domain: sdv sde

text:
Nathalie Cheviron
email: cheviron@versailles.inra.fr

Hélène Vecten

Sabine Houot

Antonio Bispo

Cécile Grand

Laurence Galsomies

Christian Mougin

MATERIALS AND METHODS

For each treatment, the Qualiagro site (INRA and Veolia Environment collaboration) comprised four replicates of plots (45 m x 10 m, 450 m 2 ). The treatments were: no organic amendment (CON), farmyard manure (FYM), biowaste compost (BIO), co-compost of green wastes and sludge (GWS) and municipal solid waste compost (MSW)(Figure 1). The present study was managed on plots with optimal mineral nitrogen fertilization. In the site, the luvisol was a silt loam comprising 15% clay, 78% silt, and 7% sand. Its initial pH value was 6.9 and initial organic matter was 1.8%. During the past twenty years, the average annual precipitations amounted to 582 mm, and the average annual temperature was 11°C. The soils were collected at different times (at T0, then 6 and 18 months after amendments from) 2011 to 2013 at a depth of 0-10 cm. Samples were sieved through a 5 mm mesh, and stored at 12 °C before use. Soil water content was determined by drying 5 g of soil at 105 °C for 48 H.0 Phosphatase, b-glucosidase, arylsulfatase, and urease activities were performed in microplate according to modified protocols [16,17,18,19] less than 48 hours after sample collection. Briefly, hydrolases were measured in triplicates in 4 g soil samples mixed during 10 min at 250 rpm with 25 ml water. For phosphatase, b-glucosidase and arylsulfatase activities, soil solutions were incubated with specific substrates (Sigma chemicals) (Tab.2). Reaction was stopped with 0,5 M CaCl 2 and 0,1 M Tris at pH 12. For urease activity, soil solutions were mixed with 0,4 M urea. NH 4 formed was revealed with ammonium salicylate and ammonium cyan rate (Hach reagents). After the reactions, each plate was centrifuged 5 min at 2000 g, and absorbance was measured on a microplate reader Genius (SAFAS). The amounts of p-nitrophenol (PNP) and NH 4 Cl formed were obtained by the absorbance at 405 nm and 610 nm.

Cheviron Nathalie 1,P , Marrauld Christelle 1 , Grondin Virginie 1 , Vecten Helen 1 , Houot Sabine 2 , Bispo Antonio 3 , Grand Cécile 3 , Galsomiès Laurence 3 , Mougin Christian 1998,2000,2002,2004,2006,2007,2009

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  fresh weight; dw, dry weight; OC, organic carbon content; ON, organic nitrogen content ; C/N tot , organic C to total N ratio b The excess mineralized carbon after 91 days in the soil-organic amendment mixtures was expressed as a percentage of the amendment organic C applied during incubation (after subtracting mineralized C of the control treatment)CONCLUSIONSOrganic amendments have noticeable long-term-effects on enzymatic activities. Soil microbial processes are sensitive indicators for monitoring changes of soil biological properties due to amendments by organic waste and for assessment of Organic Matter status . with (*). QUALIAGRO (Feucherolles -Yvelines; France), is a long-term experiment network to observe and understand the effects of OWP amendments on cropped soils to investigate the environmental impacts of recycling organic residues on soil biochemical properties, as well as the possible resilience of the agrosystems, to assess cost/benefit balance, and to improve agricultural practices. Since 1998The level of b-Glucosidase, Phosphatase and Urease activities increases depending on the nature of the amendment, more especially on the status of organic matter added, its availability and degradability. -Arylsulfatase activity does not respond to these amendments -Organic amendments could promote stability of enzymes which could strongly affect soil quality FYM exogen microorganism input, very degradable MWS high biodegradability, very thin, rapidly incorporated and mineralised GWS: pool of degradable (sludge), and few degradable (Green Wastes) BIO: contains lots recalcitrant organic coumponds; thin mineralisation, poor disponibility of nitrogen


Ioannis D Pantelis
email: ioannis.pantelis@postgrad.manchester.ac.uk

Mike C Smith

Anastasia N Vasileiou

Matthew J Roy

Measuring the effect of post-weld heat treatment on residual stress relaxation in electron beam welds made of low alloy pressure vessel steel using the contour method

Keywords: residual stress, post-weld heat treatment, contour method, SA508 steel, pressure vessel steel ________________________________________________________________________________________________________________________________

Welding is of major importance for the nuclear industry as it affects safety, structural integrity and the life span of the nuclear power plant. The uneven thermal distribution and metallurgical misfits that develop during welding procedures introduce residual stresses, which can drive damage in safetycritical nuclear components. Post-weld heat treatment (PWHT) is known to reduce residual stresses, however, determining the PWHT parameters, such as hold temperature and soak time, needs to be standardized and not be based on judgement. Five 100mm thick electron beam welds were manufactured from SA508 Grade 3 Class 2 low alloy steel, four underwent four different PWHT schedules with differing hold temperatures and soak times, and one was retained in the as-welded (AW) state. The residual stresses in the as-welded condition were measured using incremental deep hole drilling and the contour method, while residual stresses after PWHT were measured using the contour method alone. All specimens exhibited the characteristic M-shaped stress distribution associated with solid-state phase transformation in the weld fusion zone and HAZ. The peak tensile stresses developed in untransformed parent material either side of the weld, and reached 650MPa in the as-welded state. Post-weld heat treatment resulted in significant relaxation, to between 100 and 200MPa.

