.7

Table 2 :

 2 Results obtained using a re-sampling method

	Stiffness (N/mm)	N	Mean value	Standard deviation	Minimal value Maximal value
	100 re-identifications Cross-Validation	20 30 60 20 30 60	336.2 ± 0.8 336.2 ± 0.4 336.1 ± 0.3 336.1 ± 1.8 335.9 ± 0.3 336.0 ± 0.2	14.5 ± 0.9 14.2 ± 0.5 14.2 ± 0.3 15.4 ± 1.4 14.2 ± 0.4 14.2 ± 0.3	295.7 ± 4.6 298.8 ± 1.5 298.6 ± 1.0 287.8 ± 8.6 297.9 ± 1.4 298.5 ± 0.8	376.6 ± 4.3 375.6 ± 2.3 375.6 ± 1.2 384.4 ± 9.3 376.0 ± 2.5 376.5 ± 1.0

Table 3 :

 3 Results obtained by the present method : comparison between 100 re-identification of the polynomial approximation and a re-sampling method.

Acknowledgments

This work was carried out under the AMERICO project (Multiscale Analyses: Innovating Research for Composites) directed by ONERA and funded by the DGA/STTC (French Ministry of Defense) which is gratefully acknowledged. The authors would like to express their sincere gratitude to Dr. R. Valle for valuable and helpful discussions.


J A Bennett

R J Young

A Strength Based Criterion for the Prediction of Stable Fibre Crack-Bridging

Keywords: A. Polymer-matrix composites, B. Fracture, C. Fibre bridging, D. Raman spectroscopy

The likelihood that a fibre will form a stable bridge across a propagating crack has been investigated by means of experimental measurements and the theoretical consideration of the balance between interfacial strength, fibre strength, the physical properties of the component materials and the stress transfer characteristics of bonded and debonded interfaces. A simple strength based criterion has been developed from traditional stress transfer models to predict if a fibre will form a stable bridge across a propagating matrix crack. The first step in the analysis takes into consideration a range of stress transfer parameters to predict whether the build up of stress caused by the matrix crack is sufficient for the fibre to fracture or if it will cause debonding from the surrounding matrix. This is then extended to include the subsequent build up of stress in the case where the fibre debonds from the matrix. The criterion is compared to experimental data for a range of fibre/matrix systems. Raman spectroscopy was used to measure the point to point distribution of fibre strain along Twaron, Zylon, T50u and M5 fibres embedded in an epoxy resin matrix and lying perpendicular to a matrix crack. The experimental data showed that while some fibres debonded partly from the matrix and formed a stable bridge, others fractured with only a small amount of debonding.

Introduction

When a stressed material fails by fracture, the opening of the resultant crack is opposed by the fracture resistance of the material. This resistance may be attributed to mechanisms that absorb and/or dissipate energy. When considered in the context of fibre reinforced composites it has been shown that enhanced toughness requires that a matrix crack leaves intact fibres in its wake to act as a restraint across the faces of the crack. Crack bridging, fibre fracture, interfacial-debonding and fibre sliding significantly affect the resistance to crack propagation and hence toughness of the material [START_REF] Chawla | Ceramic Matrix Composites[END_REF][START_REF] Piggott | Load Bearing Composites[END_REF][START_REF] Hull | An Introduction to Composite Materials[END_REF]. These processes, however, are all interlinked and the significance of one may depend upon the characteristics of another, for instance if the advancing crack causes immediate fibre fracture then although energy has been used in breaking the fibre there can be no crack bridging or interfacial debonding and hence no fibre sliding.

The tendency of a matrix crack to deflect along the fibre/matrix interface has been observed experimentally and has been shown theoretically to be dependent on the degree of mismatch between the elastic properties of the two materials [START_REF] He | Crack deflection at an interface between dissimilar elastic materials[END_REF]. Bennett and Young [START_REF] Bennett | Micromechanical aspects of fibre/crack interaction in an aramid/epoxy composite[END_REF] have observed fibre/crack interactions in aramid epoxy composites and noted for fibres ahead of the crack tip that the fibre strain distribution indicated interfacial failure when using untreated (weak interface) fibres in contrast to the behaviour of surface-activated (strong interface) fibres which remained fully bonded. It was noted, however, that debonding occurred in the crack bridging fibres for both types of fibre, although it was found that further propagation of the debond was more difficult in composites made with surface-activated fibres.

In the case of a semi-infinite matrix crack normal to the direction of the fibre axis and loaded symmetrically about the crack plane, that has impinged upon and advanced slightly beyond an embedded fibre, but for which no fibre/matrix debonding has occurred, two outcomes are envisaged [START_REF] Chawla | Ceramic Matrix Composites[END_REF]:a) Fibre/matrix debonding occurs and a mode II crack deflects along the fibre/matrix interface with no fibre fracture b) No debonding occurs and fracture of the fibre occurs when it is loaded beyond its ultimate tensile strength ( f * ).

In this study an effort is made to understand why matrix crack propagation causes some fibres to simply fracture, while others exhibit a degree of debonding from the matrix before subsequent fracture. This is done by establishing a link between conventional strength based theory [START_REF] Bannister | Analysis of the single fibre pull-out test by means of Raman spectroscopy: Part II. Micromechanics of deformation for an aramid/epoxy system[END_REF][START_REF] Andrews | Deformation micromechanics in highmodulus fibres and composites[END_REF][START_REF] Andrews | Analysis of the deformation of aramid fibres and composites using Raman spectroscopy[END_REF][START_REF] Young | Deformation micromechanics in high-performance polymer fibres and composites[END_REF], micromechanical considerations and experimental data obtained using Raman spectroscopy to follow the deformation micromechanics of the fibre/matrix interface in model composites where a matrix crack has been propagated.

