Alexandra Lacombe
email: alexandra.lacombe1@univ-tlse3.fr

Yann Landon

Manuel Paredes

Damien Texier

Clément Chirol

Audrey Benaben

Investigations on the impact of the hole surface integrity on the fatigue life of a 2024-T351 aluminum alloy drilled part

Keywords: aluminum alloy, fatigue strength, hardness, residual stress, surface roughness

The influence of the hole surface integrity on the fatigue life of 2024-T351 aluminum drilled parts was investigated. Fatigue tests were conducted on open-hole specimens, and a large campaign was carried out to characterize the hole surface integrity (roughness measurements, hardness measurements, metallographic observations, and residual stress analysis). An innovative technique, the Hole Opening Comparative Technique, was set up in order to analyze the residual stress state of the parts. In this study, strain hardening of the hole subsurface seems to be the main factor influencing the fatigue behavior, associated with a residual stress state of the part.

INTRODUCTION

Parts of aircraft structures are mostly assembled using fasteners (rivets or screws) introduced into holes machined by drilling. Assembling an aircraft may require several hundred thousand to several million drilling operations depending on the aircraft size. [START_REF] Girot | Perçage des structures aéronautiques: Les résultats du projet MEDOC. 1er colloque Aquitaine -Québec -Mécanique des Matériaux et des Structures pour l'Aéronautique[END_REF] These holes for fastening are critical areas where fatigue damage can be initiated because they are areas of high stress concentration. The drilling procedure used for the machining of the fastening holes can affect the fatigue life of the drilled part [START_REF] Elajrami | Effect of various drilling procedures on the fatigue life of rivet holes[END_REF][START_REF] Ralph | Fatigue performance of production-quality aircraft fastener holes[END_REF][START_REF] Everett | The effect of hole quality on the fatigue life of 2024-T3 aluminum alloy sheet[END_REF] since, according to the procedure, the hole surface is subjected to different thermo-mechanical loads that can induce differences in the hole surface integrity. [START_REF] Ralph | Effect of various aircraft production drilling procedures on hole quality[END_REF] Field and Kahles [START_REF] Field | Surface integrity of machined and ground high strength steels[END_REF] were the first to introduce the concept of surface integrity in a technical sense by defining it as the inherent or enhanced condition of a surface produced in machining or other surface generation operation. Since then, this definition has been completed by adding the notion of surface functional performance. Thus, according to Mondelin, [START_REF] Mondelin | Modélisation de l'intégrité des surfaces usinées: Application au cas du tournage finition de l'acier inoxydable 15-5PH[END_REF] the surface integrity can be considered as a set of characteristics allowing a surface to be qualified with regard to a given application. Regarding the fatigue performance of a surface, it is well known that the surface integrity includes the concepts of surface topography, residual stresses, and the metallurgical condition of the material subsurface (microstructure and microhardness).

Extending the fatigue life of aircraft is one of the aircraft industry's major concerns. This requires an understanding of the impact of the surface integrity on the fatigue strength. Thus, several works have been carried out in the past few decades to study the relationships between the machining process parameters, the resulting surface integrity, and the associated fatigue life for various materials and machining processes. The reviews by Novovic et al., [START_REF] Novovic | The effect of machined topography and integrity on fatigue life[END_REF] M'Saoubi et al., [START_REF] Saoubi | A review of surface integrity in machining and its impact on functional performance and life of machined products[END_REF] and Pramanik et al. [START_REF] Pramanik | Fatigue life of machined components[END_REF] synthesize many of these works. The main conclusions concerning the individual contributions of the surface integrity parameters for commonly used materials (aluminum alloys, titanium alloys, steel, and nickel alloys) are the following. The lower the surface roughness, the longer the fatigue life [START_REF] Suraratchai | Modelling the influence of machined surface roughness on the fatigue life of aluminium alloy[END_REF] as the micro-notches of a machined surface induce stress concentration areas where a local plastic strain field can be generated when a stress is applied, thus defining a path for the crack. Compressive residual stresses improve the fatigue strength due to the crack closure effect, which slows crack propagation. [START_REF] Maximov | Effect of slide burnishing basic parameters on fatigue performance of 2024-Т3 high-strength aluminium alloy[END_REF][START_REF] Wang | Investigation on fatigue performance of cold expansion holes of 6061-T6 aluminum alloy[END_REF][START_REF] Cuellar | Residual stress and fatigue life in laser shock peened open hole samples[END_REF] In contrast, tensile stresses reduce the fatigue strength through the crack opening effect, which facilitates crack propagation. The fatigue life increases with the work hardening of the surface, which increases the surface yield strength. [START_REF] Sasahara | The effect on fatigue life of residual stress and surface hardness resulting from different cutting conditions of 0.45%C steel[END_REF][START_REF] Yao | Surface integrity and fatigue behavior in shot-peening for high-speed milled 7055 aluminum alloy[END_REF] These surface integrity parameters have different impacts in the fatigue failure: some influence crack initiation and some influence crack propagation [START_REF] Wagner | Thermomechanical surface treatment of titanium alloys[END_REF] (Table 1).

However, a machining operation impacts all the surface integrity parameters mentioned above at the same time, and some parameters depend on one another, so it seems difficult to judge which parameter has a predominant influence on the others. Moreover, depending on the level range considered for a parameter, its influence on the fatigue strength can vary. For instance, according to Siebel and Gaier, [START_REF] Siebel | Influence of surface roughness on the fatigue strength of steels and non-ferrous alloys[END_REF] a reduction in the fatigue endurance limit occurs only above a certain critical groove depth. In addition, the impact of the surface integrity parameters varies from one material to another. Koster [START_REF] Koster | Effect of residual stress on fatigue of structural alloys[END_REF] showed that the endurance limit of steels was dependent on the surface roughness, whereas this was not the case for Ti 6-6-2 and Inconel 718.

Thus, the question of the relationship between the surface integrity and the fatigue strength seems very broad and complex, and the conclusions of the works cited in the reviews mentioned above cannot be generalized to all machining processes and to all materials. In some fields, there are gaps in our understanding of these correlations. This is the case for the hole surfaces obtained by drilling processes in aluminum alloys, which are the most widely used alloys in the aircraft industry. A clear understanding of the impact of these surfaces' integrity on the fatigue life of the drilled part is not yet available, although it is required by the aircraft industry to optimize the fatigue behavior of their assemblies.

