Thomas Corre
email: thomas.corre@ensta-paris.fr

Michel Coret

Erwan Verron

B Leblé

F Le Lay

Experimental full field analysis for dynamic fracture of elastomer membranes

Keywords: dynamic fracture, finite strain, elastomer, full-field measurements

come   L'archive ouverte pluridisciplinaire

Introduction

High speed fracture of rubber-like materials is a long lasting issue, as illustrated by past (Treloar, 1944a) and very recent [START_REF] Moulinet | Popping balloons: A case study of dynamical fragmentation[END_REF] experimental studies devoted to the emblematic problem of the bursting balloon. Theoretically, crack growth in elastomers requires extension of the standard framework of linear elastic fracture mechanics to both geometrical and material non-linearities. [START_REF] Rivlin | Rupture of rubber. I. Characteristic energy for tearing[END_REF] extended the [START_REF] Griffith | The phenomena of rupture and flow in solids[END_REF] theory of fracture to the case of elastomers, proposing an energy-based criterion for crack advance in material under large strain.

Since then, extensive studies have been devoted to the measurement of the energy release rate, first for quasi-static propagation [START_REF] Rivlin | Rupture of rubber. I. Characteristic energy for tearing[END_REF]; [START_REF] Greensmith | Rupture of rubber. X. The change in stored energy on making a small cut in a test piece held in simple extension[END_REF]; Yeoh (2001)) but also during dynamic propagation from low rate [START_REF] Kadir | Tear behavior of rubbers over a wide range of rates[END_REF]; [START_REF] Gent | Adhesion and strength of viscoelastic solids. is there a relationship between adhesion and bulk properties?[END_REF]; [START_REF] Lake | High-speed fracture of elastomers: Part I[END_REF]) up to high crack speed [START_REF] Lake | High-speed fracture of elastomers: Part I[END_REF]; [START_REF] Morishita | Velocity transition in the crack growth dynamics of filled elastomers: Contributions of nonlinear viscoelasticity[END_REF]). These studies focus on the influence of material parameters or crack speed on the energy release rate, which is computed with analytical formulas derived for simple specimen [START_REF] Rivlin | Rupture of rubber. I. Characteristic energy for tearing[END_REF]. Thus, if the crack speed is recognized as one of the main feature of interest, only few studies investigate its evolution in highly stretched membranes [START_REF] Stevenson | On the bursting of a balloon[END_REF]; [START_REF] Gent | Crack velocities in natural rubber[END_REF]) and even less for supersonic crack speed [START_REF] Petersan | Cracks in rubber under tension exceed the shear wave speed[END_REF]; [START_REF] Chen | Scaling of crack propagation in rubber sheets[END_REF]). In this case, the validity of the historical approach is not well-established [START_REF] Gent | Crack velocities in natural rubber[END_REF].

Fast cracks in elastomers are mostly observed in biaxially strained membranes. In this case, 2D digital image correlation (DIC) turns out to be a very suitable technique to measure kinematic fields. Indeed, with the plane stress assumption and the incompressibility of elastomers, the out-of-plane strain can be directly computed from 2D strain fields measured on the surface. Improvements in the DIC technique have extended its applicability to large strain [START_REF] Chevalier | Digital image correlation used to analyze the multiaxial behavior of rubber-like materials[END_REF], in particular for strain fields around static cracks in rubber [START_REF] Rublon | Multiaxial deformation and strain-induced crystallization around a fatigue crack in natural rubber[END_REF][START_REF] Caimmi | Jintegral from full field kinematic data for natural rubber compounds[END_REF][START_REF] Qi | Mapping the nonlinear crack tip deformation field in soft elastomer with a particle tracking method[END_REF]. For moving cracks, many optical techniques have been developed for fragile materials, from photoelasticimetry [START_REF] Dally | Dynamic photoelastic studies of fracture[END_REF] or caustics [START_REF] Washabaugh | A reconciliation of dynamic crack velocity and Rayleigh wave speed in isotropic brittle solids[END_REF] to DIC with high frequency cameras [START_REF] Chao | An experimental study of the deformation fields around a propagating crack tip[END_REF][START_REF] Kirugulige | Measurement of fracture parameters for a mixed-mode crack driven by stress waves using image correlation technique and high-speed digital photography[END_REF] for linear elastic fracture mechanics. In the case of elastomers, which combine high speed cracks and finite strain, very few studies consider full field measurements during dynamic crack growth. Zhang et al. (2009) measured the velocity field around a supersonic crack in natural rubber with particle image velocimetry. To the authors' knowledge, the first full-field measurements around a fast crack (2 m.s -1 ) in a brittle elastomer (polyacrylamide gels) have been recently proposed by [START_REF] Livne | The near-tip fields of fast cracks[END_REF] who track a passive tracer field printed on the gel to measure displacement and strain fields, the nominal stretch in the membrane being up to 13%.

Yet experimental analysis of dynamic fracture of elastomers is still in its infancy, no study has investigated cracks growing under complex loading. In this paper, we propose to benefit from the improvement of high-speed imaging to measure all kinematic fields during crack growth, aiming for an effective procedure for crack growth in initially inhomogeneous strain fields. In particular, monitoring both deformation and fracture of the membrane allows the push-back of kinematic fields in the reference configuration, the first necessary step for a complete mechanical analysis of the phenomenon. This method addresses various issues in large strain dynamics, such as the choice of the relevant configuration to analyse the problem and the feasibility of measuring fracture mechanics quantities. In the following, the experimental method is thoroughly presented in section 2. Then, the crack growth is characterised in section 3 with, as a main result, the various full-fields we measure, from kinematic to energetic fields. These mechanical fields allow for the direct computation of the J-integral and the discussion on the energy flux toward the crack tip provided in section 4. As a supplementary investigation, the observation of mechanical waves during crack growth further illustrates the relevance of the method. Finally, section 5 concludes both on the experimental technique itself, and its specific assets for dynamic fracture of elastomers.

2 Set-up for full-field measurements during high speed crack growth

Sample and material

Experiments are carried out with "pure shear" specimens (Yeoh, 2001), which are widely used for dynamic crack growth, sometimes with prescribed stretch ratio in both vertical and horizontal directions [START_REF] Gent | Crack velocities in natural rubber[END_REF][START_REF] Deegan | Oscillating fracture paths in rubber[END_REF][START_REF] Morishita | Velocity transition in the crack growth dynamics of filled elastomers: Contributions of nonlinear viscoelasticity[END_REF]). Such samples present the advantage to provide an analytical expression of the energy release rate [START_REF] Rivlin | Rupture of rubber. I. Characteristic energy for tearing[END_REF], but they also offer a wide area with homogeneous strain field that allows the observation of steady state crack growth over a long distance. The pure shear sample is a long and thin band of material; here, its dimensions are 200 × 40 × 3 mm 3 . In order to avoid slippage during loading, samples are moulded with two cylindrical bulges (diameter 15 mm) at the edge of their long sides to fit into specially designed grips. In the following, the "working area" of samples will refer to the above mentioned thin band situated between the bulges. So, the width/height ratio of the working area is 5; it is the smallest ratio that satisfies pure shear conditions, i.e. neither extension nor compression in the width direction at the centre of the specimen (Yeoh, 2001). Nevertheless, with such a small geometrical ratio, areas of complex strain field on each side of the specimen are growing rapidly with the prescribed global stretch ratio, reducing the size of the homogeneous pure shear deformation area. In the present study, this limitation will be overcome by the direct measurement of the strain fields with the DIC technique, dispensing with the hypothesis of pure shear state. For DIC measurements, a random black speckle pattern is sprayed on the undeformed surface of the specimens; the material being naturally clear and slightly translucent, it offers a good contrast with the black paint drops under vivid see-through lightning.

The material is a commercial polyurethane made of toluene diisocyanate and polyether. It is assumed homogeneous, isotropic, and incompressible. Then its density ρ 0 does not depend on the deformation and is equal to 1044 kg.m -3 . To determine its mechanical response, uniaxial tension experiments with constant true strain rate of 10 -3 s -1 are conducted and the corresponding nominal stress vs. stretch ratio response is shown in Figure 1. They were carried out with an INSTRON 5584 tensile machine with a 1 kN load cell. To reach high stretch ratios, the tensile sample also have cylindrical bulges. The tests are monitored with a camera (AVT Prosilica GT 6600, lens Zeiss 2/100 ZF) and the strain is measured on pictures by a point tracking method. The main source of inaccuracy is the variability in the section of the sample, explaining the error bars in Fig. 1. Constant strain rate is prescribed by applying a displacement varying exponentially with time at one grip [START_REF] Petiteau | Dynamic uniaxial extension of elastomers at constant true strain rate[END_REF].

The value of 10 -3 s -1 is a compromise between prescribing a very low strain rate, reaching high stretch ratios, and the capabilities of the machine. In the following, the stress-strain curve measured at this strain rate is considered as the quasi-static response of the material. Note that with classical uniaxial tension experiment at constant crosshead speed (displacement varying linearly, the true strain rate is not constant), the strain at failure was measured at about 800%.

For further computation of stress and strain energy, experimental data are fitted with a 2-term Ogden hyperelastic model [START_REF] Ogden | Large deformation isotropic elasticity-on the correlation of theory and experiment for incompressible rubberlike solids[END_REF],

P = 2 i=1 µ i λ αi-1 -λ -α i 2 -1 (1)
where P is the nominal stress, λ is the stretch ratio, and (µ i , α i ) i=1,2 are the material parameters. Their best fit with experimental data gives µ 1 = 6.188 × 10 -2 MPa , α 1 = 3.456 ; µ 2 = -12.73 MPa , α 2 = -0.478.

(2) Note that this model ignores any viscous effects induced by large strain rate. As experimentally observed (with the classical data of Treloar (1944b) for example), the response of elastomers in uniaxial tension and pure shear are very close (see [START_REF] Ogden | Large deformation isotropic elasticity-on the correlation of theory and experiment for incompressible rubberlike solids[END_REF], and [START_REF] Marckmann | Comparison of hyperelastic models for rubber-like materials[END_REF] among others. Then, because the deformation state in the cracked membrane is always between these two types of deformation, our fit is assumed relevant for the present problem.

