Carlos Ovalle

Christophe Dano

Pierre-Yves Hicher

Mónica Cisternas

Experimental framework for evaluating the mechanical behavior of dry and wet crushable granular

Keywords: crushable granular materials, compressibility, grain-size distribution, creep, stress relaxation matériaux granulaires broyables, compressibilité, granulométrie, fluage, relaxation des contraintes

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Introduction

Granular materials are present in many civil engineering problems, such as foundations on sands or gravels, railway ballasts and rockfills embankments. It is well known that their mechanical behavior is a function of the properties of the granular packing and its water content, as well as of the mechanical and geometrical characteristics of individual grains. Under given conditions, intergranular contact forces can exceed particle strength and a fraction of the particles will crush, changing the grain-size distribution (GSD) towards a more polydisperse packing, which creates a denser arrangement for a given stress condition [START_REF] Biarez | Influence de la granulométrie et de son évolution par ruptures de grains sur le comportement mécanique de matériaux granulaires[END_REF][START_REF] Li | Influence of grain size distribution on critical states of granular materials[END_REF]. After crushing, consequently compressibility will increase and dilatancy under shear become reduced, as well as the peak shear strength [START_REF] Vesic | Behaviour of granular materials under high stresses[END_REF][START_REF] Marachi | Evaluation of properties of rockfill materials[END_REF][START_REF] Marsal | Mechanical properties of rockfill dams[END_REF][START_REF] Coop | The influence of pore water on the mechanics of granular soils[END_REF][START_REF] Lee | The intrinsic behaviour of a decomposed granite soil[END_REF][START_REF] Lade | Significance of particle crushing in granular materials[END_REF]Ovalle et al. 2013b). In general, grain crushing occurs under high stresses, e.g., at pile tips or at the base of high dams, or under quite lower stresses in materials composed of relatively weak grains. The amount of grain crushing is normally analyzed in terms of the breakage ratio B r [START_REF] Marsal | Mechanical properties of rockfill dams[END_REF][START_REF] Hardin | Crushing of soil particles[END_REF][START_REF] Lade | Significance of particle crushing in granular materials[END_REF]Einav 2007a). In this work, we use a recent B r definition proposed by Einav (2007a), obtained as the ratio between (i) the area flanked by the initial GSD and any intermediate GSD after crushing (B t in Fig. 1) and (ii) the initial GSD and an assumed ultimate fractal GSD (B p in Fig. 1, assuming a fractal dimension D = 2.5 for a cumulated GSD given by G(d) = (d/d max ) 3-D , where d is the grain diameter and d max is the maximum grain diameter).

The experiments by [START_REF] Miura | Particle-crushing of a decomposed granite soil under shear stresses[END_REF], [START_REF] Hu | A constitutive model for granular materials considering grain breakage[END_REF], and Ovalle et al. (2013a) have shown that the amount of grain crushing is related to the total amount of plastic work. These findings have led to developing phenomenological constitutive models based on the general framework of uncrushable soils (i.e., the critical state theory). Accordingly, extra plastic strain after crushing is given by an inverse relation between the critical void ratio and the breakage ratio [START_REF] Daouadji | An elastoplastic model for granular materials taking into account grain breakage[END_REF][START_REF] Russell | A bounding surface plasticity model for sands exhibiting particle crushing[END_REF][START_REF] Muir Wood | Particle crushing and deformation behaviour[END_REF][START_REF] Daouadji | An enhanced constitutive model for crushable granular materials[END_REF][START_REF] Hu | A constitutive model for granular materials considering grain breakage[END_REF]. The alternative method proposed by [START_REF] Mcdowell | On the micromechanics of crushable aggregates[END_REF] linking the plastic work to the increment of the surface fracture energy on crushed grains has been confronted with experiments by [START_REF] Russell | A compression line for soils with evolving particle and pore size distributions due to particle crushing[END_REF] and Ovalle et al. (2013a). On the other hand, Einav (2007aEinav ( , 2007b) ) used the breakage ratio as an internal energy parameter, which opened the way to the prediction of grading and its link to the mechanical response.

In fact, the source of grain crushing comes from the breakage mechanics of individual grains at the micromechanical scale. In other words, it depends on the mechanical properties of grains and their loading conditions at intergranular contacts. Therefore, multi-scale links are conducive to predict the response of the granular assembly. Recently, the multi-scale approaches developed take into account physical parameters such as the tensile particle strength, grain-size effects, and the fracture surface energy of brittle solid grains [START_REF] Mcdowell | On the micromechanics of crushable aggregates[END_REF][START_REF] Frossard | Rockfill shear strength evaluation: a rational method based on size effects[END_REF][START_REF] Ovalle | Contribution a `l'étude de la rupture des grains dans les matériaux granulaires[END_REF]Ovalle et al. 2013c). A close look at the microscopic scale (i.e., microcracks or flaws inside individual particles) gives a physical explanation of this phenomenon, as described by classical brittle fracture mechanics. [START_REF] Griffith | The phenomena of rupture and flow in solids[END_REF] proposed that fracture in perfectly linear elastic brittle materials occurs through the propagation of existing cracks. He demonstrated that the induced tensile stress () depends only on the size of the crack (a) and on intrinsic parameters. Later, [START_REF] Irwin | Analysis of stresses and strains near the end of a crack traversing a plate[END_REF] developed the notion of stress intensity factor under uniform axial stress (K I = ͌(a)) and fracture toughness (K IC for fracture in mode I). The term K IC can be defined as the material resistance to fracture by crack propagation. Hence, a crack begins to grow when K I reaches the limit K IC .

Exposed to rain or dam leakage, for instance, granular soils undergo significant changes in water content, from an almost dry state to complete saturation. For crushable materials, an increase in relative humidity will cause more particle crushing for a given stress condition [START_REF] Marsal | Mechanical properties of rockfill dams[END_REF][START_REF] Lee | The intrinsic behaviour of a decomposed granite soil[END_REF][START_REF] Oldecop | Suction effects on rockfill compressibility[END_REF]Ovalle et al. 2013a). Consequently, the mechanical behavior of the granular material is affected as explained before. At the micromechanical scale, variations in the material water content will affect the K IC [START_REF] Oldecop | Theoretical investigation of the time dependent behavior of rockfill[END_REF]. This issue has been discussed by many authors who found that the value of K IC depends on the degree of saturation and on the physicochemical composition of both solid material and liquid. [START_REF] Westwood | Tewksbury lecture: Control and application of environmentsensitive fracture processes[END_REF], [START_REF] Atkinson | A fracture mechanics study of subcritical tensile cracking of quartz in wet environments[END_REF][START_REF] Atkinson | Subcritical crack propagation in rocks: theory, experimental results and applications[END_REF], and [START_REF] Dunning | The effects of aqueous chemical environments on crack propagation in quartz[END_REF], for example, consider that the problem is quite complex because the degradation of the structural integrity of geological materials is mainly due to the corrosive attacks of the environment. The ef-fects are essentially the adsorption of elements and mineral dilution from the environment around the microcrack tip, which causes a modification of the fracture surface energy, and the electrostatic interaction between the humid environment and the material. Moreover, if the material water content increases, the capillary suction will decrease at the microcrack tips within the grains. Therefore, the effective stress against crack opening also decreases and the mechanism is affected: crack propagation velocity increases and both particle strength and K IC decrease [START_REF] Oldecop | Theoretical investigation of the time dependent behavior of rockfill[END_REF]. Therefore, the higher the water content in the material voids, the lower the K IC value and the higher the particle crushing. However, rather than the material pore water into the entire assembly, what is relevant is how much water is retained inside rock microcracks forming the grains.

A number of models for crushable granular materials including the effect of the water content have been suggested. For instance, by using the results of compression tests on rockfills under relative humidity control, Oldecop andAlonso (2001, 2003) and [START_REF] Chávez | A constitutive model for crushed granular aggregates which includes suction effects[END_REF] developed a phenomenological constitutive model based on work hardening plasticity and critical state theory, which incorporated the effect of suction. Considering the loading collapse curve concept from unsaturated soil mechanics, their model gives numerical results based on the role of suction that is related to the relative humidity in the material voids. Similarly, [START_REF] Oldecop | Theoretical investigation of the time dependent behavior of rockfill[END_REF] suggested a conceptual model to explain the creep behavior in rockfills based on the influence of suction on K IC and on an inverse relation between K IC and the crack propagation velocity. On the other hand, [START_REF] Buscarnera | The yielding of brittle unsaturated granular soils[END_REF] proposed a model based on the thermo-mechanical approach proposed by Einav (2007aEinav ( , 2007b)), where the loading collapse curve is linked to the energy balance. The model links the assumption of energy dissipation by particle breakage and the capillary theory for solid-fluid interaction. Finally, [START_REF] Bauer | Hypoplastic modelling of moisture-sensitive weathered rockfill materials[END_REF] developed a hypoplastic model based on a solid hardness concept where the mechanical response is simply enclosed between a compression curve for the dry condition and a more compressible one for the saturated material.

