Markku Ylönen
email: markku.ylonen@tut.fi

Pentti Saarenrinne

Juha Miettinen

Jean-Pierre Franc

Marc Fivel

Cavitation Bubble Collapse Monitoring by Acoustic Emission in Laboratory Testing 1

Keywords: cavitation impact detection, acoustic emission, cavitation intensity

, where an exponential distribution of bubble collapse amplitudes was found. The results of this study prove acoustic emission as a direct and non-intrusive method that can be used to monitor cavitation impacts from outside of the cavitation field.

Introduction

The impact load induced by a cavitation bubble collapsing near a solid boundary has been studied by multiple methods, both experimental and computational. In a cavitating flow, the impact load determines material damage in a boundary caused by a single bubble or bubble cloud collapse. Another important factor in material damage is the impact frequency, as cavitation erosion tends to be a cumulative process [START_REF] Franc | Incubation Time and Cavitation Erosion Rate of Work-Hardening Materials[END_REF][START_REF] Franc | Pitting and Incubation period[END_REF][START_REF] Berchiche | A Cavitation Erosion Model for Ductile Materials[END_REF]. The impact distribution that combines the impact frequency and amplitude of the cavitation impact loads is essential in determining the cavitation intensity of a flow.

Cavitation impact loads have been measured by various methods. Franc et al. [START_REF] Franc | Impact Load Measurements in an Erosive Cavitating Flow[END_REF] measured the impact loads in the PREVERO cavitation tunnel with conventional pressure sensors flush-mounted in the cavitation closure region. Hujer et al. [START_REF] Hujer | Impact load measurements with a PVDF pressure sensor in an erosive cavitating flow[END_REF] used PVDF pressure sensors, also flush-mounted in the same tunnel. Both of them observed exponential impact distributions. Hattori et al. [START_REF] Hattori | Prediction method for cavitation erosion based on measurement of bubble collapse impact loads[END_REF] studied the impact pressures in an ASTM G-32 vibratory device and Okada et al. [START_REF] Okada | Relation between impact load and the damage produced by cavitation bubble collapse[END_REF] used the same device to calibrate pressure sensors for a Venturi nozzle test. In the vibratory test, the impact distribution also follows an exponential law. Franc et al. [START_REF] Franc | Material and velocity effects on cavitation erosion pitting[END_REF] also observed the pits formed by cavitation impacts and they observed an exponential distribution in the pit size distribution. The pit shape factor or the ratio between pit depth and pit diameter increases with increasing cavitation intensity, meaning that larger impacts lead to deeper pits [START_REF] Roy | Cavitation erosion: Using the material as a pressure sensor[END_REF].

Several authors have studied acoustic emission (AE) as a method to detect and characterize cavitation and cavitation erosion. Boorsma and Fitzsimmons [START_REF] Boorsma | Quantification of Cavitation Impacts with Acoustic Emissions Techniques[END_REF] created a cavitation monitoring method for ship rudders and propellers. Yongyong and Zaiyang [START_REF] Yongyong | Experimental Research on Cavitation Erosion Detection Based on Acoustic Emission Technique[END_REF] connected the AE-event energy to mass loss in an ASTM-G32 vibratory cavitation apparatus. Schmidt et al [13; 14] worked on a cavitation detection system based in AE on a prototype Kaplan turbine. They discovered that with properly placed sensors, cavitation leads to increase in AE root mean squared voltage value, event energy and fluctuation of both. Van Rijsbergen et al. [START_REF] Van Rijsbergen | High-Speed Video Observations and Acoustic-Impact Measurements on a NACA[END_REF] found that acoustic emission sensors in direct contact with a hydrofoil capture bubble collapses near the foil surface and with the cavitation impact towards the foil, but not those that occur in the flow far from the foil. These studies encourage further development of cavitation monitoring by AE.

In this study, the same cavitation tunnel as used by Franc et al. [START_REF] Franc | Impact Load Measurements in an Erosive Cavitating Flow[END_REF] and Hujer et al. [START_REF] Hujer | Impact load measurements with a PVDF pressure sensor in an erosive cavitating flow[END_REF] was fitted with acoustic emission sensors. The difference between previous studies with the tunnel in question is that the AE sensors are placed outside of the liquid flow by placing them in the backside of the sample. Voltage peak values corresponding to individual cavitation events were classified by their quantity and voltage. The goal was to produce similar distributions for two different operating conditions from the acoustic emission voltage peak values and to correlate these distributions with those of previous works.

Experiments

The experiments were carried out in the PREVERO high-speed cavitation tunnel in LEGI laboratory. The tunnel circulates water through a radially diverging test section. Water comes in to the test section through a Ø 16 mm inlet nozzle and the flow stagnates in the middle of the test section where the sample is located and continues to diverge radially in a 2.5 mm thick channel. The samples are 20 mm thick cylindrical disks with a 100 mm diameter and placed so that the sample center is in the middle of the test section. Cavitation inception is located in the beginning of the radial section and cavitation closure is located between 21 and 26 mm radius of the disk when operating with the typical cavitation number σ = 0.9. Cavitation number in PREVERO is defined in equation 1:

𝜎 = 𝑃 𝑑 -𝑃 𝑣 𝑃 𝑢 -𝑃 𝑑 (1)
where Pd is the pressure after the test section, Pu is the pressure before the test section and Pv is the liquid vapor pressure.

The acoustic emission sensors were fitted to the sample using a waveguide that is fixed with a screw thread to the sample. Figure 1(a) shows the sample disk flush mounted to the sample holder and the AE sensors fitted to the sample from behind. Figure 1(b) presents the test section without the sample and the sample holder. The sample and the holder were fastened to the test section so that from the inlet nozzle, the flow radially diverges to a 2.5 mm thick channel. Cavitation inception occurs at the outlet of the cylindrical nozzle, as the cross section area of the flow drops to 62.5 % of that of the inlet nozzle. Cavitation closure occurs further downstream as the cross section area and therefore the static pressure of the flow increase. Tunnel downstream tank is pressurized by nitrogen and upstream section by a pump linked to a frequency transformer. Downstream pressure varies from ambient pressure to around 3 MPa and the maximum upstream pressure is 4 MPa. The cavitation tunnel is presented in more detail in [1; 4].

The acoustic emission setup was a PAC PCI-2 two channel acquisition card fitted with PAC R15 and D9203b sensors and 20/40/60 preamplifiers. The R15 sensor is a resonance type sensor with a resonance frequency of 160 kHz. A band-pass filter from 100 kHz to 400 kHz was used. The D9203b sensor is a broadband sensor with a frequency range from 100 kHz to 900 kHz. In this study, only the data acquired with the R15 sensor were analyzed. The samples were made of a stainless steel used in Francis turbines and the waveguides were made of steel.

In this study, two samples were mirror polished with successive diamond pastes and a colloidal silica suspension. One sample was subjected to cavitation erosion for 2 minutes with a 4 MPa upstream pressure and the other for 6 minutes with a 2 MPa upstream pressure. The 4 MPa and the 2 MPa upstream pressures correspond to 89.4 m/s and 63.2 m/s cavity reference velocities, respectively. Cavity reference velocity is defined as the velocity in the test section where pressure is assumed to be the vapor pressure. A more detailed explanation of cavity reference velocity is in [START_REF] Franc | Incubation Time and Cavitation Erosion Rate of Work-Hardening Materials[END_REF]. Pits with this type of exposure covered roughly 10 % of the surface of the cavitation closure area. The cavitation impacts were thus assumed to be hitting virgin material most of the time. Acoustic emission waveforms were acquired with a sampling frequency of 5 MHz for the full duration of the tests.

Peak value distribution by an envelope function

Impacts in the cavitation sample surface were observed in the AE waveform as quickly rising voltage peaks that diminish exponentially. The AE waveforms were analyzed in time-voltage space. Wave propagation in the sample and the waveguide may affect the waveforms, but this effect is not considered in detail in this study. The frequency content of each cavitation burst or AE-event resembles that of the sensor frequency response, meaning that the impacts provoke sensor resonance. This means that the events are mostly short duration impacts compared to the sensor time scales. It is assumed that each voltage peak value in the waveform, with its resonance effects, corresponds to one cavitation impact towards the sample surface. The length of a cavitation bubble collapse is in the range of some microseconds up to some tens of microseconds [2; 16]. Through waveform observations, impact overlapping seems not to be significant.

With the assumption that each measured maximum in the voltage peak value corresponds to a single cavitation event, there is a need to filter out the sensor resonance effects when peak counting is applied. In most cases in this study, the acoustic emission waveform contains more or less isolated events with breaks between them. Each event has a distinct maximum or sometimes two or more maximums. To negate the resonance effects, an envelope function is fitted to the waveform. As the signal is approximately symmetric around zero volts, the absolute value of the signal is calculated. After this, the envelope function was fitted to the waveform. The envelope was calculated by spline interpolation over local maximums. The minimum distance between local peaks was set to 16 µs, which is about five times the distance between peaks resulting from sensor resonance. This value was found to be suitable through trial and error method. Figure 2 presents a typical AE waveform sample fitted with an envelope function. As observed in figure 2, the envelope follows the waveform, properly addressing the maximum values of each event.

If the event contains two distinct peaks, it is assumed that two events are overlapping and both are taken into account.

In the area outside the events, the envelope function may create false peaks due to noise and the formulation of the function. This effect is negated in the results, as events falling below a certain threshold are considered as either static noise or bubble collapses too weak to be erosive.

Results

The peak value distributions for both the 2 MPa and the 4 MPa upstream pressure tests are presented in figure 3. The distributions are presented as cumulative so that each rate value represents the rate of voltage peak values larger than the corresponding voltage. The bin size for peak counting was 0.02 V. The rate was expressed in [START_REF] Hujer | Impact load measurements with a PVDF pressure sensor in an erosive cavitating flow[END_REF] and [START_REF] Franc | Incubation Time and Cavitation Erosion Rate of Work-Hardening Materials[END_REF] in counts per second per area, with the area being the sensor active area. In this study, the sensor captures all events occurring in the sample, so the active area cannot be properly defined.

