Jérôme Chevalier

Bernard Cales

Jean Michel Drouin

Low-temperature aging of Y-TZP ceramics

The isothermal tetragonal-to-monoclinic transformation of a 3Y-TZP ceramic is investigated from 70°to 130°C in water and in steam by X-ray diffraction and optical interferometer techniques. Aging kinetics followed by X-ray diffraction are fitted by the Mehl-Avrami-Johnson law, suggesting nucleation and growth to be the key mechanisms for transformation. Optical interferometer observations of highly polished samples effectively reveal a nucleation and growth micromechanism for tetragonal-to-monoclinic transformation. A model based on surface change analysis is developed that fits closely to the X-ray diffraction results.

I. Introduction

T HE concept of stress-induced phase transformation in zir- conia ceramics represents one of the most remarkable innovations in the ceramic field. Indeed, it was shown in the 1970s, first by Garvie et al. [START_REF] Garvie | Ceramic Steel?[END_REF] and then by Gupta et al., [START_REF] Gupta | Effect of Stress-Induced Phase Transformation on the Properties of Polycrystalline Zirconia-Containing Tetragonal Phase[END_REF] that zirconia exhibits a transformation toughening mechanism acting to resist crack propagation. The stress-induced phase transformation involves the transformation of metastable tetragonal crystallites to the monoclinic phase at the crack tip, which, accompanied by a volumetric expansion, induces compressive stresses. Yttria-stabilized zirconia ceramics belong to this family of toughened materials. They can exhibit a strength of more than 1000 MPa with a toughness of about 6 to 10 MPa и m 1/2 . Experimental 3-5 and theoretical [START_REF] Lange | Transformation Toughening, Part I and Part II[END_REF][START_REF] Green | Transformation Toughening of Ceramics[END_REF] studies agree that transformation toughening is highly dependent on grain size and stabilizer (yttria) content. The excitement created by these materials, however, was dampened by Kobayashi et al., [START_REF] Kobayashi | Phase Change and Mechanical Properties of ZrO 2 -Y 2 O 3 Solid Electrolyte After Ageing[END_REF] who discovered a serious limitation of Y-TZP ceramics for applications near 250°C. The authors revealed that Y-TZP ceramics can suffer a slow, tetragonal-to-monoclinic (t-m) transformation at the sample surface in a humid atmosphere, followed by microcracking and a loss in strength. Following this work, a series of papers were published with an attempt to understand the basic micromechanisms of t-m transformation and to reduce this limiting phenomenon. [START_REF] Lange | Degradation During Ageing of Transformation-Toughened ZrO 2 -Y 2 O 3 Materials at 250°C[END_REF][START_REF] Sato | Transformation of Yttria-Doped Tetragonal ZrO 2 Polycrystal by Annealing in Water[END_REF][START_REF] Sato | Transformation of Yttria Doped Tetrag-onal Zirconia Polycrystals by Annealing under Controlled Humidity Conditions[END_REF][START_REF] Nakajima | Phase Stability of Y-PSZ in Aqueous Solutions[END_REF][START_REF] Tsukuma | Thermal and Mechanical Properties of Y 2 O 3 Stabilized Zirconia Polycrystals[END_REF][START_REF] Watanabe | Aging Behavior of Y-TZP[END_REF][START_REF] Feng-Li | Influence of a Small Amount of Al 2 O 3 Addition on the Transformation of Y 2 O 3 Partially Stabilized ZrO 2 during Annealing[END_REF][START_REF] Hirano | Inhibition of Low Temperature Degradation of Tetragonal Zirconia Ceramics-A Review[END_REF] The main features of this so-called low-temperature degradation (LTD) are the following:

(i) Transformation proceeds most rapidly at temperatures of 200-300°C and is time dependent.

(ii) Water or water vapor enhances the transformation.

(iii) Transformation proceeds from the surface to the bulk of zirconia materials.

(iv) Higher stabilizing content or finer grain size increases the resistance to transformation. Today there are several models which attempt to explain how the presence of water can promote transformation. Sato et al. [START_REF] Sato | Transformation of Yttria-Doped Tetragonal ZrO 2 Polycrystal by Annealing in Water[END_REF][START_REF] Sato | Transformation of Yttria Doped Tetrag-onal Zirconia Polycrystals by Annealing under Controlled Humidity Conditions[END_REF] proposed chemisorption of water to form Zr-OH at the surface, which results in the accumulation of strain energy and thus in t-m transformation. Yoshimura et al. [START_REF] Yoshimura | Role of Water on the Degradation Process of Y-TZP[END_REF] also proposed the chemisorption of water as the key mechanism but attributed the accumulation of stresses to the migration of OH -species at the surface and in the lattice. A third model, presented by Lange and co-workers, [START_REF] Lange | Degradation During Ageing of Transformation-Toughened ZrO 2 -Y 2 O 3 Materials at 250°C[END_REF] postulates a reaction between H 2 O and yttria (Y 2 O 3 ), the formation of Y(OH) 3 depleting the stabilizer content then leading to transformation. However, recently, the depletion of Y 2 O 3 during LTD has been rejected and the fundamental role of internal stresses has been demonstrated. [START_REF] Kim | Phase Transformation of Y 2 O 3 and Nb 2 O 5 Doped Tetragonal Zirconia during Low Temperature Ageing in Air[END_REF] Transformation kinetics have sometimes been fitted by the Mehl-Avrami-Johnson (MAJ) law, [START_REF] Johnson | Reaction Kinetics in Processes of Nucleation and Growth[END_REF][START_REF] Christian | The Theory of Transformations in Metals and Alloys[END_REF] which suggests nucleation and growth. However, the evidence for the real nucleation and growth mechanism has never been clearly identified at the surface of Y-TZP materials. This was the aim of the present study, to identify nucleation and growth processes occurring during LTD. Another question is the extent of isothermal transformation at low temperature, typically near room temperature. Therefore, an attempt was made to provide LTD data at low temperatures, far below 150°C, in order to predict accurately isothermal t-m transformation kinetics of zirconia ceramics for low-temperature applications.

II. Experimental Procedure

The experiments were conducted on a Y-TZP ceramic containing 3 mol% (5.2 wt%) yttria (Norton Desmarquest Fine Ceramics, France). The commercial starting powder was produced by a coprecipitation method. The material was processed by cold isostatic pressing at 300 MPa, followed by pressureless sintering at 1500°C for 2 h. Hot isostatic pressing was then conducted to achieve a density of more than 99.9%. The grain size was measured by SEM according to the ASTM E112 standard. The average grain size value by linear intercept was 0.5 m. Small disks of 40 mm diameter and 4 mm thickness were ground and polished with diamond paste to reach a surface roughness (R a ) of less than 3 nm (measured by means of an optical interferometer, Phase Shift Technology®). Such a roughness was believed to be low enough to observe any small surface uplift due to t-m transformation. No annealing was done on the specimens because it was shown that monoclinic content and residual grinding stresses were suppressed during careful polishing steps. [START_REF] Chevalier | Slow Crack Growth Behavior of 3Y-TZP Biomedical Ceramics[END_REF] Monoclinic content was measured by an X-ray diffraction (XRD) technique (CuK␣ radiation, with a penetration depth in zirconia estimated to be about 5 m from Ref. 22) and calculated from the modified Garvie and Nicholson equation. [START_REF] Garvie | Phase Analysis in Zirconia Systems[END_REF] It was systematically close to 0% before any aging test. Surface changes were recorded by means of the optical interferometer used for roughness measurements in a monochromatic wavelength mode. It is characterized by a height resolution of ±0.1 nm (X-Y resolution of about 0.3 m). Aging was achieved in distilled water at different temperatures from 70°to 100°C (thermostated baths ±2°C) and in steam at 120°C and 130°C under 2 bar.

III. X-ray Diffraction Results

The relationship between the amount of monoclinic phase and aging time at various temperatures is shown in Fig. 1. The amount of monoclinic phase increases with aging time according to a sigmoidal behavior, in agreement with previous studies conducted at high temperatures. [START_REF] Tsubakino | Tetragonal-to-Monoclinic Phase Transformation during Thermal Cycling and Isothermal Ageing in Partially Stabilized Zirconia[END_REF][START_REF] Tsubakino | Martensite Transformation Behavior during Isothermal Ageing in Partially Stabilized Zirconia with or without Alumina Addition[END_REF][START_REF] Zhu | Time-Dependent Tetragonal-to-Monoclinic Transition in Hot-Pressed Zirconia Stabilized with 2 mol% Yttria[END_REF] These studies suggested that the relationship between the amount of monoclinic phase and the aging time could be expressed by the MAJ equation: [START_REF] Johnson | Reaction Kinetics in Processes of Nucleation and Growth[END_REF][START_REF] Christian | The Theory of Transformations in Metals and Alloys[END_REF]

f ס 1 -exp[-(bиt) n ] ( 1
)
where f is the transformation fraction, t is the time, and b and n are constants. The MAJ equation is often used for describing time-transformation isotherms in metals and metallic alloys. Tsubakino et al. [START_REF] Tsubakino | Tetragonal-to-Monoclinic Phase Transformation during Thermal Cycling and Isothermal Ageing in Partially Stabilized Zirconia[END_REF][START_REF] Tsubakino | Martensite Transformation Behavior during Isothermal Ageing in Partially Stabilized Zirconia with or without Alumina Addition[END_REF] suggested that isothermal transformation occurs by a nucleation and growth process without providing experimental evidence for this mechanism. In the MAJ theory summarized by Christian, [START_REF] Christian | The Theory of Transformations in Metals and Alloys[END_REF] it is shown that the n exponent, which can be derived from the slope of the ln (ln (1/(1f ))) versus ln t plot, is related to nucleation and growth conditions. b is a parameter giving the apparent activation energy

Q by b = b 0 exp ͫ -Q RT ͬ (2)
where b 0 is a constant, R the gas constant and T the absolute temperature. b can also be calculated for each temperature from a ln (ln (1/(1f ))) versus ln t plot. This plot is shown in Fig. 2. Because the monoclinic fraction never reaches 100% even after a very long duration, f is taken as the fraction in the saturated amount of monoclinic phase (88%). A linear relationship is observed for each temperature with a constant value of n equal to 3.6 ± 0.4. According to the MAJ theory, [START_REF] Johnson | Reaction Kinetics in Processes of Nucleation and Growth[END_REF][START_REF] Christian | The Theory of Transformations in Metals and Alloys[END_REF] this value between 3 and 4 corresponds to a nucleation and threedimensional growth process. The plot of ln b versus 1/T in Fig. 3 gives an activation energy of 106 kJ/mol with excellent reliability (correlation coefficient of more than 99%).