The lack of studies on the integrity of drilled aluminum surfaces may be explained by the fact that the thinness of the layer of material affected by the machining process makes the experimental characterization of the surface integrity difficult. A preliminary finite element study simulating the lateral cutting of a drilling process in a 2024-T351 aluminum part was lead in order to precise this depth of subsurface material affected. The finite element model developed for this study was a mechanical model equivalent to the one developed by Atlati [START_REF] Atlati | Développement d'une nouvelle approche hybride pour la modélisation des échanges thermiques à l'interface outilcopeau[END_REF] and was developed with the Abaqus/explicit software. The mechanical laws of material behavior and friction, and the simulation strategy used, were based on those employed by Atlati. [START_REF] Atlati | Développement d'une nouvelle approche hybride pour la modélisation des échanges thermiques à l'interface outilcopeau[END_REF] The study results showed that the depth of the subsurface material affected by plastic strain and residual stresses after machining was a few tens of microns (Figure 1). This depth is very small compared with the one usually studied in the edge area of hole for fasteners in the literature (a few millimeters). [START_REF] Elajrami | Effect of various drilling procedures on the fatigue life of rivet holes[END_REF][START_REF] Wang | Investigation on fatigue performance of cold expansion holes of 6061-T6 aluminum alloy[END_REF][START_REF] Cuellar | Residual stress and fatigue life in laser shock peened open hole samples[END_REF] This paper investigates the impact of the hole surface integrity on the fatigue life of a 2024-T351 aluminum drilled part. This alloy was chosen because it is commonly used in the aircraft industry due to its low density and its high fatigue performance. As axial and orbital drilling processes can lead to different fatigue lives, [START_REF] Sun | Hole-making processes and their impacts on the microstructure and fatigue response of aircraft alloys[END_REF][START_REF] Deitert | Orbital drilling[END_REF] these two processes were considered in the study in order to generate potential differences in hole surface integrity. Axial drilling is the most common process for the machining of a fastening hole in the aircraft industry and is usually named "conventional drilling." Orbital drilling is a more recent process, corresponding to helical milling (Figure 2), which has many advantages over the conventional process, such as avoiding burr formation and allowing better chip evacuation. [START_REF] Pereira | A review of helical milling process[END_REF] To carry out the study, fatigue tests and surface integrity analysis (roughness measurements, hardness measurements, metallographic observations, and residual stresses analysis) were performed in order to identify the parameters that control the fatigue life.

MATERIALS AND METHODS

Test specimens

Fatigue specimens were machined from 2024-T351 aluminum plates. This aluminum alloy is an Al-Cu-Mg alloy which has been solution heat treated, control stretched, and naturally aged. The mechanical characteristics of the specimens are presented in Table 2.

Fatigue specimens were "open-hole" T-type elementary specimens (Figure 3) machined in such a way that their longitudinal axis was aligned with the rolling direction of the 2024-T351 plate. The specimen width was three times the nominal hole diameter (D), and its length T A B L E 1 The influence of surface integrity characteristics on fatigue failure [START_REF] Wagner | Thermomechanical surface treatment of titanium alloys[END_REF]

Crack nucleation

Crack propagation

Surface roughness

Accelerates No effect

Cold work Retards Accelerates

Residual compressive stress

Minor or no effect Retards was 200 mm. The holes were obtained by a one-step drilling operation followed by a deburring operation.

To generate potential differences in hole surface integrity, axial and orbital drilling processes corresponding to industrial applications were used. As different tools and cutting parameters are used with these processes from one drilling diameter to another, two drilling diameters were considered in the study: 6.35 and 9.53 mm.

The fatigue specimen thickness was chosen according to industrial applications. It was 6.35 mm for the specimens with D = 6.35 and 10 mm for the specimens with D = 9.53 mm.

According to Faurie et al., [START_REF] Faurie | Guide du dessinateur Les concentrations de contraintes[END_REF] the stress concentration factor in the working section (gross section minus hole area) of the specimens was 3.5. The maximum stress in

R p0.2 (MPa) R m (MPa) A (%) 290 430
the hole edge area is obtained by multiplying the stress in the gross section by the stress concentration factor. The value of the stress concentration factor considered in this study ensured that the onset of cracking took place at the hole edge.

Procedures

Drilling processes

Drilling tests were carried out using a DMG DMU85eVo machining center. In order to conduct orbital drilling operations, a PRECISE France -ORBIBOT orbital spindle was fixed on the Z-axis of the machining center. For this drilling process, the machining was done in up-milling (tool rotation and orbital rotation in the same sense). The cutting parameters, the tools, and the lubrication used for each drilling process and for each hole diameter corresponded to industrial applications. They are presented in Table 3.

Fatigue tests

Fatigue tests were performed using a Schenk servo hydraulic machine equipped with a 100-kN load cell, at room temperature. The load was applied in the longitudinal direction of the specimen as a cyclic (sinusoidal) tensile-tensile load with a load ratio of 0.1 and a frequency of 20 Hz. Fatigue tests were carried out for various stress levels in order to obtain S-N curves (Wöhler curves). The maximum stress level applied in the working section varied from 100 to 280 MPa.

Roughness measurements

Two-dimensional roughness parameters (Ra, Rq, Rz) were measured along the height of the hole with a profilometer. In accordance with [ISO 4288, 1996], these measurements were performed with a cut-off length of 0.8 mm.

In order to complete the roughness results obtained from the analysis of linear profiles, an analysis of the surface roughness was also carried out. Three-dimensional roughness parameter measurements (Sa, Sq, Sp, Sv, Sz, Sdq, Ssk, Sku) were made using an ALICONA Infinite Focus device, which is capable of performing noncontact optical surface topology measurements through the focus variation principle. A magnification of 10× and a vertical resolution of 200 nm were used for the measurements.

Hardness measurements

Vickers microhardness measurements were made on the hole surface of the fatigue specimens previously cut in the working section (Figure 4). The indentations were In order to complete the microhardness results and to identify the material depth affected by a variation in hardness, nanoindentation tests were also conducted. The indentations were made at mid-thickness of the fatigue specimen in the (x,y) plane. To do this, specimens were extracted from fatigue specimens and set in a resin in order to be polished. They were polished manually using emery papers with decreasing grit size (down to 4000 grit), then using diamond paste (1 μm), and finished with OP-S suspension. The applied load and the size of the indentation matrix were determined on the basis of the results of the preliminary numerical study, which predicted an affected material depth of a few tens of microns (Figure 1). The indentation matrix was a matrix of 5 × 20 indents equally spaced 5 μm apart, with the first column located around 5 μm from the hole edge (Figure 5). The indentation tests were performed, with a diamond Berkovich indenter, as load-controlled tests with a maximum load of 5 mN. Load-displacement curves obtained were analyzed using the Oliver and Pharr method [START_REF] Pharr | Measurement of mechanical properties by ultra-low load indentation[END_REF] to determine the nanohardness HIT.

SEM-EBSD analysis

Previous works had shown that the electron back-scatter diffraction (EBSD) technique could be used to localize and estimate plastic strain through the analysis of local changes in the crystal orientation. [START_REF] Kamaya | Measurement of plastic strain of polycrystalline material by electron backscatter diffraction[END_REF][START_REF] Lehockey | Mapping residual plastic strain in materials using electron backscatter diffraction[END_REF] This technique was therefore used in the study to assess the strain hardening state of the hole subsurface. The grain reference orientation deviation (GROD), which represents the misorientation between a given pixel and the average orientation of the grain, was studied as a plasticity marker.

SEM-EBSD observations were carried out using a Feg Jeol JSM 7100 F scanning electron microscope (SEM) equipped with an Oxford Instruments Nordlys Nano detector (Centre de microcaractérisation Raimond Castaing, CNRS UMS 3623, Toulouse, France). They were performed on a section at mid-thickness of the fatigue specimen in the (x,y) plane, as for the nanoindentation tests. The observed section was prepared with a cross section polisher, to preserve the microstructure and avoid polishing artefacts. For the measurements, the sample was tilted to 70 relative to the incident beam, as in conventional ESBD analysis, and a voltage of 20 kV was used. An area of 600 × 200 μm in the hole edge area was scanned with a step size of 0.5 μm.

