Vincent Robert
email: vrobert@loria.fr

Yves Laprie

Anne Bonneau

A phonetic concatenative approach of labial coarticulation

Predicting the effects of labial coarticulation is an important aspect with a view to developing an artificial talking head. This paper describes a concatenation approach that uses sigmoids to represent the evolution of labial parameters. Labial parameters considered are lip aperture, protrusion, stretching and jaw aperture. A first formal algorithm determines the relevant transitions, i.e. those corresponding to phonemes imposing constraints on one of the labial parameters. Then relevant transitions are either retrieved or interpolated from a set of reference sigmoids which have been trained on a speaker specific corpus. This labial corpus is made up of isolated vowels, CV, VCV, VCCV and 100 sentences. A final stage consists in improving the overall syntagmatic consistency of the concatenation.

Introduction

One of the most crucial aspects of the development of a talking head is the synthesis of labial movements from the knowledge of the sequence of phonemes to articulate. A good modelling of coarticulation phenomena is important to make lip reading possible.

A first approach consists in realizing the visual synthesis by manipulating video images [START_REF] Bregler | Video rewrite: Driving visual speech with audio[END_REF][START_REF] Brooke | Two-and three-dimensional audio-visual speech synthesis[END_REF][START_REF] Ezzat | Trainable videore-alistic speech animation[END_REF]. Reference images correspond to monophones, diphones or triphones. In the first two cases, the effects of coarticulation cannot be captured and then rendered during synthesis. In the case of triphones, on the one hand the corpus necessary becomes huge to cover all the existing triphones, and on the other hand, more complex coarticulation effects involved in VCCV or VCCCV cannot be captured correctly. Moreover, these methods are often available for 2D images even if there are solutions to reconstruct pseudo 3D images, and they provide a talking head linked to a given speaker only.

Our objective is closer to works concerning 3D artificial talking heads [START_REF] Beskow | Rule-based visual speech synthesis[END_REF][START_REF] Cohen | Modeling coarticulation in synthetic visual speech[END_REF][START_REF] Pelachaud | Visual text-to-speech[END_REF][START_REF] Reveret | Mother: A new generation of talking heads providing a flexible articulatory control for video-realistic speech animation[END_REF]. These works comprise the determination of control parameters as a function of the phone sequence to pronounce, as well the reconstruction of the head as a function of these parameters. This paper only concerns the determination of the labial control parameters. Previous phonetic results, as well as more recent studies (that of Meada [START_REF] Maeda | Functional modeling of the face during speech production[END_REF] on two speakers, and our investigations on a ten speaker labial database [START_REF] Robert | Inter speaker variability of labial coarticulation with the view of developing a formal coarticulation model for french[END_REF] by means of a principal component analysis) show that the main modes of labial deformations are lip protrusion, lip stretching, and jaw opening.

The model proposed by Cohen & Massaro [START_REF] Cohen | Modeling coarticulation in synthetic visual speech[END_REF] relies on the gesture theory of Löfqvist [START_REF] Löfqvist | Speech as audible gestures[END_REF] which associates one target vector to each segment. Dominance functions are used to weight target values as a function of time. Dominance functions are made up of two exponential terms, one for anticipation and the other for retention. As underlined by Beskow [START_REF] Beskow | Trainable articulatory control models for visual speech synthesis[END_REF] Cohen & Massaro's model can be considered as a "time locked" model since the duration of these exponentials are context independent. The "time locked" model stipulates that the gesture starts at a constant period of time before the segment considered. This model is not well adapted to the bilabials /p,b,m/ and labial fricatives /f,v/ because dominance functions cannot capture the resistance to coarticulation. Indeed, the complete closure of lips cannot be reached through the mixture of dominance functions. Cosi [START_REF] Cosi | Labial coarticulation modeling for realistic facial animation[END_REF] thus refined Cohen & Massaro's model by adding a resistance term that forces labial parameters when needed. However, as stressed by Reveret [START_REF] Reveret | Mother: A new generation of talking heads providing a flexible articulatory control for video-realistic speech animation[END_REF] coarticulation cannot be easily rendered by the overlapping of several gestures weighted by dominance functions. He thus proposed a model based on Öhman's theory [START_REF] Öhman | Numerical model of coarticulation[END_REF] designed to model tongue movements. According to this theory tongue movements are piloted by vowels, to which consonant gestures are superimposed. This theory belongs to the family of the "look-ahead" models because coarticulation can start as soon as possible provided that there is no antagonism. Beskow [START_REF] Beskow | Trainable articulatory control models for visual speech synthesis[END_REF] compared Cohen & Massaro and Öhman models on the same corpus and found out that Cohen & Massaro model performs better if the evaluation focuses on the deviation between predicted and real data. On the other hand, perception experiments realized with a sample of 25 subjects showed the inverse result. One explanation is that the perceptive weights of labial features are not equivalent and also depend on the phonetic context. For instance, the perceptive consequence of an erroneous lip aperture prediction for /t/ is probably less sensitive than for /p/. This explains that a rule based algorithm performs slightly better because the key points of labial coarticulation are explicitly given, even if the general result is slightly less realistic. One of these algorithms [START_REF] Beskow | Rule-based visual speech synthesis[END_REF] uses target values specific to every phonemes except some, which are not given any value so that labial coarticulation can be determined by neighbour phonemes. Keating [START_REF] Keating | The window model of coarticulation[END_REF] proposed a similar approach by defining the degree of freedom for coarticulation in the form of an authorized window. In some sense, these windows correspond to the definition of resistance to coarticulation [START_REF] Bladon | One stage and twostage temporal patterns of velar coarticulation[END_REF].

Our approach

Taking into account these results, we accepted a rule based method where rules are automatically learnt from a corpus. One expected advantage is the possibility of controlling the dynamics of each parameter, and more importantly the control strategy itself, at the phoneme level.

In a previous work [START_REF] Robert | Inter speaker variability of labial coarticulation with the view of developing a formal coarticulation model for french[END_REF] conducted on a corpus of /iCy/ articulated in a carrier sentence, and where C is a non labial consonant, we showed that there exists a wide range of anticipation strategies according to the speaker. Figure 1 shows that some speakers, like Bl, present a strategy close to a time locked model whereas others, like Od, present a strategy closer to the look-ahead model or even to the time expansionist model [START_REF] Abry | Le MEM: un modèle d'anticipation paramétrable par locuteur: Données sur l'arrondissement en franc ¸ais[END_REF] which stipulates that the gesture duration can be substantially lengthened but not shortened. Given this wide range of coarticulation dynamics we retained sigmoid curves to represent the time evolution of labial parameters (Equation 1). Vi and V f represent initial and final values of the sigmoid. c monitors the speed transition and t0 corresponds to the time of the sigmoid center. Indeed, as shown in Figure 2, sigmoid curves can easily accommodate for very different time evolution profiles: in this example, a value of 0.05 for c corresponds to the "look-ahead" model, whereas a value of c greater than 0.10 together with an appropriate choice of t0 gives rise to a "time-locked" shape. It is also possible to approximate the "expansionist" model. In addition, this kind of curve easily enables the speech rate to be changed and the hyperarticulation to be simulated by modifying Vi and Vt. Note that the curves generated by Cohen & Massaro's model are close to sigmoids.

Sig(t) = Vi + (V f -Vi) 1 + e -c.(t-t 0 ) (1)
We thus built a set a reference sigmoids from the corpus recorded by an experienced talker being used to speak to hard of hearing children, and thus probably easily lip readable. This corpus dedicated to French is made of all the vowels, CV sequences, systematic VCV for a reduced set of consonants and vowels, the most frequent VCCV in French, and 100 phonetically balanced sentences [START_REF] Combescure | 20 listes de dix phrases phonétiquement équilibrées[END_REF]. This corpus was recorded by using a stereovision system [START_REF] Wrobel-Dautcourt | A low-cost stereovision based system for acquisition of visible articulatory data[END_REF] that enables the 3D tracking of markers painted onto the speaker's face at the rate of 120 Hz and the recording of the corresponding speech signal. Protrusion, stretching, lip and jaw aperture deformation modes have been determined by calculating the geometrical deviation of well chosen points with respect to their average location (see [START_REF] Wrobel-Dautcourt | A low-cost stereovision based system for acquisition of visible articulatory data[END_REF] for further details). The time evolution of these parameters has been slightly smoothed by means of regularizing splines. Only relevant sigmoids, i.e. those corresponding to characteristic dynamics of labial parameters, have been estimated. For instance, unlike /p/ which imposes a very clear time evolution for lip aperture, /k/ does not impose any clear constraint on this parameter, and consequently no corresponding sigmoid was defined. A previous study [START_REF] Robert | Inter speaker variability of labial coarticulation with the view of developing a formal coarticulation model for french[END_REF] has been dedicated to the design of a coarticulation algorithm that finds out which are the relevant labial parameters from a formal description and from the quantification of phonemes in terms of labial parameters. This algorithm also exploits interdependencies between labial parameters (for instance lip stretching decreases when protrusion increases).