IV. Optical Interferometer Investigations

The nucleation and growth mechanism for LTD, which was suggested by XRD, was confirmed by optical interferometer observations. For instance, Fig. 4 represents the same surface of a given specimen observed with the optical interferometer after different aging durations at 130°C, 2 bar in steam. Figure 4(a) corresponds to the initial surface, with a roughness R a ס 2 nm and a monoclinic content close to 0%. After an incubation period of about 1 h, the nucleation of monoclinic sites occurs (see Fig. 4(b)). Nucleation is detected by small surface upheavals (with a height less than 10 nm and a diameter close to 1 m) which correspond to the transformation of one or a few grains. [START_REF] Wayman | Martensitic Transformation[END_REF][START_REF] Lee | Surface Crack Initiation in 2Y-TZP Ceramics by Low Temperature Ageing[END_REF] This surface expansion is linked to the increase of the crystal size and thus the grain size due to t-m transformation. At that time, monoclinic fraction measured with XRD still remains close to 0%. After 5 h (in Fig. 4(c)), it appears that the initial nucleation sites have slightly increased in size. Their shape is nearly conical, suggesting that transformation effectively occurs from one grain to the neighboring grains and so on. The propagation from the initially transformed grain to the others is related to the formation of micro-or nanocracks around transformed grains, as already suggested by several authors. In addition to the initial sites, new nucleation sites are created (Fig. 4(c)). Monoclinic content measured by XRD now reaches 10%. It is observed that the "conical" monoclinic sites grow at an approximately constant rate with time, with a constant ratio of height/diameter of about 1%. Figure 4(d) shows the same surface after 7 h for a monoclinic fraction of 20% and confirms a nucleation and growth process.

Saturation in monoclinic content measured by XRD is reached at the time when the surface is completely covered by conical spots. Transformation then proceeds in the bulk of the material (i.e., deeper than the first 5 m analyzed by X-ray diffraction analysis).

V. Analysis and Discussion

Figure 5(a) shows the diameter and the height of three particular conical surface upheavals as a function of time after their nucleation, at 130°C. The plots can be fitted with linear relationships. The nucleation rate at 130°C is followed in Fig. 5(b) for two regions of the same specimen (area of each region: 165 × 124 m 2 ) from the beginning of the LTD until the surface is totally recovered by monoclinic grains. After a short apparent incubation period of about 1 h, the number of nuclei varies linearly with time, the nucleation rate being constant. After 7 h, i.e., for a monoclinic content higher than 30%, the nucleation rate reaches a saturation level. Such a saturation level can result from compressive stresses induced by the t-m transformation. [START_REF] Chevalier | Slow Crack Growth Behavior of 3Y-TZP Biomedical Ceramics[END_REF] Another explanation for the saturation is the decrease of the probability of nucleation at a given location as LTD proceeds at sites distributed randomly. The optical analysis can therefore be summarized in a mathematical form, for monoclinic content less than 30%:

d ͑t͒ = ␣ d ͑t -͒ (3)
h ͑t͒ = ␣ h ͑t -͒ (4)
dN = N r A d (5a͒
where d (t) and h (t) represent the diameter and the height of a spot originating at time t ס , and where dN corresponds to the number of nuclei created between and +d , ␣ d and ␣ h are constants characteristic of the diameter and height growth rates, and N r is the nucleation rate per unit time and area. A is the area observed for the analysis. The surface uplift observed at the surface corresponds to the volume expansion due to t-m transformation, known to be accompanied by a volume expansion of about 4%, [START_REF] Kelly | Crystallography of Stress-Induced Martensitic Transformation in Partially Stabilized Zirconia[END_REF] thus by a linear expansion, k, of about 1.3%. Thus, the volume of a given monoclinic region originating at t ס is given by

͑t͒ = 12
␣ d 2 ␣ h k ͑t -͒ 3 (6)
During the initial period of LTD, when nuclei are widely spaced and the interaction between them is negligible, the monoclinic volume at time t resulting from nuclei created between and +d is dV m ס v (t)d N. Thus, the total transformed volume at time t is equal to

V m = ͐ 0 t 12 ␣ d 2 ␣ h k N r A͑t -͒ 3 d (7)
The integration of Eq. (7) gives

V m = N r 48 ␣ d 2 ␣ h k At 4 (8)
The following constants can be derived from Figs. 5(a) and (b): ␣ d ס 2.7 m/h, ␣ h ס 2 7×1 0 -3 m/h, and N r ס 3.7 × 10 -4 /h/m 2 . The monoclinic fraction, f, calculated for a layer of material of thickness, l, is equal to

f = V m Al = N r l 48 ␣ d 2 ␣ h k t 4 (9a͒
Therefore, with k ס 1.3%, and with l ס 5 m (to compare with the monoclinic fraction obtained with the XRD analysis), the monoclinic fraction at 130°C is

f (%) ס 0.0072t 4 (9b)
The monoclinic fraction calculated at 130°C using Eq. (9a)i s plotted in Fig. 6 and compared with the experimental X-ray diffraction results. The correlation between the two curves is excellent when the monoclinic content does not exceed 30%. Above 30%, the decrease in nucleation rate is not considered in the analysis. For a more correct analysis we have to consider also the decrease in the tetragonal content, thus the decrease in the probability of nucleation at a given location, with time. A simple refinement then replaces Eq. (5a)b y

dN = N r A͑1 -f ͒ d (5b)
The combination of Eqs. (5b) and ( 6)

gives df 1 -f = ␣ d 2 ␣ h N r 12kl ͑t -͒ 3 d (10)
The monoclinic fraction is then obtained by integration of Eq. ( 10):

f = 1 -expͫ-ͩ ␣ d 2 ␣ h N r 48kl ͪt 4 ͬ (11)
The monoclinic fraction f calculated from Eq. ( 11) is also plotted in Fig. 6 and compared to XRD results. The agreement is very good. Also remarkable is the similarity between Eq. ( 11) from the present analysis and Eqs. ( 1) and ( 2) from the MAJ treatment, with b ס [␣ 2 d ␣ h N r /48kl] 1/n . The present analysis leads to a value of 4 for the n exponent, while expeirmental X-ray diffraction results lead to a value of about 3.6. The slight difference can result from the compressive stresses induced by t-m transformation and which hinder the nucleation and, maybe, the growth rate. These compressive stresses are not considered in the present analysis.

It is of prime interest then to determine the temperature dependence of the constants ␣ d , ␣ h , and N r . This was done by recording the nucleation rate and the nuclei growth rates at four temperatures: 80°, 100°, 120°, and 130°C. The variation of ␣ d , ␣ h , and N r versus temperature is recorded in Fig. 7 in Arrhenius plots to derive the activation energy. It is not surprising to obtain three, parallel, straight lines, giving each an activation energy of about 100 kJ/mol, similar to that calculated from XRD results by the MAJ equation.

VI. Conclusion

The low-temperature degradation behavior of a 3Y-TZP ceramic was investigated from 70°to 130°C. X-ray diffraction results suggested an incubation-nucleation-growth mechanism for transformation because of sigmoidal variation of the monoclinic fraction with time for all temperatures. The MAJ equation was used to fit the experimental data, with n ס 3.6. Nucleation and growth micromechanisms were then confirmed with optical interferometer observations on highly polished surfaces. A model was used that correlated well with the MAJ law. This equation allows an accurate prediction of the monoclinic fraction at the surface of the aged 3Y-TZP ceramic for a given time and temperature and particularly for low-temperature applications.

This study was focused on a given zirconia ceramic. Any extrapolation to other zirconia ceramics could be subjected to large error because LTD is primarily dependent on the ceramic microstructure (grain, size, yttrium content, density, etc.). Therefore, LTD parameters should be specifically determined for each Y-TZP ceramic.

Fig. 1 .

 1 Fig. 1. Monoclinic content measured by X-ray diffraction versus time for temperatures ranging from 70°to 130°C. Solid lines represent the fitting of the experimental results by the Mehl-Avrami-Johnson law with n ס 3.6.

Fig. 2 .

 2 Fig. 2. Plot of ln (ln (1/(1f ))) versus ln t for the determination of n (slope) and ln b (ordinate origin) in the MAJ equation.

Fig. 3 .

 3 Fig. 3. Plot of ln b versus 1/T for the determination of the apparent activation energy for the t-m phase transformation process.

Fig. 4 .Fig. 5 .

 45 Fig. 4. Optical interferometer observation of the same area (165 × 124 m 2 ): (a) before aging for a monoclinic fraction f close to 0%, (b) after 3 h in steam at 134°C, 2 bar with f < 5%, (c) 5 h with f ס 10%, and (d) 7 h with f ס 20%.

Fig. 6 .

 6 Fig. 6. Variation of monoclinic content with time at 130°C, 2 bar in steam from X-ray diffraction results (᭹), MAJ equation with n ס 3.6 (heavy line), Eq. (9a) (dotted line), and Eq. (11) (single line) of the nucleation and growth model.

Fig. 7 .

 7 Fig. 7. Variation of ␣ d , ␣ h , and N r with temperature.

Acknowledgments:

We would like to thank O. Filippini for his help with the experimental work and Ed Levadnuk for helpful comments on the manuscript.


Christophe Finot
email: christophe.finot@u-bourgogne.fr

John M Dudley

Guy Millot
email: guy.millot@u-bourgogne.fr

Generation of dark solitons by interaction between similaritons in Raman fiber amplifiers

We present a new method to generate dark soliton trains by exploiting the interaction between two time-delayed optical similaritons with the same wavelength. The temporal overlap of two similariton pulses creates a sinusoidal beating which subsequently evolves into an ultrahigh repetition-rate train of dark solitons through the combined effects of normal dispersion, non-linearity and adiabatic Raman gain. The experimental results are in good agreement with numerical predictions. We also investigate how the the repetition rate of the dark soliton train depends on the time separation between the initial pulses, the initial pulse energy, and the Raman gain. Finally, we numerically study the interaction between three similaritons of the same wavelength.

