Clément Canonne
email: clement.canonne@ens-lyon.fr

Nicolas Garnier
email: nicolas.garnier@ens-lyon.fr

A Model for Collective Free Improvisation

Keywords: Free Improvisation, Cognitive Model, Non-linear Dynamic Systems, Emergent Structure

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Introduction

Collective Free Improvisation (CFI) is a musical phenomenon produced by at least two persons improvising simultaneously and freely, i.e. trying to leave undecided every compositional aspects until the very moment of the performance.

When talking about "free" improvisation, one should carefully distinguish between two time scales. CFI is not deprived of all the automatized behaviors that can generate the improvised musical output on a short-term time scale: Embodied patterns and learned gestures are present as much as in other kinds of improvisation. In this regard, free improvisation is not to be confused with an illusory "pure" improvisation, which would account for instantaneous ex nihilo creation.

In return, CFI can be defined as a referent-free improvisation. According to Pressing [START_REF] Pressing | Cognitive Processes in Improvisation[END_REF], a referent is an underlying formal scheme or guiding image specific to a given piece, used by the improviser to facilitate the generation and editing of improvised behavior on an intermediate time scale. In CFI, as opposed to referent-based improvisation (like straightforward jazz), there is no founding act (like the common choice of a standard) that confers a given set of musical or extra-musical data the status of common knowledge in a group.

In a broader perspective, we are not either denying the importance for CFI of cultural backgrounds or musical knowledge, especially if they are shared among the group. CFI can include idiomatic borrowings: A given CFI can sound, at times, as a be-bop piece (with swing articulation, chords, tonal progression) or as a meditation on a raga (with a scale and a specific ornamentation) ; but a free improviser is someone that has no pre-commitment when the performance begins. His production is of course determined by several self-imposed restrictions, even stylistic restrictions, but he can modify these restrictions at any time.

In CFI, improvisers face two specific problems. First, the generation of improvised musical output on an intermediate time scale is not regulated. The formal unfolding is thus totally undetermined. Second, improvisers' musical coordination is not regulated and free improvisers' simultaneous production is much more difficult to control than in referent-based improvisation. 3 The fact that the way improvisers interact in CFI is not predetermined (roles and places in the ensemble can be redefined by anyone at anytime) makes it even harder.

We propose a model for CFI seen as this set of phenomenons. 4 An important inspiration for this model was the formalization of the improvisation's process proposed by Pressing [START_REF] Pressing | Improvisation: Methods and Models[END_REF] for a solo improviser. As CFI is a very interesting case of interaction, where shared information and pre-existing structures are almost nonexistent (each improviser can be described as "agnostic" before the interaction begins), it can be seen as paradigmatic. Besides the understanding of basic musical and cognitive processes in CFI, this model can be useful in both understanding social phenomena requiring effective coordination between agents (coordination problems) and in reinforcing the intuitive link between improvisational disposition and an agent's efficiency inside a complex system, following the steps of Borgo [START_REF] Borgo | Sync or Swarm: Improvising Music in a Complex Age[END_REF] that highlighted the numerous links that one can make between free improvisation's understanding and the study of complexity. This model can therefore be of interest in fields absolutely not related to music.

In this paper, the model is used to determine if, and within which conditions, a collective structure can emerge from CFI. This collective structure can be seen as a direct consequence of coordination's effectiveness or, to put it in another way, of group flow, as presented by Sawyer [START_REF] Sawyer | Group Creativity: Music, Theater, Collaboration[END_REF] and defined by some, if not all, of the following features: heightened consciousness, clear vision of a common goal, close listening, confidence in each other competency, experience of time dilatation, intuitive understanding of other musicians' intentions, sense of balance, highenough risky situations to excite each musician's virtuosity and creativity... In the following, we consider that if a collective structure emerge, it is because group flow has happened: as the two notions are strongly correlated, we only focus in this paper on the question of CFI's collective structure.

Definitions and Model

Time Scales

The improvised production extends over several time scales, that we decompose as:

-The shortest time scale, which is the scale of the musical or acoustical signal, depending on the point of view from which the signal is described: either in a musical way (encoding the signal in "notes" of different durations on a score) or in an acoustical way (following the very evolution of the same timbre through time)... This scale is not explicitly used in our model. -Short time scale τ s , of the order of seconds: it is the scale of the "clusters of events" [START_REF] Pressing | Improvisation: Methods and Models[END_REF]. -Long time scale τ l , of the order of minutes: it is the scale of the "sequences" [START_REF] Pressing | Improvisation: Methods and Models[END_REF].

-The scale of the complete improvisation piece. For a real-life improvisation, the duration is not a priori established, but in our model, it will be. We detail now the two notions of clusters of events and sequences which define relevant time scales for our model.

The cluster of events. A cluster is a cognitive chunk that gathers a set of musical, acoustical, cinetical events that were decided at the same point in time; it can be pictured as the subsequent execution of a micro-plan, over its duration of the order τ s . Pressing [START_REF] Pressing | Cognitive Processes in Improvisation[END_REF] noted that the generation of improvised behavior on a short time scale is primarily determined by previous training and embodied patterns, and is not very piece-specific. τ s is a short time, of the order of seconds. It is short because of the agent's cognitive limitations (one can't decide too much at the same time) and the improvisation's interactive dimension (one does not want to decide too much at the same time, to be able to react quickly to changes in the environment).

The sequence. The sequence is related to the long time scale τ l . A sequence is defined by a set of processes and/or a number of features (acoustical, cinetical,musical...) holding for a given length. Improvisers try to establish successive identities and stable points in musical's stream. These identities are then developed, played with or eventually negated, until new identities finally emerge. This organization in successive sequences is probably an endogenous feature of CFI [START_REF] Pressing | Improvisation: Methods and Models[END_REF][START_REF] Nunn | Wisdom of the Impulse: On the Nature of Musical Improvisation[END_REF][START_REF] Canonne | L'improvisation Collective Libre: De l'Exigence de Coordination âĂą la Recherche de Points Focaux[END_REF].

Signal and Information

What we call "signal" in this paper is not the real musical signal produced by the musician. Its realistic description would require a huge number of variables. On the contrary, our description of the signal is extremely simplified and doesn't contain anything about the acoustic representation. In fact, we consider a real number x, related to the relative complexity of the signal (a cluster on the piano is more complex than a triad; a multiphonic on the clarinet is more complex than a traditionnaly-produced sound; a sub-division in septuplet is more complex than a sub-division in sextuplet; a stretch of music with very quick changes of pitches is more complex than a stretch of music with only one pitch...). As a direct consequence, our model's focus will not be on the signal per se but rather on more high-level phenomenas (interaction, long-term intention...). Extension of our model to more realistic signals would imply the consideration of a vector x with as many components as needed (to describe, e.g., pitch, intensity, timbre, duration).

We write N the number of musicians and x k (t) the temporal signal of musician k (1 ≤ k ≤ N ). We write x(t) = (x 1 (t), ..., x N (t)) the set of signals produced at a given time t.

The quantity of information is directly related to the signal. We want that:

the larger the signal, the larger its information. So we define a static information by analogy with energy in physics I k s = 1 2 (x k ) 2 , the larger the signal varies, the larger its information. Correspondingly we define a dynamical information by analogy with kinetic energy

I k d = 1 2 τ 2 i dx k dt 2
, where τ i is a normalization time.

Then I k = I k s + I k d is the information delivered by player k, whereas the total information seen by any musician of the group is:

I = k I k = 1 2 x 2 + 1 2 τ 2 i dx dt 2 2.

Signal and Intention

Besides the signal x k produced by musician k, we define the intention ω k of this musician, which represents the ideal signal that the musician would like to deliver. The intention is a priori more complex than the signal produced, because it contains information that the musician may not be able to actually play, due to, e.g., lack of technicality, lack of time, etc. The signal x k is deduced from the intention ω k at given discrete time steps by projection, expressing the possible loss of information between the intention and its actualization in the signal:

x k = g(ω k )
The function g expresses the projection, and for the sake of simplicity in the present paper, we suppose that the musician is "perfect", so that g is identity, i.e. x k = ω k , when the projection occurs.

The intention ω k evolves on the short time scale τ s and we choose a continuous dynamics for it. On the contrary, we impose that x k is constant and equals x k n during a cluster of events, i.e. between two projections of the intention separated by the time lag d k n which is the duration of the cluster of events. The index n labels the time. Durations d k n of clusters are of the order of the short time scale τ s . To make our model deterministic, we impose

d k n = τ s -a(x k n ) 2
where a > 0 is a constant. Clusters of events are shorter when the signal is large, i.e. contains more information. Conversely, if the signal is poor in information, the corresponding cluster of events is longer. We choose a = 0.3 in the remaining of the paper.

Because the signal x k is piecewise constant, the information I k is constant on clusters of events, while there are peaks of dynamic information at the boundaries between clusters (see Fig. 2). For coherence, we choose the normalization time τ i = 2dt where dt is the time step we use in numerical integration; this way, I k d depends on the signal's amplitude variation when there is a change of cluster of events rather than on the time derivative of this amplitude which is infinite.