Loading conditions and procedure

Practically, the experimental procedure is split into two steps as depicted in Figure 2. prescribed stretch ratio. Second, a manual cut with a razor blade initiates the crack at the centre of one edge. More specifically:

-During the first step, the specimen is stretched in the tensile machine until the displacement corresponds to the prescribed stretch ratio, the crosshead speed being set to 20 mm/min. This part of the experiment is recorded with a high resolution camera STEMMER AVT GT 6600 (29 Mpix), referred to as HR camera in the following. The spatial resolution is about 0.04 mm/pix and the frame rate is 0.5 Hz. -The second step starts with the cut of the crack as illustrated in Figure 3.

The cut is made with a blade fixed on a sliding rail that is operated manually. Crack propagation is recorded with a high speed camera Photron SAZ, referred to as HS camera throughout the rest of the paper. The frame rate is adapted for each test, and its maximum value is 40, 000 Hz for full resolution (1024 × 1024). The spatial resolution is about 0.2 mm/pix. It is to note that the cut is made while the crosshead is still in motion. However, due to the short duration of this second step, i.e. between 16 and 300 ms, it is assumed fixed.

Grip razor blade Grip

2.3 Full-field measurements

Measurement of the displacement field

The commercial software Vic-2D is used to measure the displacement field by DIC during the two steps of the experiments. This software uses a local DIC method based on matching n × n pixel squares of a regular grid. This grid serves as the FEM support for our analysis as explained in the following.

Matching is defined by the conservation of the optical flow between two images whose random speckle is the digital signature [START_REF] Sutton | Image correlation for shape, motion and deformation measurements: basic concepts, theory and applications[END_REF]. Due to both thickness of the specimens and in-plane loading conditions, the problem is assumed in a plane stress state such that membrane thinning is easily computed from in-plane displacements and the incompressibility assumption.

As mentioned above, pictures are recorded with two different cameras, resulting in two independent measures of the displacement field. The displacement field measured during the second step does not take the loading step into account. To compute the total displacement field, data from HS camera (second step) are combined with those from HR camera (first step) and projected onto HS images. Adopting a continuous kinematic description, the first image of the HR camera corresponds to the stress free reference configuration of the sample, where the coordinates of the material points are denoted X. During loading and crack growth steps, the coordinates of the material points are denoted x l and x c respectively. Along the first step, the HR camera provides the displacement field directly from the definition:

u l = x l -X.
(3)

During crack growth, the HS camera measures the displacement from the end of the first step:

u c = x c -x l , (4)
which leads to the total displacement field at any time t after crack initiation at t c :

u(t) = u l (t c ) + u c (t). (5)
Practically, the DIC software computes the position of the nodes of a grid covering the area of the specimen; but the grids of the two cameras do not coincide. However, with a low crosshead speed, it is assumed that the last picture of the HR film and the first picture of the HS film, just before cutting the sample, represent the same mechanical state of the sample and therefore correspond to the same displacement field. Between these two pictures, the displacement field u l measured on the HR grid can be projected onto the HS grid [x c ] (in the following, the projection of a given continuous field * onto the HS grid is denoted [ * ]). To do so, the grids of the corresponding images are superimposed (homothetic transformation and positioning from specific points of the sample) and the displacement u l is projected on the nodes [x c ] (linear interpolation from the first grid mesh). The grid from the HR camera is about 30% more dense than the HS grid, which favours the precision of the projection. Then, the total displacement u is computed at the nodes [x c ] thanks to Eq. ( 5), and the positions of

[x c ] in the reference configuration, [X c ], are [X c ] = [x c ] -u([x c ]). ( 6
)
Remark 1 On the one side, the reference configuration is defined here as the undeformed and uncracked sample at the beginning of the loading step. The grid [X c ] maps the sample in this configuration and the initial rectangular shape of the working area is recovered. On the other side, the material configuration refers to an abstract stress and strain free configuration of the cracked sample [START_REF] Maugin | Crack tip and associated domain integrals from momentum and energy balance[END_REF]. In this configuration, the crack is a line that gets longer with time. So, the material configuration only differs from the reference configuration by the presence of the crack: X c provide the position of material points in both reference and material configurations. In the following, the term material configuration will be used when mentioning mechanical fields projected onto the [X c ] grid.

Remark 2

The two cameras cannot be placed right in front of the sample at the same time: we have chosen to adjust the HS camera and then to set the HR camera as close as possible but with a little angle (see Fig. 2 right). This angle, computed from the recorder pictures, equals 6.5±0.5 o . It leads to an error on the horizontal displacement measured during the first step of approximately 0.6 % and does not impact the vertical displacement measurements.

Space and time derivatives

Once the material configuration and the displacement field are determined, the deformation gradient F is computed for any image of the second step (crack growth):

F = I + ∂u ∂X c , ( 7
)
where I is the second order 3 × 3 identity tensor. For each image, Eq. ( 7) is derived using a finite element scheme. The material configuration is meshed with linear triangular elements whose nodes are defined by the positions of the points of the [X c ] grid provided by the DIC, as shown in Figure 4 Finally, thanks to the high frame rate of the HS camera, time derivatives of any mechanical field can be accurately computed during crack growth: the time evolution of any strain field is given by the series of frames during crack growth. In practice, the evolution of fields to be derived at a given node is first fitted by a polynomial of degree 2 over 5 time steps, then the time derivative is computed with this polynomial in order to reduce the noise induced by numerical derivation. Note that first and second time derivatives of the displacement field naturally give velocity and acceleration fields, respectively; whereas the time derivative of F provides strain rate tensors.

3 Results for dynamic crack growth under large strains

Global analysis of crack propagation

After cutting, the crack freely grows, faster than the blade. Time evolution of crack length is measured from the remaining length of the sample in the deformed configuration; an example of such evolution is drawn in Figure 5(a). The high contrast of pictures allows measurement of the crack tip position with a simple binarization. Then, the crack speed is computed using a local polynomial fit of the time evolution of the remaining length (degree 2 over 5 time steps). The corresponding results for different prescribed stretch ratios are shown in Fig. 5(b). All curves are similar and three stages are exhibited: the crack speed rises, then reaches a plateau, and finally decreases. As expected, the pure shear sample leads to a steady state propagation of the crack which corresponds to the plateau; it is due to the wide homogeneously deformed area in the middle of the sample.

Steady state growth Figure 6(a-d) presents some possible shapes of the crack tip opening during steady state crack growth for different stretch ratios; Fig. 6(e) depicts the corresponding crack speeds. The evolution of the steady state speed, i.e. a strong increase at low stretch ratios followed by a milder slope, is very similar to what has been reported for natural rubber by [START_REF] Gent | Adhesion and strength of viscoelastic solids. is there a relationship between adhesion and bulk properties?[END_REF][START_REF] Gent | Crack velocities in natural rubber[END_REF]Zhang et al. (2009). The change of the crack tip shape, from a parabolic tip in Fig. 6(a-b) to a wedge-like crack opening in Fig. 6(c-d), has been also observed by Zhang et al. (2009), and [START_REF] Morishita | Velocity transition in the crack growth dynamics of filled elastomers: Contributions of nonlinear viscoelasticity[END_REF]. In every case the crack growth should be considered dynamic, the slowest crack still growing at 4.71 m.s -1 . But for a stretch ratio higher than 2, the crack lips are retracting fast enough to induce compression of the membrane. For even higher stretch ratios, some out-of-plane displacement and even wrinkles induced by buckling are observed (Fig. 6(d)). As expected, the DIC is then functioning only before the crack tip arrival.

Local mechanical fields

For moderate initial stretch ratios, mechanical fields can be computed on the whole working area. To illustrate the method and allow comparability, all the fields shown in this section are issued from a test with a stretch ratio of 1.936±0.002 (2 was prescribed). The instant chosen corresponds to the picture in Fig. 3 where the crack is growing at 33±0.5 m.s -1 . In addition, sample sizes are indicated by the vertical and horizontal axes, graduated in mm.

Mechanical configurations

The first field obtained from DIC is the displacement field. As an example, Figure 7(a) shows the horizontal displacement field in the deformed configuration. As explained above, the material configuration can be retrieved and the same field can be drawn in this configuration (Fig. 7(b)). In the deformed configuration, the black cross is the position of the crack tip measured directly on the picture (see Fig. 3). The set-up cannot provide displacement measurements at less than 2 mm away from the crack tip (depending on the picture). Then, in the material configuration, the white line refers to the elements where DIC broke down along the crack path.

Time derivatives

The high working frequency of the high speed camera allows the accurate derivation of the velocity field during crack growth. Figure 8 presents both horizontal and vertical components of the velocity field, highlighting the trajectory of material points above the crack tip. The method is precise enough to capture both high vertical retraction velocities behind the crack tip ( 30 m.s -1 ) and slow horizontal movements.

Spatial derivatives Knowing the reference configuration, any strain field can be computed thanks to the deformation gradient. Figure 9(a) shows the Green-Lagrange strain tensor component E yy , with E = 1 2 (F T F -I). Combined with the time derivative technique presented above, the strain rate tensor is also available: it is computed from the time derivative of the deformation gradient Ḟ,

D = sym ḞF -1 . ( 8
)
One of its component is shown in

Robustness of the measurement protocol

Repeatability The repeatability of successive experiments is satisfactory. Indeed, the test with a prescribed stretch ratio of 3 has been repeated 3 times, providing a crack growth speed in the steady state regime of 46.2 ± 0.5 m.s -1 . The repeatability decreases for stretch ratios from 1.7 to 2 because of the strong influence of a small variation in the actual prescribed stretch ratio on crack speed (see Fig. 5). Moreover, discrepancy is observed between the two batches of material used (and tested at different times), as highlighted in Fig. 5. Whether this discrepancy is due to the formulation of the material or experimental conditions, e.g. temperature, is not clear yet. The cutting blade is moved manually, but neither its speed nor its angle has a noticeable influence on the free crack growth. The speed of the cutting blade, measured afterwards on the films, is maintained between 0.3 m.s -1 and 0.7 m.s -1 .

