J P Goulmy
email: goulmy@lcts.u-bordeaux.fr

G Camus

F Rebillat

Monitoring damage evolution of ceramic matrix composites during tensile tests using electrical resistivity: Crack density-based electromechanical model

Keywords: Ceramic Matrix Composites (CMCs), electrical resistivity, acoustic emission spectra, tensile tests, damage monitoring

In order to further develop and understand the data provided by electrical resistance measurements, three ceramic matrix composites (CMCs) were characterized in tension. The observed changes of the electrical resistance were compared to the acoustic emission spectra, another commonly used damage monitoring method, as well as the classical interposed unloading/reloading cycles. A model was then proposed in order to predict the evolution of the resistance as a function of the damage state of the three composites. The proposed model provides accurate results for the three materials which, although they all belong to the CMC family, display different mechanical and physical behaviors.

Introduction

The excellent thermo-mechanical properties of ceramic matrix composites have led to many developments regarding their use inside aeronautical jet engines [START_REF] Christin | Design, Fabrication, and Application of Thermostructural Composites (TSC) like C/C, C/SiC, and SiC/SiC Composites[END_REF][START_REF] Naslain | Design, preparation and properties of non-oxide CMCs for application in engines and nuclear reactors: an overview[END_REF][START_REF] Bouillon | Engine Test and Post Engine Test Characterization of Self-Sealing Ceramic Matrix Composites for Nozzle Applications in Gas Turbine Engines[END_REF]. The non-brittle mechanical behavior of these composites made of brittle constituents is brought by various energy dissipating mechanisms collectively termed as damage, such as matrix micro-cracking and concomitant debonding and sliding. These mechanisms are enhanced and controlled by coating the fibers with a so called interphase (a compliant material being mostly carbon and boron nitride (BN)).

It is therefore essential to develop non-destructive techniques able to follow the onset and propagation of damage, which can be applied to structural components displaying complex geometries. A wide range of non-destructive techniques has been evaluated for being used with CMCs [START_REF] Sun | NDT technologies for ceramic matrix composites: Oxide and nonoxide[END_REF]. Techniques using either ultrasonic or X-ray signals as well as pulsed thermography, however, usually require the component to be taken out of service for quite long durations. The recording of acoustic emission signals has proved to be efficient for quantifying matrix cracking in CMCs but is less sensitive to debonding and sliding related crack opening [START_REF] Maillet | Feasibility and limitations of damage identification in composite materials using acoustic emission[END_REF].

After having been widely used with metallic alloys and with carbon fiber-reinforced polymer composites (CFRP) [START_REF] Schulte | Load and failure analyses of CFRP laminates by means of electrical resistivity measurements[END_REF][START_REF] Wen | Damage detection of carbon fiber reinforced polymer composites via electrical resistance measurement[END_REF], electrical resistance monitoring (ER) has been recently adapted and applied to CMCs [START_REF] Smith | Monitoring damage accumulation in ceramic matrix composites using electrical resistivity[END_REF][START_REF] Smith | Electrical Resistance as a Nondestructive Evaluation Technique for SiC/SiC Ceramic Matrix Composites Under Creep-Rupture Loading: Resistance of SiC/SiC Under Creep-Rupture Loading[END_REF][START_REF] Morscher | Electrical Resistance of SiC Fiber Reinforced SiC/Si Matrix Composites at Room Temperature during Tensile Testing[END_REF][START_REF] Mansour | Monitoring interlaminar crack growth in ceramic matrix composites using electrical resistance[END_REF][START_REF] Simon | Monitoring damage evolution of SiCf/[SiBC]m composites using electrical resistivity: Crack density-based electromechanical modeling[END_REF][START_REF] Singh | Combined acoustic emission and multiple lead potential drop measurements in detailed examination of crack initiation and growth during interlaminar testing of ceramic matrix composites[END_REF][START_REF] Simon | Electrical resistivity monitoring of a SiC/[Si-B-C] composite under oxidizing environments[END_REF][START_REF] Smith | Electrical resistance changes of melt infiltrated SiC/SiC loaded in tension at room temperature[END_REF][START_REF] Stäbler | Characterisation of electrical resistance for CMC materials up to 2000 K[END_REF][START_REF] Morscher | Monitoring tensile fatigue crack growth and fiber failure around a notch in laminate SIC/SIC composites utilizing acoustic emission, electrical resistance, and digital image correlation[END_REF]. The ER technique allows to follow the onset and propagation of damage in the material, most particularly the one associated to the most conductive phase when it is subjected to crack opening (which tends to interrupt the current flow and thus to increase the electrical resistance). For example, regarding melt-infiltrated (MI) SiC/SiC materials with a BN interphase, it has been found that the highly conductive free silicon phase present in the matrix mainly drives the resistivity of the composites [START_REF] Morscher | Electrical Resistance of SiC Fiber Reinforced SiC/Si Matrix Composites at Room Temperature during Tensile Testing[END_REF][START_REF] Smith | Electrical resistance changes of melt infiltrated SiC/SiC loaded in tension at room temperature[END_REF].

Conversely, for any C/SiC composite possessing a pyrocarbon interphase (i.e. C/C/SiC composite), both the fibers and the interphase are the most conductive phases (as it is highlighted in the next section).

In order to be able to link the damage related nonlinear mechanical behavior of these CMCs to the evolution of their electrical resistance, different models have been proposed in the literature [START_REF] Morscher | Electrical Resistance of SiC Fiber Reinforced SiC/Si Matrix Composites at Room Temperature during Tensile Testing[END_REF][START_REF] Simon | Monitoring damage evolution of SiCf/[SiBC]m composites using electrical resistivity: Crack density-based electromechanical modeling[END_REF][START_REF] Böhrk | Electrical resistivity measurement of carbon-fiberreinforced ceramic matrix composite under thermo-mechanical load[END_REF][START_REF] Sujidkul | Correlating Electrical Resistance Change with Mechanical Damage in Woven SiC/SiC Composites: Experiment and Modeling[END_REF]. These models, all inspired by those first set up with CFRP composite, were based on the coupling between electrical resistance networks and mechanical shear-lag models, introducing an "electrically inefficient length". An analytical model, firstly proposed by Sujidkul et al. [START_REF] Sujidkul | Correlating Electrical Resistance Change with Mechanical Damage in Woven SiC/SiC Composites: Experiment and Modeling[END_REF] and modified by Morscher et al. [START_REF] Morscher | Electrical Resistance of SiC Fiber Reinforced SiC/Si Matrix Composites at Room Temperature during Tensile Testing[END_REF] in order to take into account transverse matrix cracking instead of fiber failure, was applied to MI composites with a BN interphase. This model allowed to obtain excellent results up to matrix cracks saturation. More recently, work has been carried out on a SiC/C/SiBC composite [START_REF] Simon | Monitoring damage evolution of SiCf/[SiBC]m composites using electrical resistivity: Crack density-based electromechanical modeling[END_REF], using a model based on a similar electrical network but mainly distinguished by the use of Hutchinson equations to link mechanical and electrical properties. Finally, one should also mention the recent development of a numerical model, aimed at applying the finite element method to quantify the effect of specific damage states on the ER response of a melt infiltrated SiC/SiC composite [START_REF] Sullivan | Modeling the effect of damage on electrical resistivity of melt-infiltrated SiC/SiC composites[END_REF].

The objectives of this study are (i) to investigate the influence of the nature of the CMC on the changes observed in the electrical resistivity measurements performed during a tensile test and (ii) to propose a general model for CMCs relating their mechanical behavior to their electrical resistance. For this purpose, three CMCs of several natures, namely a C/C/SiC, a SiC/C/SiBC and a SiC/BN/SiC (further referred to as SiC/BN/SiC-Si to highlight the importance of residual Si), have been used. The main interest with the use of these composites is that their most conducting constituent is always different.

Materials and methods

Three different continuous fiber reinforced ceramic matrix composites, all manufactured by SAFRAN Ceramics using woven interlock fibrous preforms have been tested in tension at room temperature. The material denominated SiC/C/SiBC is based on a SiC Nicalon NL207 fiber coated with a pyrocarbon interphase, the matrix being composed of ex-polymer SiC and CVI processed [SI-B-C] layers (with self-healing capacities). A second composite, denominated SiC/BN/SiC-Si, uses Hi-Nicalon S fibers coated with a BN interphase. The matrix has been processed with the melt-infiltration technique, and is therefore composed of a SiC phase and a portion of free Si. The last composite, mentioned as C/C/SiC all along the text, uses PANEX carbon fibers, coated with a pyrocarbon interphase, the SiC matrix being fully processed by CVI.

Micrographs of the as-received composites, obtained through optical microscopy observations performed on polished samples, are displayed in Figure 1. The two materials mainly processed by CVI exhibit macropores with sharp singularities in the interbundle matrix-rich region. Conversely, the MI composite does not display such macroporosity which has been almost fully filled up by the dual SiC-Si matrix. All the specimens machined from composite plates were tested in tension on an Instron 4505 machine at a constant displacement rate of 0.3mm/min with interposed unloading/reloading cycles, using hydraulic grips. Prior to testing, samples were equipped with glass fiber-reinforced epoxy tabs, in order (i) to avoid an extra damage linked to the clamping pressure and (ii) to provide an electrical insulation from the testing system. Two extensometers (25mm) were placed on the opposite edges in order to check for possible bending/torsion components intervening during the test.