Criterion for interfacial failure 2.1 Fully-bonded case

As a starting point, a matrix crack is assumed to have propagated normal to and beyond an embedded fibre, of uniform strength causing no damage to the fibre in the process.

Further, the matrix crack is spanned by the unbroken fibre and no debonding has taken place.

The application of load to a fibre, in this case by crack opening, in order to extract it from the matrix by a 'pull-out' process, results in a stress distribution along the region of the fibre embedded within the matrix [START_REF] Piggott | Micromechanics of fibre-polymer interfaces[END_REF]. The shear lag model [START_REF] Cox | The elasticity and strength of paper and other fibrous materials[END_REF] that has been used extensively for analysis of stress distributions in pull-out/crack-bridging tests, can be used to describe the build up and transfer of fibre stress, f , from the point where the fibre enters the resin to some point along the fibre axis where the tensile stress has decayed to zero. Failure of the fibre/matrix interface occurs when the interfacial shear strength, max , is reached. The conditions which determine the likelihood of fibre/matrix interfacial failure occurring instead of fibre fracture, can be examined using the shear lag model. The shear lag distribution of strain, along a fully bonded fibre can be described by [START_REF] Cox | The elasticity and strength of paper and other fibrous materials[END_REF],

) sinh( ] / ( sinh[ e f app ns r x L n e e - = (1)
where e app is the strain acting on the fibre outside the matrix, e f is the fibre strain at a distance x inside the matrix, L e is the embedded length, r is the fibre radius, s is the fibre aspect ratio (L e /r). The Cox analysis gave a value of n in terms of fibre and matrix properties, but recent shear-lag analyses [START_REF] Nairn | On the use of shear-lag methods for analysis of stress transfer in unidirectional composites[END_REF][START_REF] Nayfeh | Thermodynamically induced interfacial stresses in fibrous composites[END_REF][START_REF] Mccartney | New theoretical model of stress transfer between fibre and matrix in a uniaxially fibre-reinforced composite[END_REF] have shown it to be inaccurate. The n parameter used here is based on the parameter derived correctly by Nairn [START_REF] Nairn | On the use of shear-lag methods for analysis of stress transfer in unidirectional composites[END_REF], where n = r [START_REF] Piggott | Micromechanics of fibre-polymer interfaces[END_REF] such that

n 2 = 2 E f E m E f V f + E m V m V m / 4G f ( )+1/ 2G m ( ) 1/V m ( )ln 1/V f ( )-1-V m /2 ( ) ( ) ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ (2)
E f and G f are the fibre Young's and shear moduli, E m and G m are the matrix Young's and shear moduli and r is the fibre radius. V f is the fibre volume fraction and V m is the volume fraction of matrix, such that if R is the cylinder of surrounding matrix then

V f = r 2 /R 2 and V m = (R 2 -r 2 )/R 2 .
The fibre strain is a maximum at the crack plane (x = 0) where the fibre enters the matrix. Following work by Cervenka et al [START_REF] Cervenka | Micromechanical phenomena during hygrothermal ageing of model composites investigated by Raman spectroscopy. Part II: comparison of the behaviour of PBO and M5 fibres compared with Twaron Composites Part A: App[END_REF] values of R used in this study reflect n values obtained by fitting the shear lag model to experimental data obtained from Raman strain mapping experiments on model composites. Similarly the corresponding interfacial shear stress, , at a distance x along the interface, given by

] sinh[ ] / ) ( cosh[ 2 e app f ns r x L n e E n - = τ (3)
is a maximum at the crack plane (x = 0). Since both the interfacial shear stress and the stress acting on the fibre, are a maximum at the crack-plane then failure should be expected to initiate from this point. For fibre aspect ratios > 15, applying the condition x = 0 to equation

3 gives f f 2 e E n = τ (4)
If the fibre deforms in an elastic manner (according to Hooke's Law) then,

f 2 σ τ n = ( 5
)
where f is the fibre stress. If fibre fracture occurs when the stress in the fibre reaches its ultimate tensile strength, f * , then setting the boundary condition at

This approach suggests that the outcome of a matrix crack impinging on an embedded fibre depends on the balance between the fibre strength and the shear strength of the interface. If max is known then it is also possible to define a critical ratio of fibre strength to fibre modulus by substituting equation 2 into equation 9 and rearranging to give

2 max f 2 f * τ σ a E > ( 10
)
where a is constant and a function of the geometric and matrix of the material parameters only given by

a = 2E m V m / 4G f ( )+1/ 2G m ( ) 1/V m ( )ln 1/V f ( )-1-V m /2 ( )
( )

E f V f + E m V m ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ (11)
Lines that demonstrate the calculated relationship between f * and E f are shown in figure 1.

Plotting f * against E f 1/2 for the above condition according to equation 10 defines the boundary between interfacial failure and fibre failure. It can be seen from figure 1 that for any given set of stress transfer conditions, if the fibre modulus increases then there is a greater tendency for fibre fracture. Conversely if the fibre strength increases then the balance shifts towards interfacial failure.

Effect of interfacial shear strength

The interfacial bonding between a fibre and matrix system can be altered by the application of physical or chemical fibre surface treatments [START_REF] Kelly | Strong Solids[END_REF]. Any change in the interfacial shear strength, max , of a fibre/matrix system will influence the crack-bridging capabilities of the fibre and consequently the toughening or structural properties of the material. Figure 1a shows the theoretical boundary conditions drawn at several levels of interfacial strength, max , for a system with a matrix of modulus, E m = 3 GPa and Poissons ratio, m = 0.35, incorporating a fibre of radius, r = 6.25 m. The interfacial geometric parameter n was set at 0.1. It can be seen that as the interfacial shear strength, max , is increased from 20 MPa to 60 MPa, through an improvement in interface strength, the balance between interfacial failure and fibre fracture shifts towards fibre fracture. Thus in a system with a high interfacial shear strength, fibre fracture will be more likely then for a system with low interfacial shear strength. The maximum recorded value of interfacial shear strength for the systems studied here, 43 MPa, is close to the shear yield stress of the matrix [START_REF] Andrews | Analysis of the deformation of aramid fibres and composites using Raman spectroscopy[END_REF] and could represent interfacial failure by either matrix yielding or fibre/matrix debonding.