Precision Firstly, field measurements predominantly depend on the displacement field computed by the DIC software. To estimate the precision of this primary measurement, the software provides the confidence interval for the position of each node. This interval varies significantly over one frame, from 4 × 10 -3 mm far from the crack to 10 -2 mm just beyond (values from the HS camera only). However, in spite of the knowledge of the uncertainties in the measured displacement field, we are unable to reliably estimate the precision of the final deformation fields, because of the numerical projection step described in section 2.3.1. Secondly, the precision of the velocity field can be estimated by the noise observed in areas of the sample supposedly at rest at the beginning of crack growth. With this technique, we estimate that the velocity field is measured with a precision of about 0.05 m.s -1 . Finally, in spite of the intrinsic precision of the measurements, the present set-up cannot provide information at less than about 2 mm away from the crack tip (see Fig. 7(a) for instance). The size of the blank zone around the crack increases with the prescribed stretch ratio for three main reasons. First, in front of the crack tip, the material is so stretched that too much light passes through the thin membrane and floods the camera (see-through lightning in Fig. 2). Second, just beyond the crack tip, the high level of shear and the proximity of free edges complicates the correlation. And third, the high retraction speed of both lips immediately after the crack tip can locally blur the pictures and expel some paint drops.

First applications of the method

Measuring the dynamic energy release rate

In the classical approaches of fracture, especially in the case of linear elastic fracture mechanics (LEFM), the crack speed V is predicted by equating the instantaneous energy release rate G with its critical value for a given crack speed Γ (V ), i.e. the fracture energy [START_REF] Freund | Dynamic Fracture Mechanics[END_REF]. Practically, predicting crack growth requires the determination of the curve Γ vs. V , assumed to be a material parameter for a given elastomer [START_REF] Greensmith | Rupture of rubber. III. Determination of tear properties[END_REF]. The J-integral provides an efficient way to compute the energy release rate [START_REF] Rice | A path independent integral and the approximate analysis of strain concentration by notches and cracks[END_REF]. However, the use of such analytical formulas in an experimental context is challenging and requires full-field measurements techniques. There have been very few attempts for elastomers (large strain). Nonetheless [START_REF] Caimmi | Jintegral from full field kinematic data for natural rubber compounds[END_REF] have used DIC to compute the J-integral around a static crack in a rubber sheet, and [START_REF] Livne | The near-tip fields of fast cracks[END_REF] have measured it around a rapid crack (2 m.s -1 ) in a brittle elastomer under moderate strain level. Using our full-field measurements technique, we now question the possibility to measure the energy release rate in a very general case which combines large strain, non-linear elasticity, and high speed crack growth at non-constant speed.

The elastodynamic version of the J-integral was proposed by [START_REF] Atkinson | The flow of energy into the tip of a moving crack[END_REF][START_REF] Atkinson | The flow of energy into the tip of a moving crack[END_REF][START_REF] Freund | Energy flux into the tip of an extending crack in an elastic solid[END_REF], and extended to the non-linear case (both geometric and material) by [START_REF] Gurtin | On the energy release rate in elastodynamic crack propagation[END_REF]. It is based on the computation of the flux of mechanical energy entering a contour encircling the crack tip in the material configuration. For a 2D problem with a crack growing along e 1 at the speed V 0 in the material configuration, the flux entering the contour Σ of outward normal unit vector N is:

Φ(Σ) = Σ (w + k)V 0 + P T • ∂u ∂t • N dΣ, ( 9
)
where w is the strain energy density per unit of volume in the reference configuration, k is the kinetic energy density, u the displacement field, and P the first Piola-Kirchhoff stress tensor (adapted from Freund (1998) Eq. ( 5.2.7)). Making use of the incompressibility of the material, the kinetic energy density is defined as follows,

k = 1 2 ρ 0 ∂u ∂t T • ∂u ∂t . ( 10
)
The J-integral corresponding to the energy flux in Eq. ( 9) is then defined by considering a contour that shrinks toward the crack tip (adapted from [START_REF] Freund | Dynamic Fracture Mechanics[END_REF] Eq. ( 5.3.2)):

G = lim Σ→0 Φ(Σ) V 0 . ( 11
)
Contrary to the quasi-static case, the integral Eq. ( 9) is not path-independent in general. However, for steady-state crack growth, it is path-independent and the computation of the limit in Eq. ( 11) is not necessary (Moran and Shih, 1987). This particular case will be considered in the following and illustrated by an experiment with a prescribed stretch ratio of 2 in which the steady-state speed of the crack is 33 m.s -1 .

Method Direct observation of crack propagation provides the crack tip kinematics in the laboratory frame. In the context of finite strain, retrieving the position and speed of the tip in the material frame requires some caution and at least two hypotheses. First, we assume that the position of the tip in the laboratory frame can be recovered from its position in the deformed configuration (denoted x ct (t)): there is no movement of the matter on the path of the crack. Experimentally, this assumption is verified except in the last millimetres of the crack path where the last portion of the sample moves because of the approaching crack and wave reflection. Second, we suppose that the displacement at the tip induced by the presence of the crack, u c (x ct (t)), is small compared to the displacement induced by the loading step u l (x ct (t)).

Explicitly, with X ct (t) and x ct (t) the coordinates of the material point where the crack tip stands in the material and current configuration, this hypothesis reads:

x ct (t) = X ct (t) + u l (x ct (t)) + u c (x ct (t)) ≈ X ct (t) + u l (x ct (t)). ( 12
)
The symmetry of the crack opening in the case of a straight horizontal propagation in mode I imposes vertical component of u c (x ct ) to be null. However, the need for such hypothesis arises from the finite strain framework in which the component of u c (x ct (t)) in the direction of the crack propagation is not strictly zero [START_REF] Long | Finite strain analysis of crack tip fields in incompressible hyperelastic solids loaded in plane stress[END_REF][START_REF] Goldman Boué | Failing softly: a fracture theory of highly-deformable materials[END_REF]. Combining these two assumptions allows the estimation of u l (x ct (t)) (linear interpolation of the displacement field before crack growth at points x ct ) and the derivation of the crack speed in the material frame:

V 0 = ∂X ct (t) ∂t . ( 13
)
The speed computed with Eq. ( 13) differs from the crack speed computed in section 3 principally close to the edges, where the horizontal stretch ratio is far from 1. [START_REF] Marder | Supersonic rupture of rubber[END_REF] uses directly the deformation gradient of the loading step to compute the material crack speed from the "current" crack speed. In our case, the two techniques provide similar results within the experimental precision.

As aimed, all fields involved in Eq. ( 9) are available through the analysis presented above. Considering the large number of hypotheses required, the integration method is not the main source of uncertainties. This method is illustrated in Figure 12 and can be summarised as follow: -The energy flux is computed on various circular domains around the crack tip in the material configuration, as shown in Figure 12(a). Note that the contours are independent from the mesh and defined analytically. -As illustrated in Fig. 12(b), some elements are missing in the integration zone around the tip: the lack of data in the 2 mm zone around the crack tip is overcome by considering contours with radii larger than this value.

-Mechanical fields involved in Eq. ( 9) are computed in each element: it leads to constant value over the element because of the linear triangular finite element mesh, or it is interpolated at its barycentre for speed and displacement fields. -The contour is discretized and the outward normal vector computed analytically at each node. The value of the vectorial fields in Eq. ( 9) are computed at each node of the contour from the corresponding value in the underlying element. -The scalar product with the outward normal vector in Eq. ( 9) is computed at each node of the contour, and the resulting curve is integrated along the contour.

Unavoidably, data are not available on a small part of the contour behind the crack tip in the neighbourhood of crack lips (see Fig. 12(a)). The significance of this contribution increases as the contour shrinks toward the crack tip. This contribution is evaluated by extending the last data available over this part of the contour on each side of the crack. Note that when out-of-plane motion is observed in the unloaded part of the sample, as in Fig. 6(d), the present computation has to be performed in the vicinity of the crack tip.

Results With the previous analysis, it is now possible to evaluate the possible domain independence of Φ by varying the radius of the integration contour for the steady-state crack growth. The corresponding results are presented in Figure 13 for the particular frame of Fig. 12(a). Fig. 13 indicates that the energy flux is decreasing when the radius of the circle decreases, and therefore it is not path-independent. Note that considering the estimated contribution of parts of the contour close to the lips (error bars) reduces this decrease. The energy flux is made of three contributions (Eq. ( 9)): flux of strain energy, flux of kinetic energy, and the "flux across Σ due to the material outside Σ working on the material inside it" (Moran and Shih, 1987). The flux of kinetic energy is surprisingly small compared to the two other contributions; it is also negative as theoretically predicted by [START_REF] Gurtin | On the energy release rate in elastodynamic crack propagation[END_REF]. Figure 14 illustrates the contributions of the integrand of Eq. ( 9) for a given contour. The use of circular contours suggests an angular description of the integrand, even if the interpretation as a direction of the fluxes obviously depends on the contour shape and has to be taken cautiously. Curves in Fig. 14 lead to a straightforward analysis of the origin of energy entering in Σ: strain energy is mainly convected in front of the crack, kinetic energy is convected outside near the crack lips, and the power contribution increases behind the crack tip (area of high velocity and high shear strain). The kinetic energy contribution is comparable to the strain energy contribution behind the crack tip, as expected from their relative distribution shown in Fig. 10(c). However, the integration over the whole contour provides a small flux of kinetic energy compared to the other terms, in accordance with Fig. 13. The power contribution is the most important one close to the lips (around ±150 • ), but its evolution toward crack lips (±π) is not known due to the lack of data in these areas. Involving the velocity field, this contribution embodies the dynamic nature of the problem that the minor role played by the kinetic energy might understate.

Discussion The previous analysis demonstrates that it is possible to compute contour integrals in a dynamic and large strain context with current full field measurement techniques. Fields are sufficiently smooth for a trustworthy in-tegration on a contour a few millimetres away from the crack tip. However, the energy flux integral depends on the contour on which it is computed in the present steady-state crack growth case. Having checked the hypothesis of steady-state crack growth and estimated the flux close to the lips, two main reasons can explain this path-dependence. First, the constitutive model of the material does not take into account the influence of the strain rate, which is very significant especially in the area behind the crack tip as illustrated in Figure 15. Indeed, the strain rate varies from 0 to about 1500 s )

Fig. 15 Components of the strain rate tensor D (Eq. ( 8)) along the circular path of Fig. 14.

such large strain rates induce a substantial stiffening of elastomer response (see for example Fig. 4 in [START_REF] Roland | High strain rate mechanical behavior of polyurea[END_REF]). Therefore, the present constitutive equation probably underestimates the stress in the area behind the crack, reducing the power contribution in the energy flux. Second, due to the viscoelastic nature of elastomers, some viscous dissipation takes place in this area; it also relates to highly variable strain rates. Located far from the crack tip, this dissipation can occur outside some of the contours considered, explaining some of the decrease in energy flux as the contour radius decreases.