Introduction

The Reactor Pressure Vessel (RPV) has both operational and safety functions, retaining the high pressure primary circuit fluid and forming one of the multiple barriers to fission product release in a light water reactor. A very low RPV failure probability is required during normal operation and postulated accident conditions (USNRC 1986, ASME 1986) [START_REF] Odette | Nuclear Reactors: Pressure Vessel Steels[END_REF]. The RPV should be made of materials that provide adequate levels of strength, ductility, fracture toughness and neutron-irradiation resistance [START_REF] Joshi | Materials for Nuclear Reactors[END_REF]. Typically, the individual forged sections are made of low-alloy ferritic steels, such as ASME SA508, and these are joined using multi-pass arc welding techniques [START_REF] Guo | Microstructure and mechanical characteristics of a laser welded joint in SA508 nuclear pressure vessel steel[END_REF]. Electron beam welding has the potential to increase productivity (a single penetration pass to join large thick components), so is under active development for such applications [START_REF] Baufeld | Electron Beam Welding of Large Components for The Nuclear Industry[END_REF] [START_REF] Vadolia | Electron Beam Welding: Study of process capability and limitations towards development of nuclear components[END_REF][6] [START_REF] Sanderson | Advanced welding processes for fusion reactor fabrication[END_REF].

Residual stresses arise as a consequence of the heterogeneous application of energy and uneven temperature distributions during welding [START_REF] Withers | Residual stress part 1 -Measurement techniques[END_REF] [START_REF] Francis | Welding residual stresses in ferritic power plant steels[END_REF] and have the potential to be a major contributor to in-service degradation of nuclear components [START_REF] Joshi | Materials for Nuclear Reactors[END_REF] [START_REF] Guo | Microstructure and mechanical characteristics of a laser welded joint in SA508 nuclear pressure vessel steel[END_REF]. In addition, welded ferritic structures undergo Solid State Phase Transformation (SSPT) in the weld fusion zone and adjacent heat-affected zones during welding [START_REF] Vasileiou | The impact of transformation plasticity on the electron beam welding of thick-section ferritic steel components[END_REF]. This has the potential both to alter the as-welded residual stress state and to introduce undesirable micro-constituents into weld and HAZ, adversely affecting the mechanical properties. Post-weld heat treatment (PWHT) is normally performed on low alloy pressure vessel steel components, both to reduce and redistribute residual stresses, and to restore the toughness of the weld region, by tempering any unwanted martensitic microstructures [START_REF] Ayres | Development of reduced pressure electron beam welding process for thick section pressure vessel welds[END_REF] [START_REF] Rathod | Residual stresses in arc and electron-beam welds in 130 mm thick SA508 steel: Part 1 -Manufacture[END_REF]. It is of vital importance to control the impact of PWHT parameters, like the holding temperature and time, in order to optimise both stress relaxation and mechanical properties. Electron beam welds differ from conventional arc welds in that they are both single-pass and autogenous, with no weld filler metal. It is not self-evident that post-weld heat treatment schedules developed for multi-pass arc welds are optimal for an EB weld. The present work addresses this issue, by comparing different residual stress measurement techniques used to estimate the residual stresses both after EB welding and after four different PWHT regimes.

Experimental procedures

The Nuclear Advanced Manufacturing Centre (NAMRC) manufactured five 100mm thick electron-beam welded butt welds made of SA508 Grade 3 Class 2 low alloy pressure vessel steel. The final components had dimensions 300mm x 300mm x 100mm.

Materials

The chemical composition and mechanical properties of SA508 Gr3 Cl2 steel are given in Table 1 and Table 2, respectively.