Residual stress analysis

Current techniques for residual stress evaluation (X-ray diffraction and incremental hole drilling) were first considered for the study of the residual stress state of the part. However, these techniques were difficult to implement due to the cylindrical shape of the hole, the large grain size of the material, and the thinness of the material layer affected by residual stresses in the hole edge area.

Therefore, a novel strategy, the Hole Opening Comparative Technique (HOCT), which is suitable for the analysis of residual stresses for thin material layer affected in the hole subsurface, was set up. This method is based on the splitting method [START_REF] Walton | Deflection methods to estimate residual stress[END_REF] which is usually used to assess the residual stresses in thin-walled tubes. [START_REF] Baldwin | Residual stresses in metals[END_REF] The HOCT consists in introducing a slot along the half of the working section of the fatigue specimen that opened the hole and measuring the specimen deformation induced by this material removal which indicates the level and sign of residual stresses (Figure 6A). The specimen deformation level indicates the residual stress level: the greater the residual stress state in the hole edge area, the greater the specimen deformation after the opening of the hole. The shape of the deformation profile indicates the residual stress sign: a convex profile corresponds to tensile residual stresses, whereas a concave profile corresponds to compressive residual stresses (Figure 6B).

Thus, the HOCT allows to evaluated the residual stress state in the hole edge area in drilled open-hole T-type specimen in a qualitative way and can be used as a quick comparative test. Moreover, the quantitative analysis of the residual stresses could also be possible through the development of a numerical model which could determine, using the inverse method, the residual stress state corresponding to the specimen deformation induced by the HOCT.

The different steps of the HOCT are given below:

1. Drilling of a hole in a specimen assumed to be free of residual stresses; 2. Measurement of the profile of the specimen after drilling; 3. Machining of the slot by Electrical Discharge Machining (EDM); 4. Measurement of the profile of the specimen after machining of the slot; 5. Analysis of the residual stress state of the specimen through the measured specimen deformation.

In this work, the material studied is the aluminum alloy 2024-T351. With the treatment T351, the material is stretched a controlled amount for stress relief after being solution heat treated. Therefore, in this study, the fatigue specimens can be assumed to be free of residual stress before the drilling operation. Besides, with the HOCT, the machining of the slot is done by EDM which avoids introducing significant levels of residual stress in the specimen. Thus, the specimen deformation measured is assumed to be essentially related to the residual stresses introduced during the drilling operation.

In the HOCT procedure, the specimen deformation is evaluated through the measurement of a profile shape before and after the material removal. As the residual stresses are assumed to be much greater in the radial and orthoradial directions than in the axial direction, the specimen deformation induced by the HOCT is measured in the (x,y) plane. The measured profile is oriented along the longitudinal direction of the specimen and located at mid-thickness of the specimen on the side opposite the machined slot (Figure 6A). In this study, the profile measurements were performed with an ALICONA Infinite Focus device. (Wöhler curves), with S corresponding to the normalized stress in the working section of the specimen and N corresponding to the number of cycles leading to failure. Power trend curves are plotted on these graphs. In the same way, the values of the surface integrity parameters presented in the following sections (roughness, microhardness, and nanohardness) are also normalized by their maximum value. S-N curves obtained for D = 6.35 mm show no difference in fatigue strength between the two drilling processes. In contrast, S-N curves obtained for D = 9.53 mm show a significant increase in fatigue life for the specimens drilled by the axial technique. This gain in fatigue life can be observed for all stress levels but is greater for the lower ones.

RESULTS

Fatigue life

In the aircraft industry, the fatigue performance of a structure is commonly evaluated by the Airbus Fatigue Index (AFI), which corresponds to the stress S associated with a forecast fatigue life of 100,000 cycles. Thus, AFI values were determined for all configurations from S-N curves. They are shown in Figure 7C,D (as values normalized by the maximum AFI value obtained).

For D = 6.35 mm, similar AFI values can be observed for both drilling processes, whereas for D = 9.53 mm, a gain in AFI of 15% is visible with the axial drilling process.

The fracture surfaces were observed for all drilling configurations. No difference in failure mode was noted between the different configurations. For all specimens, the crack initiated in the hole edge area and the fracture surface was divided into two distinct zones: the crack propagation zone and the final fracture zone (Figure 8).

Thus, the drilling configuration can have a significant influence on the fatigue strength of the drilled part, but it has no impact on the failure mode of the specimen. The differences in fatigue life observed between the drilling configurations were probably related to differences in the hole surface integrity. Characterization test results of the hole surface integrity are presented in the following sections.

Roughness measurements

Among all the roughness parameters measured, only the results of arithmetic average height of profile (Ra), which gives an average indication of the overall surface topological state, are presented here. They are shown in Figure 9. For all drilling configurations, the measured Ra value complies with the aeronautical specification. No correlation can be established between the Ra results and the fatigue test results. A significant difference in Ra is observed between the two drilling processes for D = 6.35 mm whereas no difference in fatigue life is noticed between the two drilling techniques for this hole diameter. This is probably related to the relatively low Ra values measured compared to the aeronautical specification. The Ra values are not high enough to have an impact on the fatigue strength of the part.

The possible relationship between the hole roughness and the fatigue life results was also studied for all the other 2D and 3D roughness parameters measured. No obvious correlation between the roughness parameters and the fatigue strength of the drilled part can be highlighted from this analysis. As for Ra, this is explained by the low measured values of the 2D and 3D roughness parameters which do not indicate a degraded surface texture. In this study, the drilled hole surfaces seem to have a fatigue behavior corresponding to the first domain of the Kitagawa diagram. [START_REF] Kitagawa | Applicability of fracture mechanics to very small cracks or the cracks in the early stage[END_REF] This domain, which is associated to small defect sizes, corresponds to a negligible influence of the defect size on the fatigue strength and to

F I G U R E 8
Example of a fracture surface a crack initiation controlled by the microstructural state of the material.

Thus, in this study, the hole roughness is not a major factor influencing the fatigue life of the drilled part. That is why the further characterization of the hole surface integrity focuses on the internal aspects of surface integrity.

Hardness measurements

After analysis of the topographic aspects of the hole surface integrity, the metallurgical and mechanical aspects were investigated. The Vickers microhardness measurements carried out in this framework are presented in Figure 10.

For D = 6.35 mm, holes obtained by axial and orbital drilling processes have similar microhardness levels, whereas for D = 9.53 mm, a significant difference in microhardness level is observed between the two drilling techniques. For this diameter, a gain of 29% in microhardness level can be seen for axial drilling.

These results show a correlation of fatigue life results with microhardness measurements: the increase in hole microhardness for axial drilling at D = 9.53 mm seems to explain the gain in fatigue life observed for the same configuration (Figure 7). To study this result further, nanoindentation tests were performed. The test results are presented in Figure 11 as graphs showing the variation of the nanohardness according to the distance from the hole edge.

