Davy Dalmas

Claudia Guerra

Julien Scheibert

Daniel Bonamy
email: daniel.bonamy@cea.fr

Damage mechanisms in the dynamic fracture of nominally brittle polymers

Keywords: Dynamic fracture, amorphous polymers, fracture energy, fractography, high-deformation rate, microcracks, PMMA

Linear Elastic Fracture Mechanics (LEFM) provides a consistent framework to evaluate quantitatively the energy flux released to the tip of a growing crack. Still, the way in which the crack selects its velocity in response to this energy flux remains far from completely understood. To uncover the underlying mechanisms, we experimentally studied damage and dissipation processes that develop during the dynamic failure of polymethylmethacrylate (PMMA), classically considered as the archetype of brittle amorphous materials. We evidenced a well-defined critical velocity along which failure switches from nominally-brittle to quasibrittle, where crack propagation goes hand in hand with the nucleation and growth of microcracks. Via postmortem analysis of the fracture surfaces, we were able to reconstruct the complete spatiotemporal microcracking dynamics with micrometer/nanosecond resolution. We demonstrated that the true local propagation speed of individual crack fronts is limited to a fairly low value, which can be much smaller than the apparent speed

Introduction

Dynamic crack propagation drives catastrophic material failure and is usually described using the Linear Elastic Fracture Mechanics (LEFM) framework [START_REF] Freund | Dynamic Fracture Mechanics[END_REF][START_REF] Chandar | An experimental investigation into dynamic fracture-ii microstructural aspects[END_REF]. This theory considers the straight propagation of a single smooth crack and assumes all energy dissipating processes to be entirely localized in a small zone at the crack tip, so called fracture process zone (FPZ). Crack growth velocity v is then selected by the balance between the mechanical energy that flows within the FPZ per time unit and the dissipated energy within the FPZ over the same time unit. This yields [START_REF] Freund | Dynamic Fracture Mechanics[END_REF]:

Γ (1 -v/C R ) K 2 (c)/E, (1)
where C R and E are the Rayleigh wave speed and the Young modulus of the material, respectively, Γ is the fracture energy, and K(c) is the Stress Intensity Factor (SIF) for a quasi-static crack of length c. K only depends on the applied loading and the cracked specimen geometry, and entirely characterizes the stress field in the vicinity of the crack front.

LEFM predictions agree well with observations as long as the crack growth is sufficiently slow [START_REF] Bergkvist | Some experiments on crack motion and arrest in polymethylmethacrylate[END_REF]. However, large discrepancies are reported at high speed (Ravi- [START_REF] Chandar | An experimental investigation into dynamic fracture-ii microstructural aspects[END_REF][START_REF] Bouchbinder | The physics of simple cracks[END_REF]. In particular, the maximal crack speeds attained experimentally in amorphous materials are far smaller (typically by a factor of two) than the limiting speed C R predicted by Eq. 1.

The existence of a micro-branching instability [START_REF] Fineberg | Instability in the propagation of fast cracks[END_REF]) at a critical velocity v b (typically of the order of 0.35 -0.4C R ) permits to explain part of this discrepancy: Beyond v b , the crack front splits into a multiple crack state that cannot be described by Eq. 1 anymore. However, it is commonly stated [START_REF] Sharon | Confirming the continuum theory of dynamic brittle fracture for fast cracks[END_REF] that LEFM works well below v b , or more generally when microbranching instabilities are absent. As a matter of fact, most recent works focused on failure regimes beyond v b (see e.g. [START_REF] Gumbsch | Molecular dynamics investigation of dynamic crack stability[END_REF][START_REF] Adda-Bedia | Dynamic instability of brittle fracture[END_REF][START_REF] Henry | Dynamic instabilities of fracture under biaxial strain using a phase field model[END_REF][START_REF] Bouchbinder | Branching instabilities in rapid fracture: Dynamics and geometry[END_REF][START_REF] Spatschek | Phase field modeling of fast crack propagation[END_REF][START_REF] Henry | Study of the branching instability using a phase field model of inplane crack propagation[END_REF]). Still, several observations reported at lower speeds remain puzzling. In particular, even for velocities much lower than v b , the measured dynamic fracture energy is generally found to be much higher than that at crack initiation [START_REF] Sharon | Confirming the continuum theory of dynamic brittle fracture for fast cracks[END_REF][START_REF] Kalthoff | Dynamical stress inetnsity factors for arresting cracks in dbc specimens[END_REF][START_REF] Rosakis | The determination of dynamic fracture toughness of aisi 4340 steel by the shadow spot method[END_REF][START_REF] Fond | Dynamic fracture surface energy values and branching instabilities during rapid crack propagation in rubber toughened pmma[END_REF].

The experiments reported here were designed to better understand the mechanisms that select crack velocity in dynamic fracture. In this context, we developed an experimental setup (presented in section 2) that permits to characterize over a wide range of crack speeds, in particular speeds below v b , the dissipative and damage processes that develop in polymethylmethacrylate (PMMA), classically considered as the archetype of brittle amorphous materials. This setup allowed us to evidence a novel critical velocity, smaller than v b , beyond which crack propagation goes hand in hand with the nucleation and growth of microcracks ahead of the main crack front (section 3). Via accurate post-mortem analysis of the patterns let on fracture surfaces, we were able to reconstruct the full spatio-temporal dynamics of these microfailure events (section 4). Their statistics and their dependency with crack tip loading have been characterized (section 5). In section 6, we will show how microcracking acts and selects the apparent velocity measured at the continuum-level scale.

Experimental setup

Fracture tests were performed in the wedge splitting geometry [START_REF] Bruhwiler | The wedge splitting test: A method for performing stable fracture mechanics tests[END_REF][START_REF] Karihaloo | Higher order terms of the crack tip asymptotic field for a wedge-splitting specimen[END_REF] depicted in Fig. 1. Specimens were prepared from PMMA parallelepipeds of size 140 × 125 × 15 mm 3 in the propagation (x-axis), loading (y-axis), and thickness (z-axis) directions, respectively. Table 1 gives the main characteristics of the PMMA used in this study. Subsequently, a notch was shaped (i) by cutting a 25 × 25 mm 2 rectangle from the middle of one of the 125 × 15 mm 2 edges; and (ii) by subsequently adding a 10-mm groove deeper into the specimen. Two steel blocks equipped with rollers were then placed on both sides of this notch and the specimen was loaded by pushing a wedge (semi-angle of 15 • ) at a small constant velocity (40 µm/s) between these two blocks. This permits (i) to spread the loading force over a large contact area and prevent any plastic deformation of PMMA at the loading contacts; and (ii) to suppress friction in the system. As a result, the vicinity of the crack tip can be assumed to be the sole dissipation source for mechanical energy in the system.

In such a wedge splitting geometry, the SIF decreases with crack length. To obtain dynamic failure, we then introduced a circular hole of tunable diameter from 2 mm to 8 mm at the tip of the seed crack. This hole blunts the seed crack and delays the propagation onset. This increases the mechanical energy stored in the sample at the onset of crack propagation. It allowed us to widen the range of SIF (from 0.9 MPa. √ m to 4.5MPa.

√ m) and that of crack velocity (from 75 m/s ≈ 0.08C R to 500 m/s ≈ 0.57C R ) accessible in the experiments.

The apparatus used to carry out the experiments is homemade (Fig. 1). One side of the machine is fixed while the other moves via a stepper motor (Oriental motor EMP400 Series) allowing incremental displacements Specimen loading is achieved by translating horizontally the slide link, from (0) to (1), over a distance u wedge selected so that the horizontal force applying on the slide link is half that measured experimentally. Bottom-Left: Zoom on the meshing in the transition region (red) between coarse meshing (1 mm mesh size) in the bulk and fine meshing close to the crack tip. On the right is shown part of the circular hole at the seed crack tip. Bottom-Right: Zoom on the crack-tip region (green), meshed with a size of 1 nm. as small as 40 nm. The compressive force is measured, up to 20 kN via a S-type Vishay load cell 363 Series. In all performed tests, both the displacement and the force are recorded via a computer with a time resolution of 1 s. In particular, the value of the applied force at propagation onset is recorded and later used in the finite element analyses described thereafter.

To measure the instantaneous crack velocity, we used a modified version of the electrical resistance grid technique [START_REF] Cotterell | Fracture propagation in organic glasses[END_REF][START_REF] Anthony | The crack branching velocity[END_REF][START_REF] Boudet | Dynamics of crack propagation in brittle materials[END_REF][START_REF] Fineberg | Instability in the propagation of fast cracks[END_REF]. A series of 90 parallel conductive lines (2.4-nm-thick Cr layer covered with 23-nm-thick Au layer) were deposited on one of the two 140 × 125 mm 2 sides of the specimen using magnetron sputtering deposition through a polymer mask glued on the sample surface. Stripes width and length are 500µm and 100 mm, respectively, and they are separated by gaps of 500µm. Once the sample is installed in the compression machine, each line is connected in series with a resistor of 30 kΩ -this value was chosen to be extremely high with respect to that of the strip (about few ohms) so that the latter is negligible. As the crack propagates throughout the specimen, it successively cuts the conductive lines, which, each time, makes the global system resistance increase by a constant increment. The time locations of these jumps are detected via an oscilloscope (acquisition rate: 10 MHz) through a voltage divider circuitry. In order to increase the accuracy of jump detection, we used the four channels of the oscilloscope in cascade with different offsets, so that each channel is triggered as the previous one gets out of its range (Fig. 2:A andB). As a result, we obtained the crack length c(t) as a function of time with space and time accuracies of 40 µm and 0.1µs, respectively (Fig. 2:C). The profile of instantaneous velocity v(c) as a function of crack length can then be easily deduced (Fig. 2:D).