Weldment Manufacture and Non-Destructive Evaluation

The initial 300mm x 150mm x 100mm half coupons were electron beam welded using the NAMRC's K25 EBW facility. Initially, the half plates were tack-welded at the beam entry and exit points and a gutter bar was attached below the weld seam on the beam entry face. In order to diminish the probability of weld pool sagging, welding took place in the 2G position, and in order to prevent weld seam opening during welding, a sealing pass, at 10% beam current, took place, followed by a single penetration EBW pass. The penetration pass was produced by a 40kW gun, operated at 80kV, using an elliptical beam oscillation pattern with X and Y amplitudes of 2mm and 3mm, respectively and a 141Hz frequency. The temperature histories were measured using K-type thermocouples, spot-welded on the top and bottom surfaces. The welding parameters used are presented in Table 3. All welds underwent ultrasonic inspection and all were found to be free of volumetric defects. Slope-in and slope out defects were present at the beam entry and exit points, but these are of no significance for this study. Areas with lack of penetration defects were identified for two of the welds, but as these were located far from weld mid-length, they were also ignored. Acceptable areas of root concavity (less than 2mm deep) were also observed.

Post-weld Heat Treatment

With the aim of understanding, examining and optimising how residual stress relaxation occurs, four of the five specimens underwent four different post-weld heat treatments (PWHT), presented in Table 4.

Residual Stress Measurements

Residual stress measurements were made using the contour method (CM), deep hole drilling (DHD) and incremental deep hole drilling (iDHD), for the specimen in the as-welded condition, and using only the CM for the specimens in the PWHT condition. Two independent methods were applied to the as-welded plate to minimise uncertainties in the measured data, as advised by the R6 procedure [START_REF]Assessment of the integrity of structures containing defects[END_REF], and to benchmark the performance of the contour method prior to its application to the heat treated plates.

The contour method is a destructive method for measuring and mapping residual stresses, proposed by Prime in 2001 [START_REF] Prime | Cross-Sectional Mapping of Residual Stresses by Measuring the Surface Contour After a Cut[END_REF] based on Bueckner's elastic superposition principle [START_REF] Bueckner | The propagation of cracks and the energy of elastic deformation[END_REF] and it involves the following steps:

 Rigidly clamping and cutting the specimen in half  Estimating the resulting deformation by measuring the cut surfaces  Processing the surface profile data (averaging, fitting and smoothing)  Calculating the original residual stress distribution by conducting an FE analysis

The contour cuts were made in between pre-drilled 3mm pilot holes (ligaments), in order to minimize cutting-induced plasticity artefacts in the as-welded condition [START_REF] Traore | A novel cutting strategy for reducing plasticity induced errors in residual stress measurements made with the contour method[END_REF], and to simplify specimen clamping and support in the heat treated plates, using a wire electric discharge machine (EDM). The surfaces were measured by acquiring 7700 points per surface using a spacing between points equal to 1-1.2mm, using a Mitutoyo Crysta-Apex 776 coordinate measurement machine (CMM). The pre-and post-processing were conducted using Python Contour (pyCM), introduced by Roy et al [START_REF] Roy | pyCM: An open-source computational framework for residual stress analysis employing the Contour Method[END_REF]. The aligned and averaged surfaces were fitted to a bi-variate cubic spline with a knot spacing of 6mm in the transverse and normal direction, to reduce the noise, and the resulting 2D contour maps were produced using an FE mesh of quadratic elements, generated by 716 nodes spaced uniformly every 1.12mm along the perimeter of the surface, with a Young's Modulus E=191GPa and Poisson's Ratio ν=0.294. Displacement boundary conditions were set to prevent translation and rotation. Review of the measured stress fields suggested that the measurement spacing might have been insufficient to capture full details of the short wavelength M-shaped distribution expected in an EB-welded plate [START_REF] Vasileiou | Residual stresses in thick-section electron beam welds in RPV steels[END_REF][21] , so to improve the measurements, one half-plate of the AW specimen was remeasured in the CMM machine with a higher point density corresponding to 204400 points and minimum distance between points equal to 0.23mm.

The deep hole drilling method is a semi-destructive, strain relief method, where strains are calculated during strain relief, produced by material removal [START_REF] Smith | Deep Hole Drilling[END_REF]. A reference hole is gundrilled in-between reference bushes attached to the front and back surfaces. A cylinder of material containing the reference hole is cut, allowing for stress relaxation to occur, and the residual stresses are calculated by measuring the reference hole diameter prior and after the cut. To account for plasticity-induced errors, which will render the reference hole diameter measurements non-representative and thus, the measured residual stresses inaccurate, the incremental deep hole drilling method is employed, where the measurement of the reference hole diameter prior and after the EDM cut, occurs incrementally [START_REF] Mahmoudi | A new procedure to measure near yield residual stresses using the deep hole drilling technique[END_REF]. The DHD and iDHD measurements were performed transverse to the weld plane, for one of the AW half plates, at the facility of VEQTER LTD.