Effect of shear lag fitting parameter n

The fibre shear lag fitting parameter n also has an effect on the boundary between fibre fracture and interfacial failure. If the parameters are as above with both matrix modulus, E m , and interfacial shear strength, max , are kept constant at 3 GPa and 40 MPa respectively, then it can be seen from figure 1b that changing the value of n has a significant effect. As the value of n is decreased from 0.1 to 0.08 then it can be seen that the boundary shifts in favour of fibre fracture.

Partially-debonded case

In many fibre/matrix systems the propagation of a primary matrix crack will lead to debonding in the form of a mode II crack propagating along the fibre/matrix interface over part of the fibre embedded length. The strain distribution in this case can be described by a

combination of f f 2 E r x e τ = (12)
in the debonded region, where x is the distance long the fibre, and equation 1 in the bonded region.

Consideration of the theoretical strength criterion for crack-bridging can be extended to account for propagation of interfacial failure. Once debonding has been initiated, if the fibre properties are uniform along its length and debonding causes no damage to the fibre, then the fibre sliding is opposed by friction alone. Further crack-opening will cause debonding to propagate along the fibre/matrix interface. In this case two components of the micromechanical stress distribution must be considered, the lower strain elastic bonded region and the higher strain debonded region. The limiting value of interfacial shear strength, which is encountered at the beginning of the bonded region must be continually overcome for further failure of the fibre/matrix bond and hence debond propagation. In addition the transfer of interfacial frictional shear stress causes a build up of axial fibre stress in the debonded region, which is a maximum at the crack-plane. As a consequence of the build up of frictional shear stress, it is inevitable that as debonding proceeds the axial fibre stress will reach the tensile strength of the fibre.

Debonding along a continuous fibre is assumed to proceed in a stable manner, at a constant shear stress, max , and a constant fibre stress, f , at the point of debonded/bonded transition, x = L d . As the debond crack advances, the maximum value of interfacial shear stress is always found where the strain gradient is greatest; this has been shown to be at the debonded/bonded transition [START_REF] Bannister | Analysis of the single fibre pull-out test by means of Raman spectroscopy: Part II. Micromechanics of deformation for an aramid/epoxy system[END_REF]. Generally, frictional stress transfer in the debonded region causes an increase in the axial fibre stress which tends towards the tensile strength of the fibre, f * . However, one case where this does not apply is for an interfacial frictional shear stress of zero, where debonding would continue unchecked over the whole of the fibre length.

If partial-debonding conditions are applied, then the overall stress in the fibre is the sum of the stress built up in the debonded and bonded region. This can be written as

b f, d f, f σ σ σ ∆ + ∆ = (13)
where f,d is the build up of stress in the debonded region and f,b is the build up of stress in the bonded region.

The build up of stress in the debonded region is determined by the rate of change of stress along the fibre and the length of the debonded region L d , such that

d f d f, d d = L x σ σ ∆ (14)
where L d is the debonded length and d f /dx in the debonded region can be found by substituting i = frict into a stress balance equation [START_REF] Nayfeh | Thermodynamically induced interfacial stresses in fibrous composites[END_REF] 2 d

d f f r x e E = τ (15) to give r 2 d d frict f τ σ = x ( 16
)
where frict is the frictional sliding stress in the debonded region. The build up of stress in the debonded region can be determined by substituting equation 16 into equation 14 to give,

frict d d f, 2 τ σ r L = ∆ (17)
Finally, substituting f = f * , equation 5 (for = max ) and equation 17 into equation 13 yields,

frict d max f 2 2 * τ τ σ r L n + = (18)
which simplifies to,

⎟ ⎠ ⎞ ⎜ ⎝ ⎛ + = frict d max f 2 * τ τ σ r L n (19)
Plotting f * against E f 1/2 for the above condition produces curves (figure 2) that define the boundary between continuing interfacial failure and fibre fracture along a partially-debonded fibre.

Effect of interfacial frictional shear stress

The interfacial frictional shear stress that can be observed to vary between both systems with different fibre/matrix combinations and systems where the fibre surface has been modified to optimise the level of adhesion, determines the rate at which stress is transferred to the fibre along the debonded length, and hence any change will affect the eventual failure of the crack bridging fibre. Figure 2a shows the theoretical boundary conditions drawn at several levels of interfacial frictional shear stress, frict , for a partially debonded system with a matrix of modulus, E m = 3 GPa and Poissons ratio, m = 0.35, incorporating a fibre of radius, r = 6.25 m. The interfacial geometric parameter n was set at 0.1 and the interfacial shear strength, max = 40 MPa. In the failed/debonded region of the interface, the debond length, L d , is set at an arbitrary value of 400 m.

Effect of debond length

The length of the failed/debonded interface, L d , also has an effect on the build up of axial fibre stress. As the length of the failed/debonded region increases then the distance over which the axial fibre stress builds by friction is increased. The effect of this with respect to fibre fracture can be seen in figure 2b where the boundary has been drawn for several lengths of interfacial failure/debonding. The parameters used are the same as above, except the interfacial frictional shear stress is kept constant at a typical mean value of 8 MPa. It can be seen that as the distance over which interfacial failure occurs increases from 200 m to 800 m then the fibre strain builds up, moving the balance between fibre fracture and interfacial failure so that fibre fracture becomes more likely.