Once the algorithm has been applied to a sequence of phonemes, a labial parameter may remain indefinite over a sub-sequence because corresponding phonemes are neutral for this parameter. For instance, protrusion is indefinite for the phoneme /k/ of the sequence /iky/ and only one sigmoid, that for the transition /iy/, will be considered. Unlike other algorithms the mutual influence between labial parameters is taken into account.

Training and reconstruction process

The whole corpus comprises isolated vowels, systematic CV and VCV for /i,a,u,y/, 70 VCCV or VCCCV, and 100 phonetically balanced sentences. All the logatoms and 70 sentences, i.e. 85% of the corpus recorded, have been used to extract sigmoid parameters as well as minimum, maximum and mean values of labial parameters for every phoneme. Firstly, standard values of opening, stretching and protrusion parameters for vowels articulated by our subject were determined from the isolated vowels recorded. Then, the formal coarticulation algorithm was applied to find out which are the relevant sigmoids that need to be learnt for the CV, VCV, VCCV sequences and 70 phonetically balanced sentences of the training corpus.

The remaining corpus, i.e. 30 phonetically balanced sentences, was used to evaluate quality of the coarticulation synthesized.

The synthesis comprises the following stages. The first consists in segmenting the sentence to be synthesized into consecutive overlapping CV, VCV, VCCV sequences. These sequences are then recovered from the training corpus if they have been recorded, or reconstructed from existing sequences otherwise.

Completion of missing phonemes

Completion of missing vowels

Since it was not possible to record all the V CV , V CCV , V CCCV encountered in French, only /a, i, u, y/ vowels were considered. Other sequences are thus linearly interpolated from /a, i, u, y/ sequences. Weights are the barycentric coordinates of the unknown vowel with respect to /a, i, u, y/ . For example, rebuilding the sequence /εtu/ requires the calculation of the barycentric coefficients αi for /ε/ compared to /a, i, u, y/ using data obtained during the recording of the isolated vowels. So, sigmoid parameters are estimated by α1.f

([atu]) + α2.f ([itu]) + α3.f ([utu]) + α4.f ([ytu]).
In the case of a V1C...CV2 sequence where neither of the two vowels belongs to /a, i, u, y/, /aC..CV2/, /iC..CV2/, /uC..CV2/, /yC..CV2/ are first reconstructed by using the technique exposed above. Then , V1C...V2 is linearly interpolated from /aC..CV2/, /iC..CV2/, /uC..CV2/, /yC..CV2/.

Completion of missing consonants

All the CV and V CV with V in /a, i, u, y/ were recorded, but only 90 V CCV were retained to limit the size of the corpus. They were selected from a statistical analysis of a large French corpus made up to maximize the phonetic coverage of French [START_REF] Lamel | BREF, a Large Vocabulary Spoken Corpus for French[END_REF].

V C1C2V not in the training corpus are rebuilt by overlapping V C1V and V C2V sequences. This technique separates the role of vowels and consonants in agreement with Öhman's Theory [START_REF] Öhman | Numerical model of coarticulation[END_REF] which stipulates that the time varying shape of the vocal tract gradually changes from V1 to V2, onto which a consonant gesture is superimposed.

The sequence /ikty/, for instance, does not belong to the training corpus, and is thus reconstructed by using /iky/ and /ity/ sequences which, on the contrary, belong to the corpus. Even if protrusion increases from /i/ to /y/ in both sequences, /ity/ and /iky/ present different protrusion profiles. Fig. 3 shows how this VCCV is reconstructed from these two sequences.

Temporal adaptation

Durations of phonemes recorded in the corpus are not necessarily identical to their target values in the sentence to be synthesized. It is thus necessary to adapt sigmoid durations. For each sigmoid, the relative position of the sigmoid center (t0) is kept invariant according to the end of the first phoneme and the onset of the last phoneme of the sequence approximated by this sigmoid.

Amplitude adaptation

The reconstruction of labial parameters of a sentence depends on sigmoids concatenated. It is important to guarantee the syntagmatic coherence, i.e. to keep distinctive contrasts between the sounds of the sentence to be synthesized, as well as paradigmatic coherence, i.e. the intrinsic characteristics of the sounds. During the reconstruction process, it is thus necessary to adjust sigmoids to one another. For that purpose, a multidimensional minimization method (Powell type) is applied to determine the optimal shift to apply to each sigmoid so that the deviation to the neighbours is minimal (syntagmatic axis). In parallel, in order to take into account the paradigmatic axis, minimization integrates the deviation of vowel labial parameters with respect to their values for the corresponding isolated vowel. Equation 2details the expression to be minimized where Sigi(tmin) and Sigi(tmax) are the initial and final values of the sigmoid number i. The fist part of the equation represents the difference between the final value of one sigmoid and the average of the onset values of the sigmoids immediately following it (the starting phoneme of these sigmoids thus corresponds to the final phoneme of the current sigmoid). It thus corresponds to the syntagmatic axis. The second and third terms of the equation are the differences between sigmoid extremities (corresponding to vowels) and the mean values of isolated vowels. These terms are intended to capture the paradigmatic axis.

We only apply this minimization to phonemes having a certain degree of resistance to the coarticulation, i.e. those presenting characteristic values for labial parameters. αi are factors weighting the paradigmatic axis versus the syntagmatic axis and are phoneme dependent. α2 is intended to represent the paradigmatic axis and is given a non zero value only for vowels. After these corrections, a spline curve is generated to approximate parameters trajectories as shown by Figure 4.

Concluding remarks

Two measurements are used to provide an objective evaluation of our results: RMSE (root mean squared error) and correlation between observed and predicted data. RMSE values were calculated as a percentage of the full range of the target parameter. RMSE found on the test corpus is 11.66% and correlation 0.67, i.e. slightly not as good as those found in [START_REF] Beskow | Trainable articulatory control models for visual speech synthesis[END_REF] by comparing several coarticulation methods. However, a further examination of the results shows that our algorithm, unlike other coarticulation algorithms, primarily focuses on phonetically relevant features. For example, curves presented in [START_REF] Beskow | Trainable articulatory control models for visual speech synthesis[END_REF] show that the jaw opening is almost always fairly underestimated for open vowels. On the contrary, in our case, the first step of the sigmoid construction searches for the relevant labial parameters. This means that the opening of open vowels, or any other labial salient characteristic, will receive a specific attention during the training. This thus guarantees that the characteristic labial parameters are well rendered by our approach. On the other hand, this is achieved to the detriment of the overall fitting between predicted and observed labial profiles. A perceptive experiment will thus be conducted in a near future to evaluate the benefit of preserving the most salient labial characteristics.

The other very strong point of our approach is that syntagmatic and paradigmatic axes are explicitly taken into account through an optimization stage covering the whole sentence to be synthesized. This allows the most salient labial values to be preserved while removing the shifts between sigmoids concatenated, which originate in the discrepancies between phonetic and prosodic contexts encountered in the training corpus.

Figure 1 :

 1 Figure 1: Comparaison between two coarticulation stategies.

Figure 2 :

 2 Figure 2: Influence of parameter c on the shape of sigmoids.

Figure 3 :

 3 Figure 3: A VCCV reconstruction of the stretching profile of /ikty/ using consonant completion

2 )Figure 4 :

 24 Figure 4: Example of an amplitude adaptation for protrusion

Figure 5 :

 5 Figure 5: Comparison between real data and simulated data for protrusion (The sequence is "Des lièvres jouent à l'orée du bois")

August 27-31, Antwerp, Belgium


Emmanuelle De Gentili

Fabrice Bernardi
email: bernardi@univ-corse.fr

Jean-François Santucci
email: santucci@univ-corse.fr

DEVS-Based Modeling and Simulation of the CORBA Portable Object Adapter

Keywords: Modeling, Simulation, DEVS, CORBA, POA

We present in this paper our approach for the modeling and the simulation of distributed architectures based upon Common Object Request Broker Architecture (CORBA). We point out a global methodology for the modeling of algorithmic functions using a DEVS-based approach, and we apply it to a function of the CORBA Portable Object Adapter: find_POA(). We will expose the methodology we defined, which allows the taking into account of all the computer control structures, and then will introduce three types of ports allowing us to model the internal function sequencement. The originality of this paper lies in the fact that we use DEVS theory in order to simulate a Software Object. The modeling is started from the source code of the function and the simulation is validated performing the verification of the followed path.