For D = 6.35 mm, nanohardness results show a similar material depth affected by hardness variation, around 15 μm for both drilling processes. In contrast, for D = 9.53 mm, a significant difference in material depth affected is observed between the two drilling techniques. The material depth affected for orbital drilling seems to be very small (less than 5 μm), whereas it is around 40 μm for axial drilling. Also, during indentation tests for the specimen corresponding to axial drilling at D = 9.53 mm, the first indents column induced quite small, irregular prints from which no hardness value could be calculated. This indicates that the nanohardness value of the hole subsurface obtained for this configuration seems to be high.

These results highlight the thinness of the subsurface material layer affected by the drilling operation of a 2024-T351 aluminum part, in agreement with the numerical model. The nanohardness results are in accordance with the microhardness results. The hole hardness seems to be a major factor influencing the fatigue behavior of a drilled part with a positive influence.

SEM-EBSD analysis

In order to identify the phenomena responsible for the difference in hole surface hardness, a metallographic analysis was carried out. In this framework, GROD maps of the hole edge area obtained through SEM-EBSD analysis (Figure 12) were analyzed.

The GROD map obtained for axial drilling at D = 9.53 mm shows a significant material depth affected by grain misorientations, around 40 μm depth. This affected material layer varies slightly according to the grains. In contrast, almost no grain misorientation, or only some a few microns deep, is observed for the other drilling configurations. A material layer affected by grain misorientation corresponds to a strain hardened material layer. This means that axial drilling at D = 9.53 mm induces a significant strain hardening of the hole subsurface compared to other drilling configurations.

These results explain the microhardness and nanohardness results. The increase in hardness in the hole surface and subsurface, for axial drilling at D = 9.53 mm, seems to be due to the significant strain hardening of the material in the hole edge area. This assumption is strengthened by the fact that the affected material depths identified with the nanoindentation tests and with the SEM-EBSD tests are the same.

Residual stress analysis

The residual stress state of the drilled part, which, according to the literature, can also have a significant impact on the fatigue life, [START_REF] De Castro | An overview on fatigue analysis of aeronautical structural details: open hole, single rivet lap-joint, and lap-joint panel[END_REF][START_REF] Sticchi | Review of residual stress modification techniques for extending the fatigue life of metallic aircraft components[END_REF] was investigated through the novel HOCT technique. Specimen deformations obtained with HOCT are presented in Figure 13.

No deformation of the specimen is noticed after the material removal for the case of axial and orbital drilling at D = 6.35 mm and the case of orbital drilling at D = 9.53 mm. However, significant specimen deformation is observed for axial drilling at D = 9.53 mm. This means that this drilling configuration induces significant residual stress fields in the hole edge area compared to the others.

These results show a correlation of the fatigue test results with the residual stress state of the material. A high fatigue quality index is associated with the presence of significant residual stresses in the hole edge area. Furthermore, a high residual stress state is associated with high hole hardness.

DISCUSSION

This study focuses on drilling processes corresponding to industrial applications, that is to say, using optimum tools and cutting parameters in order to obtain high surface quality in accordance with the industrial specification. For this study case, the hole roughness is not a major factor influencing the fatigue life of the drilled part. In accordance with Siebel and Gaier, [START_REF] Siebel | Influence of surface roughness on the fatigue strength of steels and non-ferrous alloys[END_REF] it seems that, below a given limit, which could be the aircraft specification here, the hole roughness does not significantly impact the fatigue behavior. However, if the roughness reaches high values, its influence could be significant. This confirms the importance of the roughness specification for these aluminum alloys. In contrast, the hole hardness has a significant positive influence on the fatigue life of the drilled part in this study. A variation in hardness can be related to several factors. It can be related to mechanical loading inducing the strain hardening of the hole subsurface and to thermal loading inducing material transformations, and for nanohardness, it can also be related to the residual stresses induced by thermomechanical loading. Regarding the impact of the thermal loading, the temperatures reached during an aluminum drilling operation do not seem to be high enough to induce a phase transformation. [START_REF] Girot | Perçage des structures aéronautiques: Les résultats du projet MEDOC. 1er colloque Aquitaine -Québec -Mécanique des Matériaux et des Structures pour l'Aéronautique[END_REF] Besides, an SEM analysis of the precipitation state of the hole subsurface, conducted as part of the study, showed no increase in the number or size of the precipitates, which could reflect precipitation hardening, for the case of axial drilling at D = 9.53 mm. Thus, it seems that the thermal loading involved during drilling of a 2024-T351 aluminum part has no significant influence on the hole surface integrity. Regarding the impact of the mechanical loading, the SEM-EBSD analysis showed a significant material depth affected by strain hardening for the axial drilling process at D = 9.53 mm. Thus, it seems that, in our study case, the mechanical loading involved in drilling controls the hole surface integrity.

Therefore, the increase in hole hardness for the aluminum alloy studied is essentially related to the strain hardening of the hole subsurface, which is induced by the severe mechanical loading applied on the machined surface during the drilling process. The positive influence of the hole hardness on the fatigue life can then be explained by the increase in yield strength in the hole edge induced by the strain hardening, which slows damage generation (crack initiation) in this area.

HOCT results showed that a significant hole hardness induced by strain hardening is associated with a significant residual stress state of the part. Severe mechanical loading applied to the machined surface simultaneously generates high plastic strain and high residual stresses. The residual stresses can remain in the part because of the strain hardened material layer. Thus, it seems difficult to dissociate these two aspects. In order to study the impact of the residual stress on the fatigue strength in greater depth, it would be interesting to develop a numerical model capable of using the inverse method to determine the residual stress state corresponding to the fatigue specimen deformation obtained with the HOCT.

For the aircraft industry, it would be interesting to identify a surface integrity characteristic that would be quite easy to evaluate, as an indicator of the fatigue strength, in order to be able to assess the fatigue life of aircraft structures easily. The correlation between the hole surface microhardness and the fatigue strength of the drilled part has been demonstrated in this study, and the measurement of this surface integrity characteristic seems relatively accessible: it does not require surface preparation, the test time is short, and the cost of the device is moderate. Thus, the Vickers microhardness could be identified as an indicator of the fatigue strength for aluminum 2024-T351 drilled parts.

CONCLUSIONS

In order to identify the surface integrity parameters that control the fatigue strength of a 2024-T351 aluminum drilled part, fatigue tests were carried out on open-hole specimens and the surface integrity of the holes was characterized with a study scale and techniques suitable for aluminum drilled surfaces. An innovative technique for the residual stress evaluation of aluminum drilled parts, the HOCT, was implemented. The main conclusions of the study are as follows:

• For hole roughness levels obtained with industrial drilling processes, the hole roughness has no significant influence on the fatigue strength. • The hole hardness has a significant positive influence on the fatigue life and could be identified as a fatigue strength indicator for 2024-T351 aluminum aircraft drilled parts. • The increase in hole hardness for a 2024-T351 aluminum drilled part is mainly due to the strain hardening of the subsurface material, induced by severe mechanical loading during the machining process. • A high hole hardness is associated with a significant residual stress state of the part in the hole edge area.