Despite the many advances made in understanding the phenomenon of grain crushing, the role of the breakage ratio as a governing parameter for the effect of the water content on crushable soils is still not very clear. Experimental data on the subject are generally not available. We, therefore, carried out an extensive experimental program of the effects of flooding in crushable granular material by running a series of tests including different stress paths, stress magnitudes, with stages of stress relaxation and creep, and by checking test repeatability to obtain a consis- tent framework based on grading evolution. The main objectives are to study the influence of stress path, water content and time, as well as the link between the mechanical behavior and the evolution of the breakage ratio. Then, based on experimental evidences, we suggest an integrated analysis of the material behavior with the perspective of enhancing constitutive models based on grading, including the effects of water content and time.

Tested material

The granular material tested is an angular sand obtained from the grinding of a quartzite shale rock from the Trois Vallées quarry, in the north of France. Samples were prepared with an initial uniform GSD sieved between 2 and 2.5 mm. Figure 2 presents a photo of the material before the test. It can be seen that the particles are mainly angular and flat, which makes them vulnerable to crushing. Solid grains have a volumetric weight of 27 kN/m 3 . Figure 3 presents electron microscope scans of fine fractions of crushed grain fragments sieved after a test. For finer grains in Fig. 3a, it clearly appears that the shale rock is composed of thin sheets. Once crushing occurs, a significant amount of fines is produced essentially by small pieces of detached sheets. The analysis of X-ray diffraction (XRD) of a fine fraction of the material (grain sizes finer than 0.08 mm) shows that the powder is composed mainly of oxygen and silicon, with also significant amounts of aluminum and iron. The XRD measurement results show the presence of three main phases: 64% of quartz (SiO 2 ), 26% of albite (NaAlSi 3 O 8 ), and 10% of sodium aluminium silicate (Na 6 Al 6 Si 10 O 32 ).

To avoid crushing during the preparation, loose samples of dry material were prepared in cylindrical moulds by pluviation at low height. The material was previously air dried for at least 1 week in a constant clime chamber at 20 °C and 60% of relative humidity, where all tests were finally carried out. Here, we consider the material as dry after air drying, even if a water content of 0.5% was measured (obtained by weight changes after 24 h of oven drying at 105 °C). Under this condition, no capillary bonds were observed between particles and the apparent cohesion was negligible. As explained below, some samples were saturated or flooded with demineralized water after a phase of compression. Water was slowly added to dry samples through a pipe connected to its base, a constant water level being maintained slightly above the upper level of the sample. Due to the low water content of the specimen after drying, we simply used the notation of "dry" and "saturated" for the two opposite initial states of the specimens prior to loading. However, such observation applies to the entire sand sample, rather than into microcrack within the grains. After saturation or flooding, no chemical reaction or mineral dilution of wetted sand grains was observed. After each test, the samples were dried for 24 h in a 105 °C oven and then accurately sieved between 2.5 and 0.08 mm.

Experimental results

Triaxial and isotropic compression tests

Loose samples of dry material were prepared at an average void ratio (volume of voids/volume of solids) of 0.937 ± 0.035 in a cylindrical mould 110 mm high and 70 mm in diameter, using a 1 mm thick latex membrane. See tests D1 to F8 in Table 1, where code D stands for dry, S for saturated, and F for flooded. Strain controlled isotropically consolidated drained triaxial compression tests at effective confining pressures ( 3 ) of 0.4 and 0.8 MPa were performed using a vertical displacement rate of 0.2 mm/min (i.e., a vertical strain rate of about 2%/min). Since we considered dry and saturated cases (i.e., referring to macropores between grains) and drained tests on rather coarse uniform sand, no suction or excess pore pressure were measured during testing and the definition of the effective stress used here coincides with the classical Terzaghi's principle. The volume changes of the samples were measured through the confinement water volume control system. Thus, the triaxial cell deformation under isotropic water pressure was firstly calibrated without the sand sample to isolate the volumetric strain of the cell from the sample one.

Some tests were flooded after the isotropic consolidation phase and others after a triaxial compression up to 7% to 12% of axial strain ( a ) (see tests F4, F5, F6, and F8 in Table 1). First, the vertical strain was stopped at a given level and then flooding was carried out by imposing a water flow through the sample from the bottom to the top face by using a volume pressure controlled system. Back pressure was carefully maintained lower than the water level between the device and the top of the sample to avoid any change in the confining pressure. Once water started to appear on the top pipe connection of the sample, the water flow was stopped. This process took approximately 5 min. Then, a period of deviatoric stress (q = 1 -3 ) relaxation of 1-2 h was applied (i.e., at both constant a and 3 ), before a further increase of the axial strain was imposed. The final a for all triaxial tests was 20%. Figure 4 shows the time evolution of the void ratio, e, and of the deviatoric stress, q, during the tests. It can be seen in tests F4, F6, and F8 that the relaxation phase leads to a significant drop of q. Similarly, there is also a slight decrease in the void ratio during the relaxation phase, due to creep at constant 3 .

To study separately the deviatoric stress relaxation in the dry material and the effect of flooding, two stages of relaxation were applied in test F5. Firstly, the test was conducted up to a = 10% and then stopped for 1.5 h of stress relaxation. Then, the sample was flooded and an extra hour of relaxation was applied. Figure 5 shows the evolution in time of the void ratio and of the deviatoric stress for test F5. It can be seen that the relaxation of the dry material is in the order of ⌬q = -0.23 MPa and, after the stabilization of q in the dry condition, a further decrease of q of about 0.4 MPa occurred once the sample was flooded. That is to say, 1/3 of ⌬q corresponds to the behavior of the dry material and the rest is added after flooding, possibly because grain crushing was triggered in the presence of water.

Similarly, dry and flooded drained isotropic compression tests at 3 of 0.4 and 0.8 MPa were carried out using the same sample preparation method as for the triaxial tests. The average void ratio was 0.980 ± 0.088 (see tests D9 to F13 in Table 1). Isotropic tests were performed under stress controlled conditions. For a given 3 , the dry sample was firstly compressed and then allowed to creep for 1 h at constant stress (tests D9 and D10). In tests F11, F12, and F13, the samples were flooded and kept for an extra hour of creep.

Figure 6 shows the stress-strain response after the triaxial tests in terms of the stress ratio q/p= (where p= = ( 1 + 2 3 )/3) and the void ratio, and Table 1 presents the breakage ratio. It can be seen that, regardless of the point of flooding, the material is more compress- ible after flooding. Moreover, all samples flooded at a given 3 in a triaxial test joined approximately the same volumetric response and almost the same B r (i.e., the same final GSD). For instance, see B r and the e-p= plane for tests F3 to F6 in Table 1 and Fig. 6, respectively.

Oedometer tests

Loose samples of dry material were prepared at an average void ratio of 1.018 ± 0.05 in a cylindrical oedometric device of 19 mm high and 70 mm of diameter (see tests D14 to F25 in Table 1). Stress controlled oedometric compression tests were performed using the following effective vertical stress levels: v = 0.15, 0.40, 0.60, 0.85, 1.15, 1.50, and 2.10 MPa. To reach a given stress level, each aforementioned stress value was allowed 1 h of strain stabilization. Thus, even if the full stabilization was not attained after 1 h, this time of creep allowed comparing the results with the previous triaxial and isotropic tests. To obtain the evolution of the GSD during crushing, the samples were unloaded at various maximum effective stress ( v max ), as shown in Table 1. The tests were performed for three sample conditions: dry, saturated, and flooded. In this case, a saturated material stands for a sample that was initially flooded, i.e., before the initial loading stage of v = 0.15 MPa.

Figure 7 presents the oedometric compression curves, where it can be seen that a good repeatability was obtained. Hence, two main compression curves can be identified, one for the dry condition and the other for the saturated and flooded conditions. For example, tests F22, F23, F24, and F25 were flooded at v of 0.40, 0.40, 0.85, and 1.5 MPa, respectively. It can be seen that these tests followed the dry compression curve before flooding, and after the addition of water they behaved in the same way as the saturated material. Then, after flooding and 1 h of creep in tests F22 and F25, v was increased up to 2.10 MPa and the compression curve followed the saturated material response. Consequently, regardless of the point of flooding, the oedometric compression curve is unique for saturated and flooded samples. This result agrees with the experimental evidence from the literature [START_REF] Oldecop | Suction effects on rockfill compressibility[END_REF] and also with our own triaxial tests results. Figure 8 presents the GSD only for oedometer tests at v max = 2.1 MPa. These results clearly show that after flooding in test F25, the material reaches almost the same GSD as the one of the saturated sample (test S21), with more grain crushing compared to the dry case (test D18).