Intention's dynamics. We propose the following dynamics for ω k , inspired from dynamical systems' theory:

τ c dω k dt = α k x k + l =k β k,l x l -g ω k 2 ω k (1)
Each of these equations is a Landau equation from phase transition theory. Parameter g > 0 is a constant in front of a non-linear that prevents the solutions from diverging; we choose g = 1. Note that the evolution of intention ω k depends on signals {x l }, l = k, from other musicians, not from their intentions which are of course not known by musician k. Solutions of Eq. ( 1) vary on time scale τ s and its autocorrelation function decreases to zero over this time scale. Parameter α k > 0 expresses the self-sensitivity of musician k. Parameters β k,l express the influence over musician k of signals x l l =k from other musicians and therefore quantify the interactions between musicians. β k,l are of order 1, they can have any sign or vanish. the intention ω evolves continuously in time, while the signal x is discrete on short time periods coresponding to cluster of events. Boreness b is reset to zero when a sequence change occurs (here, around time 200 s). (b) Information I k is constant over a cluster of events, and presents a peak when a new cluster of events begins.

Objective

We define the objective Ω k of the musician k as the set of parameters α k and {β k,l } l =k that caracterizes the linear part of the equations: Ω k = (β k,l ) , where we have written β k,k = α k for l = k. The objective of musician k is a Ndimensional vector that plays the role of the control parameter in the Landau theory. The set of objectives of all players defines a N × N matrix. The objective defines the value towards which the intention tends on a time scale larger than τ s . This is obvious for a solo improvisation (N = 1 and all β k,l are zero): We then have a Landau equation and its solution ω tends to α/g. But the objective is also a measure of the interaction with the other musicians; in a collective improvisation, any improviser interacts with the others, and the intention tends towards a value given by an appropriate combination of components of the objective. Following Pelz-Sherman [START_REF] Pelz-Sherman | A Framework for the Analysis of Performer Interactions in Western Improvised Music[END_REF], we distinguish some paradigmatic cases and define the corresponding coupling of musician k with musician l:

-If β k,l
1, then ω k tends to x l . Player k is willing to imitate player l: "imitation".

-If β k,l -1, then ω k tends to -x l . Player k is willing to have a signal opposite from signal of player l: "contrast".

-If β k,l 0, then player k is not paying any attention to the signal from player l: "independency". These are limit cases -obvious for N = 2 and α k = 0 -and of course any intermediate situation can occur. Note that the couplings are non-symmetrical: β k,l = β l,k , i.e., musician k can for example "imitate" musician l while l is "independent" from k.

Contrary to classical Landau theory, we make the objective evolve in time with a specific dynamics, on the long time scale τ l . We choose a discrete dynamics, and any change in the objective of a musician defines a new sequence for this musician. This dynamics requires the introduction of the cognitive load and the boreness.

Cognitive load. In CFI, a musician's attention is shared between two tasks: generating his own signal and monitoring other musicians' signals. We introduce the cognitive load to account for finiteness of a musician's attention. Cognitive load bounds possible values of the objective components. We write a first part of the cognitive load as the part devoted to monitoring the signals:

C k monitor = 1 2 α k x k 2 + l =k 1 2 β k,l x l 2 = 1 2 Ω k .x 2
Another component of the cognitive load is related to the production of the signal. We suppose that difficulty for musician k to produce a signal is proportional, with a proportionality coefficient (a k ) 2 , to the quantity of static information in his signal:

C k prod = 1 2 (a k ) 2 I k s = 1 2 (a k x k ) 2
The total cognitive load of the musician k will be noted C k = C k monitor + C k prod , and we require that this variable be bounded from above by a constant C k max representing the maximal cognitive capacity of the musician k.

Boreness. When a sequence is lasting too long, the musician gets bored and ultimately breaks it. We define the boreness b k (t) of the musician k to quantify this effect. Boreness grows in time, until it reaches a limit b k max ; then a change of objective occurs (see Fig. 2), which is also a change of sequence. We simply choose:

db k dt = C k
with initial condition b k = 0 at t = 0. When the objective is changed, at the end of a sequence, boreness is reset to 0. Maximal boreness b k max is related to maximal cognitive charge; we choose:

b k max = τ l 3τ c C k max
Objective's dynamic. We choose that the objective remains constant as long as b k (t) < b k max . On the contrary, when the boreness b k becomes larger than the maximal value b k max , we choose a new objective such that the cognitive charge remains bounded from above. All possible choices of α k and β k,l are therefore not possible. For the sake of simplicity, the new components of the objective are choosen randomly, between -1 and +1 for β k,l , and between 0 and 1 for α k ; If the new cognitive charge C k resulting from these new values is larger than C k max , we apply the factor C k max /C k to all components of Ω k . We also decide to project ω k into x k at the very same time such a change of sequence occurs.

Results and Discussion

Collective sequences and their articulation

To analyse the complete production of the group, we define collective sequences, not to be confused with individual sequences simply defined above by a given objective. This is a way to probe and quantify coordination efficiency in the group. We call collective sequence a time frame during which each improviser maintain a relative musical identity (i.e. his intention stays more or less constant). If we find a lot of collective sequences, and if collective sequences are long enough, we will say that coordination amongst musicians in the group is good. One of the main interest of this model is to show the existence of collective sequences.

If the position of each improviser in the group stays more or less constant, i.e. all objectives are contant, then we expect a collective sequence. Nevertheless, a constant objective is not a sufficient condition for the occurence of a collective sequence. Also, a collective sequence can be composed of a series of individual sequences; if after a change of individual sequence, each musican's position into the group is not strongly altered, then the same collective sequence will continue.

This collective structuring in successive sequences is one of CFI's great challenges. Fig. 3 shows that it is not always possible to detect collective sequences in CFI. One can clearly see two types of local structure in our model of CFI:

-A stable solution which can be seen as a "collective sequence" (labelled 1,2,3 in Fig. 3); this corresponds to a fixed point in the phase space of the system. -An oscillating solution which can be seen as a phase of discoordination among the musicians (labelled B in Fig. 3); this corresponds to a limit cycle, and it is obtained when some of the eigenvalues of the matrix of objectives are not real numbers but complex conjugates.

Contributing factors to CFI's structuring in collective sequences

The model is now used to see within which conditions the emergence of collective sequences is facilitated. Fig. 3. Signals from a team of 3 musicians, all with a k = 0, as produced by our model. We can discriminate 3 collective sequences (labelled 1, 2, 3). At the begining, a short transient period is observed before collective sequence 1. Between collective sequences 1 and 2, resp. 2 and 3, we observe a chaotic behavior A, resp. periodic behavior B.

Effects of improvisers' features. We consider here two specific features:

-Virtuosi produce high-information signals at a lower cognitive cost. This is represented by a low value of a k . -Leaders have a superior cognitive capacity. As a direct consequence, they tend to get bored more slowly (i.e. from a musical perspective, they try to "work out" the different ideas and situations).

Fig. 4, where a k = 0.4 for all 3 musicians, has to be contrasted with Fig. 3, where we selected a k = 0 for all 3 musicians (highly virtuoso improvisers). It clearly shows the impact of this feature on improvisers' coordination.

Fig. 5 shows that the existence of leaders enhance the organization of CFI in collective sequences.

Number of improvisers.

As one could have expected, the fewer the musicians, the easier the collective organization, as shown on Fig. 6 where we increase the number of musicians to 5, and see the difficulty to define clear collective sequences.

Emergence of sub-teams. To adress the issue of obtaining collective sequences in large groups of musicians, we allow our improvisers to seek for the creation of sub-teams in CFI. This can be done in two different ways: -Improvisers are looking for symmetrical interactions: if A is in imitation with B, B will also try to imitate A, thus unifying a sub-team by introducing reciprocal actions: β k,l has the same sign as β l,k and is of the same order of magnitude. Results are illustrated in Fig. 7. -In large groups, improvisers do not interact with every other musicians.

On the contrary, they focus on one or two specific musicians, so that they interact only with them. This is obtained by imposing a maximum of two non vanishing β k,l for every musician k. A typical case is depicted in Fig. 8.

As shown in Fig. 7 and Fig. 8, this "sub-team" kind of reasoning is efficient to organize CFI in collective sequences.

Future plans

The model's first results presented above are encouraging. In particular, the model is successful in showing the possibility of self-organization in CFI, despite the absence of a priori structures. But this self-organization depends on several features: the musicians' virtuosity, their leadership quality, their team and subteam reasoning... and probably other features still to discover. Next step is to quantify the effects of improvisers and sub-teams' features on CFI and its organization; for this purpose, a statistical approach will be used. Some assumptions we have made can be relaxed. For example, we have tried a different projection of intention into signal, by adding some noise, thus making the musician imperfect: This does not affect the observed behavior of the model. Future plans also include the study of more realistic dynamics for the objective, and possible inclusion of a long-term memory: in this regard, Dubnov's works [START_REF] Dubnov | Unified View of Prediction and Repetition Structure in Audio Signals with Application to Interest Point Detection[END_REF] could be useful. This should provide a finer understanding of the way a collective structure can emerge from CFI. Confrontations of our model with laboratory human production is also under consideration, which may suggest in return some non-trivial modifications of our signal's conception.

Fig. 1 .

 1 Fig. 1. Separation of the different time scales in the model of CFI.

Fig. 2 .

 2 Fig.2. Example of intention, signal, boreness and information for one musician. (a) the intention ω evolves continuously in time, while the signal x is discrete on short time periods coresponding to cluster of events. Boreness b is reset to zero when a sequence change occurs (here, around time 200 s). (b) Information I k is constant over a cluster of events, and presents a peak when a new cluster of events begins.

Fig. 4 .Fig. 5 .Fig. 6 .

 456 Fig.4. Signals from a team of 3 musicians, all with a k = 0.4. Although collective sequences still exist, they occur less often.

Fig. 7 .

 7 Fig.7. Signals from a team of 5 musicians (a k = 0.4). Here, musicians tend to have symetrical interactions.

Fig. 8 .