Finite element analysis was used to estimate the SIF evolution during failure. Figure 3:Top represents the meshing of the complete system. The average mesh size is 1 mm and reduces logarithmically down to 1 nm at the crack tip (Fig. 3:Bottom), in order to resolve in space the crack tip opening. Only half of the system is required for symmetry reasons. Young modulus and Poisson ratio are set to 2.8 GPa and 0.36, respectively (see Tab. 1). The boundary conditions are the following: (i) The left edge is perfectly adhesive to a polymeric layer of thickness 5 mm, Young's modulus 3 GPa and Poisson ratio 0.41; (ii) The left edge of the layer has a no-displacement condition; (iii) On the right, the notch is loaded through a L-shaped block, of thickness 5 mm, Young modulus 300 GPa and Poisson ratio 0.4, perfectly adhering to the sample; (iv) The L-shaped block is bounded to a virtual roller, the center of which can only move along a line parallel to the wedge side, at a distance equal to the roller radius; (v) The crack edge is stress free; and (vi) Vertical displacement is forbidden along the uncracked part of the symmetry plane (top line of Fig. 3:Top). For each sample, the equilibrium position of the wedge yielding the measured applied load at the onset of crack propagation was determined using a plane stress static finite element code (Cast3M). Quasi-static crack propagation is then simulated by increasing the crack length while imposing the wedge position to remain constant. This latter assumption is experimentally justified. Indeed, it has been observed that the load cell signal started to be modified only after a few hundreds of microseconds (typical time of an experiment) after crack initiation1 . This is attributed to the time required for the sound waves to travel from the crack tip to the load cell. As a result, the wedge position can be assumed to be practically constant during crack propagation. Static SIF is then determined, for each value c of crack length, using the J-integral method [START_REF] Rice | A path independent integral and the approximate analysis of strain concentration by notches and cracks[END_REF]. The validity of our SIF calculation was benchmarked against literature results for wedge-splitting configurations: Our code agreed to better than 2% with the results of [START_REF] Karihaloo | Higher order terms of the crack tip asymptotic field for a wedge-splitting specimen[END_REF] when using their system parameters.

To image the post-mortem fracture surfaces, we used a Leica DM2500 microscope. Most of the images were taken with a ×5 objective under polarized light. The resulting images are 1.4 × 1 mm 2 in area, with a pixel size of 677 nm. We also imaged the fracture surfaces with an interferometry profilometer (FOGALE Nanotech) which allows one to gather topographical information. These 3D images were taken with a ×5 objective. Their area is 1.4 × 1 mm 2 with a pixel size of 1.86 µm. 3 Selection of fracture energy with velocity and evidence of a brittle/quasibrittle transition In each of the experiments performed, both the SIF profile K(c) and the instantaneous velocity profile v(c) have been determined (Figs. 4:Left and center). From these curves, one can derive the profiles of fracture energy, Γ (c), using Eq. 1 (Fig. 4:Right). We note here that Eq. 1 is strictly valid in the case of crack propagation in an elastic half-space, i.e. in the absence of waves emitted by the crack and coming back to it after reflection on sample boundaries [START_REF] Freund | Dynamic Fracture Mechanics[END_REF]; [START_REF] Goldman | Acquisition of inertia by a moving crack[END_REF]). We argue that our experiments are essentially unaffected by such reflected wave for at least two reasons: i) as measured by [START_REF] Boudet | Experimental study of the instability of crack propagation in brittle materials[END_REF], the sound energy emitted by a crack in PMMA lies within the range 1 -4 J/m 2 , a very small value compared to the energy required to fracture the material (a few kJ/m 2 , see below); and ii) this radiated energy is quickly dissipated within the material's bulk -The acoustic attenuation coefficient has been measured to be 0.67 dB/(cm.MHz).

Figure 5 superimposes the various curves Γ vs. v measured in the various experiments performed at various levels U 0 of the initially stored mechanical energy. Several regimes can be evidenced:

-For small v, Γ roughly remains constant, close to K 2 c /E, as expected within standard LEFM; -As v increases and reaches a first critical velocity v a ≈ 165 m/s ≈ 0.19C R , Γ suddenly increases to a value about four times larger than

Γ (v = 0) = K 2 c /E; -Beyond v a , Γ slowly increases with v; -As v reaches a second critical velocity v b ≈ 350 m/s ≈ 0.36C R , Γ starts to increase rapidly again. It seems to diverge at v ≈ 450 m/s ≈ 0.5C R .
Note also the fairly good collapse of the different curves Γ (v) below v b , and the large dispersion above. This suggests that Eq. 1 is relevant for v ≤ v b , -provided a suitable velocity dependence Γ (v) is ascribed-, but not beyond. The second critical value v b ≈ 350 m/s ≈ 0.36C R is found to correspond to the onset of microbranching instability widely discussed in the literature (see e.g. [START_REF] Fineberg | Instability in the propagation of fast cracks[END_REF][START_REF] Sharon | Confirming the continuum theory of dynamic brittle fracture for fast cracks[END_REF]). The first critical value v a ≈ 165 m/s ≈ 0.19C R was observed in our series of experiments and reported in [START_REF] Scheibert | Brittle/quasi-brittle transition in the dynamic fracture of nominally brittle materials: An energetic signature[END_REF] for the first time. This observation was made possible by the use of the wedge-splitting geometry, which is a decelerating crack configuration. It offers many data points at relatively low velocities, contrary to most other devices used before. The rapid increase of Γ with v around v a provides a direct interpretation for the repeated observations of cracks that span a large range of Γ , while keeping a nearly constant velocity about 0.2C R (see e.g. (Ravi- [START_REF] Chandar | An experimental investigation into dynamic fracture-ii microstructural aspects[END_REF][START_REF] Ravi-Chandar | On the role of microcracks in the dynamic fracture of brittle materials[END_REF]).

To shed light on the nature of the transition at v = v a evidenced on the curve relating Γ to v, we examined the post-mortem fracture surfaces and their evolution as crack speed increases (Fig. 6). For v smaller than v a , fracture surfaces remain smooth at optical scales. But above this threshold, conic marks start to be observed. These marks remain confined within a roughness scale of few micrometers at the fracture surfaces, contrary to the microbranches that deeply develop into the specimen bulk above v b (see side views and topographical images in Fig. 7).

Similar conic marks were reported in the fracture of other brittle materials, among which polystyrene [START_REF] Regel | O mekhanizme kheupkogo razusheniya plastmass[END_REF], oxide glasses [START_REF] Smekal | Zum bruchvorgang bei sprodem stoffverhalten unter ein-and mehrachsigen beanspruchungen[END_REF][START_REF] Holloway | The fracture of glass[END_REF], cellulose acetate (Kies and abd G. R. [START_REF] Kies | Interpretation of fracture markings[END_REF], polycrystalline materials [START_REF] Irwin | Fracturing and fracture dynamics[END_REF], or homalite (Ravi- [START_REF] Ravi-Chandar | On the role of microcracks in the dynamic fracture of brittle materials[END_REF]. They are classically associated [START_REF] Smekal | Zum bruchvorgang bei sprodem stoffverhalten unter ein-and mehrachsigen beanspruchungen[END_REF][START_REF] Ravi-Chandar | On the role of microcracks in the dynamic fracture of brittle materials[END_REF][START_REF] Rabinovitch | Origin of mist and hackle patterns in brittle fracture[END_REF] with the presence of microcracks that nucleate and grow ahead of the main crack front, and subsequently coalesce with it.

Figure 8 shows the surface density ρ of conic marks as a function of crack speed v. Below v a , no mark is observed, irrespectively of the chosen magnification (from ×5 to ×50). Above this value, ρ increases almost linearly with v -v a . The precise correspondence between the critical velocity v a at which the curve Γ (v) exhibits a kink and that at which conic marks start to be observed suggests that both phenomena are the signature of the same transition. Above v a , PMMA failure switches from nominally brittle to quasi-brittle and goes hand in hand with microcracking that develop ahead of the main front. In the following, we will use the term "Damage Zone" (DZ) to refer to the zone where microcracks develop in the vicinity of the main crack tip. We will distinguish this zone from the FPZ (smaller than the DZ) that embeds dissipative mechanisms (crazing for instance) at the tip of each (micro)crack front.

Note that the region of the fracture surface that bears conic markings also has a characteristic aspect to the naked eye, as it scatters ligth very efficiently. This is markedly different from the high reflectivity of the region that bears no microcrack. In fractography these regions are classically referred to as the mist and mirror zones, respectively (see e.g. [START_REF] Hull | Fractography: Observing, Measuring and Interpreting Fracture Surface Topography[END_REF][START_REF] Rabinovitch | Mirror-mist transition in brittle fracture[END_REF]). Our observation therefore strongly suggests that the mirror/mist transition is simply the morphological counterpart of the brittle/quasi-brittle transition that occurs at v a

Fractographic reconstruction of individual microfailure events

In order to further characterize and understand the dynamics of microcracking events which develop during Interpretation of the conic marks: Red dots sketch the successive positions of the fronts that, by interacting, will be giving rise to the green branch of the conic mark. These two microcracks form at t = 0 and t = τ , then radially grow at speeds c 1 and c 2 . Branch fitting via Eq. 2 permits to measure c 2 /c 1 . Note that a conic mark can be made of several branches. C: When c 2 = c 1 , branches are (mathematical) true conics and the nucleation distance d n between the front of the incident microcrack (1) and the nucleation point of the forming one ( 2) is twice the apex-focus distance 0 2 P 12 (Taken from Guerra et al ( 2012)).

fast fracture in PMMA (v > v a ), we proposed a numerical reconstruction based on the post-mortem analysis of the traces left by these events [START_REF] Guerra | Understanding fast macroscale fracture from microscale post-mortem patterns[END_REF]. The typical time between two successive microcracking events is about 10 ns, which makes them inaccessible using standard techniques as e.g. fast imaging or acoustic emission analysis.