Results and discussion

Initial Residual Stress measurements using the contour method

Line plots on a transverse line at mid-thickness, were extracted and compared in Figure 1 for the AW and PWHT specimens using the initial full measurement data sets. We would expect to see a characteristic M-shape distribution, with peak tensile stresses just outside the HAZ and reduced stress levels within the HAZ and FZ, caused by the Solid State Phase Transformation during the EB welding procedure. There is some evidence of this in the heat-treated plates, but the as-welded plate appears to show a noticeable asymmetry, with peak stresses reaching 620MPa on the right side of the weld and only ~400MPa on the left side. It is tempting to attribute this to cutting induced plasticity effects, but it is also possible that the measurement point density or other aspects of the processing pipeline have created this effect. Outside the weld region, the residual stresses are small, with the exception of regions adjacent to the ligaments, where ''non-physical'' stress perturbations are observed due to measurement and fitting artefacts, produced by the pilot-holes. Removing the pilot holes from the analysis did not produce any significant improvement on the resulting data.

The effect of PWHT is also evident from the comparative plot with the RS dropping, for all four PWHT scenarios into the range of 100-200MPa. The characteristic M-shape distribution and pilot-hole artefacts, were also observed, and small ''non-physical'' tensile mean stresses, ranging from -50 to 100MPa, were present in the parent material. The reasons for this are not yet clear, although it should be remembered that the measured stresses outside the central weld region are close to the likely uncertainty in the technique.

Residual stress measurements using the DHD/iDHD method

Both DHD and iDHD measurements were performed on the AW specimen on a transverse line at weld mid-thickness, recovering both longitudinal and normal stresses, as seen on Figure 2. Both techniques revealed an M-shaped distribution in both components, with peak tensile stresses in parent material just outside the HAZ, a significant reduction in weld and HAZ caused by SSPT during welding, and a fall into compression further from the weld centerline. The iDHD measurements, as expected, recovered slightly higher peak tensile stresses, and both techniques showed some asymmetry in the peak stresses either side of the weld. These features are normally attributed to plasticity effects, which are less important in iDHD measurements, which thus show higher peak stresses and less asymmetry. Comparison with Figure 1 shows relatively poor agreement with the contour method measurement, so it is important to understand why this might be the case.

Improvement of residual stress measurements using the contour method

For the purpose of improving the residual stress measurement of the AW specimen, and improve the accuracy of the method, only one half of the AW specimen was re-measured, without re-cutting, in the CMM machine with a higher point density, following the same preprocessing pipeline for the measured surface. A comparative plot, between the CM (coarse and refined point cloud) and DHD/iDHD methods, for the AW specimen, is presented in Figure 3.

It is observed that the refined point density, significantly changes the CM results. The asymmetry in the measured peak tensile stresses disappears, and the peak values are close to those measured using iDHD. Some disagreement remains, as the depression in stresses in weld and HAZ remains much less significant in the contour measurement, and it shows no evidence of a fall into compression outside the HAZ. Clearly the revised measurement is not perfect, as only one face of the cut was re-measured. However, it does show that the initial asymmetry was not due to cutting plasticity, but due to inadequate measurement point density. The density originally used would have been adequate for a multi-pass arc weld, which exhibits a longer wavelength residual stress distribution.

Conclusions

The following conclusions can be drawn from this study:  The residual stress distribution, for all the welds (AW and PWHT) and all the methods (Contour Method, Deep Hole Drilling and incremental Deep Hole Drilling) followed the characteristic M-shape distribution, with tensile peak stresses present just outside the Heat Affected zone, that very rapidly fall down almost to zero in the far field.

 The contour method measurements are affected by the point density utilized during the cut-surface measurement step and a point density between 0.20mm and 0.25mm is found necessary to produce accurate results.

 The contour method, with the coarse point density, produced highly asymmetrical results, with tensile stresses of the order of ~450MPa in the FZ and peak tensile stresses of the order of ~620MPa just outside the HAZ. The peak tensile stresses, exceeded the offset 0.2% yield strength (576MPa).

 The contour method, with the fine point density, produced significantly less asymmetrical results and is in accordance with the incremental deep hole drilling measurements, with the residual stresses peaking at the range of 450-500MPa. The peak tensile stresses, did not exceed the offset 0.2% yield strength.

 The major difference between the CM, with the refined point density, and iDHD method, for the AW specimen, is the failure of the CM measurement to fall into compression outside the HAZ on each side.

 All four PWHT specimens present a significant residual stress relaxation, after heat treatment, with the stresses dropping in the range of 100-200MPa.

 After PWHT, significant pilot-hole artefacts were present, with the stresses at this region surpassing the measured stress in the FZ and HAZ. This is not realistic.

 After PWHT, the stresses in the parent material are ranging between -50MPa and +100MPa. This is also not realistic.

 It is not possible to rank the PWHT regimes based upon the current state of measurements. Measurements with a refined point density need to be conducted for the pilot-hole-related perturbations to be minimized in the data censoring and analysis

  Figure Amplitude X & Y (mm) Figure Oscillation Frequency (Hz) Sealing Pass 50 2185 5 295 0 0 0

Figure 1 .