Introduction

Nowadays, more and more applications are build in order to be executed over a network. These applications are called "distributed applications", and are based upon a distributed architecture allowing the dialog between objects over such a network. The CORBA Architecture (Common Object Request Broker Architecture) is one the most important distributed architecture used. This technology provides a flexible, extensible, platform-independent and language-independent framework that is suitable for large-scale deployment. It uses standard CORBA Interface Definition Language based Application Programming Interfaces and CORBA Object Request Broker (ORB) to provide the necessary location transparency and language independence (see [START_REF] Acreman | Développer avec CORBA en Java et C++[END_REF]). These CORBA components allow running distributed applications on different platforms and written in different languages just by writing Object-Oriented Applications.

In order to study the behaviour of a CORBA-based distributed application, and more precisely of its objects over the network, we initiate an original approach based on the discrete event modeling and simulation. This approach will allow us to simulate the behaviour of objects involved in the CORBA Architecture. We presents in details in this paper the modeling of one of the functions of the Portable Object Adapter (POA) which is a fundamental of CORBA (You can note that all the other functions have been modelled too, but we can not present them here). The modeling of the behaviour has been performed using the DEVS Formalism (Discrete EVent Specification) starting from the source code written in C++. The first part of the paper presents the fundamentals of the DEVS modeling and simulation. The second part presents the global methodology we developed for the modeling of algorithmic functions. This methodology is illustrated with a little example. Then, we will present very briefly the CORBA Portable Object Adapter and the modeling and the simulation of one of its functions: the find_POA() function. The two last parts will be devoted to the presentation of the implementation of the simulation and to a conclusion and some perspectives of work.

Behavioral DEVS Modeling

The basic modeling approach is based upon the multiviews notion. This concept allows a gradual introduction of the complexity of a system by its analysis according to various points of view. Each model obtained represents a particular portion of the specifications of the global model. The two main kept views are the behavioural view and the structural view. In this paper, we will use only the behavioural view. The behavioural elements, as the structural components, are inter-connected so as to be able to exchange information through inputs-outputs sets. For our modeling approach using DEVS formalism (Discrete EVent System specification), they are of two types:

Atomic Models: They are the basic components of the behavioural view. They provide a local description of the dynamic behaviour of a system. They are described by the following elements: Starting from these definitions, we can perform a simulation of the modeled discrete-event elements using a simulation tree created automatically. Full concepts are provided in [START_REF] Aiello | Environnement Orienté Objet de Modélisation et de Simulation à Evènements Discrets de Systèmes Complexes[END_REF], [START_REF] Delhom | Modélisation et Simulation Orientées Objet, Contribution à l'Etude du Comportement Hydrologique d'un Bassin Versant[END_REF], [START_REF] Zeigler | Theory of Modeling and Simulation[END_REF], [START_REF] Zeigler | Object-Oriented Simulation with Hierarchical, Modular Models[END_REF] and [START_REF] Zeigler | Theory of Modeling and Simulation, 2 nd Edition[END_REF].

The next part of the paper presents how these concepts are used in order to find a global methodology for the modeling of algorithmic computer functions.

Global Methodoloy for Algorithmic Functions Modeling

Overview of the Methodology

Algorithmic functions can be seen as discrete-event systems, and then can be theorically modeled and simulated using DEVS approach. However, we found that a generic methodology can be applied, starting from the source code of the function. The basic approach is to consider algorithmic functions as coupled models, only composed by atomic models sequentially connected. We consider that an atomic model represents the modeling of the behaviour of the function between two calls to other external functions.

Internal variables of the function are binded to state variables inside an atomic model, and these variables are carried through specialized ports as long as the functions, and therefore the other atomic models, need them. This is viable since we start the modeling from the source code of the function.

For a better understanding, we introduce three types of ports allowing the connection between two or more models (Figure 3.1):

Sequencement Ports: these ports are atomic models ones, and are used to carry the state variables between two models. They can connect only atomic models inside a coupled model. Call Ports: these ports are used by atomic and coupled models. They carry the name of the function to be called. They connect two coupled models, therefore two functions.

Parameter Ports: these ports are used to carry the parameters needed by a function. They are similar to Call Ports, since they connect two functions and are used by the two kinds of models.

Input ports of the coupled models are constituted by the ports binded to the function parameters (Parameter Ports) and by a Call Port used to ensure that the function is really the one to be activated. Output ports are composed by a port binded to the return variable of the function and another dealing with a possible computer exception. Atomic models are more complicated since they must deal with the parameters of the function, but also with the sequencement of the state variables.

Control Structure Modeling

Most algorithmic functions use control structures such as "if...then...else" or the "for" statement.

the "if" statement (Figure 3.2): AM1 contains the test result as a state variable. The orientation will be performed using the output function of the Atomic Model. The "case" statement is similar. The "for" statement (Figure 3.3): AM1 contains the test result and the increment value as state variables. The "while" statement (Figure 3.4): AM1 contains the test value as state variable. The update is performed using the AM2 lambda function. The "do" statement (Figure 3.5): The code between "do" and "while" is executed at least at one time.

Then, the test result is placed as a state variable of AM2. AM1 declares an "a" state variable, initialises it and send it through a sequencement port. Following models will need the "b" variable, so it is sent too. AM2 contains the result of the test in line 3 and will orientate the variables towards AM3 or AM6. AM3 contains the result of the test in line 4 and will send an empty message towards AM4 or AM5 in order to continue the simulation, even if there are no variables to be passed.

CORBA Portable Object Adapter (POA) Modeling

The CORBA Portable Object Adapter

Figure 4.1 shows the significance of the Portable Object Adapter (POA) in a CORBA-based distributed architecture (see [START_REF] Geib | CORBA: Des concepts à la pratique[END_REF] and [START_REF] Omg | Corba 2.4, Specifications of the Portable Object Adapter[END_REF]). The CORBA architecture is based upon the Object Request Broker (ORB), which carry the requests from the client object to the server one. The interface between the client and the ORB is complex and we will note it ''Proxy''. The POA is the interface between the server objects and the ORB.

A POA is connected to many servants, which are object implementing interfaces of operations defined using the Interface Definition Language (IDL). In object-oriented languages (C++, java), servants are implemented using one or more objects. A client never interacts directly with a servant, but always through an object. Each servant is referenced thanks to an Object Identificator (ObjectId), identifying an object within the scope of its own Object Adapter. The mapping is made using an Active Object Map (AOM) maintained by each POA and providing the way to access the currently active objects using servants. Functionnalities of the POA can be described as follows (see [START_REF] Schmidt | Object Interconnexions[END_REF] and [START_REF] Schmidt | An Overview of the CORBA Portable Adapter[END_REF]):

Generating Object References: the POA is responsible for generating object references for the CORBA objects registered with it. Object references identify a CORBA object and contains adressing informations that allow clients to invoke operations on that object in a distributed system.

Activation and deactivation of servants: the POA can activate CORBA objects to handle client requests. Similarly, it can deactivate objects and destroy their corresponding servants when they are no longer needed.

Demultiplexing requests to servants: when the ORB receive a request from a client, it collaborates with the POA to ensure that the request reaches the proper servant. The POA parses the request to locate the ObjectId of the servant, which it uses to locate the correct servant and invoke the appropriate operation.

Invoking servant operations: the operation name is specified in the CORBA request. Once the POA locates the target servant, it dispatches the requested operation on it.

One of the main advantages of the POA is that it allows programmers to construct servant that are portable between different ORB implementations. Another is that it allows servants to assume complete responsability for an object's behavior.

A server application may create multiple POAs to support different kinds of CORBA objects. In this case, a POA Manager encapsulates the processing state of one or more of these POAs. When a request is received for a child POA that does not yet exists, the ORB will invoke an operation on an \Adapter Activator which will decide whether or not to create the required POA on demand.

In the next section, we present the modeling of one the numerous functions composing the OMG Portable Object Adapter: the find_POA() function.

Modeling of the find_POA() Function

Our main hypothesis for modeling is to consider that the POA is created and correctly supplied by the ORB. Global variables of the implementation can then be considered as state variables of the ORB coupled model.

We choose to present in this paper the find_POA() function. This function returns a pointer to a POA adapter name if it is a child of the target POA. If the target POA has no child of the specified name and activate is set to TRUE, find_POA() invokes the target POA's adapter activator, if one exists. The adapter activator attempts to restore POA adapter; if successful, find_POA() returns the specified POA object. If no POA is returned, the function raises the exception AdapterNonExistent.