Materials

The fibres used in this study were a Twaron (PPTA), Zylon (PBO), T50u (carbon) and M5 (PIPD). Each fibre was used in the as-received condition i.e. without removing the manufacturers proprietary sizing or surface finish. The mechanical properties of the fibres are detailed in Table 1. The matrix material, a two part, cold-curing epoxy resin from Ciba-Geigy (LY5052/HY5052), has a Young's modulus, E m ~ 3 GPa and a Poisson's ratio, m ~ 0.35.

Composite Compact Tension (CCT) Specimens

The CCT specimens were prepared using a window-card and a square 'picture-frame' mould. A continuous fibre was mounted across a hole on the window-card. The windowcard was then fixed into the mould that was subsequently filled with resin. The mould was then left to cure at room temperature (22 2 C) for a minimum of 7 days. Once cured the 3 mm thick resin sheet was machined into compact tension specimens such that the notch ran perpendicular to the direction of the fibre axis. Two holes of diameter 6 mm were drilled to allow the insertion of pins for loading of the sample as indicated in figure 3. The notch was then loaded using a sharp wedge (razor blade) until a sharp matrix crack propagated beyond the fibre.

Raman Spectroscopy

Single Fibre Deformation in Air

For many high-performance fibres the peak positions of certain Raman bands have been demonstrated to shift to lower wavenumber by the application of tensile stress [START_REF] Bennett | Micromechanical aspects of fibre/crack interaction in an aramid/epoxy composite[END_REF][START_REF] Bannister | Analysis of the single fibre pull-out test by means of Raman spectroscopy: Part II. Micromechanics of deformation for an aramid/epoxy system[END_REF][START_REF] Andrews | Deformation micromechanics in highmodulus fibres and composites[END_REF][START_REF] Andrews | Analysis of the deformation of aramid fibres and composites using Raman spectroscopy[END_REF][START_REF] Young | Deformation micromechanics in high-performance polymer fibres and composites[END_REF]. For all the fibres used in this study there is an approximately linear shift of the peak wavenumber, , with axial fibre strain, e f . The values of d /de f used here are listed in Table 2 [START_REF] Bennett | Fibre/Crack Interaction in Polymer Matrix Composites[END_REF].

Strain Mapping

The CCT specimen was mounted on a miniature materials straining rig, which was in turn mounted on the microscope stage of the Spex 1000M Raman spectrometer and Raman spectra were obtained from the fibre within the resin using the 632. the faces of the crack, at a point corresponding to that where the fibre bridged the crack. The profiles of Raman band positions were converted to fibre strain profiles using the shift values in Table 2.

Scanning electron Microscopy

After the Raman investigation the CCT specimens were broken into two pieces along the crack-plane and sputtered with a thin coating of gold. The points at which the fibre had bridged the crack were examined in a Philips XL30 scanning electron microscope with a field emission gun. The microscope was operated at 1kV.

4 Fibre bridging -experimental data

Twaron

Figure 4a shows the distribution of fibre strain along a Twaron fibre embedded in a crack epoxy resin CCT specimen, derived using measurements of peak shift and the calibration value of the 1610 cm -1 Raman band (Table 2), for the deformation of Twaron in air. Each point represents the axial fibre strain calculated from a single Raman spectrum.

The fibre strain distribution shown in figure 4a is for a crack opening displacement (COD) of 12 m, which is the distance between the faces of the crack at the point where it is bridged by the fibre under examination. The point x = 0 defines the location of the crack plane; x is positive to the right and negative to the left of the crack plane.

It can be seen that opening of the matrix crack has created a distribution of strain along the fibre to both the right and the left hand side of the crack. The fibre strain, which is maximum at the crack-plane, decays in both directions, tending to zero at some distance, x, along the fibre. The fibre strain distribution (on each side of the crack-plane) can be divided into two regions. The first is a higher strain region with a linear strain gradient, which is characteristic of strain transfer being controlled by friction, hence the fibre is sliding and fibre/matrix debonding has taken place. This region itself can be divided into of two subregions, which have different strain gradients, indicating the fibre is subjected to different levels of friction. It can be seen that the fibre has debonded ~ 0.4 mm to the right and ~ 0.4 mm to the left of the crack plane, giving an overall debonded length of ~0.8 mm. The strain in the fibre at the end of these regions is 0.71 % and 0.92 % respectively.

The end of the linear region marks the transition from the debonded to a fully-bonded region and is characterised by a sharp change in the gradient of the strain profile. The variation of fibre strain in this region is non-linear and shows an exponential decay; this is characteristic of elastic strain transfer, indicating that the fibre and matrix in this region remain bonded. In the bonded region, strain decays and tends towards zero over a distance of ~ 0.5 mm.

The distribution of fibre strain is approximately symmetrical about the crack-plane indicating that the same processes are taking place on either side of the crack-plane. One notable deviation from the symmetrical distribution can be observed to the right hand side of the crack where there is an immediate drop or discontinuity in strain over a distance of x = 0 -0.1 mm. This phenomenon has been observed for other systems, but the origins are not fully understood. It has been proposed that the most likely explanation is damage of the fibre skin at the point where the fibre enters the matrix [START_REF] Marotzke | The elastic stress field arising in the single fibre pull-out test[END_REF].