 8 Fig.8. Signals from a team of 5 virtuoso musicians (a k = 0) with one leader. Here, all improvisers are interacting with at most 2 other musicians.

For example, if the referent is a chord progression, everyone knows at each moment what pitches he can or can't play.

A lot of the music called free improvisation by the musicians, the audience or the record labels does not fall strictly into our category of CFI because it has some minimal form of referent, or because it takes place into an established group, with its own conventions and so on. Maybe Derek Bailey's Company Week is the most famous example of CFI in "real life".

Acknowledgements

This work has been supported by the Fonds Recherche of ENS-Lyon.


Clément Canonne
email: clement.canonne@u-bourgogne.fr

Nicolas B Garnier
email: 2nicolas.garnier@ens-lyon.fr

Cognition and Segmentation In Collective Free Improvisation: An Exploratory Study

Collective Free Improvisation (CFI) is a very challenging form of improvisation. In CFI, improvisers do not use any pre-existing structure (like the standard in straight-ahead jazz), but try anyway to produce together coherent music. This can be seen as a coordination problem: musicians' production must converge to collective sequences, defined as time frames during which each improviser achieves relative stability in his musical output while judging the overall result satisfying. In this paper, we report on an exploratory study made with free improvisers in December 2011 in order to understand the cognition of musicians placed in a CFI context, in particular the role played by their representations of the improvisation under different type of sequences into the explanation of both their behaviors and the coordination success or failure.

I. INTRODUCTION

We propose here a study of musical cognition in the context of Collective Free Improvisation (CFI). CFI is a musical phenomenon produced by at least two persons, improvising simultaneously and freely, i.e., trying to leave undecided every compositional aspects until the very moment of the performance. [START_REF] Pressing | Cognitive Processes in Improvisation[END_REF] defined CFI as a referent-free improvisation. A referent is an underlying formal scheme or guiding image specific to a given piece, used by the improviser to facilitate the generation and editing of improvised behavior on an intermediate time scale. In CFI, as opposed to referent-based improvisation (like straightforward jazz), there is no founding act (like the common choice of a standard) that confers a given set of musical (or sometimes even extra-musical) data the status of common knowledge in a group. Moreover, CFI is often seen by its practitioners as a type of experimental music, in which musicians are trying to avoid clichés or too-predictable formulas, rhythms or harmonic progressions : musicians playing CFI often seek for uncharted musical territory.

Free improvisers thus try to satisfy two constraints:

• To achieve and maintain coherence throughout the performance despite the lack of a priori collectively agreed-upon structures or abstract schemes.

•

To satisfy individual aesthetic preferences, e.g. by avoiding "low-complexity regions" [START_REF] Borgo | Sync or Swarm: Improvising Music in a Complex Age[END_REF]. As put by Borgo, musicians in CFI are continuously "surfing the edge of chaos" by "ensuring continual development and excitement" [START_REF] Borgo | Sync or Swarm: Improvising Music in a Complex Age[END_REF] while finding a way to plainly "play together" in the absence of pre-existing common rules, predetermined structures or arrangement, and internal or external conducting. CFI can thus be seen as a typical coordination problem: musicians try to achieve relative stability in their musical outputs, while evaluating the overall result satisfying. When this is collectively achieved during a long enough time frame, it defines a collective sequence.

Collective sequences can be seen as attractors or fixed points in the musical stream. Improvisers try to converge to such a fixed point which is then developed, played with or even negated, until it is finally discarded, which ends the sequence. This "sectional" organization is probably an endogenous feature of CFI [START_REF] Nunn | Wisdom of the Impulse: on the Nature of Musical Improvisation[END_REF]. Of course, the coordination problem is at its highest difficulty at these very moments of articulation between sequences, when improvisers must pass seamlessly and collectively from one attractor to another, which also often means finding a new organization of the interactions "on the fly".

In an earlier paper [START_REF] Canonne | A Model for Collective Free Improvisation[END_REF], we presented a model for CFI, using non-linear dynamics to describe the signal produced by improvisers and the evolution of cognitive parameters labelled intention and objective. One of the main results of this model was to predict the existence of two types of collective behavior in CFI:

• A stable behavior, where each improviser's signal remains relatively constant: this corresponds to a fixed point in the phase space of the system and can be seen as a collective sequence.

•

An oscillating behavior, which corresponds to a limit cycle in our system. It can be seen as a phase of discoordination or, minimally, as a phase without fixed attractors. More generally, it showed that a self-organization of the collective signal was possible, in spite of the absence of a priori structures: a given CFI can manifest, sometimes quite prominently, an emergent structure. This paper's goal is to present some findings on this organization process by examining real-life CFI, paying attention to the following questions:

• How improvisers temporally segment a given CFI? • Do they agree on the segmentation? • Do they make a distinction between different types of sequences?

•

How is made the articulation between successive sequences?

•

What are the strategies used by musicians to ensure coordination?

•

What is the role of shared representations in the emergence of collective sequences?

II. DESCRIPTION OF THE STUDY

The experimental setting we used was similar to the one proposed in [START_REF] Sansom | Musical Meaning: a Qualitative Investigation of Free Music[END_REF]. We asked groups of 2 to 4 musicians to improvise freely together for 10 minutes. They were then immediately isolated in separated cabins where they could listen to the improvisation's recording. They were given two tasks:

•

To comment on the improvisation from their own point of view, focusing on their own signal, their interaction with the other musicians, their individual decisions, and their overall feeling during that improvisation. They could stop and rewind the recording at any time if they needed to. A few guidelines were provided for the commentary, e.g.: can you point out the moments where you have tried to introduce a new musical idea? Did you have a specific reason for introducing this very musical idea? What did you think of your partners' propositions? Can you point out the moments you liked/disliked?

•

To propose a segmentation of the improvisation piece in successive sequences, if possible, and to justify their choice (i.e., which feature(s) they used to determine the transition from one part to another). Recordings were done in Paris in December 2011. 7 groups were recorded: 1 duo, 1 trio and 5 quartets, with a total of 20 different musicians. Musicians were aged from 22 to 35 years and were all thorough practitioners of free improvisation. Half were graduate students, and half were professional musicians and/or teachers.

The first part of the study is a qualitative one: we wish to describe as precisely as possible the musicians' thoughts during the performance. In this purpose, musicians were asked to avoid a posteriori analytic considerations and on the contrary to try to remember what they were thinking while they were improvising. Moreover, subjects were asked to be totally honest in the way they restitute whatever feelings and appreciations they might have had while improvising. Asking the musicians to listen to and comment the improvisation's recording was just a way for us to gain access to a "photograph", unfortunately reconstructed and biased, of the improviser's mind during the performance.

The second part of the study is more analytic. Our goal was to confront these data to the predictions of our model. In particular, it appeared clearly in our model that the coordination task was the most difficult when players start to change their objectives (i.e., when they introduce a new idea, or change the way they interact with the rest of the group), which often results in a change of the collective behavior. Our hypothesis was that the articulations between the different time segments defined by the musicians would be points of special interest for the study of musicians' cognition in CFI (in terms of strategies, intentions, goals, preferences, representations, etc). Moreover, we thought that the behavior of the musicians would depend on how they evaluate and describe the ongoing situation (i.e., if they feel that they were in a phase of coordination or a phase of dis-coordination); and that coordination itself probably depends on the existence of a shared representation of the ongoing situation.

III. SEGMENTATION AND ARTICULATION

We not only compared together segmentations proposed by musicians from the same group, but compared also the segmentation proposed by a musician with his first-person commentaries. This led us to four observations detailed below.

A: the segmentation appears to be intersubjective. B: a typical sequence length can be extracted. C: there are 4 different kinds of sequences. D: three different cues are used to decipher the transition between consecutive sequences.

A. Intersubjectivity in the segmentation

First, we compare the segmentations of an improvisation proposed by the musicians who performed it. In Figure 1 is represented such a segmentation for a quartet. Although musicians proposed different segmentations, we observed that they agree on the roughest one. For example, the euphonium proposed a rough segmentation in 6 sequences, whereas the clarinet proposed a finer segmentation in 12 sequences; nevertheless, within a margin of about 5 to 12s (see the vertical lines in Fig. 1), all segmentations match the rougher one. The thick vertical line at time 650s is described in paragraph D.

B. Sequence length

As seen in the previous paragraph, sequence lengths vary from one musician to another. And so they do from one improvisation to the other. Surprisingly, we observed that all sessions were segmented by musicians in sequences of comparable lengths. Results are reported in table 1. The duo session and the trio sessions are perfectly comparable to all quartets sessions. In Fig. 2 is plotted the histogram of sequence lengths, aggregating data from all sessions and musicians. The typical sequence duration (the one with the maximum probability) is 60 seconds, although the average duration is slightly larger (75 seconds), while the standard deviation is about 30 seconds. It is interesting to note that although musicians were providing possibly very different segmentations of the pieces, including long sequences as seen by the large upper tail of the distribution, a typical sequence length emerges and can be measured. This typical length might be explained by both the fragility of the coordination in CFI and by a lassitude factor (the musicians' tendency to change their proposition after a certain time).

C. Describing the sequences : partition in 4 species

Commentaries of improvisers on their collective output clearly revealed that they were defining sequences according to two orthogonal criteria : stability (presence of permanent acoustical, musical, gestural or interactional features over a certain laps of time) and desirability (presence of a sustained aesthetic interest over a certain laps of time).