The study focused on the fatigue life of open-hole specimens as a first step, in order to avoid the phenomena associated with the introduction of a fastener (impact of the tightening torque, impact of the assembly clearance or interference level, etc.). In future works, it would be interesting to study the fatigue life of assemblies of drilled parts with fasteners, to assess the influence of the hole surface integrity in this case.

Finally, in order to be able to improve the fatigue life of aircraft structures, the influence of the drilling process (tool geometry, cutting parameters, associated thermomechanical loading) will have to be studied in more detail in order to develop a drilling process generating optimum hole surface integrity.

F

  I G U R E 1 Results of the lateral cutting simulation: chip morphology (A), equivalent plastic strain of the subsurface (B) and residual stress of the subsurface in the tangential direction (C) F I G U R E 2 Kinematics of the orbital drilling process 22 T A B L E 2 Minimum mechanical properties for 2024-T351 aluminum plate (NF EN 485-2+A1, 2018)

F I G U R E 3

 3 Fatigue specimen geometry T A B L E 3 Drilling parameters D = 6.35 mm D = 9.53 mm

F

  I G U R E 4 Vickers microhardness test set-up F I G U R E 5 Nanoindentation test set-up (A) and example of a nanoindentation matrix (B)

For

  reasons of confidentiality, fatigue test results are presented in Figure 7A,B as semi-log S-N curves F I G U R E 7 Wöhler curves obtained for D = 6.35 mm (A) and D = 9.53 mm (B) and corresponding AFI results (C,D)

F I G U R E 9

 9 Roughness average results for D = 6.35 mm (A) and D = 9.53 mm (B) F I G U R E 1 0 Vickers microhardness results for D = 6.35 mm (A) and D = 9.53 mm (B)

F

  I G U R E 1 2 GROD maps obtained with the SEM-EBSD analysis F I G U R E 1 3 Specimen deformation obtained with the HOCT for D = 6.35 mm (A) and D = 9.53 mm (B) 3.5

F I G U R E 6 Specimen for the residual stress analysis with the HOCT (A) and shape of the deformation profile according to the residual stress sign (B)

F I G U R E 1 1 Nanohardness results for D = 6.35 mm (A,C) and for D = 9.53 mm (B,D)

AUTHOR CONTRIBUTIONS

Development of the original idea and the protocol: Alexandra Lacombe, Yann Landon, and Manuel Paredes. Analysis and interpretation of data: Alexandra Lacombe, Yann Landon, Manuel Paredes, and Damien Texier. Writing original draft: Alexandra Lacombe. Writing, review and editing: Alexandra Lacombe, Yann Landon, Manuel Paredes, Clément Chirol, and Audrey Benaben.

ACKNOWLEDGMENT

The authors would like to thank Airbus France for its financial and technical support.

DATA AVAILABILITY STATEMENT


Siavash Ghabezloo
email: ghabezloo@cermes.enpc.fr

Ahmad Pouya

Numerical modelling of the effect of weathering on the progressive failure of underground limestone mines

INTRODUCTION

Many underground limestone mines exist in France particularly in the Parisian and Normandy regions. The failure of these underground mines causes damages on infrastructures and buildings on the ground surface and sometimes their total ruin. An overview of the existing methods for modelling the failure of underground mines shows that they cannot predict adequately the progress rate and the occurrence time of this phenomenon. The objective of this paper is first to present a set of equations governing different hydro-chemo-mechanical aspects of the progressive failure process occurring in these mines. Then, the feasibility of numerical modelling of this process will be studied and a simple example of application will be presented.

FAILURE MECHANISM

Observations on various cases show a typical type of failure that emerges brutally to the ground surface by creating a pseudo-circular crater (Fig 1 d), called "fontis", begins with a localized failure at the roof of the gallery (Fig 1a). The rise of the bell shaped rupture to the roof of the gallery generally follows the local ruptures of the roof. This process can evolve to stability due to presence of a resistant and stiff layer that is opposed to the development of failure. The rise of the rupture thus stops as long as there are not any more new degradations on the level of the highroof. With time, the resistance of the stable layer could be degraded and the failure process continues until the rupture emerges on the surface, in the form of a crater [START_REF] Tritsch | Evaluation des aléas liés aux cavités souterraines[END_REF][START_REF] Fayad | Etude de stabilité de fontis au toit des carrières souterraines et traitements apportés aux conséquences induites en surface[END_REF]). The diameter and the depth of the "fontis" can vary between a few meters and several tens of meters according to the geometry of the mine, nature and the thickness of the covering layers and the presence of a water table in these layers. The results of in-situ experiments of [START_REF] Morat | Heat and water transport by oscillatory convection in an underground cavity[END_REF] can help to understand the mechanism of gradual degradation and failure in the underground limestone mines. They observed that if mirrors are placed near the floor, walls (or pillars) and roof of a cavity, it is apparent that water condenses on the roof and the upper part of the pillars but not at the bottom of the pillars nor on the floor where mirrors remain dry. Furthermore, deposits of calcite on the floor and lower part of the pillars constitute indirect evidence of water evaporation. They carried out the experiments in the Mériel quarry, 46 Km northwest of Paris, dug according to the 'rooms and pillars' method. The porosity of limestone was in the 25 -45% range [START_REF] Jouniaux | Etude des proprétés physiques du calcaire de la carrière de Mériel[END_REF]) and the water saturation was between 60 and 80%. Temperature, air ABSTRACT: The observations show that the collapse of underground limestone mines results from a progressive failure due to gradual weathering of the rockmass. The following stages can be considered for the limestone weathering and degradation process in underground mines: condensation of the water on the roof of the gallery, infiltration of the water in the porous rock, migration of the air CO 2 molecules in the rock pore water by convection and molecular diffusion, dissolution of limestone by CO 2 rich water and consequently, reduction of the strength properties of rock. Considering this process, a set of equations governing different hydro-chemo-mechanical aspects of the weathering phenomenon and progressive failure occurring in these mines is presented. Then the feasibility of numerical modelling of this process is studied and a simple example of application is presented.