Creep in oedometer tests

To study the creep behavior of the material, a second series of oedometer tests was carried out. Using the same aforementioned method, several loose samples at a void ratio of 1.019 ± 0.046 were prepared. Table 2 shows a summary of the test conditions on loose samples (test D26 to F48). The following effective vertical stress levels were imposed: v = 0.15, 0.4, 0.8, 1.3, and 2.1 MPa. This time, the load was kept for 24 h at each level. Thus, to reach a given v max in a test, a time of 24 h of creep was allowed at each mentioned stress level and for each test condition. Therefore, at least three tests for each stress level were carried out on dry, saturated, Figure 9 shows the compression curves for tests D26 to F48. Thus, the same conclusions drawn for 1 h of creep in oedometric compression can be obtained in this case. Namely, the existence of two separated compression curves, one for the dry material and the other for both saturated and flooded samples, resulting in a more compressible behavior in the presence of water. Moreover, the same final GSD for saturated and flooded samples was attained for a given v max , as shown in Fig. 10 for v max = 2.1 MPa. This finding is consistent with our results in triaxial and previous oedometer tests results and, as far as we know, has not been reported in the literature to this day.

Figure 11 presents the compression curves on effective stress log-scale for selected oedometric tests D18, S20, F22, and F25. It can be seen that a yielding point, normally associated to the onset of crushing, cannot be clearly identified. In fact, as shown in Fig. 12, breakage starts occurring even at the lowest stress value used here, which means that the material behaves in an inelastic regime almost from the beginning of the test.

Figure 13 presents the volumetric strain evolution for 24 h of creep for (a) dry, (b) saturated, and (c) flooded samples, for every stress increment. Figure 14a presents the time dependent compressibility index defined as ⌬ v /⌬[log(t)] (where t is time in minutes, measured from 1 min to 24 h). As shown previously by [START_REF] Oldecop | Theoretical investigation of the time dependent behavior of rockfill[END_REF], it is clear that the dry samples have a lower index and the flooded ones joined the index of the saturated samples after flooding. This last statement can be clearly appreciated in Fig. 14b for three representative tests at v max = 2.1 MPa.

Effect of water on the fracture toughness of rock samples

To study the water effect at the source of the particle crushing phenomenon (i.e., at the individual grain scale), we performed crushing tests on core rock samples. We used the cracked chevron notched Brazilian disc (CCNBD) method to obtain K IC , as suggested by the International Society for Rock Mechanics [START_REF] Fowell | Suggested method for determining mode I fracture toughness using Cracked Chevron Notched Brazilian disc (CCNBD) specimens[END_REF]. This method gives consistent fracture toughness values and allows testing rock anisotropy since the crack orientations can be easily arranged. The geometry of the CCNBD samples is shown in Fig. 15, where the chevron or V-shaped notch was cut using a circular diamond saw. The notch cut along the core diameter causes crack propagation to start at the tip of the V alignment and to proceed radially outwards in a stable mode until the point where the fracture toughness is evaluated [START_REF] Fowell | Suggested method for determining mode I fracture toughness using Cracked Chevron Notched Brazilian disc (CCNBD) specimens[END_REF].

For the quartzite shale rock tested in this study, we machined rock cores of D = 52 mm in diameter from large rock aggregates. The cores had both parallel and orthogonal orientations with respect to the rock anisotropies composed by the sheeted structure of the shale rock (see Fig. 16). Rock samples with thickness of B = 21.5 mm were cut from the cores. As shown in Fig. 15, the notch characteristic dimensions were a o = 6.53 mm, a 1 = 19 mm, and R s = 19.25 mm. The fracture toughness tests were carried out using a loading frame of 50 kN of maximum capacity, driven by controlled displacement at 0.1 mm/s. Figure 17 shows the setup of the sample for testing, with vertical displacement sensors and force measurements. The tests were performed in the same clime conditioned chamber used for the compression tests on granular material. We conducted tests on air-dried samples, as well as on samples submerged in water for 24 h before testing. The core samples were machined from two coarse rock aggregates (see aggregate No. 2 in Fig. 16). Due to the heterogeneity of the shale rock and to clearly identify the effect of water in K IC , we compare only the cores coming from the same aggregate.

Tables 3 and4 present K IC values measured for orthogonal and parallel core orientations, respectively. Our tests results are in the same order of magnitude as typical values of K IC for shale rocks reported in the literature of approximately 0.5-1.0 MPa•m 0.5 [START_REF] Whittaker | Rock fracture mechanics principles, design and applications[END_REF][START_REF] Ashby | Engineering materials[END_REF]. In particular, for dry orthogonal oriented cores coming from aggregate No. 1 we obtained K IC ≈ 1.5 MPa•m 0.5 and an average reduction of approximately 60% in saturated specimens with K IC ≈ 0.6 MPa•m 0.5 . Similarly, for orthogonal oriented cores from aggregate No. 2, we measured a reduction in K IC of approximately 20% to 30% (0.54-0.63 to 0.42 MPa•m 0.5 ) from dry to saturated specimens. On the other hand, in parallel oriented samples K IC drops significantly compared to the previous case, due to the induced chevron notch being in the weakest plane of the rock. Moreover, the difference between dry and saturated specimens is less important and K IC is almost the same in aggregate No. 1, with a slight increase of 5% from dry to saturated tests (0.22 to 0.23 MPa•m 0.5 ). In aggregate No. 2, the water effect reduces the K IC from average values of 0.21 to 0.14 MPa•m 0.5 (i.e., approximately 33%) from dry to saturated specimens.

Discussion

The experimental data presented in this paper show that saturated and flooded crushable granular material samples are more compressible than dry samples, mainly due to the increase of grain crushing in the presence of water. The phenomenon assumes that, for samples prepared with air dried grains, suction in capillary bonds of intergranular contacts is negligible, which is normally the case in rather coarse materials. To validate this hypothesis, an oedometric compression test on a stronger material (where no crushing is expected) was carried out. An air dried sample (19 mm high and 70 mm in diameter) of 2 mm uniformly sized glass beads was compressed, as shown in Fig. 18. After flooding at v = 2.1 MPa, the plastic strain was negligible and no crushing was observed after the test. Therefore, regarding the results on crushable granular materials presented in this paper, it can be reasonably assumed that flooding triggers particle crushing. According to previous works aforementioned and also regarding our results for the effect of water on K IC , this phenomenon could be explained by a drop in capillary suction at the microcracks scale inside the particles, as well as corrosive attacks of water, reducing their crushing strength.

Numerical analyses by the discrete element method have shown that the stresses applied on an assembly of grains are divided into a limited number of strong force chains in the same principal direction as the macromechanical stresses, as well as several weak force chains stabilizing the system [START_REF] Cundall | A discrete numerical model for granular assemblies[END_REF][START_REF] Voivret | Multiscale force networks in highly polydisperse granular media[END_REF]. As for the phenomenon of particle crushing, when the coordination number is low (e.g., in uniform graded samples as the material used in this study), it is reasonable to consider that a particle belonging to a strong chain is likely to be crushed in mode I, like in a CCNBD test for K IC evaluation. However, for well-graded materials the stress conditions could be quite different and, due to the heterogeneity of the sample, microtensile stress fields can be generated within the particles to initiate the grain fracture and fragmentation.

Nevertheless, the effect of saturation on particle crushing and on the compressibility could be more or less significant depending on the material properties and its sensitivity to the presence of water. For instance, [START_REF] Coop | The influence of pore water on the mechanics of granular soils[END_REF] investigated differences between the amounts of breakage in dry and saturated (flooded) sands and, while they found substantial differences for a decomposed granite and also for a carbonate sand, they did not find substantial differences for a quartz sand.

Our results show that, for given initial conditions (GSD and density) and compression state (stress path and p max ), the compression curve in the e-p= plane of flooded and saturated samples is virtually the same, regardless of the loading-flooding sequence. Hence, after flooding in creep tests we have observed a collapse from the dry to the saturated material compression curve. Likewise, when we reloaded a sample previously flooded during a stress relaxation phase, the compression curves did not return to the earlier dry material response, but rather followed the one of the saturated material. As shown experimentally by [START_REF] Biarez | Influence de la granulométrie et de son évolution par ruptures de grains sur le comportement mécanique de matériaux granulaires[END_REF] and numerically by [START_REF] Voivret | Space-filling properties of polydisperse granular media[END_REF], the void ratio attained at a given stress state is linked to the GSD: the more polydisperse the GSD is, the higher the packing density will be. Therefore, the uniqueness of the compression curve of saturated and flooded sand tests can be explained because they have reached the same GSD, which was made possible due to grain crushing. Moreover, the uniqueness of the mechanical response for flooded and saturated samples for a given stress path is also observed in the behavior under creep. The creep compressibility index increases with the mean stress, and, for a given stress, its magnitude is higher for saturated samples compared to dry ones. After flooding, this index increases to the same value obtained in saturated samples, analogous to the results on both the compression curve and the breakage ratio. As reported in the literature, this could be explained due to the increment of crack propagation velocity inside the grains in the presence of water [START_REF] Oldecop | Theoretical investigation of the time dependent behavior of rockfill[END_REF].