Hence, a musician can perfectly judge a stable sequence uninteresting while an unstable one can be interesting. Coordination is thus non-reducible to the common goal of achieving stability, for example by sharing some acoustical/musical/gestural parameters or by establishing a clear enough interaction pattern. Conversely, unstable or transitory sequences, if maintained long enough (achieving a duration comparable to the one of a stable sequence) can be highly valued by improvisers, probably because it is in these very moments of interpolation, confrontation or suspension that some of the most typical formal features of CFI come in the forefront. Indeed, it is in these very sequences that the complexity of CFI is best manifested.

We thus isolated four types of sequences:

1) Erratic sequence (unstable and uninteresting). This is what we call a "phase of dis-coordination". It can result from multiple reasons (some of which we are discussed below) and it is symptomatic of the following cases: high density of changes in the musical material; conflict between at least two different ideas over a long lap of time; lack of proposition (no musician is willing nor capable of proposing a stable and interesting-enough idea; this is often a result of a certain form of politeness well identified by [START_REF] Siron | L'Improvisation dans le Jazz et les Musiques Contemporaines: l'Imparfait du Moment Présent[END_REF]. We can also note that it is when a musician identified a sequence as "erratic" that we observed a peak in the commentaries' density: unsurprisingly, when the musicians detect a discrepancy or a significant problem, they try to understand why it has arisen and thus spend more time commenting it.

2) Unstable but interesting (or non-disturbing) sequence.

While not the most common (instability being often associated with dis-coordination by the musicians), some sequences were nevertheless clearly described as both unstable and interesting by the improvisers. We distinguish three different cases:

• the sequence is a long transition, with a sense of progressively getting somewhere (in a way, there is a stability or a cohesiveness in this kind of sequence, i.e., a stability in the process at work);

•

the "indeterminate" nature of the sequence is valued because it generates aesthetic qualities like suspension or fragility: this is however a very delicate balance, which necessitates a full agreement on the evaluations made by the improvisers (otherwise, it can quickly fall into an "erratic" sequence).

• "accepted instability": improvisers find perfectly normal to encounter instability at certain moments in the improvisation, and thus do not try to prevent voluntarily this instability; rather, they prefer to let things "happen by themselves". A typical case of such a sequence is the beginning of the improvisation: it seems to be wellaccepted that things need a little time to "fall in place".

3) Stable but uninteresting sequence. It happens when the improvisers fall in a "low-complexity region". There is an attractor, but it is too conventional and makes the musicians unsatisfied. For example, in one of the improvisation (a trio with saxophone, euphonium and percussions, see Sound Example 1), the musicians find themselves playing a quasijazz, groove-oriented sequence (with the percussion landing the pulse, the euphonium doing a sort of ostinato, and the saxophone acting as the soloist). There is thus a clear attractor for the musicians (with a sense of pulsation and shared motives), who all maintain a relative stability in their own signal for at least two minutes. Nevertheless, all the musicians agree that this sequence is both too connoted and not enough inspired to acquire an interest of its own. This explains why an improviser can sometimes want to create instability, by problematizing a texture he finds too simple; as put by [START_REF] Sawyer | Group Creativity: Music, Theater, Collaboration[END_REF], improvisation is sometimes more a "problem-finding" situation than a "problem-solving" one.

4) Stable and interesting sequence.

This is what we have labelled "collective sequence" and it is the paradigmatic case of coordination in CFI. In such a sequence, improvisers find an point of convergence that allow each musician to develop with confidence his signal while maintaining a certain level of excitement in the performance. Two factors can end such a sequence: the lassitude of one or more of the improvisers, who find the need for change and introduce an unilateral interruption; or the chaotic aspect of CFI, by which the accumulation of small variations (resulting from the improvisers' development of their own signal) finally produce a significant collective deviation, even if nobody wanted to quit the current sequence (i.e., the collective organization is quite often emergent over the individual decisions). As an example, we give in Figure 3 the segmentation of an improvisation (the one of Figure 2) into the different types of sequences proposed above, as given by one of the musicians in his commentaries. We notice that the number of sequences is larger, so the segmentation is more detailed. It is interesting to note that the musician defined as erratic the sequence of length 44s represented in Figure 1 as a wide transition (hatched vertical line at time 650s), thus supporting the interobjectivity of the segmentation.

D. Articulation between sequences

Nunn (1998) has presented an extensive typology of the different types of transitions between two parts of an improvisation. While finding this typology very informative, [START_REF] Canonne | L'Improvisation Collective Libre: de l'Exigence de Coordination à la Recherche de Points Focaux[END_REF] has argued that a transition often becomes a sequence of its own in CFI, which is supported by our study: musicians described some transitions as autonomous parts. Therefore, he proposed to focus instead on the notion of "articulation", either between two sequences or between a sequence and a transition.

Answers provided by the subjects suggest that three major kinds of cue are used to detect a change of sequence, both from an "insider" and an "outsider" perspective (i.e., both in the first and in the second part of our study).

1) Monitoring the signal's energy profile: cadential movements and silences. The simplest articulation one can consider is the juxtaposition of two sequences. Although it is common and easy to perform in solo improvisation, it is much more difficult to do in CFI. In all our experimental recordings, juxtaposition happens only in particular situations, when all musicians have identified without any ambiguity that the ongoing sequence has come to its "natural" ending. To gain this knowledge, musicians monitor the variations of the energy level of the performance: a large deceleration and/or a diminuendo, a sustained period of low-density signals and, of course, collective silence. Such situations happened at least once in every recording. We can interpret this by saying that it is extremely likely that, over a ten minutes-period of CFI, improvisers are not able to prevent a severe decrease in the energy profile; i.e., the collective energy seems to entirely dissipate over a certain amount of time, which characterizes the fragility of CFI's situations. We also noted that in the segmentation task, these moments were always used to define an articulation between two sequences. One must also note that in these situations, although the musicians know that an articulation must be made (if not, the improvisation will end), it may not be easy to do. In fact, once the musical stream has been interrupted, it is often difficult to continue in a way that does not sound artificial (see Sound Example 2, where, after a silence, improvisers seem to struggle to find a new stable idea). Hence, silences are not only clear clues helping in the identification of possible articulation for the musicians, but also sources of problems in the formal task inherent to CFI.

2) Detection of Salient Events. Examining the gametheoretical literature on pure coordination situations, Canonne (in press) has shown the general importance of saliency in CFI and the ability of expert free improvisers to single out salient musical events in order to face CFI's coordination problem. In a general context where unpredictability is at its foremost (due to the absence of common referent and to the non-idiomatic way of improvising), it is extremely important for improvisers to be able to create (or to discover) points of convergent expectations in order to create some stability in the flux of musical events. A salient event, by its very saliency, is transparent: it can procure something like common knowledge in the group (everyone has noticed a peculiar event and everyone supposes that everyone else has noticed it). This is probably the reason why such events play a very important role in the articulation of sequences, i.e., in operating a significant change in the collective behavior. Some events will thus "hook" all the players together and produce either abrupt or gradual changes. We observed four different types of such events: a contrasting event inside a given context (e.g. : a new timbre, a clear pitch in an otherwise noisy environment); an "accident", either at the individual level (a sound with an attack which was illmastered) or at the collective level (a sudden unison or an unexpected simultaneous attack); an event with an "immediate" saliency (a loud and/or high-pitched sound: as Juslin ( 2009) puts it, "Brainstem reflex refers to a process whereby an emotion is induced by music because one or more fundamental characteristics of the music are taken by the brainstem to signal a potentially important and urgent event. All other things being equal, sounds that are sudden, loud, dissonant, or feature fast temporal patterns induce arousal in the listener", p. 136); the entrance of a previously silent player in the improvisation, or conversely, the disappearance of a player. The following examples taken from our recordings were all noticed and reported without ambiguities by the musicians in their commentaries: a sul ponticello cello's harmonic that provokes a progressive stabilization on sustained textures (Sound Example 3); an entrance of saxophone that precipitates the end of the improvisation (Sound Example 4); an encounter between contrabass and clarinet on a common E, that brings a new sequence in crescendo (Sound Example 5); or a sudden nail-glissando on a low piano string, that stops the group for just a quick respiration, before a new sequence actually begins (Sound Example 6).

3) Detection of Transitory zones. Improvisers sometimes identify a time interval ---a "zone" ---where the music is changing and a new sequence finally emerges, but they can't precisely locate it in time. Moreover, this zone is often too short or too heteroclite to be identified by the musicians as an autonomous sequence. In such a situation, we usually observed that the musicians placed the mark between these two parts at various places on a given laps of time. The reason is that there is a chain of causes and effects taking place at that moment, and the musicians choose whatever event seems relevant for themselves to propose a sharp date. This accumulation of variations is symptomatic of CFI and defines one of its features: the musicians display a very high sensitivity to the others signal's variation. In a situation where there is no common referent, musicians tend to give each other proof that they are interacting together: and for an improviser, the better way to acknowledge that a change in a co-improviser's signal has been perceived is to introduce a variation in his own signal. Figure 1 gives an example of such a case (the entire improvisation can be heard on Sound Example 7): in this improvisation, the guitarist and the clarinetist are playing alone together from 7'40''; the tubist and the saxophonist remained silent during this duo, because they thought the improvisation would finish at the beginning of the guitar/clarinet duo, and they could not figure out a way to "come back" into the music. At 9'10'', the guitarist proposes a variation in his electronic texture, in order to "make something happens" (as noted by the guitarist himself). At 9'20'', the saxophonist makes windy quasiflute sounds, which fit nicely with the guitar texture (this articulation is noted by the saxophonist): but it does not change the music dramatically. So at 9'40'', he comes back with loud and repeated pitches (this articulation is noted by the tubist). Finally, at 10'', the tubist introduces a sustained low-tone pedal that increases the collective tension (noted by the clarinetist). This example also clearly establishes the distinction between a transitory zone and an "erratic" sequence: the erratic (i.e., incoherent and unpleasant) sequence depends on the absence of a common clear direction or the absence of something that "glues together" musicians' outputs, while the transitory sequence depends on the variation frequency observed on a given laps of time. In the example above, the musicians feel that an "erratic" sequence has begun at 10'30'' (i.e., after the last variation has been observed) when after a silence, the tubist come back with the same low-tone as before : all other musicians clearly do not know what to do with it (to play the same thing as before or to do something else). We therefore isolated three main categories of articulation in CFI: cadential articulation (drop in the collective energy), interruptive articulation (use of a salient event) and interpolative articulation (emergence of a transitory zone).