*Corresponding Author, Markku Ylönen: markku.ylonen@tut.fi The cumulative peak value distribution has a linear and a non-linear part in the linear-logarithmic scale. It was assumed that the non-linear part consists of static noise and insignificantly small bubble collapses. This study focuses only on the linear part of the distributions, corresponding to the expected exponential behavior of the cavitation impacts. The exponential law is presented in equation 2:

𝑁 ̇= 𝑁 ̇0𝑒 - 𝑈 𝑈 0 (2)
where 𝑁 ̇ is the peak rate, 𝑁 ̇0 is the reference peak rate, U is the voltage peak value and U0 is the reference voltage.

The reference values U0 and 𝑁 ̇0 are presented in table 1.

Upstream pressure/cavity reference velocity

Reference peak rate 𝑁 ̇0 Reference voltage U0 The linear parts of the cumulative distributions in figure 3 are essentially parallel. The reference voltages U0 are calculated from the slope of the linear fits and they are found to be almost equal. The reference peak rate 𝑁 ̇0 follows the cavity reference velocity V with a relation of 𝑁 ̇0~𝑉 7.4 . As the slope of the linear fit in linear-logarithmic scale does not change with the change of operating point, it means that the ratio between cumulative impact rates remains constant and independent of voltage peak value.

In this study, the connection between AE voltage peak values and the impact magnitude or impact damage is not addressed. Cavitation pitting in a virgin sample tends to produce pit sizes following a same type of exponential distribution as results in this and previous studies such as [START_REF] Franc | Material and velocity effects on cavitation erosion pitting[END_REF]. This fact is encouraging and shows that the results presented in this study somehow represent the physical phenomenon of cavitation pitting.

Conclusion

A method to monitor and characterize cavitation impacts by acoustic emission was presented. Acoustic emission was measured for two different operating conditions and envelope functions were fitted to the resulting waveforms. Voltage peak values were counted from the fitted envelope functions and cumulative distributions were calculated.

Reference peak rates and reference voltages were calculated for both operating points. The reference voltage was about the same for both operating points and the reference peak rate had a relation of 𝑁 ̇0~𝑉 7.4 .

Using the presented method, it is concluded that the cumulative distributions for acoustic emission voltage peak rate value may be calculated for any operating condition by calculating the reference peak rate 𝑁 ̇0 and reference voltage U0 and then applying the relation between reference peak rates, if the distribution for one point is known. To validate the results, multiple operating points should be tested. With the assumption that the relation holds as stated, the acoustic emission distributions can be used in estimating cavitation pitting in the cavitation tunnel. The advantage of the presented method is that it is non-intrusive, as it does not require direct access to the flow. Future studies will address these issues in more detail. These results show the potential of using acoustic emission in cavitation detection and characterization in laboratory testing and eventually in actual hydro machines.

Figures 1 (

 1 Figures 1(a) and 1(b). 1(a): Sample mounted to the sample holder and fitted with an AE sensor and a waveguide. 1(b): PREVERO test section opened and with the inlet nozzle in the middle. The sample is flush mounted so that it forms a part of the other wall in the test section.

Figure 2 .

 2 Figure 2. Typical AE waveform detail and the envelope function fitted to the waveform.

Figure 3 .

 3 Figure 3. Cumulative peak rate vs. voltage peak value in linear-logarithmic scale. A linear fit was applied to the linear part of the curve.

Table 1 .

 1 Cumulative distribution reference values

	2 MPa / 63.2 m/s	1232 1/s	0.063 V
	4 MPa / 89.4 m/s	15958 1/s	0.065 V

Acknowledgements

The authors would like to thank Tekes (the Finnish Funding agency for Technology and Innovation), Fortum Power and Heat Oy, Sandvik Mining and Construction Oy, Valtra Oy and Teollisuuden Voima Oyj for funding the research and for providing technical support.


Hyung Jun Chang

Marc C Fivel

Jean-Loup Strudel

Micromechanics of primary creep in Ni base Superalloys

The initial stages of creep in nickel base superalloy deserve a special attention since they prevail in most of the service live of critical jet engine components. In nickel base superalloys, a very sharp minimum in creep rate is observed at strains less than 0.5% in a wide range of temperatures: the lower the applied stress the steeper the minimum, often referred to as incubation period. In this paper, the primary creep stage of Ni superalloy is investigated using 3D discrete dislocation dynamics modeling. The simulated volume corresponds to a 3D periodic cell of the cubic arrangement of γγ' Ni-Al loaded along the (001) direction. Misfit stresses induced by the lattice mismatch (= -3.10 -3 ) between the precipitate and the matrix are taken into account through a coupling of the dislocation dynamics code with a finite element code. Frank-Read sources are positioned in the horizontal channel. It is shown that most of the dislocations propagate in the horizontal channels by repeated cross-slip on activated {111} glide planes and a few invade the vertical channels. This behavior is rationalized by the model. In the γ-γ' interfaces, the zigzagging lines of dislocations straighten out into [100] and [010] directions leading to a dislocation microstructure similar to the ideal misfit dislocation network needed to compensate the lattice mismatch. Calculations of internal stresses in the channels show that this dislocation microstructure induces biaxial planar stresses in the horizontal γ channels leading to an elastic contraction of the sample along the loading direction. Three-dimensional DDD simulations show that the plastic strain induced, in the loading direction, by the movement and multiplication of dislocations is less important than the elastic relaxation it produces along that same direction. Hence, for very low applied tensile stresses, Ni superalloys display an abnormal behavior often referred to as negative creep that can be evidenced experimentally, even in polycrystalline materials. This last observation clearly implies that the first active batches of dislocations, during creep, or static high temperature ageing are mobilized to cancel out the numerous and pervading internal elastic strain fields generated within the crystals by the γ' precipitates due to their misfit with the γ matrix.

Introduction

Nickel base superalloys, hardened by large volume fractions of 'precipitates, exhibit an outstanding creep resistance at elevated temperatures (400°C to 1150°C) under large stresses and for extended periods of time; therefore, they are commonly used in the hottest parts of aeroengines, as well as in land turbines. In advanced disk alloys the ' volume fraction can reach 40 to 50% and in alloys designed for single crystal turbine blades 70%. In most nickel base superalloys the 'misfit, defined as a a a δ γ γ'  

, is negative and takes values as large as -3.10 -3

; it varies widely (± 2.10 -3 ) with the chemical composition of the alloy [START_REF] Pyczak | The influence of different alloying elements on the development of γ-γ' microstructures of nickel base superalloys during high temperature annealing and deformation[END_REF][START_REF] Pyczak | Influence of lattice misfit on the internal stress and strain states before and after creep investigated in nickel-base superalloys containing rhenium and ruthenium[END_REF] and the temperature [START_REF] Lu | Elastic microstrains during tension and creep of superalloys: results from in situ neutron diffraction[END_REF][START_REF] Royer | Temperature dependence of the lattice mismatch of the AM1 superalloy. Influence of the γ' precipitates morphology[END_REF] and, more subtly, with plastic strain and internal stress fields as pointed out by [START_REF] Kuhn | An X-ray study of creep deformation induced changes of the lattice mismatch in the γ'-hardened monocrystalline Nickel base alloy SRR 99[END_REF] and experimentally measured with great details by [START_REF] Dirand | Measurement of the effective γ-γ' lattice mismatch during high temperature creep of a Ni-based single crystal superalloy[END_REF].

The tensile and creep behavior of superalloys has been abundantly studied in a large range of temperatures and stresses; results and interpretations are reviewed in a number of articles [START_REF] Pollock | Dislocations and high temperature deformation of superalloy single crystals in "Dislocations in Solids[END_REF][START_REF] Pope | Mechanical properties of Ni3AI and nickel-base alloys with high volume fraction of γ[END_REF]most comprehensive and also Link et al., 2009[START_REF] Zhang | The effect of lattice misfit on the dislocation motion in superalloys during high-temperature low-stress creep[END_REF][START_REF] Viswanathan | Deformation mechanisms at intermediate creep temperatures in the Ni-base superalloy René 88 DT[END_REF]. Yet, the rather peculiar primary creep stage they all exhibit, and its interpretation still need to be clarified. The objective of this work is to present an interpretation of experimental observations based on a micromechanical model combining a FEM approach to describe the local stress and strain fields around γ' precipitates, with a DDD approach to account for the dislocation activity. Thus, the movements of individual dislocation segments moving in the γ matrix, within these stress fields and in mutual interaction with their updated values can be described, as plasticity progresses during the initial stages of creep.

