J P Oliveira

Z Zeng

S Berveiller

D Bouscaud

F M Braz Fernandes

R M Miranda

N Zhou

Laser welding of Cu-Al-Be shape memory alloys: Microstructure and mechanical properties

Successful laser welding of Cu-Al-Be shape memory alloys • The joints exhibit excellent superelastic properties. • Superelasticity is preserved between -40 and 100 °C. • The joints can be used as energy absorptions devices.

Introduction

Shape memory alloys have the ability to undergo a reversible thermoelastic phase transformation from the parent phase austenite to a product phase martensite. This particular behaviour is responsible for two main functional properties: superelasticity and shape memory effect [START_REF] Jani | A review of shape memory alloy research, applications and opportunities[END_REF]. In the former one, the martensitic transformation is induced by applied thermomechanical loading of the austenitic phase, leading to important inelastic strains which are recovered upon unloading due to the reverse phase transformation to austenite. Austenite can transform into martensite during mechanical loading, leading to an inelastic deformation that can be recovered spontaneously upon unloading [START_REF] Patoor | Shape memory alloys, part I: general properties and modeling of single crystals[END_REF].

For the shape memory effect, the material is deformed in the martensitic state, leading to permanent inelastic strains upon unloading. These can be removed by heating up the material to the austenitic phase again: the initial shape is restored triggering the shape memory effect [START_REF] Oliveira | Welding and joining of NiTi shape memory alloys: a review[END_REF].

Cu-based shape memory alloys are considered to be possible substitutes of NiTi, the most used shape memory alloy, owing to their lower production costs and recent improvements on their mechanical properties [START_REF] Sutou | Characteristics of Cu-Al-Mnbased shape memory alloys and their applications[END_REF][START_REF] Sutou | Effect of grain size and texture on pseudoelasticity in Cu-Al-Mn-based shape memory wire[END_REF]. There are different Cu-based systems, with the most important ones being Cu-Al-Mn [START_REF] Sutou | Effects of ageing on bainitic and thermally induced martensitic transformations in ductile Cu-Al-Mn-based shape memory alloys[END_REF], Cu-Al-Ni [START_REF] San Juan | Superelastic cycling of Cu-Al-Ni shape memory alloy micropillars[END_REF] and Cu-Al-Be [START_REF] Lopez-Ferreño | Thermal treatments and transformation behavior of Cu-Al-Be shape memory alloys[END_REF]. The latter one presents, aside from the superelastic and shape memory effects, other interesting properties: strong sound, vibration and mechanical damping capability; high mechanical strength; and resistance to corrosion [START_REF] De Albuquerque | Evaluation of grain refiners influence on the mechanical properties in a CuAlBe shape memory alloy by ultrasonic and mechanical tensile testing[END_REF]. These alloys exhibit also a particular technological interest for low and intermediate temperature applications [START_REF] Lopez-Ferreño | Thermal treatments and transformation behavior of Cu-Al-Be shape memory alloys[END_REF]: as a result of the introduction of small amounts of Be, the transformation temperatures decrease drastically, allowing for the occurrence of superelasticity at very low temperatures. In fact, some empirical formulae have been proposed to determine the transformation temperatures in Cu-Al-Be alloys based on their composition [START_REF] Montecinos | Superelastic behavior and damping capacity of CuAlBe alloys[END_REF]. Thus, these alloys can be used in a wide range of temperatures, at ambient temperature or above, but they can be considered also for cryogenic applications [START_REF] De Albuquerque | Evaluation of grain refiners influence on the mechanical properties in a CuAlBe shape memory alloy by ultrasonic and mechanical tensile testing[END_REF].

One of the potential applications of Cu-Al-Be alloys is in seismic devices [START_REF] Sepulveda | Metastable effects on martensitic transformation in SMA[END_REF] as they combine several properties: high fatigue resistance, coupled with a low sensitivity for changes in the mechanical behaviour for near-ambient temperatures; they are not very sensitive to frequency changes in the 0.1 to 5 Hz regime. Moreover their properties suffer slow degradation due to the surrounding environment [START_REF] Araya | Temperature and grain size effects on the behavior of CuAlBe SMA wires under cyclic loading[END_REF].

In order to allow for the use of these materials in complex shaped applications, it is often required the use of joining techniques [START_REF] Zeng | Laser welding of NiTi shape memory alloy wires and tubes for multi-functional design applications[END_REF]. For shape memory alloys, laser welding is considered as the prime choice owning to its characteristics, such as high precision, reproducibility and reduced extension of the thermally affected regions [START_REF] Steen | Laser Material Processing[END_REF]. Laser welding is the most used technique to join shape memory alloys [START_REF] Song | The influence of laser welding parameters on the microstructure and mechanical property of the as-jointed NiTi alloy wires[END_REF][START_REF] Sadati | Investigation of mechanical properties in welding of shape memory alloys[END_REF]. This technique has been already successfully employed to preserve the functional properties in similar and dissimilar joints of Cubased shape memory alloys, namely CuAlMn [START_REF] Oliveira | Laser welded superelastic Cu-Al-Mn shape memory alloy wires[END_REF][START_REF] Oliveira | Dissimilar laser welding of superelastic NiTi and CuAlMn shape memory alloys[END_REF] and CuAlZn [START_REF] Biffi | Microstructure and calorimetric behavior of laser welded open cell foams in CuZnAl shape memory alloy[END_REF]. Furthermore, post-weld processing of the laser welded joint can improve the mechanical properties of the material [START_REF] Oliveira | Improvement of damping properties in laser processed superelastic Cu-Al-Mn shape Fig. 11. a) Evolution of the accumulated irrecoverable strain with the number of load/ unload cycling tests; b) Evolution of the stress onset for the forward martensitic transformation with the number of load/unload cycling tests; c) Evolution of the absorbed energy with the number of load/unload cycling tests. memory alloys[END_REF]. Aside from obtain defect-free joints, the main concern after welding of such class of materials is to preserve their functional properties, so that they can be used in a given required application. With the more massive use of the aforementioned Cu-based systems, weldability studies on both similar and dissimilar combinations are necessary to understand the potential of these materials in different applications.

Laser welding of Cu-based materials is not easy due to the high reflectivity of these alloys for wavelengths higher than 550 nm [START_REF] Yao | Interface microstructure and mechanical properties of laser welding copper-steel dissimilar joint[END_REF][START_REF] Kuryntsev | Fiber laser welding of austenitic steel and commercially pure copper butt joint[END_REF]. Industry-relevant lasers are based on fiber or Nd:YAG, with associated wavelengths of 1064 nm. To make it suitable for industry to join these materials, it is necessary to optimize the process parameters so that these lasers can be used.

In this work the microstructure and mechanical properties of similar laser welded Cu-Al-Be shape memory alloy joint are analysed based on the induced microstructural modifications by the laser/material interaction. This is the first work reporting the effect of laser welding on the mechanical properties of Cu-Al-Be shape memory alloys.

Materials and methods

Laser welding

Single crystal-like Cu-11.5Al-0.5Be (wt%) shape memory alloy wires produced by modified Bridgman growth technique, 1.25 mm in diameter, were used in this investigation. Differential scanning calorimetry measurements of the base material has confirmed that it was fully austenitic at room temperature.

Prior to welding, the oxide layer of the base material wire was gently removed using 2500 fine SiC paper, which lead to a final diameter of 1.24 mm. Acetone and ethanol were used to remove any impurities left in the base material before welding.

Laser welding of the single crystal wires was performed on bead on plate in order to remove joint fit-up as a variable, since this was the first investigation on laser welding of these materials. A Nd:YAG laser (1.064 μm wavelength), from Miyachi Unitek LW50 operating in pulsed mode was used. The laser beam was focused on the surface to a spot diameter of 600 μm. A schematic representation of the welding setup is depicted in Fig. 1. The welding process was selected in order to obtain full penetration joints, with minimum heat input. After preliminary tests, a 6 ms pulse profile, which included a 1 ms up and downslope, and a peak power of 5 kW, ensured complete penetration. To avoid oxidation, Ar was inserted at the face and root of the welds at a flow rate of 0.6 m 3 h -1 .

Microstructural and mechanical characterization

After mechanical polishing, the welded joints were prepared for metallographic observations to reveal the microstructure, using an electrolytical polishing a solution of 250 ml of phosphoric acid, 5 g of urea, 250 ml of ethanol, 50 ml of isopropanol and 500 ml of distilled water, followed by chemical etching with a solution of 10 ml of hydrochloric acid, iron chloride and 100 ml of distilled water. Optical microscopy and scanning electron microscopy (SEM) were performed in the fusion zone and the base material.

The crystallographic texture was determined by X-ray diffraction using a Seifert XRD 3003 PTS 4-circle goniometer. Measurements were performed with a Cu-Kα radiation operating at 40 kV and 40 mA. The phi (Φ) angle was varied between 0 and 360°, whereas the psi (Ψ) angle was varied from 0 to 70°. For both angles, the incremental step was of 2°. The β austenitic phase has a complex DO3 cubic structure with a cell parameter a 0 = 5.825 Å [START_REF] Montecinos | Thermomechanical behavior of a CuAlBe shape memory alloy[END_REF]; pole figures were determined for the {440} planes. Local crystallographic orientations were determined using Electron Back-Scattered Diffraction (EBSD) maps with a step size of 2 μm using a JEOL field electron gun SEM equipped with an Oxford EBSD CCD camera fitted with a FSD detector. EBSD data was treated with the software CHANNEL 5 from Oxford.

Hardness measurements were performed to evaluate the effect of the welding process across the welded joints. A load of 300 g and a space between indentations of 150 μm were used. Three hardness measurement lines were performed at one third, half and two thirds of the sample height.