It is worthy of note that, although test samples had somewhat various shapes and sizes since they were supplied at different times for different purposes, the similar slow displacement rate used experimentally always ensured quasi-static loading conditions. This point, added to the fact that both the gage length of the extensometers and the crosssection of the samples were largely coherent with the elementary representative volume (ERV) of the composites, has allowed to directly compare the stress-strain behaviors.

It should also be mentioned that, because of their limited availability, only two samples of each composite material were tested. Since results of the tests, displayed and discussed in the next paragraph, have shown no significant scattering (see figure 2), the behavior of only one of these two samples is systematically reported.

Electrical resistance (ER) was measured using a four-point probe method in order to minimize contact resistance. Two rings were first pasted around the specimen with CW2400 conductive epoxy, in order to collect the surface current on the whole perimeter of the samples. Shallow grooves were machined in the edges at the extremities of the samples so that the outer wires could be set in the right place, and then fixed using CW2400 conductive epoxy. A constant direct current was applied through the outer wires whereas the associated voltage was measured between the inner wires [START_REF] Simon | Monitoring damage evolution of SiCf/[SiBC]m composites using electrical resistivity: Crack density-based electromechanical modeling[END_REF].

Typical resistivity values of the material constituents are reported in table 1. The resistivity ρ is deduced from the gage resistance R according to the formula = ⁄ (where S is the average cross-section between the gage leads and Lg is the length between the two contacts). This formula reveals a fair way of compiling normalized values, which may thus be directly compared. It appears that the pyrocarbon interphase is the most conductive constituent in the case of SiC/C/SiBC composite whereas, conversely, the matrix is the most conductive constituent in the case of the SiC/BN/SiC-Si material. On the other hand, carbon fibers and pyrocarbon interphase are the most conductive constituents in the case of the C/C/SiC composite. Since these constituents account for more than 50% of the composite volume fraction, a low electric resistance increase was expected, which justifies that a different measurement method was developed for this material.

Recording of resistance was performed with an Agilent 3644A provided a current of 500 mA for the C/C/SiC composite and a current of 100 mA for both SiC/C/SiBC and SiC/BN/SiC-Si composites, and a voltmeter (HP 3852) measured the induced voltage. The stability of the measurements was checked during 24 hours once the samples were fixed in the hydraulic jaws and left at room temperature without any loading being applied. With this system, the electrical resistance was measured with an accuracy of 5 10 -3 mΩ.

It is finally worth mentioning that electric resistance variations are always reported in a relative manner all along the text (i.e. ∆ ⁄ as a function of strain), which allows to get rid of possible problems when comparing values obtained from samples of different shapes and sizes.

Acoustic emission signals were additionally recorded, using a MISTRAS Group SA system, equipped with a PICO sensors pasted on the specimens. Data were recorded at a rate of 2 MHz, and the detection threshold was fixed to 45 dB.

C/C/SiC

Results and discussion

The initial resistance values of the three materials, R0, are given in Table 2, along with their initial resistivity . It may be seen that the SiC/BN/SiC-Si composite presents an initial resistivity ten times lower than that of the SiC/C/SiBC composite, most certainly because the conductive silicon is present in a significant proportion in the matrix, in contrast with the proportion of C in the case of the SiC/C/SiC composite. It may also be seen that the C/C/SiC composite has a much lower resistivity than the other two materials (of in the order of 10 -5 Ω.m), in accordance with what was expected from the values of its constituents. increase in the acoustic emission rate is observed at the end of the test, which may be attributed to progressive fiber breakage. It is also important of note that no activity is present during the unloading/reloading cycles, which indicates that no damage takes place upon cycling.

For the three materials, the monotonic relative evolutions of the electrical resistance during the tests, displayed in Figure 3, are comparable. After a small increase mainly related to the variation in strain and therefore to the piezoelectric character of the material, a strong increase up to rupture is always observed. This increase is undoubtedly linked to the multiplication and above all to the opening of the matrix micro-cracks.

Permanent electric resistances are observed upon complete unloading for the three materials, thus indicating that there are micro-cracks which are incompletely closed. The remarkable fact is that, while this is coherent with the permanent strains exhibited by the C/C/SiC and SiC/C/SiBC composites, this is apparently no longer the case with the SiC/BN/SiC-Si composite for which these permanent strains are negligible. There is therefore a damage entity which remains active from an electrical point of view but which has no influence on the longitudinal strains. In the case of these two composites, all the studies previously performed have shown that most of the micro-cracks are perpendicular to the loading direction so that their opening/closure may be followed by the longitudinal contact extensometers presently used. Conversely, it has been shown that micro-cracks longitudinal to the loading direction are largely present in the SiC/BN/SiC-Si composite [START_REF] Tracy | Multi-scale Investigation of Damage Mechanisms in SiC/SiC Ceramic Matrix Composites[END_REF].

Even if the opening of these micro-cracks cannot be evidenced by longitudinal extensometers, a discontinue interface is present when this crack is fully closed and may be responsible for the noticeable permanent electrical resistances of the SiC/BN/SiC-Si composites.

Modeling of the electrical resistance changes 4.1.

The electromechanical model

Electromechanical models have been previously proposed in the literature, particularly for the SiC/C/SiBC material used in this work [START_REF] Simon | Monitoring damage evolution of SiCf/[SiBC]m composites using electrical resistivity: Crack density-based electromechanical modeling[END_REF][START_REF] Sullivan | Modeling the effect of damage on electrical resistivity of melt-infiltrated SiC/SiC composites[END_REF]. The model of the elementary cell and its application to a domain consisting of a finite number of these cells, see Figure 4, resulted in the following equation which expresses the relative resistance variation as a function of strain [12,19][12,19] :

With α0 being the piezoresistance factor of the undamaged composite, ld half the length of the debonded zone in the unit cell, δ the length of the unit cell, ɸs the resistivity of the undamaged unit cell, ɸd the resistivity of the fully damaged unit cell and ρc the matrix crack density (0< ρc <1).

Remark: it is worth mentioning that, when considering (i) the relationship between the resistance and the resistivity (i.e. = / ) and (ii) the extremely small changes encountered by the transverse strains during the tensile tests (i.e. the cross-section S remains close to its initial value S0), changes in the relative resistance and in the relative resistivity remain almost identical.

By considering the parameter Ld which represents the debonding density, proportional to ld/δ, a constant term K was then introduced so that Eq. 1 could be rewritten as:

∆ ( ) = (1 + . ) . ( ). ( ) + 1 -1

Eq. 2

∆ ( ) = (1 + . ) 2 - . ( ) + 1 -1
Eq. 1 This is the hypothesis of a unique type of microcracking as usually encountered in 1D composites, which allowed to consider coefficients K and α0 being constant (i.e. unaffected by damage). This last coefficient was estimated from the initial slope of the resistivity/strain curve. In order to identify K, by reversing Eq. 2 with the help of the experimental values found for the evolution of the electrical resistance, one had first to estimate Ld(ε) and ρc(ε).

This was done by using values extracted from the unloading/reloading cycles of the stress/strain curve and a micromechanical model derived from the work of Hutchinson el al. [START_REF] Simon | Monitoring damage evolution of SiCf/[SiBC]m composites using electrical resistivity: Crack density-based electromechanical modeling[END_REF][START_REF] Hutchinson | Models of fiber debonding and pullout in brittle composites with friction[END_REF][START_REF] He | Inelastic strains due to matrix cracking in unidirectional fiber-reinforced composites[END_REF]. It has be noticed that these so-called Hutchinson's equations, only qualitatively suitable for being used with 1D composites (in fact microcomposites constituted by a single fiber coated with a matrix, since no mathematical homogenization was added). However, they provided good results with the 2D interlock SiC/C/SiBC composite used in the study of Simon et al. One should however consider that this is mainly because the electrical resistance of this composite is strongly governed by the pyrocarbon interphase of the longitudinal tows [START_REF] Simon | Monitoring damage evolution of SiCf/[SiBC]m composites using electrical resistivity: Crack density-based electromechanical modeling[END_REF]. This fact is however unlikely to be true with any multi-directionally reinforced CMC when considering their general damaging process [START_REF] Guillaumat | Microfissuration des CMC : relation avec la microstructure et le comportement mécanique[END_REF]. More particularly, for the C/C/SiC and the SiC/BN/SiC-Si of the present study, carbon fibers on one hand and residual Si on the other hand are also present elsewhere than in longitudinal tows: the electrical resistance should therefore be also impacted by the damage phenomena taking place in the inter-tow matrix and in the transverse tows.

It is therefore proposed to setup a generalization of the electromechanical model, suitable to different CMCs (at least those used in the present study). The basic conjecture is that Eq.

2 remains valid if one uses non-constant terms K and α, i.e. being functions of the strain ε.