The solid line in figure 4a represents the theoretical partial debonding model [START_REF] Piggott | Micromechanics of fibre-polymer interfaces[END_REF] fitted to the data using equation 12 to model the linear strain distribution in the debonded Starting from x = 0, for the theoretical curve to the right hand side of the crack plane, it can be seen that there are two regions of constant shear stress, reflecting the change in strain gradient along the debonded region of the fibre/matrix interface. The shear stress then rises sharply to a singular peak at the bonded/debonded transition, before decaying exponentially in the fully-bonded region. This is exactly the form expected for a partially-debonded fibre [START_REF] Piggott | Micromechanics of fibre-polymer interfaces[END_REF]. The values of frictional shear stress, frict , in the debonded regions are between, 0 and 10

MPa. In the fully-bonded region max to the left and to the right max of the crack plane is ~ 36 MPa.

Zylon/epoxy

Figure 5a, shows the micromechanical strain distributions along a crack bridging Zylon fibre, at COD = 35 m. It can be seen that for the particular fibre chosen, the strain profile to the left of the crack plane is approximately similar to that to the right, as was the case for all Zylon fibres studied. The distribution of strain along the fibre is qualitatively similar to that for Twaron, hence the micromechanical deformation process during crackbridging is similar for the different materials. The distribution of strain extending to the left and to the right, from the crack-plane, along the fibre embedded in the matrix can be seen to incorporate several regions where the distribution of strain varies linearly, with a debonded length of 2.2 mm to the left of the crack plane and 2.7 mm to the right. The strain at the end of the debonded regions on both sides of the crack plane is ~ 0.48 %.

At the end of the linear regions the transition from the debonded to the fully-bonded region is marked by the characteristic sharp change in the gradient of the strain profile in the form of exponential decay, typical of elastic strain transfer which takes place when the fibre and matrix remain bonded. The strain in the bonded region decays and tends to zero over a distance of ~ 0.5 mm.

There is good correlation between the theoretical model (shown as a solid line) and the experimental data for the Zylon epoxy system. The interfacial shear stress distributions which are derived from the theoretical strain distributions are presented in figure 5b. The interfacial shear stress in the debonded regions varies but is generally between -5 and 5 MPa (the sign indicates direction of stress). The shear stress at the debonded-bonded transition, max , on both sides of the crack plane is found to be 30 -33 MPa indicating that for the Zylon/epoxy system debonding occurs well below the shear yield stress of the matrix which is 43 MPa [START_REF] Andrews | Analysis of the deformation of aramid fibres and composites using Raman spectroscopy[END_REF]; the interfacial shear stress decays from these points with distance x along the fibre.

T50u/epoxy

Figure 6 shows the micromechanical strain distribution along a T50u carbon fibre where matrix crack propagation has caused the fibre to break; the COD was maintained at COD = 5 m. The strain distribution indicates that the fibre is subjected to a slight residual compressive strain over an approximate distance of 0.35 mm. It can be seen that for the particular fibre chosen, the strain profile to the left of the crack plane approximately mirrors that to the right; this was the case for most carbon fibres observed.

A point with error bars, has been plotted in figure 6, to indicate the scatter associated with an undeformed fibre in an undeformed matrix. The error bars are based on the scatter of data from a carbon fibre embedded in an undeformed fragmentation specimen [START_REF] Yallee | Single fibre composite micromechanics for alpha-alumina and carbon/epoxy systems[END_REF]. The fibre strain distribution is outside the limits of the scatter and hence the strain gradient which corresponds to a frictional shear stress of between 6 and 9 MPa (derived from the solid line fitted to the data) is real, indicating interfacial failure must have occurred during the matrix cracking process. The fibre around the crack plane is in compression probably as a result of reverse sliding caused by fibre recoil following fracture.

M5/epoxy

Figure 7 shows the micromechanical strain distribution along a broken M5 (PIPD) fibre after matrix crack propagation, at COD = 13 m. The strain distribution, indicates that the fibre is subjected to residual strain caused by the matrix cracking over a distance of approximately 0.2 mm. For the particular fibre chosen, the strain profile is slightly asymmetric.

As for carbon, a point with error bars, has been plotted in figure 7 to indicate the scatter associated with an undeformed fibre in an undeformed matrix. The error bars are based on the scatter shown in for an M5 fibre embedded in an undeformed fragmentation sample [START_REF] So | Compressive behaviour of rigid rod polymer fibres and their adhesion to composite matrices[END_REF]. It can be seen that the fibre strain distribution, as with T50u (above), is outside the limits of the scatter and hence represents interfacial deformation. Thus the fibre strain gradient which corresponds to a frictional shear stress of between 10 and 17 MPa (derived from the solid line fitted to the data) is real, indicating interfacial failure must have occurred during the cracking process. Again, the fibre around the crack plane is in compression probably as a result of reverse sliding caused by fibre recoil.

Fibre recoil

The evidence from the fibre strain distributions from the T50u and M5 systems shown in figures 6 and 7 respectively, indicates that some interfacial failure has occurred. The nature of the failure, however, could not be determined as it was not possible to reload the fibres by crack opening. It can be seen that the fibre in both cases is in compression along the failed region. One possible reason for the compression could be fibre recoil along the failed region.

The proposed fibre recoil process depicted schematically in figure 8, is envisaged to proceed as follows; (a) the fibre is stretched across an opening matrix crack, (b) just after fibre fracture there is excess strain energy stored in the fibre, (c) the fibre overshoots during the recovery of the excess strain energy and is pinned or locked in place by the frictional constraints of the matrix.

SEM Investigations

Twaron/epoxy

Generally for Twaron fibres it was found that failure occurred close to the crack plane. In isolated cases, however, it has been possible to observe fibre failure away from the crack plane. This has made it possible to examine the state of the fibre surface and has help to gain some insight into the interfacial failure mechanisms occurring. Figures 9a,b show SEM micrographs of a broken crack-bridging Twaron fibre which has fractured some distance inside the matrix (away from the crack plane) and the matrix cavity from which it has been extracted. Examination of the fibre surface, figure 9a, reveals a smooth surface similar to that of the as-received Twaron fibre [START_REF] Kelly | Strong Solids[END_REF]. There is however, some evidence of fibre surface damage and it can be seen that some skin has been peeled from the fibre. There is also evidence of defibrillation of the fibre skin. These observations are supported by the corresponding micrograph of the cavity, figure 9b, where a sliver of skin can be seen protruding from the hole along with the fibrillated skin.