Mechanical properties of the base material and welded joints were measured during a tensile test at room temperature using an Instron 5548 micro-tensile tester. The gauge length of the samples was of 20 mm; the displacement rate was set at 0.5 mm/min and the measurement accuracy is of ±0.5 μm. Tensile tests were performed on the welded joints. Three samples were tested for reproducibility. Scanning electron microscopy was used to analyse the surface fracture of the joints. Based on the tensile tests performed, the cycling behaviour of the Cu-Al-Be laser welded joints was analysed by performing a total of 10 load/unload cycles. Load/unload cycles up to different maximum strains ranging from 10 to 47.5% were performed at room temperature, -40 °C and 100 °C to evaluate the effect of temperature on the cycling response.

Results and discussion

Fig. 2 depicts the microstructure of the laser welded joint. No defects, such as pores or cracks are observed within the fusion. A columnar-like subgrain structure is observed in the fusion zone. This large columnarlike subgrain structure, with lengths of nearly 400 μm, develops by epitaxial grain growth starting on solid grains of base material on the fusion line [START_REF] Kou | Welding Metallurgy[END_REF] and progresses perpendicular to isothermal lines. The grain growth is larger on the areas subjected to higher temperatures and longer residence times [START_REF] Kou | Welding Metallurgy[END_REF].

Crystallographic orientations of the base material and the welded joint were deduced from the {440} pole figures measurement: both zones exhibit a strong 〈100〉 texture in the wire axis. The columnar grains seem to have the same orientation as the initial single crystal. Therefore, the welding process does not contribute to the development of different crystallographic texture within the fusion zone. In order to confirm this result, the welded zone was studied by EBSD (Fig. 3).

EBSD observations confirm that the welded zone has the same orientation as the initial crystal. Misorientation between columnar grains is also very weak: it does not exceed 3°.

In both FCC and BCC metals, the easy growth direction is along the ⟨100⟩ direction [START_REF] Kou | Welding Metallurgy[END_REF][START_REF] David | Welding: solidification and microstructure[END_REF]. Two factors contribute for the development of the ⟨100⟩ texture in the fusion zone: this is the easy-growth direction for Cu-Al-Be alloys as these have a complex BCC structure [START_REF] Montecinos | Martensitic transformation and grain size in a Cu-Al-Be alloy[END_REF]; the base material is oriented in that direction and supports the onset for solidification, thus further favouring such crystallographic orientation [START_REF] David | Correlation between solidification parameters and weld microstructures[END_REF][START_REF] Park | Stray grain formation in single crystal Ni-base superalloy welds[END_REF].

Hardness measurements performed at one third, half and two thirds of the sample height show that the fusion zone has the same hardness as the base material (Fig. 4), with respectively 321.6 ± 2 HV and 324.5 ± 3 HV. This is in agreement with previous works performed on other laser welding of Cu-Al-based alloys where no significant hardness changes were found between the fusion zone and the base material [START_REF] Oliveira | Laser welded superelastic Cu-Al-Mn shape memory alloy wires[END_REF][START_REF] Oliveira | Improvement of damping properties in laser processed superelastic Cu-Al-Mn shape Fig. 11. a) Evolution of the accumulated irrecoverable strain with the number of load/ unload cycling tests; b) Evolution of the stress onset for the forward martensitic transformation with the number of load/unload cycling tests; c) Evolution of the absorbed energy with the number of load/unload cycling tests. memory alloys[END_REF].

The tensile behaviour of both base material and welded joints is depicted Fig. 5. It can be observed that the tensile strength of the base material is slightly higher than that of the welded joints (686 ± 8 MPa vs 631 ± 19 MPa, respectively). On the other hand, the fracture strain of the joints is higher (50.8 ± 0.2% vs 54.1 ± 1.2%). Fracture of the joints occurred in the fusion zone. The higher fracture strain of the welds is probably due to fact that with the refined grain structure of the fusion zone any crack that initiates there will not propagate straightforward along the wire diameter (as it would occur in the single crystal base material), hence allowing the material to deform more. It is interesting to note that the overall tensile properties of the welded joints do not differ significantly from those of the base material, which appears to be a common feature for welded joints of Cu-based shape memory alloys (CuAlMn) [START_REF] Oliveira | Laser welded superelastic Cu-Al-Mn shape memory alloy wires[END_REF] but not for NiTi [START_REF] Song | The influence of laser welding parameters on the microstructure and mechanical property of the as-jointed NiTi alloy wires[END_REF][START_REF] Tuissi | Effect of Nd-YAG laser welding on the functional properties of the Ni-49.6at[END_REF][START_REF] Gugel | Laser welding of NiTi wires[END_REF][START_REF] Chan | Effects of process parameters upon the shape memory and pseudo-elastic behaviors of laser-welded NiTi thin foil[END_REF], which is the most studied shape memory alloy.

From Fig. 5 it can also be observed that the stress onset for the martensitic transformation is similar for both base material and welded joints: approximately 400 MPa. Although it is known that Cu-Al-Be shape memory alloys have a dependence between grain size and the stress for the onset of the superelastic plateau (the higher the grain size, the lower the stress onset) [START_REF] Montecinos | Grain size and pseudoelastic behaviour of a Cu-Al-Be alloy[END_REF], the volume fraction of the welded region, compared to the remaining base material during tensile testing, is not enough to promote any significant change in respect to this mechanical property.

Fracture analysis of the welded joints was performed using scanning electron microscopy (Fig. 6). Typical ductile-like fracture, characterized by the massive presence of dimples, can be observed. Additionally, it can be detected the presence of minor cracks (marked with white arrows). The presence of those cracks in the fusion zone justifies the higher deformation until fracture of the welds: while in the single crystal base material any crack that form will propagate without any impingement, such as grain boundaries, the same does not occur in the welded joints, as the refined grain structure can delay crack propagation and also enables to accommodate stress.

Due to their lower production costs when compared with NiTi alloys, Cu-based shape memory alloys can be used in energy-absorption devices [START_REF] Sutou | Effect of grain size and texture on pseudoelasticity in Cu-Al-Mn-based shape memory wire[END_REF]. For such applications, it is required that the materials in use are able to absorb significant amounts of energy. For shape memory alloys, such dissipation is usually performed in the superelastic regime so that the imposed external deformation can be (ideally) totally restored. For this reason, is important to understand the load/unload cycling behaviour of the welded joints in order to assess the feasibility of their use in such applications.

Based on the tensile behaviour of the welded joints, their cycling behaviour was analysed by a load/unload cycle up to different maximum strains: 10, 20, 37.5 and 47.5% (Fig. 7). It can be observed that the superelastic plateau extends up to about 40% strain, as a result of the unique characteristics of the single crystal base material. This means that the transformation strain of the welded joints (the length of the superelastic plateau) is of nearly 35%. Such high transformation strains were also reported in Cu-based alloys [START_REF] Mussot-Hoinard | Influence of Resistance Welding on the Properties of a CuAlBe Single Crystal Shape Memory Alloy[END_REF], but also in the Ni-Ti-Hf system [START_REF] Wu | Ultrahigh tensile transformation strains in new Ni 50.5 Ti 36.2 Hf 13.3 shape memory alloy[END_REF]. After the end of the superelastic plateau (corresponding to the applied strain of 47.5%) an increasing stress is required to proceed with deformation, where detwinned martensite is first elastically deformed and, subsequently, plastic deformation occurs [START_REF] Otsuka | Physical metallurgy of Ti-Ni-based shape memory alloys[END_REF].

From the load/unload behaviour of the joints it is possible to determine the evolution of irrecoverable strain and absorbed energy of the Cu-Al-Be welds as a function of the applied strain, as depicted in Fig. 8. These results were obtained only for the first load/unload mechanical cycle. The very high superelastic recovery of the welds is clearly demonstrated with only 0.6, 0.8 and 1.6% of irrecoverable strain for maximum applied strains of 10, 20 and 37.5%, respectively. The increase of irrecoverable strain with applied strain within the superelastic plateau, may be due to the unfavourable microstructure of the fusion zone of the welds, when compared to the single crystal base material. That is, the slight misorientation of the fusion zone compared to the crystallographic orientation of the base material can account for part of the irrecoverable strain. Ideally, in any deformation performed within the superelastic plateau, the irrecoverable strain, upon unloading, would be null. However, the presence of a refined grain structure in the fusion zone may inhibit part of the reverse transformation from martensite to austenite during unloading, thus promoting a non-ideal superelastic behaviour, as evidenced in Cu-Al-Be shape memory alloys [START_REF] Kaouache | Stress analysis of martensitic transformation in Cu-Al-Be polycrystalline and single-crystalline shape memory alloy[END_REF]. However, during the stress induced transformation it is possible that some grains exceed locally the stress for slip to occur [START_REF] Sedmak | Grain-resolved analysis of localized deformation in nickel-titanium wire under tensile load[END_REF], which will result in an irrecoverable strain upon unloading [START_REF] Oliveira | Effect of laser welding parameters on the austenite and martensite phase fractions of NiTi[END_REF]. It must be pointed though, that these results are the best, in terms of superelastic recovery during mechanical cycling, for any welded shape memory alloy, when compared to the available literature on the subject [START_REF] Oliveira | Welding and joining of NiTi shape memory alloys: a review[END_REF][START_REF] Krooß | Electron beam welding of Fe-Mn-Al-Ni shape memory alloy: microstructure evolution and shape memory response[END_REF][START_REF] Oliveira | High strain and long duration cycling behavior of laser welded NiTi sheets[END_REF].

When the applied strain is increased to 47.5%, the irrecoverable strain increases significantly to 12.3% as a result of mechanical stabilization of martensite, which is known to occur after the superelastic plateau as results of plastic deformation of martensite [START_REF] Otsuka | Physical metallurgy of Ti-Ni-based shape memory alloys[END_REF]. Nonetheless, even for this maximum applied strain, the welded joints were able to recover more than 35%, which is a remarkable result as far as the superelastic behaviour of these joints.