To study the effect of the initial density, three denser samples were prepared by vibration and compressed at an average initial void ratio of 0.920, using the same previous methodology for 24 h creep oedometer tests (see tests F49 to D51 in Table 2). Figure 18 presents the compression curves for dense samples and also a summary of the most representative compression curves for loose ones. It can be seen that all dry, saturated, and flooded dense samples follow lower compression curves in comparison to loose ones. However, dense and loose samples tend to produce a unique response at high stresses (not completely reached at v = 2.1 MPa). This is clearer for saturated and flooded samples, where the void ratio at v = 2.1 MPa is slightly lower than the one for loose samples. For every test loaded at v max = 2.1 MPa (from Table 2), the final GSD is shown in Fig. 19. It can be seen that the GSDs after the tests for saturated and flooded samples are almost the same for both loose and dense cases. However, the dry dense sample (test D51) presents clearly more crushing than do the dry loose samples (tests D32 and D33), which is in agreement with the denser state reached at v = 2.1 MPa in Fig. 18.

Figure 20 presents B r as a function of the plastic work (W p ) for all loose samples (tests D1 to F48). In the calculation of the plastic dissipation, we have assumed all the unloading strains to be elastic. Therefore, W p in a test was estimated as the sum of all increments of the total work (W T ) done in compression ⌬W = p=(⌬ v ) + q(⌬ s ) (loading), minus the recovered elastic strain during unloading (where ⌬ v and ⌬ s are the volumetric and the deviatoric strain increments, respectively). For instance, in oedometric tests the unloading stiffness, computed as C s = ⌬ v /log( v max / v min ), varies between 0.004 at low stresses and 0.007 at high stresses, as shown in Table 2 and in Fig. 21 (for tests D26 to F48). On the other hand, in dry samples C s resulted in slightly lower values for dry samples compared with saturated and flooded ones. Finally, the elastic strain recovery has resulted to be negligible compared with the accumulated plastic strain. For instance, Table 2 shows that the elastic work, obtained as W E = W T -W p , is less than 2% of W T .

According to Fig. 20, dry materials (squares symbols) require more W p to reach a given B r , compared to saturated and flooded cases, which can be considered as equivalent. This result does not depend on the stress path and could be explained because, unlike in dry condition, wet grains have lower particle strength due to the reduction of the fracture toughness when the water content increases. Therefore, more energy dissipation is needed in a dry sample to reach a given GSD, compared with saturated or flooded samples. Consequently, for a material with a given initial density and initial GSD, the amount of particle breakage depends only on the plastic work and on the water content. At the same time, the plastic dissipation in crushable materials has different sources. It has been proposed that, once a breakage event occurs, the elastic energy stored within the grain is converted in fracture surface energy and in kinetic energy in fragments and also in surrounding grains, due to pore collapse and fragments rearrangement [START_REF] Nguyen | The energetics of cataclasis based on breakage mechanics[END_REF][START_REF] Russell | A compression line for soils with evolving particle and pore size distributions due to particle crushing[END_REF]. Then, the system will reach a new static equilibrium state, where the kinetic energy will be divided into some recovered elastic energy and dissipation by friction at newly formed contacts. Finally, the internal energy distribution plays a major role compared to surface area creation (Ovalle et al. 2013a;[START_REF] Russell | Energy dissipation from particulate systems undergoing a single particle crushing event[END_REF]. The relation presented in Fig. 20 does not make the difference of those contributions and simply suggests that there is coupling between breakage dissipation and plastic dissipation, since both phenomena evolve simultaneously.

Figure 22 presents the relation between B r and the lowest void ratio attained for each test presented here (tests D1 to D51). For the material used here, this relation can be regarded as unique and does not depend on the loading condition (stress path, creep, or relaxation) or on the test particular condition (dry, saturated or flooded). However, it should depend on the initial density and on intrinsic parameters such as the particle shape and the initial GSD. This finding means that, irrespective of the stress path and the water content, the granular packing at static equilibrium only depends on its GSD.

Analogically, [START_REF] Biarez | Influence de la granulométrie et de son évolution par ruptures de grains sur le comportement mécanique de matériaux granulaires[END_REF] have shown that, for a given grain shape, reference void ratios e max and e min are linked with the uniformity coefficient. Moreover, they stated that, to highlight the influence of grain crushing, it appears that the most significant parameter, for a given mineralogy and grain shape, is the GSD, with noticeable effects on the volume change amplitudes during compression. This evidence can be useful for constitutive modeling, since the prediction of the final GSD could be based on calibration tests along only one given stress path. Moreover, constitutive models based on a critical void ratio function of the amount of grain crushing, as in the pioneering work of [START_REF] Daouadji | An elastoplastic model for granular materials taking into account grain breakage[END_REF], could be enhanced to model the effect of water on grain crushing. Nevertheless, only three dense samples were tested in this study. Therefore, more results are needed to assess the effect of the initial packing on this material. For instance, [START_REF] Altuhafi | Changes to particle characteristics associated with the compression of sands[END_REF] have shown that the denser the initial packing is, the less amount of breakage will result (for silica sand, carbonate sand, and basaltic sand tested in oedometric compression up to v = 30 MPa). On the other hand, the experimental assessment of initial density effects is a complex problem because during dense sample preparation some crushing eventually occurs. Therefore, both grading and density effects could be mixed.

The results provided in this paper concern stress magnitudes that could be of interest in civil engineering. However, for high stresses and (or) large strains, grading evolves to an ultimate distribution that might be defined by a given uniformity coefficient [START_REF] Bard | Comportement des matériaux granulaires secs et a `liant hydrocarboné[END_REF]Biarez andHicher 1994, 1997) or by a fractal exponent of approximately D = 2.5 [START_REF] Sammis | Self-similar cataclasis in the formation of fault gouge[END_REF][START_REF] Coop | Particle breakage during shearing of a carbonate sand[END_REF]). Therefore, the results shown in Figs. 20 and 22 should not be extrapolated to outlying range values.

Conclusions

With the aim to explore and discuss the effect of water on the mechanical behavior of crushable granular materials, we have presented the results of a series of compression tests under isotropic, oedometric, and triaxial stress paths. We have compared the data of dry, saturated, and flooded samples, to establish an empirical framework based on the breakage ratio. Our results show that the amount of particle crushing increases in the presence of water and, as a consequence, the material becomes more compressible. Furthermore, for a given initial density and stress path, a dry sample when flooded reaches the equivalent behavior of an initially saturated one in terms of the compression curve, the breakage ratio and the creep compressibility index, regardless of the point of flooding. We have also found a unique relation between the breakage ratio and the lowest void ratio in a compression test at a given initial density, regardless of the stress path, the water content, the point of flooding, nor the time of creep or stress relaxation.

These findings could enhance future constitutive models based on grading to predict the influence of the water content on parti- cle breakage and time effects. This work could also be used for further study on water and time effects on the micromechanics of granular media and a better understanding of the corrosive attack of water in microcracks as a coupled phenomenon of (i) a decrease of K IC and (ii) an increase of crack propagation velocity could be attained. The focus could be the effect of the rock mineralogy, the chemical composition of the water, and other factors of influence. Moreover, it would be interesting to have a general framework based on measurements of the humidity retained into solid grains. Such results could be a base for physical-based models depending on suction in microcracks.

Fig. 1 .

 1 Fig. 1. Definition of the breakage ratio B r (modified from Einav 2007a, with permission from Elsevier).

Fig. 2 .

 2 Fig. 2. 2 to 2.5 mm grains of shale quartzite sand from Trois Vallées.

Fig. 3 .

 3 Fig. 3. Crushed grains of quartzite shale sand after uniaxial compression test sieved between (a) 0.08 and 0.1 mm, (b) 0.315 and 0.4 mm, and (c) 0.5 and 0.6 mm.

Fig. 4 .Fig. 6 .Fig. 7 .

 467 Fig. 4. Stress relaxation during flooding in tests (a) F4, (b) F6, and (c) F8. Fig. 5. Stress relaxation during flooding in test F5.

Fig. 9 .

 9 Fig. 9. Oedometric compression curves for 24 h of creep.

Fig. 10 .

 10 Fig. 10. Grain-size distribution before and after oedometric tests (24 h of creep) at v max = 2.1 MPa for dry (D32), saturated (S40), and flooded (F47) samples.

Fig. 11 .

 11 Fig. 11. Compression curves for selected oedometric tests on log-scale.

Fig. 12 .

 12 Fig. 12. Breakage ratio after oedometric tests (1 h and 24 h of creep)for dry and saturated samples.

Fig. 13 .

 13 Fig. 13. 24% of creep during oedometric compression.

Fig. 14 .Fig. 15 .