The algorithmic/C-based form of find_POA() can be written as following (see [START_REF] Ooc | Orbacus for C++ and Java[END_REF] and [START_REF] Ooc | CORBA/C++ Programming with Orbacus, Student Workbook[END_REF] This function accepts two parameters (adapter and activate) and sends back a POA_ptr. Starting from this, we can build the Coupled Model ports: X=<adapter, activate, find_poa()>. "adapter", "activate" are parameter ports while "find_poa()" is a call port. To these ports, we add two new ports (poacontrol and children) that are used as global variables and which we consider as state variables of the ORB. Finally, the input ports of the Coupled Model find_POA() appear to be the following: X=<adapter, activate, children, poacontrol, find_poa()>. The output ports are very easy to define; we create a port for the return value and a port for a possible exception during the execution: Y=<Exception, POA_ptr>. The Coupled Model input ports are binded with the ports of the first met Atomic Model. Line 3 defines an "if" statement using the result of a getDestroyed() function as test. So, we build an atomic model (AM1) containing the test result as a state variable. The input ports are binded to the Coupled Model ones and the output ports are sequencements ports that will carry the state variables along the following Atomic Models. In order to evaluate the test, AM1 is binded to another Coupled Model called "getDestroyed" which will return a Boolean value. We add then two new ports to the find_POA() Coupled Model, one output call port towards getDestroyed() and one input parameter port for the returned value. Line 5 defines a Boolean "check". This is done in our modeling approach using an Atomic Model (AM2) that will have to pass this variable through its ports. AM3 is the Atomic Model corresponding to line 6 ("if" statement). It is linked to two other Atomic Models using sequencement ports (AM4 and AM6) and to another Coupled Model containsKey(), using two parameter ports and a call one.

Implementation of the Simulation

Implementation of the simulation based upon the modeling scheme presented before has been performed using a Java-based simulation engine. The simulation allowed us to compare the behaviour of the model to the behaviour of the simulated function. The validation is achieved by a complete path comparison, and results appeared to be very acceptable. We presented in this paper only the find_POA() function, but we performed the simulation for all the Portable Object Adapter methods, and all appeared very conclusive.

Conclusions and Perspectives of Work

This paper deals with one the first parts of our CORBA Portable Object Adapter modeling and simulation. We choose to present here the find_poa() function which is one the methods of the POA object. This function is a good example since it allows to show a great part of the generic methodology we defined.

The DEVS-based approach for modeling and simulation is usually applied to physical systems. We built a methodology allowing to use this approach for computer functions. This is very interesting for the simulation of distributed applications based upon the CORBA architecture which will be more and more used. Our first step is to model the whole CORBA architecture, in order to be able to model then the higher level applications and simulate them. Then, we will be able to add a physical network simulation tool in order to simulate the distributed execution of an application over a network. The modeling and the simulation are achieved starting from the source code of the functions to be simulated, using the methodology described previously. The main problem we meet is for the validation. Two ways are exploitable: the first one is to use a debugger software to follow the function running step by step and comparing the results to those obtained from the simulation. The second one is to determine all possible paths and to follow them. When all paths give good results, we can consider that the model of the function is validated. The main inconvenient of our approach can appear to be the building of big coupled models for a simple function. This is not a real problem in effect, since we use some graphical tools providing us a fast implementation of the whole models. Another point is that the atomic models are numerous, but also usually quite simple. We have a lot of perspectives of work. The first one is to achieve and validate the complete Portable Object Adapter model and the ORB one. Starting from these two basic elements of the CORBA architecture, we will be able to model and simulate distributed applications over a network using a commercial physical network simulation tool. However, the main objective of our work is based upon a collaboration with ALCATEL-Marcoussis, and will consist in the modeling and the simulation of new CORBA services.

Figure 3 . 1 :

 31 Figure 3.1: Illustration of the global methodology

Figure 3 . 2 :

 32 Figure 3.2: Illustration of the "if" Statement Modeling

Figure 3 . 3 :

 33 Figure 3.3: Illustration of the "for" Statement Modeling

Figure 3 . 4 :

 34 Figure 3.4: Illustration of the "while" Statement Modeling

Figure 3 . 5 :

 35 Figure 3.5: Illustration of the "do" Statement Modeling3.3. Simple ExampleFigure3.6 presents a very simple function and its modeling using our approach. The function is associated with a Coupled Model presenting two input ports (a parameter port and a call port) and an output parameter port. It can be modeled using six very simple Atomic Models.

Figure 3 . 6 :

 36 Figure 3.6: Modeling of a Simple Function

Figure 4 . 1 :

 41 Figure 4.1: Position of the POA in the CORBA Architecture

Figure 4 .

 4 Figure 4.2 gives the whole Coupled Model. It shows all the Atomic Models defined and the links to the external needed Coupled Models. The input and output ports of the main coupled model are supposed to be coupled with the whole ORB model.

Figure 4 . 2 :

 42 Figure 4.2: Modeling Scheme of the find_POA() Function


A Thionnet

J Renard

MODELLING OF FATIGUE BEHAVIOUR IN COMPOSITE LAMINATES

Keywords:

INTRODUCTION

The fatigue behavior of a composite laminate is first characterized by initiation and multiplication of the cracks rather than by their quasi-instantaneous growth. Cracks multiply during the cycles until reaching a saturation state. This phenomenon was first modeled by Reifsnider [START_REF] Reifsnider | Some fundamental Aspects of Fatigue and Fracture Response of Composite Materials[END_REF] and Garret and Bailey [START_REF] Garret | Multiple Transverse Fracture in 90° Cross-Ply Laminates of Glass/Fiber reinforced Polyester[END_REF]. In this approach to transverse cracking under fatigue loading, the cracks are assumed to have the same geometric characteristics as those occurring under quasi-static loading [START_REF] Thionnet | Laminated composites under fatigue loadings : a damage evolution law of transverse cracking[END_REF]. In particular, it is assumed that they go through the thickness of the plies and that their propagation is quasi-instantaneous.

WRITING OF THE FATIGUE EQUATION

The criticism that can be made of the conventional expression for the fatigue equations is that they include loading explicity through the mean or maximum stress, the amplitude, etc ... This remark applies both to metals and to composites. For instance, the evolution equation proposed by Beaumont [4] uses the amplitude of the external loading. Another equation, written by Ye [START_REF] Ye | On the Damage Accumulation and Material Degradation in Composite Materials[END_REF], uses the maximum longitudinal load applied to the edges of the structure. Such evolution equations are therefore very difficult to apply in the case of complex structures, when the loadings vary over time both in intensity and in point of application. If the behavior of the volume element depends on external factors, the equations do not have an intrinsic character. Also most of the cases are based on uniaxial tests providing a uniaxial model whose extension to the multiaxial case is not necessarily justified. A more logical approach is to formulate a multiaxial model developed and identified on tests which could be uniaxial.

We therefore directly construct an evolution equation from an energy formulation making it unnecessary to have a 3D generalization which is difficult to achieve. In addition, the aim is to take into account the fatigue effects as well as static effects. Thus, we attempt to write an evolution equation as follows dα = g(ε, α) dε + h(ε, α) dN, where ε denotes the strain state in the volume element when the loading reaches maximum value, N the number of cycles and α the scalar damage variable [START_REF] Thionnet | Laminated composites under fatigue loadings : a damage evolution law of transverse cracking[END_REF].

It is important to point out that the validity of the damage variable (α = crack density / thickness of the damaged ply) is experimentaly show in the case of fatigue loading. Thus, the intrinsic character of this variable is now demonstrate in the case of quasi-static loading [START_REF]Habilitation à diriger des recherches[END_REF] as well as in the case of fatigue loading [START_REF] Chambon | Endommagement de fatigue dans les composites stratifiés[END_REF].

COMPARISON BETWEEN EXPERIMENTAL RESULTS AND NUMERICAL SIMULATIONS FOR TWO LEVELS LOADING TESTS

We compare the numerical and experimental response of a (0°2, 90°4, 0°2) carbon/epoxy laminate submitted to two level loading tests : • first, a weak loading followed by a strong one (0.5 σ r during 1000 cycles followed by 0.8 σ r during 1000 cycles); • second, a strong loading followed by a weak one (0.7 σ r during 1000 cycles followed by 0.5 σ r during 1000 cycles).

In the first case (Fig. 1), we notice an increase of damage due to the application of the strong loading. In fact, the amount of damage at the end of the weak loading is less than the damage occured by the first cycle of the second loading level. In the second case (Fig. 2), we do not record an increase of the damage due to the second loading level. This is explained by the fact that the damage which is created by the first loading is higher than the damage which should be created by the first cycle or the the static loading at the level of the second one. In the both cases, the good correlation between numerical simulations and experiments.