Recently, a new textured Fe28Ni17Co11.5Al2.5Ta0.05B (at. %) shape memory alloy was found to absorb a significantly high amount of energy during a hysteric load/unload cycle [START_REF] Tanaka | Ferrous polycrystalline shape-memory alloy showing huge superelasticity[END_REF], which would be very useful in damping applications, for example. The impressive results presented in [START_REF] Tanaka | Ferrous polycrystalline shape-memory alloy showing huge superelasticity[END_REF] are due to the very high stress onset for the beginning of the stress induced transformation and the high mechanical hysteresis between the forward and reverse transformations. In order to compare the absorption behaviour of both the Fe-based shape memory alloy and of the Cu-Al-Be laser welded joint the evolution of the absorbed energy and irrecoverable strain as a function of the applied strain were determined and are presented Fig. 9. The selection of this Fe-based shape memory alloy for comparison with the experimental results obtained for the welded joints is related to the fact that, to the best of the author's knowledge, no other shape memory alloy is known to exhibit such significant energy absorption.

The very high transformation strain of the Cu-Al-Be laser welded joint can exceed the already high absorbed energy of the Fe-based shape memory alloy. However, the high energy absorption capability of the Cu-Al-Be weld comes at an expense of a slight irrecoverable strain, when compared to the Fe-based shape memory alloy.

Nonetheless, these cycling results show the potential for such laser welded joints to be used in energy-absorption applications, such as seismic devices, where large amount of energy need to be absorbed.

Cu-based shape memory alloys have potential applications as seismic devices for energy absorption [START_REF] Araki | Rate-dependent response of superelastic Cu-Al-Mn alloy rods to tensile cyclic loads[END_REF][START_REF] Araki | Feasibility of tension braces using Cu-Al-Mn superelastic alloy bars[END_REF]. However, for these materials to be used in such applications it is necessary that the superelastic behaviour is stable for a wide range of temperatures [START_REF] Zhang | Mechanical properties of superelastic Cu-Al-Be wires at cold temperatures for the seismic protection of bridges[END_REF], below -10 °C and up to 50 °C, ideally.

The laser welded Cu-Al-Be joints were also subjected to a total of 10 load/unload cycles up to 37.5% strain at -40 °C, 23 °C (room temperature) and 100 °C. These results are depicted in Fig. 10, showing only the 1st, 5th and 10th cycles for clarity purposes.

One of the most remarkable features observed from these cycling tests at different temperatures is the double step plateau which occurred at -40 °C (Fig. 10 a). However, this two-step plateau is not observed in the cycling tests performed at both room temperature and 100 °C. Cu-Al-Be shape memory alloys have their stress induced martensitic transformation occurring in the following sequence: β austenite ➔ 18R martensite ➔ 6R martensite. However, the presence of the two-step plateau is evidenced or not depending on the test temperature. In [START_REF] Sade | Pseudoelasticity of Cu-Al-Be single crystals: unexpected mechanical behavior[END_REF], the authors performed load/unload testing on Cu-Al-Be shape memory alloys at different temperatures and observed that Fig. 9. Evolution of the absorbed energy and irrecoverable strain as a function of the applied strain during superelasticity cycling: comparison between the Cu-Al-Be laser welded joint and the Fe28Ni17Co11.5Al2.5Ta0.05B (at. %), noted as FeNiCoAlTaB, shape memory alloy [START_REF] Tanaka | Ferrous polycrystalline shape-memory alloy showing huge superelasticity[END_REF].

at lower testing temperatures the two step plateaus were clearly distinguished. With temperature increase, the first martensitic transformation (β ➔ 18R) starts at significantly higher stress levels, whereas the second martensitic transformation (18R ➔ 6R) starts at a lower stress level. Further increase of the testing temperature would cause for both plateaus to occur at the same stress level. Similar observations were performed in fundamental studies on the mechanical behaviour of CuAlBe shape memory alloys [START_REF] Sade | Pseudoelasticity of Cu-Al-Be single crystals: unexpected mechanical behavior[END_REF].

Of special relevance on the martensitic transformations of Cu-Al-Be shape memory alloys is the dependence of both transformations, β ➔ 18R and 18R ➔ 6R, with temperature: the β ➔ 18R has a positive Clausius-Clapeyron relation of 1.86 MPa/K, while for the 18R ➔ 6R transformation the relation is of -0.28 MPa/K. This means that the temperature effect on the stress onset for the start of the martensitic transformation on Cu-Al-Be shape memory alloys in extremely important for the β ➔ 18R transformation, whereas in the 18R ➔ 6R transformation the temperature effect is almost negligible and tends to decrease the stress onset for the transformation to occur.

In the cycling tests performed at 23 °C and 100 °C only a unique plateau is visible meaning that overlapping between the two stress induced transformation has occurred at these temperatures.

The irrecoverable strain upon unloading in the first load/unload cycle varies with no apparent influence of the temperature. This is related to the fact that during the first superelastic cycles at which shape memory alloys are subject material accommodation occurs. In this case only one load/unload cycle is required for the following cycling tests to present a stabilized behaviour, that is, the irrecoverable strain and mechanical hysteresis between consecutive load/unload cycles is minimal. In fact, the evolution of the accumulated irrecoverable strain and the stress onset for the forward martensitic transformation do not vary significantly after the first load/unload cycle.

Cycling behaviour analysis is laser welded NiTi shape memory alloys showed an increase in the accumulated irrecoverable strain due to introduction of dislocations that stabilize martensite, preventing the transformation to austenite upon unloading, thus not participating further in the superelastic behaviour [START_REF] Oliveira | High strain and long duration cycling behavior of laser welded NiTi sheets[END_REF][START_REF] Oliveira | Martensite stabilization during superelastic cycling of laser welded NiTi plates[END_REF]. Additionally, the introduction of internal residual stresses during superelastic cycling of shape memory alloys justifies the decreasing external stress required for the martensitic transformation to occur [START_REF] Miyazaki | Effect of cyclic deformation on the psedoelasticity characteristics of Ti-Ni alloys[END_REF], as depicted in Fig. 11 b).

The temperature effect on the necessary stress for the martensitic transformation to occur is clearly observed in Fig. 11 b): with increasing temperature, higher stress is required for the transformation to occur, which correlated well with a positive Clausius-Clapeyron relation when the β ➔ 18R transformation occurs.

The evolution of the energy absorbed as function of the number of load/unload cycles at the different testing temperatures is depicted in Fig. 11 c). It can be observed that the first cycle has the higher absorbed energy for all testing temperatures. This is related to the fact that in the following cycles the a given irrecoverable strain has already occurs which reduces the total superelastic strain of the welded joint. At 100 °C, the absorbed energy is slightly higher than at the other two testing temperatures. This is related to the Clausius-Clapeyron effects which rises the stress onset for the stress induced transformation, which translates into a higher capacity for energy absorption. At -40 °C and room temperature the absorbed energy does not vary significantly, despite the two step plateau observed at -40 °C.

Conclusions

Defect-free joints were obtained when laser welding of Cu-Al-Be shape memory alloys and this study shows the potential to use such high energy density welding process for joining these alloys. The following major conclusions can be drawn:

-The crystallographic texture of the fusion zone was not modified by the welding processes, following the same orientation as the base material; -No significant hardness changes were determined between the fusion zone and base material; -The tensile strength of the laser welded joints is slightly lower than the base material (686 vs 631 MPa), whereas the tensile strain was higher for the welds (50.8 vs 54.1%). This was attributed to the microstructure in the fusion zone, which sustain a higher amount of deformation owing to the presence of obstacles that oppose to crack propagation. -Superelastic behaviour of the welded joints was preserved with associated high energy absorption capabilities and transformation strain. Moreover, in the temperature range of -40 to 100 °C, the laser welded joint exhibits superelastic behaviour. These features make these joints potential candidates for applications requiring large damping capabilities, such as in seismic systems.

Fig. 1 .

 1 Fig. 1. Schematic representation of the experimental setup for laser welding of CuAlBe shape memory alloy (not to scale): a) side view; b) top view. The laser beam was focused on the surface to a spot diameter of 600 μm.

Fig. 2 .

 2 Fig. 2. Microstructure of Cu-Al-Be laser welded joint.

Fig. 3 .

 3 Fig. 3. a) Orientation map determined by EBSD; inverse pole figure in the tensile direction (wire axis) for the initial wire (b) and welded zone (c).

Fig. 4 . 5 .

 45 Fig. 4. Hardness measurements results performed at one third (top), half (middle) and two thirds (bottom) of the height of the Cu-Al-Be laser welded joint. Fig. 5. Tensile behaviour of the Cu-Al-Be base material and laser welded joints.

Fig. 6 .

 6 Fig. 6. Fracture surface of the fusion zone the welded joint. The white arrows show other cracks which developed inside the fusion zone.

Fig. 7 .

 7 Fig. 7. Load/unload cycling behaviour of the welded joints.

Fig. 8 .

 8 Fig.8. Irrecoverable strain and absorbed energy as a function of applied strain during one load/unload cycle for the welded joint.

Fig. 10 .

 10 Fig. 10. Cycling tests up to 37.5% strain at different temperatures: a) -40 °C; b) room temperature (23 °C); c) 100 °C.

Acknowledgements

The authors gratefully acknowledge Professor Toshihiro Omori from Tohoku University for providing the raw data of the cycling tests performed on the Fe28Ni17Co11.5Al2.5Ta0.05B at. % shape memory alloy. JPO and FMBF acknowledge funding by National Funds through the Portuguese Foundation for Science and Technology (FCT-MCTES), Reference UID/CTM/50025/2013 and FEDER funds through the COMPETE 2020 Programme under the project number POCI-01-0145-FEDER-007688. RMM acknowledges FCT-MCTES for its financial support via the project PEst-OE/EME/UI0667/2014 (UNIDEMI). JPO acknowledges FCT/MCTES for funding PhD grant SFRH/BD/85047/2012 and stay at Waterloo University (Canada) through MIDAS Project No 612585 "MI-DAS -Micro and Nanoscale Design of Thermally Actuating Systems" Marie Curie Actions, FP7-PEOPLE-2013-IRSES. ZZ and JPO acknowledge National Natural Science Foundation of China (51775091).