pressure and relative humidity measurements in the gallery and self-potential recordings in the walls of gallery evidenced the oscillatory convective motions of the air of the gallery, driven by the geothermal gradient, which transports water and heat. The temperature difference between the floor and the roof of the gallery is lower than that which would give the normal geothermal gradient, indicating that the transport of heat is more effective in the air of the gallery than through the wall-rock. Consequently, they noted that the water, which evaporates on the floor of the gallery, is transported by convection to the roof where it condenses and enters the rock. This water attacks the rock, preferentially along the existing cracks. The water can invade these cracks, enlarge them by dissolution and cause the failure of the roof. The process will repeat itself and in some cases can cause the appearance of the fontis on the ground surface. The mechanism of the gradual failure is summarized in the figure [START_REF] Fayad | Etude de stabilité de fontis au toit des carrières souterraines et traitements apportés aux conséquences induites en surface[END_REF]. The time needed for the completion of this process depends on the thickness of the competent layer. For limestone quarries in the Paris region, it seems to vary from a few tens of years to a few centuries [START_REF] Morat | Heat and water transport by oscillatory convection in an underground cavity[END_REF]). 3 PHYSICO-MECHANICAL PROPERTIES OF LIMESTONE [START_REF] Massieu | Essais d'identification et essais mécaniques sur le calcaire grossière de la carrière de Villiers-Adam[END_REF], [START_REF] Thierry | Problèmes posés par un calcaire hétérogène dans l'étude de la stabilité d'une carrière souterraine[END_REF] and [START_REF] Pothérat | Stabilité des carrières souterraines abandonnées, Etude et suivi instrumental du laboratoire souterrain de Mériel-Villiers-Adam (Val d'Osie)[END_REF] studied the physico-mechanical properties of the limestone of the underground quarry of Villiers-Adam, located in Val d'Oise. Measurements of degree of saturation show that the samples taken in the roof have the highest degrees of saturation, between 85 and 100%. The degree of saturation of the pillars ranges between 58 and 72% and the separated blocks on the floor have the smallest degree of the saturation, located between 49 and 57%. These results are coherent with the evaporation and condensation process, described by [START_REF] Morat | Heat and water transport by oscillatory convection in an underground cavity[END_REF]. The petrographic analysis shows that the limestone of the underground quarry of Villiers-Adam is constituted approximately of 90% calcite. More than four hundred porosity measurements show that the values of porosity are between 28 and 46% with an average value of 37,7%. The results of the uniaxiale compression tests show the strong influence of the degree of saturation on the compressive strength of limestone (figure 2). The results of the tests on the samples with different porosities show the decrease of the compressive strength of the saturated rock with porosity increase (figure 3). An increase in the porosity from 30 to 40% involves a reduction in the compressive strength from 13 to 4.5 MPa. The results of Brazilian tests show the same influence of the porosity on the tensile strength of saturated limestone (figure 3). R c and R t in the figures (2) and ( 3) are respectively uniaxiale compressive strength and tensile strength. The values measured for the Young's modulus were relatively dispersed, varying between 4600 and 12000 MPa. The results shown in figure [START_REF] Dreybrodt | Basic processes and mechanisms governing the evolution of karst. Speleogenesis and evolution of karst aquifers[END_REF] indicate that this modulus decreases for higher porosities.

PROPOSED METHOD: EFFECT OF WEATHERING

As described before, the observations show that the collapse of underground limestone mines results from a gradual phenomenon of weathering of the rockmass leading to a progressive failure.

Weathering process

Considering the in-situ experiments of Morat et al. (1991) in the limestone quarry of Mériel and the studies of [START_REF] Massieu | Essais d'identification et essais mécaniques sur le calcaire grossière de la carrière de Villiers-Adam[END_REF], [START_REF] Thierry | Problèmes posés par un calcaire hétérogène dans l'étude de la stabilité d'une carrière souterraine[END_REF] and [START_REF] Pothérat | Stabilité des carrières souterraines abandonnées, Etude et suivi instrumental du laboratoire souterrain de Mériel-Villiers-Adam (Val d'Osie)[END_REF] on the physico-mechanical properties of the limestone of the underground quarry of Villiers-Adam, we can consider the following mechanism for the weathering of limestone in underground cavities. [START_REF] Hajna | Chemical Weathering of Limestones and Dolomites in A Cave Environnement, Speleogenesis and evolution of karst aquifers[END_REF] observed a more or less identical mechanism in several natural cavities of limestone and dolomite in Slovenia.

Water evaporates on the floor and condenses on the roof of the cavity. Consequently, water is always present on the surface of the roof and infiltrates in the unsaturated rock. Infiltration of water in the rock causes the increase of the degree of saturation and as shown before, it decreases the strength properties of the rock. The pure water condensed on the roof dissolves the air's CO 2 and becomes acidic. The quantity of CO 2 dissolved in unit volume of water depends on its partial pressure in the cavity air and the temperature. So the water infiltrating the rock is rich in CO 2 and acidic and causes dissolution of calcite. Besides, the CO 2 concentration on the surface is higher than within the rockmass. This concentration gradient causes an inward migration of CO 2 from the rock surface by molecular diffusion. Consequently, the concentration of saturation of Ca 2+ ions in the rock pore water increases and the limestone continues to dissolve in its pore water. By this mechanism of limestone dissolution, the porosity increases and consequently the mechanical degradation continues even without more water infiltration. To include all these mechanisms in the model, we consider four stages in the weathering process of limestone: 1) in-filtration of water in the rock which increases the degree of saturation, 2) inward migration of dissolved CO 2 from the roof surface by advection and molecular diffusion, 3) dissolution of limestone, 4) mechanical degradation or reduction in the strength parameters of limestone due to the increase in the degree of saturation and also in the porosity of the rock.

By introducing the kinetic model and characteristic times of these different processes in the global model, we can elaborate a model for the weathering process of limestone and gradual failure of the underground cavities including the temporal aspect. Thus, it is necessary to study four different phenomena and try to model them: 1) water transfer, 2) diffusion-advection, 3) limestone dissolution, 4) mechanical degradation. In the following sections we describe separately these phenomena and their models.

Water transfer phenomenon

The hydraulic head is deduced from the potential of water and is written:

w S H z h γ z = + = + (1)
Where H is the hydraulic head, S is the suction, γ w the specific weight of water, z is the elevation and h is the pressure head. To calculate the suction according to the water content, the following equation is used [START_REF] Kaufmann | Modelling unsaturated flow in an evolving karst aquifer[END_REF]):

1 ln h r S e h n α θ α ⎛ ⎞ = ⇒ = ⎜ ⎟ ⎝ ⎠ (2)
Here θ is volumic water content, S r is degree of saturation, n is porosity and α is an experimental constant.

r S n θ = (3)
Darcy's law is assumed to remain valid in unsaturated medium and the permeability is supposed to be constant:

V K H = -∇ (4)
Where V is velocity and K is permeability. The continuity equation expresses the mass conservation for the water in the case of incompressible unsaturated medium:

0 V t θ ∂ ∇ ⋅ + = ∂ (5)

Diffusion-advection phenomenon

The transport of the solids dissolved in rock pore water can occur via two mechanisms, advection and diffusion. The advection results from the movement of water while diffusion occurs because of gradients of concentration in the medium [START_REF] Fetter | Contaminent Hydrogeology[END_REF]. The total mass of the dissolved materials per unit of area, transported per unit of time is described by the following equation:

v F C D C θ θ = - ∇ (6)
Where v is mean velocity, C is molarity (M/L 3 ) and D is coefficient of diffusion (M 2 /T). By writing the equation of mass conservation for a representative elementary volume and considering the mass produced by chemical reactions we find:

( )

0 C F R t θ ∂ + ∇ ⋅ + = ∂ (7)
Where R is the reaction rate of calcite dissolution defined as the mass produced by chemical reactions in unit volume of water per unit of time.