The first step is to identify where microcracks have initiated. On the optical images of post-mortem fracture surfaces (Figs. 9:A andB), many well defined points of high optical reflectivity can be seen. These bright/white areas are believed to come from large plastic deformations accompanying the nucleation of microcracks. Thus, they allow us to determine precisely the position of the different nucleation sites. For some microcracks, the precise location of their nucleation site is further constrained by the convergence of fragmentation lines onto it.

The second step is to infer, from the conic patterns, the velocities at which the various microcracks have grown. It is commonly admitted in the literature [START_REF] Smekal | Zum bruchvorgang bei sprodem stoffverhalten unter ein-and mehrachsigen beanspruchungen[END_REF][START_REF] Ravi-Chandar | On the role of microcracks in the dynamic fracture of brittle materials[END_REF][START_REF] Rabinovitch | Origin of mist and hackle patterns in brittle fracture[END_REF] that the conical marks indicate the intersection points between two interacting microc-racks. Let us then consider two microcracks that propagate radially along slightly different planes with velocities c 1 and c 2 (Fig. 9:B). The intersection between the initial (micro)crack front and the secondary microcrack leaves a visible trace (i.e. a tiny height difference) on the fracture surfaces. In the coordinate system (e x , e y ) centered on the site of nucleation of the first microcrack and chosen so that the x-axis passes through the two centers of nucleation, the equation describing the mark left postmortem can be written as [START_REF] Guerra | Understanding fast macroscale fracture from microscale post-mortem patterns[END_REF]:

y ∆ = ± 2c 2 (c 2 -1) 2 c 1 τ ∆ 1 - 2x ∆ (c 2 -1) + c 2 c 1 τ ∆ 2 - x ∆ 2 + 1 (c 2 -1) 2 + c 2 + 1 (c 2 -1) 2 c 2 c 1 τ ∆ 2 - 2x ∆ 1/2 , ( 2
)
where c = c 2 /c 1 , ∆ is the distance between the two nucleation sites, and τ is the time interval between two microcracking events. The analysis of this equation shows that the number c 1 τ /∆ sets the aspect ratio of the fractographic trace, while the ratio c = c 2 /c 1 fixes its form (see [START_REF] Guerra | Understanding fast macroscale fracture from microscale post-mortem patterns[END_REF] for details). We used Eq. 2 to directly extract the ratio c = c 2 /c 1 for each couple of interacting microcracks (∼ 400 couples per image) on different optical pictures obtained at different speeds (Fig. 10). Regardless of the macroscopic crack velocity, we obtain for c a Gaussian distribution centered on 1 with a standard deviation of about 0.03 (Fig. 10). This analysis permits to demonstrate that all microcracks grow at the same velocity c m inside the DZ. Note that, up to now, nothing prevents c m to vary with v or K. The mechanism that fixes the value for c m will be discussed later in this paper (see section 6). By using c 2 = c 1 = c m in Eq. 2, the following relation can be obtained:

y ∆ = ± 4 d n (2∆ -d n ) (∆ -2d n ) 2 x ∆ 2 -4 d n (2∆ -d n ) (∆ -2d n ) 2 x ∆ + d n (2∆ -d n ) (∆ -2d n ) 2 -4 d n ∆ + 4 d n ∆ 2 1/2 , (3)
where d n = (∆ -c m τ ) represents the critical distance between the front of the incident microcrack and the center of the nucleated microcrack at the instant of its nucleation. This distance is twice the distance between the apex and the focus of the considered microcrack and, as such, d n can be directly extracted from fractographic images (Fig. 9:C).

We are now in a position to reconstruct the full dynamics of microcracking events from the knowledge of two series of parameters that can be directly extracted from fractography: The position of nucleation sites and the critical nucleation distance d n associated to each conic mark. We analyzed several areas of the fracture surfaces, corresponding to various macroscopic velocities above the microcracking onset (v > v a ). In all cases, nine optical images with partial overlap were recorded and gathered into a single large image -This ensures adequate statistics in the following analyses. The methodology to extract the data is the following:

-We record the coordinates x and y of the nucleation sites; -We estimate the nucleation distance, d n , by inferring, for each nucleation site, who was the parent microcrack from the relative position of the conics apex with respect to the nucleation site (family criterion). Then, d n is twice the apex-nucleation center distance.

Following Ravi- [START_REF] Ravi-Chandar | On the role of microcracks in the dynamic fracture of brittle materials[END_REF], the reconstruction is initiated with an originally straight crack front positioned at the left of the zone of analysis. This front propagates from left to right, at a constant velocity of 1 pixel/timestep. Once the distance between one of the microcrack nucleation sites and the primary crack front reaches d n , a secondary microcrack starts to √ m (right). The blue part corresponds to uncracked material and we ascribed an arbitrary gray level to each of the growing microcrack. From this, one can deduce the intersection points between microcracks that coincide with the observed conics marks on the fracture surfaces. The size of the two zones of analysis is 2.5×2.5 mm 2 . The macroscopic crack propagates from left to right.

grow radially at the same velocity. The front is now the combination of both the translating straight front and the radially growing circular front. Beyond a given time, these two fronts intersect and define the position of the conic mark. When the growing front reaches again a distance d n associated with another site, a new micro- crack is nucleated. The procedure is repeated until all the nucleation sites have been triggered.

Figure 11 shows two examples of reconstruction for two different macroscopic velocities and Fig. 12 superimposes the conic marks obtained by the simulation to those really observed in fractography. We note the fairly good agreement between experiment and reconstruction away from the edges (in the center of the images). This permits to validate the method. Thanks to it, we are able to reconstruct deterministically the dynamics of micro-damage during fast fracture in PMMA from the observation and analysis of the patterns observed on the post-mortem fracture surfaces. We emphasize the resulting resolution of ∼ 1 µm (1 pixel) in space, and of 10 ns in time (pixel size/c m with c m ≈ 200 m/s as it will be demonstrated further in the paper). This is far better than what can be obtained via conventional experimental mechanics methods as e.g. acoustic emission or fast imaging methods.

Note however some discrepancies between experiments and reconstructions on the left, top and bottom edges of the analyzed zones. Such finite size problems are unavoidable. Along the top and bottom sides of the image, the influence of microcracks outside the field of view are naturally ignored. Also, a non-realistic straight vertical front has been used, as we cannot predict the precise instants at which the leftmost centers have turned to microcracks during the reconstruction. In order to test the sensitivity of the reconstruction to the initial front shape, we have performed different simulations with the same inputs except the initial front shape. In Fig. 13, three different cases are tested : (i) a straight vertical front, (ii) a sinusoidal front with an amplitude and a period equivalent to the mean value of a wavy front far from the first step in previous reconstruction and (iii) the same sinusoidal front vertically shifted by half a period. We can observe that, except of course at the very beginning of the simulation (i.e. on left side of the images), the difference between these three cases are very small. Moreover, these small differences tend to disappear as the front to propagate, i.e. as more and more microcracks are involved. In some cases, we have observed differences propagating over the whole image. Perfect reconstructions would be reached only by analyzing the whole fracture surfaces, and not some partial areas. Knowing all the history of crack propagation is the only solution for the determination of the real initial front shape which could allow a perfect reconstruction.

Quantitative selection of the microdamaging state by crack loading

The reconstruction method, described in the previous section, shows that the dynamics of micro-damage is entirely determined by the position of nucleation sites, the nucleation distances d n , and the microscopic velocity of the fronts c m . To understand what selects these three parameters, they will be explored in more detail.

Spatial distribution of nucleation centers

To characterize the spatial distribution of nucleation sites, we computed the distribution of the number of sites that can be observed on a square of a given size a. Figure 14 shows these distributions for two different fractographic pictures, i.e. for two values of K. These distributions follow a Poisson distribution with a single parameter ρa 2 where ρ is constant over a given fractographic picture but varies with K. This reflects a ho- mogeneous and uncorrelated distribution of the sites all along the post-mortem fracture surface. Then, ρ can be identified with the surface density of nucleation sites and fully characterizes their spatial distribution.

Figure 15 presents the evolution of ρ with loading K. The density is naturally zero for K smaller than K a (SIF value at microcracking onset, i.e. when v = v a ) as there is no microcracking. Beyond this value, it increases with K. Also, this curve exhibits large fluctuations and seems to saturate when the density reaches a value ρ sat ≈ 45.5 mm 2 , beyond K b . Note that this latter value coincide with microbranching onset and, therefore the ρ(K) curve does not really make sense above K b since the front has splitted into various microbranches, which prevents us from correlating the macroscopic K value to the number of conic marks on the main crack only.

To account for the curve ρ(K), let us consider (Scheibert et al, 2010) that a discrete population of weak localized zones is present within the material with a bulk density ρ v . Let us then assume that a microcrack can nucleate in these areas provided the two following conditions are fulfilled:

-The local stress at the considered zone is large enough, i.e. larger than a given threshold value σ * (smaller than the yield limit σ Y of the material); -The considered zone is far enough from the main crack front to allow the microcrack to grow, i.e. at a distance greater than d a .