Data availability

The raw/processed data required to reproduce these findings cannot be shared at this time as the data also forms part of an ongoing study.


A Coche

T Babey

A Rapaport

P Garnier

Interaction of porosity structures and

Keywords: pesticide remediation, transport processes, microbial uptake

teaching and research institutions in France or abroad, or from public or private research centers.

Interaction of porosity structures and microbial uptake dynamics in the degradation of pesticides at µm and mm scales

Soil microorganisms perform a major ecosystem function by degrading many chemical compounds and contaminants including agricultural pollutants such as pesticides, thus mitigating their dispersion in the environment and contributing to Earth's biochemical cycles. Deciphering the implied mechanisms at the microbial scale appears to be a key point to explain the high variability of remediation efficiency among soils, which could lead to ecologicallybased improvements of agricultural practices. Soils have indeed a great uncertainty on their bulk activity as they are among the most heterogeneous microbial habitats on Earth, with separation distances between pesticides and pesticide degraders large enough to see this habitat as a "desert" for microorganisms. In such context, besides physicochemical and catabolic limitations, spatiotemporal accessibility of pesticide degraders to their substrate might be a primary limiting step of biodegradation.

The objective of this work is to formalize the coupling between chemical transport processes resulting from the porosity structures, sorption processes and microbial processes involved in biodegradation.

We develop two reactive transport models, at µm and mm scales [START_REF] Babey | Spatiotemporal simulations of 2,4-D pesticide degradation by microorganisms in 3D soil-core experiments[END_REF], aimed at investigating the impacts of i) initial distributions of microorganisms and pesticide ii) diffusion modalities, and iii) water input modalities (amplitude and temporality) on biodegradation (fig. 1). The chemical and transport processes are calibrated on cm-scale experiments performed on the degradation of 2,4D in natural repacked soil cores under controlled distributions, diffusion and advection conditions [START_REF] Pinheiro | The millimetre-scale distribution of 2,4-D and its degraders drives the fate of 2,4-D at the soil core scale[END_REF]3], while potential microbial processes (regarding growth, physiology, mortality, mobility, nutrient recycling…) are explored to understand how they interact with the controlled conditions. We frame our results within unifying spatial descriptors of microbial uptake of substrate.

Strong differences between advectively-dominated and diffusively-dominated conditions indicate strong spatiotemporal couplings between biological, chemical and transport processes. We show how the spatial parameters (initial distributions and transport conditions) affect exposure in a scale-dependent way. The interplay of microbial uptake capacities and microbial exposure to pesticide brings out situations with contrasted biodegradation efficiencies.

Strong biodegradation heterogeneities can emerge from the interaction of the non-linear spatial and biological processes. Results give guidelines to design future experiments for validation of the explored biological processes.

  Coche, Tristan Babey, Alain Rapaport, Laure Vieublé-Gonod, Patricia Garnier, Jean-Raynald de Dreuzy To cite this version: Alexandre Coche, Tristan Babey, Alain Rapaport, Laure Vieublé-Gonod, Patricia Garnier, et al.. Interaction of porosity structures and microbial uptake dynamics in the degradation of pesticides at µm and mm scales. Computational Methods in Water Resources XXII (CMWR), Jun 2018, Saint-Malo, France. 2 p. hal-01827620

Figure 1 :

 1 Figure1: Overview of the mm-scale model[START_REF] Coche | SOM2017 conference[END_REF]


Romain Guiheux

Sophie Berveiller

Régis Kubler

Denis Bouscaud

Etienne Patoor

Quentin Puydt

Martensitic transformation induced by single shot peening in a metastable austenitic stainless steel 301LN: Experiments and numerical simulation

During conventional shot peening on metastable austenitic steels, martensitic transformation occurs in addition to plastic straining. In this work, the impact of a single spherical steel shot on a AISI 301LN steel was studied. The volume fraction of martensite, residual stresses in both phases were determined in the vicinity of the dent as a function of the shot velocity and diameter. An elasto-plastic two phase model that includes martensitic phase transformation was adapted to model mechanical and microstructural fields and implemented in Abaqus Explicit for the 2D simulation of a single shot impact. It was found, for instance, that the martensitic transformation takes place only under the dent and that martensite is in tension at the surface while austenite is in compression. Simulation results of stress levels showed a good agreement with experimental stresses determined by X-ray diffraction.

Introduction

Metastable austenitic stainless steels combine very good mechanical properties and high formability. They are good candidates for lightweight materials in automotive and other industrial sectors like energy and aviation. This remarkable combination of properties results from transformation induced plasticity. Formation of strain-induced martensite during the deformation process increases the strength level of the material. Mechanical and structural components must exhibit high mechanical resistance and high formability, but also high fatigue resistance. Shot peening surface treatment is commonly used in mechanical industry to increase fatigue resistance from the introduction of high compressive residual stresses that delays fatigue failure of component [START_REF] Lillamand | Cyclic modelling of the mechanical state produced by shot-peening[END_REF]. Evaluation of these stresses is of first importance to optimize component design and process parameters. The numerical simulation of the influence of shot-peening process on the internal stress field distribution on the near surface is a useful but difficult task because shot-penning is a complex process involving several mechanisms [START_REF] Wohlfahrt | The influence of peening conditions on the resulting distribution of residual stress[END_REF]. Numerous analytical models and numerical approaches based on finite element analysis have been developed to simulate the shot peening process and determine the relationships between process parameters, geometrical and material behavior characteristics of the target and the resulting residual stress field. Reviews of the wide variety of numerical models developed since the first simulation proposed by [START_REF] Edberg | Shot peening simulated by two different finite element formulations[END_REF] can be found in [START_REF] Rouhaud | Modeling of residual stress, shot peening[END_REF] and [START_REF] Sherafatnia | Experimental and analytical determination of shot peening residual stresses considering friction and real unloading behavior[END_REF]. Analytical models allow determining the stress distribution and the depth of the affected zone but finite element simulations provide a better understanding of the process enabling parametric studies. Firstly, two-dimensional axisymmetric finite element models were used to investigate the perpendicular impact of a single shot against an elastic-plastic surface [START_REF] Mori | Finite element analysis of peening process with plasticity deforming shot[END_REF][START_REF] Schiffner | Simulation of residual stresses by shot peening[END_REF]. Three-dimensional model was later developed by Meguid et al. for single and multiple impacts [START_REF] Meguid | Three-dimensional dynamic finite element analysis of peening induced residual stresses[END_REF][START_REF] Meguid | 3D FE analysis of peening of strain-rate sensitive materials using multiple impingement model[END_REF] and Guagliano found a way to relate the residual stress field determined from finite element simulation to Almen intensity [START_REF] Guagliano | Relating Almen intensity to residual stresses induced by shot peening: a numerical approach[END_REF]. A parametric study performed by [START_REF] Majzoobi | A three-dimensional simulation of shot peening process using multiple shot impacts[END_REF] has investigated the influence of shot number and impact velocity on the residual stress profile, finding that the simulation of 25 impacts allows obtaining a uniform stress state. Influence of thermal effect resulting from plastic dissipation was established [START_REF] Rouquette | Coupled thermo-mechanical simulations of shot impacts: effects of the temperature on the residual stress field due to shot-peening[END_REF]. Recent studies concentrated on the description of superficial shot peening damage [START_REF] Frija | Finite element modelling of shot peening process: prediction of the compressive residual stresses, the plastic deformations and the surface integrity[END_REF], peening treatment of rough surfaces [START_REF] Yang | Effect of initial surface finish on effectivness of shot peening treatment using enhanced periodic cell model[END_REF] and effect of a preexisting initial residual stress field on the residual stress field MARK distribution generated by shot peening [START_REF] Mahmoudi | A comprehensive experimental and numerical study on redistribution of residual stresses by shot peening[END_REF]. A review of the different symmetries and reference cell shapes used in the literature to reduce the computation time can be found in [START_REF] Miao | On the potential application of a 3D random finite element model for the simulation of shot peening[END_REF]. However, as shot peening involves a huge number of shots, finite element analysis fails to consider the particle dynamics inside the shot stream and its interactions with the target. Multiphase flow in computational fluid dynamics was used to simulate the coverage under multiple impacts and overlapping indentations [START_REF] Nguyen | Predicting shot peening coverage using multiphase computational fluid dynamics simulations[END_REF], but this approach provides no information about the induced residual stress field. A semi-analytical approach was developed by [START_REF] Chaise | On the effect of isotropic hardening on the coefficient of restitution for single repeated impacts using a semi-analytical method[END_REF][START_REF] Chaise | Modelling of multiple impacts for the prediction of distortions and residual stresses induced by ultrasonic shot peening (USP)[END_REF] to solve impacts problem and the plastic strain field obtained was transferred to a finite element model to determine the residual stress field in a peened thin structure. This eigenstrain method has been used for a long time in micromechanics and it appears also efficient for shot peening analysis as demonstrated by [START_REF] Musinski | On the eigenstrain application of shot-peened residual stresses within a crystal plasticity framework: application to Ni-base superalloy specimens[END_REF] who used it for imposing shot-peened residual stresses into a computational crystal plasticity framework. [START_REF] Hong | A numerical simulation to relate the shot peening parameters to the induced residual stresses[END_REF] proposed a combined numerical analysis based on finite element and discrete element methods. Such a coupling was used by [START_REF] Murugaratnam | A combined DEM-FEM numerical method for Shot Peening parameter optimization[END_REF] to run an extensive parametric study, investigate the coverage built-up and analyze energy transfer between the shot stream and the target surface. Based on this approach, a methodology was developed to simulate the process at minimal computing cost, determining representative elementary volumes for the target [START_REF] Jebahi | Robust methodology to simulate real shot peening process using discrete-continuum coupling method[END_REF]. Recently, the residual stress field due to shot peening was determined as a function of the process parameter within a gear tooth through the development of a dedicated software to determine the coverage and the velocity of the impacts on a mechanical component with complex geometry [START_REF] Gallitelli | Simulation of shot peening: from process parameters to residual stress fields in a structure[END_REF].

