Wei Hu

Christophe Dano
email: christophe.dano@ec-nantes.fr

Pierre-Yves Hicher

Jean-Yves Le Touzo

François Derkx

Erick Merliot

Keywords: triaxial cell, granular materials, rockfill behavior, coarse soil, size effect

Natural soils or coarse granular materials containing large particles such as rockfill or ballast are often an issue for geotechnical projects. However, since it is usually complicated or impossible to set up a big enough experimental apparatus to test these materials, their mechanical behavior remains poorly known. To circumvent this difficulty, we have developed a set of triaxial cells of various sizes to study the experimental behavior of granular soils containing particle sizes of up to 160 mm. This paper presents the first results of a two-part experimental study. The first part consists of triaxial tests on sand samples of different sizes in order to examine specimen scale effects. Pre-peak behavior is not affected by the specimen size, whereas post-peak behavior depends on the test conditions that control the development of strain localizations. In the second part, we present drained triaxial compression tests on large samples of calcareous rockfill materials that have parallel grain size distributions. The results show a nonstandard evolution of the friction angle in relationship to grain size if compared to similar studies on coarse granular materials. Although grain crushing occurs during compression, the strain-stress curves of the different materials with different grain sizes do remain close. We explain this phenomenon by the strength evolution of the individual grains with respect to their sizes.

Introduction

The study of granular materials containing large particles such as rockfills, ballasts, etc, requires suitable laboratory facilities whose dimensions must be in proportion to the maximum size of the grains. In order to satisfy the concept of Representative Elementary Volume, a current assumption in the geotechnical community is to test specimens six to ten times bigger than the particle maximum size [START_REF] Holtz | Triaxial Shear Tests On Pervious Gravelly Soils[END_REF]. Under this condition, the physical and mechanical properties do not depend on the specimen size. Besides, the sample height is usually twice its diameter in triaxial tests in order to lessen the effects of cap and base friction [START_REF] Bishop | The Influence of End Restraint on the Compression Strength of a Cohesionless Soil[END_REF][START_REF] Bishop | Shear Strength Parameters for Undistubed and Remoulded Soil Specimens[END_REF].

In recent years, the construction all over the world of new and sometimes very high rockfill dams (200 m and more) using natural and local fill materials corresponds to the need for sustainable energy and water resources. The optimum design of such dams involves a better understanding of the behavior of the used materials in order to guarantee the safety of its structural behavior during its lifetime. Other engineering issues (stone columns, ballasts, stone beds above rigid inclusions, etc,) justify the creation of new research programs on the currently ill-determined behavior of coarse granular materials.

The pioneering work on the behavior of coarse soils was done at the University of California, Berkeley [START_REF] Marachi | Strength and Deformation Characteristics of Rockfill Materials[END_REF][START_REF] Marachi | Valuation of Properties of Rockfill Materials[END_REF][START_REF] Becker | Strength and Deformation Characteristics of Rockfill Materials in Plane Strain and Triaxial Compression Tests[END_REF]) and at the University of Mexico [START_REF] Marsal | Research on the Behaviour of Granular Materials and Rockfill Samples[END_REF][START_REF] Marsal | Large-Scale Testing of Rockfill Materials[END_REF][START_REF] Marsal | Plane Strain Testing of Rockfill Materials[END_REF] in the 1960s and 1970s. The construction of large facilities then accompanied the scientific objective to extrapolate the behavior of natural rockfills used in dam construction from triaxial tests on finer fractions of the same material. Those experimental programs clearly highlighted the effect of grain crushing on the value of the shear resistance and more particularly on the peak friction angle [START_REF] Marsal | Large-Scale Testing of Rockfill Materials[END_REF][START_REF] Marachi | Strength and Deformation Characteristics of Rockfill Materials[END_REF][START_REF] Leps | Review of Shearing Strength of Rockfill[END_REF][START_REF] Marachi | Valuation of Properties of Rockfill Materials[END_REF][START_REF] Becker | Strength and Deformation Characteristics of Rockfill Materials in Plane Strain and Triaxial Compression Tests[END_REF]. Extensive triaxial testing programs were completed with either one dimensional compression tests [START_REF] Marsal | Large-Scale Testing of Rockfill Materials[END_REF] or plane strain compression tests [START_REF] Marsal | Plane Strain Testing of Rockfill Materials[END_REF][START_REF] Becker | Strength and Deformation Characteristics of Rockfill Materials in Plane Strain and Triaxial Compression Tests[END_REF] on the same rockfill granular materials, and all of the test results led to the conclusion that (1) grain breakage is enhanced when stress level, grain size, and sample dimension are increased; and (2) the peak friction angle value decreases if the amount of grain crushing increases, accompanying a more compressible behavior. Such large experimental facilities are still very scarce around the world, although some new devices, among which is the cell described in this paper and others [START_REF] Verdugo | Strength and Stiffness of Coarse Granular Soils[END_REF], attempt to reproduce and extend the previous research programs in light of the present knowledge and technology. Nevertheless, some smaller but still unusual cells have provided interesting results about the behavior of unsaturated gravels and rockfill materials [START_REF] Chávez | A Constitutive Model for Crushed Granular Aggregates Which Include Suction Effects[END_REF][START_REF] Chávez | Estudio del comportamiento triaxial de materiales granulares de tamaño medio; con énfasis en la influencia de la succión[END_REF]), about the dynamic behavior of ballasts [START_REF] Indraratna | Shear Behaviours of Railway Ballast Based on Large-Scale Triaxial Tests[END_REF][START_REF] Lackenby | Effect of Confining Pressure on Ballast Degradation and Deformation Under Cyclic Triaxial Loading[END_REF]), or about the pseudo-static behavior of rockfills [START_REF] Charles | The Influence of Confining Pressure on the Shear Strength of Compacted Rockfill[END_REF][START_REF] Barton | Shear Strength of Rockfill[END_REF][START_REF] Indraratna | Large-Scale Triaxial Testing of Grey Wacke Rockfill[END_REF][START_REF] Varadarajan | The Role of Nature of Particles on the Behaviour of Rockfill Materials[END_REF].

In this paper, we present the results of an experimental study based on the development of a set of triaxial cells with different dimensions, which allowed us to address the issue of the maximum grain size effect on the behavior of rockfill material. This set of triaxial cells comprises a conventional triaxial cell for samples of 70 or 100 mm in diameter, a second cell allowing testing samples of 250 mm in diameter, and a very large triaxial cell for samples with a diameter between 500 and 1000 mm.

Experimental Devices

The small and intermediate triaxial cells, designed to accommodate samples whose diameter is either 70 or 100 mm for the small one or 250 mm for the intermediate one, are not so different from many commercial devices, so that they are not described here. Experiments on relatively coarse soils merely imply that the testing procedure must be adapted as presented below. The description of the very large triaxial cell will be emphasized. Contrary to the peculiar design of Marsal's spherical cell, the principle of the new large apparatus resembles a smaller conventional cylindrical triaxial cell similar to the large test facility at the University of California, Berkeley.

Initially, the cell was designed to support vertical loading up to 1000 kN (first configuration). All the test results on sand samples presented herein refer to this first configuration. Some elements (base of the cell, sample platens, top cover, pre-tension tie rods, piston, and ball joint) were subsequently reinforced to sustain vertical loading up to 2000 kN. The driving hydraulic and electronic system was also simultaneously upgraded; this second configuration was used to carry out triaxial tests on the rockfill material presented in the last section. General sketches of the two configurations are shown in Fig. 1.

The confining chamber (Fig. 2) stands on a pre-stressed concrete slab that can support its total weight (about 150 kN) without any significant deformation. It is placed under a 100 kN travelling crane high enough to pass over the 4.5 m high triaxial device. The cell is composed of a very stiff and fixed steel base 1.60ϫ 1.60 m large, on which all the cell elements are assembled.

Depending on the sample size and nature, a suitable inner load cell (among three submersible load cells of respective capacities 500, 1000, or 2000 kN and for all of them a linearity lower than ±0.5 % of the measure scale) is chosen and installed on the cell base, just below the sample bottom platen directly screwed on the load cell. The use of an inner load cell gives a more accurate measurement of the shear force applied to the sample; it particularly does not integrate the frictional force between the cell cap or the cover plate and the loading piston.