The density ρ of nucleated microcracks per unit area of fracture is then given by ρ v {h ⊥ -d a } where h ⊥ is the thickness of the layer where the stress has exceeded σ * (h ⊥ is measured perpendicular to the mean plane of fracture). The singular form of the stress field at crack tip leads to write h ⊥ ∝ K 2 /σ * 2 , which yields [START_REF] Scheibert | Brittle/quasi-brittle transition in the dynamic fracture of nominally brittle materials: An energetic signature[END_REF]:

ρ = 0 for K ≤ K a ρ ∝ K 2 -K 2 a for K ≥ K a , (4)
with K a ∝ σ * √ d a . This equation, plotted in red on Fig. 15, reproduced quite well the experimental data below K b .

As the determination of ρ from fracture surface is much more accurate and consensual than the determination of K (based on FEM calculation with inherent approximations), in the following, we will use ρ rather than K as the control parameter as a function of which the various quantities further defined in this paper will be plotted. Indeed, ρ fully characterizes the spatial distribution of nucleation sites and monotonically evolves with K within the relevant range K a ≤ K ≤ K b .

Chronology of microfailure events

Figure 16 shows the cumulative distribution of nucleation distances d n measured on two typical fracture surfaces. The probability density function (derived from the cumulative distribution) is zero for d n ≤ d min , maximum at d min , subsequently linearly decreases, and is finally zero as d n exceeds d max . This distribution form was observed for all the fracture surfaces, irrespectively of ρ. d max is found to increase with ρ, while d min decreases with ρ and remains very close to zero [START_REF] Guerra | Understanding fast macroscale fracture from microscale post-mortem patterns[END_REF].

The variation of the mean value dn of d n with ρ exhibits two regimes (Fig. 17:Top): (i) A linear increase with ρ followed by (ii) a saturation ( dn sat ≈ 50 µm)

when ρ is greater than the value ρ sat associated in Fig. 15 with the microbranching onset (at K = K b ).

To understand the saturation origin, we plotted, on Fig. 17:Bottom, the evolution of the mean nearestneighbor distance ∆r with ρ as it is predicted for a 2D Poissonian spatial distribution ( ∆r = 1/2 √ ρ). The plotted errorbars also indicate the associated standard deviation (σ ∆r = (4 -π)/(4πρ)). When ρ reaches ρ sat , in few cases, the distance between two neighbouring nucleation centers is of the order of the nucleation distance. As a result, two centers can nucleate almost at the same time (avalanche effect). The number of microcracks involved in such avalanches, also plotted in the figure, increases in the vicinity of ρ sat . This strongly suggests that the observed saturation in the d n evolution (Fig. 17:Top) results from this steric effect.

Velocity of microcrack growth

The fractographic analysis performed in Sec. 4 showed that, within the DZ, all microcracks grow at the same velocity c m . Unfortunately, it does not allow us to directly measure the value of c m and its possible dependence with ρ. Indeed, Eq. 2 shows that the form taken by the conical brands depends only on the position of the nucleation sites and on the nucleation distance but not on the value of c m .

We propose to use the deterministic reconstructions obtained in Sec. 4 to connect the macroscopic crack speed v to c m . Figure 18:A shows the evolution of the mean front position measured from reconstructions of different fracture surfaces (i.e. for different densities of nucleation centers) as a function of numerical time step normalized by c m (homogeneous to a distance, corresponding to the distance travelled by a crack front at a velocity c m in absence of micracking events). All the curves are linear with a slope A that defines the ratio v/c m . At this point, one should recall that the reconstruction procedure starts with an unrealistic initial straight crack front. We checked that this does not affect the measured value A [START_REF] Guerra | Understanding fast macroscale fracture from microscale post-mortem patterns[END_REF]. Then, since v is known, c m can be deduced and plotted, on Fig. 18:B as a function of ρ. This microscopic velocity is found to be a constant, c m ≈ 217 m/s = 0.24C R , independent of the loading K, the density ρ, and the macroscale velocity v. The implications of this result will be further discussed in the next section.

The fracture energy measurements Γ (v) made in Sec. 3 will now allow us to understand why such a constant velocity c m is selected within the microcracking regime. Figure 19 shows the evolution of Γ with the dynamic SIF K d . In the low speed regime (i.e. for low K d ) where crack propagates without involving any microdamaging or microbranching, Γ scales with K 2 d . We can also expect that the FPZ size R c scales with K 2 d : R c = K 2 d /ασ 2 Y where α is a constant of order 1, and σ Y is the material's yield stress. The scaling of Γ with K d thus indicates a linear variation between Γ and R c and, following [START_REF] Scheibert | Brittle/quasi-brittle transition in the dynamic fracture of nominally brittle materials: An energetic signature[END_REF], a constant dissipated energy per volume unit within the FPZ. The fracture energy then writes:

Γ = ασ 2 Y K 2 d + 1 E - ασ 2 Y K 2 c ( 5
)
As long as v ≤ v a (thus without microdamage), one can relate K d to K [START_REF] Freund | Dynamic Fracture Mechanics[END_REF]:

K d = K × (1 - v/C R )/ 1 -v/C D
where C D refers to the speed of dilational waves (cf. Tab. 1). By replacing K with K d in Eq. 1, and by subsequently injecting the result in Eq. 5, one gets the following expression for Γ vs. v: This expression yields a divergence of Γ at a finite velocity c ∞ given by:

Γ = 1 -E ασ 2 Y 1 -E ασ 2 Y 1-v/C R 1-v/C D K 2 c E (6)
c ∞ = E ασ 2 Y -1 C R C D E ασ 2 Y C D -C R (7)
In this expression, the only unknown quantity is /ασ 2 Y . This can be evaluated by fitting the first linear part (i.e for K d ≤ K a d ) of the curve Γ vs. K 2 d /E plotted in Fig. 19 with Eq. 5. One then gets:

c ∞ 204 m/s = 0.23C R (8)
This value sets the maximum crack growth velocity in the absence of microcracking: Beyond c ∞ dissipation diverges within the FPZ. Assuming that the same dissipation mechanisms are involved within the FPZ of each microcrack when v > v a , we expect that the limiting speed of microcracks is also determined by c ∞ , so that c m ≈ c ∞ .

6 Role of microdamaging in the selection of continuum-level scale failure velocity

The previous section has allowed us to better understand how the different variables that characterize the microdamage developing within the DZ are selected when v exceeds v a . The most surprising observation concerns the propagation velocity of microcracks c m that remains constant, of the order of 217 m/s, and significantly lower than the macroscopic velocity v. In other terms, damage does not slow down the macroscopic crack tip as it was commonly believed until now (Ravi- [START_REF] Chandar | An experimental investigation into dynamic fracture-ii microstructural aspects[END_REF][START_REF] Ravi-Chandar | On the role of microcracks in the dynamic fracture of brittle materials[END_REF][START_REF] Washabaugh | A reconciliation of dynamic crack velocity and rayleigh wave speed in isotropic brittle solids[END_REF], but on the contrary it accelerates it. And this acceleration factor is all the more important as ρ (or equivalently K) increases (Fig. 20).

The temporal evolution of a given point on the main front, shown in Fig. 21, permits to better understand the effect of microdamage on the dynamics of macroscopic cracking. The trajectory of this point exhibits random and sudden jumps. These jumps actually correspond to the coalescence of a microcrack with the main front. Between these events, the local front velocity remains close to c m (i.e. the slope is equal to 1). Such an acceleration effect can be captured by a simple mean-field model, detailed in [START_REF] Guerra | Understanding fast macroscale fracture from microscale post-mortem patterns[END_REF], which simply consists in counting the number of coalescence events as the main front propagates over a unit of length when the nucleation sites are located at the nodes of a square lattice. One then gets:

A = 1 1 -b dn √ ρ (9)
where b is a numerical factor close to 1. This equation, combined with the evolution of dn with ρ (red line in Fig. 17

Conclusion

The series of experiments reported here were designed to shed light on the dissipation mechanisms that develop during fast crack growth in PMMA. They reveal that, above a well-defined critical velocity, PMMA stops to behave stricto senso as a nominally brittle material and crack propagation goes hand in hand with microcracks that nucleate and grow ahead of the main front. Those let characteristic marks (referred to as conic marks) on post-mortem fracture surfaces, the morphological analysis of which allowed us to reconstruct the full dynamics of microfailure events. The simultaneous space and time accuracies were of the order of the micrometer and the tens on nanoseconds, i.e. far better than what was reachable up to now.

Analysis of the reconstructions demonstrated that the true local propagation velocity of single cracks remains limited to a fairly low value c m 0.23c R in PMMA, while the apparent fracture velocity measured at the continuum-level scale (e.g. via potential drop method) can be much higher (up to twice larger!). Such an anomalously high measured velocity results in fact from the coalescence of microcracks with each other and with the main front -all of them growing at c m . In other words, the main effect of microdamaging is not, as commonly believed up to now [START_REF] Washabaugh | A reconciliation of dynamic crack velocity and rayleigh wave speed in isotropic brittle solids[END_REF][START_REF] Chandar | An experimental investigation into dynamic fracture-ii microstructural aspects[END_REF][START_REF] Ravi-Chandar | On the role of microcracks in the dynamic fracture of brittle materials[END_REF], to slow down fracture by increasing dissipation within the DZ, but on the contrary to boost the propagation of the main crack.