Introduction

The application of self-similarity techniques to the study of nonlinear pulse propagation has been the subject of much recent interest in the context of parabolic pulse generation in optical fiber amplifiers with normal group-velocity dispersion (GVD) [START_REF] Anderson | Wavebreaking-free pulses in nonlinear optical fibers[END_REF][START_REF] Tamura | Pulse compression by nonlinear pulse evolution with reduced optical wave breaking in erbium-doped fiber amplifiers[END_REF][START_REF] Fermann | Self-Similar Propagation and Amplification of Parabolic Pulses in Optical Fibers[END_REF][START_REF] Kruglov | Self-similar propagation of high-power parabolic pulses in optical fiber amplifiers[END_REF][START_REF] Kruglov | Self Similar Propagation of parabolic pulses in normal-dispersion fiber amplifiers[END_REF][START_REF] Boscolo | Self-similar parabolic optical solitary waves[END_REF][START_REF] Peacock | Solitary pulse propagation in high gain optical fiber amplifiers with normal group velocity dispersion[END_REF][START_REF] Finot | Experimental generation of parabolic pulses via Raman amplification in optical fiber[END_REF][START_REF] Peacock | Numerical study of parabolic pulse generation in microstructured fibre Raman amplifiers[END_REF][START_REF] Kruglov | Self-similar solutions of the generalized Schrödinger equation with distributed coefficients[END_REF][START_REF] Kruglov | Exact self-similar solutions of the generalized nonlinear Schrödinger equation with distributed coefficients[END_REF][START_REF] Finot | Asymptotic characteristics of parabolic similariton pulses in optical fiber amplifiers[END_REF][START_REF] Finot | Numerical and experimental study of parabolic pulses generated via Raman amplification in standard optical fibers[END_REF][START_REF] Chang | Dependence of parabolic pulse amplification on stimulated Raman scattering and gain bandwith[END_REF][START_REF] Hirooka | Parabolic pulse generation by use of a dispersiondecreasing fiber with normal group-velocity dispersion[END_REF][START_REF] Chen | Chirped self-similar solutions of a generalized nonlinear Schrödinger equation model[END_REF][START_REF] Finot | Influence of the pumping configuration on the generation of optical similaritons in optical fibers[END_REF][START_REF] Billet | Intermediate asymptotic evolution and photonic bandgap fiber compression of optical similaritons around 1550 nm[END_REF]. Such pulses, also called optical similaritons, represent a new class of solutions to the non-linear Schrödinger equation (NLSE) with gain. They are asymptotically generated in the amplifier, independently of the shape or noise of the input pulse [START_REF] Fermann | Self-Similar Propagation and Amplification of Parabolic Pulses in Optical Fibers[END_REF][START_REF] Finot | Asymptotic characteristics of parabolic similariton pulses in optical fiber amplifiers[END_REF], and propagate self-similarly subject to exponential scaling of their pulse amplitude, temporal duration and spectral width [START_REF] Kruglov | Self Similar Propagation of parabolic pulses in normal-dispersion fiber amplifiers[END_REF]. In this case, the interplay of normal dispersion, nonlinearity and gain produces a linearly chirped pulse with a parabolic intensity profile [START_REF] Fermann | Self-Similar Propagation and Amplification of Parabolic Pulses in Optical Fibers[END_REF].

Similaritons are of wide-ranging practical significance since their intrinsic resistance to the deleterious effects of the wave breaking [START_REF] Anderson | Wavebreaking-free pulses in nonlinear optical fibers[END_REF] allows the scaling of amplifiers to higher-power regimes [START_REF] Limpert | High-power femtosecond Yb-doped fiber amplifier[END_REF][START_REF] Malinowski | Ultrashort-pulse Yb3+ fiber based laser and amplifier system producing > 25 W average power[END_REF]. Moreover, their linear chirp facilitates efficient temporal compression [START_REF] Fermann | Self-Similar Propagation and Amplification of Parabolic Pulses in Optical Fibers[END_REF][START_REF] Billet | Intermediate asymptotic evolution and photonic bandgap fiber compression of optical similaritons around 1550 nm[END_REF][START_REF] Limpert | High-power femtosecond Yb-doped fiber amplifier[END_REF][START_REF] Malinowski | Ultrashort-pulse Yb3+ fiber based laser and amplifier system producing > 25 W average power[END_REF]. Combination of a similariton amplifier with an optical feedback has resulted in a new regime of laser mode-locking that is likely to have major implications for the development of high-power ultrashort pulse laser oscillators [START_REF] Ilday | Self-similar evolution of parabolic pulses in a laser[END_REF]. Recent experimental studies have also taken advantage of the similariton characteristics to propose new methods for optical pulse synthesis [START_REF] Finot | Synthesis of optical pulses by use of similaritons[END_REF], for 10 GHz telecom multiwavelength sources [START_REF] Ozeki | Generation of 10 GHz similariton pulse trains from 1,2 km-long erbium-doped fibre amplifier for application to multi-wavelength pulse sources[END_REF] or for optical regeneration of telecom signal [START_REF] Finot | Regenerative 40-Gb/s wavelength converter based on similariton generation[END_REF]. The experimental works have mainly relied on rare-earth doped fiber amplifiers with dopants such as Erbium [START_REF] Tamura | Pulse compression by nonlinear pulse evolution with reduced optical wave breaking in erbium-doped fiber amplifiers[END_REF][START_REF] Ozeki | Generation of 10 GHz similariton pulse trains from 1,2 km-long erbium-doped fibre amplifier for application to multi-wavelength pulse sources[END_REF] or Ytterbium [START_REF] Fermann | Self-Similar Propagation and Amplification of Parabolic Pulses in Optical Fibers[END_REF][START_REF] Limpert | High-power femtosecond Yb-doped fiber amplifier[END_REF][START_REF] Malinowski | Ultrashort-pulse Yb3+ fiber based laser and amplifier system producing > 25 W average power[END_REF][START_REF] Ilday | Self-similar evolution of parabolic pulses in a laser[END_REF]. However, it has been shown that a resonant amplification is not a necessary requirement for parabolic pulse generation and that Raman scattering could be efficiently used as amplification process [START_REF] Finot | Experimental generation of parabolic pulses via Raman amplification in optical fiber[END_REF][START_REF] Peacock | Numerical study of parabolic pulse generation in microstructured fibre Raman amplifiers[END_REF][START_REF] Finot | Numerical and experimental study of parabolic pulses generated via Raman amplification in standard optical fibers[END_REF]. This opens up the possibility to provide gain at wavelengths where no conventional doped amplifiers are possible.

To date, most of the theoretical and experimental studies have been devoted to the dynamics of single similariton pulses. The dynamics of the amplification of a pair of pulses is more complex. In particular, during the amplification process, the exponential self-similar scaling of the temporal duration of each individual pulse leads to a temporal overlap and interaction of the pulses after sufficient propagation distance. As similariton-based techniques have been recently demonstrated in the field of telecommunication [START_REF] Ozeki | Generation of 10 GHz similariton pulse trains from 1,2 km-long erbium-doped fibre amplifier for application to multi-wavelength pulse sources[END_REF][START_REF] Finot | Regenerative 40-Gb/s wavelength converter based on similariton generation[END_REF] it has become crucial to better understand the various phenomena which can be involved in the case of a similariton overlap during the propagation [START_REF] Finot | Interaction between optical parabolic pulses in a Raman fiber amplifier[END_REF].

We present in this paper a detailed study of the interactions occurring in the overlap region of a pair of optical similaritons of the same wavelength generated in a Raman amplifier where the amplification is nearly-adiabatic. Our amplifier at 1550 nm relies on a normally dispersive Non-Zero Dispersion-Shifted Fiber (NZ-DSF) in association with a commercial wattlevel CW pump source [START_REF] Finot | Experimental generation of parabolic pulses via Raman amplification in optical fiber[END_REF][START_REF] Finot | Numerical and experimental study of parabolic pulses generated via Raman amplification in standard optical fibers[END_REF]. We first detail the numerical predictions obtained using a model based on the Non-Linear Schrödinger Equation (NLSE) with a constant gain term. In particular, we demonstrate that the pulses evolve independently to generate self-similar parabolic profiles, only interacting in their overlap region. At the beginning of their overlap the interaction between the two linearly chirped parabolic pulses is almost linear and leads to a frequency beating of the intensity profile. After the first stage of overlap, however, the beating, subject to the combined effects of non-linearity, normal dispersion and adiabatic Raman gain, evolves into a train of dark solitons. We show that dark solitons also appear when taking into account the longitudinal and frequency dependences of the gain, as well as higher order linear and non-linear terms in the NLSE. We then compare quantitatively the results of the computations with experimental data.

We also carry out a detailed study of the dependence of the repetition rate of dark-soliton train on the characteristics of the input pulse pair and the gain of the amplifier. Finally we briefly consider the interaction between three similaritons of the same wavelength.

Numerical simulations

Basic model

We first present results from simulations that demonstrate numerically the generation of a train of dark solitons under typical experimental conditions. Evolution of the slowly varying complex envelope ψ of an initial pulse during its propagation in a normal-dispersion Raman amplifier can be modeled by the NLSE with a constant longitudinal and spectral gain g :

2 2 2 2 2 2 g i i z t β ψ ψ ψ γ ψ ψ ∂ ∂ = + - ∂ ∂ (1)
where β 2 >0 and γ are the second-order dispersion and Kerr nonlinear coefficients, respectively.