At the present time, the cell may host samples either 1 m in diameter and 1.5 m high (height-to-diameter ratio: 1.5) or 0.5 m in diameter and 1 m high (height-to-diameter ratio: 2). Very stiff distance-pieces are used to adjust the position of the sample inside the cell according to its original height. Of course, the different slenderness of the largest samples imposed by the available total height within the cell may have an effect on the stress-strain responses of the tested material. It has been largely shown that the peak friction angle and the tangent initial stiffness are greater in case the slenderness is reduced or no lubrication techniques are used [START_REF] Lam | Triaxial Compressive and Extension Strength of Sand Affected by Strength Anisotropy and Sample Slenderness[END_REF][START_REF] Darve | Les géomatériaux: theories, experiences et modèles [Geomaterials: Theory, Experiment and Modelling[END_REF][START_REF] El Bied | Etude expérimentale et modélisation du comportement des roches granulaires: écrouissage, radoucissement et rupture en mode localize [Experimental Study and Modeling of the Behavior of Granular Rocks: Hardening, Softening and Failure in a Localized Mode[END_REF] because shear stresses are generated due to friction between the platens and the soil sample. Such an effect must be taken into account when comparing the subsequent test results on sand. On the contrary, comparative triaxial tests on rockfill samples were conducted on samples with the same slenderness and the comparison is therefore assumed to be straightforward.

The bottom and top platens are made of steel circular elements 120 mm thick and 1100 mm in diameter for samples 1 m in diameter, and 100 mm thick and 600 mm in diameter for samples 0.5 m in diameter. The inside faces of the platens contain concentric grooves in order to allow drainage. They were also fitted with a metallic screen protected by two fine layers of a textile geomembrane that act as a porous stone and a filter in order to avoid any intrusion of fine particles into the drainage network. No lubrication is used to prevent boundary effects on the platens; experimental observations, however, show that sliding along the end platen is not prevented.

The lateral cylindrical envelope, commonly transparent in most standard triaxial cells, is here replaced by a 2 m high steel doublewall chamber and two heightening elements on both sides of this chamber (Fig. 2). All electrical cables for transducers or nylon hydraulic hosepipes pass through those heightening elements.

The double-wall chamber is composed of • a steel outer cylinder 12 mm thick and 1256 mm in diameter and • a high density polyethylene (HDPE) inner cylinder 8 mm thick and 1200 mm in diameter clamped at its tips. The outer and inner volumes are both filled simultaneously with water and can withstand a confining pressure of 1.5 MPa (it is practically limited to a serviceability pressure of 1 MPa). The driving system continuously controls the equality of the pressures in the two chambers so that the inner HDPE cylinder remains undeformed during the test. Therefore, the inner compartment containing the sample can be assumed to be at constant volume during the test, which enhances the accuracy of the volume change measurements. Two 1 MPa pressure transducers located at the base of the double-wall chamber are connected to the inner and outer volumes, respectively. In case of a difference of pressure on both sides of the HDPE envelope, a servo-valve is automatically opened by the driving system to balance the pressures.

Water volume changes are detected by an electromagnetic flowmeter with an accuracy of ±0.2 %, coupled to an electronic counter inserted in the cell pressure line. During shearing at constant cell pressure, the water volume changes are mainly due to the penetration of the loading ram into the inner volume, to the water compressibility, to the membrane penetration effects, and to the sample volume changes [START_REF] Georgopoulos | Corrections on the Specimen Volume Changes and Axial Force in the Wykeham Farrance Triaxial Cell[END_REF].

Although assumed to be an incompressible fluid, water contracts or dilates whenever a pressure or temperature variation is applied. At temperatures of 0 and 20°C, the coefficient of compressibility of water c w is, respectively, 5.01ϫ 10 -7 kPa -1 and 4.57 ϫ 10 -7 kPa -1 . The water volume change can be evaluated as (Georgopoulos and Vardoulakis 2005)

⌬V w = c w • ⌬ c • V w (1)
where: ⌬ c = variation in the cell pressure and V w = current water volume in the cell (1.24 m 3 for a 1 m large sample and 2.35 m 3 for a 0.5 m large sample).

A variation in the cell pressure value of 2 kPa (because of a de-layed response of the hydraulic control system for instance) induces a water volume change of 1.13 cm 3 for a 1 m sample or 2.15 cm 3 for a 0.5 m sample at a temperature of 20°C, i.e., a sample volumetric strain of about 10 -5 in the extreme case. Volume changes due to a variation of several Celsius degrees are of the same order of magnitude.

The membrane penetration effects can be assessed following two different approaches: First, by assuming that large samples of coarse soils represent a rigorously homothetic picture of small samples with small particles, the approach by [START_REF] Baldi | Membrane Penetration Effects in Triaxial Testing[END_REF] based on triaxial tests on fine sands ͑d 50 Յ 0.5 mm͒ can be considered. According to their study, the volume change due to membrane penetration, which mainly occurs during the drained consolidation phase, depends on the mean diameter d 50 of the grains, of the sample size (diameter d sample = 1000 mm and volume V sample ), of the effective cell pressure Ј c , and of the membrane properties (Young's modulus E membrane ϳ 10 MPa; thickness e membrane = 1.4 mm) and can be expressed by the following relation

V m = d 50 2 • d sample • V sample • ͱ 3 Ј c • d 50 E membrane • e membrane (2)
An alternative approach was developed by [START_REF] Frydman | The Membrane Effect in Triaxial Testing of Granular Soils[END_REF], [START_REF] Ramana | Membrane Penetration in Triaxial Tests[END_REF][START_REF] Ramana | Membrane Penetration in Triaxial Tests[END_REF][START_REF] Nicholson | Measurement and Elimination of Membrane Compliance Effects in Undrained Triaxial Testing[END_REF]. They suggested that the volumetric membrane compliance per unit membrane area is proportional to the log cycle of the effective confining pressure by a constant S called normalized membrane penetration.

Various correlations have been established between S and a characteristic grain size, either d 50 or d 20 . Based on a large collection of test results, [START_REF] Nicholson | Measurement and Elimination of Membrane Compliance Effects in Undrained Triaxial Testing[END_REF] proposed a relationship between S and d 20 that tends towards a limiting value of S = 0.4 ͑cm 3 /cm 2 ͒ / ⌬ log Ј 3 when d 20 increases up to values higher than 50 mm. This value has been used to correct the initial void ratios of the 1000 mm samples. For the 250 mm samples, according to the relationship proposed by [START_REF] Nicholson | Measurement and Elimination of Membrane Compliance Effects in Undrained Triaxial Testing[END_REF], a value of S = 0.1 ͑cm 3 /cm 2 ͒ / ⌬ log Ј 3 has been assumed. No correction has been applied in the case of sand samples. The approach of [START_REF] Baldi | Membrane Penetration Effects in Triaxial Testing[END_REF] is suitable for small samples (70 mm) but leads to an overestimated value of the membrane correction (some percent for the volumetric strain) inconsistent with the experimental observations of the volumetric contractancy during isotropic compression for large samples. The second approach considering a value of S = 0.1 or 0.4 ͑cm 3 /cm 2 ͒ / ⌬ log Ј 3 gives a more realistic value of the volumetric strain during consolidation, close to 1 %, and is therefore used for samples of 250 and 1000 mm in diameter.

Let us now consider a drained shearing phase of a coarse sample at a constant confining pressure. Volume changes due to membrane penetration effects are assumed to be negligible even if the key parameter, namely the characteristic grain size, evolves slightly due to grain crushing. Therefore, the sample volume changes are calculated in the following way

⌬V sample = ⌬V FM -⌬V piston (3)
where:

⌬V FM = water volume-entering or exiting the cell-measured by the flow meter installed in the console, ⌬V sample = sample volume change, and ⌬V piston = variation of the cell water volume corresponding to the penetration of the loading piston into the cell (⌬V piston = . ⌬H . r 2 where ⌬H is the relative movement of the loading piston within the cell determined at the hydraulic jack level, r = 40 mm is the piston radius in the first cell configuration, and r = 50 mm in the second one).

12 steel tie rods 60 and 1560 mm in diameter and height, respectively, are pre-stressed at a value of 10 MPa to ensure the complete water tightness of the cell once it is finally mounted. They also facilitate the placement of the cover plate by guiding it. The steel circular cover plate, 145 mm thick and 1600 mm large, which closes the cell (Fig. 2), is fitted with a vent valve so that the air can flow out the chamber when it is filled by water and also supports the loading frame, as shown in Fig. 1.

The first configuration of the cell uses a one-way hydraulic jack whose capacity is 1000 kN and displacement course 250 mm. More recently, a new hydraulic jack was acquired in order to apply loading up to 2000 kN statically (from 0 to 50 mm/h) and cyclically with frequencies in the range 0-5 Hz for a course of 300 mm. The hydraulic jack applies a vertical load (in compression or in extension) on the loading steel piston, which slides through the cover plate. The lower tip of the loading piston consists of a ball joint (Fig. 2) attached to the top platen of the sample. Note that the vertical shortening of the sample is measured by an external LVDT bound to the hydraulic jack. The maximum axial strain is limited to 13 % using the 1000 kN configuration of the cell and to 20 % with the 2000 kN configuration.