To conclude, and contrary to the case of brittle gels where the path independence of the J-integral was experimentally retrieved by [START_REF] Livne | The near-tip fields of fast cracks[END_REF], our quasi-static hyperelastic model may not be sufficient to precisely compute the dynamic energy release rate of elastomer. However, being able to identify a strain rate dependent model for strain rates varying from 0 to 1500 s -1 remains a difficult challenge.

Observing mechanical waves during crack growth

Quite surprisingly, the precision of the velocity field measurements offers another application to our set-up: observation of mechanical waves during crack growth. Indeed, the velocity field highlights the small displacement of the material points situated away from the crack tip, revealing the mechanical waves travelling across the sample as illustrated in Figure 16. The top picture of Fig. 16 presents the velocity norm in full scale, underlying the high retraction speed of the membrane right beyond the crack. This is the expected picture of the velocity field in this context, with a stationary pattern in the case of constant crack growth speed [START_REF] Chen | Scaling of crack propagation in rubber sheets[END_REF]. As the velocity field is measured with a precision of about 0.05 m.s -1 , it is legitimate to reduce the scale between 0 and 1 m.s -1 as depicted in the bottom picture of Fig. 16. It reveals some of the complex phenomena occurring in front of the crack tip (see supplementary videos). In a non-exhaustive manner, it exhibits:

the influence of the material retraction ahead of the crack tip with possible wave reflection on the clamped edges (velocity larger than 1 m.s -1 close to the clamped edges ahead of the tip), the wave propagation along the horizontal axis of the specimen and reflection at the free edge (velocity up to 0.6 m.s -1 at the free edge), a point of zero velocity in front of the crack tip due to a change in the velocity sign along the horizontal axis: negative horizontal velocity close to the tip and positive far away.

The negative velocity along the direction of propagation and close to crack tip is always observed, as noticeable in Fig. 8(a). Indeed, in front of the crack, the material is in a state of deformation close to uniaxial tension whereas it is closer to pure shear far away [START_REF] Long | Finite strain analysis of crack tip fields in incompressible hyperelastic solids loaded in plane stress[END_REF]. With both Poisson effect and incompressibility of the material, the strain in the horizontal direction is large enough to induce a measurable movement of material points towards the tip, hence a measurable velocity. This effect is particularly noticeable at the beginning of the test when the crack is initiated with a blade ; the sudden appearance of the crack induces this low velocity in the initially steady membrane as shown in Figure 17 velocity is observed over a few images in some tests (HS camera frequency over 20000 Hz). This pulse of horizontal displacement propagating along the horizontal direction corresponds to a longitudinal wave, and its speed can be measured by following a threshold value of the velocity as plotted in Fig. 17(b).

In the case taken as example (prescribed stretch ratio of 3), the pulse propagates at about 105 m.s -1 (Fig. ,17(c)) for a threshold of -0.07 m.s -1 (the minimum speed detectable confidently given the uncertainty of ±0.05 m.s -1 at such low values). The velocity strongly depends on the threshold chosen, retrieving the dispersion of wave propagation in elastomers [START_REF] Mason | Finite elastic wave propagation in rubber[END_REF], and the value of 105 m.s -1 is the maximum observed for this prescribed stretch ratio. Similar results could be obtained using directly the displacement field, highlighting even more the dispersion phenomenon. However, the temporal derivative increases the sensitivity which helps the definition of a clear threshold.

Analysing the small-scale displacement and velocity fields provides useful information to understand crack motion. The importance of waves is yet recognized in the case of brittle fracture [START_REF] Goldman | Acquisition of inertia by a moving crack[END_REF], and their influence on both crack speed and path in the particular case of large strain has to be confirmed. The longitudinal wave speed provides a reference value to characterise the crack speed. For instance, cracks in elastomer can grow faster than the shear wave speed but slower than the longitudinal wave speed: it corresponds to a propagation regime referred to as "intersonic" by [START_REF] Petersan | Cracks in rubber under tension exceed the shear wave speed[END_REF]. This value strongly depends on the anisotropy induced by stretching; it is therefore difficult to compute it analytically. [START_REF] Petersan | Cracks in rubber under tension exceed the shear wave speed[END_REF] used a specific apparatus to measure it with a time-of-flight method. Our set-up provides a direct way to measure it, with an equivalent "optical" time-of-flight method. Even without the crack growth step, a similar set-up might offer the possibility to investigate the wave speed dependence on both its direction (induced anisotropy) and its amplitude.

Conclusion

The two-camera full-field technique proposed in this study offers broad new possibilities in the understanding of fracture mechanisms in materials under large strain. Crack growth can be observed in a very general framework, involving dynamic, large strain, and material non-linearities. Besides the mechanical framework, the set-up permits relaxing the hypotheses that have previously governed the study of dynamic cracks in elastomer membranes, such as the use of particular simple specimens. The use of DIC during the loading phase of the sample provides the membrane strain field in which the crack will grow, rendering possible complex crack paths and not only straight propagation at constant speed. Full-field measurements during crack growth also promise to bridge the gap between theoretical developments and experimental observations and measurements. Even if a local approach would require a more precise measurement technique at crack tip [START_REF] Long | Crack tip fields in soft elastic solids subjected to large quasi-static deformation-a review[END_REF], the energetic approach of fracture, from the direct energy balance to the complex J-integral formula, is well-suited to an experimental comparison. For instance, the controversy as to the use of the pure shear analytical formula in dynamic problems [START_REF] Chen | Scaling of crack propagation in rubber sheets[END_REF] advocates a closer look at crack behaviour through fullfield measurements. In spite of the strict validation of theoretical predictions, very general information, yet not easily obtainable, are available such as the longitudinal wave speed.

Fig. 1

 1 Fig.1Nominal stress vs. stretch ratio in uniaxial tension at constant true strain rate (10 -3 s -1 ). Comparison between experimental data and the 2-term hyperelastic Ogden model.

Fig. 2

 2 Fig.2Left: the two step procedure -loading of the sample and crack initiation with a blade. Right: Scheme of the set-up with the relative location of the two cameras (top view).

Fig. 3

 3 Fig. 3 Crack growth in pure shear specimen. Prescribed stretch ratio : 2. Instantaneous crack velocity: 33.1 m.s -1 . HS camera: 20000 fps, shutter speed 1/160000 s. Note the random black speckel sprayed on the specimen.

  Fig. 4 (a) Mesh in the material configuration and (b) zoom around the crack tip where DIC breaks down (white area). The scales are in mm.

Fig. 5

 5 Fig. 5 (a) Time evolution of crack length in a sample with a prescribed stretch ratio of 2. Time origin is arbitrary. (b) Crack speed vs.normalized crack length, i.e. distance travelled by the crack divided by sample width, for different prescribed stretch ratios (λy).

  Fig. 6 (a-d) Shape of the crack tip during steady state propagation. The initial stretch ratios are (a) 1.46, (b) 1.94, (c) 2.89 and (d) 3.84. The corresponding crack speeds are (a) 4.71, (b) 32.8, (c) 51.4 and (d) 57.3 m.s -1 . (e) Steady state crack speed vs. prescribed stretch ratio (circles and squares refer to different series of tests with different material batches)

Fig. 7

 7 Fig. 7 Horizontal displacement field during crack growth (a) in the deformed configuration and (b) projected onto the material configuration. The black cross in the deformed configuration (a) marks the actual crack tip position.

FigFig. 8 Fig. 9 Fig. 10 Fig. 11

 891011 Fig. 8 Velocity field in the deformed configuration (m.s -1 ): (a) horizontal velocity, (b) vertical velocity. The black cross shows the position of the crack tip, moving at 33.1 m.s -1 in this frame.a

  Fig. 12 (a) Smallest and largest contours used to compute the energy flux integral, centred on the crack tip location in the material configuration (radii in mm). (b) Close-up on the 4 mm contour. The field is the strain energy density.

Fig. 13

 13 Fig.13Energy flux (Eq. (9)) and its contributions vs. the contour radius. Error bars: values obtained with an estimate of the contributions close to crack lips.

Fig. 14

 14 Fig.14Evolution of each contribution of the energy flux along a circular contour (radius 7 mm). Small sketches show the positions along the contour with respect to the crack direction.

Fig. 16

 16 Fig. 16 Norm of the velocity field (m.s -1 ) superimposed on the picture from the HS camera. Top: full scale, 0 to 70 m.s -1 . Bottom: reduced scale, 0 to 1 m.s -1 . Prescribed stretch ratio: 3 -crack speed: 51.s -1 .

  Fig. 17(a 1 -a 5 ) Propagation of the pulse -horizontal velocity field just after the cutting (velocity less than 0.05 m.s -1 in norm is blanked for clarity). (b) Shape of the pulsehorizontal velocity along the centreline of the sample in picture a 2 . Dotted line represents the threshold of -0.07 m.s -1 . (c) Position of the front -position on the centreline where the threshold value is reached versus time; dotted line shows the linear fit giving a pulse speed of 105 m.s -1 .

  Fig. 17(a 1 -a 5 ) Propagation of the pulse -horizontal velocity field just after the cutting (velocity less than 0.05 m.s -1 in norm is blanked for clarity). (b) Shape of the pulsehorizontal velocity along the centreline of the sample in picture a 2 . Dotted line represents the threshold of -0.07 m.s -1 . (c) Position of the front -position on the centreline where the threshold value is reached versus time; dotted line shows the linear fit giving a pulse speed of 105 m.s -1 .

Supplementary material

Four short videos are proposed to illustrates some of the results presented in the paper.

-Crack growth slow ×700.avi presents the film recorded by the HS camera before any DIC treatment; its actual duration is about 12 ms. It corresponds to Fig. 3. -Strain reference configuration.avi presents the evolution of two components of the Green Lagrange strain tensor during crack growth, in the material configuration. It corresponds to Fig. 9. -Speed field.avi shows the norm of the velocity vector, first in full scale then in a reduced one in order to reveal the complex phenomena during dynamic crack growth. It corresponds to Fig. 16. -Wave horizontal speed.avi is the film from which is extracted Fig. 17(a).

Conflict of interest

The authors declare that they have no conflict of interest.