This hypothesis is justified by the fact that the model of the elementary cell, illustrated by Figure 4, remains pertinent for any kind of damaging process (i.e. within the inter-tow matrix, within the transverse tows and within the longitudinal tows) but with different values for α0 and for the term

!" # $ % & # '& ( & (
).

The following equation has thus to be considered:

∆ ( ) = (1 + ( ). ) ( ). ( ). ( ) + 1 -1

Eq. 3

The electrical resistance variation, ΔR(ε), being experimentally determined, one has then to identify the remaining functions.

In this respect, it has been assumed, as a second hypothesis, that the microcracks density ρc is proportional either to the evolution of the usual damage parameter d related to the elastic modulus (i.e. d = 1 -E/E0), identified with the unloading/reloading cycles (at the beginning of unloading [START_REF] Hild | Matrix cracking and debonding of ceramic-matrix composites[END_REF][START_REF] Mei | Measurement and calculation of thermal residual stress in fiber reinforced ceramic matrix composites[END_REF]) or from the acoustic emission signal *+ (which may be recorded and used either in term of number of hits or of energy).

( ) = -*+ ( ) Among these two possibilities, the second one appears to be more attractive for the acoustic signal is continuously recorded, as is the electrical resistance variation. However, if the parameter k is constant by definition (i.e. d is set up with the elastic modulus, whose changes are related to the onset and multiplication of microcracks), this is not so obvious with parameter q. A rearrangement of the two terms of Eq. 4 easily leads to (of course in a domain starting slightly above the yield point):

( ) = . /( ) Eq. 4 -= . /( ) *+ ( )

Eq. 5
Thus q being constant should result in the ratio d/SAE being also constant.

Application of the model

In order to estimate the value of the piezo-resistivity of the composites, which may be chosen either constant or evolving with the strain, α(ε) is written as follows:

( ) = 12〈 -4 〉6 + Eq. 8

With : α0 the piezoresistance factor of the undamaged composite, εy the yield strain of the composite (i.e. the point at which damage starts to take place), <.> McAuley brackets defined by "∀x ϵℝ, <x> = max(0,x)", f a function, positive and equal to zero at the origin.

From Eq. 7, it may be seen that, when no damage takes place (or does not evolve again), SAE(ε) is equal to zero as well as the term <ε-εy>, so that one obtains:

∆ ( ) =

Eq. 9

Thus, α0 can be defined from the initial slope of the electric resistance/strain curves.

Measured values are reported in the following 3. Initial piezoresistance factor of the three composites along with their tensile yield strain.

In order to determine function f, the beginning of the reloading part of the interposed unloading/reloading cycles (in terms of electrical resistance as a function of strain) were used. This means that, at each unloading sequence (down to zero stress), one has to consider a new material, damaged but with a microcracks density remaining constant up to a certain reloading threshold, which obeys to the scenario schematically displayed in Figure 4.

The evolution of the piezoelectric coefficients as a function of strain, plotted according to Eq. 8, is exhibited in Figure 6. Once the piezoelectric coefficient having been identified, it was then possible to use Eq. 7 in order to plot the evolution of K', expressed with the help of ΔR, α and SAE, as a function of the strain. This has been done for the three materials with the two possibilities retained for α (i.e. either constant or evolving with the strain). Results have shown that, in both case, a quadratic function could also be used to fit the results (Figure 7). In other word, one always has:

For ε < ε 4 , ′( ) = 9 0 〈 -4 〉 ! + : 0 〈 -4 〉 + ?′

Eq. 11

As ascertained from Figure 8, both assumption made on the value of the piezoelectric factor α, i.e. either constant or evolving with the strain, leads to a similar possibility of fitting the relative evolution of the electric resistance. Therefore, one has to find another way of distinguishing between these two assumptions. This has been done by considering the average slope of the hysteretic loops induced by the unloading/reloading cycles derived from the electric resistance-strain curves. Comparisons between predictions and experiments reported in Figure 9, show that, not surprisingly given the way that it has been assessed, the best fits are always obtained with an evolving piezoelectric factor α. The necessity of considering α et K' being sensitive to the strain, i.e. in fact to the damage present in the materials, is coherent with the basic hypothesis initially made regarding the use of elementary cell models whose parameters are sensitive to the various damaging processes. These results also confirm that the method used to identify α(ε) is suitable. The parameter K', which in fact allows to fit together the increases of both electric resistance and acoustic emission signal (see Eq. 6 and Eq. 7), logically evolves with the strain, for these two physical measurements are differently affected by damage, the former being more sensitive to crack opening [START_REF] Morscher | Electrical Resistance of SiC Fiber Reinforced SiC/Si Matrix Composites at Room Temperature during Tensile Testing[END_REF][START_REF] Simon | Monitoring damage evolution of SiCf/[SiBC]m composites using electrical resistivity: Crack density-based electromechanical modeling[END_REF]. The fact that K' always increases whatever the material considered highlights the progressive preponderance taken by crack opening (and related interfacial sliding) in the damage phenomena. This is in accordance with the observations made in the literature for the CMCs [START_REF] Naslain | Design, preparation and properties of non-oxide CMCs for application in engines and nuclear reactors: an overview[END_REF][START_REF] Smith | Electrical resistance changes of melt infiltrated SiC/SiC loaded in tension at room temperature[END_REF].

Concluding remarks

The evolution of the electrical resistance of three ceramic matrix composites has been characterized in tension at room temperature. It has been shown that both the mechanical behavior and the evolution of the resistance as a function of the strain of these materials are strongly distinct.

A generic model linking the evolution of damage with that of the electrical resistance has then been proposed. It has been shown that this model is largely pertinent for the three different CMCs used in the present study, which provides strong hopes for its application to any kind of CMC. Although being mostly a fitting method aimed at conciliating acoustic emission signals and electrical resistance measures, this model is, yet quite simple, based on physically meaningful considerations. However, the present development of this model is only suitable for monotonic tension tests or for materials whose electrical resistance barely evolve upon interposed unloading/reloading cycles. It has thus been previously reported that this is not the case for SiC/C/SiBC composite, because of the modification of the electrical contacts that occur in the interphase during the cycles. Modelling the influence of interposed unloading/reloading cycles thus seems to be particularly interesting to investigate and may be the subject of further developments.

Figure 1 .

 1 Figure 1. Microstructural characterizations of the as-received composites, a) C/C/SiC composite, b) SiC/C/SiBC composite, and c) SiC/BN/SiC-Si composite.

Figure 2

 2 Figure2shows the stress/strain curves along with the evolution of the acoustic emission signal for the three materials (normalized by the maximum value reached during the test).All these curves display a more or less extended linear domain followed by a non-linear behavior up to rupture. Permanent strains are observed upon complete unloading for the SiC/C/SiBC and the C/C/SiC composites whereas, in the case of the SiC/BN/SiC-Si composite, they are almost non-existent. These curves highlight two other important facts observed for the three materials: (i) a gradual decrease of the average modulus provided by the unloading/reloading cycles interposed in the nonlinear domains and (ii) a hysteretic behavior of these cycles, in relation with the various frictional phenomena accompanying matrix micro-cracking. The permanent strains observed for the C/C/SiC and SiC/C/SiBC composites result from transverse micro-cracks (i.e. perpendicular to the loading direction) which are incompletely closed, because of the unrecoverable energy, i.e. the one dissipated by the damage processes and because of the release of the residual stresses introduced by the various processing steps. In the case of the C/C/SiC composite, the difference in thermal expansion coefficients between the carbon fibers and the matrix is sufficiently elevated to generate significant tensile residual stresses during cooling from the high temperature manufacturing process. The partial release of these residual stresses resulted in (i) the presence of cracks inside the matrix (see Figure1) and (ii) a concurrent point of the slopes of the equivalent modulus (derived from the unloading/reloading cycles) situated in the compression domain[START_REF] Camus | Development of damage in a 2D woven C/SiC composite under mechanical loading: I. Mechanical characterization[END_REF]. On the other hand, nearly no permanents strains are observed in the case of the SiC/BN/SiC-Si composite whereas the slopes of the equivalent modulus provided by the unloading/reloading cycles happen to be concurrent at a single point situated in the tensile domain. This information indicates that residual compression strains are likely to be present within the matrix. All these observations are also in good agreement with the low yield stress of 25 MPa observed for the C/C/SiC composite as compared to the value of 275 MPA observed for the SiC/BN/SiC-Si composite.

Figure 2 .

 2 Figure 2. Tensile behavior of the three materials with interposed unloading/reloading cycles and the normalized acoustic emission signal. Comparison of the two tests performed for the composite SiC/C/SiBC.

Figure 3 .

 3 Figure 3. Evolution of the electrical resistance as a function of the strain level, for tension tests performed on the three composites with interposed unloading/reloading cycles.

Figure 4 .

 4 Figure 4. Schematic representation of (a) the damaged SiC/C/SiBC material loaded at a strain of around 0.5% with crack propagation in the seal-coat, in the inter-tow matrix, in the transverse tows and in the longitudinal tows, (b) an electrical network of multiple parallel cells in series representing the cracked composite, (c) the initial unit cell and (d) the same cell with a matrix crack and a debonded interfaces [12].