Primary creep, the stage that immediately follows the loading of the test piece is usually described as a stage of decreasing strain rate, until a steady state regime, characterized by ἑ(t) ≈ cte, is reached (blue curve, Fig. 1). Instead, in nickel base superalloys (Fig. 1), a very sharp minimum strain rate is observed at around 2 to 5.10 -3 strain, followed by a steep rise; then creep is progressively turning into a slowly accelerating stage that eventually merges into tertiary creep and rupture. This peculiar creep behavior, although already reported in the 90's by Provendier-Aubourg and Strudel (1995), who studied the creep and relaxation behavior at 650° C of alloy N-18, was improperly related to the formation of stacking faults developing in the matrix and extending into the γ' precipitates. At that time, these mechanical tests were performed at a rather low temperature and under high stresses (850 to 1000 MPa) in an alloy with high cobalt content (15.7 at %). The TEM observations were then biased by the overwhelming presence of stacking faults, and the role of perfect dislocations, located in the γ-γ' interfaces, was screened by this abundance. Furthermore, the appropriate simulation tools that would have led to the correct interpretation were missing. Later, this marked minimum creep rate was observed in all superalloys containing a misfitting γ' phase (Epishin and Link, 2004a[START_REF] Pyczak | The influence of different alloying elements on the development of γ-γ' microstructures of nickel base superalloys during high temperature annealing and deformation[END_REF][START_REF] Wu | Double minimum creep of single crystal Ni-base superalloys[END_REF]. Its amplitude seems to be increasing with the volume fraction of precipitates. It has been reported in AM1 [START_REF] Fredholm | High Temperature Creep Mechanisms in Single Crystals of Some High Performance Nickel Base Superalloys[END_REF], in the CMSX series [START_REF] Matan | On the kinetics of rafting in CMSX-4 superalloy single crystals[END_REF][START_REF] Reed | Creep of CMSX-4 superalloy single crystals: Effects of rafting at high temperature[END_REF][START_REF] Svoboda | Model of creep in <001>-oriented superalloy single crystals[END_REF], in SRR 99 [START_REF] Hammer | High temperature creep and microstructure of the monocrystalline nickel base superalloy SRR 99[END_REF] in RR-1000 [START_REF] Hardy | Developing damage tolerance and creep resistance in a high strength nickel alloy for disc applications[END_REF], in René 88 (Fig. 1) and even in Alloy 800H under low stress, which contains only 2% volume fraction of γ' precipitates (El-Hazim, 1996). In CMSX 4 (Fig. 2), this sharp minimum creep rate was reported for <001> oriented crystals tested at 1200° C under 75 and 50 MPa with strain rates in the range 10 -7 to 10 -9 s -1 . Under such conditions, even in a low stacking fault energy alloy such as CMSX 4, perfect matrix dislocations alone are observed moving in matrix channels normal to the applied stress, by a repeated cross slip mechanism, as described by [START_REF] Carry | Direct observation of <110>-{1-10} in FCC single crystals of a Ni base superalloy[END_REF] (Fig. 3). Fig. 2: Creep of CMSX-4 at 1200°C under low stress [START_REF] Reed | Creep of CMSX-4 superalloy single crystals: Effects of rafting at high temperature[END_REF] Notice that the originally zigzagging, mixed 60° lines left by the leading screw segment in the γγ' interfaces, later straighten out into pure edge, straight lines oriented into the cube directions. Notice also that, on the average, the leading screw segments, by their repeated cross-slip behavior (Fig. 3), are moving in that same direction. Thus, it appears that the best conditions to observe and study this marked minimum creep rate is to submit a <001> oriented single crystal to a tensile creep test, at fairly high temperature (750° C to 1200° C) under an applied stress, low enough to place that minimum creep rate around 10 -9 s -1 , or below. It is difficult experimentally, to collect data so close to the background noise of the sensing and recording devices (10 -10 s -1 in a carefully air conditioned room) but this regime is actually most important, as it encompasses most service life or may even be considered as an upper bound to it. Indeed, the minimum creep rate under 75 MPa on the CMSX-4 test (Fig. 2), yields a value of 6.10 -8 s -1 . If we assume that this minimum creep rate would be maintained for 45 hours, the material would already reach a plastic strain of 1%, an underestimated value, considering the assumption. In service, even for military engines, strain rates, at least, two orders of magnitude lower are usually required. Similarly, [START_REF] Ram | On the origin of creep dislocations in a Ni-base, single-crystal superalloy: an ECCI, EBSD, and dislocation dynamics-based study[END_REF], testing a high creep resistant single crystal at 900°C under 450 MPa report strain rates in the range ≈10 -8 s -1 at 1% strain and a time to rupture of ≈200 h. As a consequence, they observe and model a scenario of general plasticity with dislocation avalanches affecting several families of channels simultaneously and involving interactions and junctions of dense populations of dislocations.

a) b)

Fig. 3: Repeated cross-slip of the leading screw segment of a perfect dislocation leaving zigzagging 60° segments which straighten out along cube directions, by climb in the γ-γ' interfaces. a) Schematic; b) TEM observation in alloy AM-1 at 850°C/ 190 MPa [START_REF] Carry | Direct observation of <110>-{1-10} in FCC single crystals of a Ni base superalloy[END_REF].

Under high stresses, the details of the stress fields introduced by the γ-γ' misfit need not be taken into account as shown by [START_REF] Huang | DDD modeling of mechanical deformation of nickel based single crystal superalloys[END_REF], who simply introduced a uniform average internal stress field in each matrix channel where the multiple slip systems are crisscrossing and interacting with one another. They even claim that the coherency stress field has negligible influence on the stress-strain response of the material. Similarly, [START_REF] Rao | Discrete dislocation simulations of precipitation hardening in inverse superalloys[END_REF], using a DDD simulation model to obtain an estimate of the yield stress of superalloys, indicate that, when introducing a coherency stress induced by a misfit δ= -0.3%, the yield stress increases by only about 10%.

In this paper, the dislocation configurations examined are those similar to service conditions; i.e. creep under intermediate and high temperature, under applied stresses low enough to produce rupture after several thousand hours and creep rates, after 1 % strain, 200 to 500 times lower than those previously described in the literature [START_REF] Reed | Creep of CMSX-4 superalloy single crystals: Effects of rafting at high temperature[END_REF][START_REF] Ram | On the origin of creep dislocations in a Ni-base, single-crystal superalloy: an ECCI, EBSD, and dislocation dynamics-based study[END_REF]. When applied stresses are low, the local internal stress fields generated by precipitates or pre-existing dislocation arrangements must be taken into account when trying to understand the movement of dislocations. In nickel base superalloys, the misfitting γ' phase is generating nonuniform 3-D stress fields that strongly distort and pre-strain the matrix with an intensity increasing with the amplitude of the misfit, the volume fraction of precipitates and their morphology. A precise 2D-mapping of the stress field surrounding γ' cuboids was first published by [START_REF] Pollock | Creep resistance of CMSX-3 nickel base alloy single crystals[END_REF] and more recently in 3-D by Link et al. (2009), who could thus identify and localize specific edges of cuboids where the presence of particular stress components are favoring the repeated cross slip of dislocations moving in a quasi-periodic array of cuboids of identical size (Fig. 4a). The first part of this paper will be devoted to the presentation of the 3D stress field and its variations along and across the three matrix channels, 1) in the absence of any applied stress and 2) under a uniaxial tensile stress along <001>. Then the resolved shear stress for a given slip system, instead of the Von Mises stress, will be calculated in a FEM scheme.

When a perfectly oriented <001> single crystal is submitted to a tensile stress, four different Burgers vectors are expected to be active simultaneously. In practice, single crystals exhibit a mosaic structure and only one or two dislocation systems are activated simultaneously in any given subgrain. If the disorientation happens to be toward [011] for instance, the two activated Burgers vectors are orthogonal, i.e. a/2[101] and a/2[-101] and the dislocations have little interaction with each other; on the other hand, if the disorientation happens to be toward a <111> direction, the two activated Burgers vectors, i.e. a/2[101] and a/2[011] share a common glide plane, namely (111), where they can recombine to form a/3 configurations that can shear the γ' particles by creation of low energy superlattice intrinsic and extrinsic stacking faults (Kear et al., 1970). As reported by [START_REF] Wu | Double minimum creep of single crystal Ni-base superalloys[END_REF] and at length by [START_REF] Rae | Primary creep insingle crystal superalloys. Origin, mechanisms and effects[END_REF], such complex events can take place during primary creep of superalloys with low stacking faults energy (CMSX-4, for instance) and produce a second creep rate minimum around 4 to 5% strain [START_REF] Svoboda | Model of creep in <001>-oriented superalloy single crystals[END_REF]. In fact, it is necessarily preceded by the generation of perfect matrix dislocations moving in the γ matrix channels and interacting with the internal stress field. As described above, they are responsible for the first minimum. Only then, can the various perfect dislocations recombine into complex planar slip systems involving planar defects. In this paper, this second stage of primary creep will not be examined since it has been well described and interpreted at length by several authors [START_REF] Unocic | Superalloys 2008[END_REF][START_REF] Rae | Primary creep insingle crystal superalloys. Origin, mechanisms and effects[END_REF][START_REF] Knowles | Superlattice stacking fault formation and twinning during creep in γ-γ' single crystal superalloy CMSX-4[END_REF][START_REF] Rae | The role of stacking fault shear in the primary creep of [001] oriented single crystal superalloys at 750°C and 750 MPa[END_REF] and is taking place only under applied stresses high enough for the penetration and shearing of γ' precipitates to take place.

In situ measurements of  and ' lattice parameters during monotonic tensile tests, at temperature, were made by use of neutron diffraction techniques and their variations were interpreted as "load transfers" between the matrix and the precipitates in a PM version of RR1000 [START_REF] Preuss | The effect of γ 'particle size on the deformation mechanism in an advanced polycrystalline nickel-base superalloy[END_REF], as well as in the DS-MAR-M200 [START_REF] Lu | Elastic microstrains during tension and creep of superalloys: results from in situ neutron diffraction[END_REF] alloys. Making use of an ingenious experimental set-up for carrying out tensile creep tests at 900°C and above, [START_REF] Lu | Elastic microstrains during tension and creep of superalloys: results from in situ neutron diffraction[END_REF] clearly established that a significant reduction of the elastic strain of the  matrix channels along the tensile direction was taking place during the first 10 hours of creep at 900°C under 425 MPa, on a DS MAR-M200 sample, while it was increasing in the transverse direction. The authors correctly attributed these "load transfers" between the matrix and the'cuboids to the presence of dislocations in the ' interfaces relaxing the misfit generated elastic strain fields in the channels normal to the applied uniaxial stress and, on the other hand, aggravating them in the longitudinally oriented channels. But they omitted to take into account the effect of the misfit stresses when estimating the stress necessary for matrix dislocations to bow out, into the matrix channels, assuming simply an Orowan process, hence a purely geometric effect. This latter case of interactions between dislocations and impenetrable spherical particles is relevant in ODS type materials and was recently treated with a theoretical approach and in a comprehensive manner by [START_REF] Zalezak | High temperature dislocation processes in precipitation hardened crystals investigated by 3D discrete dislocation dynamics[END_REF].

The suggestion that the elastic relaxation of the pre-stressed matrix channels provided by the introduction of dislocations during the initial stages of plasticity could exceed the forward plastic strain produced by their generation and multiplication in the horizontal channels was first introduced by Strudel 2007. But the argument was purely qualitative; a quantitative estimate with appropriate tools is carried out here. The mechanisms of plasticity and the local stress redistributions in the presence of moving dislocations are examined quantitatively in the present paper; thus providing a tentative answer to the question of interaction between the plastic and elastic driving forces of the rafting phenomena in nickel base superalloys [START_REF] Ichitsubo | Rafting mechanism for Ni-base superalloy under external stress: elastic or elastic-plastic phenomena[END_REF].