A critical issue involves the manufacture of the membranes. In order to prevent prohibitive costs and sudden leakage due to membrane tears, a 1.5 mm thick and nonreinforced PVC membrane, usually used as a swimming-pool liner, was chosen. A perfectly rectangular piece slightly greater than the necessary size is carefully cut. For a sample 1 and 1.5 m in diameter and height, respectively, the rectangular piece is 2 m high and 3.19 m long. The ends of the membrane are then thermally glued using a heating tool at a temperature of about 200°C. The membrane, which can be slightly warmed up in order to facilitate its installation, is finally placed on a four-quarter aluminum cylindrical mold and the whole set up is installed on the bottom platen of the cell. Steel rings encircle the membrane on the platens beforehand coated with silicone grease. Because the grain angularity may cause membrane tear, a second PVC membrane and a 0.5 mm thick layer of geo-membrane whose dimensions coincide with the sample dimensions are usually superimposed when coarse soils are tested; the inner membrane is free to slide against the outer membrane, itself fixed on the sample pedestals. The cumulative thickness of the protecting membrane is thus about 3 mm, which is relatively small compared to prior experiments by [START_REF] Marsal | Large-Scale Testing of Rockfill Materials[END_REF] or [START_REF] Marachi | Strength and Deformation Characteristics of Rockfill Materials[END_REF].

Before filling the mold, the membrane is pushed against the mold by a vacuum of 50 kPa by using a vacuum pump and a vacuum transducer.

Experiments on Sand Samples

In order to evaluate the performance of the cell, to improve the experimental procedure and to prove that no material size effects can subsequently affect test results on coarse soils, a series of drained consolidated triaxial compression tests using the 1000 kN configuration of the cell was conducted on dry Loire River sand. It has been previously studied in the laboratory [START_REF] Dano | Characterization of Loire River Sand in the Small Strain Domain Using New Bender-Extender Elements[END_REF]) and huge and uniform quantities are available. A similar approach preliminary to extensive programs on coarse materials was previously carried out by [START_REF] Marachi | Strength and Deformation Characteristics of Rockfill Materials[END_REF] and [START_REF] Becker | Strength and Deformation Characteristics of Rockfill Materials in Plane Strain and Triaxial Compression Tests[END_REF], for instance.

The Loire River sand consists of recent alluvial deposits of the Loire River, close to its mouth in the west of France. It is composed of sub-rounded silica particles and elongated shell fragments. The mean grain size d 50 is 520 µm. The coefficient of uniformity C u is equal to 3.5. Therefore, Loire River sand is poorly graded sand (ASTM D2487-93). The specific weight ␥ s of the grains is 26.2 kN/ m 3 (ASTM D854-92). The minimum unit weight ␥ min (ASTM D4254-91) and the maximum unit weight ␥ max (ASTM D4253-93) are equal to 15.1 kN/ m 3 ͑e max = 0.735͒ and 18.5 kN/ m 3 ͑e min = 0.416͒, respectively.

Triaxial tests on samples 100, 500, and 1000 mm in diameter were carried out with nearly the same dry unit weight ␥ = 17.10 kN/ m 3 . Tests on 100 mm samples were carried out using a 50 kN loading device. It is worth noting that calibration tests in the largest samples, in which small aluminum boxes were randomly placed before filling, show a scattering of the unit weight of about 0.3 kN/ m 3 from one point to another, in spite of all the precautions taken to obtain a homogeneous sample. Sample reconstitution for all the configurations follows a quite similar procedure to achieve the same density state. A given mass of sand (0.6, 40, and 110 kg for the 100, 500, and 1000 mm samples, respectively) is poured into the mold then laid out and finally compacted by using a rammer. The process is repeated nine, eight, and nine times for the 100, 500, and 1000 mm samples, respectively. Such a process involves a clear anisotropy, particularly apparent in the largest sample, even through the PVC membrane. The top surface is carefully leveled and protected by a double layer of geo-membrane. The top platen is finally installed and the membrane fixed on it. A vacuum of maximum value 80 kPa is then applied inside the sample whereas the vacuum between the mold and the membrane is simultaneously released. The mold is then removed after slightly lifting the top platen. The appearance of the samples is shown in Fig. 3. The sample is of a cylindrical shape. Moreover, due to the relatively small grain size, surface defaults caused by membrane penetration are negligible.

The cell is then completely mounted and filled with water by simultaneously controlling the level of water on both sides of the inner HDPE envelope. At the end of the filling phase, the cell pressure c is gradually increased up to 100 kPa by using a pump, whereas the vacuum u a inside the sample is simultaneously released so that the sum ͑ c + u a ͒ remains always lower than the test confining effective pressure. At this stage, fully drained conditions are imposed. The cell pressure is finally increased up to the test confining effective pressure. Note that the confining pressures given in this paper are determined at the pressure transducer level, 1 m below the mid height of the samples. The shearing phase is conducted at a velocity of 25 mm/h (1.66 %/h for a sample 1500 mm high; 2.5 %/h for a sample 1000 mm high) at a constant cell pressure (among the following values: 100, 200, 300, and 350 kPa). Repeatability is assessed by carrying out two tests in the same conditions on samples 0.5 m in diameter at a confining pressure of 300 kPa. As the jack is a one-way device, the final unloading phase cannot be recorded. Therefore, at the end of the shearing phase, a vacuum is applied inside the sample, the hydraulic jack is stopped, and the cell pressure is slowly decreased.

The change of the cross section area of the specimen cannot be neglected during the triaxial test. The following formula (NF P 94-074 10/94) has been used for the correction of the cross section area

S = V 0 -⌬V s -⌬V i H 0 -⌬H s -⌬H i ϫ ͫ 1 + ⌬H i 2͑H 0 -⌬H s ͒ ͬ (4)
where:

V 0 , H 0 = volume and height of the specimen under the initial vacuum respectively, ⌬V i and ⌬H i = change of volume and height during the shearing phase, respectively, and ⌬V s and ⌬H s = change of volume and height of the specimen during the consolidation.

Because the consolidation of dry granular material can induce very small volume changes, the variations ⌬V s and ⌬H s are neglected in the calculation.

The triaxial test results are presented in Fig. 4 for each of the four confining pressures. The test result on the 1000 mm sample and at a confining pressure of 100 kPa is not relevant because of a poor control of the cell pressure, due to inappropriate proportionalintegral-derivative regulation factors.

As expected [START_REF] Habib | Effet d'Echelle sur les Sables Denses [Scale Effect in Dense Sands[END_REF]) and also previously shown by [START_REF] Marachi | Strength and Deformation Characteristics of Rockfill Materials[END_REF] on Monterey 20 sand, because the tests are conducted on representative volumes, the soil response does not significantly change with the sample size in the pre-peak domain. No grain breakage is involved in the present case; the grain size distributions were measured before and after the triaxial tests on the 0.5 and 1 m large sand samples. Only a representative mass of sand was sampled in different points of the specimens. The results are presented in Fig. 5 for the 1000 mm samples. Changes in the gradation are not significant and probably due to the sampling itself, even if some random breakages were observed on elongated shell fragments. However, the post-peak behavior exhibits clearly a dependence on the sample size. Only tests conducted at a cell pressure of 200 kPa are analyzed for this purpose. The rate of softening in-creases with the sample size. However, it is worth reminding that the height-to-diameter ratio is 1.5 for the 1 m sample and 2 for the other sample sizes. Observation of localization patterns reveals that a single shear band appears in the small sample 0.1 m in diameter, whereas in the 1 m sample, a set of parallel and cross shear bands are visible through the membrane, as shown in Fig. 3.

Figure 6 presents the evolution of the ratio max = ͑q / pЈ͒ max where q is the deviatoric stress and pЈ is the mean effective stress at peak, respectively. The variation of max for the 350 kPa cell pressure, for instance, corresponds to a variation in the value of the peak friction angle of 2°. In a first approximation, one can observe a decrease of the maximum stress ratio with the stress level, which indicates a curvature of the maximum shear strength envelope. However, the test results are quite inconsistent since samples with a slenderness of 2 show a greater maximum stress ratio and samples with a lower density (500 mm samples) show a bigger maximum stress ratio. The effect of the density heterogeneity and of the strain localization is assumed to explain the observed differences.

The parameters of the linear elastic Mohr-Coulomb perfectly plastic material are given in Tables 123, where E is the secant elastic modulus at an axial strain of 0.2 %, is the Poisson's ratio, is the peak friction angle determined from the tangent curve to the Mohr's circles, which can be seen as an average peak friction angle, ␥ d is the dry unit weight, Ј 3 is the confining pressure, e 0 is the void ratio, and is the dilation angle. The cohesion is null. The peak friction angle value for samples 100 mm in diameter has been validated by another study (Benazzouz 2005), which gives a value of the peak friction angle equal to 39.2°for similar unit weights. The comparison of test data on samples 100 and 500 mm in diameter, for which the slenderness is identical, does not reveal any difference in the peak friction angle. Therefore, we can conclude that there is no obvious sample size effect in the pre-peak behavior.