IV. COMMUNICATION, IMPROVISERS' STRATEGIES AND SHARED REPRESENTATIONS

When musicians improvise, they tend to distinguish two aspects of a given signal: its acoustic, musical and/or gestural content on the one hand; and its "intentional" content, i.e., what the signal means, what it let appear from its producer's intentions and objectives, on the other hand. As such, a great deal of the improvisers' cognitive resources is devoted to an encoding/decoding activity as identified by Pelz-Sherman (1998): the competency of musical agents in an improvised interaction depends on their ability to "convey the semantic intent of their own musical ideas to other performers in real time" and to "make accurate judgements in real time about the semantic intent of each performer" (p. 127). This "semantic" content can stand for different things which all refer to intentional states, transmitted in or deduced from the signal: the formal content (through a given gesture a musician can communicate to his co-improvisers his objective about the current situation's evolution); the interactional content (through a given gesture, a musician can communicate to one of his co-improviser's his intention to enter in a certain form of interaction with him); the evaluative content (through a given gesture, a musician can communicate to his coimprovisers his evaluation or representation of the prevailing situation); and probably others.

It is this very distinction between musical and intentional contents that makes strategic reasoning crucial in CFI: improvisers make musical decisions because they expect or anticipate some kind of response or reaction from the other improvisers, based on the implied semantic content of their signal.

CFI is what Pelz-Sherman calls heteroriginal music, i.e., a form of music where "decisions are made during the performance as the product of the relationships of multiple agents" (p. 9). In this kind of music, the separateness of the agents makes discrepancies in the situation representations and evaluations, or in the preferences about a given situation's evolution quite likely: one can thus partly relate the success of the musicians coordination to their ability in constructing a shared representation of the improvisation.

We can draw, in this regard, two significant results from our study:

• Musicians use different strategies based on the way they construct their representation of the current situation (i.e., as one of the four types of sequences described above).

•

The most patent case of dis-coordination emerges when the musicians have contradictory or quite different representations of a given situation. As such, discoordination may be seen as miscommunication between improvisers.

A. Decisions and Strategies

Not every decision is based on a strategic intention. Most common decisions of a musician are made based on the musical logic (as seen by the musician) of both his individual discourse and the collective result. In this case, the primary focus of the musician is on the musical aspects of his signal, and the way these musical aspects satisfy the requirements of both internal consistency with what has been previously played, and external consistency with what is played concurrently.

Other decisions are based on more cognitive factors, especially the cognitive load (sometimes, an improviser decides to "stop paying attention" to one of the other improvisers because he "can not follow him anymore") and the lassitude (sometimes, an improviser just get bored with his own idea or "does not believe in it anymore" and then decides to do something else).

Nevertheless, it appeared clearly in the commentaries that strategic decisions (decisions made with a specific goal and/or a collective response in mind) were quite frequent. Of course, improvisers use a very large array of strategies in CFI; but the most interesting ones are the "meta-pragmatic strategies"; they result not only in decisions taken for an intended purpose but also in decisions of which the primary motivation is precisely to convey the improviser's intention or representation of the situation. In such situations, the signal is produced primarily for its intentional aspects, and sometimes musical aspects are not even relevant. The purpose of these strategies is essentially to modify the improvisation's interaction pattern or formal contours, either radically or partially.

Here are two examples, taken from our recordings: in one improvisation, the saxophonist noticed that the flutist was not playing for a while; he thus decided to abandon the highpitched creaking sounds he was producing to play a distinct pitch (a low F), hoping to give the flutist a "passageway" into the interaction. And indeed, the flutist took advantage of the opportunity, and came back into the improvisation, with a whistle-tone that merged beautifully with this low note. But as soon as his goal was accomplished, the saxophonist returned to his high-pitched creaking sounds (Sound example 8), making clear that this distinct pitch was not the beginning of a new idea.

The other example is much more radical: because he was evaluating an ongoing sequence and finding it too long and static, the tuba player suddenly interrupted the improvisation with a loud and strident motif (much like a scream), with the hope for "producing something new". But the three other musicians did not react to this strong interruption even if, as it appeared clearly in the commentaries, they had all understood his intention. The tubist tried again a few seconds later with the same motif: the other musicians were then ready and used his motif as a signal to increase the music's intensity (Sound Example 9). This is also a nice example of the kind of implicit negotiations that can be seen in CFI. Since the original strategy has failed, its meaning has been modified between the first and the second occurrences: the absence of reaction the first time (partly because some musicians did not want to modify their proposition) caused the tuba motif to shift from a clear interruptive signal to a mere stimulation signal. This change of status was obvious in the way the tubist was playing his motif: three short outbursts the first time; a longer and continuous stream the second time, as if it was more a way to enrich the polyphony, by adding a new, contrasting layer, than a way to radically modify the collective organization.

Aside from these meta-pragmatic strategies, we have observed four significant types of strategy used by the musicians in CFI:

1) Stabilization strategy. It is certainly the most recurrent type of strategy used by the musicians. This indicates clearly that the stabilization of the improvisation (i.e., finding a way to make all musicians' signals "work together" for a while) is one of the musicians' important goals in CFI. However, this is not surprising if we remember that CFI is "referent-free": achieving formal stability in the absence of a given pre-existing structure is precisely one of CFI's great challenges. In the commentaries, we observed that when a musician describes the sequence as "erratic", chaotic" or "unstable" he also gives an explicit description of his own reaction and actions in terms of a "stabilization strategy". Two sub-types can be distinguished: the musician simply repeats a pattern, strictly or with little variations (creating an ostinato or a loop), or holds a sound/texture/pitch for a certain time, proposing de facto an attractor for the others to join in; or he tries to operate as a mediator between two too heterogeneous ideas presented by other improvisers, thus proposing a possible synthesis and future attractor.

2) "Wait and see" Strategy. This can be seen as a variant of the stabilization strategy. It is mainly used by the musicians when they describe the situation as transitory. The musician deliberately stops playing or maintains a very lowinformation signal, in order to, quoting one commentary, "see where things are going": he is waiting for a new idea to emerge. When musicians are trying to find a new attractor, this strategy can be very helpful to clarify the situation.

3) "Playing along" strategy. It appeared clearly in the commentaries that the musicians are fully aware of the fragility of CFI, and the fact that it can "fall apart" so easily. Hence, the cooperative aspect of the interaction always seems to be stronger than its agonistic aspect: it is more important to find a way to play together than to impose a personal and preferred idea. When musicians feel that an idea is not very interesting but can nevertheless work as an attractor, or when they feel that the group is going towards an inevitable direction, musicians generally "play along", i.e., they do their part in the interaction, play what they are implicitly expected to play. There is of course a place for implicit negotiations and confrontations between different propositions, but typically at the beginning of a new sequence. Once the music starts to flow, the negotiation margin quickly decreases. The "playing along" strategy is a way to acknowledge this state of affairs. In such a situation, the musician decides to play his part, but in a minimal way, just enough for the music to continue, but not enough to remain the same for very long.

4) Densification strategy. This kind of strategy is typically used by a musician when he described the improvisation as boring, predictable, uninteresting... In such a situation, the musician deliberately creates complexity by adding a new layer in the improvisation. He hopes to provoke a transition, or a crystallization, by the dialectical confrontation of two contrasting elements, of a new, more exciting attractor. A typical example is given by a saxophonist commenting about the beginning of one of the improvisations: "It is my first sound in this improvisation and I think to myself: I'm going to break this situation, which is either too stable or too boring, but for sure without any clear direction. As a consequence, some diversified, non-congruous events start to appear and are added to this continuous texture maintained by the euphonium. In that sense, my intervention was rather successful".

B. Coordination and shared representations

The fact that improvisers maintain similar representations of a given situation plays a great role in the success of the coordination task. If all improvisers frame a situation as erratic, for example, it will not be long before an attractor appears and a new collective direction emerge. Conversely, an erratic sequence emerge typically when improvisers have very contrasting representations of the situation. Similarly, a difficult articulation between two sequences often results from different appreciations on the improvisation's formal conduct.

It follows from the commentaries' analysis that three different kind of representations should be distinguished:

• The musicians' representations of a given situation: this includes the aesthetic interest of the situation (is the music good?), its stability/instability (is there a clear attractor?), its directionality/absence of direction (do we know where the music is going? Is it a transition?), its place in the improvisation as a whole (e.g.: can it be seen as a recapitulation of a previous sequence?), its place in the actual sequence (is the sequence already long enough or should it be developed little more?), the way the interaction are structured in it (e.g.: has the saxophonist taken a soloist spot?)...

•

The musicians' representations of an other musician intentions and objectives.