Figure 5 , 6 1 Eq. 7 Figure 5 .

 56175 Figure 5. Evolution of the damage parameter d as a function of the acoustic emission signal SAE for the three materials investigated.

10 Figure 6 .

 106 Figure 6. Evolution of the initial reloading elastic resistance obtained from the unloading/reloading cycles as a function of the strain (beyond the damage threshold).

Figure 7 .

 7 Figure 7. Evolution of K' as a function of the maximum strain reached at each cycle for either α constant or α(ε).

Figure 8 .

 8 Figure 8. Relative variation of the electric resistance as a function of strain, for the three materials, along with their two possible representations.

Figure 9 .

 9 Figure 9. Average slope of the electric resistance-strain unloading/reloading hysteretic loops as a function of strain for the three materials, according to the model used.

Table 1 . Typical resistivity values of the materials constituents as found in the literature [21,22,10,20,12].

 1

	Constituents	C fiber	SiC matrix	PyC
	Resistivity (Ω.m)	1.7 10 -5	10 3 -10 5	1.5 10 -5 -4.5 10 -5
	Typical surface in the
	composite relative to PyC	10	20	1
	surface (order of magnitude)
	ρ/ρ(PyC) (order of magnitude)	0.5	10 3	1
		SiC/C/SiBC
	Constituents	NL207 fiber	SiC matrix	PyC
	Resistivity (Ω.m)	9 -10	10 3 -10 5	1.5 10 -5 -4.5 10 -5
	Typical surface in the
	composite relative to PyC	20	20	1
	surface (order of magnitude)
	ρ/ρ(PyC) (order of magnitude)	10 4	10 3	1
		SiC/BN/SiC-Si
	Constituents	NL207 fiber	SiC-Si SiC matrix	BN
	Resistivity (Ω.m)	9 -10	7.45 10 -5 [20]	0.6 10 -3 -3
	Typical surface in the
	composite relative to MI SiC/Si	1.15	1	0.2
	surface (order of magnitude)
	ρ/ρ(MI SiC/Si) (order of magnitude)	10	1	10 3

Table 2

 2

	Composite material	C/C/SiC	SiC/C/SiBC	SiC/BN/SiC-Si
	Length between contacts / cross-section (mm -1 )	5	4	2.5
	Initial resistance R0 (Ω)	0.0507	1.07	0.08
	Initial resistivity (Ω.m)	3 10 -5 ± 5 10 -6	1.08 10 -3 ± 2 10 -4	1.02 10 -4 ± 1 10 -5
	R at failure (Ω)	0.0528	1.72	0.30
	R-R0 at failure (Ω)	0.00210	0.65	0.21246

also displays an average measure of the resistance values reached by the three specimens slightly before failure. It clearly appears that the variations in resistance observed during the tests are dependent on the type of composite, the greatest variation being obtained with the SiC/BN/SiC-Si material (about 250%) while the SiC/C/SiBC composite has a 60% increase in resistance and the C/C/SiC displays only 4% of increase.

Table 2 . Electrical values (resistance and resistivity) before and after testing the three composites.

 2

table :

 : Table

	Composite material	C/C/SIC	SiC/C/SiBC	SiC/BN/SiC-Si
		0.41 ± 0.1	2.30 ± 0.1	12.04 ± 0.2
	4 (%)	0.04	0.1	0.1

Acknowledgements

This work has been supported by the Centre National de Recherche Scientifique (CNRS) and the Commissariat à l'énergie atomique et aux énergies alternative (CEA) through a grant given to one of the authors (J.-P. G.). The authors are grateful to Safran Ceramics (SAFRAN GROUP) for supplying the samples and also thank Alexandre Allemand, Patrick David and Christophe Tallaron (CEA) for valuable insights and Bruno Humez (LCTS) for his technical support.


Bernard Raffaelli
email: bernard.raffaelli@unice.fr

Prepared for

Keywords: Black holes, quasinormal modes, Hawking radiation, particle statistics

In this paper, we consider a massless field, with spin j, in interaction with a Schwarzschild black hole in four dimensions, focusing mainly our study on the s-wave scattering. First, using a Fourier analysis, we show that one can have a simple and natural description of the Physics near the event horizon without using any conformal field approaches. Then, within the same "scattering picture", we derive analytically the imaginary part of the highly damped quasinormal complex frequencies and, as a natural consequence of our analysis, we show that thermal effects and in particular Hawking radiation, can be understood through the scattering of an ingoing s-wave by the non null barrier of the Regge-Wheeler potential associated with the Schwarzschild black hole. Finally, with the help of the well-known expression of the highly damped quasinormal complex frequencies, we propose a heuristic extension of the "tripled Pauli statistics" suggested by Motl, some years ago.

Introduction

Scattering by black holes (BH) have been studied intensively since the pioneer work of Matzner, in 1968 [START_REF] Matzner | Scattering of massless scalar waves by a schwarzschild singularity[END_REF]. There have been a lot of works using different approaches. One of them uses the complex angular momentum (CAM) theory, the S-matrix formalism and the associated Regge pole techniques. In this framework, the authors have shown [START_REF] Decanini | Complex angular momentum in black hole physics and quasinormal modes[END_REF][START_REF] Decanini | Unstable circular null geodesics of static spherically symmetric black holes, regge poles and quasinormal frequencies[END_REF] that the weakly damped quasinormal mode (QNM) complex frequencies of a wide class of static, asymptotically flat and spherically symmetric BH of arbitrary dimension, can be understood as Breit-Wigner type resonances generated by the interferences and damping of a family of "surface waves" lying close to their photon sphere, whose existence is of fundamental importance. It should be noted that, in [START_REF] Decanini | Quasinormal modes of the btz black hole are generated by surface waves supported by its boundary at infinity[END_REF], they have shown that this "surface waves" interpretation can also be applied to the BTZ BH, a non-asymptotically flat spacetime, i.e. in a framework where the notion of an S-matrix does not exist, by extending a powerful formalism introduced several decades ago by Sommerfeld [START_REF] Sommerfeld | Partial Differential Equations of Physics[END_REF]. Finally, by noting that each "surface wave" is associated with a Regge pole of the corresponding S-matrix, this approach permits to construct analytically the spectrum of the weakly damped QNM complex frequencies of the corresponding BH, beyond the leading order term. Physically, it gives a powerful and quite elegant way to obtain a semiclassical interpretation of BH resonance phenomena. However, even if they have computed the absorption cross section (beyond the eikonal order) and the greybody factor in such scheme, the WKB approximation used to obtain analytically the expression of the Regge poles restricts the study to "high frequency" regime, i.e. for frequencies of order ω ≫ (2M ) -1 , where M is the mass of the Schwarzschild BH and thus limits the analysis to a scattering description of weakly damped QNM, formally associated to high angular momenta. Thus, highly damped QNM, Hawking radiation, and more generally, the physics near the BH horizon seems to be out of such framework. Nevertheless, in this paper, we claim that one could have a physical description of the near horizon physics without using any conformal field theories but by focusing, precisely, on the scattering of an ingoing s-wave by the non null barrier of the Regge-Wheeler potential of the Schwarzschild BH. In section 2, we introduce some notations, assumptions and well-known results linked to a scattering analysis and the Smatrix formalism. In section 3, after having discussed the "Rindler approximation", we will compute some results which have been assumed or derived by Solodukhin but within a conformal field framework [START_REF] Solodukhin | Horizon state, hawking radiation and boundary liouville model[END_REF]. In section 4, we will focus on the "far horizon limit" and we will show that one can easily compute the relevant physical quantities needed for a scattering analysis. They will allow us to compute the exact expression of the imaginary part of highly damped QNM complex frequencies. In section 5, using the results obtained in section 4, we will derive explicitly the reflection coefficient related to the scattering of the ingoing s-wave by the non null barrier of the Regge-Wheeler potential of the BH which is exactly the Boltzmann weight associated with the Hawking radiation, giving, in a purely scattering picture, another possible explanation of BH thermal effects. Finally, in section 6, extending some reflexions due to Motl [START_REF] Motl | An analytical computation of asymptotic schwarzschild quasinormal frequencies[END_REF], we will claim by an heuristic approach that the real part of these highly damped QNM complex frequencies is intimately linked to some "exotic" statistics, going beyond the Bose-Einstein and the Fermi-Dirac ones, which could be expected near any BH horizon where a theory of quantum gravity is maybe unavoidable.