Dissolution phenomenon

Dissolution mechanism

Limestone dissolves in water and the presence of carbon dioxide increases the calcite solubility. The dissolution of calcite in water in presence of carbon dioxide is given by the following reaction [START_REF] Plummer | The kinetics of calcite dissolution in CO 2 -Water system at 5° to 60°C and 0.0 to 1.0 atm CO 2[END_REF]

2 2 2 3 1 3 1 2 2 4 C H CO Ca eq Ca HCO K K K C P K γ γ + + - ⎛ ⎞ ⎜ ⎡ ⎤ = ⎣ ⎦ ⎜ ⎟ ⎝ ⎠ ⎟ 2 (9)
In this equation K 9) are taken equal to 1, then this equation and (10) allow us to write:

2 2 3 1 2 4 C CO Ca eq K K C K + C ⎡ ⎤ ⎡ ⎤ = ⎣ ⎦ ⎣ ⎦ (11
C dC R k dt C + + ⎛ ⎞ ⎜ = = - ⎜ ⎟ ⎡ ⎤ ⎣ ⎦ ⎝ ⎠ ⎟ ( 12
)
Where k cal is the reaction coefficient (M/L 3 T) and n cal is the order of reaction. C Ca2+ and [C Ca2+ ] eq are respectively molarity and saturation molarity of dissolved calcite in water (M/L 3 ). The ratio of these terms is called saturation ratio. The values of n cal and k cal change with saturation ratio, temperature and CO 2 partial pressure.

Degradation phenomenon

The dissolution of calcite increases the volume of voids and thus the porosity of rock. The porosity increase is equal to the volume of dissolved calcite per unit of fluid volume, multiplied by the volumic water content:

2 v c a l Ca t t dV dV m dn dC V V M θ + = = = (13)
In these relations, dV v is the variation of volume of the voids, V t is total volume and dV cal and dC Ca2+ are respectively the variation of volume and the number of moles of dissolved calcite. m is mass of a mole of limestone and M is mass of one litre of limestone.

Using the equations ( 12) and ( 13), we obtain:

2 Ca dC dn m m R dt M dt M θ θ + = = (14)
The mechanical degradation in our model has two different origins. The first one is the increase in degree of saturation. The second one is the porous matrix dissolution and porosity increase. As a matter of fact, in laboratory experiences if the water injected in a rock sample is not chemically saturated with the rock minerals, then it can also dissolve these minerals. In this case the two effects mentioned above contribute simultaneously to mechanical degradation of the rock. To separate the parts of strength reduc-tion due to these mechanisms, specific testes have to be realised with chemical control of the water injected to the sample or passing through and leaching it. In the data given in figures ( 2) and (3) the effects of the two mechanisms of mechanical degradation are superposed. Since we have not other data allowing us to separate the part of the two mechanisms, we admitted as a work assumption, that the reduction in the mechanical parameters can be attributed to the first effect (degree of saturation increase) for 80%, and to the other one (dissolution) for 20%. Supposing the partial pressure of carbon dioxide, P CO2 , equal to 0,003 atm [START_REF] Palmer | Origin and morphology of limestone caves[END_REF]) and using equation ( 9) we can deduce from the data of the figure (2) the reduction of the compressive strength versus the dissolved mass of limestone (Figure 5). We calculated from the data of the figure (2) for n=0.37, the quantity of dissolved calcite corresponding to the volume of water that enters the rockmass and increases the degree of saturation. Considering an exponential function for the strength reduction and using the assumption described above for attributing 20% of the strength reduction to dissolution phenomena, we can calculate the rate of variation of compressive strength as a function of the mass of dissolved calcite: 0.2 10000 6.9 2000

C C C C dR R e d R -× = ⇒ =- C (15)
According to the data of figure (3), the tensile strength is about 12% of the compressive strength; therefore we can also use the equation [START_REF] Plummer | The kinetics of calcite dissolution in CO 2 -Water system at 5° to 60°C and 0.0 to 1.0 atm CO 2[END_REF] to evaluate the tensile strength reduction.

APPLICATION TO THE CAVITY FAILURE

To study the effect of weathering on the gradual failure of the underground limestone mines we consider an underground cavity at 20m depth. The geometry of the model is presented in figure [START_REF] Hajna | Chemical Weathering of Limestones and Dolomites in A Cave Environnement, Speleogenesis and evolution of karst aquifers[END_REF]. For the numerical modelling we use the part delimited by the dashed lines (square of 9m).

Initial conditions

To define the initial conditions of the molarity of Ca 2+ in the limestone mass before digging of the cavity, we consider the dissolution in a closed system with the initial partial pressure of CO 2 equal to 0.003 atm [START_REF] Palmer | Origin and morphology of limestone caves[END_REF]. The initial value of molarity of Ca 2+ is calculated equal to 1.607×10 -4 mole/litre. The initial value of the molarity of CO 2 in the mass of limestone is null, but to avoid the numerical difficulties, we take a very low value. The initial porosity of limestone in the solid mass is constant. To define the hydraulic initial conditions, we calculate the values of the volumic water content such that the water transfer velocity in the equation ( 4) is null, in other words, there is no flow of water before the digging of the cavity.

Boundary conditions

As it was mentioned above, the in-situ measurements of [START_REF] Morat | Heat and water transport by oscillatory convection in an underground cavity[END_REF] show that water evaporates on the floor and condenses on the roof of the cavity. Consequently, water is always present in the roof surface. This corresponds to a boundary condition of degree of saturation equal to 1 on this surface, or the volumic water content is equal to the porosity.

On the roof of the cavity, carbon dioxide dissolves in pure water. The dissolved quantity of CO 2 per unit volume of water depends on its partial pressure in the atmosphere and on the temperature of the cavity. Thus we calculate the molarity of CO 2 in the boundary of the model by using equation [START_REF] Massieu | Essais d'identification et essais mécaniques sur le calcaire grossière de la carrière de Villiers-Adam[END_REF]. Considering the partial pressure of CO 2 equal to 0.003 atm and the temperature equal to 10°C, we obtain the CO 2 molarity equal to 1.611×10 -4 mole/litre. Since the Ca 2+ ions cannot leave the roof, we impose the condition of null flow in the vertical direction on the lower limit of the model and thus the term F given by the equation ( 6) is equal to zero and we obtain:

2 2 2 r r z Ca Ca Ca Ca r z C C VC n VC n n r z D θ + + + 2 z n + ∂ ∂ + + = ∂ ∂ (16)
We impose the conditions of null gradient on the higher boundary and the boundaries on the left and on the right of the model, therefore the gradients of water content and Ca 2+ and CO 2 molarities are null also on these boundaries.

Final equations

For the numerical modelling, it is initially necessary to finalize the equations of the problem. Four phenomena were considered for the process of weathering of limestone: water transfer, diffusion-advection, dissolution and degradation. For the phenomenon of water transfer we use the equations of Darcy and the conservation of mass. Using the equations ( 1) to ( 5), we can calculate the flow velocity and the water content in all of the nodes. We assumed that the Ca 2+ and HCO 3 ions are always in equilibrium. That means, according to [START_REF] Kaufmann | Karst landscape evolution[END_REF], the molarity of HCO 3 ions is always twice the molarity of Ca 2+ ions. With this assumption, we write the equations of diffusionadvection only for the ions Ca 2+ and the molecules of CO 2 . Using the equations ( 5), ( 6), ( 7), ( 11) and ( 12), and considering the variables C 1 and C 2 respectively for the molarities of CO 2 and Ca 2+ , we obtain:

( ) ( )

1 1 1 1 1 0 C VC D C C V R t θ θ ∂ + ∇ ⋅ -∇ ⋅ ∇ -∇ ⋅ + = ∂ (17) ( ) ( ) 2 2 2 2 2 0 C VC D C C V R t θ θ ∂ + ∇ ⋅ -∇ ⋅ ∇ -∇ ⋅ -= ∂ (18)
These equations and initial and boundary conditions constitute a complete set of equations and conditions allowing us to do a numerical modelling.