The value c m that limits the true local velocity of single crack fronts in PMMA in the dynamic regime is set by the physico-chemical dissipative and non-linear processes that develop within the FPZ, e.g. thermal [START_REF] Estevez | Modeling of the competition between shear yielding and crazing brittle polymers[END_REF], viscoelastic effects [START_REF] Boudet | Dynamics of crack propagation in brittle materials[END_REF][START_REF] Persson | Crack propagation in viscoelastic solids[END_REF] or hyperelastic processes [START_REF] Bouchbinder | Weakly nonlinear theory of dynamic fracture[END_REF][START_REF] Livne | The near-tip fields of fast cracks[END_REF][START_REF] Buehler | Hyperelasticity governs dynamic fracture at a critical length scale[END_REF]. Hence, it will depend on the considered material. On the other hand, the boost effect induced by microdamaging takes the form of a purely geometrical factor controlled by two quantities: the density ρ of nucleation sites and the mean distance d n at nucleation. These two variables fully characterize the damaging state within DZ and evolve with K. Once ascribed, they permit to relate the continuum-level fracture speed v to the true local propagation speed c m . Further work is required to unravel how PMMA internal structure selects the material-dependent quantities c m , ρ(K), and d n .

The presence of microcracks forming ahead of a dynamically growing crack has been evidenced in a variety of materials, e.g. in most brittle polymers [START_REF] Ravi-Chandar | Dynamic fracture of nominally brittle materials[END_REF][START_REF] Du | Experimental observation and computer simulation of conic markings on fracture surfaces of polymers[END_REF], in rocks [START_REF] Ahrens | Impact-induced tensional failure in rocks[END_REF], in some nanophase ceramics and nanocomposites [START_REF] Rountree | Atomistic aspects of crack propagation in brittle materials: Multimillion atom molecular dynamics simulations[END_REF], in oxide glasses [START_REF] Rountree | Fracture in glass via molecular dynamics simulations and atomic force microscopy experiments[END_REF], in metallic glasses [START_REF] Murali | Atomic scale fluctuations govern brittle fracture and cavitation behavior in metallic glasses[END_REF], etc. The boost mechanism demonstrated here on PMMA is expected to hold in this whole class of materials. Note that a similar effect has also been invoked [START_REF] Prades | Nano-ductile crack propagation in glasses understress corrosion: spatiotemporal evolution of damage in the vicinity of the crack tip[END_REF] in oxide glasses under stress corrosion, where ultra-slow crack propagation (down to few tenths of nanometers per second) was found to involves "nano"cracks forming ahead of the main crack tip [START_REF] Célarié | Glass breaks like metal but at the nanometer scale[END_REF][START_REF] Celarié | Surface fracture of glassy materials as detected by real-time atomic force microscopy (afm) experiments[END_REF][START_REF] Bonamy | Nanoscale damage during fracture in silica glass[END_REF][START_REF] Ferretti | An espi experimental study on the phenomenon of fracture in glass. is it brittle or plastic?[END_REF]. This yields us to conjecture that failures with anomalously high apparent velocities measured at continuum-level scale may arise in all situations involving propagation-triggered microcracks, including e.g. shear fracture in compressed granite [START_REF] Moore | The role of microcracking in shear-fracture propagation in granite[END_REF] , thermal failure in shale [START_REF] Kobchenko | 4D imaging of fracturing in organic-rich shales during heating[END_REF][START_REF] Panahi | A 4D Synchrotron X-Ray-Tomography Study of the Formation of Hydrocarbon-Migration Pathways in Heated Organic-Rich Shale[END_REF], and more generally failure of so-called quasi-brittle materials. One of the most robust observations in this field is the power-law form followed by the distribution in size and in time interval between two successive microfailure events (see [START_REF] Bonamy | Intermittency and roughening in the failure of brittle heterogeneous materials[END_REF][START_REF] Deschanel | Mechanical response and fracture dynamics of polymeric foams[END_REF] for recent reviews). It would be interesting to see whether or not the microcracking evidenced here in dynamic failure regime share the same scale-free features and how this affects the boost effect. Work in this direction is under way.

Fig. 1

 1 Fig. 1 Sketch (top) and photo (bottom) of the experimental setup.

Fig. 2

 2 Fig. 2 Measurement of instantaneous crack velocity. a: Voltage as a function of time as recorded via the oscilloscope. b: Zoom that permits to distinguish the individual jumps yielded by the successive cuts of the conductive lines as crack advances. c: Resulting position of crack tip as a function of time. d: Final curve showing the variations of crack velocity with crack length.

Fig. 3

 3 Fig.3Computation of SIF. Top: Typical mesh used for finite elements calculations, in order to access the stress/strain fields in the experiments. Red: polymeric layer. Black: sample. Blue: L-shaped block. Green line: cracked line. The slide link connected to the L-shaped block is used to model the motion of the contact point between the pushed wedge and the roller. Specimen loading is achieved by translating horizontally the slide link, from (0) to (1), over a distance u wedge selected so that the horizontal force applying on the slide link is half that measured experimentally. Bottom-Left: Zoom on the meshing in the transition region (red) between coarse meshing (1 mm mesh size) in the bulk and fine meshing close to the crack tip. On the right is shown part of the circular hole at the seed crack tip. Bottom-Right: Zoom on the crack-tip region (green), meshed with a size of 1 nm.

Fig. 4

 4 Fig. 4 Variation of SIF K (left), instantaneous crack velocity v (center) and fracture energy Γ (right) as a function of crack length c for five experiments with different stored mechanical energy U 0 at propagation onset: 2.0 J (A, A' and A"), 2.6 J (B, B' and B"), 2.9 J (C, C' and C"), 3.8 J (D, D' and D"), and 4.2 J (E, E' and E").

Fig. 5

 5 Fig. 5 Fracture energy Γ as a function of crack velocity v for five different experiments with different stored mechanical energies U 0 at crack initiation. The two vertical dashed lines correspond to v a and v b . The two horizontal dashed lines indicate the confidence interval (95%) for the measured fracture energy K 2 c /E at crack initiation (Taken from Scheibert et al (2010)).

Fig. 6 1

 6 Fig. 6 1 × 1.4 microscope images of fracture surfaces created at various speed v.

Fig. 7

 7 Fig. 7 Profilometric 1.4 × 1 mm 2 surface (Top) and optical 0.7 × 0.25 mm 2 side-view (Bottom) images taken at velocity smaller than v a (left), between v a and v b (center), and above v b (right). Crack has propagated from left to right.

Fig. 8

 8 Fig. 8 Surface density ρ of conic marks as a function of crack velocity for five different experiments with different stored mechanical energies U 0 at crack initiation (Taken from Scheibert et al (2010)).

Fig. 9

 9 Fig. 9 A: Typical microscope image of the post-mortem fracture surface. Bright regions indicate the nucleation centers. B:Interpretation of the conic marks: Red dots sketch the successive positions of the fronts that, by interacting, will be giving rise to the green branch of the conic mark. These two microcracks form at t = 0 and t = τ , then radially grow at speeds c 1 and c 2 . Branch fitting via Eq. 2 permits to measure c 2 /c 1 . Note that a conic mark can be made of several branches. C: When c 2 = c 1 , branches are (mathematical) true conics and the nucleation distance d n between the front of the incident microcrack (1) and the nucleation point of the forming one (2) is twice the apex-focus distance 0 2 P 12 (Taken from[START_REF] Guerra | Understanding fast macroscale fracture from microscale post-mortem patterns[END_REF]).

Fig. 10

 10 Fig. 10 Direct extraction of the relative speed between two interacting microcracks at three different microcrack densities.(Top) Zones of investigation. Each conic branch has been attributed a given color and the nuclei of the two corresponding interacting microcracks has been joined by a dotted segment of the same color. Note that a conic-like mark is often made of several of these conic branches. The ratio c 2 /c 1 is the only adjustable parameter in equation Eq. 2 to determine the branch geometry once the nuclei position and the branch apex are set. (Bottom) Corresponding distributions for c 2 /c 1 : In the three cases, the distributions are found to fit normal distributions of mean value 0.98 -0.99 and standard deviation 0.03 -0.04, irrespective of ρ (Taken from the supporting information of[START_REF] Guerra | Understanding fast macroscale fracture from microscale post-mortem patterns[END_REF]).

Fig. 11

 11 Fig. 11 Reconstructed sequence of crack propagation and microfailure events at the microscale for two different values of K, namely K = 2.53 MPa. √ m (left) and K = 4.18 MPa.√ m (right). The blue part corresponds to uncracked material and we ascribed an arbitrary gray level to each of the growing microcrack. From this, one can deduce the intersection points between microcracks that coincide with the observed conics marks on the fracture surfaces. The size of the two zones of analysis is 2.5×2.5 mm 2 . The macroscopic crack propagates from left to right.

Fig. 12

 12 Fig. 12 Qualification of the procedure: Superimposition of the conic pattern as obtained from reconstruction onto that observed by fractography. A: K = 2.77 MPa. √ m. B: K = 4.18 MPa. √ m. Red dots indicate nucleation centers (Taken from the supporting information of Guerra et al (2012)).

Fig. 13

 13 Fig. 13 Influence of initial crack front morphology on the reconstruction for ρ = 45.0 mm -2 (K = 3.65 MPa. √ m). Three different initial conditions were used. (A) Straight vertical line. (B) Vertical sinusoidal shape with a period of 186 µm and a peak-to-peak amplitude of 242 µm. (C) Same sinusoidal shape, but translated vertically over half a period (Taken from the supplementary information of Guerra et al (2012)).