 1 Figure 1. Comparative line plots of longitudinal component (S33) of residual stress in the EBW, in the AW and PWHT conditions, as measured by the contour method

Figure 2 .

 2 Figure 2. Incremental and conventional deep hole drilling measurements on AW plate on a transverse line at mid-thickness

Figure 3 .

 3 Figure 3. Comparative plot of longitudinal residual stress distribution on AW plate on a transverse line at mid-thickness, using the contour method (with coarse and fine point density), the deep hole drilling and the Incremental deep hole drilling method.

Table 1 .

 1 Chemical composition of SA508 Gr3 Cl2 steel (wt.%)

		C: 0.18	Si: 0.24		Mn: 1.32 P: 0.019	S: 0.022
		Cr: 0.24	Mo: 0.53	Ni: 0.47 Cu: 0.052 V: 0.003
		Nb: 0.004 Ca: 0.0004 B: 0.0001 Ti: 0.001	Al: 0.003
	Table 2. Mechanical Properties of SA508 Gr3 Cl2 steel
	Temperature	0.2% Proof Stress	Tensile Strength	Elongation	Reduction of Area
	( o C)	(MPa)		(MPa)	%	%
	Room Temp.	576			701	24	60

Table 3 .

 3 Electron Beam welding parameter set

	Beam	Lens	Travel	Working
	Current	Current	Speed	Distance
	(mA)	(mA)	(mms -1 )	(mm)

Table 4 .

 4 Experimental PWHT matrix for 100mm thick EB welds of SA508 Gr3 Cl2

		Temperature	Hold time	Controlled cooling and	Cooling below
		(+/-10°C)		heating rates above 290°C	290°C
	A:	595°C	4 hr.	18°C/hr	Air cooling
	B:	610°C	4 hr.	18°C/hr	Air cooling
	C:	610°C	1 hr.	18°C/hr	Air cooling
	D:	625°C	4 hr.	18°C/hr	Air cooling

Acknowledgements

The authors are grateful to the NAMRC for producing the welded specimens and to VEQTER ltd for conducting the DHD and iDHD measurements. The authors would also like to thank Ian Winstanley and Paul English, at the University of Manchester, for technical assistance. Finally, the authors would like to acknowledge Rolls Royce Plc, for funding the PhD project, where this research was based on.


L Millán

G Carro-Sevillano

G Kronberger

O Garnica

I Collado

G Bokuchava

R Fernández

J I Hidalgo

P Halodova

A Sáez-Maderuelo

G González-Doncel

Genetic Programming and FEM simulation for a microscopic residual stress description in polycrystals using neutron diffraction and EBSD data

Keywords: Residual Stresses, Genetic Programming, FEM, aluminium alloys, microstructure _____________________________________________________________________________________________________________________________ ____

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

Introduction

Metallic material components undergo in-service failure due, in many cases, to the presence of residual stresses, RS. It is known that the damage caused by these stresses manifests at different levels and in varying degrees of severity. The most worrying nowadays are microscopic stresses, m-RS, because they are generated at the microstructural scale, and their magnitude is not yet known. In particular, inter-granular stresses, those existing among neighboring grains, may be the origin of cracks across grain boundaries. These stresses are generated during material/component manufacture and ulterior thermo-mechanical processes. While the determination of RS fields at the macroscopic scale, M-RS, is somewhat overcome using analytical diffraction methods (direct use of Bragg´s law) and classical linear elasticity theory, undertaking this task at the microscopic level remains to be done.

Recent studies have attempted to quantify stresses at the grain level [START_REF] Everaerts | Evaluation of macro-and microscopic residual stresses in laser shock-peened titanium alloy by FIB-DIC ring-core milling with different core diameters[END_REF][START_REF] Salvati | An analysis of macro-and micro-scale residual stresses of Type I, II and III using FIB-DIC micro-ring-core milling and crystal plasticity FE modelling[END_REF][START_REF] Kumagai | Depthprofiling of residual stress and microstructure for austenitic stainless steel surface treated by cavitation, shot and laser peening[END_REF]. However, the invasive characteristics of the techniques used mean that the potential errors associated with the measurements remain high. Diffraction methods are, nowadays, the most powerful tool to investigate RS. The RS, specifically the M-RS, are determined from the displacement of the diffraction peaks obtained in the diffraction pattern; i.e., peak shift from the relaxed position to the stressed one. However, it is known that RS, as a whole, also affect the peak shape, width, and maximum height. The problem relies in that the extraction of the specific stress held by individual grains from the information supplied by the diffraction peaks is, unlike the M-RS, of great complexity. In other words, the conventional analytical methods used for the M-RS cannot be used in this case. To achieve this purpose, a very detailed description of the microstructure, including parameters such as, grain morphology, size, close neighbors, crystal orientation, etc., seems to be needed; It is reasonable that the stress of individual grains is dependent not only on the thermomechanical process underwent by the material, but also on the surrounding grain microstructure. Using this information, together with that supplied by the diffraction experiments, seems a realistic starting point to address this problem.