 1415 Fig. 14. Creep compressibility index.

Fig. 16 .

 16 Fig. 16. Core rock samples for two directions: orthogonal and parallel; machined from rock aggregate No. 2.

Fig. 18 .

 18 Fig. 18. Oedometric compression curves for loose and dense samples.

Fig. 19 .Fig. 20 .

 1920 Fig. 19. Grain-size distribution before and after oedometric tests of loose and dense samples at v max = 2.1 MPa.

Fig. 21 .

 21 Fig. 21. Unloading stiffness C s in oedometric tests.

Fig. 22 .

 22 Fig. 22. Breakage ratio versus the minimal void ratio for all tests.

Table 1 .

 1 Summary of triaxial, isotropic, and oedometric compression tests.

		Test	Initial	v max or	B r	W p
	Test	No.	void ratio	3 (MPa)*	(%)	(MPa)	Test conditions
	Triaxial	D1	0.933	0.80	22	0.372	Dry
		F2	0.929	0.40	25	0.361	Flooded after isotropic consolidation
		F3	0.935	0.80	25	0.345	Flooded after isotropic consolidation
		F4	0.945	0.80	23	0.338	Flooded at a = 12%
		F5	0.903	0.80	24	0.335	Flooded at a = 10%
		F6	0.948	0.80	26	0.359	Flooded at a = 8%
		D7	0.949	0.40	17	0.199	Dry
		F8	0.957	0.40	18	0.195	Flooded at a = 7%
	Isotropic	D9	1.069	0.40	4	0.010	Dry
		D10	0.955	0.80	6	0.019	Dry
		F11	0.991	0.40	4	0.009	Flooded after 1 h of creep at 3 = 0.4 MPa
		F12	0.952	0.80	8	0.018	Flooded after 1 h of creep at 3 = 0.8 MPa
		F13	0.967	0.80	7	0.025	Flooded after 1 h of creep at 3 = 0.8 MPa
	Oedometric †	D14	1.066	0.40	6	0.033	Dry
		D15	1.015	0.60	7	0.032	Dry
		D16	0.995	0.85	8	0.040	Dry
		D17	1.014	2.10	13	0.118	Dry
		D18	0.990	2.10	14	0.123	Dry
		S19	1.035	0.40	8	0.016	Saturated (or initially flooded)
		S20	1.029	0.85	12	0.046	Saturated
		S21	1.051	2.10	22	0.174	Saturated
		F22	1.008	2.10	21	0.173	Flooded at v = 0.40 MPa and reloaded to 2.1 MPa
		F23	1.009	0.40	8	0.018	Flooded at v = 0.40 MPa
		F24	1.000	0.85	12	0.052	Flooded at v = 0.85 MPa
		F25	1.003	2.10	21	0.201	Flooded at v = 1.50 MPa and reloaded to 2.1 MPa

Note: W

p , plastic work.

* v max for oedometric tests and 3 for triaxial and isotropic tests. † 1 h of creep at each stress level.

Table 2 .

 2 Summary of creep tests under oedometric compression.

	Test	Initial
	No.	void ratio	v max (MPa)	B r (%)	W p (MPa)	W E /W T (%)	C s	Test conditions
	D26	0.998	0.15	2	0.003	0.9	0.004	Dry
	D27	1.010	0.40	6	0.014	0.3	0.005	Dry
	D28	1.034	0.80	8	0.037	1.1	0.004	Dry
	D29	1.065	0.80	9	0.043	1.2	0.005	Dry
	D30	1.014	1.30	10	0.076	1.2	0.006	Dry
	D31	1.011	1.30	11	0.077	1.1	0.005	Dry
	D32	1.037	2.10	12	0.000	0.1	0.006	Dry
	D33	1.010	2.10	13	0.129	1.3	-	Dry
	S34	0.994	0.15	3	0.005	0.7	0.004	Saturated
	S35	1.005	0.40	7	0.017	0.3	0.005	Saturated
	S36	1.010	0.40	8	0.018	0.2	0.004	Saturated
	S37	1.026	0.80	11	0.000	0.1	0.006	Saturated
	S38	1.016	1.30	12	0.000	1.4	0.006	Saturated
	S39	1.020	1.30	13	0.105	1.5	0.006	Saturated
	S40	0.994	2.10	20	0.191	0.9	0.006	Saturated
	S41	0.995	2.10	20	0.189	0.0	0.006	Saturated
	F42	1.010	0.15	-	0.006	0.6	0.005	Flooded at v = 0.15 MPa
	F43	1.007	0.40	8	0.021	0.2	0.005	Flooded at v = 0.40 MPa
	F44	1.036	0.80	12	0.064	0.7	0.005	Flooded at v = 0.80 MPa
	F45	1.006	1.30	14	0.118	0.9	0.006	Flooded at v = 1.30 MPa
	F46	1.045	2.10	21	0.286	0.0	0.007	Flooded at v = 2.10 MPa
	F47	1.062	2.10	17	0.257	1.1	0.006	Flooded at v = 2.10 MPa
	F48	1.029	2.10	8	0.248	1.2	0.007	Flooded at v = 2.10 MPa
	F49	0.920	2.10	20	0.216	1.9	0.006	Flooded at v = 2.10 MPa
	S50	0.906	2.10	18	0.163	1.4	0.007	Saturated
	D51	0.934	2.10	13	0.123	1.4	0.006	Dry

Note: C s , unloading stiffness (= ⌬ v /log( v max / v min )); W E , elastic work; W T , total work.

Table 3 .

 3 Mode I fracture toughness of CCNBD specimens with orthogonal core orientation.

	Rock aggregate	Sample No.	Condition	K IC (MPa•m 0.5 )
	1	1	Saturated	0.58
		2	Saturated	0.59
		3	Dry	1.48
		4	Dry	1.57
		5	Dry	1.49
	2	1	Saturated	0.42
		2	Dry	0.54
		3	Dry	0.63

Table 4 .

 4 Mode I fracture toughness of CCNBD specimens with parallel core orientation.

	Rock aggregate	Sample No.	Condition	K IC (MPa•m 0.5 )
	1	1	Saturated	0.23
		2	Dry	0.22
	2	1	Saturated	0.16
		2	Saturated	0.11
		3	Dry	0.21
		4	Dry	0.22
		5	Dry	0.19
		6	Dry	0.23

Acknowledgements

The authors express their appreciation for the scientific collaboration with Erich Bauer from Graz University of Technology. His useful comments and encouragement to publish our results have contributed to the writing of this article. Carlos Ovalle gratefully acknowledges CONICYT Chile, the Embassy of France in Chile, and the financial support of the Vicerrectoría de Investigación VRI, Pontificia Universidad Católica de Chile, through projects PIA 2014 3519-301 and INICIO 7/2014.


Carlos Ovalle
email: covalle@ing.puc.cl

Etienne Frossard

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

Wei Hu

Siegfried Maiolino

Pierre-Yves Hicher

Pierre-Yves Hicher

P.-Y Hicher

The effect of size on the strength of coarse rock aggregates and large rockfill samples through experimental data

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

Introduction

The subject of the size effect on brittle material strength is in itself not new, it has been experimentally treated in the past, and it is well known that strength is inversely proportional to sample size. A physical explanation was given by Griffith [START_REF] Griffith | The phenomena of rupture and flow in solids[END_REF], who stated that brittle failure occurs through the crack opening at the tip of internal flaws, or discontinuities, in the solid material. Therefore, the probability of finding these flaws within a specimen increases with its size, which results in a decrease in the strength. Afterward, based on the weakest link concept (WLC), Weibull [START_REF] Weibull | A statistical theory of the strength of materials[END_REF] proposed a statistical distribution for strength as a function of specimen size, which can predict the size effect through the data scatter of experimental results. The comprehension of the phenomenon of the size effect is essential in order to predict the mechanical behavior of a prototype when one characterizes the problem by means of tests on small-scaled samples in the laboratory.