As pointed out by [START_REF] Fredholm | High Temperature Creep Mechanisms in Single Crystals of Some High Performance Nickel Base Superalloys[END_REF] who studied the rafting process in AM-1 and more recently emphasized by [START_REF] Zhang | The effect of lattice misfit on the dislocation motion in superalloys during high-temperature low-stress creep[END_REF]2010), the movement, initially by glide and cross-slip, later by climb, together with the organization of dislocations into dense networks in the ' interfaces of the horizontal channels are the precursor events of rafting of the ' microstructure. These phenomena minimize the elastic strain energy initially stored in optimally heat treated superalloys [START_REF] Ratel | Predicting the onset of rafting of γ' precipitates by channel deformation in a Ni superalloys[END_REF] and ensure a marked minimum creep rate for alloys with a large negative misfit [START_REF] Zhang | The effect of lattice misfit on the dislocation motion in superalloys during high-temperature low-stress creep[END_REF][START_REF] Pyczak | Influence of lattice misfit on the internal stress and strain states before and after creep investigated in nickel-base superalloys containing rhenium and ruthenium[END_REF]. Recently, [START_REF] Tanaka | Effect of elastis driving force on the evolution of microstructures in the secondary creep rate stage[END_REF] pointed out that the initial elastically strained state of the  matrix channels along the tensile direction 33, was drastically altered during the first stages of a uniaxial creep test. By use of the concept of eigenstrain of the  matrix submitted to the stress resulting from the ' misfit and to the plastic strain provided by creep, they show that the optimal resistance of the rafting ' microstructure is obtained when the density of interfacial dislocations barely reaches the optimal value for which the misfit is globally balanced in the horizontal channels. It is one of the goals of this paper to suggest a rational and quantitative interpretation of this matrix stress relaxation based on the introduction of interfacial dislocations in preferential matrix channels under the applied stress, during the initial stages of plastic deformation.

Three dimensional Discrete Dislocation Dynamics (DDD) simulation is an ideal tool when investigating the initial stage of creep behavior of Ni base superalloys. Indeed, the amount of plastic strain and the size of the typical representative cell of an idealized single crystal of superalloy are both limited so that the required computation time is not too demanding. For this reason a few DDD simulations were conducted in 3D in the last few years. As an example, [START_REF] Devincre | Mesoscopic simulations of plastic deformation[END_REF][START_REF] Vattré | Dislocation dynamics simulations of precipitation hardening in Ni-based superalloys with high ' volume fraction[END_REF][START_REF] Vattré | Orientation dependence of plastic deformation in nickel-based single crystal superalloys: Discrete-continuous model simulations[END_REF]) used a coupling with Finite Element (FE) to account for the misfit stresses and their Discrete-Continuum model was able to account for elastic anisotropy. They computed the creep behavior and the dislocation microstructure obtained under [001] and [111] tensile loading conditions [START_REF] Vattré | Orientation dependence of plastic deformation in nickel-based single crystal superalloys: Discrete-continuous model simulations[END_REF] and also the shearing process of the ' precipitates [START_REF] Vattré | Dislocation dynamics simulations of precipitation hardening in Ni-based superalloys with high ' volume fraction[END_REF]. Practically, most of the DDD simulations performed so far focused on the precipitate shearing process which corresponds to the creep behavior under high stress [START_REF] Yashiro | Discrete dislocation dynamics simulation of cutting of γ' precipitate and interfacial dislocation network in Ni-based superalloys[END_REF][START_REF] Vattré | Dislocation dynamics simulations of precipitation hardening in Ni-based superalloys with high ' volume fraction[END_REF][START_REF] Huang | DDD modeling of mechanical deformation of nickel based single crystal superalloys[END_REF][START_REF] Yang | Modeling dislocation cutting the precipitate in nickel-based single crystal superalloy via the discrete dislocation dynamics with SISF dissociation scheme[END_REF][START_REF] Li | Parametric dislocation dynamics simulation of precipitation hardening in Ni-based superalloy[END_REF]Gao et al., 2015a-b;[START_REF] Gao | 3D discrete dislocation dynamics study of creep behavior in Ni-base single crystal superalloys by a combined dislocation climb and vacancy diffusion model[END_REF][START_REF] Rao | The strength and dislocation microstructure evolution in superalloy microcrystals[END_REF]. Consequently, only a few of them account for the misfit stresses, either using FEM [START_REF] Devincre | Mesoscopic simulations of plastic deformation[END_REF][START_REF] Liu | Interfacial dislocation motion and interactions in single-crystal superalloys[END_REF] or FFT approaches (Gao et al., 2015b). [START_REF] Liu | Interfacial dislocation motion and interactions in single-crystal superalloys[END_REF] were able to investigate the early stage of high-temperature low-stress creep in Ni-base superalloys using a DD framework where the misfit stress was implemented as an ad-hoc bi-axial stress field as also done by [START_REF] Huang | DDD modeling of mechanical deformation of nickel based single crystal superalloys[END_REF]. They focused their study on the motion of the interfacial dislocations and their local reactions at the -' interface but nothing was said on the resulting mechanical response. [START_REF] Yashiro | Discrete dislocation dynamics study on the interaction between prismatic dislocation loop and interfacial network dislocations[END_REF] also presented a DDD investigation of the stability of the interfacial dislocation network and its interaction with a prismatic dislocation loop but they did not take into account the misfit stress. More recently, [START_REF] Ram | On the origin of creep dislocations in a Ni-base, single-crystal superalloy: an ECCI, EBSD, and dislocation dynamics-based study[END_REF] used ECCI observations to define arrangement of dislocations that served as the initial configuration of a DDD simulation. Neglecting the misfit stress, they observed how the dislocations could invade the different channels and they conclude that dislocation sources should exist somewhere outside the considered low-angle boundaries in order to account for the observed creep behavior. In the case of Ni-based superalloy with low stacking fault energy, Huang and Li, 2013 took into account dislocation dissociation in their DDD model to investigate the dislocation microstructure developed in the channels.

Thus, the aim of this paper is to use 3-dimensional DDD simulations in order to address the creep anomaly evidenced in the early stage of creep deformation of Ni superalloys, in two steps: first by computing, for a [001] oriented single crystal, the dislocation paths and resulting patterns, induced by the misfit stresses superimposed to a [001] tensile loading, typical of in service turbine blade conditions; secondly by estimating the elastic and plastic strain induced by the generation and multiplication of dislocations, belonging to one or several slip systems, getting trapped in the γ-γ' interfaces where they climb and rearrange into straight lines along the cube directions. Finally, a stress analysis is conducted inside an ideal interfacial network aiming at perfectly compensating the misfit stress fields.

Simulation setup 2.1 Dislocation Dynamics modeling

The DD code used in this paper is the 3D edge-screw model TRIDIS as described in [START_REF] Verdier | Mesoscopic scale simulation of dislocation dynamic in fcc metals: Principle and applications[END_REF]. In this model, the dislocation lines are decomposed into sets of edge and screw segments connected together. The DD code is coupled to the Finite Element (FE) software CAST3M (CASTEM) in order to enforce complex boundary conditions. For each dislocation segment, the Peach Koehler force is computed at its middle point and projected along the glide direction. This effective force results from the superposition of the internal stress field generated by all the dislocation segments and the applied stress field obtained by the Finite Element correction. This gives the effective resolved shear stress, *, from which the dislocation velocity is evaluated as

B b τ v *  , [1]
where b is the Burgers vector magnitude and B is a phonon drag coefficient.

At high temperature, one should account for recovery mechanisms like cross-slip and eventually climb. In this paper, climb mechanism is not taken into account but cross-slip is implemented as proposed by many authors [START_REF] Verdier | Mesoscopic scale simulation of dislocation dynamic in fcc metals: Principle and applications[END_REF]) (Weygand et al. 2001)(Zbib and Diaz de la Rubia, 2002) (Kubin). In the case of screw segments, the cross-slip mechanism is computed via a thermally activated probabilistic algorithm aiming at modelling the constriction-expansion mechanism proposed by (Bonneville et al., 1988). First we compute the probability function, P, using the following formula

         V kT t t III CS / exp L L P 0 0     , [2]
with L the dislocation length; t  the waiting time;

CS  the shear stress resolved in the deviate slip system; III  the critical shear stress at the onset of Stage-III hardening; T, the temperature and V, the activation volume. L0 and 0 t  are reference length and time respectively and  is a prefactor normalizing the range value of function P into [0;1] interval. At each time step, the value of P is compared to a computer generated random number, r, and the change of plane is accepted if r<P.

The set of parameters used for the DDD simulations are given in Table 1.

b [Ang] B [Pa.s]  L0 [µm] t0 [s] V/b 3  [MPa] t [s] µ [MPa] T [K]
2.54 1.5 10 -5 6.10 4 1.0 10 -9 350 200 10 -11 81000 800

Table1: Parameters used for DD simulations (Equ. ( 1) and ( 2)).

Geometry and boundary conditions

Assuming periodic boundary conditions, the elementary simulation box consists of a 500 nm cubic cell containing one horizontal channel and two vertical channels (Fig. 4). The channel thickness is fixed at 80 nm leading to a volume fraction of ' precipitates of 59.3% with a precipitate size of 420 nm. Notice that only 1/8 of the volume of the 8 ' cuboids appears in Figure 4 since the center of the elementary box is chosen at the meeting point of the 3 matrix channels, in order to observe more easily the movement of dislocations in the matrix channels. In practice, periodic boundary conditions are enforced in the DD code by re-introducing dislocation segments in the opposite surface it escaped from. For each dislocation segment, the internal stress field is computed as if the segment was at the center of the periodic box. This means that the number of replicas taken into account for the dislocation stress field is N=1. The entire volume is meshed into 6912 twenty-node elements (27873 nodes) locally refined at the -' interface. In order to better represent the actual shape of the ' precipitates, we designed cuboids with round corners of 15nm radius; they match closely the shapes predicted by phase field method (Gao et al., 2015a). The periodic boundary conditions in the FEM are enforced by imposing that the distance between two nodes belonging to two opposed surfaces is a constant throughout all the nodes of the face using CAST3M operator @CLPC.

Using the superposition principle, the dislocations are moving in the matrix according to an effective stress computed as the summation of the internal stress induced by all the dislocations included in the simulation box and the finite element stress accounting for the boundary conditions. Here the boundary conditions are of four types: (i) the misfit stress induced by the lattice mismatch between the  matrix and the ' precipitate; (ii) the load applied as a tensile stress along the [001] direction; (iii) the periodic boundary conditions along the (100), ( 010) and ( 001) surfaces of the simulated box and (iv) the non-penetrable facets located at the -' interface.