Generalities and notations

We consider first a static spherically symmetric four-dimensional spacetime with metric

ds 2 = -f (r)dt 2 + dr 2 f (r) + r 2 dσ 2 . (2.1)
Here dσ 2 denotes the line element on the unit sphere S 2 . It should be noted that a metric such as (2.1) does not describe the most general static spherically symmetric spacetime but it will permit us to easily apply the following results to the Schwarzschild case. In Eq. (2.1), we shall assume that f (r) is a function of the usual radial Schwarzschild coordinate r satisfying the properties:

• (i) There exists an interval I =]r h , +∞[⊂ R with r h > 0 such as f (r) > 0 for r ∈ I. • (ii) r h is a simple root of f (r), i.e. f (r h ) = 0 and f ′ (r h ) = 0 and f (2) (r h ) > 0. (2.2)
These assumptions indicate that the spacetime considered has an single event horizon at r h , its exterior corresponding to r ∈ I. If we want to work with an asymptotically flat spacetime, in order to define an S-matrix, we need to impose the condition

lim r→+∞ f (r) = 1. (2.
3)

The Klein-Gordon wave equation for a massless scalar field propagating on a general gravitational background is given by

Φ = g µν ∇ µ ∇ ν Φ = 1 √ -g ∂ µ √ -gg µν ∂ µ Φ = 0. (2.4)
If the spacetime metric is defined by (2.1), after separation of variables and introduction of the radial partial wave functions Φ ℓ (r) with ℓ = 0, 1, 2, . . . , this wave equation reduces to the Regge-Wheeler equation

d 2 Φ ℓ dr 2 * + ω 2 -V ℓ (r) Φ ℓ = 0. (2.5)
Here we have assumed a harmonic time dependence exp(-iωt) for the massless scalar field. The variable r * = r * (r) is the well-known tortoise coordinate defined, for r ∈ I, by the relation dr * /dr = 1/f (r). Moreover, it is worth noting that the previous definition of r * provides a bijection r * = r * (r) from I to ] -∞, +∞[. In Eq. (2.5), V ℓ (r) is the Regge-Wheeler potential associated to the massless scalar field.

V ℓ (r) = f (r) ℓ(ℓ + 1) r 2 + 1 r f ′ (r) . (2.6)
In four dimensions, we recall the well-known cases of non zero spin j fields propagating on the BH background described by (2.1). Indeed, one can be interested, for example, in the propagation of an electromagnetic test-field (j = 1) or a linearized perturbation of the metric (j = 2). Thus, for more general situations in four dimensions, the Regge-Wheeler potential can be written as

V ℓ (r) = f (r) ℓ(ℓ + 1) r 2 + J r f ′ (r) (2.7)
where J = 1 -j 2 , with j the spin of the massless field under consideration. We recall that this result is valid when the BH does not carry any electric or magnetic charge. However, we would like to refer the reader to [START_REF] Decanini | Unstable circular null geodesics of static spherically symmetric black holes, regge poles and quasinormal frequencies[END_REF] for a d-dimensional generalization of the scattering of a massless scalar field by a static and spherically symmetric BH through the semi-classical CAM techniques. According to (2.7), it should be noted that lim r→r h V ℓ (r) = 0 and lim r→+∞ V ℓ (r) = 0 and therefore the solutions of the radial equation (2.5) have a exp(±iωr * ) behavior at the horizon and at infinity. In other words, for a given angular momentum index ℓ, a general solution of the Regge-Wheeler equation (2.5), satisfies the following asymptotic behaviors:

Φ ω,ℓ (r * ) ∼ r * →-∞ e -iωr * (2.8)
which is a purely ingoing wave at the event horizon and which has the following general expression at spatial infinity r * → +∞

Φ ω,ℓ (r * ) ∼ r * →+∞
A in e -iωr * + A out e +iωr * .

(2.9)

Moreover, by considering the wronskian of two linearly independent solution of Eq. (2.5) at r * = ±∞, we obtain 1

+ |A out | 2 = |A in | 2 .
(2.10)

As independent solutions we have chosen Eq. (2.9) and its complex conjugate. Furthermore, introducing the transmission and reflection amplitudes T ℓ and R ℓ defined by

T ℓ = 1 A in (2.11a) R ℓ = A out A in (2.11b)
we can rewrite Eq. (2.10) in the more familiar form

|T ℓ | 2 + |R ℓ | 2 = 1.
(2.12)

Thus, from Eq. (2.9), a QNM, which is defined as a purely ingoing wave at the event horizon and a purely outgoing wave at infinity, corresponds to A in = 0. Therefore, the part of the ingoing wave that will not be absorbed by the BH will be reflected back to spatial infinity. It should be noted that these quantities can be deduced from the S-matrix elements. Here, our problem is spherically symmetric so, the S-matrix is diagonal. We recall that the S-matrix elements, noted S ℓ (ω), are related to the reflection amplitude by

S ℓ (ω) = (-1) ℓ+1 R ℓ (ω). (2.13)
It has been shown in [START_REF] Decanini | Complex angular momentum in black hole physics and quasinormal modes[END_REF][START_REF] Decanini | Unstable circular null geodesics of static spherically symmetric black holes, regge poles and quasinormal frequencies[END_REF][START_REF] Decanini | Quasinormal modes of the btz black hole are generated by surface waves supported by its boundary at infinity[END_REF] (and references therein) that the S-matrix permits to analyze the resonant aspects of the considered BH as well as to construct the form factor describing the scattering of a monochromatic scalar wave. In this paper, we are mainly interested in the scattering of an ingoing s-wave, i.e. ℓ = 0, for which one has, on one hand

S 0 (ω) = -R 0 (ω), (2.14)
and on the other hand,

V 0 (r) = J r f ′ (r)f (r). (2.15)
For a Schwarzschild BH of mass M , i.e. f (r) = 1 -2M/r, and a massless scalar field (j = 0), the Regge-Wheeler has a maximum V 0,max for r = (4/3) r h [START_REF] Matzner | Scattering of massless scalar waves by a schwarzschild singularity[END_REF]. As illustrated in Fig. 1, we will show in section 4 that the non null potential barrier, for which we will associate an energy ω min = V 0,max , could be at the origin of the Hawking radiation and more generally of thermal effects. 3 The near horizon limit

The Rindler approximation

In order to deal with the spacetime structure in the very vicinity of the BH horizon, i.e. the "near horizon limit", we first naturally expand the function f (r) around the value r = r h .

f (r) ∼ f ′ (r h )(r -r h ) + O r→r h (r -r h ) 2 . (3.1)
Then inserting (3.1) into (2.1), we obtain

ds 2 = -f ′ h (r -r h )dt 2 + dr 2 f ′ h (r -r h ) + r 2 h dΩ 2 . (3.2)
We introduce the new variable ρ defined by

dρ = dr f ′ h (r -r h ) (3.3) such as ρ = 2 r -r h f ′ h or r -r h = f ′ h 4 ρ 2 . (3.4)
Thus, the metric reads

ds 2 = -κ 2 ρ 2 dt 2 + dρ 2 + r 2 h dΩ 2 (3.5)
which is regular at ρ = 0 and where κ = (1/2)f ′ h is the well-known surface gravity of the Killing horizon at r = r h . This "near horizon limit" form of the metric is of course a "Rindler approximation" with a corresponding constant acceleration κ. From Eq. (3.1), we can obtain the behavior of the tortoise coordinate r * near the event horizon

r * = dr f (r) ∼ 1 f ′ h dr (r -r h ) (3.6) which gives r * (r) = 1 f ′ h ln r r h -1 , (3.7)
where the constant of integration is chosen such as r = 2r h implies r * = 0. From (3.7), we write the usual radial coordinate r as a function of

r * r -r h = r h exp f ′ h r * (3.8)
which allows us to write Eq. (3.1) as

f (r * ) ∼ r h f ′ h exp f ′ h r * + 1 2 r 2 h f (2) h exp 2f ′ h r * + O r * →0 exp 3f ′ h r * , (3.9)
where f

(p) h = (d p f /dr p )| r h .
Then one can express the Regge-Wheeler equation in the "near horizon limit" as it has been done in [START_REF] Solodukhin | Horizon state, hawking radiation and boundary liouville model[END_REF], but, as we will see later, it is of no importance in this paper. In terms of the tortoise coordinate, the metric (2.1) for the Schwarzschild geometry, i.e. f (r) = 1 -2M/r, in the near horizon limit is given by

ds 2 = e 2κr * -dt 2 + dr 2 * + r 2 h dσ 2 (3.10)
where κ = 1/4M and r h = 2M . Of course, if one chooses a hyperplane such as dσ = 0, then in the (t, r * ) coordinates, the metric is conformally flat.