Fig. 14

 14 Fig. 14 Cumulative distribution of the number of nucleation centers contained in square regions of lateral size a, for two different fractographic images. Solid lines represent Poisson fits of parameter ρa 2 , where the fitted value ρ can be identified with the center density.

Fig. 15

 15 Fig. 15 Variation of the density ρ of nucleation centers with applied SIF. Black disks correspond to the eight images on which post-mortem reconstruction was performed. Red line is a fit via Eq. 4. K a and K b are associated with microcracking and microbranching onset, respectively (Taken from Guerra et al (2012)).

Fig. 16

 16 Fig. 16 Cumulative distribution of the nucleation distances d n , as measured for two different fractographic images. Solid lines are fits with P (d n ) = ((d max -d n )/(d max -d min )) 2 .

Fig. 17

 17 Fig. 17Top: Mean distance at nucleation dn as a function of ρ. the red curve is a linear fit over ρ ≤ ρ sat = 45.5 mm -2 and a plateau at dn sat = 50 µm over ρ sat . Bottom: Saturation of dn and avalanches. The red line is the same as in the top figure. It is compared to the mean nearest-neighbor distance in a Poissonian distribution (black solid line). Solid squares indicate the proportion of microcracks involved in avalanches, as computed from the reconstruction.

Fig. 18

 18 Fig. 18 Determination of the local propagation velocity c m of individual microcracks. Left: Evolution of the mean crack front position as a function of c m × t. Different curves correspond to different fractographic images with different microcrack density. Slope of these curves defines the ratio A between continuum-level scale failure velocity v and microcrack velocity. Right: Variation of c m with ρ. Horizontal red line indicates the mean value c m ≈ 217m/s.

Fig. 19

 19 Fig. 19 Variation of the fracture energy Γ with K d = K × (1-v/C R )/ 1 -v/C D . The vertical dashed line corresponds to microcracking onset K a d = 1.83 MPa √ m. Note that K d can stricto-senso be identified with the dynamic SIF in absence of microcracking, i.e. for K d ≤ K a d . The solid red line is a fit with Eq. 5.

Fig. 20

 20 Fig. 20 Acceleration factor A = v/c m as a function of the microcrack density ρ. Red line is a fit by Eq. 9 with b = 1.19± 0.02. Horizontal dot line defines v b /c m and passes through the slope breakdown observed when ρ = ρ sat (Taken from Guerra et al (2012))

Fig. 21

 21 Fig. 21 Position of a given point of the front as a function of "time" c m × t as obtained from the reconstructed simulation, applied to a fracture surface with ρ = 64.6 mm -2 . The dotted straight line indicates the expected position without microdamage (slope 1). The jumps (arrows) correspond to coalescence events with microcracks (Taken from Guerra et al (2012))

  :Top) is plotted in red in Fig. 20. It reproduces quite well both the data coming from experiments (grey triangles) and from the reconstruction (black circle). Note the change in slope observed at ρ sat . At this point, A = A b = 1.63. This corresponds to a ratio A b v b /c m where v b 0.4C R is associated with the microbranching onset. This strongly suggests that the steric effect at the origin of the saturation of d n also plays a role (still unsolved) in the micro-branching instability.

Table 1

 1 Mechanical properties of the PMMA used in our fracture experiments. E andν denote the Young modulus and the Poisson ratio, respectively. C D , C S , and C R denote the speeds of dilational, shear and Rayleigh waves respectively.

	E(GPa)	ν	Density (kg/dm 3 ) C d (m/s) C s (m/s) C R (m/s)
	PMMA 2.8 ± 0.2 0.36 ± 0.08	1.18 ± 0.02	2010 ± 60	950 ± 30	880 ± 20

To measure this time shift, we directly connected the Wheatstone bridge of the load cell to an oscilloscope, without passing through the signal conditioner. This latter, indeed, imposes a time resolution of 1 s.

Acknowledgements We warmly thank K. Ravi-Chandar (Univ. of Texas, Austin) for many illuminating discussions. We also thank T. Bernard (SPCSI) for technical support, P. Viel and M. Laurent (SPCSI) for gold deposits, and A. Prevost (ENS, Paris) for his help with the profilometry measurements at ENS. We also acknowledge funding from French ANR through Grant No. ANR-05-JCJC-0088 and from Triangle de la Physique through Grant No. 2007-46.


Magnus Larsson
email: magnus.larsson@telecom-bretagne.eu

Gilles Coppin
email: gilles.coppin@telecom-bretagne.eu

Franck Poirier
email: franck.poirier@univ-ubs.fr

Olivier Grisvard
email: olivier.grisvard@telecom-bretagne.eu

A Framework for a Priori Evaluation of Multimodal User Interfaces Supporting Cooperation

Keywords: Analysis, a priori evaluation, cooperation, multimodality, system design. ACM Classification H.1.2: Human factors, H5.2: Evaluation/methodology

In this short paper we will present our latest research on a new framework being developed for aiding novice designers of highly interactive, cooperative, multimodal systems to make expert decisions in choice of interaction modalities depending on the type of activity and its cooperative nature. Our research is conducted within the field of maritime surveillance at ATOL Laboratory and it is focused on the next generation distributed work support.

INTRODUCTION

The computer industry is on the brink of a new era. The future is not a solitary PC, but a diverse set of smart, cooperative devices interacting not only with its end users but also with each other while fully integrated in their environment. The interaction with these systems are multimodal where the tools become extensions of the human sensor and motor systems supporting the end users' cooperative execution of actions while trying to solve problems. The computer is thus no longer a system that just determines something by mathematical means, brings order (Fr. 'Ordinateur'), handles data (Swe. 'Dator'), count information (Hun. 'Bilgisayar'), or is a machine full of knowledge (Fin. 'Tietokone'). It is rather an infrastructure for multimodal human-computer interaction and cooperation. However, are we as designers equipped to meet the rapid evolution within the computer industry? In this short paper, we suggest that we need to find a way to minimize the gap between analysis and design to be able to continue delivering optimized and satisfactory systems to our customers and end users at a reasonable price.

THE DESIGNERS' CHALLENGES OF TODAY

The vast majority of today's expert designers are still novices within the design of highly interactive, cooperative, multimodal systems. However, they are still supposed and demanded to deliver intuitive, useful systems of high quality to a reasonable cost. The technology necessary to create these systems are mere a mash-up of existing technologies. Model-driven languages, methods and tools are continuously being developed and enhanced to meet the demands of the industry on adaptive, flexible and robust [START_REF] Vanderdonckt | Multimodality for Plastic User Interfaces: Models, Methods, and Principles[END_REF] systems designed and developed at a low cost. One example of such a project is the pan-European ITEA2: UsiXML project which is based on the µ7 concept, i.e. multi-device, multi-platform, multi-user, multi-linguality/culturality, multi-organization, multi-context, and multi-modality. However, due to the designers' lack of experience and know-how in designing these new complex systems, and due to the intended end users' and customers' inability to clarify and articulate their cooperative and multimodal needs in a comprehensive way, the designers often face infoglut resulting in poor choices in interaction modalities. Some of the most common challenges are:  The lowest common denominator is "easily" validated for single user systems, but not for multimodal cooperative systems with a big variety of end users  There is a disparity in activity objective and needs between who does the work (the end users) and who gets the benefit of that same cooperative work (the management team/the customer)


One way to aid the designers would be to provide a framework that can alleviate the transition from analysis to design. Today, this is a tedious time consuming work biased on deficient mental models by the designers and without any promise of quality delivered. Therefore, our intention is to help novice designers of cooperative multimodal systems to make expert decisions in choice of modality or combinations of modalities. We believe that this will not only enable the creation of new, for the end users, adequate intuitive systems supporting their cooperative work, but it will also optimize the projects' ROI. In the following paragraphs we will present an a priori evaluation framework being developed based our understanding of human behavior and cooperation and how multimodal interaction can be approached to solve these issues.

HUMAN BEHAVIOR AND COOPERATION

One of the strongest assets of human beings are their ability to interact with each other in quite complex ways in order to fulfill a great number of simultaneous tasks initiated from a wide variety of intentions [START_REF] Nowak | Five Rules for the Evolution of Cooperation[END_REF]. These interactions take place within a group of people, i.e. two or more participants, who can be considered to cooperate to the extent that they 1) consider each other cognitively in interaction, 2) have a joint purpose, 3) consider each other ethically in interaction, and 4) trust each other to act according to 1-3 [START_REF] Allwood | Cooperation and Flexibility in Multimodal Communication[END_REF]. Novices and experts meet in different groups and teams within which they take on passive, active or expansive roles, while belonging to different communities of interest and practice at the same time. [START_REF] Hoogstoel | Les répercussions du travail coopératif assisté par ordinateur sur les systèmes d'information[END_REF]. Their interactions can be collective or dispersed and they can be direct, i.e. interpersonal, or indirect, e.g. mediated by computers. Furthermore, depending on their level of involvement, one can consider them to engage in no interaction, lightweight interaction, information sharing, coordination, collaboration or cooperation. Evidently, the complexity of human interaction together with the challenges posed on the designer regarding choice of interaction modality demands a comprehensive framework to avoid infoglut in the moving from analysis to design.

Based on the work of prominent scientists during mid and late 19 th century, such as Charles Robert Darwin, Gustav Theoder Fechner and Mikhaylovich Sechenov, the Russian psychologist Lev Semyonovich Vygotsky founded cultural-historical psychology, thus closing the gap between the natural sciences and the mental sciences of human behavior. He approached behavior not as a result but rather as a process in motion and in change, i.e. by studying behavior as interaction. Vygotsky's research on activities bridged the gap between the mental and the physical contexts of human behavior and consciousness [10]. Activity Theory, an evolution of Vygotsky's research, provides a basic framework for human interaction and a useful basic unit of analysis; the activity.