Both the matrix and the precipitate are considered as elastic isotropic media with a Young modulus of E=208,000MPa and a Poisson's ratio =0.3 leading to a shear modulus G=80,000MPa. In this study, the difference in elastic moduli of the two phases is neglected so that the calculation of the image forces induced by the precipitate can be omitted. Practically, the presence of the precipitates is accounted for by the introduction of facets at the -' interface which prevent the dislocations from entering the particles [START_REF] Shin | On dislocation-impenetrable precipitates interactions: a discrete dislocation simulation analysis[END_REF].

Dislocation configuration

After optimized heat treatments, the hardening phase with simple cubic ordered L12 crystalline structure is generally arranged in nearly periodic arrays of cuboids, 0.4 µm in size, in quasi-coherency with the  matrix and aligned along the <001> cube directions (Fig. 3)The remaining  channels of the fcc matrix are narrow (0.05 to 0.1 µm) and highly stressed by the misfitting cuboids they are surrounding. The γ-γ' misfit parameter is of the order of -3.10 -3 in many nickel base superalloys, a value usually considered as offering the best creep resistance [START_REF] Pyczak | Influence of lattice misfit on the internal stress and strain states before and after creep investigated in nickel-base superalloys containing rhenium and ruthenium[END_REF].

Four dislocation sources are introduced as pinned segments of screw character located within the -matrix in the horizontal channel. Each screw segment is centered in the middle of the cuboid interfaces as shown in Figure 5. The initial length of each segment is set to 86 nm. where a=3.52 Å. Since each dislocation segment is artificially pinned at both ends, it will act as a Frank-Read source when the load is applied. Cross slip events will generate dislocations on the 8 possible slip systems using the probabilistic law given in equation ( 2). The dislocation motion is restricted to the -matrix by the introduction of the impenetrable facets delimitating the -' interfaces as depicted in Figure 5.

Results and discussion

-Misfit stress calculations

The stress fields generated by the misfit are computed using a finite element analysis (Fig. 6). Assuming that the matrix and the precipitate both has a thermal expansion coefficient =10 -5 , a temperature difference of 300 K is imposed between the -matrix and the γ'precipitates. Then, the FE model finds the equilibrium state of the system assuming periodic boundary conditions along the (100) surfaces and a similar elastic modulus for both phases. The eigenstrain induced by the thermal expansion thus corresponds to a negative misfit = -3.10 -3 between the precipitate and the matrix.

Two of the stress field components are presented in Figure 6, namely σ11 and σ33, since σ22 is identical to σ11 rotated 90°. In the matrix channel normal to [001] for instance, the two horizontal components σ11 and σ22 are maximum, compressive (blue color) and parallel to the γ-γ' interface; σ33 the component normal to the interface is much smaller and tensile (red color). The shear stress component σ12 is given in Figure 6c. It is localized in the corner of the cuboid interfaces aligned with the Z-axis. Similar profiles are obtained by permutations for components σ13 and σ23. The magnitude of the shear stresses in the horizontal and vertical channels is close to zero. Therefore, the stress field in the channels can be described as a quasi-planar compressive stress field with a peak in the center of the γ' face. By symmetry, the same is true in each of the three matrix channels. Eigenstrains induced by a negative misfit = -3.10 -3 between the precipitates and the matrix are described in Figure 7, namely ε11 and ε33. It appears that a matrix channel is elongated in the direction normal to its bounding γ-γ' interfaces, i.e. its width is increased by the misfit, with an amplitude maximal in the center of the faces of the γ' precipitates and minimal at the center of symmetry of the 8 cuboids considered in the simulation box (yellow-green colors at the crossing of channels). Hence, the prestrain of each matrix channel resulting from the γ-γ' misfit is simply a uniaxial elongation in the direction normal to the face of the cube as shown in the 2-D view (Fig. 7c).

-Activation of a slip system

A typical mapping of the shear stress resolved on the a/2(111)[-101] slip system (RSS) is given in Figure 8a, in the absence of any externally applied stress. One can see that the resolved shear stress varies between -495 and 495 MPa and that it is significantly different, on the average, in each channel. Therefore, we shall use thereafter the designation of channels for a given set of slip systems suggested by Link et al., (2009). At the position of the dislocation sources, i.e. inside the horizontal channel, or propagating channel (P channel), this shear stress is ≈ 280 MPa. In the two vertical channels, the shear stresses are not equivalent. The vertical channel normal to the [010] direction, or repulsive channel (R channel) is loaded negatively, whereas the one normal to the [100] direction, or crossing channel (C channel), experiences hardly any resolved shear stress. When a tensile stress of 500 MPa is imposed along the [001] direction, the RSS mapping is changed into the one plotted in Figure 8b. Because of crystal symmetries, 8 slip systems out of 12 can be activated simultaneously. These potentially active slip systems correspond to the 4 Burgers vectors introduced in Figure 5, each having two possible glide planes. They are forming two sets: those with Burgers vectors normal to the X direction, with RSS described in Figure 8 and those with Burgers vectors normal to the Y direction and having RSS rotated 90° around the Z direction with respect to the previous ones. Notice that the RSS in the horizontal channel is now higher than in the vertical channels. The RSS asymmetry between channels, induced by the misfit, is now amplified by the applied stress. Therefore, the main part of the plastic activity is expected to take place in the P channel, as repeatedly confirmed by numerous experimental observations (Link et al., 2009;[START_REF] Zhang | The effect of lattice misfit on the dislocation motion in superalloys during high-temperature low-stress creep[END_REF]Pollock andArgon, 1994, 1992).

-Bowing out of the dislocations into the matrix channels: analytical estimate

Let us first examine the initial movement of perfect dislocations with Burgers vectors b=a/2<110> moving under a uniaxial stress applied along the Z axis. In this simplified analytical investigation, we will not consider the dislocation mutual interaction. They are bowing out from Frank-Read sources located in the center of the simulation cell (Fig. 4) and penetrating the various channels as they come in contact with the edges of the γ' cuboids. A first-order approximation consists in considering that the movement of the leading segments is controlled by a simple geometric rule relating to the width w of the channel, as for instance an Orowan stress. It yields a single value in the center of the 100 nm wide channels:

          A A w A [3]
Where µ=80 GPa is the shear modulus of the γ matrix and  a coefficient close to 0.5.

The model presented here enables a more refined description of the local stress fields the dislocation is traveling through in channels clearly presenting drastic differences in resolved shear stress (Fig. 8). The force on a dislocation line of Burgers vector b(b1,b2,b3) resulting from the presence of these cumulated stress fields is given by the Peach-Koehler equation:

                22 1 2 11 2 1 11 3 1 33 1 3 33 2 3 22 3 2 ) ( ) ( 2 (U) F         u b u b u b u b u b u b a appl appl  [4]
where U(u1,u2,u3) is the local line vector of the dislocation,  is the local stress field combining the misfit stresses and appl the external tensile stress applied along the Z direction. Notice that the Z component of this force is independent of σappl.

The glide force gl

U F  on the segment of direction U is the projection of   U F 
on the glide plane. Hence:

  2 1 2 2 2 gl U n . F F α cos 1 F sin α F F                  [5]
where α is the angle between the force on the dislocation and the normal n to its glide plane.

And similarly; the force normal to the glide plane reads:

n . F cosα F F n U       [6]
It is the climb force on an edge segment oriented along U.

As an example, let us assume that the stress field in the channels of a [001] oriented single crystal under an applied tensile stress of 300 MPa along the [001] direction is constant and uniform. It is probably rather well verified in the central part of the channels. The computation was made for b2=a/2 [011] gliding in plane (1-11). The forces were computed in each channel, respectively along two orthogonal directions: namely U║ (center columns Table II) characterizing the attraction or repulsion of the γ-γ' interface for the dislocation line lying against it and U= (right hand columns Table II) giving an estimate of the driving force acting at the center of the bowing out segment of the dislocation. The resulting force reads:

               22 1 33 1 33 2 22 3 ) ( ) ( 2 (U) F       u u u u a appl appl  [7]
The results presented in table II are illustrated by the schematic of Figure 9d where the glide forces are represented by red vectors with a surface area proportional to their magnitude. The climb forces are normal to the glide plane, hence to the plane of the figure .   The glide forces are constant and uniform in the center part of each channel. Their amplitudes are equal on the screw segment and the bowing out arc, when it is assumed that the stress field is uniform in the channel, as is the case in this simplified model. They are exactly identical on the cross-slip plane (11-1) when the applied stress is exactly oriented in the [001] direction.

 It appears clearly that the largest glide forces are acting in channel P.

 In the C channel the glide force is 2.5 weaker than that in the P channel and the screw segments lying in the γ-γ' interfaces are submitted to a rather strong force normal to the glide plane.

 In the R channel, under this applied stress of 300 MPa, the glide force is oriented in a way that would prevent the dislocation line from moving into the R channel, or pull it out if it had intruded that channel before, as shown in Figure 9d in E and F.

 In the P channel, the extra half plane associated with the edge component of the segment lying in the interface is pointing toward the inside of the γ' precipitates (in B, Fig. 9d).

 On the contrary, in the R channel, the extra half plane associated with the edge component is pointing away from the γ' precipitates and into the γ matrix (in E and F, Fig. 9d). It is due to the presence of a repelling force exerted by the γ-γ' interface: it is made of a single component in the y direction (Eqt [7]) and is equal to The actual change in penetration of the various channels depends so strongly on the stress level that a sequence of three steps computed by a DDD simulation is presented in figure 9 a, b, and c corresponding to the extension of a glissile a/2[011](1-11) loop under respectively 300, 400 and 600 MPa applied tensile stress. Notice that in these DD calculations, the internal stress field induced by all the dislocation segments is taken into account while this was neglected in the analytical estimates. Also the misfit stress used in the DD calculations is computed by FEM and includes all the components of the stress tensor (see Fig. 6).

 In Fig. 9, the "yield stress" of 300 MPa necessary to initiate plasticity in channel P (Fig. 9a), is raised up to 600 MPa (Fig. 9c) in order to show that closed loops in Superalloys with high γ' volume fractions are more likely to be diamond shape than triangle (as often sketched in the literature) since the third channel, the R channel, requires very high stresses to be penetrated by dislocations. Exceptionally, when the slip plane of a dislocation cuts the very apex of a γ' cuboid can a triangular loop form.