Journal of Fracture 65, 97-114. Yeoh, O.H., 2001. Analysis of deformation and fracture of 'pure shear' rubber testpiece. Plastics, Rubber and Composites 30, 389-397. Zhang, H.P., Niemczura, J., Dennis, G., Ravi-Chandar, K., Marder, M., 2009. Toughening effect of strain-induced crystallites in natural rubber. Physical Review Letters 102, 245503.


F Baillot

D Durox

R Prud'homme

Experimental and Theoretical Study of a Premixed Vibrating Flame 1

An experimental study of vibrating flames above a cylindrical burner has been conducted in order to examine some of the fundamental characteristics of the flow perturbations-combustion interaction. Here the perturbation chosen is a space-time sinusoidal flow velocity. The vibrational nature of the flow has been studied with and without the flame, using a LDV setup and a tomography system. The prevalent kinematic phenomenon is an effect of flow pumping-indeed the velocity perturbation propagates along the vertical axis nearly at the mean flow velocity. A laminar flame in such a flow modulation (<200Hz) responds with a periodic distortion around its steady, almost conical, configuration. The oscillation is the same frequency as that produced by the flow pumping. The deformation and the kinematics of the flame, studied by the means of a tomography system and a photographic record, are characterized by the height, the length, the area, and at the top by the flapping, the absolute speed, the thickness of the thermodiffusive zone, and the normal burning speed. In particular, the burning speed evolves globally between 0.5 and 2 times the laminar burning speed measured at the top of the reference steady flame. The area variations evolve from 11% to 25%, for the RMS intensity of the velocity fluctuations well below 10%. Nevertheless over a cycle the mean area is approximately equal to that of the laminar steady reference flame. On average the instantaneous stretching effects arc negligible on the front structure, in spite of the presence of large instantaneous effects like at the top. An analytical model, based on small disturbances theory and a constant burning speed, shows that the space-time front deformation results from waves originating at the flame base towards its upper limit. A comparison between theoretical and experimental results is in good agreement, 1  first term of the asymptotic development of  with a Ψ progressive wave of kinematic disturbance

NOMENCLATURE

Greek Symbols a half-angle at the apex of the conic front of the steady theoretical flame of refeence ϕ velocity wave phase 1, 2 characteristic wavelengths of the theoretical wrinkled flame ω phase speed of the velocity wave  angle between the unit normal n and the horizontal at a point of the disturbed front of the theoretical flame

INTRODUCTION

The direct investigation of turbulent combustion waves remains a complex task, despite the advent of a new generation of computers and instrumentation, due to the random nature of the flows in which combustion takes place. Thus, in order to study some characteristics of the combustion-flow perturbations interaction, "simple" fundamental experiments involving the disturbance of a laminar flame have been conducted, in which the perturbation is considered to be well-known and well controlled: vibrating flames, (b1) stabilized upon a vibrating flame holder [4][START_REF] Markstein | Combustion, Flame and Explosion Phenomena[END_REF][START_REF] De Soete | [END_REF] (b2) subjected to a time sinusoidal velocity field [3][4][START_REF] Markstein | Combustion, Flame and Explosion Phenomena[END_REF][START_REF] De Soete | [END_REF][7] (b3) subjected to a time-space sinusoidal velocity field [START_REF] De Soete | [END_REF]8,[START_REF] Hoffman-Berling | Nineteenth Symposium (International) on Combustion[END_REF] In the first case (a), the experimental contributions deal with the impact of vortices from a laminar or turbulent Von Karman street, created behind a cylindrical rod placed in an unburned gas flow, on a V-shaped flame front. This interaction can be identified with an idealized two-dimensional turbulence, localized in a precise part of the front, since the perturbation spectrum is reduced to the Strouhal frequency of the vortex passage. As in the case of turbulent flow fields, characterized by a large spectral distribution of different length scales, this phenomenon can be analyzed in terms of a relative scale defined as the ratio of the vortex length over the flame thickness. Since the first experiments [4], much work has concentrated on describing and analyzing the mechanisms along the front: the formation of undulations and cusps, the evolution of the wrinkling enlargement rate, the existence of a transverse propagative disturbance speed (which is equal to the velocity component of the unburned gas flow parallel to the front), and the differentiated role of the outer or inner vortices of the Von Karman street.

In the second type of experiments (b), a sinusoidal disturbance (low frequencies) is applied to the flame, either (bl) by directly vibrating the holder [START_REF] De Soete | [END_REF], or (b2) and (b3) by creating a flow modulation.

Experiments (bl) are similar to the case (a) where the perturbation is two-dimensional and localized on the flame front. The propagative laws of the front perturbations show some similar characteristics. The first observations made by Peterson and Emmons, when the disturbance amplitude is not too large, indicate the existence of a transverse wave that runs along the front at a speed c equal to the gas driving velocity parallel to the reference steady front. As for any V-shaped configuration, the time evolution of the flame surface is difficult to obtain, because of the difficulty in defining the upper limit of the front, which is truly little luminous. Nevertheless a measurement of the area variations can be obtained for a given height limited by two steady planes.

In the case (b2), the flame is placed into a tube and subjected to a sinusoidal time modulation of the unburned gas flow, by using the frequency of the vertical duct analogous to a Helmholtz resonator [3,7]. This problem becomes one of the interactive studies of the phenomena combining combustion and acoustics. The front dynamics has a similar behavior to that observed for the space-time modulation of the free flow. In Ref. 3 some steady contractions of finite thickness are observed along the front, similar to interference nodes with an amplitude that would be zero if there were no nonlinear effects. The presence of these nodes as well as the existence of a transverse wave with a speed c have been demonstrated in a linearized calculation in Ref. 3 and a nonlinearized calculation in Refs. 8 and 10. As a reminder, we also mention some experimental and theoretical studies that have more specifically taken up the acoustic/combustion coupling by using the combustion chamber as a Helmholtz resonator [11][START_REF] Blackshear | Combustion, Flame, and Explosion Phenomena, 4th Symposium[END_REF][START_REF] Valk | [END_REF][START_REF] Poinsot | Mathematical Modeling in Combustion and Related Topics[END_REF][15][16]. These studies are initially based on some observations made by Rayleigh about the response of a diffusive flame to some standing waves developed in tubes [11]. Although they are similar to that of Boyer and Quinard [3] these studies concern purely acoustic parameters such as the power supplied by the flame and, with the exception of Ref. 12, do not examine flame deformation with respect to the front shape and the surface.

The experiments of case (b3) are based on the existence of a space-time sinusoidal velocity field characterized by a wavenumber k and frequency f. The selection of the sinusoidal modulation of the flow by a loudspeaker as a vibrating system makes it possible to isolate a single excitation frequency in a large band of use. Similarly, the disturbance amplitude is adjustable and reaches high values of velocity fluctuations, up to 30% of the mean flow velocity if desired. This problem corresponds more to an idealized combustion/ turbulence interaction because of the nature of the flow: the kinematic perturbations are convected at the speed 2  f/k, practically equal to the mean flow velocity which is similar to the case of a homogeneous turbulent flow with a constant mean velocity [START_REF] Hinze | Turbulence[END_REF][START_REF] Bradshaw | Turbulence[END_REF]. The study bas been carried out on a conical flame. This configuration has the advantage of being a closed surface, and its top can be easily followed in time and space contrary to another point of the front (or a point of a V-shaped flame too) whose coordinates vary simultaneously in time. This geometry then gives access to parameters such as the instantaneous surface of the front and the normal burning velocity at the top.

The present article exhibits experimental results obtained by using new investigative and diagnostic techniques when a premixed flame is placed in such a space-time sinusoidal flow. These results are compared with those provided by the kinematic theory of the front deformation described hereafter.

EXPERIMENTAL STUDY Experimental Setup

A premixed flame is stabilized above a burner with a convergent and cylindrical nozzle (Fig. 1) -for more details on the burner design, see Refs. 8, 19, and 25. Without any perturbation, the steady laminar flame is practically conical except at the top and in the anchorage zone (Fig. 2). Outside these two zones, the normal burning velocity vL is practically constant along the front [START_REF] Van Tiggelen | Oxydations et Combustion[END_REF][START_REF] Hirschfelder | Molecular Theory of Gases and Liquids[END_REF].

The study of the flame is carried out at low pressure (0.25 atm), which offers the following advantages:

1. A thickening of the thermodiffusive and reactive zones, allowing an easier visualization of the front thickness.

2. A better anchorage without flame holder.

3. The attenuation of an extraneous oscillation of the flame top, caused by a low-frequency disturbance fb (  11 Hz), induced by the action of gravity on the hot gases surrounding the flame [START_REF] Durox | Joint Meeting of the British and French Sections of the Combustion Institute[END_REF][START_REF] Durox | Seventh European Symposium on Materials and Fluids Sciences in Microgravity[END_REF][START_REF] Durox | [END_REF]. Since the relative effects of buoyancy and pressure act in a similar way on the convection processes through the Grashoff number to that proposed in Ref. [START_REF] Dordain | Fluid Sciences and Materials Science in Space[END_REF], the pressure reduction counteracts the disturbance action due to gravity. An excitation of variable amplitude and low frequency, generated by the movement of the membrane of a loudspeaker placed at the bottom of the burner, causes the flow to vibrate. The membrane of the loudspeaker is made of fiberglass, resistant to any corrosive effects and ensuring impermeability to gases. The flame responds with a distortion and a periodical oscillation around its steady almost conical position. The intermittency zone thus swept by the flame appears to the eye as a thick front (Fig. 2).

The studied phenomenon appears only at frequencies less than a critical frequency fc whose the value depends on the pressure used (fc  200 Hz).

For each frequency, two values of velocity fluctuation intensities of the perturbation have been chosen, which are, respectively, referred to as test 1 and test 2. The experimental conditions are given in Tables 1 and2.

The study deals as much with the kinematics of the vibrating flow as with the front deformation.