•

The musicians' representations of team preferences, i.e., of what is best for this particular band he's playing in, given the instruments, the musicians, their various cultural backgrounds, etc. Of course, there are often differences in the first kind of representations, because these representations are build on very individual-specific backgrounds of preferences, past experiences, musical knowledge, etc. As for the differences in the second kind of representations, they often result from the inability of the musician to unambiguously convey the intentional content of his signal. Representations of the third kind were less often formulated, but seemed to play a great role in the regulation of improvisers' decisions: indeed, the musicians will shape the coordination problem of CFI according to these representations of team preferences by sharply distinguishing between situations that fit the group well and situations that risk to jeopardize the group. As an example, we can quote a clarinetist at the beginning of his commentaries: "I quickly said to myself that the best thing to do, given that we don't know each other, and given the instrumentation, would be to work on lengthy things, with slowly evolving textures, trying to converge at least on the pitches".

We give below a few examples of the impact of shared representations on the smoothness of the improvisation's collective flow.

1) Misperceived intention (Sound Example 10).

In this example, the flutist says that her energetic intervention is designed to produce a quick articulation, and to avoid a collective sagging: the goal is to attract immediately the other musicians with this new proposition. But the other musicians do not perceive that intention: on the contrary, they all consider her intervention as a clear signal for the beginning of a solo. This leads to a literal dead end, manifested by an awkward silence and a few hesitations, before something else is proposed by the percussionist.

2) Transparent intention (Sound Example 11).

The percussionist says that his intervention was designed as a cut, to create a sense of synergy. This was clearly perceived as such by the other musicians. One of the musicians evoked the "breaking nature" of the percussion, capable of provoking such vivid articulations: for him, the intention of the percussion intervention was obvious. And indeed, the resulting collective articulation was efficient and very successful.

3) Shared formal representations (Sound Example 12).

Here, all the musicians said that they knew, when this sequence began, that it would probably be the last part of the improvisation. Here are two comments made by different musicians, which are remarkably similar: "It's always very difficult to abandon this kind of proposition. Sometimes, the best way is just to stop!" (Double-bass player); "We know that we are probably coming to an end, because if we continue to push things further, we know that we will arrive to a breaking point, a difficult point: the moment when you need to get out of this thing. And, aside from just stopping the improvisation, it's difficult to do such a thing" (Clarinetist). The musicians are thus all waiting for an "exit point", something that will help them with stoping the improvisation; it is the sudden appearance of higher harmonics in the cello (produced accidentally, according to the cellist) that gives the signal for everyone.

4) Contrasting evaluations of a given situation (Sound Example 13).

Here, the evaluation of the situation strongly depends on the musician: "There is a collective rarefaction that I found very interesting. I thought it was a nice moment" (Clarinetist); "It becomes suddenly more pointillistic. There! That's when things are starting to fall apart" (Tubist). The tubist clearly fears the apparition of an erratic sequence, while the others are rather satisfied with the fragile quality of the music. Unsurprisingly, the tubist adopts a stabilization strategy, by introducing a simple ostinato. But this signal is not well-accepted by the other musicians, who would have enjoyed maintaining the same atmosphere and are puzzled by the ostinato. As a result of this seminal difference of appreciation, the sequence is perceived as rather unpleasant by the musicians, as it is composed of parallel ideas that never really converge. It then takes almost 60 seconds before this initial dissonance is resolved. It appears on these examples that the emergence of shared representations plays a crucial role in the success of CFI's coordination task. It would certainly be of great interest to use paradigms and results from the field of team cognition and apply them to the case of CFI.

V. CONCLUSION

This study highlights some essential features of musical cognition in CFI. A significant part of the improvisers' cognitive resources seems to be devoted to the formal problem of CFI: what to play next and what to play together? This double-question is not trivial, and it does not receive easy answers, as testified by the number of erratic or unpleasant sequences identified by the musicians in the analysis of their actual improvisation. Two difficulties come in the forefront: the first one is about establishing and/or identifying an attractor (convergence problem), while the second one is about the transition from one attractor to another (articulation problem). We identified three relevant paths followed by the improvising musicians to cope with these difficulties: the presence of salient events; the use of strategic reasoning; and the emergence of shared representations. This last aspect opens a promising direction to explore: it should lead to new works in the field of team cognition research.

Figure 1 .

 1 Figure 1. Example of segmentation (arbitrary colors). Thin vertical lines indicates intersubjective transitions, of respective width 11s, 12s, 10s, 5s (from left to right). The thick vertical line of width 44s corresponds to an erratic (unstable and unpleasant) sequence qualitatively undistinguishable from a transition (see text). The improvisation can be heard as Sound Example 7.

Figure 2 .

 2 Figure 2.Histogram of sequence durations, for all improvisations, and all musicians.

Figure 3 .

 3 Figure 3. More detailed segmentation of the same improvisation as in Fig. 2, but extracted from the spoken commentaries of clarinet player.

Table 1 . Average sequence duration <T> in seconds and corresponding standard deviation σ in different groups : duo N=2, trio N=3, and quartets N=4.

 1

ACKNOWLEDGMENT

This work has beneficed from the support of the IXXI (Project "Improvisation and Non-linear Systems") and the ANR (Project "IMPROTECH ANR-09-SSOC-068").


Jiagui Liu

Dominique Saletti

Stéphane Pattofatto

Han Zhao
email: zhao@lmt.ens-cachan.fr

Impact testing of polymeric foam using Hopkinson bars and digital image analysis

Keywords: Polymeric foam, Impact testing, Hopkinson bars, High-speed imaging, Digital image correlation

This paper has investigated the impact testing method on the polymeric foams using digital image correlation. Accurate average stress-strain relations can be obtained when soft large diameter polymeric pressure bars and appropriate data processing are used. However, as there is generally no homogeneous strain and stress fields for polymeric foam, an optical displacement field observation is essential. In contrast with quasi-static tests where the digital image correlation (DIC) measurement is commonly used, technical difficulties still remain for an use under impact conditions such as synchronization and measuring accuracy due to rather poor quality images obtained from high speed camera. In the present paper, an accurate synchronization method based on direct displacement measurement from DIC is proposed and the feasibility of a calibration method using DIC on pressure bar ends is discussed. The relevance of the present method for establishing mechanical response of polymeric foam is also demonstrated.

Introduction

Polymeric foams have been widely used in weight-saving and energy-absorbing structural applications, such as automobile vehicles, aircraft structures [START_REF] Gibson | Cellular Solids: Structure and Properties, 2nd Edition[END_REF] and personal protections [START_REF] Mills | Polymer foams for personal protection: Cushions, shoes and helmets[END_REF].

Under quasi-static loading, analytical/numerical approaches were developed to give a reasonable prediction of foam-like material behaviors [START_REF] Zhu | Analysis of the elastic properties of open-cell foams with tetrakaidecahedral cells[END_REF][START_REF] Zhu | Analysis of the high strain compression of open-cell foams[END_REF][START_REF] Deshpande | Multi-axial yield behaviour of polymer foams[END_REF][START_REF] Gong | Compressive response of open-cell foams. part i: Morphology and elastic properties[END_REF][START_REF] Gong | Compressive response of open cell foams -part ii: Initiation and evolution of crushing[END_REF] and experimental data were obtained without difficulties. However, such approaches cannot be easily generalized in the case of impact loading, which is generally the real working conditions of polymeric foams for energy absorption applications. It is therefore very important to develop a dynamic experimental approach to characterize polymeric foams under impact loading.

The behavior of polymeric foams at relatively high strain rates has been an interest of investigators since 1960s [START_REF] Lacey | Response of several materials at intermediate strain rates[END_REF][START_REF] Traeger | Physical properties of rigid polyurethane foams[END_REF]. Experimental results using different devices such as the falling weight or impacting mass technique [START_REF] Schreyer | Anisotropic plasticity model for foams and honeycombs[END_REF][START_REF] Faruque | Strain rate dependent foam: Constitutive modeling and applications[END_REF][START_REF] Zhang | Constitutive modeling of polymeric foam material subjected to dynamic crash loading[END_REF], rapid hydraulic testing machine [START_REF] Rehkopf | Fluid and matrix components of polyurethane foam behavior under cyclic compression[END_REF][START_REF] Wagner | Modeling foam damping materials in automotive structures[END_REF][START_REF] Chang | Unified constitutive equations of foam materials[END_REF] and also split Hopkinson bar have been reported [START_REF] Rinde | Time and temperature dependence of the mechanical properties of polystyrene bead foam[END_REF][START_REF] Rinde | Dynamic shear modulus of polystyrene bead foams[END_REF][START_REF] Song | Dynamic compressive response and failure behavior of an epoxy syntactic foam[END_REF] Hopkinson bars have been widely used to experimentally characterize the mechanical behaviors of materials under impact strain rates loading, from 10 2 s -1 to 10 4 s -1 and provide more accurate measurement than drop weight and high-speed testing machine at this range. Common metallic bar with quartz-crystal piezoelectric force transducers [START_REF] Song | Dynamic compressive response and failure behavior of an epoxy syntactic foam[END_REF] or large diameter nylon pressure bars [START_REF] Zhao | On the use of a viscoelastic split hopkinson pressure bar[END_REF][START_REF] Zhao | Behaviour characterisation of polymeric foams over a large range of strain rates[END_REF] were used to improve signal/noise ratio. Typical overall stress-strain curves at high strain rates were obtained with such nylon pressure bars method for polymeric foams [START_REF] Zhao | Testing of polymeric foams at high and medium strain rates[END_REF] and metallic foams [START_REF] Zhao | An experimental study on the behaviour under impact loading of metallic cellular materials[END_REF].