Zylon/epoxy

In the case of Zylon fibres significant lengths of fibre were pulled from the matrix during fracture. as-received fibre) [START_REF] Kelly | Strong Solids[END_REF].

T50u/epoxy

For the T50u fibre failure occurred at, or in the immediate vicinity of the crack plane. This is illustrated by figures 9e,f, which show SEM micrographs of a failed fibre end at the point of exit from the crack face. Here it can be seen, figure 9c, that a very small length of fibre protrudes from the matrix, indicating it has broken away from the crack plane and supporting our earlier observation that limited fibre matrix debonding has occurred. The length of the portion of protruding fibre can be seen to be very short, probably less than one fibre diameter. This is complemented by figure 9f, which shows a very shallow matrix cavity from which the short length of fibre has been extracted. The complementary fracture surface of the broken fibre can be seen inside the matrix cavity. Examination of both the fibre protruding from the crack face and the fibre in the matrix cavity at the other crack face shows the fibre to have maintained the integrity of its skin and core.

M5/epoxy

For M5 (PIPD) fibres, complete fibre failure occurred at/or around the crack plane. This is illustrated by figures 9g,h, which shows SEM micrographs of a failed fibre end protruding from the matrix and the cavity from which it has been extracted, respectively.

Figure 9g shows a short length of ragged fibre extending from the matrix indicating that a certain amount of defibrillation has occurred. The length of the portion of protruding fibre can be seen to be fairly short, probably no more than a few fibre diameters. In figure 9h some short lengths of fibre skin can be seen protruding from the matrix cavity.

Crack-tails

The 'crack-tail' takes the form of a mark or scar on the surface of the matrix crack face. In figures 9c-h, the micrographs for Zylon, T50u and M5 fibres show a tail running away from the fibre in the direction of crack propagation, as indicated. This phenomenon [START_REF] Kinloch | Fracture Behaviour of Polymers[END_REF], indicates the matrix crack has travelled around the inclusion (fibre), hence at some point the fibre must have bridged the crack.

The mechanism which is depicted schematically in figure 10 proceeds as follows; (a) the front of a propagating matrix crack approaches an inclusion (in this case a fibre), (b) the crack is split by the inclusion and slowed in the area around it, (c) as it passes the inclusion the crack front joints up again on different levels, (d) a tail is formed running way from the fibre in the direction of the propagating crack.

Discussion

For the fibre/matrix systems studied here both fibre/matrix interfacial failure and fibre fracture have been observed. While the type of fibre used has been varied, the resin matrix has been kept constant throughout and been prepared in an identical manner for all specimens.

Twaron/epoxy

Figure 11 shows the plot of f * versus E f with the boundary between fibre fracture and interfacial debonding drawn for the Twaron/epoxy system. A limiting value of max ~ 40 MPa was used. The data point for Twaron falls above the theoretical (solid) line indicating that when a matrix crack impinges on the fibres, the interface should fail with intact fibres bridging the faces of the crack.

However, once failure has been initiated and debonding has begun to propagate then the partial-debonding conditions must be applied. It can be seen that this moves the critical condition (dashed line) in favour of fibre fracture. The partial debonding conditions applied include a debond length of 1.2 mm. The magnitude of interfacial frictional shear stress values measured for crack bridging Twaron fibres ranges between 0-10 MPa, therefore a value of 5

MPa was chosen for the purposes of this model. For these conditions it can be seen that the data point is fairly close to the boundary, indicating that debonding > 1.2 mm will be limited by a critical build up of stress in the debonded zone and will cause fibre fracture to occur across the crack plane.

Zylon/epoxy

Figure 12 shows a comparison between the tensile strength and modulus components of a Zylon fibre and the theoretical boundary conditions for this system. The plot of f * versus 12, with the boundary between fibre fracture and interfacial debonding shows that the data point for Zylon falls well above the theoretical line indicating when a matrix crack impinges on the fibres, the interface should fail with intact fibres bridging the faces of the crack.

E f 1/2 , figure
It can be seen in figure 12 that when typical partial debonding conditions for this fibre are applied, at a debond length of 3 mm and an interfacial frictional shear stress of 3 MPa, that the model predicts the Zylon fibre remains intact bridging the faces of the crack. The point for Zylon still lies a considerable distance above the line. This can be attributed to its exceptional strength (4.9 GPa). Thus it is inferred from the model that it would take a considerable increase in the interface strength, the debond length or the interfacial frictional shear stress to cause fibre fracture, which did not occur in this case.

T50u/epoxy

Figure 13 shows the plot of f * versus E f 1/2 , with the boundary between fibre fracture and interfacial debonding drawn for T50u carbon fibres embedded in the epoxy resin matrix.

Stress transfer information for this system was not directly measurable by crack-bridging experiments because stable bridging did not occur, therefore data from fragmentation experiments using Raman spectroscopy [START_REF] Yallee | Single fibre composite micromechanics for alpha-alumina and carbon/epoxy systems[END_REF] were used. The maximum interfacial strength of the system was estimated to be ~27 GPa. The modulus, E f , of T50u has been measured at 324 GPa and its tensile strength at 2.27 GPa. It can be seen for the initiation of interfacial failure the data point for T50u falls just above the theoretical curve (solid line) indicating that when a matrix crack impinges on the fibres, the interface is likely to fail with intact fibres bridging the faces of the crack. Raman experiments on the T50u/epoxy system confirmed that some limited interfacial failure does occur over an approximate distance of 0.35 mm, however the fibres were also found to have fractured.