The investigation of the flow field has been accomplished utilizing a two-component LDV system. Since only those phenomena occurring in the unburned gases (including those just at the crossing of the flame front) were of interest, we have chosen to seed the flow with oil droplets (size less than 1 m) that disappear just before the reactive zone. The selected seeding does not provide any Doppler signal in the gas zone with a temperature higher than the vaporization value of the oil droplets [START_REF] Boyer | [END_REF]. One of the channels (the green light beam) detects the front flapping. The intermittency in the reception of data causes a dropout in the P.M. signal, characterized by a plateau. The acquisition of the Doppler signal always occurred in the favorable situation where the rate of validated particles was at least 10 times higher than the excitation frequency f of the generator, which avoids the degradation of the velocity signal-to-noise ratio [28]. The other channel (the blue light beam) was used to measure the axial component V of the only unburned gas velocity from the analog signal of a counter. The signal was directly sent to an oscilloscope or a tape recorder. This technique is more precisely described in Ref. [START_REF] Baillot | International Specialities Meetings on the Use of Computers in Laser Velocimetry[END_REF]. Several pieces of information about the velocity were obtained:

Statistical data such as mean and RMS velocities. Spectral data by a FFT analyser -HP 3582A characteristics: sampling frequency = 102.4 kHz, number of points used for each FFT = 256, number of averaged data blocks = 64 (if needed, possibility 0f averaging 256 data blocks).

Instantaneous velocities Vfront just before the crossing of the flame front inside the measurement volume. These velocity Vfront were measured by means of conditional sampling of the velocity signal: the instantaneous velocity was sampled just before the front crossing as the green light beam P.M. dropped out, since the analog output of the counter of the blue beam channel remained at the level of the last Doppler burst.

The investigation of the front deformation was performed using a laser tomography system. The droplets in the unburned gases vaporize at an isotherm of about 600 K in the flame thermodiffusion zone. When the flow is lighted by a laser sheet, only the droplets in the unburned gases scatter the light, the burned gases being transparent. The line of vaporization of oil droplet then indicates the position of the front [START_REF] Boyer | [END_REF]. In order to obtain the geometrical and kinematic characteristics of the front, we recorded its evolution using a 16-mm camera. From these sequences, filmed images have been processed by means of a digital table. The coordinates of front points are obtained with an accuracy between 0.05 and 0.1 mm and processed by a microcomputer. In addition to the visualization of this isotherm line, the reactive zone has been detected simultaneously, which has given, in particular, a value of the flame front thickness at the top.

Vibratory Nature of the Velocity Field with no Flame

In the absence of flame, the undisturbed flow is steady, cylindrical, and spreads out slightly up wards. It showed no destabilization by vortices, caused by the shear layer between the jet and the ambient air, over a distance at least one and a half times longer than the space occupied by the flame. It is characterized by a nearly uniform distribution of the axial velocities over the whole jet cross-section except at the edge, in agreement with Ref. [START_REF] Van Tiggelen | Oxydations et Combustion[END_REF].

Disturbed by an external sinusoidal perturbation at the frequency f provided by the loudspeaker, the jet, still with no flame, is characterized by some oblong structures starting at the base of the burner and moving at a speed c (Fig. 3). The jet is defined by a sinusoidal velocity field, with the vertical component V expressed by [START_REF] Escudié | Lecture Notes Eng[END_REF] where x and y are, respectively, the horizontal and the vertical polar coordinates (Fig. 1), v' is the RMS intensity of fluctuations, V is the mean velocity, ϕ is the wave phase angle between the velocity fluctuations and the initial vibration of the generator, f is the forcing frequency, and k is the perturbation wavenumber related to the propagation velocity c by e = 2  f/k. We have also verified, as De Soete did, that the V(y, t) component at the burner exit was indeed sinusoidal and that the phase angle remained constant in the planes y = 2 or 5 mm, as far as the measurement domain remained outside the boundary layer of the jet.

In Fig. 4, the evolution of (ϕ-ϕ0 )/k* versus y has been plotted. The parameter k* is the wavenumber linked to the propagation speed (V ) of the velocity perturbation, where (V ) is the average of the mean velocities V(y), as defined by the equality ( ( )

 = = N i l N y V V , 1
) (N is the number of points used to estimate (V )). Whatever the flow conditions (frequency and average velocity), the results are well grouped around a unique straight line of interpolation calculated by a least squares method. With no flame at 70.4 Hz as at 35.2 Hz, the speed c of the velocity wave along the axis remains lower by about 33% than the average V . A similar study carried out with a hot wire anemometer confirms the present results [START_REF] De Soete | [END_REF]. These data are also shown in Fig. 4. This space-time velocity disturbance assumes that the prevalent phenomenon of the flow field is not as much an acoustic disturbance as an effect of forcing fluid, causing a space-time pumping of the flow. Of course, the disturbance created near the loudspeaker which propagates inside the cylindrical burner is of acoustic type as shown by the measurements of the mean axial velocity carried out by De Soete [START_REF] De Soete | [END_REF], ( = const inside the burner whatever the y value). Outside the burner, however, there is an adaptation of the flow to new boundary conditions evidenced by the existence of this phase angle ( ) y  .

Velocity Field with Flame

The velocity measurement technique with oil droplets does not allow measurement of the velocity of the burned gases, and therefore two zones of study may be investigated:

1. The zone of only unburned gases situated inside the conical envelope of the front flapping.

2. The intermittent zone.

Unburned Gases Zone

The spectral study of the velocity signal (Fig. 5) and its correlation with the generator signal show that the velocity component V can be correctly represented in the exit plan of the burner (y=2 mm) by Eq. 1, as long as the domain of measurement is outside the boundary layer, equivalent to 10% of the exit cross-section where in fact v', V, and ϕ depend on the horizontal abscissa x.

The velocity measurements along the vertical axis of symmetry show that V(y, t) still follows closely this sinusoidal evolution represented by Eq. 1, except in the vicinity of the intermittent zone. In fact, in this region the low frequency fb of the sinusoidal disturbance due to buoyancy (see § Experimental Setup) can be detected: in Fig. 5, from y = 25 mm at 35.2 Hz and from y= 20 mm at 70.4 Hz, the spectral analysis of the velocity signal shows the two predominant frequencies equal to f and fb (fb = 10.9 Hz at 35.2 Hz and fb = 11.2 Hz at 70.4 Hz). However, in the nonintermittent zone, since the amplitude of the disturbance remains very small with respect to the intensity of the forced kinematic fluctuations of the movement, the Eq. 1 remains a good approximation of V(y, t) (Fig. 6).

The cross-correlation between the generator signal and the local velocity signal makes it possible to obtain ϕ; the evolution ϕ (y) is plotted in Fig. 4 (the method used to obtain ϕ, is the same as the one with no flame). The phase angle ϕ, still varies linearly with y, but the value of the slope of the corresponding straight line of interpolation, that is, the nondimensional wavenumber k/k* us not the same as before: k/k* = 0.88 at 35.2 Hz and k/k* = 0.98 at 70.4 Hz instead of 1.33.

Therefore the presence of the flame modifies the propagation speed but does not change its nature since the velocity fluctuations remain a sinusoidal function of the frequency f and the wavenumber k. This modification can be explained both by hydrodynamic interaction of the wrinkled flame with the incident flow and by expansion of the unburned gases within the thermodiffusive zone. This influence leads to a propagation speed c that is practically equal to the average V of the mean flow velocities.

In Fig. 6, the mean velocity ( )

y V
and the intensity v' of the fluctuations decrease by 3% and by about 30%-40%, respectively, in presence of the flame, as shown by the measurements carried out along the axis. All the discrepancies observed in the cases with and without the flame can be interpreted as a display of the hydrodynamic effects of the flame upon the flow which retroacts" by diminishing the velocity amplitude.

At the burner exit (0 <y < 10 mm), the slope of the velocity V is not at all flat, as in the case with no flame; the velocity is greater at the exit and decreases with increasing y. This phenomenon may be due to a nozzle effect caused by the heating of the burner which is attenuated with y. This has been already observed in a laminar steady case [START_REF] Wagner | [END_REF] and in a turbulent case as well [31]. For y  10 mm, regard of excitation frequency used, an increase is observed, possibly due to expansion effects of gases.

In Fig. 6 we can notice a clear decrease of the fluctuation v' in the field of unburned gases as the distance from the burner increases. Then, just before the flapping zone, there is a large and sudden increase of v', with a maximum at the entrance of the intermittency zone. This v'increase zone extends over 3-10 times the thermodiffusive thickness of the flame. This effect increases with the frequency f which bas to be compared with the front kinematics whose movement becomes wider from 35.2 to 70.4 Hz. This phenomenon seems to be a turbulence increase just upstream of the flame and is probably due to a hydrodynamic influence of the flame upon the flow.

Intermittent Zone. It should be remembered that the oil droplets, used to measure the velocities, disappear at an isotherm before the reactive zone and give information only in the unburned gases. Therefore the analog velocity signals coming out of the counter present some plateaus, due to the absence of luminous signal. The duration of these plateaus increases as far as the measurement volume approaches the upper limit of the intermittent zone. The counter is then held until the arrival of the next oil droplet in the measurement volume. Such recordings of analogic velocity signals are presented in Ref. [START_REF] Baillot | International Specialities Meetings on the Use of Computers in Laser Velocimetry[END_REF]. The spectral analysis of the velocity signal always shows the two predominant frequencies equal to f and fb (see previous section). However, at 35.2 Hz the signal maintains the same regular shape as in the first zone; the amplitude of the "disturbance fb" remains lower than that of the "excitation f." The front and the velocity signal are in phase. On the other hand, at 70.4 Hz, the velocity signal [8] is somewhat modified, in particular, by a superposition of a nonnegligible modulation of the low-frequency fb perturbation (see in Fig. 5 the amplitudes corresponding to f and fb). This phenomenon leads to a fluctuation of the value of the unburned gas velocity Vfront at the crossing of the front which increases with y. Therefore the mean value front V calculated over time at a point of the front (Fig. 7), tends towards the mean velocity V measured at this same position (Fig. 6), when the frequency is increased from 35.2 to 70.4 Hz. Moreover the frequency of the vibrating flow shows a discrepancy compared to the value f, which increases with y. The front flapping and die velocity signal do not evolve completely in phase. A display of this distortion is given by the measurement of front V in Fig. 7: the maximum ( ) max front V is no longer obtained when the flame is the tallest, as f increases (this was confirmed at another frequency (106 Hz)); on the contrary, the associate position ymax moves away from the highest position Hmax of the front. In order to carry out a more precise quantitative study of the velocity signal starting from the mean and fluctuating velocities measurement, a conditional sampling would be necessary, taking into account only the velocities of the unburned gases, that is, excluding the plateaus. Therefore in die special case of v' (Fig. 6), without such an operation, we observe a large decrease, followed at the end of the intermittency zone by a rise. This sudden increase may be explained by the hydrodynamic effect due to the unsteady movement of the front, and more particularly at the top zone when the flame is the tallest. In particular, at 70.4 Hz, the flame is "pinched" close to the top, so that the combustion is practically instantaneous and cuts off the flame (a complete front deformation cycle is shown in Ref. 8).