Triaxial Tests on Calcareous Rockfill Samples

Even if the triaxial cell could be used as a calibration chamber, its main objective is to test coarse granular materials. This type of granular soil exhibits peculiar behaviors under compression and shearing that have been related to grain crushing more likely to appear in coarse granular materials. Statistically, large particles can undergo a more intense crushing than smaller particles at stress levels (some hundreds of kilopascals) currently encountered in civil engineering for the same mineralogy. The amount of crushing depends on many key factors such as stress levels, stress paths (compression or shearing), density, moisture [START_REF] Chávez | A Constitutive Model for Crushed Granular Aggregates Which Include Suction Effects[END_REF], grain morphology and mineralogy, and grain size distribution [START_REF] Hardin | Crushing of Soil Particles[END_REF][START_REF] Biarez | Influence de la granulométrie et de son évolution par ruptures de grains sur le comportement mécanique de matériaux granulaires [Effect of the grain size distribution and its evolution due to grain crushing on the mechanical behaviour of granular materials[END_REF]. At the grain scale, many fracture patterns can be created depending not only on the individual strength of the grains but also on the coordination number and the way local forces are applied to the grains [START_REF] Guimaraes | Aggregate Production: Fines Generation During Rock Crushing[END_REF]. For instance, loose assemblies have smaller coordination numbers; splitting mode fracture is then enhanced. On the contrary, in dense granular assemblies, loads at the particle scale are statistically more uniformly distributed on the grains, which therefore flake [START_REF] Guimaraes | Aggregate Production: Fines Generation During Rock Crushing[END_REF]). However, this may not always be the case since it is well known that external loads are mainly transmitted through strong force chains into the granular materials, thus enhancing the splitting mode fractures (See, for instance, [START_REF] Richefeu | Force Transmission in Dry and Wet Granular Media[END_REF].

Other parameters can also affect the material behavior. Using the large triaxial facility at Idiem, University of Santiago (Chile), De La Hoz (2007) tested different coarse soils with different break- age capabilities under drained triaxial conditions. He chose to test parallel grain size distributions of the same mineral stock, as previously done by [START_REF] Marachi | Valuation of Properties of Rockfill Materials[END_REF]. The diameter of the particles reached 100 mm. At a given stress level for some granular materials, the peak friction angle decreased when the mean grain size in-creased and grain crushing was relatively intense. On the other hand, the peak friction angle would increase when the grain size distributions were not perfectly parallel. The increase was found to be related to the quantity of fine particles in the initial grain size distribution as well as to a dependence of the angularity with the mean diameter. An ultimate case for which the peak friction angle as well as the initial tangent stiffness did not depend on the mean grain size was obtained with strictly parallel grain size distributions, a low proportion of fines, the same density (no variation in the minimum and maximum dry densities), a similar angularity of the grains for all dimensions, and mainly the absence of grain crushing. Except the last one, these conditions are those of the subsequent tests on the calcareous rockfills carried out with the three triaxial cells.

In this section, another condition for an almost constant maximum stress ratio at the same stress level and at a same strain level but with different mean grain sizes was obtained. Triaxial test results on samples of a calcareous granular material 100 (Fig. 7(a)), 250 (Fig. 7(b)), and 1000 mm (Fig. 7(c)) in diameter, respectively, are compared. The height-to-diameter ratio is 1.5 for all configurations. The tested material is a calcareous rockfill from a quarry located in Préfontaines (center of France). Initial gradations for the three samples size are parallel, as suggested by [START_REF] Lowe | Shear Strength of Coarse Embankment Dam Materials[END_REF], with a uniformity coefficient U c close to 2 (Fig. 8). Grain sizes are in the range (1.25-12.5) mm for the 70 mm samples, (5-40) mm for the 250 mm samples, and (25-160) mm for the 1000 mm samples. The angularity of the particles, classified as sub-angular (ASTM D2488-10), is shown in Fig. 9. The specific unit weight measured on small particles is 26.8 kN/ m 3 . The material is first dried by natural evaporation in a covered hall; the water content initially equal to 11 % decreases until a value of 0.3 % is reached. Equilibrium between atmospheric conditions in the hall and within the material is assumed.

Minimum and maximum void ratios cannot be experimentally determined on the coarse fraction because of the grain size. Therefore, a relative density cannot be defined and we chose to conduct the tests at a same density rather than at a same relative density. This choice was also corroborated by the fact that the grains present a similar angularity at each scale. [START_REF] Becker | Strength and Deformation Characteristics of Rockfill Materials in Plane Strain and Triaxial Compression Tests[END_REF] indicated that the reduction of the friction angle value was lower for tests at a fixed relative index than for tests at the same initial void ratio. Test characteristics are indicated in Table 4. The rockfill material for the 1000 mm samples is deposited by 15 or 17 successive layers of 100 kg. Each of them is then compacted using a hand-held vibratory needle inserted at different locations into the rockfill layer during 60 s for the unit weight of 11.9 kN/ m 3 (estimated relative density: 52 %) and more than 5 min to obtain a unit weight of about 14.3 kN/ m 3 (estimated relative density: 85 %). Note that Test 1000-200B was conducted at a lower unit weight of 14.05 kN/ m 3 . The relative densities have been estimated from e min and e max measurements on the small grain size fraction. The height of each layer is controlled so that homogeneity through the sample can be assumed. Two layers of PVC membrane and one layer of geomembrane surround the rockfill material. A similar procedure was adopted for the 70 and 250 mm samples. The 70 mm specimens were surrounded by two layers of 0.5 mm latex or 1 mm neoprene membrane while the 250 mm samples, tested under a 500 kN loading system, were surrounded by two 1 mm thick neoprene membranes.

The shearing rate was 1.67 %/h for the loosest sample (using the 1000 kN configuration) and 2 %/h for the densest samples (using the 2000 kN configuration). In order to compare grain size evolutions, the shearing phase was stopped at the same maximum axial strain of 20 % for all the tests carried out on the 2000 kN configu-

PROOF COPY [GTJ103095] 006103GTJ

ration. At the end of the test, the membrane was cut at its base and gradually opened over a height of about 0.6 m. Contrary to an uncemented sand mass, which would immediately flow, the rockfill mass did not collapse and photos of the final structure could be taken (Fig. 10). The observations show intact grains even of large diameters, some grains deeply crushed, and some grains that split into two parts. It was not possible to distinguish specific zones of strong crushing or zones not affected by grain breakage. In other words, crushing seems to be uniformly distributed within the sample volume. Accumulation of fine particles at the base of the sample was observed and associated with a possible surface degradation of the grains due to frictional attrition. The deviatoric stress versus axial strain curves and the volumetric strain versus axial strain curves are presented in Fig. 7. The comparison between the 70, 250, and 1000 mm samples does not show any significant change in the shear strength, contrary to prior experimental observations by [START_REF] Marsal | Large-Scale Testing of Rockfill Materials[END_REF] or [START_REF] Marachi | Strength and Deformation Characteristics of Rockfill Materials[END_REF]. This conclusion cannot be strictly extended to the peak friction angle since the maximum deviator stress was not reached in all of the tests. Comparing the stress ratio max = ͑q / pЈ͒ max at the same axial strain of 20 % shows a decrease of the stress ratio with confining pressure (Fig. 11). At any given confining pressure, the differences in the stress ratio values can be mainly attributed to differences in initial densities, the 250 mm samples being slightly denser than the 70 and 1000 mm ones.

Grain size distributions carried out after the triaxial compressions are presented in Fig. 8. They clearly show that for a given grain size distribution, the amount of grain crushing increases with the confining stress (Fig. 8(b)), and that at a given confining stress, the 1000 mm samples present a slightly greater amount of particle breakage related to a lower maximum stress ratio. Even if nonsignificant, the lower amount of grain crushing for denser samples, namely samples of 250 mm in diameter, can be correlated with greater shear strength.