In this work, Genetic Programming, GP, and simulations using the Finite Elements Method, FEM, are used to deepen into the RS state of individual grains in a single-phase aluminum alloy. The results obtained using both procedures are discussed to take further steps in the objective of quantifying these m-RS. On one hand, the use of Machine Learning, ML, tools, is here used to reproduce the diffraction peak profile generated by grains with a given hkl, and minimizing the error associated with the measurement. This approach is based on the use of Genetic Programming, GP, algorithms. These algorithms provide an expression which predicts the real peaks by "assigning" specific stress to specific grains. Of particular interest in this work is the sensitivity analysis of the possible microstructural parameters affecting the m-RS. On the other hand, FEM is used to analyze the RS developed in individual grains with different yield stress with respect the surrounding medium. The grains are appropriately located in the microstructure, at different locations with respect the sample central axis, and the resulting stress evaluated.

Materials and methodology

An extruded bar (40 mm in diameter) of aluminum alloy AA5083 was selected for this study. This single-phase alloy will yield more robust results due to its simple, non-ageing nature. In other words, the absence of a precipitation sequence, which would affect diffraction peak position and "distort" the calculation the stresses generated, facilitates the overall RS analysis; only thermo-mechanical treatments will be responsible of the RS state developed at the macro and microscopic scales. To simplify the problem as much as possible, a cylindrical sample, 60 mm in length and 25 mm in diameter, extracted from the center of the alloy bar, was used. The RS fields, at the macro and microscopic levels, were generated from a quenching treatment in fresh water after sample annealing at 530ºC during some 3 hours. From previous studies [START_REF] Fernández | A multi-scale analysis of the residual stresses developed in a single-phase alloy cylinder after quenching[END_REF][START_REF] Millán | Further insights on the stress equilibrium method to investigate macroscopic residual stress fields: Case of aluminum alloys cylinders[END_REF], it is known that the resulting profile of the M-RS field responds to a parabola, if observed from a longitudinal cross section of the sample (or a plane parallel to the extrusion axis, and passing through the center of the cylinder). The neutron diffraction experiments were conducted on the Fourier Stress Diffractometer, FSD, operating at the IBR-2 pulsed reactor in the Frank Laboratory of Neutron Physics, FLNP, of the Joint Institute for Nuclear Research, JINR, (Dubna, Russia). The sample was, then, sectioned to extract small cube samples along the cross section of the cylinder (at the mid-height), to study the microstructure at the center of the sample (x=0) and the external region (x=12 mm), to account for the gradient resulting from the extrusion. Electron backscattering diffraction, EBSD, was used for this purpose. The specific details of the neutron diffraction and EBSD measurements are described elsewhere [START_REF] Millán | Study of Microscopic Residual Stresses in an Extruded Aluminium Alloy Sample after Thermal Treatment[END_REF][START_REF] Millán | Estimation of Grain-Level Residual Stresses in a Quenched Cylindrical Sample of Aluminum Alloy AA5083 Using Genetic Programming[END_REF].

Genetic Programming

GP algorithms were designed with the aim of finding an expression of the lattice spacing, 𝒅 𝒊 𝒉𝒌𝒍 , that depends on microstructural parameters. These algorithms search among a set of initial expressions the one that best fit the real neutron diffraction peak. The algorithm assigns a 𝒅 𝒊 𝒉𝒌𝒍 value to every grain within the range of lattice spacing values within the peak. A maximum number of generations, a crossover ratio and mutation probability of the expressions generated, as such as the limit on their length, have been established.

The microstructural parameters considered in this procedure are: grain area (size), number of near neighbors to each grain, their aspect ratio, and the slope of the long grain axis with respect the radial direction of the cylinder. Thirty runs were performed to validate the models. Resulting sensitivity analysis using the expressions predicted by GP is summarized in the plots of figure 1. For each microstructural parameter of the above ones, the mean of the other three was calculated and these values were kept fixed, varying only the one under study. The analysis indicates that the trends described by the predicted expressions appear to be a reflection of the mechanical behavior of the grains. Figure 1A, for example, shows the most influential parameter for 111 grains (the hardest ones) appears to be the Slope. This influence, even more pronounced and with a single trend, is maintained for the outer grains (figure 1C). This suggests that high values of the slope increases 𝑑 𝑖 ℎ𝑘𝑙 , and therefore the stress. The two tendencies in the center may be attributed to grains of different nature within this group. On the other hand, while the lattice spacing decreases with the aspect ratio of grains in the center, it increases it in the outer grains. However, the number of neighbors does not seem to be relevant, as revealed by the high dispersion. The latter seems to be due to the nature of the surrounding neighboring grains. Finally, the Area does not seem to influence 𝑑 𝑖 ℎ𝑘𝑙 in the center, but it does become important in the outer grains, minimizing their stress as Area increases.