The beneficial effect of shot peening on metastable austenitic steels was recently established by [START_REF] Fargas | Effect of shot peening on metastable austenitic stainless steels[END_REF], resulting in extensive austenite to martensite phase transformation. The residual stress state was obtained from the plastic incompatible strain imposed by the process and also from the strain-induced martensitic transformation. This phenomenon, called transformation-induced plasticity (TRIP), has been first studied by [START_REF] Olson | Stress-assisted isothermal martensitic transformation: application to TRIP steels[END_REF]. It was established that martensite formation is influenced by strain rate [START_REF] Hecker | Effect of strain state and strain rate on deformation-induced transformation in 304 stainless steel: part I. Magnetic measurements and mechanical behavior[END_REF], temperature [START_REF] Talonen | Formation of shear bands and strain-induced martensite during plastic deformation of metastable austenitic stainless steels[END_REF] and the deformation mode as tension and compression [START_REF] Iwamoto | Investigation on deformation mode dependence of strain-induced martensitic transformation in trip steels and modelling of transformation kinetics[END_REF], torsion [START_REF] Lebedev | Influence of phase transformation on the mechanical properties of austenitic stainless steels[END_REF], shear [START_REF] Gallée | Mechanical behavior of a metastable austenitic stainless steel under simple and complex loading paths[END_REF] or bi-axial stress [START_REF] Perdahcioglu | Influence of stress state and strain path on deformation induced martensitic transformations[END_REF]. TRIP steels exhibit a multiphase microstructure and internal stresses are generated during the phase transition. This internal stress state is distributed among the different mechanical constituents of the microstructure, producing a high level of stress heterogeneity inside the material [START_REF] Berrahmoune | Analysis of the martensitic transformation at various scales in TRIP steels[END_REF]. The occurrence of this solid-solid phase transformation and the coupling between plasticity and transformation-induced plasticity increase the level of complexity for the development of numerical simulation of industrial processes for these alloys. Large amount of works have been published about modeling of TRIP steel behavior and reviews can be found in [START_REF] Fischlschweiger | A mean-field model for transformation induced plasticity including backstress effects for non-proportional loadings[END_REF][START_REF] Kubler | Semi phenomenological modelling of the behavior of TRIP steels[END_REF]. From a phenomenological point of view, two main problems have to be solved. The first one corresponds to the determination of the transformation kinetics as a function of the control variables (stress, strain and temperature), and the second one to the determination of the overall behaviour. About the former, [START_REF] Olson | Kinetics of strain-induced martensitic nucleation[END_REF] have related the rate of transformation with plastic strain. This physically based model was extended by [START_REF] Stringfellow | A constitutive model for transformation plasticity accompanying strain-induced martensitic transformation in metastable austenitic steels[END_REF] to include the influence of stress, by [START_REF] Iwamoto | Investigation on deformation mode dependence of strain-induced martensitic transformation in trip steels and modelling of transformation kinetics[END_REF] to get the triaxiality dependence and by [START_REF] Tomita | Constitutive modeling of TRIP steel and its application to the improvement of mechanical properties[END_REF] to take into consideration strain rate effects. For the latter, in a seminal work [START_REF] Leblond | Mathematical modelling of transformation plasticity in steels I: case of ideal-plastic phases[END_REF] and [START_REF] Leblond | Mathematical modelling of transformation plasticity in steels II: coupling with strain hardening phenomena[END_REF] have used a continuum mechanics description, assuming the transformation strain rate follows the direction of the applied deviatoric stress. A thermodynamically consistent model, including a full coupling between phase transformation and plasticity, was proposed by [START_REF] Fischer | Modelling and Simulation of Transformation Induced Plasticity in Elasto-Plastic Materials[END_REF] and extended by [START_REF] Levitas | Thermomechanical theory of martensitic phase transformation in elastic materials[END_REF] to consider finite deformation. Following the pioneering paper of [START_REF] Diani | Micromechanical modelling of the transformation induced plasticity (TRIP) phenomenon in steels[END_REF] several works have considered the modeling of the TRIP behavior at finer scales, using scale transition techniques to determine the macroscopic behavior. Among them, [START_REF] Cherkaoui | Micromechanical modeling of martensitic transformation induced plasticity (TRIP) in austenitic single crystals[END_REF][START_REF] Cherkaoui | Coupling between plasticity and martensitic phase transformation: overall behavior of polycrystalline TRIP steels[END_REF] have developed micromechanical models based on the crystallographic theory of martensitic phase transformation. [START_REF] Gallée | Mechanical behavior of a metastable austenitic stainless steel under simple and complex loading paths[END_REF] have adapted models derived from the model of [START_REF] Leblond | Mathematical modelling of transformation plasticity in steels I: case of ideal-plastic phases[END_REF] and [START_REF] Leblond | Mathematical modelling of transformation plasticity in steels II: coupling with strain hardening phenomena[END_REF] or from the one of [START_REF] Stringfellow | A constitutive model for transformation plasticity accompanying strain-induced martensitic transformation in metastable austenitic steels[END_REF] such as [START_REF] Kubler | Semi phenomenological modelling of the behavior of TRIP steels[END_REF] at the phase level using a mean field approach to describe the behavior of a multiphase material. This last class of model et al. (2016), or hydroforming of steel tube as in [START_REF] Lindgren | Simulation of hydroforming of steel tube made of metastable stainless steel[END_REF].

In the present work, the approach initially developed by [START_REF] Kubler | Semi phenomenological modelling of the behavior of TRIP steels[END_REF] for forming problems is applied to the finite elements simulations of single shot-peening.

To the knowledge of the authors, only few experimental studies have been published up to now dealing with the impact of shot peening on metastable austenitic steels. Kleber and Barroso (2010) have determined the evolution of residual stresses both in the martensitic and austenitic phases, and the martensite volume fraction for different shotpeening conditions on AISI 304L samples. They showed that the maximum compressive stress was obtained at the depth where the martensite volume fraction reaches its maximum. [START_REF] Turski | Engineering the residual stress state and microstructure of stainless steel with mechanical surface treatments[END_REF] have performed microstructural observations and residual stress analysis in AISI 304L steel for different mechanical surface treatments including shot peening, showing evidence of deformation induced martensite in the near surface, extensive work hardening and isotropic in-plane residual stress state within the austenitic phase. [START_REF] Fargas | Effect of shot peening on metastable austenitic stainless steels[END_REF] have studied the effect of different shot-peening conditions in cold rolled and annealed AISI 301LN sheets. They showed that similar or lower fatigue limits were obtained for both specimens and that a pre-existing amount of martensite in the cold-rolled specimens slowed down the transformation kinetics of martensite compared to annealed one. [START_REF] Fu | Micro-structure and surface layer properties of 18CrNiMo7-6 steel after multistep shot peening[END_REF] have studied the evolution of residual stresses both in the austenitic and martensitic phases in the 18CrNiMo7-6 steel and showed that it was higher in the martensitic phase of higher yield strength. In 2016, using a large strain formulation of transformation plasticity, Halilovic et al. proposed the very first numerical simulation of the residual stress state in AISI 304 peened steel with a single laser shot [START_REF] Halilovic | Prediction of the residual state in 304 austenite steel after laser shock peeningeffects of plastic deformation and martensitic phase transformation[END_REF]. They concluded that at low temperature the residual stress state is mainly caused by phase transformation, and it is mainly due to plastic deformation at high temperature.

Therefore, in this paper, it is proposed to investigate, both experimentally and by numerical model, the effect of a single spherical normal shot on strain-induced martensitic transformation in AISI 301 LN samples at room temperature. Different shot velocities were studied. Microgeometry of the impact was measured. Martensite volume fraction and residual stress profiles in both phases around the impact were evaluated using X-ray diffraction. Experimental results were compared to two-dimensional axisymmetric finite element simulations of a single shot impact. For the material behaviour, a semi-phenomenological model coupling the martensitic phase transformation and the elastoplastic behavior of austenite and martensite was calibrated using quasistatic tensile tests and implemented in a user subroutine in ABAQUS Explicit. Finite element simulations were performed in the same experimental conditions and results were compared with experiments.

Experiments a. Experimental method

The studied material was a hot rolled AISI 301LN austenitic stainless steel supplied as sheets of 10 mm thick by Aperam; its chemical composition (in weight percent) is the following: 0.019C-0.57Si-1.71Mn-6.60Ni-17.31Cr-0.108N-0.002S-0.030P. For microstructure observation, the samples were mechanically polished and electro-polished with a solution of perchloric acid in order to eliminate the residual surface hardening. Electrolytical etching was performed using a solution of nitric acid (60 ml) and water (40 ml) during 30 s at 2 V. As shown Fig. 1, the initial microstructure is austenitic with a grain size of about 50 mm.

Single shot tests were performed on plates with dimensions of 75 mm × 75 mm × 8 mm. The sample surface was previously ground with 600-1200 grit paper, followed by successive polishing with of 6, 3 and 1 mm diamond suspension. Residual stress were determined before shot peening to ensure that polishing has not affected the surface, as described in Section 2c. Single shot peening was performed on a inhouse device. A spherical steel (100Cr6) shot was projected from an air gun system oriented normal to the workpiece and located at a distance of 80 mm from the workpiece. Three shot diameters were used: 2, 5 and 10 mm. The shot velocity was measured during the whole test thanks to a laser beam fence and a high-speed camera; this allows a precise determination of shot speed just before and after the impact. The gun pressure was set-up to achieve two initial velocities, namely 35 and 70 m s -1 .