The asymptotic similariton solution of Eq.(1) in the limit z → ∞ is characterized by a parabolic intensity profile and a positive linear chirp, whose analytical expression is given by [START_REF] Fermann | Self-Similar Propagation and Amplification of Parabolic Pulses in Optical Fibers[END_REF]:

( ) 2 2
, ( ) 1 exp ( ) , ( )

p P p P p p t (t z) A z i z C t t T T z ψ ϕ π     = - - ≤         (2) 0 , p p (t) t T ψ = ≥
Here ϕ p is a phase offset. The solution (2) corresponds to a compactly-supported pulse whose zero-crossing points are given by

T p (z) = (6 A 0 / g) (γ β 2 / 2) 1/2 exp(g z/3), with A o a constant amplitude term A o = 1/2 [g U ini / (γ β 2 / 2) 1/2 ] 1/3
, where U ini is the initial pulse energy. These parabolic pulses have a peak-power given by |A p | 2 = A 0 2 exp(2 g z/3) and a positive linear chirp with the slope C p = g / ( 6 π β 2 ). We note that the temporal width T p of the pulse exponentially increases with the propagation length z.

Similariton propagation outside the overlap region

We now consider the amplification of a pair of identical pulses separated by a time-delay ∆T = 55.5 ps. The time origin of the copropagating frame is the temporal centre of gravity of the pair of initial pulses, so that the pulses are centered at t = ± ∆T / 2. Each initial pulse has a chirp-free Gaussian intensity profile with a duration of 5 ps, a carrier wavelength of 1550 nm and an initial energy of 2.9 pJ. The Raman amplifier parameters at 1550 nm are γ = 2.0 × 10 -3 W -1 m -1 and β 2 = 4.6 × 10 -3 ps 2 m -1 . The linear gain coefficient g is 0.972 × 10 -3 m -1 , which leads, for an amplifier length of 5.3 km, to a total integrated gain of 22.3 dB. Figure 1(a) shows the evolution of the pair of pulses as a function of the propagation distance in the amplifier, as calculated from numerical integration of Eq. ( 1). The solid lines show the total evolution of the incident pair of pulses and, for comparison, the open circles show the expected evolution of one pulse propagating on its own. We note that there is no noticeable difference between the intensity profile of a single pulse (circles) and the intensity profile of a pulse in the presence of its neighbor (solid line). Therefore, both before and after the pulses overlap, each pulse evolves separately with a parabolic profile, and only interact in the overlap region. Owing to their compact nature, parabolic pulses are clearly unaffected by their neighbor outside the overlap region. In consequence, their mean temporal positions remain constant along the amplifier. This is in contrast to the case of two interacting solitons, where both the position and shape of the pulses are modified by the interaction.

Modulation of the intensity profile in the overlap region

Figure 1 (a) clearly shows that, at a distance of 4.17 km, an oscillation appears inside the overlap. Indeed, as the parabolic pulses are linearly chirped, the frequency difference between the overlapping leading and trailing edges of the two pulses induces a beating in the resultant signal. This is shown more clearly in Fig. 1(b) (solid line).

To understand the characteristics of this beating in more detail, we first neglect the nonlinear effects which occur in the interaction process between the two similaritons. If we consider that the pulses have developed a parabolic shape as described by Eq. ( 2), the overlap region extends from t = ∆T/2

t T i C t T T (t) A t T i C t T T π ψ π     + ∆     - - + ∆             =       -∆     + - - -∆             , (3)
where the phase offset terms were neglected. We will thus see a sinusoidal beating due to the superposition of the two instantaneous frequencies separated by f s = C p ∆T. The intensity in the overlap region is then given by:

2 2 2 2 2 2 2 1 1 4 2 / 2 / 2 cos(2 ) 1 1 p pair P P p p T t T (t) A t T t T C t T T T ψ π     ∆ - +           =         + ∆ -∆ + ∆ - -                   . ( 4
)
We thus have N = f s ( 2 T p -∆T) oscillations which appear in the overlap region. At the center of the overlap region (when / 2

p t T T -∆ ≪
), the intensities of the pulses are comparable so that near 100% modulation is expected. At the inner edges of the overlap region, the contributions of the pulses become different, so that the modulation depth decreases. The circles in Fig. 1(b) represent the linear superposition of the similaritons (Eq. 4) after 4.17 km of propagation, resulting in oscillation with a modulation frequency f s = C p ∆T = 622 GHz. Figure 1(b) shows that, at the center of the overlap region, this oscillation is in good agreement with the numerical simulations of Eq. (1) (solid line). However, it is significant that the oscillations predicted by numerical simulations extend over a larger range than those obtained from simple linear superposition of the two parabolic pulses : the value of N obtained by numerical simulations is then superior to f s ( 2 T p -∆T). These additional oscillations, which originate in the wings of the similariton pulses [START_REF] Kruglov | Self Similar Propagation of parabolic pulses in normal-dispersion fiber amplifiers[END_REF][START_REF] Boscolo | Self-similar parabolic optical solitary waves[END_REF], implicitly appear from numerical integrations of Eq. ( 1), but are neglected by assuming that the pulses have an exact parabolic shape (Eq. ( 4)). In our case, the large apparent value of the wings is related to the fact that, after 4.17 km of propagation, the pulses are not completely in the asymptotic regime.

Generation of high-repetition rate trains of dark solitons

When the overlapping pulses further propagate through the amplifier subject to the combined effects of non-linearity, normal dispersion and adiabatic Raman gain, the oscillation reshapes into a train of dark solitons [START_REF] Dianov | Generation of a train of fundamental solitons at a high repetition rate in optical fibers[END_REF]. Such a reshaping effect has been previously exploited in the spatial field with the transformation of a periodic sinusoidal modulation into a spatial array of dark solitons [START_REF] Mamyshev | Generation of a periodic array of dark spatial solitons in the regime of effective amplification[END_REF]. The normalized intensity profile of a dark soliton is given by [START_REF] Thurston | Collisions of dark solitons in optical fibers[END_REF]

I(T) = A 2 (1-B 2 sech 2 T) /B 2
, where A 2 is the intensity depth of the hole, B 2 is a blackness factor related to the contrast and T is a dimensionless retarded time. Dark solitons with B 2 = 1 have been called black, whereas other dark solitons with -1 < B < 1 are called gray. The blackness parameter |B| can be calculated from the values of the power and phase jump at the dip center. For example, for the dark soliton designed as (1) in Fig. 1(c), the blackness parameter is |B| = 0.997 ≅ 1 and the phase jump is 0.97 π ≅ π, so that this hole is very close to a black soliton with a tanh 2 shape.

Indeed, Fig. Taking into account that the fundamental black soliton depth is

P o = β 2 /(γT d 2 ) = 15.9 W ≅ P d ,
we may conclude that the characteristics of dip ( 1) are very close to those of a fundamental black soliton.

The repetition rate of the dark-soliton train after a propagation distance of 5.3 km is f d = 380

GHz. Figure 2(a) shows the spectrum of the dark soliton train (solid line) after 5.3 km of propagation, compared with that of a single pulse (dashed grey line). As can be seen from Fig. 2(a), the generation of a train of dark solitons is characterized by the apparition of peaks, with a frequency spacing equal to f d . The growth of these spectral peaks can be seen as a manifestation of a four-wave mixing process. satisfy the relationship between the peak power and the duration during the amplification process [START_REF] Zhao | Generation of dark solitons under a cw background using waveguide electro-optic modulators[END_REF]. There is a global conservation of the area of the dark soliton pulse. This temporal compression is different from the spreading of the dark solitons, as described by Rothenberg [START_REF] Rothenberg | Dark soliton trains formed by visible pulse collisions in optical fibers[END_REF] in the case of the generation of a train of dark solitons by the colliding of two pulses. In the scheme of Ref. [START_REF] Rothenberg | Dark soliton trains formed by visible pulse collisions in optical fibers[END_REF], the physical phenomenon responsible for the overlap of the pulses was the temporal broadening induced in a passive fiber by the deleterious effect of optical wavebreaking.

Use of a more realistic model

The model (Eq. ( 1)) used in the previous section was based on the assumption of a constant longitudinal gain and neglects several effects which can affect the shape of the output pulse, such as the longitudinal dependence of the gain profile [START_REF] Kruglov | Self-similar propagation of high-power parabolic pulses in optical fiber amplifiers[END_REF][START_REF] Finot | Influence of the pumping configuration on the generation of optical similaritons in optical fibers[END_REF] or the dispersion of the Raman gain [START_REF]Nonlinear Fiber Optics[END_REF][START_REF] Stolen | Raman response function of silica-core fibers[END_REF]. For a more accurate modelization of our experimental results, we present now simulations using the generalized extended NLSE [START_REF]Nonlinear Fiber Optics[END_REF] that rigorously includes the Raman amplification process through an appropriate integral term, as well as effects such as higherorder dispersion terms and self-steepening :

2 3 3 2 2 3 2 0 ( , ) ( ') ( , ') 2 2 6 1 ' s z t t z t t i i i z t t i R dt t β β ψ ψ ψ α ψ γ ψ ψ ω ∞ - ∂ ∂ ∂ = - + + ∂ ∂ ∂   ∂ - +   ∂   ∫ . ( 5
)
The function R(t) = (1 -f r ) δ(t) + f r h r (t) includes the instantaneous electronic response and the delayed Raman contribution h r (t), with the fractional Raman contribution f r = 0.18. For h r (t), we used the measured Raman response of fused silica [START_REF] Stolen | Raman response function of silica-core fibers[END_REF]. The third order dispersion is β 3 = 1.0 x 10 -6 ps 3 m -1 . The wavelength dependent loss coefficient α is also included in Eq. ( 5). Here, ψ(z,t)

= ψ s (z,t) + ψ r (z,t) exp(-iΩ t), where ψ s and ψ r are the slowly varying envelopes of signal and pump fields oscillating at ω s and ω r , respectively. Ω = ω r -ω s is the pump-signal frequency detuning chosen so that near optimal Raman gain for the signal is achieved at 1550 nm. The initial conditions ψ s (0,t) for the signal are given by the intensity and chirp of the two input pulses directly measured by using a frequency-resolved optical gating (FROG) technique [START_REF] Trebino | Frequency-Resolved Optical Gating : The Measurement of ultrashort laser pulses[END_REF]. The pump ψ r is assumed to be a noise free continuous wave at 1455 nm. We use a counterpropagating amplification scheme, which has the advantage to be only slightly affected by the time-dependent pump depletion effects [START_REF] Finot | Influence of the pumping configuration on the generation of optical similaritons in optical fibers[END_REF]. In consequence, in that pump configuration, the two input signal pulses have nearly the same Raman gain. On the contrary, pump depletion effects, which would occur in a forward pumping geometry, would lead to different effective Raman gain for the two signal pulses. The pump power is adjusted in order to provide a total integrated gain of 22.3 dB for the signal. This corresponds to a power |ψ r (L)| 2 of 0.95 W.