The Activity Theory concept deals with a set of fundamental types [START_REF] Kuutti | The concept of activity as a basic unit of analysis for CSCW research[END_REF], which are:

 An object -Activities can be distinguished by their objects. It is the object and the transformation of that set object that drives the activity.

 A collective phenomenon -The activity does not take place in isolation but is always a collective phenomenon.

 A subject (agent) -The activity has a subject or a collective of subjects who understands the motive of the activity. In our research we refer to the subject as an actor or a role.

 A material environment -The activity exists in and transforms its material environment.

 A historically developing phenomenon -The activity is a process that has a shared memory.

 Contradictions -The force behind the development of an activity are contradictions.

 Actions -Participants realize an activity through conscious and purposeful actions.

 Culturally mediated relationships

These fundamental types can easily be illustrated in a diagram together with their individual relationships. Kuutti's research [START_REF] Kuutti | The concept of activity as a basic unit of analysis for CSCW research[END_REF] on Activity Theory and its fundamental types has resulted in a useful framework for research on computer-supported cooperative work. Cadier [START_REF] Cadier | Modèles Cognitifs pour les Système d'Aide à la Décision Collective: Application à la Patrouille Maritime[END_REF] extended the Activity Theory framework of Kuutti to manage both negotiation and execution of cooperative work (see Figure 1 and Figure 2, below), thus enabling analysis of cooperative activity. The model in Figure 1, above, depicts the 'playground' of an activity, whereas the model in Figure 2, below, illustrates the actual execution process of an activity and its sub-activities/tasks and operations. Furthermore, this model also illustrates the cooperative steps of an activity where the negotiation of division of labor is the starting point, but also the result. Based on this knowledge we can conclude that cooperation is heterogeneous where contradictions force activities [START_REF] Kuutti | The concept of activity as a basic unit of analysis for CSCW research[END_REF], that it is culturally and contextually situated and that it makes use of internal as well as external communication [10], both verbal and non-verbal [START_REF] Allwood | Cooperation and Flexibility in Multimodal Communication[END_REF], to organize the same activities. These activities the users later execute with the help of mediating artifacts such as computers (see Figure 3, below). We can also conclude that the level of verbal versus non-verbal communication depends on the social context of the actor/role, which are mediated via social rules and norms and the activity's division of labor. This would suggest that no person act in isolation and that one could consider all activities, if taking into account the different levels and types of human interaction and work support, as cooperative as well as multimodal.

Community

Division of Labor Rules

Object

Based on this understanding of human interaction and computer supported cooperative work we can look closer at

what implications this has on the choice of interaction modalities and how we can develop a framework suitable for designers.

A SOUND CHOICE OF INTERACTION MODALITY

Human behavior, interaction and cooperation are multimodal by origin. Activity Theory provides, as shown above, a comprehensive high-level framework for organizing cooperative activities into manageable entities. However, in order to be able to provide any insight into preferred choice of multimodality for any specific context we need to enhance and develop it further. To be able to manage the cognitive aspects of the actors/roles in cooperation we can make use of Endsley's Situation Awareness model [START_REF] Endsley | Situation Awareness Analysis and Measurement[END_REF], which, in combination with our understanding of the human sensor and motor system provide a mind and body description of human capabilities (see Figure 4, below). The actor/role has been given a physical interface in his/her motor and sensor system as well as a detailed description of his/her mental capabilities.

The mind and body together defines the activity process from situation awareness via decision to execution of the activity, either alone or in cooperation with other actors/roles. The actors'/roles' mental capabilities and properties are affected by the community's social and cultural rules and vice versa and the actors'/roles' use of artifacts to transform the object of interest into sought for outcome also transforms the actor/role in that same process. [

model construction in progress]

A computer system also has a physical interface towards the outside world and an inner "mental" core based on the intentions of the system creators as well as its users' use and re-work. The physical aspects (input and output devices) together with the logical interaction language make an interaction modality which together with e.g. CARE properties can be combined into multimodal interactive systems [START_REF] Nigay | Design space for multimodal interaction[END_REF] while providing plasticity [START_REF] Vanderdonckt | Multimodality for Plastic User Interfaces: Models, Methods, and Principles[END_REF] (see Figure 5, below) to correspond to the changing context. The understanding that human interaction with computer systems and other humans varies depending on if the actor/role is acting alone or in cooperation with other humans or computers as well as on the context of use is crucial. By comparing the actors'/roles' mental capabilities and objectives with the logical device and its intended task support of the computer system one can evaluate the mental aspect of the activity. By comparing the physical capabilities of the actors/roles with each other and with the computer system one can find constraints as well as possibilities of interaction modalities. The actors'/roles' negotiation with the group and the community regarding division of labor while executing tasks aided by a computer system changes the way a task is conducted and what interaction modalities that are suitable for the overall activity as well as the execution and negotiation of the tasks.

Unfortunately there is not enough space in this short paper to go into our models in more detail, discussing each building block and their relationships with other parts of the model. However, the continuous development of our models based on our cross-disciplinary research proves very promising and we hope to conclude everything in an extensive thesis before the end of the year. Our research on the next generation work support for tactical officers and sensor operators within maritime surveillance, who work closely within highly specialized teams, while making use of different kinds of interaction modalities or combinations of modalities to execute their work, and who negotiate their division of labor, are well suited for our research. We hope that our results will shed some light on the impact of cooperation on the preferred choice of interaction modalities. We believe that our framework will be a welcome help for novice designers of cooperative multimodal systems when making expert decisions in choice of modality or combinations of modalities.

  The intuition and decision-making of the designers regarding computer supported cooperative multimodal systems are biased by previous experiences of single-user system design  The complexity in group interactions and activities pose great challenges: o Group logistics of data collection o Number and complexity of variables o Validation of re-engineered group work  It is time and money consuming to perform evaluation of multimodal cooperation even though one focus on a smaller set of activities and well defined user groups (even for a one-user application)

Figure 1 .

 1 Figure 1. Basic structure of a cooperative activity [2] with its properties visualized and given a relation to each other with mediating artifacts, rules and division of labor being multimodal in their character.

Figure 2 .

 2 Figure 2. An iterative cooperative activity process [2]where cooperative requests are multimodal acts of negotiation and decision whilst the rest of the process is multimodal (internal as well as external) actions.

Figure 3 .

 3 Figure 3. An example of an activity decomposition via negotiation of division of labor and the work task execution, both influenced by the social rules of the community.

Figure 4 .

 4 Figure 4. Our latest actor/role model together with the activity process together with the mental as well as physical capabilities based on work of [2], [3], [4] and [5] [model construction in progress]

Figure 5 .

 5 Figure 5. Our latest artifact model of the computer interface based on the work of [6], [8], [9] and [10].


Mohamed-Ikbel Boulabiar
email: boulabiar@gmail.com

Thomas Burger
email: thomas.burger@univ-ubs.fr

Franck Poirier
email: franck.poirier@univ-ubs.fr

Gilles Coppin
email: gilles.coppin@telecom-bretagne.eu

A Low-cost Natural User Interaction Based On A Camera Hand-Gestures Recognizer

Keywords: natural gesture interaction, low-cost gesture recognition, interactive TV broadcast, ubiquitous computing

The search for new simplified interaction techniques is mainly motivated by the improvements of the communication with interactive devices. In this paper, we present an interactive TVs module capable of recognizing human gestures through the PS3Eye low-cost camera. We recognize gestures by the tracking of human skin blobs and analyzing the corresponding movements. It provides means to control a TV in an ubiquitous computing environment. We also present a new free gestures icons library created to allow easy representation and diagramming.

Introduction

HCI research focuses more attention than before on enhancing the user experience regarding human-display interaction. This area of research focus on making interaction more natural, so that it is not necessary to click on buttons and to touch screens. To do so, automatic gesture recognition is an interesting paradigm [START_REF] Bolt | Put-that-there: Voice and gesture at the graphics interface[END_REF], [START_REF] Pavlovic | Visual interpretation of hand gestures for human-computer interaction: A review[END_REF], [START_REF] Derpanis | A review of vision-based hand gestures[END_REF]. However, in everyday life, for interaction with home devices having embedded computational power, such as interactive TVs, one does not make benefit from these new interaction paradigms yet.

We present in this paper an approach that allows the detection and the interpretation of gestures of a TV spectator from a simple and low cost camera. this work takes place in the context of the French FUI 4 Project RevTV, the aim of which is to add new interaction techniques, sos that telespectators take activaly part to the TV broadcast. The final objective of this project is to control the animations of an avatar that should be inserted within a TV show beside the presenter or actor. By now, the major scenario that we rely on, corresponds to some educational game where a pupil controls her or his avatar which interacts in a broadcast program led by a "real" TV animator. During such kind of scenarios, commands interaction like pointing, selecting and moving, as well as natural gesture animation, are required.

The paper is organized as following. Section 2 presents the study held on gestures semantics, sources, and taxonomies and include the presentation of the free icons library created and used. Section 3 shows the technical details of handling the camera input for gestures recognition. Section 4 shows the modes where the gestural information generated are used with some screenshots of the running application. Section 5 answers the usability and naturelness question of the use of gestures specially in an ubiquitous environment. Finally section 6 discusses the future of the work and the possible integration of other multimodal modules.