The microgeometry of the dent was analysed, using a contact surface roughness measuring device. The 3D characteristic dimensions were extracted (see Fig. 2): diameter 2a, depth δ, height of the pile-up around the dent border H b and the curvature radius at the base of the dent, R c .

Residual stresses were analysed by X-ray diffraction, using the classical sin 2 y method (NF EN 15305); as it is presented in Section 2c, it was only possible to determine residual stress in the austenite phase using Mn-Ka radiation. For the {311} crystallographic planes, the elastic radiocrystallographic constants values are ½ S 2 = 7.18 10 -6 MPa -1 and S 1 = -1.20 10 -6 MPa -1 . For a microstructure, including austenite and martensite, the stress determined by XRD in each constituent is (σ xx -σ zz -), z being the indepth direction. Only the macroscopic σ zz stress was considered as null.

b. Microgeometry of the single impact Experimental conditions and shot velocities are presented Table 1. The coefficient of restitution is defined as the ratio between the shot velocities after impact and before impact [START_REF] Fathallah | Determination of shot peening coefficient of restitution[END_REF]. It is decreasing when increasing the initial shot velocity. This indicates that the absorbed energy during peening is increasing with shot velocity. This effect is less sensitive with the highest diameter (10 mm) as the coefficient of restitution is quite the same for both velocities.

The microgeometry of the peened samples was analysed by measuring the dent diameter 2a, its depth δ and the height of the pile-up H b as defined in the previous section. Profile measurements along four lines or paths have shown that the profile is symmetric. Therefore, in the following, only 2D profiles are presented, corresponding to averaged measurements over four lines.

For a constant shot diameter (Table 2), both dent diameter and depth increase with increasing shot velocity and, for a constant shot velocity, they increase with shot diameter. The maximum values are obtained with a diameter of 10 mm at 70 m s -1 : the dent diameter is about 4 mm and the depth is about 390 mm. These observations are in agreement with the simulation of [START_REF] Meguid | Three-dimensional dynamic finite element analysis of peening induced residual stresses[END_REF] and experimental results of [START_REF] Mann | Residual stress near single shot peening impingements determined by nanoindentation and numerical simulations[END_REF] on an aluminum 2024 alloy. [START_REF] Chaise | Modélisation mécanique par méthode semi analytique: du contact roulant élastoplastique aux impacts multiples[END_REF] has performed single shot peening on an AISI 316L steel which is a stable austenitic steel. He measured the surface profile using a contact profilometer and stereo correlation. He found that all dimensions of the dent increase for shot velocity in the range between 10 and 40 m s -1 . Then saturation was observed for impact velocity higher than 40 m s -1 . This trend is quite different from the behaviour of the 301LN steel.

Reduced dimensionless parameters which depend only on the shot velocity (v) and not on its diameter (d) were calculated for each peening condition, according to the analytical model of [START_REF] Johnson | Contact Mechanics[END_REF]. The material mechanical behaviour is assumed to be elastic perfectly plastic; the expressions of the reduced parameters of the dent depth and radius are given by Eqs. ( 1) and (2):

⎜ ⎟ = ⎛ ⎝ ⎞ ⎠ δ d Kπρ v E 4 2 * b 2 2 5
(1)

⎜ ⎟ = ⎛ ⎝ ⎞ ⎠ a d Kπρ v E 2 2 4 2 * b 2 1 5
(2)

where K = 0, 8 is a constant representing the percentage of the total energy used for elastoplastic deformation, ρ b = 7800 kg m -3 is the density of the shot, v its velocity and E* is the equivalent Young's modulus of both sample and shot defined as: c. Residual stress analysis in the vicinity of a single impact Residual stress analysis was performed by XRD at the surface of the sample. Before shot peening, the stress value in austenite was homogeneous in all directions and equal to 40 ± 30 MPa. X-Ray analysis requires a flat surface; an attempt was done to determine stress inside the dent: martensite diffraction peaks were visible but due to the curvature, their quality was not satisfactory to be exploited for stress and volume fraction. Therefore, we only analyse stress around the dent. Firstly, measurements were performed all around the crater, at a given distance from its center: the different stress values did not differ from more than 30 MPa. So, the stress state was assumed to be axi-symmetrical; the stress profile was determined along one direction in the following, as a function of the distance from the dent center, named "r". Outside the dent, no martensite was detected; martensitic transformation occurs only below the shot impact. So, outside the dent, only austenite exists. Analyses were realized both in the tangential (σ θθ ) and radial (σ rr ) directions (see Fig. 4) with respect to the dent.

= - + - E ν E ν E 1 * 1 1
We focused firstly on samples peened with a 10 mm diameter shot for both velocities, 35 and 70 m s -1 (Fig. 5). For both velocities, stresses are in tension in the tangential direction and in compression in the radial one.

After peening at 35 m s -1 , the maximum stress is observed at the edge of the dent: σ θθ = 180 MPa and σ rr = -190 MPa. By comparison with the initial state, it has increased in the tangential direction and decreased in the other one. The stresses, in absolute value, decreases away from the dent: at a distance of 13 mm from the impact center, stresses values are almost null in both directions, that corresponds to the initial value. For higher distance, no stress evolution was observed. After peening at 70 m s -1 , the same stress profile is observed. The maximum stress was also obtained at the edge of the crater in both directions: the tangential stress value is quite the same as in the previous peening conditions, (σ θθ = 170 MPa) but the radial one is much more in compression (σ rr = -310 MPa). Stresses, in absolute value, decrease away from the dent and are equal to their initial value at a Fig. 3. Evolution of the characteristic dimensionless parameters of the dent as a function of the shot velocity and comparison with the analytical model of [START_REF] Johnson | Contact Mechanics[END_REF].

distance of about 20-25 mm from the dent centre. So increasing shot velocity affects mainly the tangential stress, by increasing maximum compressive value; it influences also and the affected zone magnitude: initial stresses are modified on a higher distance from the impact with higher velocity. However, direct comparisons are not trivial as both dents do not have the same geometry.

The same analysis has been performed on the sample peened with a 2 mm diameter shot. At a velocity of 35 m s -1 , the affected zone was too small to allow obtaining a profile. The stress evolution obtained with a velocity of 70 m s -1 is presented Fig. 6. Due to the small size of the dent, it was difficult to obtain measurement close to the edge of the impact. Nevertheless, the same trend is observed as with the 10 mm diameter shot: the absolute stress values decreases with increasing the distance from the dent center. But the maxima are smaller: at 2 mm from the dent, the tangential stress is about 100 MPa and the radial one is about -130 MPa. Moreover, stresses are almost null for distance higher than about 6 mm, which is lower than the ones observed with the biggest shot.

Material modelling and finite element simulations

A semi-physical numerical model, coupling the mechanical and thermodynamical behaviours, was developed at the scale of the phases following the work of [START_REF] Kubler | Semi phenomenological modelling of the behavior of TRIP steels[END_REF]. It consists in a thermoelastoplastic model with phase transformation. A Representative Volume Element (RVE) of a set of two phases was selected in order to model the behaviour of the austenite/martensite material. The following two phases are considered in this study:

-The austenite A of volume fractionf

(1 ), -The martensite α′ which forms from the austenite transformation under thermomechanical loadings, of volume fraction f.

a) Mechanical behaviour law

The macroscopic strain rate of the RVE is given by:

= - + + + + ′ ′ Ė f ε ε f ε ε ε (1 )( ˙˙) (˙˙) ij ij epA ij thA ij epα ij thα ij T (4)
where ε˙e p and ε˙t h are respectively the elastoplastic strain rate tensor and the thermal strain rate tensor of each phase of the RVE. ε˙T is a mean phase transformation strain rate tensor. It is defined by: 4) and ( 5), the following expression of the RVE macroscopic strain rate can be obtained:

= ε f ˙˙ξ ij T ij T ( 5
⎜ ⎟ = -⎛ ⎝ + + - ⎞ ⎠ + + ′ ′ Ė f ε ε f f f ε ε (1 ) ˙˙1 ξ (˙˙) ij ij epA ij thA ij T ij epα ij thα (6)
The localization method of [START_REF] Berveiller | An extension of the self-consistent scheme to plastically flowing polycrystals[END_REF] was used in order to calculate the strain rate tensor of each phase ε˙A and ′ ε˙α . That knowledge allows calculating the elastoplastic strain rate of each phase. The stress rate in each phase is given as a function of the tangent elastoplastic modulus tensor l ep and the thermal modulus tensor m according to the Eqs. ( 7) and ( 8). The macroscopic stress rate of RVE is obtained by:

⎜ ⎟ = ⎛ ⎝ - - ⎞ ⎠ - σ l ε f f m T ˙˙( 1 ) ξ ij A ijkl epA kl A kl T ij A (7) = - ′ ′ ′ ′ σ l ε m T ˙˙i j α ijkl epα kl α ij α (8)
= - + + - ′ ′ Σ ˙f σ fσ σ σ f (1 ) ˙˙( ) ij ij A ij α ij α ij A (9)
The elastoplastic modulus l ep tensor in each constituent is obtained using the standard Von Mises criterion with the associated plastic flow rule. The martensitic transformation kinetics derives from a thermodynamic approach.