In practice, to improve the numerical efficiency (and thus reduce the computation time), assuming a continuous Raman pump, we used the following set of two coupled extended NLSE's [START_REF] Finot | Numerical and experimental study of parabolic pulses generated via Raman amplification in standard optical fibers[END_REF][START_REF] Finot | Influence of the pumping configuration on the generation of optical similaritons in optical fibers[END_REF] with the subscripts s and r indicating the signal and Raman pump wavelengths respectively:

{ } 2 3 3 2 2 3 2 2 2 2 0 0 ( ( ) ' 2 2 6 1 (1 ) 2 ) ( ') ( , ') ( , ') ( ') ( , ') ( , ') ' s
ψ α ψ β ψ β ψ γ ψ ψ ψ ω ψ ψ ψ ψ ψ ψ ∞ ∞ * -Ω + ∂ ∂ ∂ = - + + ∂ ∂ ∂   ∂ - + - +   ∂   + - - + - - ∫ ∫ (6) ( ) 2 r d r r z z α α ψ ψ + ∂ = ∂
The influence of pump depletion on the longitudinal pump power evolution is taken into account by the introduction of a term α d (z), which can be numerically determined by an iterative process based on an average power analysis [START_REF] Min | Efficient formulation of Raman amplifier propagation equations with average power analysis[END_REF]. The generalized split-step Fourier method was used to numerically solve Eq. ( 6) [START_REF]Nonlinear Fiber Optics[END_REF].

Figures 3(a) and 3(b) show the phase and intensity profiles (solid lines) obtained from numerical integration of Eq. ( 6) at a propagation distance of 5.3 km. The intensity profile of the initial pulses retrieved from FROG measurements is displayed in Fig. 3(a) by the dashed line. In Fig. 3(b), the deepest hole is in good agreement with a least-squares fit of a tanh 2 function (circles) with a temporal width of T d = 0.42 ps. We also observe in Fig. 3(b) a brutal phase jump of a value close to π at the minimum of the pulse, which is the signature of a dark (black) soliton.

The spectrum of the pair of pulses displayed in Fig. 3(c) is similar to that resulting from the model with a constant gain (Fig. 2(a)). The frequency separation between two consecutive peaks indicates a repetition rate of 446 GHz instead of 380 GHz obtained from Eq. ( 2). So we can conclude that the simple model with a constant gain (Eq. ( 1)) gives a correct qualitative prediction of the consequences of the interaction between the two similaritons. However, an accurate quantitative comparison between numerical calculations and experimental results requires the use of the extended NLSE (Eq. ( 6)).

Experimental results

Experimental setup

Figure 4 shows a schematic representation of the experimental setup. The 1550 nm input pulse was obtained from a Pritel passively mode-locked fiber laser operating at a repetition rate of 22 MHz. The initial pulse had a nearly Fourier transform-limited intensity profile with a temporal duration of 5 ps (see section 4.2). The pulses emitted from the picosecond laser were sent through a beamsplitter and recombined in a Mach-Zehnder interferometer with a variable delay between the two pulses. A variable attenuator placed in one arm of the interferometer allows us to obtain two pulses with the same energy, whereas a polarizer located at the interferometer output permits to obtain two pulses with the same linear state of polarization. The Raman amplifier consists in 5.3 km of NZ-DSF, whose parameters have been reported in section 2.2.

The fiber was pumped in a backward configuration [START_REF] Finot | Influence of the pumping configuration on the generation of optical similaritons in optical fibers[END_REF] by a continuous wave Raman laser (Keopsys), delivering a power of 1420 mW at 1455 nm. A WDM coupler with high power rating permitted the superposition of pump and signal beams. An optical isolator at 1480 nm was inserted after the polarizer to prevent that part of the residual pump in the Raman amplifier disturbs the operation of the picosecond laser. Input and output pulses were characterized by their optical spectrum and autocorrelation. When possible (according to the pulse energy and recording time), the pulse intensity and phase were characterized using a FROG device based on second harmonic generation [START_REF] Finot | Numerical and experimental study of parabolic pulses generated via Raman amplification in standard optical fibers[END_REF]. The phase and intensity retrieval was performed using the generalized projections algorithm [START_REF] Trebino | Frequency-Resolved Optical Gating : The Measurement of ultrashort laser pulses[END_REF]. The reliability of the FROG measurements was checked using the standard techniques based on comparisons of the independently-measured autocorrelation and spectrum with the FROG trace marginals [START_REF] Trebino | Frequency-Resolved Optical Gating : The Measurement of ultrashort laser pulses[END_REF].

Characterization of a single similariton

We have first characterized using FROG the initial pulses whose retrieved intensity profile is shown in Fig. 5 by diamonds. Figure 5 (circles) also shows the retrieved intensity and chirp profiles of the pulse at the amplifier output in the case of a single initial pulse. As can be seen in Fig. 5, the experimental intensity and chirp profiles are in very good agreement with parabolic and linear fits, respectively (gray dashed line). The output pulse exhibits the expected characteristics of a similariton with a total temporal width 2 T p = 73.5 ps, a chirp slope C p = 0.0118 THz/ps and a peak power of 10 W. The experimental data are also compared with numerical integration of Eq. ( 6) (solid line). As can be observed in Fig. 5, there is a very good agreement between numerical and experimental results, confirming the reliability of the model described by Eq. ( 6). It is noteworthy that the model based on the assumption of a constant gain (Eq. ( 1)) leads to significantly different parameters: 2 T p = 85 ps and C p = 0.0107 THz/ps. The assumption of a constant gain is clearly not the most appropriate to accurately reproduce the characteristics of the similariton as observed experimentally. As a result, we have only used the numerical integrations of Eq. ( 6) in what follows.

Characterization of the interaction between two similaritons

We now characterize the similariton interaction in the case of an initial pair of pulses with an energy of 2.9 pJ and a temporal separation of ∆T = 55.5 ps. We note that, as the dark solitons generated during the similariton interaction are expected to have a temporal width of only T 0 = 0.32 ps, pulse characterization using FROG would require both a large total temporal window (about 200 ps) as well as a high temporal resolution (about 20 fs) to completely define the pulse structure. This would necessitate measurement of a very large number of data points and a corresponding long acquisition time. Unfortunately, owing to the lack of long term stability of our FROG device, we were not able to process a direct FROG characterization of light at the amplifier output. It would also be very difficult to implement a cross-correlation technique similar to that used in Refs. [START_REF] Rothenberg | Dark soliton trains formed by visible pulse collisions in optical fibers[END_REF][START_REF] Emplit | Picosecond steps and dark pulses through nonlinear single mode fibers[END_REF][START_REF] Weiner | Experimental observation of the fundamental dark soliton in optical fibers[END_REF] since our initial pulses are much more broader than the dark solitons we would like to characterize. However, it was possible to characterize the generation of the dark soliton train via interaction of two parabolic pulses by direct measurements of the autocorrelation and spectrum of light at the amplifier output.

Figure 6(a) shows the autocorrelation trace recorded at the center of the overlap region (circles). The experimental trace, which exhibits 410-GHz oscillations, is in good agreement with numerical integrations of the extended NLSE (Eq. ( 6)). The experimental data are also compared with a sinusoidal fit (dotted line). We can observe from Fig. 6(a) a large disagreement between the experimental results and the sinusoidal fit, which is a clear signature of the nonlinear reshaping of the sinusoidal modulation from the onset of the similariton interaction into a train of dark solitons. The experimental output spectrum, displayed in Fig. 6(b), exhibits several sidebands separated by 407 GHz, consistent with the repetition rate measured on the autocorrelation signal. We note that that the small spectral asymmetry, which is also present in the experimental output spectrum of a single similariton (dashed gray line), is mainly due to the asymmetry of the initial pulse spectrum (not shown here), slightly accentuated by the effects of Raman intra-pulse scattering. The experimental spectrum is also in good agreement with the corresponding numerical spectrum, shown in Fig. 3(c), obtained from integration of the extended NLSE (Eq. ( 6)). In view of the excellent agreement between numerical and experimental spectrum and autocorrelation, we can conclude that our results provide clear experimental evidence for the signature of the generation of a train of dark solitons arising from the interaction between two optical similaritons.

Control of the repetition rate of the dark soliton train

In this section, we investigate both numerically and experimentally the influence of several parameters on the repetition rate f d of the dark soliton train generated from the interaction between two similaritons. We will show that the repetition rate can be controlled by specific choices of the initial time separation ∆T, pulse energy U ini or integrated gain G of the amplifier.

The repetition rate has been determined by measuring the spectral separation of the characteristic sidebands of the spectrum of the two similaritons. Figure 7 shows the evolution of the repetition rate of the train of dark solitons as a function of ∆T (a), U ini (b) and G (c). The experimental results (circles) are compared with numerical integrations of the extended NLSE (Eq. ( 6)) (solid line) and of the NLSE with a constant gain (Eq. ( 1)) (dashed line), and with the frequency f s = C p ∆T of the precursory sinusoidal oscillation resulting from linear overlap of the similariton pulses (mixed line). asymptotic regime. On the other hand, the temporal width T p of the similariton increases with the initial energy (see section 2.1), so that the distance at which the similaritons overlap decreases with U ini . In consequence, after their generation dark solitons will propagate with their own velocities |β| on a longer distance, so that the frequency of the periodic train of dark solitons will decrease (see Fig. 2(b)). Another contribution to the decrease of f d with U ini comes from the fact that the depth of the dark soliton increases with U ini (see section 2.4) and the velocity |β| increases with the soliton depth.

Influence of the integrated gain G

We now study the variation of the repetition rate of the dark soliton train as a function of the integrated gain of the Raman amplifier. The initial pulse energy and time delay between the input pulses were fixed to 2.9 pJ and 55.5 ps, respectively. We experimentally change the pump power so that the integrated gain G varies from 50 to 170 (17 to 22.3 dB). We can see from Fig. 7(c) that the experimental values of f d (circles) remain nearly constant. Those results are consistent with the evolution predicted by numerical simulations based on the Eq. ( 6). On the other hand the NLSE with a constant gain (Eq. ( 1)) predicts calculated values of f d which are slightly underestimated as the integrated gain increases. This can be explained by the fact that the effects of pump depletion become more significant as the integrated gain increases.