To understand why no size effect appeared in the present tests, a series of crushing tests on individual particles placed between two stiff plates was carried out, as previously done by [START_REF] Marsal | Large-Scale Testing of Rockfill Materials[END_REF]. The average diameter d ave of the crushed particles was in the range 10-100 mm. All the crushing tests exhibited an elastic-fragile behavior so that it was easy to measure the maximum force F max as a function of the average diameter (Fig. 12). The fitting curve gives a power law with a power coefficient of 1.90 (Fig. 12). This means that the failure stress f = F max / d ave 2 decreases only slightly with d ave . Consequently, there is no longer an obvious size effect in particle individual strength. It will be demonstrated in another paper [START_REF] Frossard | Evaluation of Rockfill Shear Strength Including Size Effect Within Granular Materials[END_REF][START_REF] Frossard | Effect of Sand Grain Shape on Interparticle Friction: Indirect Measurements by Rowe's Stress-Dilatancy Theory[END_REF], from a micro-macro analysis, that the decrease of the maximum stress ratio is directly related to the individual strength properties of the grains. Previous experimental observations by De La Hoz (2007), [START_REF] Marsal | Research on the Behaviour of Granular Materials and Rockfill Samples[END_REF], [START_REF] Marsal | Large-Scale Testing of Rockfill Materials[END_REF], [START_REF] Marachi | Strength and Deformation Characteristics of Rockfill Materials[END_REF][START_REF] Marachi | Valuation of Properties of Rockfill Materials[END_REF] confirm this finding. Several numerical models [START_REF] Daouadji | An Elastoplastic Model for Granular Materials Taking into Account Grain Breakage[END_REF][START_REF] Muir Wood | Changing Grading of Soil: Effect on Critical State[END_REF]) assume a relationship between micromechanical properties, grain crushing, and macroscopic behavior.

Conclusions

This paper presents triaxial test results by using a set of triaxial devices among which is a large triaxial cell designed for investigating coarse soil behavior. The main characteristics of the cell as well as the main difficulties in the use of this facility have also been evoked. At least two sample configurations can be tested. The most recent evolutions allow the users to apply confining pressures in the range 100-1000 kPa and axial loading up to 2000 kN. The original features of the cell beyond its dimensions can be attributed to the different possible sample sizes and the double-walled cylindrical Corrected value including membrane penetration effect based on S = 0.4 cm 3 /cm 2 per log cycle of Ј c for 1000 mm samples and S = 0.03 cm 3 /cm 2 per log cycle of Ј c for 70 mm samples [START_REF] Nicholson | Measurement and Elimination of Membrane Compliance Effects in Undrained Triaxial Testing[END_REF], assuming no particle crushing, which favours compressibility during consolidation.

envelope designed to accurately measure the volume changes of the samples during testing. The first set of tests carried out on sand samples attempt to show that size effects cannot affect the results of drained triaxial compression tests before peak. Test results on samples 100, 500, and 1000 mm in diameter were compared. The pre-peak behavior is not significantly dependent on the sample size, as assumed by the Representative Elementary Volume concept. Drained triaxial tests on a calcareous rockfill show that with the large triaxial cell we are capable of carrying out experiments on coarse soils and quantify the amount of grain crushing. In the specific case of the calcareous material, a nonstandard behavior was highlighted: The shear strength at a given axial strain level does not decrease as samples with larger particles are tested. It is the correlation with the individual strength of the grains that explains such a peculiar behavior.

FIG. 1 -

 1 FIG. 1-Design draughts of the two configurations of the large triaxial cell.

FIG. 3 -

 3 FIG. 3-Localization patterns after shearing at a cell pressure 200 kPa.

FIG. 4 -

 4 FIG. 4-Triaxial tests results on Loire River sand.

FIG. 5 -

 5 FIG. 5-Comparison of gradations before and after triaxial tests for sample 1000 mm.

FIG. 7 -

 7 FIG. 7-Triaxial test results on rockfill specimens.

FIG. 10 -

 10 FIG. 10-Final granular structure showing grain crushing.

FIG. 12 -

 12 FIG.12-Individual grain crushing tests.

TABLE 1 -

 1 Parameters of the linear elastic perfectly plastic model for Loire river sand specimens of 100 mm.

	Ј 3	␥ d		E sec (0.2 %)
	kPa	kN/ m 3	e 0	MPa		degree	degree
	100	17.13	0.53	65	0.322	11.5	39.6
	200	17.35	0.51	94	0.273	9.3
	300	17.31	0.51	85	0.258	13.3
	350	17.01	0.54	75	0.25	8.2

TABLE 2 -

 2 Parameters of the linear elastic perfectly plastic model for Loire river sand specimens of 500 mm.

	Ј 3	␥ d		E sec (0.2 %)
	kPa	kN/ m 3	e 0	MPa		degree	degree
	100	17.10	0.53	55	0.10	15.1	39.5
	200	17.06	0.54	65	0.22	13.3
	300	17.08	0.53	72	0.30	9.8
	350	17.02	0.54	63	0.21	11.3

TABLE 3 -

 3 Parameters of the linear elastic perfectly plastic model for Loire river sand specimens of 1000 mm.

	Ј 3	␥ d		E sec (0.2 %)
	kPa	kN/ m 3	e 0	MPa		degree	degree
	200	17.09	0.53	63	0.36	8.2	36.9
	350	17.00	0.54	70	0.28	8.5

TABLE 4 -

 4 Calcareous samples characteristics for triaxial tests.

		Diameter a	Height	Mass	Unit Weight
	Name	(mm)	(mm)	(kg)	͑kN/ m 3 ͒	Void Ratio b
	70-100A	66.2	105.0	0.527	14.30	0.86
	70-100B	66.0	105.9	0.530	14.35	0.86
	70-100C	65.7	105.6	0.529	14.49	0.84
	70-200A	65.5	105.9	0.522	14.34	0.85
	70-200B	65.9	106.0	0.529	14.37	0.85
	70-400A	66.1	106.0	0.544	14.70	0.80
	70-400B	65.4	103.8	0.521	14.63	0.80
	70-400C	65.8	105.4	0.531	14.55	0.81
	250-100A	245.2	379.8	26.61	14.56	0.83
	250-100B	244.9	380.1	26.51	14.52	0.84
	250-200A	245.9	377.0	26.45	14.49	0.83
	250-200B	245.9	381.0	26.40	14.31	0.85
	250-200C	245.9	379.0	26.55	14.47	0.83
	250-400A	246.0	377.5	26.55	14.51	0.82
	250-400B	246.0	382.8	26.51	14.29	0.85
	1000-100	1003	1496	1707	14.34	0.86
	1000-200A	997	1503	1526	12.91	1.06
	1000-200B	1006	1500	1687	14.05	0.89
	1000-400	1004	1506	1709	14.23	0.86

a Corrected diameter= external diameter minus four times the membrane thickness.

b

Acknowledgments

The writers wish to thank all the persons who contributed to the development of this experimental apparatus. The writers appreciate the aid of their corporate partners (Tractebel Engineering, Electricité de France, Eiffage Travaux Publics, CETE Sud-Est, LMS-Armines) as well as the financial grant to the ECHO project by the French National Agency for Research (ANR). Last but not least, the writers thank P. Denain and M. Schaba, technicians, and I.-K. Benazzouz and M. Khemakhem, graduate students, of the laboratory GeM at Ecole Centrale de Nantes for their conscientious work.


Emilien Azéma

Sandra Linero

⋆⋆ Nicolas Estrada

⋆⋆⋆ Arcesio Lizcano

Does modifying the particle size distribution of a granular material (i.e., material scalping) alters its shear strength?

By means of two dimensional contact dynamics simulations, we analyzed the effect of the particle size distribution (PSD) on the shear strength of granular materials composed of un-breakable disks. We modelled PSDs with a normalized beta function, which allows for building S-shaped gradation curves, such as those that typically occur in soils. We systematically controlled and varied the size span and the shape of the PSD, and found that the shear strength is independent both characteristics. This implies that PSD modification procedures such as material scalping (i.e., removing the smallest and/or largest particles in the sample) should not affect significantly the shear strength of the material composed of unbreakable discs. In order to explore the origins of the invariance of the shear strength with PSD, we analyzed the connectivity, force transmission, and friction mobilization in terms of anisotropies, finding that the constant shear strength is due to a subtle compensation of anisotropies.

Introduction

The study of the mechanical behavior of mine wastes containing particles of metric order is a challenging task because commercial testing devices can only accommodate samples composed of particles a few centimeters in dimension. The construction of larger devices for laboratory testing is not economical. To overcome the equipment size limitations, testing of coarse material is conducted on samples prepared by convenient modification of the particle size distribution (PSD). Parallel scaling (i.e., reducing the particles' size while keeping the PSD shape constant) or material scalping (i.e., removing the smallest and/or largest particles) are some of the technics used to prepare samples for laboratory testing [START_REF] Duncan | Soil Strength and Slope Stability[END_REF]. The tested sample may thus differ from the prototype in both the size of the particles and the form of the PSD.

Sample preparation procedures such as parallel scaling or material scalping are widely used in engineering practice. However, when and how these procedures affect the material's mechanical response (e.g., the shear strength) remains an open question. Important elements to be considered for tackling this question are: (i) the relation between particle size and fragmentation probability; (ii) the link between PSD, packing fraction, and connectivity; and (iii) the evolution of particle's shape and interaction's forces with particle size.

⋆ e-mail: emilien.azema@umontpellier.fr ⋆⋆ e-mail: slinero@srk.com.au ⋆⋆⋆ e-mail: n.estrada22@uniandes.edu.co ⋆⋆⋆⋆ e-mail: alizcano@srk.com A possible strategy for studying this problem theoretically is to analyze simplified systems (e.g., twodimensional systems, comprised of circular and unbreakable grains, with simple interaction laws such as Coulomb friction). This allows for singling out the effects of the PSD while leaving out the "noise" of other important factors. Discrete element methods are well suited for this type of simulations. Recently, this strategy has been used successfully to analyze the effect of the PSD on packing fraction and shear strength (see Refs. [START_REF] Wackenhut | [END_REF][3][4][5][6]), focusing on the comparison between bi-disperse and continuous PSDs, as well as on uniform distributions by volume fraction. Works exploring the effect of the PSD shape by means of discrete element simulations are rare (see Refs. [7,8]).

The main objective of this work was to analyze the effect of the particle size distribution (PSD) on the shear strength of granular materials composed of unbreakable disks. We build different PSDs using a normalized beta function, which allows for building S-shaped gradation curves, such as those that typically occur in soils. We varied two parameters independently over a broad parametric space, controlling both the size span and the shape of the PSD. We found that the shear strength is invariant with both the size span and the shape of the PSD. From a micromechanical viewpoint, we show that the shear strength remains constant for different PSDs thanks to a subtle compensation of anisotropies, specifically, contact and branch length anisotropies. The results support several hypotheses proposed recently and extend their validity to a larger family of PSDs.

Numerical procedures

The size span of the PSDs is defined by:

s = (d max -d min )/(d max + d min ), (1)
where d max and d min are the minimum and maximum particle diameters, respectively; s = 0 corresponds to a monodisperse packing whereas s = 1 corresponds to an infinitely large polydispersity. A β-function was use to model the shape of the PSDs. It is defined by:

β(x) = 1 B(a, b) x 0 t a-1 (1 -t) b-1 dt, (2)
where, a > 0 and b > 0 are the parameters of the distribution and B(a, b) = Γ(a)Γ/Γ(a + b), where Γ is the Gamma function. The PSD can then be described using the cumulate distribution function h(d) (also called grading curve in industrial contexts), given by In order to generate a set of particle diameters following a given distribution, the range d maxd min was divided into N c classes. Within each class, a uniform distribution by volume (area) of grains was considered. By construction, the number of particles increased with b, thus N c and the total number of particles N p were chosen in order to: (i) obtain a good fit with the theoretical curve and (ii) obtain a "reasonable" sample size, in terms of the number of particles. In the simulations presented here, the parameter a was fixed to 1 and the parameter b varied from 1 (a uniform distribution by volume fraction) to 5 (S-shaped gradation with a pronounced curvature). The total number of particles increased from 10, 000 for b = 1 to ∼ 70, 000 for b = 5. Figure 1 shows the theoretical PSDs, i.e., Eq.3, as well as the PSD of the simulated samples (points), for several combinations of s and b.

h(d) = β[x = d r (d); a, b], (3)
The simulations used the contact dynamics (CD) method [10], which assumes perfectly rigid particles interacting through mutual exclusion and Coulomb friction. We used the code LMGC90 which is a multipurpose software developed in the University of Montpellier, capable of modeling a collection of deformable or undeformable particles of various shapes by different algorithms. In each simulation, the particles were first compacted by isotropic compression inside a box. The gravity was set to 0. The friction coefficients between particles was set to µ = 0.4. The friction coefficient between the particles and the walls was set to µ w = 0, in order to obtain homogeneous and isotropic particles packing. Figure 2 shows zooms of the packings obtained for four different values of s and b at the end of isotropic compaction. Small crystalized regions can be observed for small values of s, which happens because most of the particles have the same diameter. In contrast, disorder increases with s.

The samples were then sheared in a biaxial configuration. A downward velocity v y was imposed on the upper wall while keeping a constant confining stress on the lateral walls. The strain rate v y /H was low, so that the test can be considered as quasi-static.

Macroscopic shear strength

The stress tensor σ can be evaluated from the simulation data as an average over all the contacts of the dyadic product of the contact force f c and the branch vector ℓ c : [10], where n c is the number density of contacts c and the average ... c runs over all contacts in a control volume. In 2D, we define the stress deviator as q = (σ 1σ 2 )/2 and the mean stress as p = (σ 1 + σ 2 )/2, where σ 1,2 are the principal stresses. The stress state is described by the mean stress p and the normalized shear stress q/p [START_REF] Mitchell | Fundamentals of Soil behavior[END_REF]. During shear, q/p increased initially to a peak value and then decreased to a nearly constant value (q/p) * , which we used to characterize the material's shear strength in the steady state. Figure 3 shows the evolution of (q/p) * as a function of s, for all values of b. It can be seen that, regardless of b, the shear strength is almost independent of s and close to 0.28. This has been previously observed for samples composed of circular [START_REF] Voivret | [END_REF] and polygonal [6] particles with a uniform distribution by volume fraction. Our results extend this counterintuitive observation to the family of Sshaped PSDs described by Eq. 3.

σ αβ = n c f c α ℓ c β c

Anisotropies of the contact and force network

The shear strength of dry granular materials is generally attributed to the buildup of an anisotropic structure during shear due to friction between the particles and as a result of steric effects depending on particle shape [13].This is well illustrated in Fig. 4, which shows two maps of normal forces in a portion of the sample for (s, b) = (0.2, 5) and (s, b) = (0.9, 5) after a large shear strain. The contact forces are represented by segments joining the particle centers, whose thickness is proportional to the force magnitude. Visual inspection reveals that the connectivity of the contact network varies drastically with s in terms of both the mean number of contacts per particle (i.e., coordination number) and the contacts' orientation and length.

A way to capture this microstructure is to consider the probability distribution P(n) of the contact normals n, which is generically nonuniform. In two dimensions, the unit vector n is described by a single angle θ, the orientation of the contact normal. The probability density function P(θ) of contact normals provides detailed statistical information about the fabric. It is π-periodic in the absence of an intrinsic polarity for n. At the lowest order, we have [14,15]

P(θ) = {1 + a c cos 2(θ -θ σ )}/2π, ( 4
)
where θ σ is the major principal stress direction (θ σ = π/2), and a c is the contact anisotropy. This description of the granular microstructure can be enriched by considering also the angular average of normal branch length and forces:

         ℓ n (θ) = ℓ {1 + a ℓn cos 2(θ -θ σ )} f n (θ) = f n {1 + a f n cos 2(θ -θ σ )} f t (θ) = f n a f t sin 2(θ -θ σ ), (5)
where f n and ℓ are the mean force and the mean branch length, and a ln , a f n , and a f t the branch length, normal force and tangential forces anisotropies. The anisotropies a c , a ℓn , a f n , and a f t are interesting descriptors of the granular microstructure and force transmission properties, because they underlie the different microscopic origins of shear strength. The general expression of the stress tensor leads to the following simple relation [14,15]:

q p ≃ 1 2 (a c + a ln + a f n + a f t ). ( 6
)
This expression is very useful, since it allows for describing the shear strength as the addition of several microstructural anisotropies. It is worth noting that the shear strength can be understood as the material's ability to bear anisotropic stress. Figure 5(a) shows the evolution of a c and a ℓn as functions of s, for all values of b. For small values of b (i.e., b = 1 and 2), it can be seen that a c decreases with s, indicating that the anisotropy of the contact network is decreasing. This happens because, as s increases, the coordination number also increases, and the particles are surrounded by an increasing number of contacts. On the other side, it can be seen that a ℓn increases with s, indicating that the branch lengths of the contacts that are oriented along the principal stress direction tend to be longer. These results show that the shear strength remains constant due to a compensation of anisotropies: the contact network becomes less anisotropic but the force chains along the principal stress direction tend to be captured by the larger particles. Lastly, for large values of b (i.e., b = 4 and 5), both a c and a ℓn remain approximately constant with s and b.

Figure 5(b) shows the evolution of a f n and a f t as functions of s for all values of b. In contrast to contact and branch length anisotropies, it can be seen that a f n and a f t are approximately constant with s and b. This means that the anisotropy of force chains as well as the degree of friction mobilization are not affected by changing the PSD.

Conclusions

We investigated the joint effect of the size span and shape of the particle size distribution (PSD) on the shear strength of packings composed of unbreakable disks, by means of contact dynamics simulations. For this, we independently varied the size span and shape of the distribution across a broad parametric space, and investigated the micromechanical origin of the shear strength for each simulated PSD.

On the first hand, we showed that the shear strength is independent of both the size span and the shape of the PSD. This has already been observed, also by means of discrete element simulations, for samples with uniform distributions by volume fraction [6,[START_REF] Voivret | [END_REF]. In these cases, only the PSD size span was varied in samples with uniform distributions by volume fraction. Our results extend the validity of this result by showing that the shear strength is invariant, not only with the size span, but also with the shape of the PSD, at least for those shapes that can be described by Eq. 3. These findings are in agreement with those reported recently in [8] for particle distributions described by a power law.

On the second hand, we investigated the origins of the shear strength by analyzing the contact, branch length, and force anisotropies. It was shown that the shear strength remains constant due to a compensation between the contact and branch length anisotropies, while force anisotropy and friction mobilization remain constant. As shown previously [6,[START_REF] Voivret | [END_REF], these results confirm that, as the size span of the PSD increases, the system is more and more connected and larger forces tend to be captured by larger particles.

The results suggest that the differences observed on the laboratory, in the mechanical response of prototype soil samples and its scaled representations, can be attributed to factors other than the PSD changes introduced with the scaling procedure. It should be noted that this investigation considers that particle characteristics (e.g., shape, strength, and interaction laws) do not depend on particle size. In real materials, several important characteristics, such as grain fragmentation probability, grain shape, and interaction forces, depend notably on particle size. As a consequence, modifying the PSD of a real soil change also the nature of the particles that compose the sample, which can provoke appreciable changes in the mechanical response of the granular system.

  where d r (d) = (dd min )/(d maxd min ) is the the reduced diameter. As shown by Voivret et al. in Ref. [9], this model allows for controlling both the size span and the shape of the PSD, including reference distributions such as monodisperse, power law, and S-shaped PSD curves. In our simulations, d max was 0.15 m and the size span was varied in the range s ∈ [0.2, 0.4, 0.6, 0.8, 0.9].

1 Figure 1 .

 11 Figure 1. Theoretical and simulated particle size distributions for several combinations of parameters s and b.

Figure 2 .

 2 Figure 2. Zoom for (s = 0.2, b = 1)(a), (s = 0.2, b = 5)(b), (s = 0.9, b = 1)(c) and (s = 0.9, b = 5)(d) at the end of the isotropic compression. Floating particles (i.e., particles with one or no contacts) are shown in white.

Figure 3 .

 3 Figure 3. Shear strength (q/p) * in the steady state as function of s, for all values of b.

Figure 4 .

 4 Figure 4. Maps of normal forces in a portion of the sample for (s = 0.2, b = 5)(a) and (s = 0.9, b = 5)(b) after a large shear strain. Line thickness is proportional to the normal force.

Figure 5 .

 5 Figure 5. (a) Contact and branch length anisotropies, a c and a ℓn respectively, and (b) Normal and tangential force anisotropies, a f n and a f t respectively, all averaged in the steady state, as functions of s, for all values of b.


Emilien Azéma

Nicolas Estrada

⋆⋆ Itthichai Preechawuttipong

⋆⋆⋆ Jean-Yves Delenne

Systematic description of the effect of particle shape on thestrength properties of granular media

In this paper, we explore numerically the effect of particle shape on the mechanical behavior of sheared granular packings. In the framework of the Contact Dynamic (CD) Method, we model angular shape as irregular polyhedral particles, non-convex shape as regular aggregates of four overlapping spheres, elongated shape as rounded cap rectangles and platy shape as square-plates. Binary granular mixture consisting of disks and elongated particles are also considered. For each above situations, the number of face of polyhedral particles, the overlap of spheres, the aspect ratio of elongated and platy particles, are systematically varied from spheres to very angular, non-convex, elongated and platy shapes. The level of homogeneity of binary mixture varies from homogenous packing to fully segregated packings. Our numerical results suggest that the effects of shape parameters are nonlinear and counterintuitive. We show that the shear strength increases as shape deviate from spherical shape. But, for angular shapes it first increases up to a maximum value and then saturates to a constant value as the particles become more angular. For mixture of two shapes, the strength increases with respect of the increase of the proportion of elongated particles, but surprisingly it is independent with the level of homogeneity of the mixture. A detailed analysis of the contact network topology, evidence that various contact types contribute differently to stress transmission at the micro-scale.

Introduction

In many particulate materials found in nature and industry, scientists and engineers need to quantify the effect of complex particle shapes. This is of major importance in the context of civil engineering and powder technology, where most processes need to be optimized or revised following the dramatic degradation of natural resources. Non-convex particles can be found in metallurgical and sintered powders, angular shaped particles are common in rocks and soils, whereas elongated and platy particles may occur in pharmaceutical and food products. Mixture of shapes is also used in order to reinforce soils or in civil engineering structures.

The first works dealing with particles shape parameters are not new. Indeed authors such Wadell in 1932 [1] or Krumbein Pettijohn 1938 [START_REF] Krumbrein | Manual of sedimentary petrography[END_REF] are the first to have developed methods for characterizing the shape of the particles.

⋆ e-mail: emilien.azema@umontpellier.fr ⋆⋆ e-mail: n.estrada22@uniandes.edu.co ⋆⋆⋆ e-mail: itthichai.p@cmu.ac.th ⋆⋆⋆⋆ e-mail: jean-yves.delenne@umontpellier.fr † e-mail: franck.radjai@umontpellier.fr These methods were primarily dedicated to the characterization of particles in the field of Sedimentology. Without going into details of the classification techniques, the particles can be grouped into three categories according to their angular, non-convex or elongation/platyness (also called sphericity) nature; see Figure 1(A).

Nevertheless, even if an increasing number of experimental and numerical studies have shown that the grain shape considerably affects the quasistatic behavior of granular materials [START_REF] Mair | [END_REF][6][START_REF] Mitchell | Fundamentals of Soil behavior[END_REF][START_REF] Sacanna | [END_REF][9], systematic and quantitative investigations of shape or mixture of shape effects are still elusive. One of the difficulties is that the shape parameters mentioned above need to be defined conveniently in order to be able to generate particle shapes with continuouslyvariable shape parameters. In addition, particle shape is difficult to control experimentally.

A possible way for studying this problem is to analyze simplified systems with "idealized non-spherical" particles. With this respect, Discrete Element Methods are well suited for this type of simulations even if introducing particle shape in numerical simulations give raise to various technical difficulties such as contact detection and force calculation between particles of arbitrary shape.

Typically, angular shape represents a property of polygonal particles in 2D and polyhedral particles in 3D. Elongation and platyness can be analyzed throughout the aspect ratio parameter. Along the same line, interlocking is a basic feature of non-convex particles and can be represented by aggregate composed of spheres. Thus, the main objective of this work was to analyze numerically the effect of particle shape on the strength properties by varying systematically the shape from spheres to angular, non-convex and elongated/platy shape, as well as by considering mixture of shape.

Numerical procedures 2.1 Particle shape parameters

The level of sphericity can be defined by the three ratios λ 1 = L p /h p , λ 2 = l p /h p and λ 3 = L p /l p , where L p , l p and h p are the principal lengths axis ; see Fig. 1(B-c). Consequently, spherical particles are such λ 1 = λ 2 = λ 3 = 1, In 2D, elongated particles are modeled as a juxtaposition of two half-disks of radius r and one rectangle of length L and width 2r as shown in Fig. 1(B-a). The aspect ratio λ 3 = (L + 2r)/(2r) and λ 1 = λ 2 = 0. This particles are named "Rounded-Cap-Rectangle (RCR)". In the same way, in 3D platy particles are modeled as square plates with rounded edges, built as a sphero-polyhedra resulting from sweeping a sphere around a polyhedron. In this case the diagonal of the plate is equal to R and the aspect ratio is given by λ 3 .

Non-convex particles are regular aggregates of 4fold rotational symmetry composed of four overlapping spheres of the same radius r as shown in Fig. 1(B-b). In this case λ 1 = λ 2 = λ 3 = 1. This shape can be easily characterized by the ratio λ * = l/2r, where l is the distance between the centers of spheres. This parameter varies from 0, corresponding to a sphere, to √ 3/2 corresponding to an aggregate where three coplanar spheres intersect at a single point, so that the radius R of the circumscribing sphere is given by R = r(1 + λ * √ 3/2). Elongated, non-convex and platy shape may also be characterized by their degree of distortion η from a perfectly spherical shape, defined as η = ∆R/R, with ∆R = R -R ′ and R ′ the radius of the inscribed circle [10].

We define the angularity ⟨α⟩ of a polyhedral particle as the mean exterior angle between its touching faces. In order to eliminate the effect of particle eccentricity, a strict procedure is implemented in order to isolate and control precisely the shape. First, a set of n v vertices is randomly generated on a unit sphere. The convex hull of these points is created by associating three vertices for each face. Secondly, we impose that all ratios {λ i } i∈1,3 to be greater than 0.9 ; see Fig. 1(B-c). This means that, numerically, the sets of vertices are generated until this condition is satisfied. In this simple way, for a given aspect ratio, we can control the angularity of the particles with a single continuouslyvariable shape parameter depending only to the number of faces n f .