Deformation and Kinematics oh the Flame Front

As soon as the laminar flame is subjected to the disturbed flow, it moves with an elongation phase and a collapse phase on both sides of its reference steady state. The spacetime evolution of this deformation, as compared with the reference steady state (.)0 is characterized by the height H(t), the length L(t) of the front contour corresponding to the projection of the front surface in a vertical plane of symmetry, the axisymmetrical area S(t), and at the top by the flapping nature, the absolute flame speed ws(t) and the thickness of the visualized thermodiffusive zone (Fig. 8). The parameters of the steady state of reference are reported in Table 3.

The spectral analysis of the analog signal of the top flapping shows an adaptation of the flame to the upstream disturbance: the front flapping is periodical, with a frequency identical to that of the loudspeaker vibration at 35.2 Hz. Nevertheless we observe a slight difference (less than 0.5%) between the front flapping and the loudspeaker vibration, when the frequency of the loudspeaker is equal to 70.4 Hz. This is probably due to the disturbance induced by the buoyancy effects in the gases [8,[START_REF] Durox | [END_REF]. Moreover, the frequency fb of the buoyancy effects disturbance does not clearly appear on the flapping spectrum, as is the case for the velocity signal. Therefore the sensitivity of the front to the latter remains small, contrary to the velocity field (see the previous section). Thus the flame is more sensitive to the main disturbance started at its base than to the presence of the disturbance of the frequency fb . The front is then wrinkled essentially by a progressive wave that propagates from its base up to its top.

The measurement of the distances between two successive undulations is not identical in time. Thus it is impossible a priori to define a single propagation speed. However, the existence of a propagation velocity close to the velocity of the mean flow represents the phenomenon, as suggested in Refs. 4 and 6. This result is also advanced in Refs. 1-3. The model proposed at the end of this article confirms this for a flame that is sufficiently wrinkled or tall [8] In Fig. 9, an increase of the flapping amplitude (Hmax -Hmin)/Ho is observed with the amplification of (v'/ V )y=5mm or when the frequency f is increased from 35.2 to 70.4 Hz. Nevertheless its evolution is linear neither with a nor with f. Overall, as shown in Fig. 10, the flame has a mean elongation of ± 15% on both sides of its reference position. This is important in comparison with the ratio (v'/ V )y=5mm which remains much lower than 10% (Table 2). However, during a cycle, the duration of the elongation phase is systematically longer than that of the collapse phase. At 70.4 Hz, a sudden transition from Hmax to appears due to a strong elongation effect at the top. It seems that this phenomenon is an inherent characteristic of the excitation frequency f and is independent of the disturbance amplitude within the experimental approximation. We have measured the length L(t) of the contour and deduced the area S of the flame. In Fig. 10, it appears that the variations of H and L are similar, regardless of frequency. This strong correlation between H and L is due to the absence of large frontal distortions.

The comparison of the curves S(t) and H(t) shows that the flame essentially evolves in the elongation phase (Fig. 10): the area increases during about 60% of the cycle at 35.2 Hz and 70% at 70.4 Hz; the height increases during about 55% of the cycle at 35.2 Hz and 75% at 70.4 Hz. We observe that the height extrema are a lagging behind those of the area. This seems to be due essentially to the kinematic effects, as suggested in Ref 8.

The area variation reaches 25% at 35.2 Hz and 11% al 70.4 Hz, for an intensity (v'/ V )y=5mm of 6.1% and 5.3%, respectively (Fig. 10). During a cycle, the mean area ST of these instantaneous area measurements is approximately equal to the area of the reference laminar flame S0 (the difference is less than 5%). It follows that the front propagates on average at a relative burning velocity vT equal to the adiabatic laminar burning velocity. In a purely turbulent flow condition, the same conclusion was reached in Ref. 32. The instantaneous stretching effects due to the front curvature and to the deformation rates would therefore be negligible on average on the front structure, or at the most, smaller than the measurement uncertainties. Of course these effects may be locally strong, as shown by the results of the evolution of the top normal burning velocity presented below.

From the absolute speed ws of the flame top, taken from the H(t) curve (Fig. 10), the normal burning velocity vn, at the top bas been calculated, with respect to the instantaneous velocity at the front crossing. The variables at the front top are defined as follows:

* s t n w V v - = fron , * dt dH w s = .
The velocity vn is presented in Fig. 11: it evolves globally between 0.5 and 2 limes the laminar burning velocity vno measured at the top of the reference associated steady flame (thus vno takes into account the curvature effects, and is therefore different from the plane adiabatic burning velocity vL). The values of vn, may exceed 2vno during elongation but also can decrease strongly and tend toward zero. The average n v of vn, over a cycle becomes clearly smaller than vno at 70.4 Hz, if it remains equal to vno at 35.2 Hz. The top stretching effects therefore increase, on with frequency and tend to reduce the burning velocity. The superposition of curves S(t) and vn(t) (Figs. 10 and11), shows the coincidence of the minima (maxima) of vn(t) with the maximum (minimum) areas of the instantaneous surfaces. Thus, when the flame is strongly elongated (collapsed), the stretching effects, tangential to the front, can impose a considerable decrease (increase). Similar observations have been formulated in Ref. [START_REF] De Soete | [END_REF]. Lastly, during test 1 it was possible to measure the time evolution of the top, visualizing the external part of the reactive zone (Fig. 12). This point moves in an analogous manner to that of the last visualized isotherm of the thermodiffusive zone as reference for the height calculation. Since the distance d that separates them remains of the same order of magnitude over a cycle, it can be said that the flame thickness at the top is almost uniform. We have d  1 mm, a value similar to that of the characteristic distance of the steady flame. Thus low frequencies have no in fluence upon the combustion processes for a first-order approximation.

This experiment clearly shows the flame/flow coupling.

Fig. 12. Evolution of the visualized thermodiffusion thickness versus dine.

THEORETICAL STUDY

In order to investigate this behavior, we have analytically studied the response of a thin premixed flame stabilized above a burner with a circular cross-section. The computation has been extended to the case of a rectangular geometry, that is, to a dihedric or a V-shaped flame [8].

The configuration of the reference steady flame placed in a uniform flow of velocity V0 j is conical (Fig. 13). Fig. 13. Geometric description of the steady and unsteady theoretical fronts.

The theoretical kinematic approach to the front deformation, is based on the following principal assumptions:

1. The velocity field does not appear to be a function of the front position. 2. The relative normal burning speed vL is uniform. Thus the front (disturbed or not) is characterized by the same relative normal burning speed -locally plane and adiabatic -which remains uniform if the unburned gases are at a fixed temperature, pressure, and concentration. The effects of the curvature and the stretching on vL are assumed to be negligible. This implies in particular that the velocity field, within the flame thickness, is not subjected to any local change, due to the presence of curvatures.

3. The flame is considered as an interface.

The flame is assumed to be a surface constantly attached to the edge of the burner, with the equation [START_REF] Herzberg | [END_REF] x=F(y, t)

where x represents the vector radius with respect to axisymmetrical flows. In the reference state, this evolution equation is written

(2')  tan y R x - =
where R is the radius of the burner and  is the half-angle at the apex of the conical front, the sine of which is equal to the ratio v L /V0. The unit normal to the instantaneous point of the flame, directed towards the burned gases, is at an angle  to the horizontal (Fig. 13). We have

(3)  tan - =   y F (4)   sin cos V U v + =  = ⊥ n v
and, if w is the normal flame speed,

(5) t F w   =  cos
The parameters U and V represent the coordinates of the unburned gases velocity v projected in the polar coordinate system.

The dissipative phenomena in the gases are negligible upstream and downstream of the surface. Thus the only balance equation to be considered at the interface crossing is that of the mass conservation under the uniform burning velocity assumption:

(6) ( )   ( )   const downstream upstream = - = - ⊥ ⊥ w v w v   Or (6') L v w v = - ⊥
With the notations introduced above, this leads to the new following expression:

(7)   cos tan L v V U t F - + =  
From Eqs. 3 and 7 the fundamental equation is deduced:

(8) ( ) 0 cos sin sin 2 = + +   - +         ' V cos ' U y v V t L
U' and V' are the derivatives of the components U and V with respect to y at constant t and taking Eq. 2 into account. Provided with the boundary conditions indicating that x = R for y=0 whatever the t value, the functions ( ) t , y  then F(y, t) can be determined. We introduce the nondimensional kinematic disturbance as a variation  from the reference steady uniform flow V= (1 + ) V0, with

( )   c y t f a - =   2 sin
.  is a progressive wave of kinematic disturbance, propagating at the speed c, with a frequency f such that the wavenumber k is equal to the ratio c f  2

.

[ The case ( )

t   =
, which is a velocity disturbance propagating at quasi-sonic speed, has been also studied in Refs 8 and 10.]

The computation is made with nondimensional variables, relating the length to R, the velocity to V0, and the time to R/V0. The phase velocity  is equal to f  2 . The horizontal velocity component U under the assumptions of incompressibility and axisymmetry is

U= a k x cos(w t -k y)/2.
The complete analytical solution is that of the small disturbance case, that is, a << 1; hence the input of linearized variables:

θ =  + a θ1+O(a 2 ), x = x0 + ax1 + O(a 2 ).
This velocity field does, in fact, verify the Euler equation of the order O(a 2 ). From Eq. 7 and the boundary conditions, the nondimensional expression x of the front deformation is extracted (calculations are made in de tails in Ref. 8): predominates. For long amplitudes of undulations (k small) and small flames, the term 2B sin ( t - Y/2(1/c1 + l/c2) is the most important. The predominant propagation speed is 2 c1 c2 /(c1+ c2). In the other cases, the three functions have a similar influence.

x = 1 + y tan + a x1 + O(a 2 ), x1= -2{A cos ( t - Y/2 (l/cl + 1/c2)), +B sin( t-Y/2(1/cl + 1/c2)} sin(Y/2(1/c1 -1/c2)) -YC sin( t -w Y/c1) with A = k/2( -k cos ), B = tan  ( -k cos 2  / ( k cos ) 2 = 2 Ah() tan  /k, C = A tan , Y = y/cos ,
The computation of the absolute flame speed , equal to ( )

2 1 cos a O t x a +   
, indicates the absence of steady nodes, characterized by a speed w equal to zero whatever the t value, except the anchorage point (x = 1, y= 0), contrary to the case Ψ=Ψ(t) [8,[START_REF] Baillot | [END_REF].