For the following, we introduce the "null tortoise coordinates" defined by

u = t + r * (3.11a) v = t -r * (3.11b)
and the associated Kruskal coordinates

κU = e κu (3.12a) κV = -e -κv .
(3.12b)

The time dependent "Doppler-gravitational" shift effect

Now, let us consider, near the Schwarzschild event horizon, a "local inertial frame" with coordinates (T, R), i.e. a timelike coordinate T and a spacelike coordinate R, such as the Kruskal-Szekeres coordinates are defined as "null coordinates":

U = T + R (3.13) V = T -R. (3.14)
We recall that the (U, V ) Kruskal coordinate system is analogous to the Rindler coordinates while the (T, R) coordinate system is analogous to the Minkowski one. Moreover, they cover the entire spacetime manifold of the maximally extended Schwarzschild solution, being well-behaved everywhere outside the physical singularity at r = 0. Furthermore, since we are working with an s-wave which is spherically symmetric, one could always approximate it locally as a "plane wave" propagating in the radial direction. So, in the previously defined "local inertial frame" (T, R), we can consider a locally (monochromatic) "plane wave" propagating towards the BH. Up to an amplitude coefficient, it could be written as According to the previous relations, one could conclude that an observer looking at the ingoing wave actually sees a superposition of "plane waves" (cf. [START_REF] Alsing | Simplified derivation of the hawking-unruh temperature for an accelerated observer in vacuum[END_REF]). Indeed, if we write the inverse Fourier transform, we have

Φ inertial (T, R) ∝ exp [-iω(T + R)] . ( 3
Φ(u) = +∞ -∞ dΩ Φ0 (Ω) e -iΩu . (3.18)
The expression of Φ(u) can be understood as resulting from a superposition of ingoing and outgoing "plane waves" near the Schwarzschild event horizon where Φ0 (Ω) is the amplitude of the "plane wave" of frequency Ω. In other words, we focused first on a single plane wave, i.e. a single frequency. We found that in the freely falling frame it becomes a superposition of "plane waves". Now, if we want to quantize the considered spin j field, we have to identify each plane wave with single particle states. In [START_REF] Solodukhin | Horizon state, hawking radiation and boundary liouville model[END_REF], Solodukhin talks about a transition between the "horizon state" and the outgoing propagating wave. Here, the interpretation from the scattering point of view is analogous and far more natural.

There is a transition from a single "plane wave", i.e. one particle state, to a superposition of "plane waves", i.e. superposition of single particle states, during the free-fall into the Schwarzschild BH, and vice versa by time symmetry. Furthermore, it should be noted that close to the horizon, according to eq (2.8), Eq. (3.18) is similar to Eq. ( 19) in [START_REF] Solodukhin | Horizon state, hawking radiation and boundary liouville model[END_REF] Φ

(u) = +∞ -∞ dΩ Φ0 (Ω) Φ Ω,0 (u) (3.19)
but, we can go one step further in this "freely falling wave" description. Indeed, in order to know the frequency-distribution of the locally accelerated "wave" Φ(u), we can Fourier transform it

Φ0 (Ω) = +∞ -∞ du ′ Φ(u ′ ) e iΩu ′ . (3.20)
We would like to note that what follows is not really "rigorous" in a quantum field theory sense, as it has been noticed in [START_REF] Alsing | Simplified derivation of the hawking-unruh temperature for an accelerated observer in vacuum[END_REF], but it permits us to have a physical intuition of what happens near the event horizon. According to eqs. (3.17) and (3.20) we can write explicitly

Φ0 (Ω) = +∞ -∞ du ′ e iΩu ′ exp -i ω κ e κu ′ .
(3.21) Thus, following [START_REF] Alsing | Simplified derivation of the hawking-unruh temperature for an accelerated observer in vacuum[END_REF], we change variables to y = exp(κu ′ ) and write

Φ0 (Ω) = 1 κ +∞ 0 dy y iΩ/κ-1 exp -i ω κ y . (3.22)
The integration reads

Φ0 (Ω) = 1 κ Γ iΩ κ ω κ iΩ/κ exp (-πΩ/2κ) . (3.23)
The modulus squared of the amplitude of each "plane waves", i.e. Φ0 (Ω)

2

, is naturally interpreted as a probability (density) of measuring the associated frequency Ω. Since we know from the properties of Γ functions that

Γ iΩ κ 2 = π (Ω/κ) sinh (πΩ/κ) (3.24)
then the probability (density) of measuring the frequency Ω is given by a Bose-Einstein-like distribution Φ0 (Ω)

2 = 2π Ωκ 1 exp (2πΩ/κ) -1 . (3.25)
It is worth noting that this time dependent "Doppler-gravitational" shift effect can be adapted to fermions with the formal prescription iΩ/κ → iΩ/κ + 1/2. We refer the reader to [START_REF] Alsing | Simplified derivation of the hawking-unruh temperature for an accelerated observer in vacuum[END_REF] for more details. Of course, the previous quantum point of view with superposition of particle states remains valid if one associates the frequency Ω with the energy of a quantum particle and interprets the probability of measuring Ω as the probability of finding particles of energy Ω, as we will show it in section 5.

4 The far horizon limit and the QNM complex frequencies

The far horizon limit

We start by using the transformation

Φ ω,ℓ (r) = 1 - 2M r 1/2 φ ω,ℓ (r). (4.1)
Then, after a partial fraction decomposition, the Regge-Wheeler equation becomes

1 4M 2 d 2 φ ω,ℓ dr 2 + ω 2 + J -3/4 r 2 + J + ℓ(ℓ + 1) -1/2 2M r + 8M 2 ω 2 -J -ℓ(ℓ + 1) + 1/2 2M (r -2M ) + 4M 2 ω 2 + 1/4 (r -2M ) 2 φ ω,ℓ = 0. (4.2)
We substitute the variable r by x = (r -2M )/2M such that x ∈ [0, +∞[ when r ∈ [2M, +∞[. Thus, we consider large x region, i.e. r ≫ 2M , for which we can expand Eq. (4.2) in powers of x -1

d 2 φ ω,ℓ dx 2 + ω 2 + 2ω 2 x + ω 2 -ℓ(ℓ+1) 4M 2 x 2 + 1 + ℓ(ℓ + 1) -J 4M 2 x 3 + . . . φ ω,ℓ = 0. (4.
3)

It should be noted that in Eq. ( 4.3) the spin, J = 1 -j 2 , only appears in higher orders of the expansion, starting from the third one. One way to solve such an equation is to consider the first two terms of the asymptotic expansion (4.3) but the price to pay is the lost of information concerning the spin

d 2 φ ω,ℓ dx 2 + ω 2 + 2ω 2 x + ω 2 -ℓ(ℓ+1) 4M 2 x 2 φ ω,ℓ = 0. (4.4)
Thus, the equation becomes analogous to a Coloumb differential equation which reads for an s-wave, i.e. ℓ = 0,

d 2 φ ω,0 dx 2 + ω 2 + 2ω 2 x + ω 2
x 2 φ ω,0 = 0. (4.5)

The s-wave scattering and the QNM complex frequencies

In order to find the QNM complex frequencies, we will make some changes of variables.

We introduce b and z such as

-4M 2 ω 2 = b(b -1) (4.6a) -4iM ωx = z. (4.6b)
Finally, we define a new "s-wave function"

ψ ω (z, b) = x b exp(2iM ωx)φ ω,0 (z). (4.7)
Therefore, the equation (4.5) reads

z d 2 ψ ω dz 2 + (2b -z) dψ ω dz -(b -2iM ω)ψ ω = 0 (4.8)
which is one of the well-known confluent hypergeometric equations [START_REF] Abramowitz | Handbook of Mathematical Functions[END_REF] (for the ℓ = 0 case, we refer the reader to [START_REF] Liu | On the spectrum of oscillations of a schwarzschild black hole[END_REF]). The solution of Eq. (4.8) is a combination of confluent hypergeometric functions of the first kind

ψ ω (z, b) = Γ(1 -2b) Γ(1 -b -2iM ω) F 1 (b -2iM ω; 2b; z) + Γ(2b -1) Γ(b -2iM ω) z 1-2b F 1 (1 -b -2iM ω; 2 -2b; z) . (4.9)
Then the solution for the "s-field" φ ω,0 reads

φ ω,0 (x) = x b e 2iM ωx Γ(1 -2b) Γ(1 -b -2iM ω) F 1 (b -2iM ω; 2b; z(x)) + (-4iM ωx) 1-2b x b e 2iM ωx Γ(2b -1) Γ(b -2iM ω) z(x) 1-2b F 1 (1 -b -2iM ω; 2 -2b; z(x)) (4.10)
where z(x) is defined by Eq. (4.6b).