In the case of the 200 grains (figure 1B and1D), the influence of slope seems to decrease, although it shows again two behaviors. Aspect ratio is the least important parameter at the center, while area and number of neighbors, especially the latter, seem to be the most relevant. Again, the number of neighbors shows a greater dispersion and a negative trend, possibly since as this parameter increases, the probability that these are 200 (soft) or 311 grains (intermediate hardness) also increases, as these grains are the most abundant at this location. On the outside, it is observed that if the grains are deformed, the stresses are minimized according to the variations of the aspect ratio. It should not be forgotten that these grains are the softest, so it seems reasonable that they tend to modify their spherical shape to the constraints imposed by the environment. The number of neighbors is still relevant, while the area does not influence this position.

Finite element modelling.

The Heat Transfer, Nonlinear Structural Materials and Structural Mechanics Modules of COMSOL Multiphysics software were used to simulate, using FEM, the evolution of the RS during the quenching process. The simulation considers that the sample is fully submerged in the quenching media. The thermo-mechanical behavior is simulated using a directly coupled formulation. The heat conduction problem is solved independently from the stress problem to obtain temperature history. The formulation considers the contribution of the temperature field variation with time to the stress analysis through thermal contraction, as well as temperaturedependent thermal and mechanical properties. The solution is obtained from two steps. First, the temperature distribution and its history in the quenching model are computed by the heat conduction analysis. At this point, it is critical the heat transfer coefficient, HTC. The HTC evolution with temperature calculated according with [START_REF] Chobaut | Measurements and modelling of residual stresses during quenching of thick heat treatable aluminium components in relation to their precipitation state[END_REF] has been used. The convective heat flux, q, across the external surface of the cylinder is described as:

𝑞 = h ⋅ (𝑇 𝑒𝑥𝑡 -𝑇) (1)
where h is the heat transfer coefficient and Text and T the temperature of the water reservoir and the cylinder, respectively. The temperature history is employed as a thermal load in the subsequent mechanical elastic-plastic calculation of the RS field. In this model, the length and radius of the cylinder and water reservoir are 60 mm by 12.5 mm (as the sample used in the experiments), and 400mm by 125mm, respectively, Figure 2. Because of the cylindrical symmetry, a 2D FEM is developed around the length axis of the cylinder. A triangular free mesh has been used, Fig. 2c), with 1216 edge elements and 112 vertex elements. The largest element is 4mm and the smallest one is 0.01mm. The grid is designed such that it gets finer close to grains squares. The external boundary of the cylinders is free. A square of 1x1mm 2 located at the (0,0) position is fixed during the quenching. This allows deformation of the sample and prevents from sample displacement during cooling.

Mechanical constitutive laws.

The mechanical model chosen to simulate the behavior of the material is based on previous work on RS development during quenching [START_REF] Millán | Study of Microscopic Residual Stresses in an Extruded Aluminium Alloy Sample after Thermal Treatment[END_REF]. The aluminum cylinder has been modelled as an isotropic material. In the quenching process the temperature drops suddenly from 530ºC (803K) to room temperature. As the mechanical behavior of the material depends on the temperature, its response should vary from almost pure plastic behavior at high temperature to an elasto-plastic one at room temperature. This evolution was simulated with a thermo-elasto-plastic constitutive model. The plastic behavior is defined as thermal softening; i.e. temperature dependence of yield strength and isotropic hardening. The definition of temperature dependent material properties (the elastic limit among them) allows a gradual increase of the residual stress field (Von Misses) as the temperature decreases. The total strain tensor, , is the driven variable of the mechanical problem computed in terms of the displacement field, u, as:

(2)

The constitutive model including stress equilibrium can be written as:

(3

)
where the force, F, is expressed as the as the Second Piola-Kirchoff stress S, E is (temperature dependent) elastic modulus and  is the Poisson´s coefficient [START_REF] Hailongchen | Constructing continuum-like measures based on a nonlocal lattice particle model: Deformation gradient, strain and stress tensors[END_REF]. The thermal deformation is computed to the thermal contraction during the cooling process from 803K to room temperature. The plastic strain was described by a yield stress and a power-law hardening function with exponent 0.5.

Micro residual stress analysis by FEM.