CONCLUSION

For this approach of fatigue in laminated composites we arrived at a new form of the transverse cracking damage evolution equation. The originality of the work resides in three points : • first, in the evolution equation, which can take possible static loadings into account. This makes it very flexible for simulations as regards the diversity of possible applied loadings; • next, in the process by which the equation was established. The assumption that the defects created in a structure are similar for quasi-static loading and for cyclic loading makes it natural to extend the quasi-static model to the fatigue model; • finally, the parameters of the equation do not explicitly include the external loading applied to the structure. This makes the model intrinsic to the material and therefore applies to all types of loadings and structure.

Fig. 1 -Fig. 2 -

 12 Fig. 1 -Weak loading followed by strong loading


A Thionnet

J Renard

FIBER/MATRIX DEBONDING CRITERIONS IN SIC/TI COMPOSITE. NUMERICAL AND EXPERIMENTAL ANALYSIS

Keywords: metal matrix composite, debonding, modelling

INTRODUCTION

As SiC/Ti metal matrix composite have good mechanical properties, they have been chosen these last years for aeronautical applications. In order to increase their utilization, it becomes necessary to detect and analyze the propagation of the various types of damages which occur when the composite structure is loaded.

The linear elastic behaviour of these materials is correctly simulated by homogenization models. However, the dissipative phenomenon as well as the induced non-linearities are still badly understood. The properties and the behaviour of these composite materials depend, not only on the intrinsic constituent properties but also on the interface between the fiber and the matrix. This study analyzes the influence of the debonding between the two constituents.

Our approach is carried out both experimentally and numerically. For each of these two investigations, we first analyze the phenomenon at the microscopic scale (where we distinguish the constituents of the heterogeneous material) and then its influence on the mesoscopique scale (where we do not distinguish anymore the constituents of the material now become homogeneous). Experimentally, the use of fine analysis technics (optical microscopy, spectrometry...) enables to observe the debonding phenomenon at its microscopic scale. Then, experiments allow to quantify the influence of the phenomenon at the mesoscopic scale. Numerically, the microscopic phenomenon is analyzed using a finite element calculation with a mesh having the scale of the fiber and the matrix where the debonding is realized by penalties. A homogenization process helps to reach the mesoscopic scale and evaluate the influence of the degradation on the behaviour of the equivalent homogeneous material. Finally, the numerical and the experimental response of a composite sample submitted to a transversal traction loading is compared at the macroscopic scale (the scale of the structure). This approach is realised in the both cases with or without taking into account the curing residual stresses.

The originality of the present study is to show the feasibility to get some mesoscopic quantities [START_REF] Renard | Modelling of a Damaged Composite Specimen by a Micro-Macro numerical Simulation[END_REF] from microscopic information and observations. One application is the determination of a mesoscopic debonding criterion built up from its local equivalent.

DESCRIPTION OF THE MATERIAL

The studied composite material is made by hot compaction (950 °C) of 12 SiC long fibers' layers in a titanium metal matrix. The matrix is a "super-α∀ titanium alloy. The Silicon carbide fibers, the diameter of which is about 100 µm (Fig. 1), are surrounded by a carbon deposit of approximately 2 µm of thickness (Fig. 2). The fiber / matrix debonding arises from the rupture of this zone (Fig. 3). The microstructure of the composite is approximately periodic with a fiber volume fraction of about 0.32, a carbon deposit of 0.01 and a volume fraction of matrix of about 0.67.

The difference between the coefficients of thermal expansion of the reinforcement and the matrix gives the appearance of a residual stress field. These residual stresses can be damageable for the material, as they affect the bonding between the fiber and the matrix. A solution is to introduce an interphase sufficiently smooth between the fiber and the matrix that absorbs gaps of displacement. This is the reason why the fiber presents on its periphery a protective carbon deposit which play this role.

The proposed study analyzes the rupture of this interphase leading to the debonding between the fiber and the matrix (Fig. 3).

EXPERIMENTAL ANALYSIS

At the microscopic level, we observe where and when debonding appears in the representative cell of the material. At the macroscopic level, we record the loading-unloading curve during a test on a sample representative of the elementary volume of the material and the evolution of the number of debondings for different loadings. The used specimen (Fig. 4), called T90, are trilaminar (a SiC-Ti layer between two titanium layers) and such that Sic fibers are located at 90° from the axis of the specimen. (X, Y, Z) is the cartesian frame of the specimen where the X axis is parallel to its length. After polishing the lateral faces and increasing the load, the debondings are observed and counted on a chosen zone of one lateral face (Fig. 4). The same procedure is repeated at each load increase. By this way we follow the surfacial evolution of the debonding.

Choice of the Applied Loading on to the Specimen

To observe the debondings, the specimen was loaded with a torque along the X axis. The used machine allows to apply a linear time-dependant torque. Thus we obtain the evolution of the torque C versus the angle Ω. This device has a mobile jaw which avoid some parasite stress during the torsional loading.

The main motivation for the choice of the torsional loading is that the induced state of stress is simple. Indeed, we have simulated the torsion of the T90 specimen by the finite element method using a pseudo-tridimensionnal element. The analysis of the numerical simulation on the lateral side of the specimen shows that only componant σ xy and σ xz are non zero. But, if we look more attentively, we observe that in fact only σ xz is not significantly null in the Sic-Ti ply. Thus, if we note (x 1 , x 2 , x 3 ) the local orthotropic frame of the SiC-Ti material where the x 1 axis is parallel to the fibers (Fig. 4), the mesoscopic state of stress in the composite ply of the T90 specimen is the following :

σ = 0 0 0 0 0 σ 23 0 σ 23 0 .

Superficial Evolution of the Debonding Phenomenon

After loading the T90 specimen we observe the zone of the lateral side. We plot their density D S versus the applied torque (Fig. 6) and we record the macroscopic response of the specimen giving the value of the applied torque according to the angle (Fig. 5). The main conclusion is that the debondings however important seems not affect the global behaviour. Indeed, although we observe a lot of debondings on the specimen faces, nothing proves that they propagate all along the fibers, far in the width of the specimen. To evaluate their penetration, finite element calculations with a homogenization method have been used.

NUMERICAL ANALYSIS OF THE DEBONDING INFLUENCE ON THE BEHAVIOUR

The homogenization method used suppose periodic properties of the studied composite.

Calculations are realized by finite elements method on the representative elementary cell of the material. Calculated characteristics of the homogenized behaviour calculated of our virgin material are relatively in good agreement with the values found experimentally by Robertson [START_REF] Yang | Interfacial Properties Measurement for SiC Fiber reinforced Titanium Alloy Composites[END_REF] on a close material.

Concerning the influence of the debonding on the behaviour of the material, the study is realized using two different ways. First, by simulating really the debonding by splitting the nodes of the mesh between the two primary layers, i.e. on a circle of 1 µm superior to the fiber. This approach induces locally a penetration of the constituents if we do not manage the contact in the case of the shear loadings. Second, we simulate the debonding by weakening a small part of carbon interphase (we have divided by 100 the initial characteristics of the carbon, in a zone of thickness 1.00 mm) situated around the fiber. Results obtained by this method do not give significant difference when compared to the first approach. We suppose a complete debonding what has for consequence to overestimate the stiffness losses.

NUMERICAL ANALYSIS OF THE VOLUMIC CHARACTER OF THE DEBONDINGS

Experimentally, we get the value of the torque C 0 for which appear the first debondings on the T90 specimen. The finite element calculation gives the critical debonding stress tensor σ 0 .

To estimate the penetration of these debondings into the material, we realized the simulation of the torsional test by the following steps : we load until the torque C 0 and we look in which elements a debonding critical state of stress has been reached. The damaged behaviour of these elements is calculated during the homogenization step. The torque is then increased, and so on. The cartography of damaged elements indicates the depth of penetration of the debonding (Fig. 7) and we give the torque/angle simulated curve (Fig. 5). Numerically, we observe a decrease of the specimen torsion modulus of approximately 25% while the experimental curve shows no decrease. This can be explained because decreases of material modulus have been overestimated and because the calculation by finite element method is realized with plane strain elements.