There are two distinct propagation speeds, c1 and 2 c1 c2 /(c1+ c2) each of which defines a characteristic wavelength 1 and 2, equal to c1/f and 2 c1 c2 /f (c1+ c2)Thus from a direct measurement of the wrinkling wavelength on the flame front, it is not possible to get back to 1 and 2. This calculation makes it possible to analyze the behavior of the front regardless of dispersion law k = k(). In particular an analysis of the case k = , corresponding to a propagation at the mean flow velocity, is developed in Ref. 8.

COMPARISON BETWEEN THE THEORETICAL AND EXPERIMENTAL RESULTS: CONCLUSION

The comparison of experiment with theory is possible since the experimental conditions are included in the restricted field (frequency, intensity) of the disturbance, determined under the hypothesis of a propagation at the mean flow velocity, limiting the validity of the linearized model [8].

The theoretical reference configuration has been chosen by taking the ideal cone tangent to the actual flame front. Thus we have introduced in the computation of the front the numerical conditions based on the experiment and listed in Table 4.

Although we have not explicitly taken into account the influence of the flame upon the flow, and in particular the gas expansion [33], the characteristic values of the disturbed velocity field (frequency f, amplitude a, wavenumber k, speed V ) introduced in the theoretical computation, correspond to those measured in the presence of the flame, and are therefore a priori influenced by the retroactive hydrodynamic effects of the flame on the flow of the unburned gases. For instance, compare the values of k and V obtained with and without flame (Figs. 4 and6).

An example of the theoretical and experimental front profiles, obtained by processing images, is plotted in Figs. 15 and16. The comparison of the results at 35.2 Hz, just as at 70.4 Hz, indicates a very good agreement between experiment and theory, since the overall shape of the front is maintained. As might be expected, the most important differences appear at the base and the top of the front. In fact here, the curvature and stretching effects are more important and modify the value of the normal burning speed, whose variation is not considered in the calculations. It can be considered that the experimental propagation of the deformation can be well represented by the theoretical model, and in particular by the simultaneous existence of the two velocities c1 and c2, practically equal to (V )/cos  and (V )cos  according to the experimental results of the section on Unburned Gases relative to the propagation speed at the mean flow velocity. In the case of high flames, the model shows that the apparent predominant velocity is V0/cos . The propagation speed is therefore proportional to the mean velocity V0 of the flow. This phenomenon is mentioned in other theoretical and experimental works [START_REF] Escudié | Lecture Notes Eng[END_REF][START_REF] Herzberg | [END_REF][3][4][START_REF] Markstein | Combustion, Flame and Explosion Phenomena[END_REF][START_REF] De Soete | [END_REF].

The comparison of the experimental and theoretical temporal curves of the front surface area, while confirming this agreement, shows some differences (Fig. 17). It would seem that the maxima have been undervalued. Indeed it has been demonstrated experimentally that the maximum areas are related to a maximum elongation of the flame at the top, and the assumption of a uniform normal burning velocity vL introduced in the computation quite obviously prevents the existence of such phenomena. Moreover, Fig. 17 shows an overvaluation of the theoretical minimum area Smin /S0 due to the truncation limited to the order O(a 2 ) beyond the vL invariability assumption; the discontinuity of the flame curve appears as soon as the phenomenon of "pinching" occurs. The mean area ST calculated over a cycle is equal to the steady area S0 at 35.2 Hz, but at 70.4 Hz ST exhibits a slight difference from S0. This difference observed at 70.4 Hz can certainly be explained by the linearized method which is near its limit of validity as suggested in Ref. In Fig. 17, we have also plotted the nondimensional curves of the height H(t) and the speed at the top ws(t) The theoretical expression of the speed ws is equal to w(ys, t)/sin  at the first order of the asymptotic development as shown in Ref. 8. The reference height (experimental and theoretical) and the flow speed ( <V> = V0) are used to obtain nondimensional variables.

The curves of height evolution show a good agreement between experiment and theory -The elongation and collapse phenomena are correctly represented by the model. A more accurate study shows some differences: at 35.2 Hz, the theoretical front elongation duration is longer than that which is obtained experimentally. This causes an offset of the extrema Hmax and Hmin and a more rapid collapse of the front. At 70.4 Hz, this difference is strongly reduced by the abrupt nature of the collapse. In both cases, the minimum height zone seems more spread out on the experimental curve than on the theoretical curve. The actual top of the flame has therefore a less rapid response to the collapse phenomenon than in the model. On the other hand, the amplitude of the flapping (Hmax -Hmin)/H0 has been studied theoretically. The theoretical curve (Hmax -Hmin)/H0 versus f, for a given value a, presents a curve with a maximum [8]. This represents a filter phenomenon for the highest frequency values. This behavior has been observed experimentally, and in particular the maximum of the flapping for the test 2 conditions is obtained in the vicinity of 70.4 Hz.

The theoretical curves of the top speed over a cycle are in good agreement with the experimental velocity evolution. Overall, a quasi-systematic increase of the theoretical values over the experimental ones is observed. This difference cannot be only explained by the velocity measurement process. The assumption of a uniform vertical mean field should contribute to this translation. At 70.4 Hz, the negative velocity peak is very much undervalued. The assumptions of invariance of the mean velocity field and of invariability of the normal burning speed probably hide the occurrence of these high values of ws.

This purely kinematic model of small disturbances has indeed demonstrated the predominant characteristics of the front deformation as much on a global level (form of the profiles of the front, surface variation) as locally at the top. The agreement of the experimental and theoretical results confirms the important role played by the kinematics of the unburned gas flow on the behavior of the front.

In the future it is necessary to extend the calculations in order to consider the variations of the burning speed with curvature and stretching. If this approach based on a method of small perturbations gives some information about the deformation process of the front, it would be useful then to develop a numerical scheme to complete the analysis in the presence of high perturbation amplitudes.

This study has provided the following information about the response of a premixed flame to an external sinusoidal low-frequency perturbation:

1. The front oscillates at the same frequency as the perturbation frequency. 2. The visualized thermodiffusive thickness at the top is unchanged in comparison to the laminar case: with only first-order consideration the perturbation does not seen to modify the inner structure of the flame.

3. The deformation of the front is due to progressive waves originated at its base and then convected along it proportionally to the flow velocity V . Two speeds of propagation appear, V cos  and V /cos , the latter of which predominates for sufficiently long fronts.

4. The flame is subjected during a cycle to large elongations, leading to important instantaneous variations of the front area (25% at 35.2 Hz, 1l % at 70.4 Hz) and of the normal burning speed at the top (from 0.5 to 2 times the laminar value of the normal burning speed at the top). But on average the area is equal to the area of the reference laminar flame. This suggests that the stretching and curvature effects are negligible on average, even if locally they may be particularly important like at the top.

a

  amplitude of velocity perturbation c propagation speed of velocity perturbation c1, c2 speeds of progressive waves developed along the front f frequency of the external sinusoidal perturbation fb frequency of the disturbance due to buoyancy effects of hot gases F space-time function of the flame front position H height of the flame H0 height of the steady flame in the reference state j vertical vector k velocity perturbation wavenumber (=2 f/c) k* velocity perturbation wavenumber of reference (= 2 f /<V>) L length of the tomographic flame front contour L0 length of the tomographic steady flame front contour in the

  flames disturbed by a street of vortices [1-4] (b)

Fig. 1 .

 1 Fig. 1. Setup of the facility.

Fig. 2 .

 2 Fig. 2. (a) Laminar conical flame with no perturbation. (b) Thick front visible to the naked eye with 70.4 Hz perturbation. (c) Instantaneous wrinkled flame at 70.4 Hz (processed image).

Fig. 3 .

 3 Fig. 3. Shape of a meridian section of the undisturbed and disturbed flows with no flame by laser tomography.

Fig. 4 .

 4 Fig. 4. Evolution of the phase angle ϕ of the velocity perturbation versus y in the absence and in the presence of flame. -, straight line of interpolation; --straight line characterizing the propagation at the average speed V .

Fig. 5 .

 5 Fig. 5. Frequency spectra of the analog velocity signals and of the front flapping signals along the y axis at x = 0 f= 35.2 Hz (test 1).

Fig. 5 .

 5 Fig. 5. f = 70.6 Hz (Continued)

Fig. 6 .Fig. 8 .

 68 Fig. 6. Evolution of the mean velocity and rms fluctuations versus y in the absence (E) and in the presence (•) of flame at x=0. , limit of the intermittency zone (test 1).

Fig. 9 .

 9 Fig. 9. Evolution of the tip flapping (Hmax-Hmin)/H0 versus (v'/ V )y=5mm and f.

Fig. 10 .

 10 Fig. 10. Evolution of height, length, and area versus time (test 2).

Fig. 11 .

 11 Fig. 11. Evolution of the normal burning velocity at the top versus time vno: 2.31 m/s (test 1); 2.32 m/s (test 2).

  the ordinate of point of the front in an inclined system attached to the reference front, c1 = /k cos , and c2=  cos  /k.A schematic representation of the speeds c1 and c2 is shown in Fig.14. c1 is the propagation velocity necessary for a flow disturbance covering the vertical distance y at the velocity /k, to cover the distance Y along the front; c2, is the driving velocity of the disturbance, tangentially to the reference flame front.

Fig. 14 .

 14 Fig. 14. Schematic representation of the wave speeds c1 and c2.

Fig. 15 .

 15 Fig. 15. Comparison of the theoretical fronts with the experimental fronts over a cycle at 35.2 Hz. p.f., perturbed front; f.r., reference front.

Fig. 16 .

 16 Fig. 16. Comparison of the theoretical fronts with the experimental fronts over a cycle at 70.4 Hz. p.f., perturbed front; f.r. reference front.

  8. On the other hand the equality between the two areas ST and S0 has been confirmed experimentally at 35.5 Hz as well as at 70.4 Hz.

FFig. 17 .

 17 Fig. 17. Comparison of die theoretical area, height, anti tip velocity of the front (-) with the experimental values (.) over a cycle.

Publié dans Combustion and Flame 88:149-168 (1992)

Laboratoire d'Aérothermique du CNRS, F92190 Meudon, France