Von Mises criterion gives the yield function F for each phase as:

= --= - - -- F σ σ R S X S X σ R 3 2 ( ) ( ) eq ij ij ij ij 0 0 ( 10
)
where σ eq , σ 0 , R and X are respectively the equivalent stress of Von Mises, the initial yield stress, the isotropic hardening parameter and the kinematical hardening tensor of the considered phase. Isotropic and kinematic hardenings laws are respectively given by [START_REF] Voce | A practical strain-hardening function[END_REF] and [START_REF] Armstrong | A Mathematical Representation of the Multiaxial Bauschinger Effect[END_REF] equations such as:

= -- ( ) R Q e 1 bε 0 eq p (11) = - X Cε γ X p ˙2 3 * ˙ij ij p ij (12)
The coefficients Q 0 , b, C* and γ are material constants associated to each constituent of the RVE. ε eq p is the equivalent plastic strain according to Von Mises, ε p is the plastic strain tensor and p˙defines the cumulated plastic strain rate. From Eqs. (( 10)-( 12)), the elastoplastic modulus is calculated by: . S is the deviatoric stress tensor and I is the fourth order identity tensor. The methodology is fully detailed in appendix A.

= ⎛ ⎝ ⎜ ⎜ - - - + + - - ⎞ ⎠ ⎟ ⎟ ( ) l C I S X σ μ S X σ μ C H γ S X X 3 2 ( ) 3

b) Martensitic transformation criterion and kinetics

The elastoplastic behaviour of each phase is coupled to a thermodynamical model which allows taking into consideration the transformation of austenite into martensite under thermomechanical loadings. In the case of the thermodynamics of irreversible processes and for a transformation independent from time, the driving force F f of the martensitic transformation is compared to a critical force F c as proposed by [START_REF] Cherkaoui | Micromechanical modeling of martensitic transformation induced plasticity (TRIP) in austenitic single crystals[END_REF], [START_REF] Kubler | Semi phenomenological modelling of the behavior of TRIP steels[END_REF] and [START_REF] Patoor | Micromechanical Modeling of the Thermomechanical Behavior of Shape Memory Alloys[END_REF].

The driving force for martensitic transformation F f is defined as a function of the stress in the austenitic phase, the mean instantaneous transformation strain proposed by [START_REF] Kubler | Semi phenomenological modelling of the behavior of TRIP steels[END_REF] and the temperature T, according to the following expression with B a material constant and T 0 the temperature of the thermodynamical equilibrium:

= - - F σ B T T ξ ( ) f ij A ij T 0 (14)
The critical force F c takes into consideration the influence of the martensite already formed and the austenite plastic strain. It is defined by:

= - - - - - - - F F B M T β αε f ( ) [1 exp( )] κ ln(1 ) c c s eq pA n 0 0 (15)
where F c0 is the critical force during a cooling without applied stress, M s is the temperature at which the martensitic transformation starts during a cooling without stress and f is the martensite volume fraction. β, α, n and κ are material constants controlling the martensitic transformation kinetics.

For a rate independent transformation, the transformation kinetics derives from the consistency rule such as:

⎧ ⎨ ⎪ ⎩ ⎪ < = = < = = = > F F so f F F and dF dF so f F F and dF dF so f ˙0 ˙0 ˙0 f c f c f c f c f c (16)
The expression of the kinetics of the martensitic transformation coupling the [START_REF] Koistinen | A general prescribing the extent of the austenitemartensite transformation in pure iron-carbon alloys and plain carbon steels[END_REF] and [START_REF] Olson | Kinetics of strain-induced martensitic nucleation[END_REF] laws in austenite is obtained:

= -⎧ ⎨ ⎩ - + - - - ⎫ ⎬ ⎭ - df f dσ BdT αβn αε αε dε 1 κ ξ . exp( ). [1 exp( )] . ij A ij T eq pA eq pA n eq pA 1 (17)
In this expression, σ A stands for the stress state tensor in the austenite phase and ε eq pA denotes the equivalent plastic strain within the austenite. The MITS is expressed according Kubler's model [START_REF] Kubler | Semi phenomenological modelling of the behavior of TRIP steels[END_REF] for isotropic materials considering that it depends only of the stress state within the untransformed austenite:

= + + ⎛ ⎝ - ⎞ ⎠ V V δ d S d S S δ S S ξ 1 3 Δ 9 16 [ ] 27 16 ij T ij ij γ kl γ kl γ ij ki γ kj γ 1 2 (18)
where V V Δ is the volume change associated with the martensitic transformation, S A is the deviatoric stress tensor within the austenite and d 1 and d 2 are material constants.

c) Identification of the constitutive law parameters

The different parameters of the model were identified from tensile tests in quasi-static regime (5.10 -4 s -1 ) on AISI301LN. The constitutive behaviour law was implemented in a user subroutine in Abaqus Explicit. Simulations were carried out on a single cubic element with reduced integration. The identified parameters are presented in Tables 3 and4 respectively for the plastic behaviour and for the transformation kinetics. The yield stresses of each phase are close to the ones experimentally determined by [START_REF] Jacques | Micromachanical characterization of TRIP-assisted multiphase steels by in situ neutron diffraction[END_REF], using local insitu determination of the stress vs strain curve in each constituent. Hardening parameters, the parameters of the Olson-Cohen law (κ, n, α) and β and the coefficients d 1 and d 2 of the MITS were identified after a parametric analysis. The choice of the critical thermodynamical force F C0 has an influence on the onset of martensitic transformation. Fig. 7(a) presents the comparison between the experimental and the simulated behaviour curves which are in good agreement. The martensitic transformation starts at a strain of about 12%, leading to an extra-hardening of the material: as martensite is harder than austenite, the hardness increases with increasing its volume fraction. At the constituent scale (Fig. 7(b)), as expected, martensite exhibits a higher stress state than austenite: it starts from 700 MPa for a strain of 3% and increases up to 1000 MPa for a strain of 30%. Those values are in good agreement with the experimental ones of [START_REF] Berrahmoune | Analysis of the martensitic transformation at various scales in TRIP steels[END_REF] determined on a AISI 301LN cold-rolled and annealed steel: 990 ± 20 MPa for a strain of 30%. Therefore a good prediction of the mechanical behaviour is observed at both macroscopic and microscopic scales.

d) Single shot-peening simulations Simulations by Abaqus Explicit were performed with shot peening conditions similar to the experimental ones. As the shot is spherical and is projected normal to the surface, a 2D-axisymmetric model has been used. It consists in the impact of a spherical media on a semi-infinite body, which is sufficiently large to overcome the boundaries effects. All the lateral faces of the impacted workpiece are constrained in displacement along their normal directions. The mesh is made of quadrilateral elements with reduced integration CAX4R in the upper part, and of triangular elements CAX3 in the bottom part. The shot is meshed with CAX4R elements (see Fig. 8). The CAX4R elements on the body have a size of 10 mm in the impacted zone in order to refine the description in the zone of interest.

The size of the body depends on the diameter and the shot velocity. The bigger the diameter and velocity are, the larger the size of the medium has to be in order to fulfil the semi-infinite condition. An example is given for peening with a 10 mm diameter shot at 35 m s -1 (see Fig. 8). The workpiece is 25 mm deep and 35 mm wide. The shot is meshed with 1089 CAX4R elements; for the workpiece, the first 10 millimeters are meshed with 2401 CAX4R elements, the last 20 millimeters with 1074 CAX3 elements. The contact modelling was defined frictionless.

The mesh size was chosen to ensure acceptable computational time with regards to a convergence of the residual stress profiles. The shot behaviour is assumed to be elastic.

Results and discussion

The example of a workpiece impacted by a 10 mm diameter shot at 35 m s -1 is presented Fig. 8.

Simulations were carried out and compared to the experiments for microgeometry (Table 5) and residual stresses (Fig. 9). Simulated values do not fit well the experimental ones for a given velocity. But the effect of velocity increase is well reproduced: increasing the speed by a factor 2 increases also the pile-up height by a factor 2 for experiments and simulation. A better prediction could be probably achieved in a future model taking into account strain rate effect.

The stresses in two directions were analysed along the surface, from the dent to the edge of the body: σ θθ along e θθ and σ rr along e rr . The martensite volume fraction was also extracted from the simulations: martensite is created within a distance of 2.1 mm from the impact center for a shot velocity of 35 m s -1 , and 2.8 mm for 70 m s -1 . These values are of the same magnitude as the experimental ones, confirming that martensite appears only in the dent. A good correlation is also observed for residual stress values in both peening conditions. The simulations showed that the austenite is in compression in both directions and maximum at the centre of the body: -370 MPa at 35 m s -1 and -780 MPa at 70 m s -1 . An increase by around a factor two of the maximum stress is observed when the shot velocity is increased by a factor two. In the tangential direction (σ θθ ), the stress comes from a compression state at the center of the impact to a tensile state at the edge of the dent. Outside the dent, a re-balancing of stress was observed in both directions. [START_REF] Kobayashi | Mechanism of creation of compressive residual stress by shot peening[END_REF] have performed dynamic impact test by dropping steel shots of two different sizes (50 and 75 mm) from a height of 2 m on steel plates of width of 50 and 75 mm, respectively. The corresponding shot velocity was estimated at 6.5 m s -1 . Due to the large size of the dent, they were able to perform stress measurement inside the dent, in the radial direction. Unexpectedly, they observed that tensile stress was created whereas compressive stresses are obtained outside the dent over a range of three to four times the radius of the indentation. Our results are in agreement with Kobayashi's work outside the dent, even for the magnitude of the affected zone. But a difference is noted in the dent: in their experiment, the workpiece thickness is equal to the shot diameter. In our simulation, plasticity is observed down to 15 mm depth for a shot diameter of 10 mm and the residual stress field is over the 35 mm depth. The lower ratio of the shot diameter over the workpiece depth could be responsible for a different stress field. This effect can be highlighted by the numerical results of [START_REF] Shivpuri | Elasto-plastic pseudo-dynamic numerical model for the design of shot peening process parameters[END_REF]. They simulated the impact of one single shot on a workpiece of 1.3 mm thickness. When the shot diameter varies from 0.7 to 1.3 mm, the residual stress inside the dent varies approximately from -400 MPa to +150 MPa at the surface. Comparing their simulation with experimental results of Kobayashi et al., they obtained a good agreement outside the dent but the model under estimates stress inside the dent. Following the authors, this is due to the strain-rate effect that is not taken into account in their model: it uses a rate-independent constitutive law.