 In the P and C channels the climb forces, on the segment lying in the γ-γ' interfaces (in B and D), are of opposite signs on the horizontal and vertical faces of the γ' cuboid, hence the first one is pointing into the γ' precipitate, thus pushing the dislocation into the precipitate, the latter one into the γ channel, thus pulling the dislocation away from the γ' precipitate. These screw dislocations, in channel C, connecting edge or near edge dislocations lying in the P channels were observed and well described by [START_REF] Zhang | The effect of lattice misfit on the dislocation motion in superalloys during high-temperature low-stress creep[END_REF]. They offer short distance connections between adjacent faces of the γ' cuboids and modify, in opposite directions, the chemical potentials of γ and γ' atomic constituents on neighboring faces of the γ' cuboids.

The chemical gradients between differently oriented faces, generated by the applied uniaxial stress, are thus enhanced and, by pipe diffusion, these arrays of interfacial connecting lines offer highly efficient diffusion routes to the migrating species. These chemical and diffusional processes are basic mechanisms in the implementation of the rafting of the γ-γ' microstructure, as described by [START_REF] Carry | Apparent and effective Creep Parameters in Single Crystals of a Nickel Base Superalloy, Part II: Secondary Creep[END_REF] and observed by [START_REF] Schmidt | Phase chemistry of a nickel-base superalloy after creep experiments[END_REF]. They were introduced in a DDD model simulating the octahedral glide and possible climb of dislocations during high temperature creep of a [001] oriented single crystal by Hafez [START_REF] Haghighat | Effect of climb on dislocation mechanisms and creep rates in γ' strengthened Ni based superalloy single crystals: a discrete dislocation dynamics study[END_REF]. But they remain beyond the scope of this paper that intends to deal with the micromechanical description, interpretation and numerical simulation of the elastoplastic interactions between precipitates and dislocations during the early stages of plastic deformation in nickel-based superalloys.

Misfit stress components (in MPa) This example, which can be transposed, by rule of symmetry, to each of the three other potentially active slip systems, clearly shows that the spreading out of dislocations and their bowing out into the various matrix channels is not a simple geometric and 2-D Orowan problem, as suggested in the "first order approximation". The internal stresses generated by the misfit between the two phases must be taken into account since they are similar in amplitude to the stresses applied during creep tests and definitely, during service.

U║ x 1/√2 F║ (MN/m) x a/2√2 gl U F  glide n U F  normal U= x 1/√6 F= (MN/m) x a/2√6

-Propagation, motion and multiplication of dislocations

Let us first consider a single slip system, namely a/2 [-101] matrix dislocations gliding either in the (111) or in the (1-11) planes and cross slipping freely as they move along the P channel. The elementary cell, where the computation is made, is duplicated along two directions in order to better illustrate the zigzagging progression of the slip system along the [010] direction (Fig. 10). In the present model, these zigzagging segments of mixed 60° dislocations are unable to straighten out into pure edge dislocations along cube directions, since climb is ignored. This additional moving process is most important, especially at high temperature in connection with creep processes and was included in the improved version of this simulation scheme as proposed in [START_REF] Gao | 3D discrete dislocation dynamics study of creep behavior in Ni-base single crystal superalloys by a combined dislocation climb and vacancy diffusion model[END_REF].

The tensile loading applied along the [001] direction is progressively increased from 500 MPa to 1000 MPa in a quasi-static manner: the stress is increased by 65 MPa after all dislocations have reached an equilibrium position. Figure 10 shows the dislocation microstructure reached after 4000 steps (≈0.3 % plastic strain). It confirms the adequate performance of the model since the dislocations are seen mainly traveling in the P channel and can also penetrate fairly easily the C channel, the more so the applied stress is increased, but they remain impeded from penetrating the R channel, as they are seen waiting and accumulating at its entrance. Frequent cross slip events in the P channel are well illustrated by Figure 10b where the dislocations are imaged with the Burgers vector end on. The probability for cross slip is an adjustable parameter in this model. In Figure 11b, the cross-slip ability was arbitrarily enhanced, so that many dislocation lines are now oriented along the ( 100) and ( 010) directions. Experimental observations by TEM (Fig. 3a, Pollock andArgon, 1992, Zhang et al., 2010;[START_REF] Pollock | Dislocations and high temperature deformation of superalloy single crystals in "Dislocations in Solids[END_REF] clearly indicate that the spacial frequency of cross slip is ≈200 nm, hence once or twice across a γ-γ' interface and once more at the crossing of the C channel as explained by Link et al. (2009). Whatever the detailed mechanisms of restoration are (climb, pipe diffusion of vacancies and various atomic species,…), the zigzagging dislocations will shorten their length by straightening out into <100> directions. Notice that this dislocation motion will take time since diffusion is two order of magnitude slower than glide. Yet it seems to be implemented within a couple microns behind the cross slipping and leading screw segment during creep at 850°C/190 MPa, as evidenced in C and on the left of point A on On the other hand, in the vertical channels, whatever the cross slip probability, the line orientations lie along the [101] and [011] directions. Hence, we can conclude that cross-slip has no effect on the dislocation patterning in the vertical channels, whereas it plays a major role in the movement and patterning of dislocations in the horizontal channel.

Mechanical response

During the first stages of plasticity, dislocations are emitted from sources located mainly in subboundaries in single crystals and in grain boundaries, twin boundaries and carbides in polycrystals. In nickel base superalloys, because of the large volume fraction of γ' precipitates, they immediately and almost exclusively interact with them and their stress fields rather than with each other. In so doing, they introduce their own stress field and deeply modify the preexisting internal stress fields. The present model, by combining DDD and FEM can adequately image the strongly heterogeneous stress fields resulting from these interactions. This ability is illustrated in the following micrographs taken from the previous example where two interacting sources are operating and the simulation was interrupted after 1868 steps, when the applied stress reached 875 MPa. We see that the density of dislocations in the P channel is such that their presence has more than compensated the compressive biaxial stress field produced by the γ-γ' misfit since a situation of biaxial tension (red color dominating) is now pervading in the central part of the channel (Fig. 12a). Near the γ-γ' interfaces of the P channel (Fig. 12b) the resulting stress field is less uniform than in the center of the channel, due to the vicinity of the dislocation lines, but it is well compensating the γ-γ' misfit, on the average, near the interfaces (gray color) and definitely tensile in the crossing of the vertical channels C and R, away from any immediate interface. Fig. 13: Stress components resulting from the presence of dislocations alone after applying σ appl =875MPa in 1868 steps a) σ 11 ; b) σ 33 at 0.05w and c) σ 11 ; d) σ 33 at 0.5 w, mid P channel.

If we now compare the biaxial stress field (σ 11, σ 22 ) and the uniaxial vertical stress field σ 33 in the middle of the P channel mapped across the elementary cell of the model (Fig. 13), we see that, at this point of the plastic deformation, the γ-γ' misfit is fairly well compensated in that channel at the verge of the γ' precipitates (Fig. 13a and13c) but that the applied stress (along Oz) is still weaker than the Z component of the misfit stresses within the vertical channels (compressive, blue regions in Fig. 13b and13d). At higher applied stress (1200 MPa) and larger plastic strain, the same situation prevails, despite the higher density of dislocations (Fig. 14a and14b) since their presence in the vertical channels is aggravating the misfit rather than compensating it (Fig. 9d). Pushing this model which assumes non-shearable particles having a strong misfit stress field to its upper limits, we show that the development of the dislocation microstructure, even after large cumulated plastic strain, is still governed not only by the applied stress but still very much by the local stress field that it tends to compensate. Let us consider now the strain fields: initially, the matrix channels are prestrained elastically in elongation along Oz by the γ-γ' misfit stress fields, in the direction normal to the γγ' interfaces. All channels are elastically thickened by this internal stress field distribution. According to Hooke's law, the plane stress exerted by the misfit across the horizontal channel, for instance reads:

  0 2 22 11 33            E misfit [8]
since σ 11 = σ 22 =Eδ in that channel. Note that all misfit quantities are negative, The introduction of an applied uniaxial stress σappl will break the initial symmetry between channels and initiate plasticity by introducing dislocations and their associated additional stress field. In particular, let us examine the horizontal channel, which is most exposed to their activity (forward plastic flow induced by σappl) and to their presence in the γ-γ' interfaces where they relax most of the misfit stresses.

Figure 15 shows the various contributions to the zz component of the strain during the simulation. The plastic strain is computed from the area swept by the dislocation lines during tensile straining. This component increases during the simulation up to a cumulated plastic strain of  p zz= 0.7x10 -4 (DDD plastic, pink curve, Fig. 15). Meanwhile the local arrangement of dislocations induces an elastic relaxation of the pre-thickened horizontal channel. This recovered elastic strain (DDD elastic, red curve figure 15) reaches  e zz = -1.8x10 -4 at the end of the simulation sequence.

Summing up the two contributions, we finally obtain an overall contraction of the channel of 1.1x 10 -4 along the tensile direction. The light blue line (Fig. 15) is the elastic strain of the material under 1200 MPa before any plasticity occurred and the dark blue curve the resulting contraction of the material (total strain i.e., elastic + plastic) as plasticity develops and added interfacial dislocations progressively cancel out the misfit stress fields within the P channel. Notice that only a few dislocations have moved after 500 steps, hence the small amplitude of all strains. Experimentally, this unusual behavior has been described as "negative creep" in superalloys. It was first reported in the 80's by [START_REF] Reppich | Negatives Kriechen[END_REF] who could set it aside from the volume contraction taking place in these alloys when a residual precipitation of the ' phase occurs at the test temperature. This apparently paradoxical contraction under tensile stress was correctly interpreted by Louchet (1995) as a process of minimization of the free energy of the material when placed under a tensile stress [START_REF] Ratel | Application of anisotropic inclusion theory to the energy evaluation for the matrix channel deformation and rafting geometry of Ni superalloys[END_REF] but was improperly attributed to the rafting of the microstructure, when it is simply concomitant. As creep progresses, the movement of perfect matrix dislocations is made easier by the widening of the P matrix channels resulting from the rafting and the coarsening of the microstructure. These processes are stimulated by the internal stress field distribution generated by the 'misfit and diffusion is enhanced in the core of dislocations. In nickel base superalloys, this rafting phenomenon is necessarily preceded by the micro-mechanisms described in this model. It is actually biased by the applied stress field [START_REF] Pineau | Influence of uniaxial stress on the morphology of coherent precipitates during coarsening. Elastic energy considerations[END_REF] but also requires diffusional processes, abundantly activated during creep at high temperature and enhanced by the presence of interfacial dislocation networks [START_REF] Carry | Apparent and effective Creep Parameters in Single Crystals of a Nickel Base Superalloy, Part II: Secondary Creep[END_REF][START_REF] Strudel | Nickel base superalloys: an engineering and scientific challenge[END_REF].