ANALYSIS WITHOUT CONSIDERATION OF THE RESIDUAL STRESSES

Identification of the Microscopic Debonding Criterion

By applying the mesoscopic state of stress σ 0 on the representative cell of the material, the procedure of localization associated with the homogenization gives the microscopic state of stress in all point of this cell. We can therefore access to the normal and tangential strength between the two primary layers, located close the debonding. In besides, a micrography of the debonding phenomenon (Figs. 89) for the torque C 0 allows to understand preceeding results and therefore, help to write a microscopic debonding criterion. We call ω the area constituted by the fiber and the part of the deposit of carbon that it is always interdependent and bω its frontier. By calling s the microscopic stress tensor, we note s n = s T t + s L x 1 + s N n the stress vector on bω, where the unit vectors n and t are respectively oriented with the normal to bω et with the tangent at bω in the plane of the cross section. A local debonding criterion can be written as :

s T s T c 2 + s L s L c 2 + s N s N c 2 = 1
where s T c et s L c are the critical tangential values and s N c the critical normal value of debonding. As first approximation, we can estimate s T c § V L c and here the simulations give : s T c § V L c § MPa and s N c § MPa. However, in the literature, we find s L c § MPa [START_REF] Robertson | Manufacture and properties of Sigma-Fibre reinforced Titanium[END_REF].

Although constituents and the process of elaboration of the composite are not rigorously identical, these values indicate an magnitude order. Thus, we observe that evaluated critical values with the simulation are significatively lower than the experimental values. The main reason of this difference is that residual elaboration stress that put the fiber in compression (binding) have not been taken in account. It is the next objective given to the continuation of this study.

Mesoscopic Debonding Criterion

If ε is the mesoscopic strain tensor, we can write a debonding criterion at the scale of the homogeneous material as a quadratic form of its components : where ε 11 c , ε 22 c , ε 33 c , ε 23 c , ε 13 c and ε 12 c represent the different critical debonding values. It is the finite element resolution of the six localization problems of the homogenization that allow to determine these values in seeking for each of them from what mesoscopic deformation the debonding takes place. This is obtain using the finite element similar to a spring (penalty) placed to the place of the possibly debonding, and whose behaviour of typical whole-ornothing is governed by the preceding microscopic criterion.

Simulation of Transversal Tensile Tests

The goal now is to simulate a transversal tests at room temperature on a SiC-Ti specimen (without titanium layers) called S0 where the axis of the specimen are the same as the axis of SiC-Ti material defined in figure 3. The mesoscopic debonding criterion is known. We can thus obtain the value of the stress tensor from which the debonding starts. We found that the starting value of the debonding is very different between the experiment and the simulation (error : 200%). This is because the manufacturing residual stresses have not been considered. We will see in the next section wether the result for the transverse tensile test can be improved by considering the manufacturing residual stresses.

ANALYSIS WITH CONSIDERATION OF THE RESIDUAL STRESSES Manufacturing Residual Stresses

A quarter of the cell applying symmetric conditions on the axes and thus we obtain the manufacturing residual stresses [START_REF] Thionnet | Micromechanical modelling of fibre/matrix interface effects in transversely loaded SiC/Ti-6-4 metal-matrix composites[END_REF].

Identification of a Microscopic and a Mesoscopic Debonding Criterion and Simulation of Transversal Tensile Tests

Using the same microscopic and mesoscopic criterion equations and proceeding as in the case without residual stresses, we determine first the characteristics of the local debonding criterion (s T c , s N c ) and thus those of the mesoscopic one (ε 11 c , ε 22 c = ε 33 c , ε 23 c , ε 13 c = ε 12 c ).

Using the new identified critical values in the mesoscopic criterion, the simulated value agrees well with the experimental value (Fig. 10).

CONCLUSION

The debonding phenomenon between the fibre and the matrix was studied in a long fibre SiC-Ti metal-matrix composite. The debonding between the two constituents occured by the rupture in the carbon interphase. The analysis was both numerical and experimental and realized in a qualitative and quantitative ways. It shown the feasibility of the following methodology : to determine quantities at the scale of the homogeneous material from microscopic informations and observations (i.e. at the scale of the constituents of the material). An exemple is given for the determination of a mesoscopic debonding criterion. This work is realized without and with consideration of the manufacturing residual stresses. The conclusion is that they have a very significant role in the determination of the critical debonding values.

The use of numerical techniques for the experimental analysis seems increasingly necessary because of the complexity of the microstructures of the composite materials and the complexity of the phenomena involved. Although the calculation did not solved all problems, it has partly removed some ambiguities and has led to more finest understanding of the phenomena.

Fig. 1 -Fig. 3 -

 13 Fig. 1 -Microstructure of the Sic-Ti composite Fig. 2 -Interphase

Fig. 4 -

 4 Fig. 4 -Tested specimen (♦ : observed area)

Fig. 5 -

 5 Fig. 5 -Macroscopic response torque/angle Fig. 6 -Superficial debonding / torque curve for the T90 specimen in torsion

Fig

  Fig 8 -Debonding micrography Fig. 9 -Debonding simulation

  simulated debonding start, (2) : experimental debonding start (3) : simulated point giving the elastic slope of the completely damaged material Fig. 10 -S0 specimen transverse tensile test at room temperature. Simulation with consideration of the manufacturing residual stresses

ACKNOWLEDGEMENT : The authors wish to thank the SNECMA (Société Nationale d'Etudes et de Construction de Moteurs d'Avions) for providing all the samples and the financial support. Authors want also thank Mr. Y. FAVRY and Mr. J.C. TEISSEDRE for their assistance with the realization of tests.


F Deléglise

M H Berger

A R Bunsell

MICROSTRUCTURE AND MECHANICAL PROPERTIES OF A NANOCOMPOSITE OXIDE SYSTEM : THE NEXTEL 720 FIBRE

Keywords: Alumina, mullite, fibres, nanocomposites, creep

The two phase microstructure and the mechanical properties of the 3M Nextel 720 fibre annealed at different temperatures were investigated. The as received fibre was found to be composed of pseudo-tetragonal mullite aggregates enclosing elongated or round shaped α-alumina grains. After annealing the fibre at 1400°C the mullite structure became orthorhombic while losing alumina. A small increase of the Young modulus was observed because of the change of relative amounts of alumina and mullite and a diminution of the porosity. Thanks to its original microstructure the fibre was found to have a reduced creep rate when compared to other polycristalline oxide fibres but it was found to be extremely sensitive to external conditions and to suffer from premature failure due to slow crack growth (SCG).

INTRODUCTION

The development of ceramic matrix composites capable of operating at high temperatures above 1100°C in air requires fibre reinforcements which are stable under these conditions. Oxide based fibres do offer the possibility of long term use in air above 1400°C. However at temperatures above around 1000°C single phase polycrystalline alumina fibres have been found to suffer from creep and above 1150°C weakening of the grain boundaries results in major strength loss [START_REF] Lavaste | microstructure and mechanical characterics of alpha-alumina-based fibres[END_REF]. The Nextel 720 fibre is the latest of a series of fibres produced by 3M of which have been based on alumina or mullite. This study has revealed the original two phase microstructure of this fibre and its influence on the mechanical properties at high temperatures.

I. MATERIAL

The 12 µm diameter Nextel 720 fibre produced by 3M is composed of 85% Al 2 O 3 and 15% SiO 2 in weight. The producers indicate that a sol-gel method is used to produce a fully crystallised fibre with an approximate composition of 55% mullite and 45% α-alumina. The room temperature single filament strength announced by the manufacturer is 2100 MPa and its Young modulus is 260 GPa. The 3M company claims a reduced creep rate of three orders of magnitude when compared to that of a single phase polycrystalline alumina fibre such as Nextel 610 and attributes this improvement to the two-phase microstructure consisting of mosaic and elongated grains [START_REF] Wilson | microstructure and high temperature properties of nextel 720 fibers[END_REF].

II. EXPERIMENTAL PROCEDURE

The mechanical behaviour of the fibre has been studied at room and high temperatures up to 1200°C. Creep and tensile tests at different strain rates have been carried out on as-received and heat-treated fibres (up to 1400°C). Single filaments as well as bundles of fibres were tested on an horizontal tensile machine with a maximum displacement speed of 1.5 mm/s [START_REF] Bunsell | an apparatus for fatigue-testing of fibres[END_REF]. Before each test the diameter of the fibre was measured using a Watson ocular mounted on an optical microscope. An electrical resistance oven composed of a slotted alumina tube inside a lanthanum chromite heating element allowed the fibres to be tested at high temperatures with extremely rapid heat-up rates. The jaws remained cold and the hottest part of the furnace was 25 mm long. The fibre fracture surfaces were observed using a LEO DSM 982 Gemini FEG SEM. The effects of heat treatment on the microstructure were investigated using a PHILIPS EM 430 TEM-STEM with an acceleration voltage of 300 kV. An X-ray diffractometer Siemens D500 was used to analyse the different phases and the influence of heat treatments on the crystal structures. The radiation corresponds to the K α1 line of cobalt (λ = 0.1789 nm).