The very small dependence of f d versus G is rather unexpected since the frequency of the initial beating f s increases with the linear gain g in the fiber (mixed line): f s = C p ∆T with C p directly proportional to g (see section 2.1). So, the global evolution of f d versus G can not be only explained by the dependence of f s . The velocity |β| of dark solitons and the distance over which dark solitons will propagate after their generation play a significant role in the dependence of f d versus G. Indeed, higher the gain is, lower is the propagation distance at which the overlap of the similaritons occurs and higher is δ = f s -f d . Moreover, the increase of the depth of the dark soliton with G induces an increase of the velocity |β| and consequently an increase of δ. Thus the evolution of f d versus G results in a balance between the increases of f s and δ.

Interaction between three similaritons

Finally, we investigate using numerical simulations the case of the interaction between three similaritons of the same wavelength and temporally separated by same quantity ∆T o = 35 ps. The characteristics of the three initial pulses and Raman amplifier are identical to those considered in section 2.2. Figure 8(a) shows that the three initial pulses are transformed into similaritons, characterized by a parabolic intensity profile, before overlapping. Two distinct overlap regions clearly appear at a propagation distance of about 3.3 km. Similarly to the case of the interaction between two similaritons, oscillations appear within each overlap region and further reshape into dark solitons for longer propagation distances. Figure 8(b) shows that at a distance of about 5 km both overlap regions overlap themselves and collisions of dark solitons appear. It is remarkable to observe from Fig. 8(b) that dark solitons are stable against collisions.

To illustrate this property, Fig. 8(c) shows an example of a collision between two dark solitons that occurs at a propagation distance of 5992 m (solid line) within the time window spreading from -6 ps to -2 ps. We clearly observe from Fig. 8(c) that the intensity profiles of the two colliding dark solitons before the collision (dashed line) remain unchanged after the collision (dotted line). On the other hand, Fig. 8(d) shows that a temporal shift of the dark solitons appears during the collision [START_REF] Thurston | Collisions of dark solitons in optical fibers[END_REF].

Conclusions

In conclusion, we have analyzed the dynamics of the self-similar amplification of a pair of timedelayed identical pulses of the same wavelength in a Raman amplifier. During the amplification process, due to the exponential scaling of their temporal duration, each pulse broadens, so that, after a given distance of propagation, the two pulses overlap and interact. Owing to their compact nature, similaritons are clearly unaffected by their neighbor outside the overlap region.

Inside the overlap region, the superposition of the linearly-chirped pulses creates an oscillation.

Due to the combination of non-linearity and gain, this oscillation adiabatically evolves into a finite-extent high-repetition rate train of dark solitons. Our experimental results are in good agreement with theoretical calculations and confirm the generation of dark-soliton train at high repetition rates. We have also investigated the evolutions of the repetition rate of the dark solitons as a function of the initial pulse temporal separation, pulse energy and integrated gain.

These evolutions are both affected by the intrinsic properties of the similaritons and the relative velocities of the dark solitons. Finally, we have shown that stable collisions of dark solitons appear during the interaction of three similaritons.

Although our experimental results have been obtained using Raman amplification, our conclusions can be extended to other amplification processes as long as we restrict ourselves to normal dispersion, and we are the regime of an adiabatic amplification regime. For example, dark solitons could also be generated via similariton interaction in ultra-long erbium-doped fibers with controlled dispersion properties [START_REF] Nakazawa | Ultralong dispersion-shifted erbium-doped fiber amplifier and its application to soliton transmission[END_REF].

Biographies

Figure captions

  1(c) confirms that after 5.3 km of propagation, the intensity and phase profiles of the dip (1) (solid line) are in very good agreement with the corresponding profiles of a black soliton (circles) with a temporal width T d = 385 fs, a pulse depth P d = 15.6 W and a phase jump of π.

Figure 2 (

 2 Figure 2(b) represents the evolution of the contour plot of the normalized intensity profile

Christophe

  Finot was born in Chatillon sur Seine, France, in 1978. He received the DEA degree and the engineer degree from the Ecole Superieure d'Optique, Orsay, France in 2002. In 2002, he began a Ph D. at the University of Burgundy, Dijon on optical parabolic pulses. He has published over 20 contributions in journals and conference proceedings. John M. Dudley John Dudley was born in Otahuhu, New Zealand in 1966. He received the B.Sc and Ph.D. degrees in physics from the University of Auckland, New Zealand, in 1987 and 1992 respectively. In 1992 and 1993, he carried out postdoctoral research studying femtosecond selfmodelocked lasers and optical parametric oscillators at the University of St Andrews, Scotland. In 1994, he took up a lecturing position at the University of Auckland where he studied the physics of ultrashort pulse generation and measurement, and nonlinear fiber optics. In 2000, he was appointed a Professor at the University of Franche-Comté in Besançon, France where he is currently carrying out research in ultrafast optics, high speed optoelectronics and optical communications. He has published over 170 contributions in journals, books and conference proceedings. Prof. Dudley is a Member of the IEEE and the Optical Society of America. He is a member of the Institut Universitaire de France (IUF). Guy Millot was born in Alligny-en-Morvan, France, in 1960. He received the Ph.D. degree on laser Raman spectroscopy in gases, from the University of Burgundy, Dijon, in 1986. In 1987, he carried out post-doctoral research on time-resolved infrared double resonance experiments in methane and silane at the Massachusetts Institute of Technology. Since 1988, he has been an Assistant Professor and since 1994, a Full Professor in the Physics Department at the University of Burgundy. From 1988 to 1995, he carried out research on nonlinear spectroscopy, molecular structures, time-resolved Raman-infrared double resonance experiments, collisional effects on Raman line-shapes for temperature and concentration measurements in combustion media. Since 1995, his research interests are focused on nonlinear effects in optical fibers, modulational instabilities, solitons, generation, propagation and characterization of optical pulse trains at high repetition rates, stimulated Raman scattering, frequency conversion and applications in optical communications. He has published over 180 contributions in journals, books and conference proceedings. G. Millot was a member of the Institut Universitaire de France (IUF) in the period 2000-2005. He obtained the Silver Medal of the CNRS in 2005.

Fig. 1 :

 1 Fig. 1: (a) Numerical calculations of the intensity profiles of a single pulse (circles) and a pair

Fig. 2 :

 2 Fig. 2: (a) Calculated spectra of a single pulse (dashed gray line) and of the pair of pulses (solid

Fig. 3 :

 3 Fig. 3: Numerical simulations based on the extended NLSE of the propagation of the pair of

Fig. 4 :

 4 Fig. 4: Experimental setup used for the generation of the pair of pulses and for the self-similar

Fig. 5 :

 5 Fig. 5: Intensity and chirp profiles retrieved from FROG measurements for the initial pulse

Fig. 6 :

 6 Fig. 6: (a) Autocorrelations of the pair of pulses at z = 5.3 km: experimental results (circles),

Fig. 7 :

 7 Fig. 7: Evolutions of the repetition rate of the dark soliton train as a function of (a) the time

Fig. 8 :

 8 Fig. 8: Propagation of three initial pulses in the Raman amplifier. (a) Intensity profiles at

Finot, Dudley and MillotFig.1

Finot, Dudley and MillotFig.3

Finot, Dudley and MillotFig.4

Finot, Dudley and MillotFig.5

Finot, Dudley and MillotFig.6

Finot, Dudley and MillotFig.7

Finot, Dudley and MillotFig.8

Acknowledgments

The authors would like to thank Stéphane Pitois and Cyril Billet for fruitful discussions. This work has been supported by the Fond National pour la Science (FNS) under contract "ACI-Photonique PH43", the Centre National de la Recherche Scientifique (CNRS), and the Conseil Régional de Bourgogne.

Influence of the initial time separation ∆T

In this study the initial pulse energy and integrated gain were fixed to 2.9 pJ and 22.3 dB, respectively. Figure 7(a) shows that the repetition rate f d is a quasi-linear function of ∆T. The experimental results (circles) are both in good agreement with numerical simulations based on Eq. (6) (solid line) and Eq. (1) (dotted line). As expected, the frequency f d is lower than f s due to the transverse velocity |β| of the dark solitons. The difference δ = f s -f d decreases with ∆T : for ∆T = 47 ps, δ = 270 GHz, whereas for ∆T = 70 ps, δ is only 180 GHz. Since the propagation distance at which the pulses begin to overlap increases with ∆T, dark solitons are generated after a longer distance in the amplifier as ∆T increases. In consequence, after their generation dark solitons will propagate with their own velocities |β| on a shorter distance, so that δ will be lower (see Fig. 2(b)). The results of Fig. 7(a) clearly demonstrate that the repetition rate f d of the dark soliton train can be precisely adjusted just by changing the time separation ∆T between the initial pulses.

Influence of the initial pulse energy U ini

With the time separation between the initial pulses being now fixed to 55.5 ps, we now change the value the initial pulse energy U ini . The experimental values (circles) shown in Fig. 7(b) show that f d only slightly decreases with U ini . The experimental results are in good agreement with the numerical simulations of both models (solid and dotted lines). The global behavior of f d versus U ini can be explained by similar considerations that those used in section 5.1. We recall that the frequency f s = C p ∆T of the oscillation created at the beginning of the similariton overlap does not depend on the pulse energy. Indeed the chirp parameter C p is independent of U ini in the


Jérôme Chevalier

Christian Olagnon

Gilbert Fantozzi

Study of the residual stress field around Vickers indentations in a 3Y-TZP

niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Indentation flaws are widely used for the mechanical behaviour characterization of dense ceramics. They present several advantages: (i) a very well defined geometry, (ii) a precise knowledge of crack para meters due to numerous fracture mechanics analyses [1][2][START_REF] Lawn | Fracture of Brittle Solids[END_REF], and (iii) a high reproducibility of size and shape. Indentation flaws, and especially Vickers flaws are therefore conveniently used to induce critical flaws in small 4-point bending specimens to characterize the toughness or the sub-critical crack growth behaviour of ceramic materials.