Gesture Semantics

Symbolic gestures, such as emblems, play an important role in human communication as they can fully take the place of words. These gestures are processed by the same area of the human brain as the spoken language [START_REF] Xu | Symbolic gestures and spoken language are processed by a common neural system[END_REF]. Hence, these gesures do not come alone, and they are often combined with speech, or with cognitive activities, as they are classically used to add more precision on the details of a talk.

We have extracted some semantics of such type of gestures, by the analysis of the movements of a user explaining a story, and by searching for a relation between gestures and the part of the story being told [START_REF] Mcneill | Gesture and Thought[END_REF]. After extraction, most of gestures semantics can be divided into two parts for their possible future use as:

Animation Gestures used to animate a virtual avatar. Command Gestures used to launch a specific predefined action.

Gestures taxonomies

A rapid look the possible gestures human can perform [START_REF] Streeck | Gesturecraft[END_REF], lead to the conclusion that the amount of possibilities is tremendous. Nonetheless, the gestures can be classified according to various criteria not only to simplify their recognition, but also to allow their reproducing on an avatar to putting tags on them for future analysis [START_REF] Neff | Gesture modeling and animation based on a probabilistic re-creation of speaker style[END_REF]. Here, we consider a taxonomy based on the context of the gestures as well as on their types: According to McNail [START_REF] Mcneill | Gesture and Thought[END_REF], it is possible and useful to distinguish between these gestures. Here follows a possible taxonomy :

Gesticulation is a motion that embodies a meaning reliable to the accompanying speech. It is made chiefly with the arms and hands but is not restricted to these body parts. Speech-linked gestures are parts of sentences themselves, the gesture completes the sentence structure. Emblems are conventionalized signs, such as thumbs-up or the ring (first finger and thumb tips touching, other fingers extended) for "OK."

Pantomime is dumb show, a gesture or sequence of gestures conveying a narrative line, with a story to tell, produced without speech.

In our work, we focus on Emblems for Command Gestures, and on Gesticulations for Animation Gestures.

Gestures Library

In order to identify relevant gestures we have analyzed some talk-shows, and we have extracted the most frequent gestures being performed. According to our scenario of use in the context of interactive TV (RevTV project), we need some gesture to trigger commands, as well as other gestures to implement natural interaction with the avatar. We have created a vector based gestures icons to represent them. These 4 icons, represented in figure 1, are part of the ishara library [START_REF]ishara vector based and open gestures icons[END_REF]. They represent the gestures supported by the software of our application. We provide a movement mode, a multitouch mode, a command mode, a hand fingers recognition, and, finally, a cursor movement mode (see Section 5 for detailled explanation of these modes).

Fig. 1. Family of supported gestures represented using the free gestures icons library from [START_REF]ishara vector based and open gestures icons[END_REF] These five basic gestures modes are already supported in our system as a first step. They will be extended depending on the context fulfilling the requirements of our scenario.

Technical Work

Figure 2 is a description of the pipeline of the modules used to extract gestural information. Each part is described in details in the next subsections.

Camera Input

We have chosen a low-cost camera that can be integrated with other components in the future. The PS3Eye camera matched our expectations with a 640x480 resolution combined with a 60 frames-per-second capability. This camera costs about 40$, which is a reasonable price for a potential market deployment.

Colors Transformations

The video processing which depend on the OpenCV library [START_REF] Bradski | The OpenCV Library[END_REF] is based on several modules, such as illustrated in figure 2. First, a basic but adaptive skin color segmentation of a large interval of human skin varieties is performed [START_REF] Askar | Vision-based skincolour segmentation of moving hands for real-time applications[END_REF]. We also perform histogram equalization technique to better enlarge the colors to all the space available. Then we transform the color space from BGR, which come from the camera to YCrCb, as it is the most adapted one to skin segmentation. Other color spaces like HSV or CIE-Lab can be used for skin color segmentation, but the transformation from BGR either takes more time (the transformation we use is linear) or needs more parameters in the next step [START_REF] Askar | Vision-based skincolour segmentation of moving hands for real-time applications[END_REF].

Space

Input Filtering

During this step, a 5x5 opening morphological kernel [START_REF] Soille | Morphological Image Analysis: Principles and Applications[END_REF] is applied to smooth the results. Practically, it cleans the image from isolated pixels and holes inside skin zones.

Blobs Detection

Each blob is labeled in linear time using the contour tracing technique with the linear-time component-labeling algorithm [START_REF] Chang | A linear-time component-labeling algorithm using contour tracing technique[END_REF], which requires 1 to 4 passes to recognize all blobs with their bounding contour. We identify the hands and the head by a calibration focused on their first position in the screen. We identify them and add a tag to their respective blobs.

Blobs Tracking

After having the hands and head blobs identified, an efficient tracking method based on appearance model is used [START_REF] Senior | Appearance models for occlusion handling[END_REF]. Let us note that it handles occlusion problems. We have used the cvblob implementation [START_REF] Lin | cvblob[END_REF] for that algorithm which Once the matrix of distance is defined, the following loops are executed to handle blobs changes:

-A loop to detect inactive tracks: those tracks with no blobs near.

-A loop to detect and create new tracks: those blobs without a track near.

-A loop to assign blobs to tracks. It makes clusters with blobs that are close and assign them to a track. In this step some tracks could merge in one. -A last loop which check all inactive tracks to delete the old ones.

To better handle blobs, the bounding path of the blobs is simplified to a set of few points when needed. This is used to identify picks for the counter mode (see Section 5). Finally, a phase of calibration is triggered when the logic of tracking is lost or damaged.

Gestural information and Recognition modes

The output of the video processing is the input of the recognition module, which is based on the track location on the screen. Four different modes are defined according to the hand locations. The switch between them is based on the following grammar:

The movement mode: it is activated when two hands are close to each other, so that the user selects between 4 directions. This mode can be used as input for games to select between the directions.

The multi-touch mode: where we consider the similarities between 2D gestures in a tactile multi-touch context and gestures in the 3D space. To do so, we consider the hands blobs in a manner similar to that of two finger tips in the input of a multi-touch device. Based on that, we can recognize well known multi-touch gestures, such as Drag, Pinch/Zoom and Rotate with a consideration of a interaction centroid [START_REF] Gorg | A framework for abstract representation and recognition of gestures in multi-touch applications[END_REF] but applied in our case. The counter mode: counts the number of fingers in a preselected hand (the left and right hand are automatically discriminated) by counting the peaks in the simplified contour of the blob. This mode can again be used in games for kids. The mouse mode: it is used to control a cursor to select something in an arbitrary place on the screen. With this mode, we only get the input from a sub zone in the screen and map it to the full screen. We use it in a similar way to a computer touchpad. The user can validate clicks using a tempo on another zone. The usability of our system should be compared to other systems in an ubiquitous environment. In our case, we have a TV instead of a PC. The gestures in such environment, and specially for children game scenario, are made for a short period of time, and doesn't require a good usability, even if we can recognize gestures in real-time. The Naturalness of gestures is supported by the analysis of those to be supported and the preference of gesticulations. Gesticulations are the most common gestures, so they inherit from this their ability to be produced with less complication.

Emblems gestures which englobe commands, are chosen to be produced in more difficult positions to allow easy discrimination from gesticulations. Even with these extractions, the naturalness aspect of the gestural interaction itself are still objected. [START_REF] Norman | The way i see it: Natural user interfaces are not natural[END_REF] Spatial gestural interactions always lack from the feedback for the interaction. This affects also the usability because the user can no more be guided in space as he was with surface movement or simple physical joysticks.

6 System integration of this interaction and future work Human brain Broca's area, identified as the core of the brain's language system, are not in fact committed to language processing, but may function as a modality-independent semiotic system that plays a broader role in human communication, linking meaning with symbols whether these are words, gestures, images, sounds, or objects [START_REF] Xu | Symbolic gestures and spoken language are processed by a common neural system[END_REF]. This natural handling of multimodal communications gives an argument to build recognizer taking part of most of these modalities specially to remove ambiguous gesture meaning decision cases.

Our system only supports gestural interaction at the moment but the integration of other multimodal means are possible to fine tune the input and give the user better feedback. These possible evolution of the system are possible: Facial Recognition : our system don't take care of the facial expressions recogntions. A future support in this area can be added for more reactiveness in pupil's games. Voice Recognition : To speed up commands handling, we can use short voice keywords either to move from one mode to another, or to select object. Haptic feedback : A special wearable vest can be used to allow more reactiveness with the user. But this area still lacking innovation because the haptic feedback is by area and not continue.

Conclusion

In this paper we have shown a proof-of-concept mechanism to recognize hand gestures then use them to control a TV or to take actions in a Game scenario for pupils. We have also provided a free library which can be used to represent gestures in other scenarios and other studies. The evolution of our system could be possible in the direction of a multimodal environment in the case where we can be aware of the limits and myths [START_REF] Oviatt | Ten myths of multimodal interaction[END_REF]. But this evolution is a natural choice as new devices are getting more computational power and the ubiquitous environment is becoming a reality.

Fig. 2 .

 2 Fig. 2. Pipeline of the work components showing the routing and transformation of the camera information until getting the gestural information ready to use in the scenario

7 ]

 7 Cb[77 .. 127] C[0 .. 65] Param. Max{R,G,B}-Min{R,G,B}<15 S[0.15 .. 0.75] Cr[133 .. 173] I[0 .. 14] abs(R-G)>15, R>G, R>B V[0.35 .. 0.95]

Fig. 3 .

 3 Fig. 3. A representation of different human skin color variations

Fig. 4 .

 4 Fig. 4. A sub-list of recognition modes showing pointing, number of hand fingers counting and the movement mode