For large frequencies, i.e. for ω > ω min where ω min has been defined in section 2, we can use the following approximation

b ± ≈ 1/2 ± 2iM ω
and, from the large x limit, we use the asymptotic expansions of the previous confluent hypergeometric functions in terms of Γ functions. Moreover, in order to obtain the amplitudes related to the ingoing wave, i.e. A in , and to the outgoing wave, i.e. A out , one has to note that x and r * are related by the identity e ±2iM ωx x ±2iM ω = e ±iωr * . Thus, up to numerical constants of order unity, the amplitude of the asymptotic behavior of the s-wave, for "high frequencies", reads

A in (ω) ≈ (-4iM ω) -b-2iM ω Γ(1 -4iM ω)Γ(4iM ω) √ π Γ(1/2 -4iM ω) x -1+2b b 1-2b + 1 (4.11a) A out (ω) ≈ (4iM ω) -b+2iM ω Γ(1 -4iM ω)Γ(4iM ω) π x -1+2b b 1-2b + 1 . (4.11b)
such as the "s-field" has an asymptotical behavior similar to Eq. (2.9). It should be noted that, in these expressions, after some simplifications, the value of b + = 1/2 + 2iM ω is the same for A in and for A out . Then, as seen earlier, the QNM complex frequencies which are defined by A in = 0, i.e. by the poles of the function Γ(1/2 -4iM ω), can be written as

∀n ∈ N, 1 2 -4iM ω n = -n (4.12)
or, in other words

∀n ∈ N, 8πM ω n = 2πi n + 1 2 . (4.13)
Here, the highly damped QNM complex frequencies are obtained considering a "far region" limit, r > 2M , and a "high frequency" regime, i.e. ω > ω min . It should be noted that, in such calculation, ω n has no real part, which is known to be non null and depending both on the spin j of field and on the characteristics of considered BH. More particularly, one could think that not considering the third order term in the expansion in x ≫ 1 in Eq. ( 4.3), is equivalent to consider the particular case of spin j satifying

1 -J = 1 -(1 -j 2 ) = 0. (4.14)
In this case, Eq. ( 4.3) would be equivalent to Eq. (4.5) if j = 0. But we know that this "j = 0 result" differs from the results discussed by Motl in [START_REF] Motl | An analytical computation of asymptotic schwarzschild quasinormal frequencies[END_REF] obtained with the help of the powerful monodromy techniques. In other terms, the statement "Eq. ( 4.3) would be equivalent to Eq. (4.5) if j = 0" seems to be wrong. Then, as we already noticed earlier, the exact expression of the "spin j dependent"-QNM frequency spectrum can't obviously be obtained by our second order approximation, i.e. Eq. (4.5) for which we only get the exact expression of the imaginary part.

5 From scattering to statistics

Quantum field theory: a very brief survey

In quantum field theory [START_REF] Parker | Quantum field theory in curved spacetime[END_REF], in order to compute the spectrum of outgoing particles from a BH, one usually considers the coefficients of the Bogolubov transformation relating the well-known creation (resp. annihilation) operator b Ω (resp. b † Ω ) to a ω and a † ω such as

b Ω = dω α * Ωω a ω -β * Ωω a † ω (5.1)
with the commutation relations It should be noted that using eq (5.

η ω 1 , η † ω 2 = δ (ω 1 -ω 2 ) (5.2a) [η ω 1 , η ω 2 ] = 0 = η † ω 1 , η †
2), the normalization condition for the Bogolubov coefficients reads

dω (α Ωω α * Ω ′ ω -β Ωω β * Ω ′ ω ) = δ Ω -Ω ′ . (5.4)
Then, from the standard mode expansions for the considered field operator both in the coordinates (u, v) and (U, V ), and after some calculations, it follows that the coefficients of the Bogolubov transformation obey the well-known relation

|α Ωω | 2 = exp (8πM Ω) |β Ωω | 2 .
(5.5) Thus, one can easily deduce the expectation value of the "b-particle" number operator, i.e.

N Ω = b † Ω , b Ω in the Kruskal vacuum state and, more generally, the physics of the Hawking effect.

The Hawking radiation: a scattering effect

Let us focus, now, on the scattering/reflection coefficient of an ingoing s-waves, i.e. S 0 (ω) = -R 0 (ω). We recall the main result (4.11) obtained in section 4, i.e.

A in (ω) ≈ (-4iM ω) -b-2iM ω Γ(1 -4iM ω)Γ(4iM ω) √ π Γ(1/2 -4iM ω) x -1+2b b 1-2b + 1 A out (ω) ≈ (4iM ω) -b+2iM ω Γ(1 -4iM ω)Γ(4iM ω) π x -1+2b b 1-2b + 1 .
Using Stirling's approximation, i.e. considering "large frequencies", i.e. ω > ω min , one has

Γ(1/2 -4iM ω) ≈ e 4iM ω (4iM ω) -4iM ω e -4πM ω (5.7)
then, we can easily deduce that the "ℓ = 0" reflection coefficient can be written as

R 0 (ω) = A out (ω) A in (ω) ≈ e 4iM ω × e -4πM ω .
(5.8)

From Eq. (5.8), the reflection probability off the "ℓ = 0" Regge-Wheeler potential barrier (cf. Fig. 1) reads

|R 0 (ω)| 2 = e -8πM ω . (5.9)
The equation (5.9) could be interpreted as a Boltzmann factor characterized by a temperature T = 1/8πM . With the help of Eq. ( 5.5), we can now stress the link between the quantum field theory approach and the scattering approach, i.e.

|α Ωω | 2 = |R 0 (Ω)| -2 |β Ωω | 2 .
(5.10)

Physics and thermodynamical aspects of the Schwarzschild BH

In the following, we will associate every "local plane wave" to a "single particle state". According to the above results, the infalling motion of a quantum particle is analogous to a time dependent boost along the radial direction and consequently the proper distance of the infalling quantum particle exponentially decreases with time. Moreover, if the infalling particles lose enough energy, they can be considered, for an external observer, as "trapped" between the event horizon r h and the location of V 0,max , which then will play the role of a "thermal atmosphere" of the BH. Finally, as a consequence of the time dependent boost, this thermal atmosphere will become thinner as the particles eternally fall towards the horizon. We claim that this analysis could allow to clarify the link between conformal field approaches and BH scattering and could be at the origin of BH thermal effects. Indeed, an s-wave, with energy at least equals to ω min , can escape the BH without tunneling. Reciprocally, such s-wave will be able to penetrate the barrier from the outside and fall to the horizon (cf. Fig. 1). It is worth noting that, in this case, the amplitude of the reflected wave off the "ℓ = 0" Regge-Wheeler potential barrier is exponentially small, but not null. Moreover, the mean energy of massless particles in thermal equilibrium at temperature T = 1/8πM is roughly of order of T which is bigger than ω min . Therefore, some of the s-waves, i.e. "s-states" particles, will easily escape to infinity. Unless the BH is kept in equilibrium by incoming radiation, it will lose energy to its surroundings. In other words, the BH evaporates. This is one possible explanation of the Hawking radiation in terms of s-waves scattering, without tunneling. It should be noted that it is not the case for fields (or particles) with ℓ ≫ 1 because the Regge-Wheeler potential barrier becomes large enough to reduce significantly this process. Moreover, in this case, the location of V 0,max moves away from the horizon to be closer to the location of the photon sphere which will play a central role in the analysis of weakly damped QNM [START_REF] Decanini | Unstable circular null geodesics of static spherically symmetric black holes, regge poles and quasinormal frequencies[END_REF]. Furthermore, up to a normalizing contant, the expression (5.9) tells us that the probability of finding one particle reflected by the "ℓ = 0" Regge-Wheeler potential barrier is the same as the probability of finding one particle of energy ω in a system, i.e. the thermal atmosphere, in thermodynamical equilibrium at temperature T = 1/8πM . Thus, the probability P R (N k ) to find N k particles of energy ω k in the thermal atmosphere of the BH, in thermodynamical equilibrium at the temperaure T = 1/8πM is

P R (N k ) = 1 Z |R 0 (ω k )| 2 N k = 1 Z e -8πM ω k N k (5.11)
where the normalizing constant Z is the partition function such as the probabilities sum up to one Z = 1

N k P R (N k )
.

(5.12)

The mean number of reflected particles depends of course on the nature of the considered particles and is usually given by

N (F/B) (ω k ) = 1 or ∞ N k =0 N k P R (N k ) (5.13)
where F/B stands for "Fermion/Bosons". More explicitly

N (F ) (ω k ) = 1 N k =0 N k P R (N k ) = 1 e 8πM ω k + 1
for fermions (5.14a)

N (B) (ω k ) = ∞ N k =0 N k P R (N k ) = 1 e 8πM ω k -1
for bosons.

(5.14b)

From the Fourier analysis of section 3, one then has

N (F/B) (Ω) ∝ Φ(F/B) 0 (Ω) 2 .
(5.15)

Therefore, the mean number of particles of energy Ω is proportional to the probability of measuring a frequency Ω in the spectrum of the field Φ(u).

In this sense, it is tempting to think that the Physics which has to be considered to describe the spectrum of the Hawking radiation would be a conformal field theory, in the near vicinity of the horizon, associated to the field in interaction with the considered BH. Therefore, due to the accumulation of all the Fourier components (quantum particles states) of the incoming wave spectrum on this thermal layer, it would not be a surprise to describe the near horizon physics in terms of some "other/new" degrees of freedom which would then be expected to have rather "exotic" statistics. In the following, we will give a heuristic point of view concerning these new statistics.

A naive approach

Let us introduce the usual variable β = T -1 to simplify the notations and let us consider "ℓ = 0" fermions reflected off the Regge-Wheeler potential barrier. The mean number of fermions with energy ω n is given by the well-known Fermi-Dirac statistics

N i (ω n ) = 1 e βωn + 1 = N i g i (6.1)
where g i is the degeneracy of the i th state of energy ω n . The QNM complex frequencies can be seen as poles in the complex ω-plane of the Fermi-Dirac distribution, for which one can write

e βωn/T + 1 ≈ 0 ⇒ βω n ≈ 2πi n + 1 2 (6.2)
which is the result obtain in the previous section.