The experimental recording of negative creep is rather difficult since it requires a fully stabilized material (absence of residual γ' precipitation), a high thermal stability of the extensometer and a low noise (better than 10 -9 s -1 , on the strain rate). In order to avoid the triggering of additional mechanisms of plasticity taking place during later stages of creep or under high stress levels, very low stresses must be applied for fairly long times (> 1000 h). In the development program of PM alloy N 19, three different alloy compositions, with γ' volume fractions higher than 40%, were tested in creep at 700° C under the same stress of 650 MPa. The creep curves are presented in Figure 16 where a (log t) scale is used in order to stretch out the primary stage of creep. For the two lesser resistant alloys (AD-8 and René 88) their creep rate remains quasi-null for several hundred hours, thus indicating that shrinkage due to elastic relaxation along the Z direction is just about compensated by the forward elongation associated with the plasticity of this initial stage II. On the other hand, the more resistant alloy AD-5 clearly exhibits negative creep under the selected stress of 650 MPa.

These results together with the neutron diffraction in situ measurements of lattice parameters (under applied stress and after straining of PM alloy RR 1000, [START_REF] Lu | Elastic microstrains during tension and creep of superalloys: results from in situ neutron diffraction[END_REF] clearly imply that, regardless of grain orientation the first active populations of dislocations are mobilized to cancel out some of the numerous pre-existing internal elastic strain fields generated within the crystals by the misfit of γ' precipitates of all sizes.

Ideal interfacial networks

In the absence of any externally applied stress, when ' precipitates are in the micron range and above, matrix dislocations, stored in the subgrain boundaries, move spontaneously at high temperature, during aging at 1000°C/24h in AM1 alloy for instance (Fredhom et al., 1988;[START_REF] Long | Effect of lattice misfit on the evolution of the dislocation structure in Ni-based single crystal superalloys during thermal exposure. Louchet F., 1995. A model of negative creep in nickel based superalloys[END_REF] and cover the overgrown precipitates with 3-D edge oriented networks of edge dislocations [START_REF] Lasalmonie | Interfacial Dislocation Networks Surrounding γ' Precipitates in a Nickel Base Alloy[END_REF]. Under these conditions, the mesh size d of the equilibrium edge networks that best compensate for the existing misfit, is given by Brooks formula [START_REF] Brooks | Theory of internal boundaries[END_REF]:

 b d  [9]
where b is the modulus of the projection of the Burgers vector of the dislocations onto the γ-γ' interface and the misfit parameter. For a nickel base superalloy with lattice parameter a=3.56 Å, then b=a/2 and with a misfit parameter│ δ │= 3.10 -3 , the equidistance of dislocation lines is 60 nm.

During creep at intermediate and high temperatures (650°C to 1100°C) and under a low enough applied stress where the matrix is plastically deformed but the ' precipitate are not sheared, matrix dislocations with b=a/2 <110> can move into the matrix channel normal to the applied tensile stress, the P channel (as described in § 3.4), under the combination of the applied stress and the local stress fields generated by the ' misfit. They are generally observed to progress into the channels by repeated cross slip on <111> planes and their initial zigzag aspect results from their immobilization in the directions common to their slip planes and the ' interfaces. By climb in these interfaces, they rapidly shorten their length by straightening out from their ±60° orientations into pure edge, cube orientations; a configuration that best compensates the pre-existing stress fields [START_REF] Carry | Direct observation of <110>-{1-10} in FCC single crystals of a Ni base superalloy[END_REF][START_REF] Lasalmonie | Interfacial Dislocation Networks Surrounding γ' Precipitates in a Nickel Base Alloy[END_REF]. They have been observed covering hundreds of rafted γ' precipitates after longtime creep under low stress by [START_REF] Zhang | The effect of lattice misfit on the dislocation motion in superalloys during high-temperature low-stress creep[END_REF]. Under a uniaxial stress applied along the Z axis, four different Burgers vectors can be activated. Considering that each of them is associated with an array of parallel edge dislocations, periodically spaced by 60 nm, with cube orientations [100] and [010] in the horizontal plane (Ui vectors), we end up with a square grid of dislocation lines (Fig. 17), each, lying in the γ-γ' interface, with their extra half plane pointing toward the precipitate. This idealized dislocation network is generating, in the P channel, a dipolar stress field of significant intensity. Its computed components are represented on Figure 18. Notice that:

 they are generally similar to the components of the stress fields generated by the γ-γ' misfit but of opposite sign [START_REF] Ratel | Predicting the onset of rafting of γ' precipitates by channel deformation in a Ni superalloys[END_REF].

 the vertical component of the stress field σ33 (Fig. 18b) is almost null in the channel, except in the immediate vicinity of the dislocations where it is strongly nonuniform.

 on the other hand, the planar biaxial stress field (σ11, σ22) is quite uniform and strong across the P channel (Fig. 18a), exhibiting minor variations in the vicinity of the lines, as expected.

 the width of the channels (100 nm) being close to the distance between interfacial dislocation lines (80 nm), the stress fields generated by this array of dislocations is hardly perturbed by the crossing of the vertical channels.

If we assume that the stress fields are completely uniform in the P channel, a simple estimate of the elastic and plastic components of the strain associated with the development of such ideal dislocation networks can be made. The elastic relaxation of the pre-widened channel P is given by the equation [8] and yields: On the other hand, the plastic strain generated by the movement and multiplication of dislocations crisscrossing across the P channel in order to generate this ideal array, can be calculated by estimating the total area swept:

           w a L b plastic    [11]
since ρ, the dislocation density in the elementary simulation cube (size L 3 ) is the sum of twice (2 orthogonal directions in the horizontal plane) 4 pairs of dislocations (with b=[±101]) with 60 nm distance between lines, in each interface of the channel (x2), hence:

           L L L  [12]
This simple calculus clearly confirms the result of the simulation presented in Figure 16 showing that the elastic relaxation of the pre-widened P channel, provided by the development of interfacial dislocation networks, is significantly larger than the plastic strain produced by the movement of dislocations during the first stages of plasticity, in nickel base superalloys at high temperature. However, this simple calculus yields an upper bound to the amplitude of the contraction taking place during the first stage of creep, since it is assuming that this elastic relaxation is taking place uniformly throughout the test sample and is also the sole mechanism of plasticity taking place during that stage. In reality, plasticity is nonuniform throughout the gauge length under control, some areas may still be undeformed when others are already involved in more advanced stages of creep with higher rates of plasticity which involve shearing of precipitates, migration of grain boundaries, climb, recombinations, annihilations and other restoration processes.

Finally we investigate the elastic stress field that would induce a perfect periodic square grid of a/2[±101] and a/2[0±11] edge dislocations, similar to the previously described network but located away from each other by 500 nm and centered in the middle of a large γ' precipitate (Fig. 19). Most of the general features are similar to the previously described dipolar wall, except for the uniform aspect of the fields which now spread out without much decay between the two polar walls of opposite signs. In polycrystalline materials used for discs, the volume fraction of γ' precipitates is over 40% and large precipitates (in the micron range or over) are scattered throughout individual grains where finer secondary and tertiary precipitates are also present. Although these larger precipitates are no longer closely aligned in quasi-periodic manner as in single crystals, the situation we have concentrated most of the simulation so far, nevertheless we can see that similar stress fields can develop and produce similar relaxation effects during the first stages of plasticity in polycrystalline materials. This point is confirmed experimentally, in Fig. 16 where the creep curves of polycrystalline alloys under 650 MPa tensile stress at 700°C, exhibit either a prolonged period of several hundred hours without any detectable strain or even a contraction ranging to about 10 -3 , before it starts flowing forward in the tensile direction; a behavior often and improperly described in the literature as "incubation period".

When building these idealized networks, it was assumed that the dislocations would cross each other at right angle without interacting. In reality, they do recombine and form square arrangements of lines [START_REF] Pollock | Dislocations and high temperature deformation of superalloy single crystals in "Dislocations in Solids[END_REF][START_REF] Lasalmonie | Interfacial Dislocation Networks Surrounding γ' Precipitates in a Nickel Base Alloy[END_REF] with Burgers vectors a/2[±110] and a/2[1±10]. Notice that these dislocations exhibit an extra half plane not only pointing toward the γ' precipitate but directly perpendicular to the interface, thus directly contributing to the compensation of misfit. On the other hand, they should remain inactive since their Burgers vectors are normal to the applied stress.

CONCLUSION

In nickel base superalloys primary creep is very important since it covers most of the service life of critical engineering parts such as discs or blades. Therefore, developing a reliable model describing, in details, the first stages of plasticity taking place in a wide range of temperatures (500°C to 1100°C) offers the tool for an appropriate and quantitative interpretation of the unusual minimum creep rate exhibited by these γ' strengthened alloys.

The micromechanical model presented here is a combination of FEM, which provides updated detailed mapping of local stress fields and DDD techniques which describe the multiplication and movement of perfect dislocations within the prestrained channels of the γ matrix. For the sake of simplicity, it is based on a strictly triperiodic arrangement of identical γ' cuboids having a misfit parameter δ= -3.10 -3 with the matrix.

This model confirms that the negative γ-γ' misfit induces an in-plane equi-biaxial stress in each channel parallel to the interface that results, in the orthogonal direction, in an elastic elongation of the width of the channel:

  0 2 22 11 33            E misfit for instance.
When a tensile load is imposed along the (001) axis, an asymmetry is introduced which biases the first stages of plasticity. Slip systems, involving dislocations with four different Burgers vectors are equally activated, each with equal probability of cross slip on two different planes. The channel normal to the tensile stress (channel P) is penetrated under local stresses lower than the Orowan stress, since it is attractive for these dislocations. The channel parallel to the tensile stress and the Burgers vector (channel C), where the dislocations leave the screw dipoles in the γ-γ' interface is explored next. The other vertical channel (channel R), where the edge component of the dislocations would aggravate the misfit rather than compensate for it, is repulsive to the dislocations and would only be penetrated under high applied stress. Thus, a specific arrangement of near edge dislocations develops at the γ-γ' interface of the P channel which progressively relaxes the elastic pre-strain prevailing initially in this channel. In reality, it is observed that climb in these interfaces enables zigzagging segments of 60° dislocations to straighten out into pure edge perfect dislocations, oriented along the cube directions, with their extra half plane's pointing into the γ' precipitates and thus efficiently compensating the misfit in channel P. The proposed model, in its present form, cannot describe this particular stage of primary creep since it does include any diffusional mechanism.