An explanation for the observation of both interfacial failure and fibre fracture can be arrived at by considering the partial-debonding aspect of the strength criterion. When the partial debonding conditions, figure 13, are applied (dashed line) for a debond length of 0.35 mm, typical for this system, it can be seen that there is good correlation between the data point for carbon fibre and the boundary conditions. Two factors are critical in the correlation between the actual data of the theoretical boundary conditions. Firstly, the interfacial frictional shear stress, frict = 8 MPa, which was the maximum value observed by fragmentation studies [START_REF] So | Compressive behaviour of rigid rod polymer fibres and their adhesion to composite matrices[END_REF], is higher than the values used for Twaron and Zylon thus the rate of build up of fibre stress by transfer along the frictional interface is greater. Secondly the fibre radius, r = 3.6 m, is significantly smaller than those of Twaron and Zylon. It can be seen from figure 3d that this has a significant effect on the critical conditions for debonding/fibre fracture.

M5/epoxy

Figure 14 shows the plot of f * versus E f 1/2 , marking the critical condition between fibre fracture and interfacial debonding drawn for a M5/epoxy, fibre/matrix system with max as the shear yield stress of 43 MPa. Due to fibre fracture, relevant stress transfer data for M5

was not obtainable directly from crack-bridging experiments and was obtained from fragmentation data obtained in an earlier study [START_REF] Kinloch | Fracture Behaviour of Polymers[END_REF]. The data point for M5 is above the theoretical curve (solid line) indicating when a matrix crack impinges on the fibres, the interface is likely to fail with intact fibres bridging the faces of the crack. In common with T50u fibres, Raman experiments on the M5/epoxy system confirm, even though the fibres were also found to have fractured, that some limited interfacial failure does occur over an approximate distance of 0.3 mm.

It can be seen that although the system easily satisfies the condition for initial debonding that when the partial debonding conditions, figure 11, are introduced (dashed line) using a typical debond length of 0.3 mm and frict = 25 MPa [START_REF] Kinloch | Fracture Behaviour of Polymers[END_REF] there is good correlation between the model and the data. The relatively high interfacial frictional shear stress appears to be a critical factor, causing the stress in the fibre to build up to the fibre failure stress over a very short debond length.

Fibre/interface properties and crack bridging

Table 3 presents a summary of the various properties relating to the fibre, the interface and the crack bridging behaviour of the various fibre/matrix systems studied. It is indicated, which fibres form a stable bridge across the matrix crack (i.e. without fibre fracture) with significant interfacial debonding and which fibres fracture with relatively little interfacial failure. From the model it is possible to make some observations about the crack bridging behaviour of the fibres in relation to the data presented in table 3.

Predictions from the strength criterion indicated that all the fibres studied were strong enough to remain intact such that interfacial debonding would occur as a result of matrix cracking. The results obtained from the crack bridging experiments showed that while Twaron fibres formed a stable bridge across the crack with interfacial debonding occurring over distances > 1 mm, T50u fibres which have a similar strength were found to fracture with very little interfacial debonding, ~ 0.35 mm. M5 fibres, which are substantially stronger than both Twaron and T50u also fractured with very little interfacial failure. Zylon fibres which have an exceptionally high strength (much greater than the other fibres) were shown to form a very stable bridge and were found to debond over distances > 2 mm.

In the case of T50u, which has the highest modulus of all the fibres, it can be seen from figure 13 that the higher the fibre modulus then the more likely it is that conditions will favour fibre fracture. Secondly T50u has a much smaller radius than the other fibres, hence when the debonding criterion is applied, decreasing the fibre radius will shift the critical condition in favour of fibre fracture.

The explanation for fibre fracture in the case of M5 is different from that of T50u.

The fragmentation experiments have shown [START_REF] Kinloch | Fracture Behaviour of Polymers[END_REF] that the M5/epoxy system has a comparatively high interfacial frictional shear stress after fibre matrix debonding, hence causing the stress in the fibre to build up to the fibre strength over a very short debond length.

Thus it appears that in this case the interfacial frictional shear stress is the critical factor.

Conclusions

The comparison between the modulus and tensile strength data for the various fibre matrix systems studied and the boundary conditions drawn according to the theoretical strength criterion correlate well with experimental data obtained from the Raman experiments. Since Twaron and Zylon are tough fibres that remain intact after matrix cracking, it is shown that after initial failure of the fibre/matrix interface, considerable debond propagation is possible. In the cases of T50u and M5 fibres, however, although initial debonding is possible, the fibre properties and the interfacial failure characteristics limit debonding to much smaller lengths.

The effects of frictional stress transfer in the debonded region and fibre radius are very important in determining the rate of build up of stress in the fibre and thus will be crucial in determination of a maximum possible debond length, limiting interfacial failure before fibre fracture. As the level of frictional shear stress, frict , increases then the debond length decreases before fibre fracture occurs. When all of the other material parameters are kept constant, a greater toughening effect is to be had by increasing the fibre strength.

Conversely this effect is reversed by an increase in fibre modulus.