To isolate the influence of martensitic transformation on the resulting microstructure and stress state, the same numerical simulations were realized considering now the austenite as a stable phase, so the martensitic transformation can no longer occurs. Results such obtained are compared to the previous simulations. The results are presented for a 10 mm diameter shot with a velocity of 70 m s -1 . The phase stresses have been analyzed (Fig. 10). The austenite is more in compression when the martensitic transformation is activated, the difference is about 50 MPa. Martensite is in tension in a layer of 1 mm depth from the surface, then in compression down to 2 mm depth. At larger depth, martensite stress state up to tension is observed for larger depth and associated. After this depth, martensite was found in tension, until 600 MPa at 3 mm depth. At this depth there is very few martensite (no more than 1%).

The evolution of the macroscopic stress was calculated using a mixture law (see Fig. 11). When the martensitic transformation is activated, the macroscopic stress is of -80 MPa at the surface and increase until a maximum compression stress of -280 MPa at 2.5 mm depth. When the martensitic transformation is not activated, the maximum compression stress is obtained at the surface and is slightly higher (-300 MPa). At the opposite, the area which is in compression is smaller when the transformation is switched off: 6 mm against 7 mm when it is activated.

This result confirms the theory of [START_REF] Wohlfahrt | The influence of peening conditions on the resulting distribution of residual stress[END_REF] for the case of shot-peening. Without martensitic transformation, the material is fully austenitic and its hardness is smaller than the one of the shot. So the "hammering" effect is dominant: the main part of the energy provided by the shot is consumed in the plastic stretching of the subsurface of the material: the shear and stresses are then maximum at the subsurface. But when the martensitic transformation is activated, a hardening of the material is observed, especially between 0 and 2 mm depth. The energy provided by the shot is no more essentially used in plastic stretching of subsurface but in the one of deeper layers. So the maximum stress is shifted towards deeper layers, at 3 mm in this case.

Conclusions

In this paper, experimental analyses and two-dimensional axisymmetric finite element simulations were carried out to study the influence of stress-induced martensitic transformation on the residual stress field obtained in a peened metastable austenitic stainless steel. The results of our work reveal the following:

• The microgeometry analysis of single shot impacts shows char- acteristic parameters of the dent like impact depth and radius close to those predicted, using the analytical model of Johnson whereas this analytical model was developed for elastic-perfectly plastic materials.

• The stress affected area increases when the velocity of the impacting medium is increased.

• The numerical model gives results close to the experiments for both austenitic stress and martensitic volume fraction.

The model will be used to model conventional shot peening process on TRIP steels which exhibit more complex microstructures: the advantage of the model is to take into account phase interaction on the resulting stress state.

Fig. 1 .

 1 Fig. 1. Microstructure of AISI 301LN steel.

  ν s , ν b and E s , E b are respectively the Poisson coefficient and Young modulus of both the sample and the shot. Both Poisson coefficient is equal to 0,3 and the Young Modulus of the sample and shot are respectively of 200 and 207 GPa. The evolutions of δ d and a d 2 as a function of shot velocity are presented in Fig. 3. For a given shot velocity, values are very close whatever the shot diameter. They increase with increasing shot velocity. Measurements are closed to Johnson's model value for the lowest velocity but a shift appears for the velocity of 70 m s -1 . This is not surprising: Johnson's model considers an elastic perfectly plastic behaviour of the materials. This confirms that the TRIP effect has to be considered to model shot peening of AISI 301LN steel.

Fig. 2 .

 2 Fig. 2. (a) Characteristic dimensions of the dent and (b) Example of a 3D micro-geometry with a 10 mm diameter shot at 70 m s -1 .

  ) where ξ T describes the Mean Instantaneous Transformation Strain (MITS), introduced by Kubler et al. (2011), representing the global contribution of the transformation strain of each individual variant of martensite. It takes into consideration the Bain strain and a part of the Greenwood-Johnson effect by the accommodation mechanisms of Bain in the martensite (appr. 17% compression and 12% expansion for conversion into BCC). By coupling Eqs. (

Fig. 4 .

 4 Fig. 4. Top view of the peened sample and definition of stress directions.

Fig. 5 .Fig. 6 .

 56 Fig. 5. Evolution of the austenitic residual stress as a function of the distance from the impact center, at the surface of the sample, for a 10 mm diameter shot and for two velocities.

  the elastic stiffness tensor, μ the shear coefficient and = -

Fig. 7 .

 7 Fig. 7. (a) Evolution of experimental and simulated macroscopic stress and martensite volume fraction, (b) Evolution of simulated stress in each phase.

Fig. 8 .

 8 Fig. 8. 2D-axisymmetric model of single shot-peening simulation: (a) mesh and boundaries conditions, (b) examples of the distribution of stress in the austenitic phase (σ θθ ), the equivalent plastic deformation (PEEQ) and the martensitic volume fraction (f).

Fig. 9 .

 9 Fig. 9. Comparison of experimental and simulation stress profile in the austenite for a 10 mm diameter media at: (a) 35 m s -1 , (b) 70 m s -1 .

  This class of model enables the numerical simulation of industrial processes like deep drawing of metastable austenitic stainless steel as shown in[START_REF] Gallée | Deep drawing simulation of a metastable austenitic stainless steel using a two-phase model[END_REF] and in Msolli

	Nomenclature demonstrated both a good accuracy for model prediction and reason-able computation cost.	ε eq p l ep	Von Mises equivalent local plastic strain according Tangent elastoplastic local modulus tensor
	2a	Diameter of the dent	m	Thermal local modulus tensor
	δ	Depth of the dent	a th	Thermal expansion tensor
	H b	Height of the pile-up around the dent	Σ	Total macroscopic stress tensor
	R c	Curvature radius at the base of the dent	σ	Local stress tensor
	d	Shot diameter	S	Deviatoric part of the local stress tensor
	K	Constant of the Johnson's law	σ eq	Equivalent stress of Von Mises
	ρ b	Shot density	σ 0	Initial yield stress
	E *	Equivalent Young's modulus of both sample and shot	R	Isotropic hardening coefficient
	E s , E b	Young's modulus respectively of the sample and the shot	p	Cumulated plastic strain
	ν s, ν b	Poisson's ratio respectively of the sample and the shot	Q b C γ , , *, 0	Material constants of the hardening law
	σ θθ	Stress component in the tangential direction with respect	CC	Elastic stiffness tensor
		to the dent	μ	Shear coefficient
	σ rr	Stress component in the radial direction with respect to	I	Identity fourth order tensor
		the dent	F f	Driving force for martensitic transformation
	f	Volume fraction of martensite	F c0	Critical force for martensitic transformation
	A a'	Austenite Martensite	V Δ V	Volume change associated with the martensitic transfor-mation
	E	Total macroscopic strain tensor	β, α, n, κ Material constants of the kinetics of martensitic transfor-
	ε	Elastoplastic local strain tensor		mation
	ε	Plastic local strain tensor	d 1, d 2	Material constants of the MITS
	ε	Thermal local strain tensor	MITS	Mean Instantaneous Transformation Strain
	ϵ T	Transformation local strain tensor	RVE	Representative Volume Element
	ϵ T	Mean Instantaneous Transformation Strain (MITS) tensor	XRD	X-Ray Diffraction

Table 1

 1 Results of the single shot-peening experiments.

	Shot diameter (mm)		2			5		10
	v beforeimpact (m s -1 )		3 5	7 0		3 5	7 0	3 5		7 0
	v afterimpact (m s -1 )		15	25		1	22	10		20
	Coefficient of restitution	0.43	0.36	0.4	0.31	0.28		0.27
	Table 2
	Dent diameter (mm), depth (mm) and pile-up height (mm) as a function of shot diameter
	and velocity.
	Shot diameter (mm)	2			5			10
	Velocity (m s -1 )	2 a	d	H b	2a	d	H b	2a	d	H b
	35	0.59	36	1	1.49	101	6	2.96	198	10
	70	0.78	72	5	2.00	183	15	4.04	386	20

Table 3

 3 Elastoplastic main parameters of the numerical model.

		σ 0(MPa)	Q 0 (MPa)	b	C*(MPa)	γ
	Austenite A	320	300	5	750	5
	Martensite a'	850	300	20	0	0

Table 4

 4 Martensitic transformation main parameters of the numerical model.

	V V Δ	(%)	d 1 (MPa -1 )	d 2 (MPa -1 )	F c0(MPa)	κ	n	α	β
	2		2.10 -7	0	-350	585	16	12	1000

Acknowledgement

This work was supported by the French Technological Research Institute for Materials, Metallurgy and Processes (IRT M2P) under the CONDOR project.

Appendix A. Definition of the tangent elastoplastic modulus tensor l ijmn ep for isotropic and kinematical non linear hardenings

The Von-Misès yield function F is defined by:

With the equivalent stress

The Voce isotropic hardening

3)

The Armstrong Frederik kinematical hardening formulated in the incremental form:

Applying the normality rule to the yield function F, the plastic strain rate is obtained

And the plastic multiplier defined by:

The plastic multiplier dλ is obtained using the consistency equation dF = 0:

(A.12)

Thus the expression is simplified such as:

The rate of the yield function dF becomes:

In the case of isotropic elasticity where the elastic constants are defined by C ijkl = 2μI ijkl + λ*δ ij δ kl where μ and λ* are Lamé coefficients, the following simplifications are obtained:

(A.17)

The consistency rules dF = 0 is expressed:

(A.18)

From Eq. (A.18), the plastic multiplier dλ is obtained:

The behaviour law defined by the stress rate dσ ij is obtained: