Anatoly Pashkevich

Damien Chablat
email: damien.chablat@irccyn.ec-nantes.fr

Philippe Wenger
email: philippe.wenger@irccyn.ec-nantes.fr

Kinematic calibration of orthoglide-type mechanisms

Keywords: parallel robots, kinematic calibration, model identification, joint offsets, and error compensation

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INTRODUCTION

Parallel kinematic machines (PKM) are commonly claimed to offer several advantages over serial manipulators, like high structural rigidity, better payload-to-weight ratio, high dynamic capacities and high accuracy [START_REF] Merlet | Parallel Robots[END_REF][START_REF] Tlusty | Fundamental Comparison of the Use of Serial and Parallel Kinematics for Machine Tools[END_REF]. As stated by a number of authors [START_REF] Tsai | Robot Analysis: the Mechanics of Serial and Parallel Manipulators[END_REF][START_REF] Wenger | A comparative study of serial and parallel mechanism topologies for machine tools[END_REF], the conventional serial kinematic structures have already achieved their performance limits, which are bounded by high component stiffness required to support sequential joints, links and actuators. Thus, the PKM are prudently considered as promising alternatives to their serial counterparts that offer faster, more flexible, more accurate and less costly solutions. However, while the PKM usually demonstrate a much better repeatability compared to serial mechanisms, they may not necessarily posses a better accuracy, which is limited by manufacturing/assembling errors in numerous links and passive joints [START_REF] Wang | On the accuracy of a Stewart platform -Part I: The effect of manufacturing tolerances[END_REF]. Besides, for the non-Cartesian parallel architectures, some kinematic parameters (such as the encoder offsets) cannot be determined by direct measurement. These motivate intensive research on PKM calibration.

Similar to the serial manipulators, the PKM calibration procedures are based on the minimisation of a parameter-dependent error function, which incorporates residuals of the kinematic equations. For the parallel manipulators, the inverse kinematic equations are considered computationally more efficient, since the most of the PKM admits a closed-form solution of the inverse kinematics [START_REF] Innocenti | Algorithms for kinematic calibration of fully-parallel manipulators[END_REF][START_REF] Iurascu | Geometric algorithm for kinematic calibration of robots containing closed loops[END_REF][START_REF] Huang | A general and novel approach for parameter identification of 6-dof parallel kinematic machines[END_REF]. But the main difficulty with the inverse-kinematics-based calibration is the full-pose measurement requirement, which is very hard to implement accurately [START_REF] Jeong | Kinematic calibration of redundantly actuated parallel mechanisms[END_REF] Recently, several hybrid calibration methods were proposed that utilize intrinsic properties of a particular parallel machine allowing to extract the full set of the model parameters (or the most essential of them) from a minimum set of the pose error measurements. An innovative approach for the I4 parallel mechanism has been developed by [START_REF] Renaud | Combining end-effector and legs observation for kinematic calibration of parallel mechanisms[END_REF][START_REF] Renaud | Kinematic calibration of parallel mechanisms: a novel approach using legs observation[END_REF] who applied vision to perform measurement on manipulator legs, which connect the end-effector to the base-mounted linear actuators. This paper focuses on the Orthoglide-type mechanisms, which kinematic architecture admits parallel leg motions (along the longitudinal axis). From observations the leg parallelism at prespecified postures, the proposed calibration algorithm estimates the joint offsets that are treated as the most essential parameters to be adjusted. The main advantage of this approach is the simplicity and low cost of the measuring system composed of standard comparator indicators attached to the universal magnetic stands. They are sequentially used for measuring the deviation of the relevant leg location while the manipulator moves the TCP along the Cartesian axes.

The remainder of the paper is organised as follows. Section 2 describes the manipulator geometry, its inverse and direct kinematics, and also contains the sensitivity analysis of the leg parallelism at the examined postures with respect to the joint offsets. Section 3 focuses on the measurements and parameter identification. Section 4 contains experimental results and their discussion, while Section 5 summarises the main results.

Manipulator geometry

The Orthoglide is a three degree-of-freedom mechanism actuated by three linear drives with mutually orthogonal axes [START_REF] Chablat | Architecture Optimization of a 3-DOF Parallel Mechanism for Machining Applications, the Orthoglide[END_REF]. Its kinematic architecture is presented in Fig. 1 and includes three identical parallel chains that are formally described as PRPaR, where P, R and Pa denote the prismatic, revolute, and parallelogram joints.

The mechanism input is made up by three actuated orthogonal prismatic joints. The output machinery (with a tool mounting flange) is connected to the prismatic joints through a set of three kinematic chains that further referred as manipulator "legs". The legs are made with articulated parallelograms that are oriented in a manner that the output body is restricted to the translational movements only.

A specific feature of the Orthoglide mechanism, which will be further used for the calibration, is displayed during the end-effector motions along the Cartesian axes. For example, for the x-axis motion in the Cartesian space, the sides of the x-leg parallelogram must also retain strictly parallel to the x-axis. Hence, the observed deviation of the mentioned parallelism may be used as the data source for calibration algorithms.

Modelling assumptions

Following previous studies on the parallel mechanism accuracy (Wang, andMassory, 1995, Renaud, et al., 2004;[START_REF] Caro | Sensitivity Analysis of the Orthoglide, a 3-DOF Translational Parallel Kinematic Machine[END_REF], the influence of the joint/link defects is assumed negligible compared to the joint positioning errors mainly caused by the encoder offsets. This validates the following modelling assumptions:

(i) the manipulator parts are supposed to be rigidbodies connected by perfect joints; (ii) the articulated parallelograms are assumed to be identical and perfect, which insure that their sides stay parallel in pares for any motions; (iii) the manipulator legs (composed of one prismatic joint, one parallelogram, and three revolute joints) are identical and generate a five-DOF motion each; (iv) the linear actuator axes are considered mutually orthogonal and intersected in a single point to insure a translational three-DOF movement of the end-effector.

The actuator encoders are assumed to be perfect but their location (zero position) is defined with some errors that are treated as the offsets to be estimated. Using these assumptions, there will be developed a convenient calibration methodology based on the observation of the parallel gel motions.

Basic equations

Under the above assumptions, the Orthoglide can be presented by a simplified model, which consists of three rod-links connected by spherical joints to the tool centre point (TCP) at one side and to the corresponding prismatic joints at another side. Thus, if the origin of a reference frame is located at the intersection of the prismatic joint axes and the x, y, z-axes are directed along them, the manipulator geometry may be described by the equations

[ ] 2 2 2 2 ) ( L p p p z y x x = + + ρ Δ + ρ - [ ] 2 2 2 2 ) ( L p p p z y y y x = + ρ Δ + ρ - + (1) [ ] 2 2 2 2 ) ( L p p p z z z y x = ρ Δ + ρ - + +
where p = (p x , p y , p z ) is the output position vector, ρ = (ρ x , ρ y , ρ z ) is the input vector of the joints variables, Δρ = (Δρ x , Δρ y , Δρ z ) is the encoder offset vec-tor, and L is the length of the parallelogram principal links. Hence, within the adopted model, four parameters (Δρ x , Δρ y , Δρ z , L) define the manipulator geometry, but because of the rather tough manufacturing tolerances, the leg length is assumed to be known and only the joint offsets (Δρ x , Δρ y , Δρ z ) are in the focus of the proposed calibration technique.

It should be noted that for the nominal ''mechanicalzero'' posture corresponding to the joints variables ρ 0 = (L, L, L), the x-, y-and z-legs are oriented strictly parallel to the relevant Cartesian axes. But the joint offsets cause the deviation of the "zero" location and corresponding deviation of the leg parallelism with respect to the manipulator base surface. The latter may be (i) measured and (ii) computed applying the direct kinematic algorithm for the joint variables ρ = (L+Δρ x , L+Δρ y , L+Δρ z ). However, the capability of this simple technique is limited by evaluating the offset of the z-axis encoder only, since the Orthoglide mechanical design does not allow making similar measurements for the remaining pairs of the legs, with respect to the xz-and yz-planes.

Inverse and direct kinematics

To derive calibration equations, first let us expand some previous results on the Orthoglide kinematics [START_REF] Pashkevich | Design Strategies for the Geometric Synthesis of Orthoglide-type Mechanisms[END_REF] taking into account the encoder offsets. The inverse kinematic relations are derived from the equations (1) in a straightforward way and only slightly differ from the "nominal" case

x z y x x x p p L s p ρ Δ - - - + = ρ 2 2 2 y z x y y y p p L s p ρ Δ - - - + = ρ 2 2 2 (2) z y x z z z p p L s p ρ Δ - - - + = ρ 2 2 2
where s x , s y , s z ∈ { ±1} are the configuration indices defined for the "nominal" manipulator as signs of (ρ x -p x ) , (ρ y -p y ), and (ρ z -p z ) respectively. It is obvious that expressions (2) define eight different solutions to the inverse kinematics, however the Orthoglide assembling and joint limits reduce this set for a single case corresponding to the s x = s y = s z = 1.

For the direct kinematics, the equations (1) can be subtracted pair-to-pair that gives the following expression for the unknowns p

x , p y , p z } , , { ; 2 z y x i t p i i i i i ∈ ρ Δ + ρ + ρ Δ + ρ = ( 3
)
where t is an auxiliary scalar variable. This reduces the direct kinematics to the solution of a quadratic equation

0 2 = + + c bt at with the coefficients ; ) )( ( ∏ ≠ ρ Δ + ρ ρ Δ + ρ = j i j j i i a ∏ ρ Δ + ρ = i i i b 2 ) ( ; ; 4 ) ( 2 2 ∑ - ρ Δ + ρ = i i i L c } , , { , z y x j i ∈
From two possible solutions that give the quadratic formula, the Orthoglide prototype (see Fig. 1) admits only one expressing as 2 ( 4 ) / 2 t b m b ac a = -+ -.

Sensitivity analysis

To evaluate the encoder offset influence on the legs parallelism with respect to the Cartesian planes XY, YZ, and YZ, let us derive the differential relations for the TCP deviation for three types of the Orthoglide postures: mechanical zero" posture and "maximum/minimum displacement" postures for the directions x, y, z. These postures are of particular interest for the calibration since in the "nominal" case (no encoder offsets) at least one leg is parallel to the corresponding pair of the Cartesian planes.

The desired differential relations may be derived from the Orthoglide Jacobian, the inverse of which is obtained from (1) as

1 1 ( , ) 1
1 y z x x x x x z y y y y y x z z z z p p p p p p p p p p p p ρ ρ ρ ρ ρ ρ - ⎡ ⎤ ⎢ ⎥ - - ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ = - - ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ - - ⎣ ⎦ J p ρ (4)
For the "mechanical zero" posture, the differential relations are derived in the neighbourhood of the point p 0 = (0, 0, 0) and ρ 0 = (L, L, L), which gives the identity Jacobian matrix 0 0 3 3 ( , ) × = J p ρ I . Hence, in this case the TCP displacement is related to the joint offsets by trivial equations ;

i i p ρ Δ = Δ } , , { z y x i ∈
. Taking into the account the Orthoglide geometry, this deviation may be estimated by evaluating parallelism of the legs with respect to the Cartesian planes. However, as mentioned in sub-section 2.3, this technique is feasible for z-direction only and may produce estimation of Δρ z merely.

For the "maximum displacement" posture in the xdirection, the differential relations are derived in the neighbourhood of the point ) 0 , 0 , sin ( α = L p and ) , , (

α α α + = LC LC LS L ρ
, where, α is the angle between the y-, z-legs and corresponding Cartesian axes, and

α = α = α α cos ; sin C S
. This gives the inverse Jacobian as a lower triangle matrix

1 0 0 ( , ) 1 0 0 1 T T α α ⎡ ⎤ ⎢ ⎥ = ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ J p ρ (5) where α = α tan T
. Hence, the differential equations for the TCP displacement may be written as ;

; ; z x z y x y x x T p T p p ρ Δ + ρ Δ = Δ ρ Δ + ρ Δ = Δ ρ Δ = Δ α α (6)
and the joint offset influences on the TCP deviation is estimated by factors 1.0 and T α . It is also worth mentioning that measurement of the x-leg parallelism with respect to the XY-plane gives an equation for estimating the offset Δρ x (provided that the offset Δρ z has been obtained from the "mechanical zero").

Similar results are valid for the "maximum displacement" postures in the y-and z-directions (differing by the indices only), and also for the "minimum displacement" postures. In the latter case, the angle α should be computed as

) ( asin min L ρ = α
. Hence, the leg parallelism is rather sensitive to the joint offsets and relevant deviations Δp x , Δp y , Δp z , and may be used for the offset identification.

CALIBRATION METHODOLOGY

Measurement technique

To identify the model parameters, we propose a single-sensor measurement techniques for the leg/surface parallelism (Fig. 2). It is based on fixed location of the measuring device for two distinct leg postures corresponding to the minimum/maximum of the joint coordinates. Relevant calibration experiment consists of the following steps:

Step 1. Move the manipulator to the "mechanical zero"; locate two gauges in the middle of the X-leg (parallel to the axes Y and Z); get their readings.

Step 2. Move the manipulator to the "X-maximum" and "X-minimum" postures, get the gauge readings, and compute differences

- - + + Δ Δ Δ Δ x x x x z y z y , , , .
Step 3+. Repeat steps 1, 2 for the Y-and Z-legs and compute corresponding differences.

Calibration equations

The calibration equations can be derived using expressions from sub-section 2.5. First, it is required to define the gauge initial location. For the X-leg, it is the midpoint of the line segment bounded by the TCP (Δρ x , Δρ y , Δρ z ) and the centre of the X-joint (L+Δρ x , 0, 0). This yields the following expression for the X-leg midpoint: ( 2 ; 2; 2)

x y z L ρ ρ ρ + Δ Δ Δ .
For the "X-maximum" posture, the X-leg location is also defined by same two points, but their coordinates are ) ; ; (

z x y x x T T LS ρ Δ + ρ Δ ρ Δ + ρ Δ ρ Δ + α α α and ) 0 ; 0 ; ( x LS L ρ Δ + + α
respectively. Then, it may be written the equations of a straight-line passing along the X-leg and computed the point corresponding to

x L x ρ Δ + = 2
. Hence, finally, the deviations of the X-leg measurements may be expressed as

z x x y x x S T S z S T S y ρ Δ + ρ Δ + = Δ ρ Δ + ρ Δ + = Δ α α α + α α α + ) 5 . 0 ( ; ) 5 . 0 ( (7)
Similar approach may be applied to the "Xminimum" posture, as well as to the equivalent postures for the Y-and Z-legs. This gives the system of twelve linear equations in three unknowns and the angles α 1 , α 2 are computed for the maximum and minimum postures respectively. The reduced version of this system may be obtained if to assess the leg/plane parallelism by the difference between the "maximum" and "minimum" postures. The latter leads to the system of six linear equations in three unknowns.

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ Δ Δ Δ Δ Δ Δ Δ Δ Δ Δ Δ Δ = ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ ρ Δ ρ Δ ρ Δ ⋅ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢

EXPERIMENTAL RESULTS

Experimental setup

The measuring system is composed of standard comparator indicators attached to the universal magnetic stands allowing fixing them on the manipulator bases. The indicators have resolution of 10 μm and are sequentially used for measuring the X-, Y-, and Z-leg parallelism while the manipulator moves between the Max, Min and Zero postures. For each measurement, the indicators are located on the mechanism base in such manner that a corresponding leg is admissible for the gauge contact for all intermediate posters (Fig. 3).

For each leg, the measurements were repeated three times for the following sequence of motions: Zero → Max → Min → Zero→ …. Then, the results were averaged and used for the parameter identification. It should be noted that measurements demonstrated very high repeatability (about 0.02 mm).

Calibration results

The first calibration experiment produced rather high parallelism deviation, up to 2.37 mm, which in the frames of the adopted kinematic model was expected to be reduced down to 1.07 mm only (Table 1). The corresponding residual norm was also rather high and yielded unrealistic estimate of the measurement noise parameter σ ≈ 1.0 mm. It impels to conclude that the mechanism mechanics requires more careful tuning. Consequently, the location of the joint axes was adjusted mechanically to ensure the leg parallelism for the Zero posture.

Fig. 3. Experimental Setup

The second calibration experiment (after mechanical tuning) yielded lower parallelism deviations, less than 0.70 mm. For these data, the developed calibration algorithm yielded the joint offsets that are expected to reduce the deviation down to 0.28 mm. Besides, the estimated value of σ ≈ 0.28 mm is rather realistic taking into account both the measurement accuracy and the manufacturing/assembling tolerances. Accordingly, the identified vales of the joint offsets were input into the control software.

The third experiment (validation of the identified parameters) demonstrated good agreement with the expected results. In particular, the maximum parallelism deviation reduced down to 0.34 mm (expected 0.28 mm), while its root-mean-square decreased down to 0.21 mm (expected 0.20 mm). On the other hand, further adjusting of the kinematic model to the new experimental data gives both negligible improvement of the r.m.s. and rather small alteration of the model parameters. It is evident that further reduction of the parallelism deviation is bounded by the manufacturing and assembling errors.

Hence, the calibration results confirm validity of the proposed identification technique and its ability to tune the joint offsets from observations of the leg parallelism.

CONCLUSIONS

This paper proposes a novel calibration approach for the parallel manipulators, which is based on observations of manipulator leg parallelism with respect to some predefined planes. Presented for the Orthoglide-type mechanisms, this approach may be also applied to other manipulator architectures that admit parallel leg motions along the longitudinal axis.

The proposed calibration technique employs a simple and low-cost measuring system composed of standard comparator indicators attached to the universal magnetic stands. They are sequentially used for measuring the deviation of the relevant leg location while the manipulator moves the TCP along the Cartesian axes. From the measured differences, the calibration algorithm estimates the joint offsets that are treated as the most essential parameters to be tuned. The validity of the proposed approach and efficiency of the developed numerical algorithm were confirmed by the calibration experiments with the Orthoglide prototype, which allowed reducing the residual r.m.s. by three times.

To increase the calibration precision, future work will focus on the development of the specific assembling fixture ensuring proper location of the linear actuators and also on the essential theoretical issues such as calibration experiment planning, expanding the set of the identified model parameters and their identifiably analysis, and compensation of the nongeometric errors non detected within the frames of the adopted kinematic model.

Fig

  Fig. 1. The Orthoglide mechanism 2. KINEMATIC MODEL

Fig. 2 .

 2 Fig. 2. Measuring the leg/surface parallelism

Table 1 .

 1 Experimental data and expected improvements of accuracy(model-based)

	Data Source	Δx y mm	Δx z mm	Δy x mm	Δy z mm	Δz x mm	Δz y mm	r.m.s. mm
			Initial settings (before mechanical tuning and calibration)
	Experiment #1	+0.52	+1.58	+2.37	-0.25	-0.57	-0.04	1.19
	Expected improvement	-0.94	+0.63	+1.07	-0.84	-0.27	+0.35	0.74
				After mechanical tuning (before calibration)
	Experiment #2	-0.43	-0.37	+0.42	-0.18	-1.14	-0.70	0.62
	Expected improvement	-0.28	+0.25	+0.21	-0.14	-0.13	+0.09	0.20
					After calibration
	Experiment #3	-0.23	+0.27	+0.34	-0.10	-0.09	+0.11	0.21
	Expected improvement	-0.29	+0.23	+0.25	-0.17	-0.10	+0.08	0.20


M Souley

N Lafrance

C Auvray

V Labiouse

T Belem

An elastoplastic and viscoplastic model for porous geomaterials

Chalk rocks have been used for centuries as strong building materials and for the production of lime. The shallow underground room-and-pillar quarries where chalk rocks were extracted are now abandoned and are enduring the effects of time and weathering which increases their risk of collapse. A large campaign of laboratory tests under different Relative Humidity values (from 90 to 98%) was undertaken to describe the short and long term behavior of two chalks. These laboratory tests clearly show an elastoplastic behavior with strain-hardening/softening, a relative humidity-dependent strength, a pore collapse and matrix densification even if the stress level is much higher than that prevailing in situ as well as a time-dependent shear and volumetric strains. A multi-mechanism (tensile, shear, pore collapse/compaction, creep) model of the mechanical behavior for highly porous geomaterials is proposed based on the laboratory characterization and constitutive models published in the literature. The proposed model was implemented in FLAC 3D and used to simulate laboratory tests in order to verify and validate the implementation.

INTRODUCTION

Characterization and modeling of the mechanical behavior of compressible materials are well-known issues, and have been studied for a long time: (a) in soil mechanics, including the foundations on soft ground or stability of embankments, structures on compressible soils etc.; (b) in the field of metallurgy for the behavior of metal foam containing voids (one of the first ductile damage model has been proposed by [START_REF] Gurson | Continuum theory of ductile rupture by void nucleation and growth: Part I -Yield criteria and flow rules for porous ductile media[END_REF], and widely used in the literature for geomaterials) or for compaction of metal and pharmaceutical powders, snow or crushed salt compaction (for instance) or in the ceramic industry. The common thread for these materials is that when they are subjected to loadings, significant modifications of their microstructure (compaction, stiffening, increase of their cohesion, etc.) are experimentally observed.

Chalk rocks have been used for centuries as strong building materials and for the production of lime. The shallow underground room-and-pillar quarries where chalk rocks were extracted are now abandoned and are enduring the effects of time and weathering, which in turn increases their risk of collapse. The risk of instability, involving natural or anthropogenic underground structures, is an issue for more than 10 000 towns in France. In order to investigate the causes of underground quarry collapse and to aid preventive strategies, we carried out a study of the physico-mechanical behaviour as well as the water-rock interaction of chalks sampled from abandoned pillars in two underground quarries, in situ-instrumented and monitored by INERIS (Institut National de l'Environnement Industriel et des Risques), in the Parisian Basin, France. There are two partially flooded underground chalk quarries, subjected to water table fluctuations, at Estreux (EX) and St-Martin-Le-Noeud (SM) in northern France [START_REF] Gombert | In-situ and laboratory tests to evaluate the impact of water table fluctuations on stability of underground chalk mines[END_REF]), located at an average depth of 20 to 25 m.

A large laboratory characterization was conducted to understand the short and long terms behavior of these two chalks depending on the relative humidity, although in the context of this paper we will limit ourselves to the relative humidity levels ranging between 90% and 100% which are the most representative of the in situ conditions of abandoned quarries. The ageing aspects of these chalks as well as the impact of weathering on the macro-mechanical properties are described in [START_REF] Lafrance | Impact of weathering on the macro-mechanical properties of chalks[END_REF].

Indeed, several models of mechanical behavior for highly porous materials (chalk for the oil industry, metal powders or PVC in metallurgy or pharmaceutical industry, ceramics, etc.) can be found in the literature [START_REF] Homand | Mechanical Behaviour of a Porous Chalk and Water/Chalk Interaction, Part II: Numerical modelling[END_REF][START_REF] Azami | 3D computational modeling of powder compaction processes using a threeinvariant hardening cap plasticity model[END_REF][START_REF] Wu | Modelling the mechanical behaviour of pharmaceutical powders during compaction[END_REF]. However, many of these models are limited to the short-term response. Based on our previous laboratory tests, a new constitutive model (extending previous works) is proposed to account for deformations and the role of relative humidity in the short-term and long-term behavior. Typically, the time-dependent behavior is governed by deformation processes corresponding to different physical mechanisms for changing material microstructure.

BRIEF REVIEW OF MODELS FOR HIGHLY POROUS GEOMATERIALS

The mechanical behavior of very porous rocks is widely varied and depends on the mean stress and the loading path. At low stress levels, porous rocks break by the development of one or more shear bands accompanied by a softening behavior: strains at the microscopic scale consist of microcracks dilatancy and grains rotation. Under high mean stresses, porous rocks undergo a hardening behavior which is microscopically associated with volumetric strains due to compaction. Inelastic compaction of the pore space occurs on the debris and/or sliding through the contacts between grains whereas brittle failure dominates the microscopic processes. As a function of loading path, the material may eventually break by localization of shear strain.

A pioneering extension of the theory of plasticity for metals to the plastic behavior of soils was conducted by Drucker and Prager (1952): they extended the failure criterion of von Mises to take into account the confining pressure in granular media. One of the first criterion including closed yield surfaces (commonly called "cap" surfaces) with their numerical implementation without smoothing the edges was presented by [START_REF] Dimaggio | Material model for granular soils[END_REF]. Even if such "cap" models were originally developed for sands, they were used successfully for other materials such as clay and concrete.

For the modeling of porous geomaterials, significant advances have been made in recent decades. Several plastic models have been proposed to account for the influences of porosity, expansion and collapse of pores [START_REF] Seifert | Plastic yielding in cyclically loaded porous materials[END_REF]. Other studies have investigated the propagation of microcracks, the strain localization and viscous deformation in porous materials [START_REF] Mroginski | A thermodynamical gradient theory for deformation and strain localization of porous media[END_REF]. Based on the working reference of [START_REF] Gurson | Continuum theory of ductile rupture by void nucleation and growth: Part I -Yield criteria and flow rules for porous ductile media[END_REF], a number of advances have been made on the micromechanic analysis for the purpose of determining the macroscopic plastic yield function and plastic potential through a homogenization scheme [START_REF] Magous | Micromechanical approach to the strength properties of frictional geomaterials[END_REF]. All these models have clearly shown that the macroscopic behavior of porous materials depends on the porosity and its evolution.

Most short term models of porous geomaterials behavior are based on a "cap" yield surface whose shape can be spherical, elliptical or composite and taken into account other mechanisms: shearing, pore collapse and/or tensile. In the case of chalks, one can refer to the constitutive and numerical developments achieved in the framework of the European program PASACHALK "Mechanical Behaviour of Partially Saturated chalks and Multiphase Fluid-skeleton" (CONTRACT NO: ENK6-2000 -00,089). However, these models do not generally take into account the effect of relative humidity and creep which are predominant phenomena regarding the long-term stability of abandoned quarries.

PHENOMENA CONSIDERED FOR THE TWO CHALKS

A large campaign of laboratory tests was undertaken to characterize the short and long terms behavior of these two chalks. It includes several uniaxial and triaxial compression tests performed on dry and saturated samples as well as for different relative humidity values ranging from 90 to 98% typically encountered in situ, hydrostatic tests aimed at characterizing the volumetric yield surface and uniaxial creep tests with numerous stress levels and relative humidity values.

Figure 1 illustrates the typical short and long terms behavior for a given relative humidity and given chalk, the impact of a specific relative humidity stage and of the sample flooding during the last stress and relative humidity levels. These laboratory tests clearly indicate an elastoplastic behavior with strain-hardening/softening, the relative humidity-dependent strength, the pore collapse of chalk even if the last stress level tested is much higher than that prevailing in situ, as well as time-dependent shear and volumetric strains. More precisely, for the short term behavior: (a) a linear behavior and an appearance of plastic strains at low deviatoric stress (the elastic limit is low); (b) under low confining pressures, the samples failure corresponds to the formation of a shear band inclined with respect to the sample axis. There is a dependence of the mechanical behavior on the relative humidity; (c) a post-peak behavior has been captured with the external displacement transducer. The hydrostatic compression carried out on the two chalks clearly indicates a similar behavior than other geomaterials (sandstone and limestone: Wong & Baud 2012) under hydrostatic compression path; that is to say, two to three distinct domains of behavior i.e. elasticity, pore collapse and densification.

For the long-term behavior (a) simultaneous measurements of axial and lateral strains during creep tests show that volumetric strains are not negligible according to the relative humidity and stress levels; (b) the creep strain rate increases in a non-linear manner with applied deviatoric stresses and relative humidity.

PROPOSED CONSTITUTIVE MODEL

Although laboratory tests have shown that the short and long terms behavior is strongly influenced by the relative humidity, in this paper we restricted the rheological development to isothermal conditions and a given fixed relative humidity. A more general model will be addressed in the near future.

The rheological model proposed is based on results obtained from different laboratory characterizations. We recall the main features of the short-term mechanical behavior observed on the chalk samples under triaxial tests. They can be summarized as follow: the behavior is linear under low deviatoric stresses; the loss of linearity of lateral and axial strain-stress curves begins at a very low value of deviatoric stress. Depending on the considered chalk, the volumetric behavior according to the triaxial loading path is contracting and/or dilatant.

Figure 2 summarizes the proposed modeling of each mechanism of the short-term response (ST) which is characterized by (a) an elastic yield, a peak strength and a residual strength based on the Drucker-Prager's criterion. The advantage of this criterion compared to Mohr-Coulomb or Hoek-Brown ones is the dependency on the intermediate principal stress σ 2 ; (b) a strain hardening based on the test data and strain softening which is assumed to be symmetrical and homothetic to the hardening due to the lack of experimental results. The strain hardening and softening are modelled by an exponential function with respect to the internal plastic variable (plastic distortion).

For the pore collapse mechanism (volumetric cap yield), the yield function is based on the Tvergaard (1990) criterion, that is to say, an incorporated elliptical yield surface for both pore collapse and compaction. The main advantage of this micromechanicsbased criterion is that the yield function depends explicitly on the porosity of the medium. The hardening due to pore collapse is governed by a local hardening variable which depends on the macroscopic plastic deformation with respect to the pore collapse mechanism (e.g. volumetric plastic strain due to the hydrostatic failure) and determined by the principle of equivalence plastic energy.

Finally, for the short-term response, failure is based on a combination of a cut-off in tensile, a yield function based on Drucker-Prager's criterion in shear (including hardening and softening) and an elliptical yield surface for both pore collapse and compaction.

The mathematical expressions of each of the mechanisms considered in the short term are summarized in Table 1. In this table, the subscript 0 expresses the mechanism initiation (shear or pore collapse), the subscript m is related to parameters at failure, q and p are respectively the deviatoric stress and the mean stress, φ corresponds the total porosity.

The time dependent behavior is governed by deformation processes corresponding to different physical mechanisms for change in material microstructure. Using the formalism proposed by [START_REF] Perzyna | Fundamental problems in viscoplasticity[END_REF], it is possible to express the total viscoplastic strain rate.

More precisely, based on the main observations from creep tests, a creep model is proposed by introducing a creep threshold function depending on the relative humidity in the well-known Lemaitre's law. By taking into account all the observed phenomena (detailed in the beginning of this section), the following expression was proposed for the viscoplastic strain rate tensor: whereA is the material viscosity depending on the relative humidity, σ 0 the reference stress, n a dimensionless exponent corresponding to the deviatoric stress power factor, m the exponent of hardening work, ε eq the viscoplastic strain (or viscoplastic distortion), q 0 the deviatoric stress threshold, q the current deviatoric stress, σ the stress tensor.

Although creep tests showed the occurrence of deferred volumetric strains, in this first approach they will be neglected. the model with the aim of its use to other deeped chalks (petroleum applications) or other porous and compressible materials.

NUMERICAL IMPLEMENTATION AND VERIFICATION

The proposed multi-mechanism model was implemented by coupling the viscoplastic and short-term response in the three-dimensional explicit finitedifference code, FLAC 3D . At each step, the computation of new stresses is achieved by testing firstly the current deviatoric stress with respect to the threshold and then calculating the appropriate viscoplastic strain increment. The strain increment for instantaneous response are computed, leading to correct the stress when failure in tensile or in shear and/or pore collapse is detected.

In order to verify the model (constitutive equations and numerical implementation), several triaxial compression tests with confining pressures of 0, 1, and 3 MPa are simulated. Figure 3a presents the deviatoric stress -axial, lateral and volumetric strain curves for different confining pressures.

The resulting curves display four regions (elastic, hardening in pre-peak, softening in post-peak and residual phases). These curves are qualitatively similar to the experimental ones (which were not reported herein). In addition, the elastic limit, the peak and residual strengths derived from these simulations are compared with the theoretical envelopes with a good agreement. Similarly, simulation of multistage (deviatoric stress and relative humidity) creep test, not shown here, indicates a good agreement with the experimental curves.

On the Figure 3b we present the evolution of hardening during a hydrostatic compression test carried out on one of the two chalks. For this test, the mean stress at the beginning of pore collapse is about 21 MPa. As a result, a linear fitting as a function of the volumetric plastic deformation can reproduce this hardening. Finally, it may be noted that even for a hydrostatic pressure of 60 MPa, there is no densification of the chalk (stiffening of modules).

The model is now applied to simulate some creep tests performed under uniaxial compression condition. The evolutions of axial stress with time for the four tests are given in Figure 1b. The numerical simulation carried out with the model is shown in Figure 4. Qualitatively and quantitatively, the model response is in good agreement with the experimental curves.

CONCLUSION

This paper presents the development of a multimechanism constitutive model for porous geomaterials, its numerical implementation in the threedimensional code FLAC 3D , as well as its verification. Firstly, a non-linear elastoplastic model based on the interpretation of laboratory test performed using a cutoff in tension, the Drucker-Prager's failure criterion for shear, an elliptical yield surface for both pore collapse and compaction based on [START_REF] Tvergaard | Material failure by void growth to coalescence[END_REF]'s criterion, is presented.

Simulation of triaxial compression tests at different levels of confining pressure provides a verification of the implemented model. The resulting curves display four regions (elastic, hardening in pre-peak, softening in post-peak and residual phase) in accordance with the laboratory tests.

The next step is to incorporate the changes in viscosity as a function of relative humidity and the deferred volumetric strains observed on samples in the proposed model and finally to perform some long-term stability applications at pillar and quarry scales.

Figure 1 .

 1 Figure 1. Typical behavior of the studied EX chalks: (a) triaxial compression test (deviatoric stress vs strains) at 98% of RH, (b) uniaxial creep test showing the effects of stress and hygrometry (90% and 98% RH) and flooding.

Figure 2 .

 2 Figure 2. Schematic representation of initial yield surfaces in tension, shear and pore collapse as well as the hardening and main stress pathway for conventional tests.

Table 1 .

 1 Figure 3. (a) Triaxial compression test simulations using the proposed model: deviatoric stress vs strains; (b) hydrostatic test and corresponding simulation showing the pore collapse hardening (pore collapse stress σ M0 = 21 MPa).

Figure 4 .

 4 Figure 4. Viscoplastic shear strain for a multi-stage uniaxial creep test and simulation (1st stage σ 1 = 0.5 MPa and RH = 90%; 2nd stage σ 1 = 1 MPa and RH = 90%; 3rd stage σ 1 = 2 MPa and RH = 90%; 4th σ 1 = 3, 5 MPa and RH = 90% & RH = 98%).


C Auvray

N Lafrance

M Souley

V Labiouse

Impact of weathering on macro-mechanical properties of chalks

Chalk has long been used as a strong building material. The shallow underground room-andpillar quarries where the chalk was extracted now lie abandoned and endure the effects of time and weathering, increasing their risk of collapse. The risk of instability is an issue for more than 10 000 towns in France. We have studied the physico-mechanical behaviour and ageing of two types of chalk in underground quarries in the Parisian Basin, France. The dolomitic chalk (Saint-Martin-Le-Noeud) exhibits variations in mechanical and physical properties with horizontal pillar depth, and both density and compressive strength appear to increase outwards from the core to the wall of the pillar. SEM analysis reveals progressive degradation and increasing homogeneity of the grains from the edge to the inside of the pillar. In contrast, in physico-mechanical tests and SEM analyses performed on the Estreux glauconitic chalk, no significant variation was observed between the pillar wall and pillar core.

INTRODUCTION

Alteration can be defined as a modification of the physico-chemical properties of minerals and therefore rocks, by atmospheric agents, groundwater or thermal waters [START_REF] Foucault | Dictionnaire de géologie[END_REF]. The process is dependent on climate, water and temperature, as well as on the nature and degree of fissuring of the rock. Alteration generally leads to less-coherent rocks, promoting their deterioration and eventual failure [START_REF] Gupta | Weathering effects on the strength and deformational behaviour of crystalline rocks under uniaxial compression state[END_REF], Massuda 2001). Macroscopic signs of alteration include the appearance of cracks, microfractures and macroporosity, as well as the presence of hard deposits that result from chemical transformations [START_REF] Oyama | Weathering rate of mudstone and tuff on old unlined tunnel walls[END_REF]. The alteration of rocks by dissolution (limestones, tuffs, sandstones, crystalline rocks), with or without mineral neoformation, has been studied by numerous authors [START_REF] Gupta | Weathering effects on the strength and deformational behaviour of crystalline rocks under uniaxial compression state[END_REF].

The term "ageing" refers to weathering of a rock within a medium influenced by human activity. In the context of underground structures, the term is used to describe the range of mineralogical and physical modifications of the rock over time. These modifications then lead to a deterioration of the hydraulic or mechanical properties.

The effect of time on the behaviour of an unlined underground structure encompasses various phenomena responsible for the variations in the material properties: (i) degradation of minerals in the rock as a result of physico-chemical action, leading to a reduction in the mechanical characteristics; (ii) differed deformation due to constant loading; (iii) variations in humidity (often linked to the airing conditions in the underground quarry) and thus in the saturation and suction conditions that affect the rock mass, over time. These cyclical variations in the hydric boundary conditions may generate damage due to hydro-mechanical coupling.

The influence of all of these phenomena must be taken into account when studying the temporal evolution of underground quarries, with the aim of being able to provide optimum remediation strategies for ensuring the long-term stability of the quarries with regards to public safety and the protection of built resources.

In the present study, we restrict our assessment of ageing to two types of chalk sampled in two different underground quarries. Following a short description of the quarries and sampling methodology employed, the scanning electron microscope (SEM) observations of the chalk samples are discussed in detail. The trends in a number of physico-mechanical parameters (porosity accessible to water, water content, density and uniaxial compressive strength) are then highlighted.

DESCRIPTION OF THE UNDERGROUND QUARRIES

Chalk is an easily-crafted material that has been used for centuries in building and construction. The shallow underground room-and-pillar quarries where the chalk rocks were extracted have since been abandoned and now endure the effects of time and weathering, potentially increasing their risk of collapse. In order to prevent this risk, the "Institut National de l'Environnement Industriel et des Risques" (INERIS) has been studying the behaviour of the rock in two partially flooded underground chalk quarries, subject to water table fluctuations, at Estreux (EX) and St-Martin-Le-Noeud (SM) in northern France [START_REF] Gombert | In-situ and laboratory tests to evaluate the impact of water table fluctuations on stability of underground chalk mines[END_REF]). These quarries, located at an average depth of 20 to 25 m, were exploited by the room and pillar method, with an extraction rate of 60-75%. The quarries were abandoned over a century ago. In situ instrumentation was implemented in these quarries in 2004 and 2009, with the aim of determining the impact of water-table fluctuations on their stability. These fluctuations lead to chemical interactions between the chalk and the water, thereby affecting the mechanical behaviour of the chalk through changes in the weathering conditions [START_REF] Talesnick | The elastic deformability and strength of a high porosity, anisotropic, chalk[END_REF][START_REF] Schroeder | Du coccolithe au réservoir pétrolier: approche phénoménologique du comportement mécanique de la craie en vue de sa modélisation à différentes échelles[END_REF][START_REF] Schroeder | Mechanical behaviour of partially and multiphase saturated chalks and fluid skeleton interaction: Main factor of chalk oil reservoirs compaction and related subsidence[END_REF][START_REF] Risnes | Chalk-fluid interactions with glycol and brines[END_REF], 2005[START_REF] Duperret | Effect of groundwater and sea weathering cycles on the strength of chalk rock from unstable coastal cliffs of NW France[END_REF][START_REF] Nguyen | Influence des interactions eau-roche sur le comportement à long terme de cavités souterraines dans la craie[END_REF][START_REF] Gombert | In-situ and laboratory tests to evaluate the impact of water table fluctuations on stability of underground chalk mines[END_REF][START_REF] Lafrance | Ageing of chalk rocks in an underground quarry[END_REF]. Preliminary laboratory characterization (e.g., mineralogical studies and uniaxial and triaxial compression tests) has also been undertaken [START_REF] Gombert | In-situ and laboratory tests to evaluate the impact of water table fluctuations on stability of underground chalk mines[END_REF][START_REF] Lafrance | Ageing of chalk rocks in an underground quarry[END_REF].

Nevertheless, the hydro-chemo-mechanical behaviour of the chalk remains poorly understood due to complex interlocking phenomena that act on the mechanical behaviour of chalk. It therefore remains necessary to address this issue in the framework of multi-scale (SEM, laboratory and in situ scales) and multi-physics (mechanic, hydraulic and chemical) approaches.

The Saint-Martin-le-Noeud underground quarry (SM) is located on a hillside to the south of Beauvais (Oise, Picardie, France). The chalk was quarried by the room-and-pillar method, leaving behind galleries of between 3 and 5 m width, originally 4 m in height, and irregular pillars of between 2 and 4 m width. The estimated rate of extraction is 50% to 67% [START_REF] Gombert | In-situ and laboratory tests to evaluate the impact of water table fluctuations on stability of underground chalk mines[END_REF]. Based on the overlying rock thickness (25 m), the average vertical stress in the pillars is about 1 to 1.5 MPa. The stratigraphy of SM has a near-tabular profile with a gentle dip (≈4-10 • ). Due to this shallow dip, the deepest parts of the underground quarry reach the shallow aquifer, resulting in the formation of several "underground lakes". The Santonian-age chalk is notable for its relatively high dolomite content (14%). The average porosity is about 42%, with a tight range of pore-size distribution.

The Estreux underground quarry (EX) is located in the town of Saint-Saulve, close to the city of Valenciennes in northern France. A total surface area of approximately 10 5 m 2 has been excavated in underground quarries that date from the 19th century. The quarry has been abandoned for more than a century.

Like the SM underground quarry, the EX quarry was exploited by the room-and-pillar method and contains galleries of 1.5-to 2-m width supported by rectangular pillars measuring 1.5 to 4 m from side to side (rate of extraction ≈78%, overlying rock thickness ≈20 m, estimated vertical stress ≈1.8 MPa; Nguyen 2010). Backfilling of the entire exploited area with mining waste has resulted in a lowering of the height of the gallery, from an initial height of three meters to its present height of two meters. The stratigraphy has a tabular profile. The Turonian-age chalk contains a relatively high amount (24%) of glauconite, a clay mineral of the muscovite family, and is not as homogenous as the dolomitic chalk in terms of its structure. The porosity is around 35% and displays a tight range of pore-size distribution.

Chalk samples were collected from both sites by drilling 120-mm diameter, 2 m long boreholes horizontally into the pillars. Cylindrical plugs (20 mm diameter, 40 mm height) were taken from each of the cores at intervals of approximately 10 cm. The longitudinal axis of each plug sample is therefore parallel to the axis of the pillar. The same sampling approach has been successfully applied to gypsum and iron ore in previous studies [START_REF] Auvray | The ageing of gypsum in underground mines[END_REF][START_REF] Grgic | Ageing of Lorraine (France) abandoned iron mines. Rock Mechanics: a challenge for society[END_REF][START_REF] Grgic | Impact of chemical weathering on micro/macro-mechanical properties of oolithic iron ore[END_REF].

SCANNING ELECTRON MICROSCOPE OBSERVATIONS

Samples from both quarries were examined by scanning electron microscopy (SEM). In the SM chalk, structural differences were observed between samples collected from the pillar-walls and those taken from the middle of the pillars. At the pillar walls, we observed euhedral crystals of 5 to 10 microns in size micro-calcite that presented no signs of neoprecipitation (Fig. 1a). In contrast, crystals at the centre of the pillars were micro-to submicron-sized in size and xenomorphic (anhedral) (Fig. 1e), probably due to partial dissolution. It was not possible to identify any particular zoning of these differences; both the decrease in crystal size and the change from euhedral to anhedral crystal shape appeared to be continuous from the walls to the centres of the pillars. It was not possible to distinguish or measure the dimensions of any dissolution fringe.

In contrast to the EX chalk, no variation in crystal size with horizontal core-depth was observed in the Estreux chalk pillars. A number of crystals in each sample plug exhibited clear dissolution marks (Figs. 1f and1j).

PHYSICO-MECHANICAL CHARACTERIZATIONS

Different physical and mechanical parameters were determined for all cores of up to 2 m length: rock density (dry and at saturation) (ρdry/ρsat); porosity accessible to water (nw); water content at saturation (Wsat); and uniaxial compressive strength under dry/saturated conditions (Rcdry/Rcsat). The water used to saturate the samples was saturated (in terms of calcite) prior to the experiments in order to avoid any dissolution of the rock. The compression tests were performed using a hydraulic press of 5 kN capacity. Some internal variation was observed in the data acquired from laboratory tests performed under dry and saturated conditions, however a discussion of these variations is beyond the scope of our study. We can nevertheless discuss the main variations observed in the physico-mechanical properties of the chalk with horizontal core-depth.

The results of the tests performed on samples from the SM underground quarry are presented in Figure 2: (i) the observed variations in the four physical properties (densities, water content and porosity) clearly indicate the existence of two distinct domains in the pillars: one between the wall and a depth of 75 cm, and the second from a depth of 75 cm to the middle of pillar. In the outer domain, wet and dry densities decrease with depth while the wateraccessible porosity and water content increase. In the inner domain, despite some internal variation, the physical properties generally appear to form plateaus; (ii) in terms of the mechanical properties (uniaxial compressive strength in saturated and dry conditions), two domains of variation can again be identified in the pillars but the boundary between them occurs at 50 cm horizontal depth rather than at 75 cm. In the outer domain, uniaxial compressive strength does not vary and can be considered constant. Approximately the same mean value was obtained for both dry and saturated states. In the second (innermost) domain, the dry uniaxial compressive strength increases with depth whereas the strength under saturated conditions forms a plateau with a value that is 2-3 times lower than that obtained at the edge of the pillar (in the outer domain). Thus, a significant drop in saturated uniaxial compressive strength occurs at the transition between the two domains.

Results for the tests on samples from the EX underground quarry are presented in Figure 3: (i) the trends in the physical and mechanical properties of the EX chalk are all continuous with core-depth. (ii) the resistance of dry and saturated samples under uniaxial compression shows some variation, but there is no clear overall trend with core depth. However, the variation in uniaxial compressive strength is identical to that seen in the second domain of the SM chalk, i.e., increasing from the wall to the middle of pillar in the dry state and forming a near-plateau in saturated uniaxial compressive strength; (iii) overall, the trends are more clearly defined in the EX chalk than they are in the SM chalk, but the main variations are of smaller magnitude at EX.

INTERPRETATION & DISCUSSION

In the chalk from the EX underground quarry, the intensity of the traces of dissolution does not vary along the length of the core [START_REF] Lafrance | Ageing of chalk rocks in an underground quarry[END_REF], however the porosity accessible to water decreases from the pillar wall to the middle of the pillar. Water (vapour and/or liquid) will progressively dissolve crystals and penetrate a pillar towards its centre. An increase in porosity accessible to water is therefore a direct effect of dissolution. Consequently, the amount of empty space, such as cracks and pores, should be greater at the pillar walls than in the middle of the pillars. From the SEM observations (Fig. 1), these empty spaces appear to be in the form of dissolution cavities. The dissolution does not appear to be zoned, however, and in fact, the entire pillar appears to have experienced partial dissolution. According to these hypotheses then, the strength of the material should increase towards the centre of a pillar (the edges of the pillars being more damaged than their centres and being composed of more porous material). However, the data obtained from the uniaxial compression tests suggest that the strength of the chalk is in fact highly variable, masking any correlation with core depth if it does indeed exist. It is possible that the anthropogenic changes, synonymous with ageing, are too advanced in these pillars to allow either a dissolution fringe or progressive degradation to be distinguished. Investigation of samples from a longer, horizontal borehole drilled into the rock face could shed further light on this.

In the SM underground quarry samples, no dissolution marks were observed at the edges of the pillar at the scale of our observations [START_REF] Lafrance | Ageing of chalk rocks in an underground quarry[END_REF]), but dissolution marks become increasingly visible towards the centres of the pillars. In addition, a large decrease in the size of calcite crystals was observed from the edge to the centre of the pillar. However, the explanation put forward for the EX chalk does not hold for the SM chalk. Even though the density of the rock decreases from the pillar wall to its centre, the other parameters -water-accessible porosity, water content at saturation and strength under uniaxial compression in saturated conditions -all show an overall decrease towards the centre of the pillars. In other words, the centres of the pillars appear to be more damaged than their walls.

In an attempt to explain these variations, a number of hypotheses can be put forward. Sorted by their reaction rate, these include (in descending order): (i) vertical drainage that is greater down the interior of the pillar than down its edges; (ii) aggrading neomorphism or Ostwald ripening; (iii) an oxidizing environment, which would favour recrystallization at the pillar edges: i) Vertical drainage through the pillar: [START_REF] Barhoum | Transferts dans la craie. Approche locale: la carrière de Saint-Martin-le-Noeud. Approche régionale: le Nord-Ouest du Bassin de Paris[END_REF] demonstrated that a significant proportion of the water circulation in the SM quarry study area takes place vertically. The pillars therefore appear to act as giant stalactites [START_REF] Gombert | Bilan des études et travaux réalisés dans les carrières souterraines de craie de Saint-Martin-le-Noeud (60) et d'Estreux (59)[END_REF]. Stalactites are formed by the crystallization of rings of calcium carbonate towards the exterior during carbon dioxide outgassing, forming a tube through which water can percolate. Crystallization around the pillar edges is therefore feasible. As well as playing a role in chemical weathering via dissolution, water also plays a part in erosion. The dissolved ion content of the percolating waters is usually equal to or lower than that of the lake waters. Because the core of the pillar is in a permanent state of saturation, oversaturation/oxidation of the environment does not occur. As the residence times of fluids (a mixture of primarily percolating water and underground lake water [START_REF] Barhoum | Transferts dans la craie. Approche locale: la carrière de Saint-Martin-le-Noeud. Approche régionale: le Nord-Ouest du Bassin de Paris[END_REF] are longer in the centre of the pillar than at its edges, the loading surface of the crystals will change in the presence of crystals, and dissolution as well as particle transport will dominate over crystallization. ii) Aggrading neomorphism (Ostwald Ripening):

In the SM chalk, the largest crystals at the edges are well-formed rhombohedra, very different in form and size from the minute calcite shells encountered in the chalk. No dissolution cavities were observed in these large crystals whereas some were observed in the smaller crystals. The fact that dissolution marks are present in the small crystals but not in the large crystals suggests that the latter are more stable and more resistant to dissolution. This discrimination between crystal size is characteristic of aggrading neomorphoism or the Ostwald 'ripening' phenomenon, which is driven by differences in surface energy [START_REF] Ratke | Growth and Coarsening: Ostwald Ripening in Material Processing[END_REF][START_REF] Voorhees | Solution to the multi-particle diffusion problem with applications to Ostwald ripening[END_REF]. Ostwald ripening is a spontaneous process in which minute solid particles evolve through time in a lightly oversaturated liquid. The grains all have the same chemical composition but differ in size. The ripening process is driven by the dependence of the solubility of a crystal on crystal size, and is expressed by the Gibbs-Kelvin relationship. The SM quarry is situated directly at the level of the water table (or unsaturated zone) [START_REF] Barhoum | Transferts dans la craie. Approche locale: la carrière de Saint-Martin-le-Noeud. Approche régionale: le Nord-Ouest du Bassin de Paris[END_REF]) and waters derived from the lakes that have formed around the pillar are in chemical equilibrium with the calcite, at a pH level of around 7.30. The edges of the chalk pillars are therefore subject to quite frequent wetting/drying cycles, which are largely synchronous with fluctuations in the water table [START_REF] Gombert | Bilan des études et travaux réalisés dans les carrières souterraines de craie de Saint-Martin-le-Noeud (60) et d'Estreux (59)[END_REF]. iii) Evolution in an oxidising environment predisposed to recrystallization: Of the different accessory minerals found in chalk, silica, like clay, has a tendency to inhibit the dissolution/recrystallization of calcite, which is in equilibrium at a pH of around 8.0.

The pillar in the underground quarry at SM, appears to have evolved in an oxidising environment. The pH is stable at around 7.30 (and does not vary over the course of the year; Barhoum 2014) and, like for hydrogen carbonates, the water is saturated in oxygen in a cyclical manner according to the period of the year. However, sufficiently oxidising conditions in the chalk will lead to the breakdown of organic matter and the clearing of crystal growth sites. Recrystallization will take place once a site is completely free of organic matter and the environment is enriched in calcium and hydrogen carbonate ions [START_REF] Belova | The effect on chalk recrystallization after treatment with oxidizing agents[END_REF]al. 2012, Robinson &[START_REF] Robinson | An experimental study of crust development on chalk download soils and their impact on runoff and erosion[END_REF]. The oxidising properties, the cyclical oversaturation (via fluctuations in the water table) and the chemistry of the water, together represent a set of conditions that are particularly favourable to the recrystallization of calcite at the edge of a pillar. Such conditions do not appear to exist at the core of the pillars, which are continually saturated (see the following section).

The chalk at EX exhibits traces of dissolution but no recrystallization (according to the SEM images). The pH of the groundwater in the surrounding area is clearly lower than that required for calcite equilibrium and is therefore potentially oxidizing. However, the chalk also contains a relatively high amount of glauconite (a green ferrous alteration mineral that forms in a reducing environment during diagenesis) which will act as a reducing agent. If the reduction potential is greater than the oxidation potential, as seems to be the case here, then the environment will remain in a reducing state and the clay matter will retain its role as an inhibitor of recrystallization and have a buffering effect on the oxidising properties of the water. However, because the pH is lower than the equilibrium pH of the calcite, the calco-carbonic equilibrium is displaced and dissolution marks may begin to appear.

Any one of these mechanisms, or indeed a combination of two or all three, may explain the observed decrease in the physical and mechanical properties of the rock towards the centres of the Saint-Martin pillars. Elsewhere, a dataset comprising measurements of resistance under compression for horizontal boreholes drilled into a phosphate chalk underground quarry at Malogne at Cuesmes, near Mons in Belgium [START_REF] Funcken | Procès verbal -Stabilité de la carrière de la Malogne[END_REF]) is currently available. The resistance values measured in the Malogne chalk appear to decrease towards the interior of the pillar. This supports the findings of our study and allows us to favour the hypothesis of preferential vertical drainage down the middle of a pillar. It also suggests that the trend shown for SM underground quarry is not merely an isolated case.

CONCLUSION

Evidence for temporal changes to chalk in underground quarries where human activity has long ceased, which can be interpreted as ageing, are observed at the underground quarry at SM. The density and intensity of dissolution marks in calcite grains appears to decrease outwards from the inner-core of the pillar towards the pillar walls. However, rock density decreases in the opposite direction, from the wall towards the centre, and parameters such as wateraccessible porosity, water content at saturation, resistance to uniaxial compression under saturated conditions also present an overall decrease from the wall towards the middle of the pillars. Three hypotheses, sorted by their reaction rate in descending order, can be put forward to explain the observed variations: (i) vertical drainage that is greater down the interior of the pillar than down its edges; (ii) aggrading neomorphism or Ostwald ripening; (iii) an oxidising environment which favours recrystallization at the pillar edges.

At the Estreux site, the ageing of the glauconitic chalk already appears to be at an advanced stage. The level of uniformity in the data suggests that there is currently no intermediate stage between a 'healthy' and an aged state (with dissolution marks). This may explain why we were unable to distinguish any zoning of either dissolution marks or resistance to uniaxial compression.

Figure 1 .

 1 Figure 1. Structural and morphological variations (⇒: dissolution marks) in the Saint-Martin-Le-Noeud and Estreux chalks.

Figure 2 .

 2 Figure 2. Variations in physico-mechanical parameters with horizontal core-depth (SCx: horizontal core, nw: porosity accessible to water, w: water content, ρ: density, Rc: uniaxial compressive strength, sat: in a saturated state, sec: in a dry state) in the Saint-Martin-Le-Noeud chalk.

Figure 3 .

 3 Figure 3. Variations in physico-mechanical parameters with horizontal core-depth (SCx: horizontal core, nw: porosity accessible to water, w: water content, ρ: density, Rc: uniaxial compressive strength, sat: in a saturated state, sec: in a dry state) in the Estreux chalk.