Packing construction

We prepared different packings composed of large number of angular or non-convex or platy particles with n f ∈ [596, .., 8] or η ∈ [0, 1[, respectively. In order to avoid long-range ordering, we introduce a size polydispersity by varying the circumradius R of the particles in the range [r min , 2r min ], where r min is the minimum radius with a uniform distribution by volume fractions.

In order to analyse the effect of mixture by particle shape, we built also various packings composed with both discs and rcr particles with η = 0.7 in 2D. The mixture ratio parameter α is the proportion of elongated particles. We varied α from 0 for a packing composed of only disks to 1 for packing composed of only elongated particles. Mixture homogeneity is measured throughout Lacey mixture indice, which is directly proportional to the standard deviation from a given mean property of one of the constituents of a mixture. Knowing the fully segregated state where all elongated particles belongs in a band of thickness ϵ located at the center of the packing, and homogenous state, the Lacey parameter M * is given by

M * = S 2 r -S 2 S 2 r -S 2 0
, where S r and S 0 are the values of the variance of the position of elongated particles calculated from the homogenous and fully segregated systems, respectively. Hence, numerically we constructed various packings with M * varies from 0, for fully segregated system, to 1, for fully homogeneous mixture. Figures 2 displays snapshots of the packings for several values of M * and α at the end of isotropic compaction (see below).

By means of contact dynamic simulations (CD) [11] the particles are first compacted by isotropic compression inside a box. The gravity is set to 0 as well as friction coefficient µ and µ w between particles and with the walls. A the end of isotropic compression, the connectivity number, defined as the mean number of contacts per particle (see Sec. 4) is nearly 12 for all packings which is in full agreement with the isostatic nature of our initial packings. These samples are then used as initial configuration for biaxial, respectively triaxial, quasi-static compression tests, for which friction between particles is set to 0.4. In granular systems, the stress tensor σ is given by

Strength properties

(q/p) * (q/p) * (q/p) * M * (q/p) * (a) (b) (c) (d)
σ αβ = n c ⟨ f c α ℓ c β ⟩ c
, where n c is the number density of contacts c, f c and ℓ c refer to force and branch vector at the contact c, and the average ⟨...⟩ c runs over all contacts in a control volume [5]. In 3D with axial symmetry we define the stress deviator q = (σ 1σ 3 )/3 and the mean stress p = (σ 1 + σ 2 + σ 3 )/3, where σ 1,2,3 are the principal stress values. In 2D, we set q = (σ 1 -σ 2 )/2 and p = (σ 1 +σ 2 )/2. For our system of perfectly rigid particles, the stress state is characterized by the mean stress p and the normalized shear stress q/p [START_REF] Mitchell | Fundamentals of Soil behavior[END_REF]. During shear, the shear stress jumps initially to a high value before decreasing to a nearly constant value in the steady state. The steady-state shear stress (q/p) * characterizes the shear strength of the material.

Figure 3 shows the evolution of (q/p) * averaged in the steady state as a function of particle platyness/elongation (a), non-convexity (b), angularity (c), and as function of packing homogeneity for all values of mixture ratio (d). A general observation is that the shear strength increases as shape deviate from spherical shape and also as the proportion of non-spherical particle increases too. But surpris-ingly, the shear strength saturate at larger values of particles angularity and he remains independent with packing homogeneity. In this last case, this means that homogeneous mixtures have the same strength properties as segregated packings of the two particle shapes.

Particle connectivity and origins of strength

The main effect of particle shape is to allow for multiple contacts between particles as shown, for instance, in Fig. 4 for mixture of two shapes and non-convex aggregate. For non-convex particles as those used in this study we can identify seven different types of contacts : 1) "simple" contacts (s), 2) "double-simple" contacts (ds) defined as two simple contacts between two pairs of spheres, 3) " double contacts" (d) defined as two contacts between one sphere of one aggregate and two spheres belonging on the other aggregate, 4) "triple" contacts (t) defined as a combination of simple and double contacts or one sphere of one aggregate and three spheres of another aggregate or three simple contacts, 5) "quadruple" contacts (q) defined as a combination of two double contacts, and 6) five or six contacts with a negligible proportion (below 1%) compared to other contact types. With polyhedral particles, contacts can be grouped in terms of face-face( f f ), face-vertex ( f v), face-edge ( f e) and edge-edge (ee) contacts. With elongated particles we have cap-cap (cc), capside (cs) and side-side (ss) contacts. With mixture of two shapes we can consider contact between disks (dd), rcr (rr)and disk-rcr (dr) particles. To identify the impact of each contact type on the shear strength, we proceed by additive decomposition of the stress tensor according to contact type by σ = γ σ γ , where γ stands alternatively for { f f, f v, f e, ee} for polyhedral particles, {s, d, sd, t, q} for non-convex particles, {cs, cc, cs} for elongated particles and for {dd, dr, rr} contacts for mixture of two shapes. The corresponding stress deviators q γ , are then calculated and normalized by the mean pressure p such (q/p) * = γ q γ /p. The corresponding normalized stress deviators q γ /p averaged in the residual state are shown in Fig. 5 as a function of shape parameters, for packings of (a) elongated, (b) non-convex and (c) angular particles, as well as for mixture of two shapes as a function of mixing index M * for packing composed by the same proportion of disk and rcr. We see that contact types contribute differently on strength properties as function of shapes. For instance, the rapid increase of the shear strength with angularity is due to a rapid increase of strength support by f f contact at low angularity, but the plateau observed for langer angularity is due to a rapid fall off of q f f /p compensated by a rapid increase of q f e /p and q f e /p. For elongated particles (q/p) * follow q cs /p. And, for mixture of two shapes, we see that q dd /p is nearly constant, whereas remarkably q dr /p follows a trend opposite to that of q rr /p so that the decrease of q dr /p with M * is exactly compensated by the increase of q rr /p, which explains the independence of (q/p) * with respect to M * .

Conclusions

In this paper, a systematic numerical analysis of the effect of particle shape properties on the quasistatic rheology of sheared granular media, as well as packings composed of mixtures of two shapes was presented in the framework of Contact Dynamics simulations. The shape of the particles is systematically varied from spherical shape (as referential shape) to angular, non-convex and elongated or platy shape. We considered also binary mixture of disks and elongated particles characterized by the mixture ratio which varies from 0 for a packing composed of only disks to 1 for a packing composed of only rcr particles, and with an homogeneity parameter M * which varies from 0 for an homogeneous packing to 1 for a fully segregated packing.

A major result of this work is that the particle shape effects are nonlinear and counterintuitive. We show that the shear strength increases as shape deviate from spherical shape. But, for angular shapes it first increases up to a maximum value and then saturates to a constant value as the particles become more angular. For mixture of two shapes, the strength increases with respect of the increase of the proportion of elongated particles, but surprisingly it is independent with the level of homogeneity of the mixture. At the micro-scale we show that various contact types, depending on the specificities of each shape, contribute differently to stress transmission. A general observation is that multiple contact are at the origins of the increase of strength.

In this investigation, the shape was systematically varied along "one" direction. Indeed, we used convex and smooth particles when varying elongation, convex and spherical particles for studying angularity, and spherical and smooth particles for analyzing the effect of nonconvexity. However, it should be interesting to vary the shape considering combined effect of elongation and angularity, or elongation and non-convexity, or angularity and non-convexity. In the same way, by analogy with polydisperse packing by particles size, it will be very instructive to consider mixture of various shape, which imply the definition of higher order parameters such particle shape distribution function in order to systematically investigate such effects.

Figure 1 .

 1 Figure 1. (A) 3D-diagram of particles shapes following Santamarina [3]. (B-a) Platy (elongated in 2D), (B-b) Non-convex and (B-c) Angular shapes .

Figure 2 .

 2 Figure 2. Examples of the generated packings at the initial state, here for binary mixture with α = 0.5 and M * = 0 (a) and M * = 1(b). The aspect ratio is fixed at η = 0.7.

Figure 3 .

 3 Figure 3. Normalized shear strength averaged in the steady state as a function of particle platyness/elongation (a), non-convexity (b), angularity (c), and as function of packing homogeneity for all values of mixture ratio (d).

Figure 4 .

 4 Figure 4. Snapshot of normal forces in (a) packing composed of non-convex particles (here η = 0.6), and (b) packings composed of mixture of two shape for homogeneous (up) and segregated (down) packings. The forces are plotted in different colors depending on contact types. The inset shows contacts type for non-convex aggregate whereas in mixture we can distinguish disks-disks, discs-elongated and elongated-elongated contacts.

Figure 5 .

 5 Figure 5. Normalized shear strength as a function of particle shape parameter or Mixing index, for (a) cs, cc and cs contacts in packing composed of elongated particles, (b) s, d, sd, t and q contacts in packing composed of non-convex particles, (c) f f , f v, f e and ee in packing composed of non-convex particles, and (d), dd, dr and rr contacts for mixture of two shapes composed by the same proportion of disk and rcr.