III. RESULTS

A. Room temperature characterisation

A-1. Microstructure

The external aspect of the fibre is shown in Fig. 1. The fibre is circular in cross section. The average diameter is 12.5 µm with a standard deviation of 0.55µm. The fracture surface shows "grains" of around 0.5 µm the nature of which will be explained by TEM observations. As shown in Fig. 2 the whole fibre is composed of large mosaic grains or aggregates of mullite with wavy contours consisting of several grains separated by grain boundaries of low disorientation. When tilting an area of the sample over a large range of orientations it became appearent that each part of this area shows an aggregate for a given angle of incidence. The aggregates are isotropic with an average size of 500 nm which is larger than the thickness of the thin foil (less than 100 nm) so that only one aggregate is present on the thickness of the foil. We can then conclude that the whole fibre is composed of these mosaic grains. As shown in Fig. 2 these mosaic grains enclose some round or elongated α-alumina grains identified by an energy dispersive X ray analysis system coupled with the T.E.M. and by HRTEM. The average size of the rounded α-alumina grains is between 50 and 100 nm and the elongated ones have lengths of 50 of 100 nm with aspect ratios of up to 6. XRD revealed that the fibre was composed of α-alumina. The comparison of the remaining peaks with those of the orthorhombic mullite shows that the 401 peak of the orthorhombic mullite was not present in the Nextel 720 fibre and that the peak 041 was shifted (Fig. 4). It was also observed that peaks 120 and 250 were missing. A-2. Influence of a heat treatment on the microstructure XRD acquisition after heat treatments have shown, after annealing for 4 hours at 1400°C, the splitting of the mullite peaks 210/120 (Fig. 4), 520/250, 420/240, 410/140 as well as the increase of the α-alumina 012 (Fig. 4), 018, 116, 211, 214 peaks. of 4 hours at 1400°C.

After 5 hours at 1200°C the microstructure had the same appearance as that of the as received fibre: the sizes and shapes of the aggregates had not changed. The microstructure begun to grow from 1300°C. After 4 hours at 1400°C, the shape of the mullite aggregates had changed and they appeared to be more faceted as seen on Fig. 6. The size of alumina grains had increased and their interfaces were much more angular than those of the as-received fibre (Fig. 5 and6). The average length of these grains was 200 to 300 nm.

A-3. Mechanical properties

Considerable variation in fibre diameter was observed so that the diameter of each fibre was measured prior to tensile testing. Ceramic fibres show considerable scatter in their mechanical properties: failure strength depends on the gauge length and is controlled by the presence of defects and their distributions. In order to determine the room temperature properties 30 tests were conducted at each gauge length. Failure strains were small and never exceeded 1%. However they were superior to those of the FP pure α-alumina fibre. There was no influence on the stress rate on the mechanical properties at room temperature.

A-4. Fracture analysis at room temperature

The fibres showed linear elastic behaviour with brittle failure. The fracture surfaces of all fibres broken at room temperature had the same appearance and were planar. Crack propagation was intragranular. Fig. 1 shows a failure surface of an as-received N720 broken at room temperature with a failure strength of 1.30 GPa.

B. High temperature characterisation

B-1. Tensile tests

Up to 1000°C the fibres showed perfectly linear elastic behaviour. At higher temperatures (1100°C to 1200°C) fibres broke very quickly at loads less than 2g. No plasticity was observed during high temperature tensile testing and the fibres still exhibited brittle failure surfaces. Stress rate (MPa/s) Tensile strength (MPa) 1100°C, as received 1100°C, after 1200°C, 3h 1200°C, as received Fig. 8: Tensile strength at a constant strain Fig. 9: Failure strengths at 1100°C and rate of 6.10 -5 /s as a function of temperature 1200°C of as-received and heat treated fibres as a function of stress rate. For each temperature 35 fibres were tested. Around 1000°C the strength suddenly dropped as shown in Fig. 8. At 1200°C the failure strength was 0.24 GPa at a strain rate of 6.10 -5 /s which is only 14% of the room temperature failure strength. The dependance of the fracture strengths on the strain rates as shown in Fig. 9 indicating subcritical crack growth. Up to 1000°C, the fibre fracture surfaces had the same appearance as those at room temperature however from 1100°C the fracture surfaces were drastically modified. Failure was seen to be always initiated at a large defect (up to 1.5µm) consisting of α-alumina platelets. As can be seen in Fig. 10 the fracture surfaces present an area which fans out symmetrically from these platelets and consists of two zones. The fracture surface is at first rough due to intergranular failure and then smoother due to intragranular failure. The room temperature tensile strength dropped by approximatively 10 % after a four hour heat treatment at 1200°C at a strain rate of 6.10 -5 /s. An increase in Young's modulus with heat treatment can be seen to have occured. SEM observations show that fracture surfaces have the same appearence than that of as received fibres.

B-2. Fracture analysis

C. Influence of a heat treatment on the mechanical properties

D. Influence of impurities combined with a heat treatment

The sudden fall in strength around 1100°C coupled with the presence of large grain growth at the fracture surfaces, which was not seen after heat treatment, suggested that the fibres were being weakened by the combined effects of the applied load as well as external effects possibly amplified by impurities in the fibres. Lanthamum chromite was first investigated as it was the composition of the furnace heating element. Fibres, heat treated without load in contact with lanthanum chromite, showed a drop of 35% of their room temperature properties. Furthermore, Fig. 11 shows the growth of small α-alumina platelets on a fibre heat treated at 1200°C for 24 hours in the vicinity of a lanthanum chromite tube. EDX SEM analysis of the fibre surface showed no impurities of a concentration > 0.2 %. However these platelets were smaller than the one observed on high temperature tensile fracture surfaces. Many creep tests were conducted on monofilaments at different stresses and temperatures. It was very difficult to observe steady state creep as the fibres rapidly broke because of 1200°C subcritical crack growth. At high temperature or stress (above 0.2 GPa at 1100°C) catastrophic failure during primary creep occurred in less than 3 hours. Only one fibre was found to creep and survive after 12 hours. For this test at 1100°C and 0.23 GPa a steady state creep rate of 7.10 -8 /s was measured (Fig. 12). Shrinkage was observed from 800°C to 1000°C in Fig. 12. As monofilaments broke because of SCG some creep tests were performed on bundles of 400 fibres. The applied stress indicated in Fig. 13 are the minimum stresses applied as it is assumed that all the fibres in the bundle support the load. This is unlikely to have been the case. In these tests the fibres were found to survive for several days at 1200°C whereas a monofilament never survived more than five minutes at 1200°C whatever stress was imposed.

E. Creep properties

IV. DISCUSSION

A. Microstructure

As schematised on Fig. 14 the Nextel 720 fibre is composed of aggregates of mullite composed of grains separating by low angle disorientations and enclosing round or elongated α-alumina grains. The elongated alumina grains have got aspect ratios of up to 6 and they are mainly orientated in the fibre axis direction.

B. Cell parameters of the mullite

Mullite is the only stable crystalline compound in the Al 2 O 3 -SiO 2 system under normal atmospheric pressure. It is commonly described as having a composition ranging from 3Al 2 O 3 :2SiO 2 (60 mol % Al 2 O 3 ) to 2Al 2 O 3 :SiO 2 (66 mol % Al 2 O 3 ) and crystallising in the orthorhombic system. However it has been shown [START_REF] Cameron | Mullite:a substituted alumina[END_REF][START_REF] Kriven | Solid Solution Range and Microstructures of Melt-Grown Mullite[END_REF] that solid solution mullite can occur over a larger range of compositions from 58 to more than 75 mol % alumina when crystallised from a melt. Furthermore, Ossaka and Schneider [START_REF] Ossaka | Tetragonal millite like phase from Co-precipitated Gels[END_REF]7] showed the occurrence of a new mullite-like phase. In the diffraction diagram of this new phase the pairs of 120/210, 420/240, 041/401, 250/520 reflections of ordinary mullite were found to merge into each other forming a single peak. The lack of splitting of these reflections means that the length of the a and b constants became equal and that the symmetry was tetragonal instead of orthorhombic. This phase was described as pseudo-tetragonal [START_REF] Ossaka | Tetragonal millite like phase from Co-precipitated Gels[END_REF] because its tetragonal character is thought to be due to twinning and/or domain formation of orthorhombic structural units in an elementary cell scale. This mullite is usually formed from highly reactive metal organic compounds. The phase is metastable and transforms gradually to orthorhombic mullite at temperatures above 1000°C. The lack of the 401, 120, 250 peaks of the Nextel 720 mullite in the as received conditions indicates that this phase is the pseudo-tetragonal phase described by

C. Nextel 720 composition

Cameron [START_REF] Cameron | Composition and Cell Dimensions of Mullite[END_REF] and later Ban [START_REF] Ban | Structure Refinement of Mullite[END_REF] expressed numerically the linear increase of the alumina content in mol % of the mullite with the augmentation of the lattice parameter a. From Ban's expression and from the value of the parameter a calculated before, the mullite of the Nextel 720 fibre was calculated to contain 68 mol % alumina. As the fibre is composed of 15% weight silica and 85% alumina, we can calculate its composition which is 69% alumina. From the densities of alumina and mullite, the volume composition of the Nextel 720 fibre was calculated to be 74 % mullite and 26 % α-alumina. Finally the as received Nextel 720 fibre is composed of aggregates of a metastable pseudo-tetragonal mullite phase enclosing α-alumina grains. The volume of the mullite phase is 3 times that of the alumina. The alumina content in the heat treated orthorhombic mullite was calculated to be 59.6 mol % using the same relationship. The mullite lost alumina while recristallising in the orthorhombic system and that is the explanation for the increase of the α-alumina peaks. The volume composition of the fibre after a heat treatment of 24 hours at 1400°C was 52.6 % mullite and 47.4 % α-alumina. The lost of alumina occured gradually during heat treatment as indicated by the value of a calculated after the heat treatment at 1200°C.