Model parameters

The values of the model parameters are presented in Table [START_REF] Fayad | Etude de stabilité de fontis au toit des carrières souterraines et traitements apportés aux conséquences induites en surface[END_REF]. We have not found sufficient data in the literature to determine the value of the coefficient k cal in equation [START_REF] Palmer | Origin and morphology of limestone caves[END_REF] for the dissolution process in the pore structure of limestone. The results are very sensible to this coefficient. For high values of this coefficient the dissolution rate is very high and the chemical weathering is concentrated in very vicinity of the roof surface. The value given here for this coefficient was chosen as to find a zone of active weathering process of about a few tens of centimeters depth from the roof surface.

Modelling results

A 2D program is developed using MATLAB software and the Finite Differences method. A grid containing 2025 nodes is created. The smallest distance between the nodes in the horizontal and vertical directions is respectively 10 and 5 cm near the roof of the cavity. By using the model equations we can calculate the rate of strength reduction with the time.

To have a simple idea of the failure zone, a linear elastic mechanical modelling is done using CESAR-LCPC software and the values of tensile vertical stress are calculated in the nodes corresponding to the nodes of current model. By calculating the ratio of tensile stress to tensile strength in each node in the model we can have an idea of the failure zone. We ignore the stress field changes due to cracking and also the reduction of the elastic parameters with dissolution process. The first step of the modelling was stopped when the first clear zone of failure is appeared in the 80 th year and the results are as follows: Figure [START_REF] Jouniaux | Etude des proprétés physiques du calcaire de la carrière de Mériel[END_REF] shows the volumic water content. It shows an increase in the first few meters of depth of the rock, where the water condenses on the roof. Figures [START_REF] Kaufmann | Karst landscape evolution[END_REF] and ( 9) show respectively the CO 2 and Ca 2+ molarities. The transport of the dissolved CO 2 molecules and produced Ca 2+ molecules in the rock with the advection and diffusion mechanisms is clearly shown. Figure [START_REF] Massieu | Essais d'identification et essais mécaniques sur le calcaire grossière de la carrière de Villiers-Adam[END_REF] shows the porosity increase in the model. As it can be seen, the maximum increase is in the order of 10 -4 that is a very low value. The more important phenomenon that we can see in this figure is the higher porosity increase in the corner of the roof, where we have the instantaneous changes in the boundary conditions. This is a numerical problem that we could not completely resolve and causes the appearance of the first failure zones in this zone, which is not compatible with the observations in underground mines. Figure [START_REF] Morat | Heat and water transport by oscillatory convection in an underground cavity[END_REF] shows the values of tensile strength. We can see the significant decrease in the tensile strength in the first meter of the roof depth, where it can be a possible zone for the failures. Figure [START_REF] Palmer | Origin and morphology of limestone caves[END_REF] shows the ratio of tensile vertical stress to tensile strength. The maximum value of the ratio is considered as 1 and we assume the failure is happen in these nodes. After 80 years, we could see the failure zone showed in the figure [START_REF] Palmer | Origin and morphology of limestone caves[END_REF] with the maximum depth of about 20cm and we assume it as the first step of the gradual failure of cavity.

For the modelling of the second failure step, another mechanical modelling is done with the new geometry of the model and the vertical tensile stresses were calculated. By transferring the boundary conditions to the new boundaries the modelling was continued until the appearance of another failure zone. That happens in the 125 th year with the failure zone showed in figure [START_REF] Palmer | Pattern of dissolution porosity in carbonate rocks. Speleogenesis and evolution of karst aquifers[END_REF]. These results show that by using the presented equations it is possible to model limestone weathering and gradual failure of the underground mines with the progress rate and failure shape more or less compatible with the observations of the different cases of failure in these mines.

Obviously with a completely coupled finite element model, a more refined mesh and specially with the more precise parameters we can have a better model to predict the progressive failure of the underground cavities. The progressive failure in underground mines is partly due to weathering processes that take place under natural conditions within and in the vicinity of the rockmass. The weathering process consists of chemo-physical mechanisms that have a mechanical degradation effect for the rock. We showed that by taking into account the kinetics of the weathering process, the transport of chemical constituents in the rockmass and the model of mechanical degradation caused by chemo-physical interactions, it is possible to give a model of progressive failure for underground structures. The work presented in this paper is based obviously on very sharp simplification assumptions. The aim of this work was only a first approach of this problem and to show the feasibility of such a modelling. This work must be continued by using more adequate data and models and also compared to in situ observations. The very interesting result of such an approach would be a kinetic model of the progressive failure and so an access to the temporal aspect of the failure in risk assessment of underground mines.

Figure 1 .

 1 Figure 1. Mechanism of gradual failure of underground limestone mines.

Figure 2 .Figure 3 .Figure 4 .

 234 Figure2. Reduction of compressive strength by degree of saturation[START_REF] Thierry | Problèmes posés par un calcaire hétérogène dans l'étude de la stabilité d'une carrière souterraine[END_REF]

) 4 . 4 . 2

 442 Dissolution kineticsThe rate of dissolution of calcite depends mainly on chemical saturation of water and can be shown by this empirical equation[START_REF] Kaufmann | Modelling unsaturated flow in an evolving karst aquifer[END_REF], Palmer 1991& 2003[START_REF] Dreybrodt | The kinetics of the reaction CO 2 +H 2 O → H + +HCO 3 as one of the rate limiting steps for the dissolution of calcite in the system H 2 O-CO 2 -CaCO 3[END_REF]):

Figure 5 .

 5 Figure 5. Variation of the compressive strength due to dissolution of calcite deduced from the data of figure (2)

Figure 6 .

 6 Figure 6. Geometry of model

Figure 7 .Figure 8 .Figure 9 .Figure 10 .Figure 11 .Figure 12 .Figure 13 .

 78910111213 Figure 7. Volumic water content -80 th year

Table 1 .

 1 Model parameters

	Parameter	Value	Unit	Parameter	Value	Unit
	n ini	0.4	-	C 2ini	1.607×10 -4	mole/litre
	C 1ini	3.982×10 -7 mole/litre	K	0.005	m/year
	C 1lim	1.611×10 -4 mole/litre	D 1	0.3	m 2 /year
	dt	0.002	Year	k cal	1.0×10 -3 mole/litre-year
	D 2	0.3	m 2 /year	α	0.03	m -1
	n cal	4	-	m	100	g
	M	2500	g	K 1	10 -6.46	mole/litre
	K C	10 -8.41	mole 2 /litre 2	K 2	10 -10.49	mole/litre