Meanwhile, it is noticed that aforementioned testing results (under static as well as impact loading rate) are only the average stress-strain relations with an assumption of homogeneous stress and strain fields within the foam specimen, which is unfortunately not true for most of the foam-like materials. This non-uniform or even localized deformation has been observed by a number of studies [START_REF] Gong | Compressive response of open cell foams -part ii: Initiation and evolution of crushing[END_REF][START_REF] Zhao | On the use of a viscoelastic split hopkinson pressure bar[END_REF][START_REF] Wang | The deformation habits of compressed open-cell solid foams[END_REF]. Thus, field observation techniques have been added in more recent tests, often associated with digital image correlation (DIC) [START_REF] Sutton | Image correlation for shape, motion and deformation measurements: basic concepts, theory and applications[END_REF][START_REF] Grédiac | Mesures de champs et identification en mécanique des solides[END_REF]. In contrast with rather common applications of photography and DIC to the foam testing under static loading [START_REF] Zhang | Compression strain measurement by digital speckle correlation[END_REF][START_REF] Wang | Full-field measurements of heterogeneous deformation patterns on polymeric foams using digital image correlation[END_REF][START_REF] Heinz | Uniaxial compression analysis of glassy polymer networks using digital image correlation[END_REF], only a few preliminary studies with high speed imaging were reported under impact loading rates [START_REF] Gilchrist | Impact deformation of rigid polymeric foams: experiments and fea modelling[END_REF][START_REF] Viot | Polymeric foam behavior under dynamic compressive loading[END_REF][START_REF] Nemat-Nasser | Experimental investigation of energy-absorption characteristics of components of sandwich structures[END_REF][START_REF] Othman | Identification of non-homogeneous stress fields in dynamic experiments with a non-parametric method[END_REF][START_REF] Liu | Impact testing of cellular materials with field measurement-a review[END_REF] because of synchronization difficulties and poor image resolution from high-speed camera.

In this paper, an optical field measurement along with large diameter nylon Hopkinson bars is highlighted in the experimental setup for the polymeric foams characterization. Nylon Hopkinson bar methods and the associated data processing are briefly recalled in the section 2. Afterwards, technical points of high speed strain field measurement such as time synchronization accuracy as well as DIC method with poor resolution images is discussed. Finally, testing of a polymeric foam is given as an example to illustrate the needs of an accurate field measurement.

Conventional tests on the polymeric foams

Foam specimen and quasi-static tests

Conventional quasi-static compressive tests on polymeric foams can be performed using a universal testing Machine. A polymeric foam supplied by EADS company is studied as an example. The density is 70 kg/m 3 , the diameter of unit cell is about 0.8 mm and the thickness of the cell wall is about 50 µm from microscopy observations. The specimen tested in this study is a cylinder of 60 mm in diameter and 40 mm in height.

MTS 810 material testing system is used. The specimen is centrally sandwiched between the two platens. The lower one rises at a constant speed of 0.2 mm/s during loading, which ensures the nominal strain-rate of 5×10 -3 s -1 . Three tests are carried out and repeatability is checked (see in Fig. 1). This mechanical response is typical for cellular materials which exhibits a linear Figure 1: Three nominal stress -nominal strain curves from quasi-static loading increase or elastic period, followed by a very long plateau and a densification phase.

Impact tests with SHPB

The dynamic tests are performed on nylon Hopkinson bar. A typical SHPB is made of long input and output pressure bars with a short specimen sandwiched between them. The impact between the projectile and the input bar generates a compressive longitudinal incident pulse ε i (t) in the input bar. Once this incident pulse reaches the specimen-bar interface, one part is reflected as the reflected pulse ε r (t) in the input bar and the other part transmitted as the transmitted pulse ε t (t) in the output bar. With the gauges glued on the input bar and output bar, one can record these three pulses.

The transmitted wave can be shifted to the output bar-specimen interface to obtain the output force and velocity, at the same time, the input force and velocity can be determined via incident and reflected waves shifted to the input bar-specimen interface. Forces and velocities at both faces of the specimen are then given by the following:

F input = S B E(ε i (t) + ε r (t)), F output = S B E(ε t (t)) and V input = C 0 (ε i (t) -ε r (t)), V output = C 0 ε t (t) (1)
where F input , F output , V input , V output are forces and particle velocities at the input and output interfaces, S B , E and C 0 are the cross-sectional area, Young's modulus and longitudinal wave speed in the pressure bars respectively. ε i (t), ε r (t) and ε t (t) are the strain signals at the barspecimen interfaces.

SHPB system used for this study consists of two 3 m long nylon bars of 62 mm diameter (density 1200 kg/m 3 and wave speed 1800 m/s). The use of the nylon bars multiply by 20 the impedance ratio between the foam and the bar material(thus the signal/noise ratio), in comparison with a classical steel bar (density being 7850 kg/m 3 and wave speed 5000 m/s). However, with the use of big diameter visco-elastic bars, the reasoning in Eq. ( 1) should be amended by considering wave dispersion effect in the shift from measuring points to the bar/specimen interfaces [START_REF] Zhao | A three dimensional analytical solution of the longitudinal wave propagation in an infinite linear viscoelastic cylindrical bar. application to experimental techniques[END_REF][START_REF] Zhao | On the use of shpb techniques to determine the dynamic behavior of materials in the range of small strains[END_REF]. The correction of this dispersion effect on the basis of a generalized Pochhammer's wave propagation theory is consequently performed in data processing [START_REF] Zhao | On the use of a viscoelastic split hopkinson pressure bar[END_REF].

As the coefficients between the strain and recorded tension may be influenced by many factors, such as gauges factors, temperature, humidity or gain of the amplifier, calibration of these coefficients is very important for the overall precision. The principle of calibration for the input bar is on the basis of the energy conservation, i.e., the kinetic energy of the impactor is equal to the sum of the kinetic energy and the elastic energy carried by the incident impulse in the input bar. This is valid if the input bar and the impactor have the same properties (diameter, material).

The calibration of the output bar is realized by the force balance during a test without specimen. Here, the forces can be calculated as Eq. ( 1) and should be exactly the same because they indicate the force at the same interface (input bar/output bar interface). Therefore, the calibration coefficient for the gauge on the output bar is ascertained.

From the forces and velocities at both bar-specimen interfaces, nominal stress and nominal strain can be obtained following the standard formulas of SHPB analysis, which assumes the uniformity of stress and strain fields:

σ s (t) = Foutput Ss ε s (t) = t 0 Voutput(τ )-Vinput(τ ) ls dτ (2)
The present Hopkinson bars experiments followed this procedure with wave dispersion correction and gauge calibrations. Before every test, the interfaces of bar-specimen are lubricated to decrease the frictional effect as much as possible. Input and output forces history is examined to check whether the dynamic equilibrium is achieved. An example is given in Fig. 2 for a test at 10 m/s. The equilibrium is established around 0.3 ms. The mechanical response of the foam at 18 m/s is represented in Fig. 3 and is compared to the quasi-static case. One can notice that, for dynamic loading, the densification phase is not reached, due to the impulse length generated in the SHPB system.

Following the same procedure, seven specimens are tested using nylon SHPB with the projectile speeds ranging from 7 m/s to 19 m/s. The plateau stress remains nearly constant in the range of impact loading speeds (Fig. 4), from 5×10 -3 s -1 to 4.5×10 2 s -1 . However the average strength increased of 8% from quasi-static loading to impact loading.

Digital photography and image correlation

As mentioned in the introduction, digital photography and the associated digital image correlation is nowadays widely used to observe the deforming mode of the specimen, to monitor the loading condition and to calculate the strain field. Even though the equilibrium state within the specimen is reached, heterogeneous strain field is still possible with a localized area such as the case of honeycombs for example.

Direct image observations

For quasi-static loading, a Canon EOS 50D camera is used and triggered with an equal timeinterval signal (3.5 s). Fig. 5 shows the deforming states of the specimen at different crush displacements. The deformation of the foam is highly localized in the darker areas, visible next to the platens. A simple image processing is made with such images. By locating the boundary of the darker area, the length of the uncollapsed part can be obtained, which is plotted in Fig. 6. The x-axis is the actual specimen length obtained by subtracting the prescribed crush displacement from initial length of the specimen. In this figure, the markers * are corresponding respectively to the last three images of Fig. 5 (the darker band is too small to be processed on the first image). It can be seen that the length of the uncollapsed part decreases linearly.

One can find that the strain is highly localized in the collapsed areas and the crush displacement is mainly obtained by the increase of these areas. Along with the stress-strain curve in Fig. 3, one may conclude that the specimen collapses little by little at the ends as long as the crush displacement goes and it thus manifests a plateau in the stress-strain curve. Whether this conclusion is correct or not, it needs further examination of the strain field on the surface of the specimen.

Principle of digital image correlation

More sophisticated image processing can be used to match the material points in the image sequence and to calculate displacement fields. Consequently, a quantitative strain fields can be offered.