The description of the material around a Vickers indentation was shown to be complex. It is generally considered as a hemispherical plastic zone beneath the indenter, surrounded by the elastic body. The elastic/ plastic contact generates residual stresses around the indentation, acting as an additional crack driving force during bending (or any other applied stress) tests.

Theoretical models and experimental investigations have allowed the understanding of the impact of the residual stress on this additional driving force. It is now well established [START_REF] Lawn | Fracture of Brittle Solids[END_REF] that the residual stresses lead to a residual stress intensity factor on each crack in the form of: (1) where a is the crack size induced by the indentation, P is the load applied to the indenter and x is a con stant of the material which depends on the Y oung's modulus to hardness ratio. The value of x has been the subject of many hypotheses to evaluate the toughness.

Recently, Pajares and co-workers [START_REF] Pajares | Pro ceedings of the Second Euro Ceramics[END_REF][START_REF] Pajares | Proceedings of the Third Euro Ceramics[END_REF] and Kurth et al. [START_REF] Kurth | Proceedings of the Third Euro Ceramics[END_REF] have developed a novel method to determine the toughness using the stable propagation of indenta tion cracks. An important advantage of this method is to directly measure the x parameter without _ restrictive assumption. This allows one to take into account the residual crack driving force and to obtain accurate values of the toughness.

For sub-critical crack growth measurements (static fatigue, dynamic fatigue and cyclic fatigue), in which indentation flaws are introduced, the analysis gener ally does not take into account this residual stress intensity factor. It is however clear that owing to this residual stress around the indent no accurate lifetimes and consequently no relevant propagation law can be obtained [START_REF] Liu | [END_REF].

An established method to reduce this residual stress intensity factor is to recover it by a thermal treatment. In the case of ceramics that do not exhibit a phase transformation (e.g. Al 2 0 3 , Si 3 N4, SiC, mullite, etc.) the recovery is due to both dislocation movement and/or grain boundary sliding that can be assisted by diffusion. Therefore, long duration and high temper ature annealings are required to suppress or reduce this residual stress. The case of zirconia-based ceram ics is different since transformation plasticity occurs and acts as an additional plastic deformation con tribution. Transmission electron microscopy (TEM) characterization of an indented polycrystalline Y-PSZ [8] has shown that the microstructure around the indentation consists of a transformed zone and another region having a high dislocation density. Experiments on a yttrium-stabilized zirconia single crystal [9] have shown that many slip systems are activated, but it has also been shown in a polycrystal line 3Y-TZP that a large transformed zone was pres ent after indentation.

Pajares et al. [START_REF] Pajares | Proceedings of the Third Euro Ceramics[END_REF] have clearly demonstrated that short times and relatively low temperatures ( < 400 °C) can reduce the residual stress intensity factor. I t has however not been proved that the resid ual stress was actually annealed for these short time ageings and no correlation with microstructure has been made. On the other hand, Wang and Stevens [10] have demonstrated that a 1250 °C for 20 min heat treatment altered the indentation prints. They con sidered that it was not only due to annealing of the residual stress but that crack healing could occur by diffusion mass tran s port.

The aim of this paper is to better understand the effect of annealing on the residual stress intensity factor and on the microstructure modification and to correlate them. The aim is also to determine a heat treatment that could reduce the residual stress inten sity factor without healing the crack.

For this purpose, different methods have been used to characterize the stress field around indentation cracks in 3Y-TZP materials. The method proposed by Pajares et al. [START_REF] Pajares | Pro ceedings of the Second Euro Ceramics[END_REF] was used to determine K1c and x after heat treatments between 20 to 1200 °C. The measurement of the apparent toughness in a fast frac ture bending test was conducted after the same treat ments. The residual stress has been directly measured by a micro-indentation method [11].

Experimental procedure 2.1. Material

The material studied in this work was a 3Y-TZP sintered at 1450 °C. A density of 6.06 gcm-3 was measured by Archimedes method and the average grain size was 0.58 µm. Bending bars of dimensions 3 mm x 4 mm x 40 mm were ground and polished down to 1 µm grit. The specimens were subsequently heat treated at 1200 °C for 1/2 h. Indentations were subsequently made on one face of the specimens. The post-indentation treatments were conducted at various temperatures from 20 up to 1200 °C for 12 min.

Controlled propagation of indentation

cracks [START_REF] Pajares | Proceedings of the Third Euro Ceramics[END_REF][START_REF] Kurth | Proceedings of the Third Euro Ceramics[END_REF]] Indentation flaws were made on polished 4-point bending bars by a Vickers indenter by applying a load of 100 N. Indentation cracks were shown to be ap proximately halfpenny-shaped and the length meas ured by an optical microscope was found to be of 144 ±2 µm.

When loading these bars in bending the cracks are submitted to a two-fold stress field: (i) the applied stress cr• PP ' and (ii) the residual stress cr" They are therefore submitted to the total driving stress intensity factor K1tot of the form:

K1 to t = K + K = xPa-3 12 + Ycr a 112 (2) r app app
where Y is a parameter that depends on the type of crack and loading. Its value for a halfpenny crack has been evaluated by Smith et al. [12]. The crack propa gation occurs if the total stress intensity factor K1tot matches the material toughness K1c, but a stable propagation is expected as long as the condition:

d K1to 1 0 --< da (3)
is fulfilled. Under these conditions and the assumption of a constant Y parameter, Equation 2 can be re arranged to a linear form as:

Y 112 _ K _ K p -3; 2 CT app a - app - le -X a (4)
Therefore by plotting the applied stress intensity fac tor versus xP a-3 12 one can obtain the two parameters X and K1c. This can be achieved by recording the applied stress and the crack length during a stable propagation experiment. For this purpose, indented specimens were loaded at high loading rate (8000 N si ) to prevent sub-criti cal crack growth, and subsequently unloaded at the same rate. The crack length was further measured by optical microscopy with a precision of ± 2 µm. The same procedure was repeated several times by increas ing the load until specimen failure due to crack insta bility. This measurement has been achieved both on as-indented specimens and after annealing.

Apparent toughness measurement

The toughness is generally measured by recording the fracture stress on a specimen containing a defect of known size and shape, by applying the standard formula:

K1c = Ycra 112 (5)
This can be done in bending on an indented specimen after measuring the crack length. However, since the above equation does not take into account the influ ence of the residual stress, only an apparent toughness lower than the real value will be measured. The apparent toughness has been measured on a series of indented specimens tested in 4-point bending after various annealing treatments (as defined in Section 2.1). The displacement rate was 5 mm min-1 , and the apparent critical stress intensity factor evaluated by Equation 5with Y = 1.34 [12] and by considering the initial crack length before the test, the sub-critical crack growth being considered as negligible. Three specimens were used for each heat treatment. The measured apparent toughness was compared to the real value obtained from Section 2.2.

Stress measurement by micro-indentation method

Indentation flaws can be used to analyse residual stress fields in materials. The basis of the method is that indentation induced cracks are submitted to a two-fold stress field: the stress due to the indentation itself and the residual stress CTres• Recalling that the stable crack propagation stops at K1t01 = K1c (for dK1t01/da < 0), the probe indentation crack length will be given by the equation:

K1c = Ycrresa112 + xP'a-312 (6)
where P' represents the probe indentation load. The residual stress to analyse can therefore be expressed as:

O'r es = K1c -xP'a-3;2 Ya11 2 (7)
The toughness can be expressed as a function of the crack length obtained in a stress free material, a0, as K1c = xP a0 3 12 which can be substituted in Equation 7 leading to: (8) This method proposed by Chantikul et al. [13] has recently been used to characterize the stress field around an indentation (macro) in glass [START_REF] Chevalier | Proceedings of the 3rd European Ceramic Society Conference[END_REF] by using small indentations (micro) as a probe.

In this work a similar procedure has been achieved to investigate the evolution of the residual stress as a function of the heat treatment.

Macro-indentations were made on a polished specimen at a load of 625 N and several annealing treatments were conducted. Micro-indentation probes were subsequently made at a load of 50 N around the macro-indentation and the residual stress, considered as constant has been evaluated from Equation 8.

Previous analyses [START_REF] Lawn | Fracture of Brittle Solids[END_REF] of the residual stresses around an indentation have shown a central sym metry at the surface characterized by the two principal stresses, radial ( crR) and tangential ( crT ). By carefully aligning the probe indentations along the radial direc tion (cf. Fig. 1) the two components could be directly obtained from the different crack lengths. This proced ure requires the knowledge of the toughness which was selected as Kic = 7.2MPa m1 12 (value presented in the Sections 2.2 and 2.3).

3. Results and discussion 3.1. Stable propagation of indentation cracks Fig. 2 shows the plot of (Y cr a 112) as a function of (Pa -312) for different bending bars. Different behav iours are observed.

In the case of as-indented specimens, the curve is linear indicating that the stress field due to indenta tion follows the expected form given by Equation 1. It can also be considered that Y is constant over the crack length investigated. This graph also shows that the onset of crack propagation begins for a threshold stress intensity factor Kapp= 1.9 MPa m11 2 , for an ini tial measured crack length of aim = 144 µm. This offset value which indicates that the starting total stress intensity factor (K1t01) is lower than the toughness (K1c ) will be explained further in the paper. The fitting of the curve by a straight line gives: K1c = 7.25 MPam11 2 and x = 0.0925.

In the case of the specimens annealed up to 200 °C, a stable crack propagation could be obtained. This indicated that the residual stress was still present in the specimen. The data define a straight line which shows that the residual component can again be de scribed by Equation 1, but with a lower slope when expressed by Equation 4. It is also important to note that the stress intensity factor offset is more prominent with a value of Koffset = 4 MPa m1 12 . The fitting of the data by a straight line leads to K1c = 7.25 MPa m 11 2 and x = 0.0626. There is therefore an apparent stress relaxation suggested by the reduction of x and the increase of Koffset•

An annealing treatment at room temperature for 7 days plays a similar role as a 200 °C heat treatment. This suggests an apparent stress relaxation around the indentation at ambient temperature.