Although this present study has been confined to model single fibre composite systems, it has important implications for the behaviour of fibre-reinforced composites in general. It is anticipated that the similar factors such as the balance between interfacial shear strength, fibre strength and the physical properties of the component materials will control the fracture characteristics and toughening behaviour of high volume fraction composite systems.

  equation 5 gives a relationship between the strength of the fibre and the interfacial shear stress such that if n the fibre will fracture. Similarly if interfacial debonding/yielding is considered to occur when the interfacial shear stress reaches its shear strength max equation 8 into equation 5 a boundary condition for fibre/matrix interfacial failure can be established whereby,

  figure 3. The COD was recorded as the distance measured (at the sample surface) between

region and equation 1

 1 to model the shear-lag exponential decay in the elastically bonded region. Using the fitting parameters listed in the figure caption it is possible to attain good agreement between the theoretical and experimental data in the bonded region. In the debonded regions the gradient of the theoretical fit was optimised by using equation 12 and adjusting the interfacial frictional shear stress, frict ; the frictional shear stress values used can be found in figure4b(as described below).The interfacial shear stress, , along the fibre was calculated by fitting the partial debonding model to the experimental data and subsequently substituting the derivative of the model curve, de f /dx, into equation 17. The variation of , with x is presented in figure 4b.

  Figure 9c & d, shows FEGSEM micrographs of a broken crack-bridging fibre protruding from the crack face and the matrix cavity on the opposing crack face, from which it was extracted. Ultimate failure of the fibre has occurred within the matrix, resulting in a length of fibre being pulled out. Examination of the fibre at the point of exit from the matrix, figure 9c, shows that the fibre skin is ravelled up or kinked and that a strip or strips of skin have been torn from the fibre. Conversely, part of the surface of the fibre skin has remained relatively 'undamaged' and appears to be very smooth (not much different from an

Figure 1

 1 Figure 1 Dependence of fibre fracture/interfacial debonding upon the relationship between fibre strength, f * , and fibre modulus E f with (a) variation in interfacial shear strength, max , and (b) variation in matrix modulus E m . Except where given, the relevant physical properties used were as follows: E m = 3 GPa, max = 40 GPa, m = 0.35, n = 0.1.

Figure 2

 2 Figure 2 Dependence of fibre fracture/interfacial debonding upon the relationship between fibre strength, f * , and fibre modulus E f with (a) increasing debond length, L d , and (b) variation in interfacial frictional shear stress, frict , in the failed/debonded region. Except where given, the relevant physical properties used were as follows: E m = 3 GPa, max = 40 GPa, m = 0.35, n = 0.1, L d = 0.2 mm, r = 6.25 m, frict = 8 MPa.

Figure 3 Figure 4 Figure 5 Figure 6

 3456 Figure 3 Schematic representation of a crack-bridging fibre in a CCT specimen

Figure 7 Figure 8

 78 Figure 7 Distribution of (a) fibre strain, e f , and (b) distribution of interfacial shear stress, , along a broken, crack-bridging M5 fibre in an epoxy resin matrix

Figure 9 SEM

 9 Figure 9 SEM Micrographs showing corresponding pairs of fractured (a,b) Twaron, (c,d) Zylon, (e,f) T50u and (g,h) M5 fibres at the matrix crack plane.

Figure 10

 10 Figure 10 Schematic illustration of a crack front (a) approaching a fibre, (b) splitting and slowing, (c) rejoining and (d) forming a tail.

Figure 11

 11 Figure 11 Comparison of the relationship between fracture strength, f * and fibre modulus, E f , with the boundary conditions for fibre fracture/interfacial failure in an Twaron fibre/epoxy composite material.

Figure 12

 12 Figure 12 Comparison of the relationship between fracture strength, f * and fibre modulus, E f , with the boundary conditions for fibre fracture/interfacial failure in a Zylon fibre/epoxy composite material.

Figure 13

 13 Figure[START_REF] Nayfeh | Thermodynamically induced interfacial stresses in fibrous composites[END_REF] Comparison of the relationship between fracture strength, f * and fibre modulus, E f , with the boundary conditions for fibre fracture/interfacial failure in a T50u-carbon fibre/epoxy composite material.

Figure 14

 14 Figure 14 Comparison of the relationship between fracture strength, f * and fibre modulus, E f , with the boundary conditions for fibre fracture/interfacial failure in a M5-PIPD fibre/epoxy composite material.

  Figure 1

  Figure 3

  Figure 5b

2 )Figure 14

 214 Figure 10

Table 1

 1 Mechanical properties of Twaron, Zylon, T50u and M5 fibres; determined from single fibre specimens of 50 mm gauge length.

	Fibre	E f (GPa)	f * (GPa)	e f * (%)	r ( m)
	Twaron	104 4	2.1 0.2	2 0.1	12.4 0.3
	Zylon	228 16	4.9 0.5	2.1 0.2	13.5 1
	T50u	320 26	2.3 0.4	0.7 0.1	7.2 0.3
	M5	304 17	3.7 0.3	1.6 0.2	13.4 0.8

Table 2

 2 Raman band shift rates for Twaron, Zylon, T50u and M5 fibres; obtained from single fibres in air[START_REF] Mccartney | New theoretical model of stress transfer between fibre and matrix in a uniaxially fibre-reinforced composite[END_REF].

	Fibre	Strain sensitive band		d /de f , (cm -1 )/%
				(cm -1 )
	Twaron		1610		3.7 0.3
	Zylon		1618		5.5 0.4
	T50u		1580		12.5 0.5
	M5			1615		15.3 0.3
		Table 3 Fibre, interfacial shear stress and crack bridging
			properties of Twaron, Zylon, T50u and M5 fibres
	Fibre	E f	f *	max	f	r	L d	Stable
	(GPa)	(GPa)	(MPa)	(MPa)	( m)	(mm)	bridge
	Twaron	104	2.1	33	0-10	6.2	1.2
	Zylon	228	4.9	33	0-5	6.75	> 2
	T50u	325	2.3	27	8	3.6	0.35
	M5	304	3.7	40	20	6.7	0.3

Acknowledgments

The authors would like to extend thanks to the EPSRC who funded this work (Grant number EP/C002164/1).

Fully bonded region

Fully bonded region

Distance along fibre, x (µm)