The size effect plays a role in brittle sand grain and rock aggregate crushing strength [START_REF] Alonso | Scale effects in rockfill behaviour[END_REF][START_REF] Mcdowell | The application of Weibull statistics to the fracture of soil particles[END_REF][START_REF] Mcdowell | On the micromechanics of crushable aggregates[END_REF]. As a consequence, crushable granular materials composed by brittle grains also present an effect of particle size. For instance, Marachi et al. [START_REF] Marachi | Strength and deformation characteristics of rockfills materials[END_REF] have shown that for a given shearing condition, the coarser the granular material is, the higher the grain breakage ratio, the higher the volumetric contraction and the lower the peak friction angle will be. For coarse granular materials commonly used in civil engineering, such as rockfills, this phenomenon presents difficult experimental problems because samples formed by large rock particles (i.e., 200 mm in diameter or more) cannot be tested inside shear or oedometric devices in the laboratory. However, small-scaled samples of around 1,000 mm in diameter with maximal particle size (d max ) of around 100-200 mm have been tested with very large triaxial devices [START_REF] Bard | Waste rock behavior at high pressures: dimensioning high waste rock dumps[END_REF][START_REF] Becker | Strength and deformation characteristics of rockfill materials in plane strain and triaxial compression tests[END_REF][START_REF] Hu | Effect of sample size on the behavior of granular materials[END_REF][START_REF] Marachi | Strength and deformation characteristics of rockfills materials[END_REF][START_REF] Marsal | Large-scale testing of rockfill materials[END_REF][START_REF] Marsal | Mechanical properties of rockfill dams[END_REF][START_REF] Nobari | Effect of reservoir filling on stresses and movements in earth and rockfill dams[END_REF][START_REF] Verdugo | Strength and stiffness of coarse granular soils[END_REF]. Nevertheless, due to the high cost of such large tests, not much experimental data are available and only few authors have so far also tested the rock aggregate strength [START_REF] Hu | Effect of sample size on the behavior of granular materials[END_REF][START_REF] Marsal | Mechanical properties of rockfill dams[END_REF], which is the origin of crushing and size effect. Extending the work of McDowell and Bolton [START_REF] Mcdowell | On the micromechanics of crushable aggregates[END_REF] on the micro-mechanics of particle breakage, Frossard [START_REF] Frossard | Scale effects in granular fill shear strength and in stability of large rockfill structures[END_REF] treated theoretically the problem of the size effect in granular materials. This method predicts the size effect on the rockfill shear strength with nonlinear envelope, based on a statistical representation of rock aggregates crushing strength and assuming that the same amount of particle breakage of scaled materials is obtained at different shear stress magnitudes, which will allow the aggregate strength to be linked with the shear envelope.

The aim of this paper is to contribute to the discussion on the phenomenon of size effect and the connection between the breakage mechanics of individual particles and the granular packing, supported by new data on two rockfills. The impact of size effects obtained from the experiments is analyzed in terms of the factor of safety (FS) of rockfill slope stability. In addition, the experimental results validate the multiscale method of Frossard, which was originally based on partial data from the literature.

Size effects on rock aggregates strength

Empirical evidence

Analogous to the tensile failure of a spherical particle under compression, the tensile strength (i.e., failure in mode I) of rock aggregates or sand grains can be indirectly obtained by diametral compression between two stiff parallel platens [START_REF] Mcdowell | On the micromechanics of crushable aggregates[END_REF]. According to Jaeger [START_REF] Jaeger | Failure of rocks under tensile conditions[END_REF], the characteristic tensile stress induced within the particle may be defined as

σ = f d 2 (1)
where d is the characteristic size of the particle and f is the diametral compression force. Therefore, using Eq. ( 1), the tensile strength σ f can be obtained from the crushing force f f . Following Lee [START_REF] Lee | The angles of friction of granular fills[END_REF], a power law expressing the size effect on σ f can be obtained (where α is an empirical parameter):

σ f ∝ d α . ( 2
)
Typical values of α are negative, indicating that the larger the particle size, the lower the tensile strength. For instance, Lee [START_REF] Lee | The angles of friction of granular fills[END_REF] reported α values of -0.357, -0.343 and -0.420 for Leighton Buzzard sand, oolitic limestone and carboniferous limestone, respectively, using grains from d = 1 to 50 mm. Similarly, using crushing tests on rock aggregates of d = 20 to 80 mm reported by Marsal [START_REF] Marsal | Mechanical properties of rockfill dams[END_REF], α values are given by -0.43, -0.83, -0.59 and -0.26, for Pinzadaran gravel, El Infiernillo diorite, San Francisco basalt and Mica granitic gneiss, respectively. [START_REF] Mcdowell | On the yielding and plastic compression of sand[END_REF] Leighton Buzzard silica sand 2.34-3.44 Lim et al. [START_REF] Lim | The application of Weibull statistics to the strength of railway ballast[END_REF] Ballasts (granodiorites, granite, porphyritic felsite) 2.06-3.42 McDowell and Amon [START_REF] Mcdowell | The application of Weibull statistics to the fracture of soil particles[END_REF] Quiou limestone sand 1.16-1.93 Lobo-Guerrero and Vallejo [START_REF] Lobo-Guerrero | Application of Weibull statistics to the tensile strength of rock aggregates[END_REF] Biotite gneiss 2.75 Gray quartzite 4.23

Weibull's theory on rock aggregates

The statistical theory of the strength of materials proposed by Weibull [START_REF] Weibull | A statistical theory of the strength of materials[END_REF] gives the following distribution for the survival probability of a material of size d subjected to a tensile stress σ :

P s (d) = exp - d d o n d • σ σ o m ( 3
)
where σ o is the characteristic strength (P s = 37 % for a sample of size d o ) and m controls the amplitude of data scatter on crushing strength. Equation ( 3) is based on the WLC, which is supported by the notion that, for a chain of length d to survive under uniaxial stress, every element of length d o must also survive. The term (d/d o ) n d is the scale ratio indicating how many links form the chain. According to Bažant and Planas [START_REF] Bažant | Fracture and Size Effect in Concrete and Other Quasibrittle Materials[END_REF], n d is the geometric similarity of the mechanical problem: n d = 1, 2 or 3, for linear, surface or volume similarity, respectively. For a given survival probability and known empirical parameters, a size effect relation is obtained between the induced tensile stress at failure (σ f ) and the characteristic size of the sample (d):

σ f ∝ d -n d /m . ( 4
)
Empirical validation of Weibull's theory is possible by verifying the validity of the relation α = -n d /m (from Eqs. ( 2) and ( 4)). For soil particles and rock aggregates under compression between two stiff parallel platens and assuming an isotropic homogeneous material, it has been generally assumed that failure occurs under bulk induced tensile stress, which gives n d = 3 [START_REF] Lobo-Guerrero | Application of Weibull statistics to the tensile strength of rock aggregates[END_REF][START_REF] Mcdowell | On the yielding and plastic compression of sand[END_REF][START_REF] Mcdowell | The application of Weibull statistics to the fracture of soil particles[END_REF][START_REF] Mcdowell | On the micromechanics of crushable aggregates[END_REF].

Table 1 shows m values for different materials reported in the literature. It can be seen that higher m values of around 3 to 4 correspond to sound heterogeneous materials such as quartz sands, biotite and ballast. On the other hand, for feldspars, limestones and carbonates, m varies between 1 and 3. McDowell and Amon [START_REF] Mcdowell | The application of Weibull statistics to the fracture of soil particles[END_REF] tested the application of Weibull's theory on weak limestone sand grains of similar geometry (i.e., rather spherical shape) by checking the ability of Eq. ( 4) for n d = 3 and obtained a very good agreement. McDowell [START_REF] Mcdowell | On the yielding and plastic compression of sand[END_REF] showed similar conclusions for silica sand and also Lobo-Guerrero and Vallejo [START_REF] Lobo-Guerrero | Application of Weibull statistics to the tensile strength of rock aggregates[END_REF] for rock fragments of biotite gneiss and gray quartzite.

However, tensile failure of rock samples or soil grains does not occur always under bulk induced stress. For instance, Ratigan [START_REF] Ratigan | A statistical fracture mechanics approach to the strength of brittle rock[END_REF] proposed geometric similarities of n d = 2 to represent size effects in granite disks after Brazilian tests, where failure was generally induced in weak surface flaws. Moreover, other authors have reported that Weibull's theory cannot always fit the empirical size effect on soil grains. For instance, Lim et al. [START_REF] Lim | The application of Weibull statistics to the strength of railway ballast[END_REF] tested a number of railway ballast particles of d = 20 to 50 mm, and they did not found a good agreement between α and the theoretical exponent -3/m. They explained this fact arguing that the petrographic analysis showed an anisotropic heterogeneous material, so part of the hypotheses of Weibull's theory was unfulfilled. Based on analytical results of internal stress distributions in elastic spheres under diametral compression, Lim et al. [START_REF] Lim | The application of Weibull statistics to the strength of railway ballast[END_REF] assumed that the maximum internal tensile stress could occur near the surface. In this case, and assuming the presence of surface flaws, they used a two-dimensional geometric similarity (i.e., n d = 2) and obtained a better agreement when comparing α to the exponent -2/m. However, in this paper, it is shown that even an n d value smaller than 2, say for instance n d = 1, gives a better fitting with the empirical results. Figure 1 presents the results of Lim et al. [START_REF] Lim | The application of Weibull statistics to the strength of railway ballast[END_REF] on railway ballast particles for α and -n d /m, and it is clear nd nd Fig. 2 Effect of particle size on crushing strength (results from [START_REF] Nakata | A probabilistic approach to sand particle crushing in the triaxial test[END_REF])

that n d = 1 gives the closest solution for α = -n d /m, when compared to n d = 2 or 3. However, as discussed later, this comparison may not be related to the physical sense of the WLC.

Nakata et al. [START_REF] Nakata | A probabilistic approach to sand particle crushing in the triaxial test[END_REF] also reported an example of an unsuccessful prediction of size effects using Weibull's theory with statistics of compression tests on sand grains. They carried out crushing tests on quartz and feldspar sand grains of d = 0.84 to 2.0 mm. For a constant survival probability of 37 %, using Eq. (3) (with n d = 3), they obtained the theoretical size effect on the characteristic particle strength σ c as

σ c (d) = d d o -3/m σ o . ( 5
)
Figure 2 presents the comparison made by Nakata et al. [START_REF] Nakata | A probabilistic approach to sand particle crushing in the triaxial test[END_REF] between σ o obtained by a Weibull statistical analysis and the prediction of Eq. ( 5). The authors noted that there was not a good agreement, particularly for the feldspar, presumably-they argued-because the flaw distribution may not have the geometric similarity assumed in Eq. ( 5).

On the other hand, in the absence of experimental data on rock aggregates' crushing strength to perform a statistical analysis, others authors have assumed that Weibull's theory is applicable for n d = 3. For instance, Frossard et al. [START_REF] Frossard | Rockfill shear strength evaluation: a rational method based on size effects[END_REF] used published results of particle strength by Marsal [START_REF] Marsal | Mechanical properties of rockfill dams[END_REF] in order to fit the values of m by using the relation α = -3/m (where α is obtained from Eq. (2) for the average crushing particle strength shown in Fig. 10). Thus, the authors obtained values of m from 4 to 15 on rock aggregates of El Infiernillo diorite and San Francisco basalt. Using the same approach, McDowell and Bolton [START_REF] Mcdowell | On the micromechanics of crushable aggregates[END_REF] have indirectly estimated m values between 5 and 10 in particle crushing tests performed by Lee [START_REF] Lee | The angles of friction of granular fills[END_REF] on oolitic limestone, carboniferous limestone and Leighton Buzzard sand. These values of m obtained indirectly (i.e., not by a Weibull statistical analysis) seem to be too high in comparison with those reported in Table 1, and the results shown later in this paper for rock aggregates, probably because Weibull's theory is not always valid for n d = 3.

In general, if no information is available on material flaw distribution or internal stress distribution on particles, the physical sense of n d may be missed. As aforementioned, Weibull's theory assumed that the structure is equivalent to a uniaxially stressed bar, which can be extended to a three-dimensional analogy like a spherical particle under diametral compression. Afterward, if one considers an alternative structure with random geometry, all information about the mode of failure could be lost and become irrelevant [START_REF] Bažant | Statistical size effect in quasi-brittle structures-Part 1: Is Weibull theory applicable?[END_REF]. This could be in the case of soil grains and rock aggregates with significant data scatter in particle shape. Moreover, if the material has a heterogeneous structure, the hypotheses of Weibull's theory are not satisfied; therefore, the physical sense of the WLC may be completely missed. As discussed before, this is probably the case in randomly shaped grains (e.g., high angularity) composed by nonhomogeneous or anisotropic materials, where one has no control of the internal stress distribution. Nevertheless, the Weibull distribution can still be a powerful statistical tool for practical applications. In this case, the parameter n d has to be fitted with the empirical results. For instance, in this work the best fittings of n d are obtained for the experimental results of Nakata el al. [START_REF] Nakata | A probabilistic approach to sand particle crushing in the triaxial test[END_REF], which are presented in Fig. 2: n d = 1.46 and 0.19 for quartz and feldspar, respectively, following a purely phenomenological approach.

Rock materials tested

Experimental results on two kinds of rock aggregates were used in this work. Firstly, the results of Hu et al. [START_REF] Hu | Effect of sample size on the behavior of granular materials[END_REF] on a calcareous rock (CP) from a quarry located in Préfontaines, in the center of France, have been analyzed. Thin sections of CP rock samples shown in Fig. 3a reveal that the material is composed by micrite, that is to say, formed by consolidation of lime mud. The material is porous and presents traces of shell debris and bioturbation figures, as a result of reworking due to biological activity. As presented in Fig. 3b, CP rock aggregates are white and have a subangular shape, a structure characterized by an open porosity within the grains and a number of inclusions. Solid grains have a density of solids of 26.54 kN/m 3 , and the porosity of the rock is about 10 %, measured on cylindrical rock cores of 14 mm in diameter.

Secondly, rock aggregates from a quartzite shale rock (STV) from Trois Vallées quarry, in the north of France, have been tested. Figure 4a presents a picture of the material before the test, and it can be seen that particles are quite angular and flattened. The rock has an anisotropic sheeted structure with a clear main orientation, typically found in shale rocks. Figure 4b presents electron microscope scans on fine fractions of crushed grain fragments sieved after a test. It appears clearly that the shale rock is composed by thin sheets. Once crushing occurs, a significant amount of fines is produced essentially by small pieces of detached sheets. The analysis of X-ray diffraction (XRD) scans of a fine fraction of the material (sieved under 0.08 mm) shows that the powder is composed mainly of silica, with also significant amounts of aluminum and iron. The XRD measurement results show the presence of three main phases: 64 % of quartz (SiO 2 ), 26 % of albite (NaAlSi 3 O 8 ) and 10 % of sodium aluminum silicate (Na 6 Al 6 Si 10 O 32 ). Solid grains have a density of solids of 27 kN/m 3 .

Rock aggregates crushing tests

Hu et al. [START_REF] Hu | Effect of sample size on the behavior of granular materials[END_REF] carried out a series of 144 rock aggregate crushing tests on CP rock aggregates, using a range of particle sizes from 7 to 80 mm. Crushing tests were performed between two stiff parallel platens in a 10 kN capacity device with a constant displacement rate of 0.1 mm/min. Particles were located over the fixed bottom plate and placed along its longer size in order to ensure stability. In this paper, these experimental data of particle crushing were separated by size fractions for a statistical analysis. Table 2 presents a summary of the analysis, where d o is the average size, and the induced tensile strength was computed according to Eq. (1) for f f .

As proposed by Weibull [START_REF] Weibull | A statistical theory of the strength of materials[END_REF], for a given size fraction (i.e., d = d o ), m and σ o were obtained by writing Eq. (3) in the following form and fitting the best straight line between ln[ln(1/P s )] and ln(σ ):

ln ln 1 P s = m • ln σ σ o . ( 6
)
Figure 5 presents the plottings of Eq. ( 6) for all tests of Hu et al. [START_REF] Hu | Effect of sample size on the behavior of granular materials[END_REF] on CP aggregates. Likewise, Fig. 6 shows that the results are in very good agreement with Weibull's distribution for an average value m avg = 2.18. Similarly, 100 crushing tests were carried out on STV rock aggregates, using a range of particle sizes from 15 to 70 mm. The same aforementioned statistical analysis was performed, and the results are shown in Table 2 and Figs. 7 and8. A very good fitting with Weibull's distribution was also obtained with an average modulus m avg = 1.54. A clear size effect appears on particle strength for both CP and STV particles, and m values are in the lower range of those reported in Table 1. Hence, data scatter is relatively high, presumably due to the random shape of subangular and angular grains. Figure 9 shows the crushing strength for all tests in CP and STV aggregates, as well as σ o for each size fraction. It can be seen that the size effect is stronger on STV in comparison with CP.

For P s = 37 % and given reference parameters d o and σ o , Eq. ( 3) gives the following expression for the characteristic strength σ c as a function of d: In order to obtain a prediction of the size effect using Eq. ( 7), it was assumed that the reference parameters for each rockfill are as follows: (i) the average value for Weibull's modulus (m avg ) and (ii) the coarser size fraction as a reference for d o and σ o . Figure 9 shows the fitting of Eq. ( 7) for both materials, with fitted n d of 0.63 and 0.81 for CP and STV, respectively. These results show that, as aforementioned, the geometric similarity represented by n d is far from the expected values (i.e., n d = 2 or 3), and the physical notion in Weibull's theory is no longer valid for the current heterogeneous and randomly shaped rock samples. Figure 10 presents the average particle crushing strength for each size fraction for CP, STV and four other materials used in rockfill dams and tested by Marsal [START_REF] Marsal | Mechanical properties of rockfill dams[END_REF] under similar conditions. It can be seen that CP and STV are in the lower and upper ranges, respectively, considering all aggregates.