It is shown that the development of this square grid of dislocations in the P channel produces an elastic relaxation along the tensile axis, the amplitude of which is larger than the forward plastic strain in that direction, assuming that this mechanism is acting alone and uniformly across the creeping part or test piece. These latter conditions are rarely met in practice, but when they are, under carefully applied low stresses for instance, these findings explain the negative creep results observed experimentally during the initial stage in Ni based superalloys, single crystals and polycrystals as well.

The bipolar stress field generated by an idealized square network of straight and pure edge dislocations of opposite signs lying in two parallel planes was computed in order to compare it with the stress field generated by the γ-γ' misfit in the matrix channel, assuming similar elastic moduli of both phases. Making use of Brook's formula, the distance between dislocations was chosen to exactly compensate for the selected misfit of -3.10 -3 . As expected, the field is rather heterogeneous at distances from the dislocations equivalent to their equidistance, but rapidly becomes uniform, with a predominant plane stress tensile nature, exactly opposite to the plane stress compressive nature prevailing in the matrix channels. This clearly indicates that the quasiperiodic dislocation networks observed after creep in the γ-γ' interfaces are effectively compensating for the misfit and that their interconnections between P channels, promoting paths for pipe diffusion become main channels for microstructural rearrangements taking place during creep and called rafting. The micro-mechanisms described in the present model are the forerunners of this phenomenon, widely observed during creep in nickel base superalloys.

Fig. 1 :

 1 Fig. 1: Creep rate of René 88 at 700°C (Augustins-Lecallier, 2012).

Figure 4 :

 4 Figure 4: Geometry of the periodic simulation box and the corresponding finite element mesh used for the DD modelling. A typical ' cuboid particle is superimposed to better illustrate the 3D geometry.

Figure 5 :

 5 Figure 5: Initial dislocation configuration for the DD simulation and facet description. The upper part of the simulated box was removed for clarity.

Fig. 6 :

 6 Fig. 6: Misfit stress components in the matrix channels, computed for δ = -3.10 -3 .

Fig. 7 :

 7 Fig. 7: Misfit strains in the matrix channels, computed for δ = -3.10 -3 .

Fig. 8 :

 8 Fig. 8: Resolved shear stress induced on slip systems [011] (1-11) or (11-1) and [0-11](111) or (-111): a) by misfit alone. b) by misfit and tensile stress of 500 MPa along the vertical axis.

  attractive if the applied stress would overcome the local compressive component of the γ-γ' misfit in the Z direction.

  Misfit stress components alone in the first column. Forces, under appl =300 MPa, on the dislocation segments lying in the γ-γ' interfaces U║, or bowing out into the channels U=. 10 -3 N/m taking a=3.56 Å. Locations A, B, …F refer to Fig.9d.

Fig. 9 :

 9 Fig. 9: Equilibrium dislocation configuration for a b=a/2[011] shear loop expanding in a (1-11) plane, within the misfit internal stress field and under different applied tensile stresses, before any cross slip event takes place. Snapshots of DDD simulations: a) under 300 MPa; b) under 400 MPa; c) under 600 MPa; d) Schematic with projected forces F║ and F= in red.

Fig. 10 :

 10 Fig. 10: Cross slip scene. The elementary computation box is drawn in red in the lower left hand corner. a) View along [001]; b) View along [-101].

  Fig. 3b. Hence the cross slip ability retained for the model was adjusted here to fit the experimental observations as can be seen on fig. 10 and fig. 11a.

Figure 11 :

 11 Figure 11: Dislocation microstructure formed from two screw dislocation sources a/2 [-101] (in blue) and a/2 [011] (in red) when (a) cross-slip is limited or (b) enhanced.

Fig. 12 :

 12 Fig. 12: Stress component σ 11 resulting from the presence of dislocations alone, after applying 875MPa in 1868 steps. a) at mid P channel and b) at 0.05w.

Fig. 14 :

 14 Fig. 14: Stress components resulting from the presence of dislocations alone after applying σ appl =1200MPa. a) σ11 or σ22, at mid P channel and b) σ33 at mid P channel.

Fig. 15 :

 15 Fig. 15: Evolution of zz strain components in the horizontal channel, during the first stages of plasticity under uniaxial loading along Oz.

Fig. 16 :

 16 Fig.16: Total strain (elastic + plastic) during creep of three PM disc alloys at 700°C under 650 MPa. Notice the long "incubation period" of alloys AD8 and René 88 and the negative creep rate of the most resistant alloy AD5 (Augustins-Lecallier, 2012)

Fig. 17 :

 17 Fig. 17: Details of Burgers vectors bi, line vectors ui and extra half plane orientations used in the description of an ideal γ-γ' interfacial dislocation network.

Fig. 18 :

 18 Fig.18: Ideal interfacial dislocation network in channel P in absence of misfit stress fields and associated internal stress field a) 11 or 22 ; and b) 33.

Fig. 19 :

 19 Fig. 19: Ideal γ-γ' interfacial dislocation network at large distances (w=500 nm) and associated stress field components a) 11 or 22 and b) 33. Arrows corresponds to the Burgers vectors of each dislocation line.

Acknowledgements

This research was carried out under Project ANR-07-MAPR-0023-04 CAT-SIZE Matériaux et Procédés which provided the financial support for Dr H.J. Chang. We also acknowledge the financial support of DGA, through the PROMETHEE program, that offered a framework for the alloy development program N-19 launched and guided by SAFRAN Motors. The authors also gratefully acknowledge the fruitful and stimulating discussions and the enthusiastic support of S. Forest.


Rachel Nadif

Steven R Kleeberger

Francine Kauffmann

Polymorphisms in manganese superoxide dismutase and catalase genes: functional

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

RE "Polymorphisms in manganese superoxide dismutase and catalase genes: functional study in Hong Kong Chinese asthma patients" Rachel Nadif 1,2 , Steven R. Kleeberger 3 , and Francine Kauffmann 1,2 1 INSERM, U780, Villejuif, France. 2 Université Paris Sud, Faculté de Médecine, IFR69, Villejuif, France. 3 It is interesting to note that Mak et al (1) observed a protective effect of the CAT -262 T allele in non-smoking asthma patients only. Regardless of the disease, an oxidant-rich environment has been found to increase risk only in subjects with the low-activity catalase CAT -262 T allele [START_REF] Ahsan | Susceptibility to arsenic-induced hyperkeratosis and oxidative stress genes myeloperoxidase and catalase[END_REF]. Factors previously found to be protective, such as an antioxidant-rich dietary environment or a low level of circulating proinflammatory cytokines, seemed to be beneficial only in subjects with the high catalase activity CC genotype [START_REF] Nadif | RE "Associations between breast cancer risk and the catalase genotype, fruit and vegetable consumption, and supplement use[END_REF]. Similar to the oxidant-rich environment in our occupational study, asthma may generate reactive oxygen species, and catalase activity significantly increases in asthma patients compared to controls [START_REF] Mak | Polymorphisms in manganese superoxide dismutase and catalase genes: functional study in Hong Kong Chinese asthma patients[END_REF]. As catalase is involved in the first response to oxidative stimuli, the increase in its activity may be interpreted as a compensatory response to the changes in the oxidant/antioxidant imbalance occuring in asthmatics. In the study of Mak et al. [START_REF] Mak | Polymorphisms in manganese superoxide dismutase and catalase genes: functional study in Hong Kong Chinese asthma patients[END_REF], the level of the oxidant state related to asthma may partly mask the a priori deleterious effect of the CAT -262 T allele.

Further, the CAT -262 T allele frequency in this Chinese population (1) was five times lower than that observed in Caucasians (mean 26 percent [20-56] from 7 studies), and was similar to the frequency observed in 40 African Americans (6 percent) by Zhou et al. [START_REF] Zhou | Polymorphisms in the promoter region of catalase gene and essential hypertension[END_REF]. The differences in reported CAT -262 T allele frequency between studies indicates significant variation between populations, and could also partly explain the a priori deleterious effect of the CAT -262 T allele, as patterns of linkage disequilibrium across populations were different.

In summary, these results highlight the complexity of the potential role of the CAT -262 polymorphism in the etiology of oxidant-related diseases.

words

  Laboratory of Respiratory Biology, National Institute of Environmental Health Sciences, Research Triangle Park, NC. We read with interest the paper by Mak et al. (1) on asthma in a Hong Kong Chinese population. They reported a protective effect of the catalase (CAT) -262 T allele in nonsmoking asthma patients. This result is of importance, as it suggests for the first time a potential role of the CAT -262 polymorphism in the etiology of asthma. At variance with previous publications was the lack of association of erythrocyte catalase activity with the CAT -262 polymorphism within controls and cases. In a cohort of 196 coal miners, we observed lower erythrocyte catalase activities in miners heterozygous and in those homozygous for the CAT -262 T allele, heterozygotes having an intermediate activity (2). Although the difference was not statistically significant, a 28 percent decrease in catalase activity was observed in the non-smoker T carrier controls by Mak et al. (1), similar to the 31 and 33 percent of decrease reported by us (2), and by Ahn et al. (3) in 18 women from a population-based case-control study respectively. As Mak et al. (1) observed a decrease in catalase activity related to CAT -262 polymorphism in non-smokers only, we stratified on smoking status in our coal miner cohort. We found again a significant decrease in catalase activity with the CAT -262 T allele in current and in ex-or non-smokers (see figure), as we found previously in miners with current high and low exposure to coal dust (2), which demonstrates the consistency of the association. The lack of significance of the association in the study by Mak et al. (1) may be due to a low statistical power as 28 subjects were heterozygous and only one was homozygous for the CAT -262 T allele out of the 308 Chinese controls.

ACKNOWLEDGMENTS

Conflict of interest: none declared.