D. Rule of mixtures for physical properties of a two-phase material

Young modulus of a two phase material is not dependant exclusively on the amounts of its constituents so there is no exact expression for the elastic modulus as a function of constituent fraction and morphology. However, Paul [START_REF] Paul | Prediction of elastic constants of multiphase materials[END_REF] has proposed a semi-empirical expression that gives a good approximation of the modulus for uniforme distributed microstructures. Fig. 15 shows the Paul expression of E plotted for the mullite-alumina system. The Young's modulus given by the Paul relation for the as received Nextel 720 fibre, would be 270 GPa without any porosity. The actual value is around 251 GPa. From these values and the Mackenzie [START_REF] Mackenzie | Proc. Phys. Soc[END_REF] relationship we can calculate the porosity of the as received fibre which gives a value around 4%. After a heat treatment at 1400°C, the proportion of the two phases was found to have changed and the E modulus given by Paul's relationship was 304 GPa. The

E E EE E V E E E V V c = = = = + + + + - -- - + + + + - -- - - -- - [ ( ) ]/[ ( ) ( )] / / / 1 2 1 2 1 2 2 2 3 1 2 1 2 2 3 2 1 3
1 measured value was 288 GPa (see Table 2) suggesting a porosity after heat treatment of less than 3%. That explains the shrinkage observed during creep tests.

E. Subcritical crack growth and occurrence of alumina platelets

Failure of ceramics at elevated temperatures usually occurs by either slow crack growth (SCG) or creep rupture depending on the stress level and temperature conditions. Failure due to SCG takes place at high stresses within a relatively short time, and generally involves the extension of a single dominant processing flaw to the critical size. Wiederhorn and Hockey [START_REF] Wiederhorn | Nucleation and Growth of Cracks in Vitreous-Bonded Aluminium Oxide at Elevated Temperatures[END_REF] observed the nucleation and growth of cracks in vitreous bonded aluminium oxide at elevated temperatures for strains of 0.08% to 0.12% and suggested that the crack distribution was a consequence of heterogeneous deformation within the material. Once nucleated crack propagation occured along grain boundaries as long as the stress for crack propagation was maintained until failure occured. Dey and al [START_REF] Dey | Tensile Creep Behaviour of a Vitreous-Bonded Aluminium Oxide under Static and Cyclic Loading[END_REF] assumed that in ceramic polycristals with viscous grain boundary, since both creep deformation and SCG may be governed by the same mechanism, eg. cavitation in the grain boundary phase, they are interrelated so that failure strain may simultaneously be controlled by the creep strain rate and the SCG rate. It was shown in Fig. 9 that failure strengths of the Nextel 720 fibres were dependent on the stress rate. This suggests that failure occurs by SCG. Furthermore alumina platelets (Fig. 10) were present in each tensile test as soon as the temperature reached 1200°C independently of the stressing rate and even for tests lasting for only one second in the furnace with a heating element of lanthanum chromite. These platelets did not appear after a heat treatment of 24 hours at 1400°C without any stress in an alumina tube. However Fig. 11 shows the growth of small platelets after a 1200°C heat treatment with no applied stress in a lanthanum chromite tube. It seems that platelets formation requires external contaminants and is greatly enhanced by stress. This hypothesis was confirmed by the creep tests performed on bundles of fibres. These bundles were found to survive as, presumably, the inner fibres were protected from the external contaminants by the outer fibres. As the growth of platelets is an extremely rapid and local phenomena the driving mechanism must be diffusion through a liquid phase due to the presence of local heterogeneities in the fibre microstructure. This has been studied by Change and Page [START_REF] Wiederhorn | Nucleation and Growth of Cracks in Vitreous-Bonded Aluminium Oxide at Elevated Temperatures[END_REF] who have shown that creep crack growth in ceramics can occur by a direct mass transport process dictated by surface and grain boundary diffusion through a liquid phase. This may be the phenomena involved in the growth of the platelets in the Nextel 720 fibre. Indeed the presence of few ppm of cations can drasticaly reduce the cristallisation temperature of silica so that external contamination could be responsible for the formation of a silica liquid phase at grain boundaries.

F. Creep behaviour

The most common creep mechanism of polycrystalline oxide fibres is grain boundary sliding accommodated by diffusional flaw and interfaces dislocation mobility. The particular microstructure of the Nextel 720 fibre reduces the grain boundary surface because of the aggregates which act as single big entities. Therefore the density of interfaces and so the grain boundary diffusion paths are reduced. Furthermore the complex cell structure of mullite as well as the wavy boundaries of the aggregates are two important parameters in the reduction of the strain rate of this fibre. As a consequence the creep resistance of Nextel 720 is better than that of other polycrystalline oxide fibres. Indeed the creep rate at 1100°C under 0.23 GPa is seven times smaller than that of FP fibres (pure alumina) with the same conditions (5.10 -6 /s) [START_REF] Lavaste | microstructure and mechanical characterics of alpha-alumina-based fibres[END_REF]. However SCG occurs from 1000°C during creep tests on monofilaments and causes premature failure of the fibre before reaching the secondary creep stage. Much longer test were possible with fibre bundles than with single fibres and led to lower rates suggesting that single fibres were not completely in secondary creep.

CONCLUSION

The two phase microstructure of the as-received Nextel 720 fibre was found to be composed of pseudo tetragonal mullite aggregates enclosing elongated or round shaped α-alumina grains. After annealing the fibre at 1400°C the mullite structure became orthorhombic while losing alumina. A small increase of the Young's modulus was observed because of the change of relative amounts of alumina and mullite and a diminution of the porosity. Thanks to its original microstructure the fibre was found to have a reduced creep rate when compared to other polycristalline oxide fibres but was found to be extremely sensitive to external contaminants and to suffer from premature failure due to SCG. Even though the Nextel 720 fibre has a good creep resistance creep the effects of different environements on its SCG behaviour should be more accurately investigated because it will be the most limitating parameter for its use in high temperature composites.

Fig. 1 :

 1 Fig.1: SEM micrograph of the N720 fibre

Fig. 3 : 4 :

 34 Fig.3: Part of the XRD acquisition Fig.4: Part of the XRD acquisition for for the as received fibre the as received and heat treated fibres.

Fig. 5 : 6 :

 56 Fig.5: Microstructure after a heat Fig.6: Grain growth after a treatment treatment of 4 hours at 1400°C of 4 hours at 1400°C.

Fig. 7 :

 7 Fig.7: Median tensile strength as a function of the gauge length.

Fig. 10 :

 10 Fig.10: Fracture surface at 1200°C Fig.11: microstructure after a heat treatment in the presence of lanthanum chromite.

Fig. 12 :

 12 Fig.12: Creep curves of monofilaments Fig.13: Creep curves of bundles of fibres.

Fig. 14 :

 14 Fig.14: Schematisation of the microstructure.

Fig. 15 :

 15 Fig.15: Expression of E as a function of the mullite content.

Table 1 .

 1 Room temperature properties for different gauge lengths.

	Gauge length (mm)	250	100	50	25	10	5
	failure strain (%)	0.52	0.62	0.67	0.66	0.81	0.95
	Young modulus (GPa)	249	250	255	256	256	243
	Median tensile strength (GPa)	1.29	1.52	1.60	1.74	1.9	1.9

Table 2 :

 2 E modulus at room temperature after heat treatment of 5 hours.

	Temperature of heat treatment	1100°C	1200°C	1300°C	1400°C
	Young modulus (GPa)	282	284	291	290
	STD	16 %	18 %	17 %	9 %

µm 2 µm