Even if the behavior is correct for large n, this approach does not give, obviously, the correct answer with the well-known constant real part, i.e. ln(3)/8πM . How can one have access to the latter? One way to answer this question is to assume that near the horizon, there may be another, "exotic" statistics satisfied by the considered quanta which would not behave neither like fermions or bosons. A first very naive approach is to consider a non-null chemical potential µ, or a corresponding fugacity z = e µ/T , for a thermodynamical system with a conserved number of fermions.

Then the Fermi-Dirac distribution reads

N i (ω n -µ) = 1 ze βωn + 1 = z -1 e βωn + z -1 . (6.3)
One could consider the fugacity z as an universal degeneracy g = z -1 for each state of an equivalent thermodynamical problem where the number of particles is now not conserved and, by definition, which statistics would be given by Ñi

(ω n ) = 1 g N i (ω n -µ) = 1 e βωn + g . (6.4)
Then, the QNM complex frequencies, seen as the poles of the statistical distribution are

e βωn + g ≈ 0 ⇒ βω n ≈ ln |g| + 2πi n + 1 2 . ( 6.5)
The exact analytical result suggest that for a Schwwarzschild BH, g = 3. In other words, each energy state could have 3 possible sub-levels. We are conscious that such explanation is just a mathematical trick and that the underlying physics is still expected.

More on statistics: the fractional statistics

However, one could assume that near the event horizon of the Schwarzschild BH, particles behave neither like bosons or fermions but they satisfy a more general statistics which could be the Polychronakos' fractional statistics [START_REF] Polychronakos | Probabilities and path-integral realization of exclusion statistics[END_REF], the Haldane's fractional statistics [START_REF] Haldane | fractional statistics" in arbitrary dimensions: A generalization of the pauli principle[END_REF] or even the infinite statistics [START_REF] Altherr | Thermal field theory and infinite statistics[END_REF], introduced some years ago, which all start by a generalization of the Pauli exclusion principle (highly suggested by the Motl's "tripled Pauli statistics" [START_REF] Motl | An analytical computation of asymptotic schwarzschild quasinormal frequencies[END_REF]). In the former case, one could use the simple Eq. ( 22) of [START_REF] Polychronakos | Probabilities and path-integral realization of exclusion statistics[END_REF] which reads

N i (ω n , α) = 1 e βω k + α . ( 6.6)
Fermions and bosons corresponds respectively to α = 1 and α = -1. If we introduce the variable g such as α = 2g + 1, then

N i (ω n , g) = 1 e βωn + 1 + 2g (6.7)
where fermions and bosons corresponds now respectively to g = 0 and g = -1. We deduce naturally

βω n = ln |1 + 2g| + 2πi n + 1 2 . ( 6.8)
One can noticed that the choice g = 1 (α = 3) allows us to recover the intriguing "tripled Pauli statistics". Let us note that the different expressions of highly damped QNM complex frequencies can be seen as statistics obtained for different kind of BH which could be embedded in the "richness" of these statistics [START_REF] Polychronakos | Probabilities and path-integral realization of exclusion statistics[END_REF][START_REF] Haldane | fractional statistics" in arbitrary dimensions: A generalization of the pauli principle[END_REF][START_REF] Altherr | Thermal field theory and infinite statistics[END_REF]. In particular, we think that the fractional statistics could be compatible with the "general" expression defining the quasinormal frequencies given in [START_REF] Skakala | Quasi-normal modes, area spectra and multi-horizon spacetimes[END_REF].

Conclusion

In our scattering picture, the Hawking radiation and thermal effects for a Schwarzschild BH are associated with the scattering of the s-waves by the "ℓ = 0" Regge-Wheeler potential barrier. A very interesting result is that the scattering/reflection amplitude is exponentially small but not null and is exactly the Boltzmann factor related to Hawking effect. We have seen, from an external observer perspective, that incoming waves (quantum particles) can be "trapped" between the location of V 0,max and the horizon. Moreover, this "thermal atmosphere" becomes thinner as a consequence of the time dependent boost, i.e. the freefall of the waves (quantum particles) into the BH. Then one could ask: is there a thickness limit for the "thermal atmosphere"? According to the well-known thermodynamical results, the answer would probably be linked to the existence of a cutoff, maybe the Planck length, but this is way too far from the scope of our analysis. Furthermore, we claim that this thin thermal atmosphere could be at the origin of new exotic statistics, due to the change of degrees of freedom during the transition between the incoming field in spacetime and the accumulation of all its Fourier components (quantum particles states) on the "thermal layer". It is worth noting that this thermodynamics, and consequently all the associated quantities, is usually associated to the BH, seen as one thermodynamical system. We have thus two different origins for the highly damped QNM complex frequencies. The first is the damping of the QNM due to the scattering/reflection off the "ℓ = 0" Regge-Wheeler potential barrier in the "far region limit", directly described by the imaginary part. The second would be the thermodynamics of the "thermal layer" associated with the real part.

Our scattering analysis seems to be in agreement with the Maggiore's interpretation [START_REF] Maggiore | The physical interpretation of the spectrum of black hole quasinormal modes[END_REF].

It should be noted that the Hawking effect and the highly damped QNM are not linked to a very low frequency limit [START_REF] Matzner | Scattering of massless scalar waves by a schwarzschild singularity[END_REF] but simply to the interaction of s-fields with the considered BH, whatever their energy is. Moreover, in addition to the existence of a thermal atmosphere, we have also recovered some results obtained in [START_REF] Solodukhin | Horizon state, hawking radiation and boundary liouville model[END_REF] but through conformal field frameworks, suggesting naturally the use of such approaches to describe the physics near the event horizon. Furthermore, one has to bear in mind that the statistics of particles should be thought as a property of spacetime and particularly near the horizon of a Schwarzschild BH. Therefore, if spacetime is quantum by nature, the commutation or anticommutation relations for creation and annihilation operators, for the considered fields, may change [START_REF] Swain | Exotic statistics for ordinary particles in quantum gravity[END_REF]. Thus, exotic statistics and thereby a modification of the spin statistics theorem could be deeply related to the very quantum nature of spacetime and could emerge from an underlying theory of quantum gravity in the near vicinity of a BH horizon. Finally, it is worth noting that a generalization to d-dimensional Schwarzschild-Tangherlini BH in interaction with a scalar (j = 0) field is trivial, at least for the imaginary part of the QNM complex frequencies. Indeed, in this case, the "dimension parameter" in the Regge-Wheeler potential is multiplied by the angular momentum ℓ (for more details, see [START_REF] Decanini | Unstable circular null geodesics of static spherically symmetric black holes, regge poles and quasinormal frequencies[END_REF]). Thus, for the scattering of an s-wave, the results will be identical. Concerning the real part, there is no "simple" argument except that the "ℓ = 0" Regge-Wheeler potential is not modified and thus one could perform the same calculations and recover the same scattering/reflection amplitude as in section 4. Actually, the real part seems to not be affected either by the dimension of spacetime as it has been noticed by Motl in [START_REF] Motl | An analytical computation of asymptotic schwarzschild quasinormal frequencies[END_REF] (and references therein).

Figure 1 .

 1 Figure 1. "ℓ = 0" Regge-Wheeler potential; For the Schwarzschild BH (2M = 1), the maximum of the Regge-Wheeler potential is evaluated at V 0,max = (27/256) (2M ) -2 ≈ 0.11 (2M ) -2 and located near the event horizon, i.e. at r = (4/3) (2M )

ω 2 ( 5

 25 .2b) where η = a (resp. b) and (ω 1 , ω 2 ) = (ω, ω ′ ) (resp. (Ω, Ω ′ )). The annihilation operators b † Ω and a † ω define respectively the Boulware vacuum state (in the (u, v) coordinate system) and the Kruskal vacuum state (in the (U, V ) coordinate system) by b † Ω |0 B = 0 Boulware vacuum (5.3a) a † ω |0 K = 0 Kruskal vacuum. (5.3b)

  ) coordinates where it is obviously not a monochromatic plane wave. It is worth noting that keeping the variable u = t + r * , the field reads

					.15)
	Thus, according to eqs. (3.11) and (3.12), the locally (monochromatic) ingoing plane wave
	Φ inertial (T, R) becomes	Φ(t, r * ) = exp -i	ω κ	e κ(t+r * )	(3.16)
	in the (t, r Φ(u) = exp -i	ω κ	e κu	(3.17)

*

which is analogous to the "horizon state" φ H introduced by Solodukhin

[START_REF] Solodukhin | Horizon state, hawking radiation and boundary liouville model[END_REF]

in the framework of a conformal field theory, if one identifies ω/κ with the parameter µ H . Physically, the ingoing wave Φ(u) is not monochromatic simply because of the time dependent Doppler shift effect due to the gravitational field of the Schwarzschild BH.

Statistical "heuristics"As we have just seen, in thermodynamical terms, the system in thermal equilibrium at temperature T = 1/8πM is the thin thermal atmosphere located close to the event horizon.

Acknowledgments

I would like to thank Professors Antoine Folacci, Yves Decanini, Jean-Pierre Provost, Christian Bracco, Thierry Grandou and Sergey Solodukhin for very stimulating discussions and support.