DIC code Correli Q4 , developed in LMT-Cachan is used. It is a global approach of image correlation based on the conservation of optical flow. Considering a reference image f and a deformed image g, the conservation of optical flow can be written as (in 1D for brevity):

f (x) = g [x + u(x)] (3)
where u(x) is the motion from the reference image to the deformed image. When the motion is small, the first order Taylor expansion of g [x + u(x)] yields to:

g [x + u(x)] ≈ g(x) + g ′ • u(x) (4)
To measure the motion u(x), a minimization of the gray level residual is therefore introduced on the region of interest (ROI):

η 2 = ROI {f (x) -g [x + u(x)]} 2 dx (5)
Assuming that the motion u(x) is expressed in a space of basis functions, {ψ p }, i.e.

u(x) = u p ψ p (x) (6)
where u p is the unknown degrees of freedom. By substituting Eq. ( 6) to Eq. ( 5), it thus obtains:

η 2 = ROI f (x) -g x + u p ψ p (x) 2 dx (7)
And considering the first order approximation, Eq. ( 4), the minimization of the object function becomes:

η 2 lin = ROI f (x) -g(x) - u p ψ p (x)g ′ (x) 2 dx ( 8
)
An iterative scheme is followed to obtain the best estimation of u p , and thus u(x). Furthermore, the strain field in the ROI is obtained ( [START_REF] Grédiac | Mesures de champs et identification en mécanique des solides[END_REF][START_REF] Besnard | finite-element" displacement fields analysis from digital images: Application to portevin-le chatelier bands[END_REF][START_REF] Hild | Digital image correlation: from displacement measurement to identification of elastic properties-a review[END_REF]). In this case, Q4P1-shape functions are chosen as the basis functions so that the measured displacement field is consistent with FE analysis computations.

Strain field analysis

Displacement and strain fields on the surface of the specimen are calculated by DIC. Only the uncollapsed part is chosen as ROI. Indeed, the highly densified dark zone leads to convergency problem in DIC algorithm because the gradient of gray is nearly zero in this area. It might be worth noting that 2D image correlation generally requires the object moving in a plane parallel to the camera sensor, which is not the case with the present cylindrical specimen. Nevertheless, the strain in transversal direction is not our interest here. Moreover, as Poisson ratio is low, transversal strain is limited and does not affect the longitudinal strain. As a result, the strain analysis with DIC on the cylindrical specimen is still acceptable to provide the longitudinal strain.

32 pixels × 32 pixels element size is adopted for quasi-static cases. The ROI and the strain fields are shown in Fig. 7, in which the strain fields correspond to the images in Fig. 5. The results show the non-uniformity of strain even within this apparently uncollapsed part.

It might be interesting to compare the nominal strain with the DIC strain measurement of this uncollapsed part of specimen. The evolutions of the global nominal strain and the average value of the DIC strain are plotted with respect to the image number in Fig. 8. Great disparity is found between these two curves. The nominal strain increases rather linearly and the average local strain remains constant at a value of about 2.2%. The DIC strain measurement confirms that a localized deformation area is created when the specimen is compressed to a critical strain around 2.2%.

High Speed digital image system

For the Hopkinson bar tests, a Photron Fastcam SA5 high-speed camera is used to record the deforming process of the specimen during impact tests. Image sequences with a 384 pixels×432 pixels resolution at 40,000 f ps is acquired. Shutter speed is set to 1/70,000 s and aperture f5.6 so that the obtained images are sharp enough. Thanks to two powerful cold spotlights, the images are ensured to a suitable contrast in despite of the small aperture, high shutter speed and frequency.

Compared to the static case, there are no apparent dark areas in the image sequences from dynamic experiments. An example is given in Fig. 9 for a prescribed velocity of 10 m/s. Instead, a lot of fragments and localized bands are found.

d=1.99 mm d=7.57 mm d=11.82 mm Indeed, as u and v share the same source, synchronization could be established by shifting them in time axis. The procedure is as following: 1) interpolating one of them acquired with lower frequency to the higher one, i.e. making sure that the two series have same time interval between two adjacent points, u(i) and v(i) are the interpolated results; 2) make a convolution between u(i) and v(i), and find the maximum location of convolutional result. The convolution is realized in detail as:

w(k) = j u(j) • v(n -k + j) (9)
where n is the length of series v(i). The maximum convolutional result, w(k), is corresponding to the location where u(i) and v(i) are best superposed and the time is thus needed to shift in order to achieve the synchronization. To trace the displacement of one point on the bar (the cross located in the central element in Fig. 12), a small element of 8 pixels × 8 pixels (about 1 mm × 1 mm in reality) is adopted to calculate the displacement of the bar (see the solid square in Fig. 12). The 8 surrounding elements help for this calculation. It is stressed that the strain of the small element on the bar is ignored, thus the displacement of the bar is calculated by averaging displacements of its four nodes. Generally, the point traced is very close to the bar-specimen interface, usually less than 2 mm. Therefore, the displacement of the bar in this point is regarded as the displacement of the interface. It is worthwhile to note that when the strain can be accurately measured in the bars with camera technical evolution, a direct stress/force measurement at the ends of the bars will be available. The synchronization method introduced in section 4.2 gives a correspondance between images and experimental data from Hopkinson bars. Therefore, the displacements of both specimen-bar interfaces synchronized are plotted in Fig. 13(a). The two solid lines are the displacements of bar-specimen interfaces measured by SHPB, while the dash lines are the displacements calculated with DIC tracking method (section 4.3). A significant gap is observed in the displacements at the input end, the displacement at the output end is too small to observe an obvious gap. However, the superpositions are well achieved if two images shift on the basis of the synchronization by MCDL method, as shown in Fig. 13(b).

Consequently, the conclusion can be made as that: firstly, both the displacements from SHPB and DIC tracking method are credible because of the possibility of good superpositions, there is no error of the coefficient calibrations, nor DIC parameter mistake; secondly, the synchronization method with MCDL box is not very accurate. In fact, the delay of several images in synchronization exists in all the experiments, not only in Nylon pressure bars presented in this paper but also in tensile bars in our laboratory. As a result, the examination of synchronization with DIC analysis is necessary in the case of high speed camera.

DIC measurements

For the uncollapsed part of the specimen, the DIC measurement is possible. In contrast to static case, elements size of 8 pixels × 8 pixels (1 mm × 1 mm in reality) are employed in DIC algorithm. The chosen ROI along with the undeformed reference image is shown in Fig. 14.

These strain fields corresponding to the images shown in Fig. 9 are also presented in Fig. 14. They manifest that even the uncollapsed part of the specimen is not compressed uniformly because each of these three strain fields shows color difference or color strips. Some highly deformed spots scatter in the fields, which reveals that some cells are largely compressed or collapsed. Moreover, the color difference becomes larger for image from left to right (from small crush displacement to the large one) and the number of the highly deformed spots increases as well. Similarly to the quasi-static case, the global nominal strain and average value of the DIC local strain are compared, see in Fig. 15. The same conclusion can be made as the quasi-static one while the plateau strain is about 1.9%. With the knowledge of both the strain localization feature and the plateau of the mean local strain, the real deforming process of this polymeric foam is unveiled. At the beginning of the loading, the foam is firstly compressed as a whole where the stress increases linearly. With the increase of the strain, the limit of failure stress of cells is reached and they begin to collapse. From then on, the collapse front propagates while the uncollapsed part stops being further compressed, which displays the plateaus in the mean local strain and in nominal stress.

Conclusion

This paper deals with impact testing on the polymeric foams using large diameter nylon Hopkinson bar technique and optical DIC strain field measurement. Even though the overall stress -strain curves under impact loading can be already obtained by SHPB measurement, a field measurement is still necessary because of heterogeneous fields within the foam specimen.

High speed images together with the digital image correlation techniques is a complementary measurement to SHPB technique. They are not only used to evaluate the strain field of the foam specimen, but also to obtain the displacement history at the ends of pressure bars. By comparing this measurement with the ones from Hopkinson bars, it allows to correct the time synchronization error and to check the calibration of the strain/tension coefficients.

For the studied brittle polymeric foam, an overall rate sensitivity of 8% is found from quasistatic loading to impact loading with a speed of 20m/s. Even though the equilibrium state (between forces of two bar/specimen) is reached rapidly in SHPB test, there is no homogeneous strain field. The DIC strain field measurement is the only mean to reveal this crushing mechanism like a honeycombs, i.e., the crush displacement (nominal strain) is mainly due to the enlargement of the very localized area/bands within the specimen and the uncollapsed part remains at a constant small strain.

Figure 2 :

 2 Figure 2: Forces history at specimen interfaces for a test at 10 m/s. Force equilibrium is checked.

Figure 3 :

 3 Figure 3: Comparison of nominal stress -nominal strain behavior under impact loading and quasi-static loading

Figure 4 :

 4 Figure 4: Mean plateau stress vs impact speed under both dynamic and quasi-static loading

Figure 5 :

 5 Figure 5: The deforming states at different crush displacements under quasi-static loading

Figure 6 :

 6 Figure 6: Evolution of the uncollapsed central part length with the current length of specimen, where the stars mark the values corresponding to the last three images of Fig. 5

Figure 7 :

 7 Figure 7: ROI calculated in quasi-static loading and the longitudinal strain fields

Figure 8 :

 8 Figure 8: Comparison of the global nominal strain and the mean local strain in the quasi-static loading

Figure 9 :

 9 Figure 9: The deforming states at different crush displacement under impact loading

Figure 10 :

 10 Figure 10: Diagram of SHPB system and the digital image photography system

Figure 11 :

 11 Figure 11: Diagram of the relationship between stress-strain curve and images via MCDL

Figure 12 :

 12 Figure 12: The illustration of the points on the bars traced by DIC

Figure 13 :

 13 Figure 13: Comparison of displacements-time profiles obtained by SHPB and DIC tracking method: a) synchronized directly using the method (MCDL) detailed in section 4.2; (b) two images shifted to make coincidence.

Figure 14 :

 14 Figure 14: ROI calculated in impact loading and the longitudinal strain fields which are corresponding to the images in Fig. 9

Figure 15 :

 15 Figure 15: Comparison of the global nominal strain and the mean local strain in the dynamic loading

Acknowledgements

The authors acknowledge the support of China Scholarship Council (CSC) as well as NSFC funding no11228206, and appreciate the kindness of EADS in supplying the specimens.