The behaviour of the specimens annealed at a tem perature in the range 400 to 800 °C is different. No stable propagation could be obtained, but it can be noted that failure is obtained at a stress intensity factor lower than the fracture toughness. The recorded offset values of the stress intensity factor were 6.0 MPa m 1 12 at 400 and 800 °C. Theoretical analyses show that the initial crack length ai obtained just after indentation is given by: (9)

The extrapolation of the graph Y cr a 112 = f(Pa -3 12 ) for null stress values theoretically leads to the value of a; given by Equation 9which is here equal to 116 µm, i.e. lower than the experimental measurement. Anstis et al. [2] have pointed out that Equation 9is not valid if sub-critical crack growth occurs after indentation. In previous studies [START_REF] Chevalier | Proceedings of the 3rd European Ceramic Society Conference[END_REF][START_REF] Knechtel | [END_REF], it has been shown that Y-TZP was sensitive to sub-critical crack growth in air. Thus the presence of the Koffset at room temper ature could be explained by the sub-critical crack growth which would lead to a longer initial crack submitted to a stress intensity factor due to residual stress lower than K1c. The condition of controlled crack propagation would therefore not be obtained at the onset of loading. It could also be argued that a partial stress relaxation occurs at room temperature, which also would lead to an offset.

The first assumption has been verified by indenting a specimen in silicon oil which effectively prevents extensive sub-critical crack growth. The measurement of this crack after indentation gave a value of 114 µm. This proves that the cause of the offset at room tem perature is the immediate crack propagation sub sequent to indentation.

For samples heat treated between 400 and 800 °C, no stable propagation occurs, even with careful load ing. However, the residual stress is not fully relaxed since the offset stress intensity factor was below the toughness. This apparent contradiction already men tioned by Pajares et al. [START_REF] Pajares | Proceedings of the Third Euro Ceramics[END_REF] can be explained as follows:

It is first assumed that the residual stress intensity factor is still of the form given by Equation 1. For each heat treatment temper, ature, x can be determined either by linear regression for room temperature and 200 °C or from the measurement of the Koffset for temperatures of 400 and 800 °C. In this last case, the x value is obtained from the offset value (correspond ing to unstable crack propagation) and from Equation 4. The values are reported in Table 1.

The crack stability analysis in this type of test shows that an instability point (i.e. dK1t01/da ;::o 0 and K1to1 = K1c) is obtained for a critical value of crack length ac given by [2]: (10) By replacing the different values of x corresponding to the different temperature treatments one can evaluate I). It appears that for the temper atures of 400 and 800 °C these critical values are smaller than the initial measured value aim• This ex plains why no stable propagation could be obtained although a residual stress is still present. This shows that the presence of a residual stress is not a sufficient condition for having a stable crack propagation in this test, but that the initial crack length obtained after indentation must be smaller than the critical crack length ac.

Apparent toughness measurements

Fig. 3 represents the measured apparent toughness as a function of the heat treatment temperature. The shape of the curve is rather complicated and two plateaus are observed, respectively, between 400 and 800 °C, and between 1000 and 1200 °C. It immediately appears that for temperatures lower than 1000 °C the apparent toughness is lower than the toughness indicating that the residual stress has not been relaxed. On the other hand, heat treatments up to 1400 °C lead to an apparent toughness higher than the real tough ness. As already mentioned by Wang and Stevens [10], this phenomenon can be due to crack healing by diffusion because the temperature is close to the sintering temperature. This can be confirmed by microscopic observations which show a crack length decrease.

Heat treatments between 1000 and 1200 °C seem however relevant for effective stress relaxation. The understanding of the behaviour has been achieved by optical microscopy in Normarsky interferences which reveals the surface displacement due to indentation (Fig. 4). It should be noted that heat treatments result in a surface displacement around the indents. Behrens et al. [START_REF] Behrens | [END_REF] have shown that this displacement is a con sequence of a stress assisted tetragonal to monoclinic (t � m) transformation around the indent which re sults in a volume increase.

Up to 400 °C, an increase in the monoclinic content around the indentation is observed.

The first two parts up to 800 °C of the curve in Fig. 3 can therefore be explained by this t � m trans formation. Indeed, it might be supposed that the --------------------.-Temperature ( °C) t --+ m transformation leads to a compressive stress that hinders the initial residual stress. It can also be assumed that this transformation does not lead to a real stress relaxation, but only acts as a super imposed compressive.stress. This will be addressed in the following section. The shape of the curve between 800 and 1200 °C can not be explained by the transformation induced stress at the surface since the monoclinic content decreases. The possible mechanisms will be discussed in the fol lowing section. No, te however that a real stress relax ation occurs.in this temperature range because the apparent toughness matches the real toughness value and the surface monoclinic content is negligible.

Residual stress measurements

Fig. 5 shows the stress profiles around indentations. It appears that the radial stresses are compressive while the tangential stresses are tensile components, in agreement with theoretical analyses. It should be noted that the compressive component is higher but does not contribute to the crack advance. The main limit of this method is the impossibility to measure stresses in the region very near to the indentation. Indeed, in this case, the tangential secondary cracks are closed and one of the radial secondary cracks is connected to the macro-indentation, which prevents any measurement (cf. Fig. 6a).

The stress profiles for specimens annealed between 200 and 800 °C are also represented in Fig. 5. The radial component of the stress is unchanged in the explored range. Note also that measurements of aT were not possible in the high monoclinic content region. Tangential stresses have been modified in this range 9f heat treatments. It is important to r�call that this corresponds to heat treatments that induced a strong t --+ m transformation that led to an additional compressive stress. The tangential stress is therefore the sum of two components, a tensile stress resulting from the indentation and a compressive stress resulting from the temperature induced phase transformation. This can be confirmed by noting that the compressive stress is only present in a limited area around the centre of the indent. Outside this area the tangential component is the same as it was before the heat treatment. This can also be illustrated by Fig. 6b where it is observed that an indentation made in the compressive region induced no crack neither in the radial nor in the tangential direction.

The increase of apparent K1c with temperature in the range of 200 to 600 °C in the previous experiments is therefore clearly due to the increase of this addi tional compressive stress and it is definitely not a re laxation phenomenon.

The stress measurement profiles of specimens heat treated at temperatures over 800 °C are also repre sented in Fig. 5. It is observed that the compressive radial stress due to the t � m transformation has dis appeared. In this temperature range, the tetragonal phase is stable. Both the radial and the tangential components of the residual stress have been reduced. The tangential component is even fully relaxed be tween 1000 and 1200 °C. This therefore explains why the apparent toughness is equal to the real toughness since there is no residual stress. This is illustrated in Fig. 6c.

The mechanism responsible for this real relaxation can be either due to dislocation movement or to m � t transformation. Indeed, Pajares et al. [8] have ob served on an indented Y-TZP that the microstructure associated with indentation consists of a large hemi spherical transformed monoclinic zone UJ:/-der the in dentation print, and a region containing a high density of dislocations. Behrens et al. [START_REF] Behrens | [END_REF] have shown on a 3Y-TZP that a significant t � m transformation occurs below the indenter after indentation. Stress relaxation can therefore be due to the reversible m � t in the range 1000 to 1200 °C, or to the dislocation movement as in other non-transforming ceramics. It can be noted that at 1200 °C and for the relatively limited treatment ageing times (12 min) conducted in this work, dislocation movements are probably lim ited [9] and the relaxation is probably due to the m � t transformation below the indent. Furthermore the diminution of the amount of monoclinic phase at the surface corresponds to the phase diagram trans formation.

Conclusion

An analysis of indented Y-TZP specimens that have been heat treated for 12 min over a wide range of temperatures has been carried out. Three different tests have been conducted: apparent toughness measurement fast fracture, stable crack propagation and measurement of the residual stress by indentation method.

Experiments conducted on as-indented samples have shown that the residual stress intensity factor Kr could be expressed by the relation Kr = xP a-312, in which the constant x has been determined.

Heat treatments up to 800 °C lead to an apparent stress relaxation which actually is due to the appear ance of a t � m transformation induced compressive stress near the indentation. The resulting stress field leads to a decrease of the residual stress intensity factor and therefore to an increase of the apparent toughness.

Between 800 and 1200 °C, the reversible m � t transformation and possibly dislocation movements lead to real stress relaxation.

Above 1200 °C, crack healing due to diffusion oc curs and modifies the crack pattern.

Finally an annealing treatment at 1200 °C for 12 min leads to a total stress relaxation around the indent without crack healing. Such a treatment is therefore relevant to prepare indented specimens in Y-TZP materials without residual stress for any subsequent mechanical tests such as static or cyclic fatigue.

Figure J

 J Figure J Schematic drawing showing the procedure conducted to evaluate the stress field around a macro-indentation, by the micro indentation technique.

  8 .---�.,---�-r--�-.--�--..-�---r�--...�---.�---.

Figure 2

 2 Figure 2 Plot of the applied stress intensity fa ctor as a function of Pa-312 during a controlled propagation of an indentation flaw on a bending bar after several treatments:(•) as-indented; (0) annealed at 200 °C, ( +) annealed at 400 °C; ( *) annealed at 600 °C (L) annealed at room temperature for 1 week.

Figure 3

 3 Figure 3 Apparent toughness as a function of the annealing temper ature, measured in bending using the crack size measured just after indentation.

.Figure 4

 4 Figure 4 Optical micrograph in Normasky interference contrast of Vickers indentations after different heat treatment temperatures: (a) no treatment, just after indentation, (b) 200 °C, (c) 400 °C, (d) 800 °C, (e) 1000 °C, and (f) 1200 °C. Bar scale represents 50 µm.

Figure 5 Figure 6

 56 Figure 5 Surface tangential (a) and (b), and radial (c) and (d) stresses as a function of the distance from the centre of an indentation after different annealing treatments. Key: + as-indented; 0 200 °C; D 400 °C; x , * 600 °C; esoo °C; • 1OOO°C; D. 1200 °C.

TABLE I

 I Values of the residual stress constant and the critical crack length for both stable and unstable crack propagation

		x	ac(µm)
	Ambient	0.0925	296
	200°C 400°C 800°C	0.0626 0.023 0.023	229 117

117

the critical crack length leading to unstable crack propagation (Table

Acknowledgement

The authors wish to thank Dr. Bernard Cales and Ceramiques Techniques Desmarquest (France) for supplying the material and for their financial support.