FEM has been also used to describe the influence of some microstructural parameters on the m-RS generated from the quenching. Among the possible ones, the following parameters are considered here: grain size, aspect ratio, number of neighbors, slope, and texture. In particular, the results concerning the effect of texture and grain position on the intergranular stresses are presented. The present work does not intend to carry out an exhaustive study on the influence of microstructural parameters on the residual stress state. The aim is to show that the use of a combined GP and FEM strategy is effective in unravelling the highly complex correlation between microstructure and the m-RS field. First, a microstructure composed by three grains with a high yield strength located along the radial direction of the cylinder (figure 3a) is considered. The rest of the material is considered as a continuous medium. The yield strength of the blue grains is 25% higher than that of the polycrystal (average value). Figure 3b) shows the axial component of the m-RS between blue grains and those with the average value. The m-RS values obtained are small, < 45MPa. This result is not relevant, because the absolute values are not representative of the microscopic stress Axial inergranular stress (MPa)

Cylinder Water

state as the microstructure used in this case is not real. In fact, the cylindrical symmetry used in this FEM study provides sound conclusions of the axial and radial components. However, these results anticipate relevant information when the relative value of these m-RS at different positions along the radial direction of the sample are compared. In particular, the high yield strength grains located close to the symmetry axis or close to the outer surface, present a considerably higher stress value than those located at positions around r=6mm. In particular, the m-RS at the outermost position is at least three times higher that of the inner ones. This result shows that the quenching is more aggressive on the surface in contact with the medium, resulting in high m-RS.

In the following, a microstructure closer to the real one is although it is still a simplified one (figure 4a). Specifically, the microstructures represented by the EBSD maps of figure 4 have been divided into grids of 5x5 elements. Each of these grid elements has been identified with the population of grains showing the highest volume fraction in this grid. Thus, there is a prevalence of high yield strength, blue grains, in the inner and outer zones of the cylinder, and average grains in the middle (figure 4a). The average grains have the same mechanical properties as the rest of the material, which is considered as a continuous medium. The blue grains in figure 4b) have, then, a yield strength 25% higher than the average and the red ones 25% lower. In this case, it is also observed that the intergranular stresses are strongly dependent on the position in which the grains are located. Figure 4c) shows the axial component of the m-RS among grains with high, low, and average yield strength. The m-RS are small, < 30MPa. As in figure 4b) the relevance of the results revealed in figure 4c), however, is related with the stress achieved at different positions along the radial direction. In particular, the grains with a high yield strength located close to the symmetry axis, present a considerably higher m-RS than in the central position. This result shows that, although the hardening aggressiveness near the symmetry axis of the specimen is low, high stresses develop if certain spatial arrangements of grains of different yield strength are present.

Figure 1 .

 1 Figure 1. Sensitivity analysis resulting from GP, after 30 runs. A) grains with 111 parallel to the extrusion axis, at x=0 mm (sample central region). B) grains with 200 parallel to the extrusion axis, at x=0 mm. C) grains with 111 parallel to the extrusion axis, at x=12 mm (sample outer region). D) grains with 200 parallel to the extrusion axis, at x=12 mm.

Figure 2 .

 2 Figure 2. Geometry of the FEM system showing a) real comparison between the cylinder and the water reservoir (blue) and b) detailed distribution of grains along radial direction and c) meshing elements used for the central part (mid-height) of the cylinder.

Figure 3

 3 Figure 3. a) Distribution of high yield stress grains along sample dimeter and b) axial intergranular stress profile along sample radial direction.

Figure 4 .

 4 Figure 4. a) 5x5 grid over the real microstructure of the material investigated b) distribution of the high yield stress grains along sample radial direction, inverse pole figure, IPF, coloring reference for the extrusion axis and b) axial intergranular stress profile along radial direction.

Acknowledgements

This work has been supported by Spanish This work has been supported by Madrid Regional Goverment-FEDER grants Y2018/NMT-4668 (Micro-Stress-MAP-CM), MAT2017-83825-C4-1-R and B2017/BMD3773 (GenObIA-CM). Thanks also to the FLNR-JINP for the beam time allocated on FSD instrument, and the support by the Christian Doppler Research Association within the Josef Ressel Center for Symbolic Regression.

Conclusions

The analysis of RS at a microstructural scale, m-RS, is a problem of extraordinary complexity. Even in very simplified cases, with good control of the thermomechanical process and the simplicity and high symmetry of the microstructure selected for the study, severe approximations are still needed to delve into the stress developed in individual grains. The m-RS depends not only on the specific characteristics (parameters) of the grains and the thermomechanical process undergone by the component, but also on the specific location of the grain in the component and the parameters of the surrounding neighbors. Techniques such as GP and FEM are used here to investigate these m-RS. The combination of both tools, together with microstructural data of the material, is revealed as an appropriate strategy to achieve this goal, although there is still a long way to go to achieve it.