σ c (d) = d d o -n d /m σ o . ( 7
)
3 Size effects on the mechanical behavior of rockfills 3 presents all triaxial test conditions. It is assumed that particle shape and mineralogy are the same for two scaled materials of each rockfills because prior to sieving, each one was obtained from the same grinding process in the quarry. Tests were run in two large triaxial devices, one for samples of 250 mm in diameter and 375 mm in height and the larger one for samples of 1,000 mm in diameter and 1,500 mm in height. Due to geometrical limitations in the large triaxial cell, the aspect ratio (height/diameter) of the samples tested in both triaxial cells is 1.5, which is lower than the standard value of 2. Even if under these conditions a nonhomogeneous stress distribution can be expected, both sample sizes present the same constraint, and it is assumed that the results are therefore comparable for size effects assessment. The volume change was measured by volume changes in the confining water. To allow this, the cylindrical envelope of the large triaxial cell (1,000 mm samples) is made of a 2 m high double wall chamber. During the test preparation, both outer and inner volumes were filled simultaneously with water, maintaining the same pressure in both chambers. Therefore, the inner cylinder remains nondeformable during the test, and the volume change in the compartment is due only to volumetric deformation of the specimen and penetration of the loading piston. An exhaustive description of both triaxial devices and testing methodology can be found in Hu et al. [START_REF] Hu | Effect of sample size on the behavior of granular materials[END_REF]. Figure 12 presents a general scheme of the larger device, and Fig. 13 shows a general picture of both devices. For dry samples preparation, materials were compacted by vibration in several layers and the average initial void ratios were equal to 0.85 ± 0.03 for CP and 0.53 ± 0.04 for STV. Figure 14 shows the mechanical response during drained triaxial testing for σ 3 between 100 and 400 kPa. Figure 11 shows the GSD before and after the test at σ 3 = 400 kPa for both materials. In Fig. 15, it can be seen that size affects both rockfills, with (a) a higher breakage ratio and (b) a slightly lower maximum friction angle for the coarser materials. For the breakage ratio, the definition proposed by Marsal [START_REF] Marsal | Large-scale testing of rockfill materials[END_REF] was used, given by the sum of positive differences between the percentage of the total sample contained in each size fraction before and after the test. The results Fig. 12 Configuration of the large triaxial device [START_REF] Hu | Effect of sample size on the behavior of granular materials[END_REF] Fig. [START_REF] Frossard | Rockfill shear strength evaluation: a rational method based on size effects[END_REF] Large triaxial devices at the Ecole Centrale Nantes laboratory: a 1,000 mm diameter and 2,000 kN of max. vertical load; b 250 mm diameter and 500 kN of max. vertical load in Fig. 15 are consistent with the size effect on the single particle crushing strength in Fig. 9. Nevertheless, as shown in Fig. 14, no strong size effect takes place in the volume changes, and probably small differences may be attributed to experimental data scatter. Marachi et al. [START_REF] Marachi | Strength and deformation characteristics of rockfills materials[END_REF] and Becker et al. [START_REF] Becker | Strength and deformation characteristics of rockfill materials in plane strain and triaxial compression tests[END_REF] showed similar results when comparing two rockfills of homothetic GSD and with d max of 50 and 150 mm (Pyramid Dam material rockfill, crushed basalt, Oroville Dam rockfill, sandstone). A possible explanation for these systematic results could be given by the relatively high ratio between d max and the rockfill specimen diameter D sample (around d max /D sample = 1/6). Then, after a test to 20 % axial strain, coarse particles could not displace themselves enough in order to dislocate after breakage and to allow for more contraction. Thus, full packing rearrangement may not be allowed under these conditions, and this could be of course more likely with a limited aspect ratio for cylindrical samples, as the results presented in this paper (i.e., 1.5/1). In fact, many broken coarse particles observed after the triaxial tests seem to stand without fragment dislodgment. In other words, some coarse grains actually broke but kept almost the same initial volume because all fragmented pieces stayed together forming the original particle shape, without enough relative displacement. Moreover, some of these grains were easily entirely recovered after the test, as shown in Fig. 16. Consequently, according to the results presented here, as well as the evidence from similar tests on rockfills, it is believed that the proposed testing methodology is not appropriate for size effects assessment on the material compressibility due to grain crushing.

According to published experimental results of shear tests on sand and glass beads where no crushing occurred [START_REF] Gang | Influence of grain size distribution on critical states of granular materials[END_REF][START_REF] Liu | Influence of grading on the undrained behavior of granular materials[END_REF], as well as numerical simulations of uncrushable granular materials using the discrete element method [START_REF] Liu | Influence of grading on the undrained behavior of granular materials[END_REF][START_REF] Muir Wood | Changing grading of soil: effect on critical states[END_REF][START_REF] Voivret | Multiscale force networks in highly polydisperse granular media[END_REF], the critical state friction angle (i.e., constant shear at constant volume when shear strain increases) is not affected by the GSD, provided that the shape of the grains remains unchanged. Moreover, in crushable materials, it has also been shown that the critical friction angle does not depend on the breakage ratio [START_REF] Coop | Particle breakage during shearing of a carbonate sand[END_REF][START_REF] Luzzani | On the relationship between particle breakage and the critical state of sands[END_REF]-i.e., for samples tested at different confining pressures and/or different straining. Therefore, it could be expected that its value is neither affected by size effects caused by particle crushing. Nevertheless, in our tests, the samples did not reach a pure critical state, and, in order to study size effects on the mobilized shear resistance, we focused our analysis on the shear strength envelope obtained from the maximum friction angle in each test. Figure 17 shows the shear strength envelopes for CP, STV and rockfill materials tested by Marsal [START_REF] Marsal | Mechanical properties of rockfill dams[END_REF], which are made of the grains shown in Fig. 10 for the particle crushing strength. For comparison with the results presented in this paper, only the stress domain corresponding to the CP and STV triaxial tests has been analyzed, even if in most of Marsal's results the normal stresses vary up to 4 MPa. It can be seen that the CP is on the lower shear strength range, as well as Mica granitic gneiss rockfills. This is coherent with the particle crushing strength values from Fig. 10, where it is shown that those materials have the weakest particle strength. In the same way, STV is among the strongest rockfills in Fig. 17, consistently with its relatively high particle strength. However, this link between particle strength and the granular assembly strength envelope has to be considered with some caution, because of the influence of gradation and particle shape, so conclusions cannot be drawn by simply considering particle strength. The shear envelopes of CP and STV were fitted to the expression suggested by De Mello [START_REF] De Mello | Reflections on design decisions of practical significance to embankment dams[END_REF] for a nonlinear envelope in rockfills:

τ = A (σ n ) b (8)
where A and b are empirical parameters, τ is the rockfill shear strength, and σ n is the normal stress. Table 4 presents the best fitting of A and b for CP and STV, and Fig. 18 shows the experimental shear envelopes, as well as the fitted curve envelopes according to Eq. ( 8). Following Charles and Soares [START_REF] Charles | Stability of compacted rockfill slopes[END_REF], the FS for the limit equilibrium slope stability based on the method of Bishop [START_REF] Bishop | The use of the slip circle in the stability analysis of slopes[END_REF] can be expressed as

FS = A (b, β) (γ H ) (1-b) ( 9
)
where H is the height of the slope, γ is the bulk density of the compacted rockfill and is the dimensionless stability number-depending on the slope inclination (β) and b. Equation 9 assumes a strong bedrock founda- tion and a circular failure through the rockfill mass. For different material parameters and slope configurations, Charles and Soares [START_REF] Charles | Stability of compacted rockfill slopes[END_REF] produced the chart in Fig. 19, which gives values for slope stability assessment. Then, if one considers two rockfill slopes, each one built with different materials G1 and G2, with heights H 1 and H 2 and considering shear envelopes characterized by empirical parameters A G1b G1 and A G2b G2 , respectively, the FS ratio between G2 and G1 is given by

FS G2 FS G1 = (γ 2 H 2 ) b 2 -1 (γ 1 H 1 ) b 1 -1 • A G2 A G1 • G2 G2 , β) G1 (b G1 , β) . ( 10
)
Therefore, using rockfill slopes with the same geometry but composed by size scaled rockfills, Eq. ( 10) can be used to compare the impact on the FS due to shear strength reduction with the increase in particle size. For instance, considering arbitrary geometrical parameters tan β = 1/1.5 and H 2 = H 1 = 20 m, a constant density for size scaled samples γ 1 = γ 2 (average values were γ = 14.3 and 17.8 kN/m 3 for CP and STV, respectively) and fitted parameters A and b (see Table 4), the FS reduction between a fill built with CP1 compared to CP2 is about 30 % (FS CP2 /FS CP1 = 0.69). Likewise, for STV, the difference is negligible (FS STV2 /FS STV1 = 0.99). Nevertheless, this simplified hypothetical analysis does not necessarily reveal the case occurring in fullscale rockfill slopes. As shown in Fig. 10, CP is a relatively weak rock, and normally in the field such a material is not used for embankments construction. Moreover, the material is very vulnerable to grain crushing because its initial GSD is quite uniform [START_REF] Hardin | Crushing of soil particles[END_REF], which is generally not the case in real projects, where the use of well-graded materials is preferred in order to reach high in situ densities. However, rockfills in real civil works can be easily of one order of magnitude coarser than the size scaled materials tested in this work. Hence, size effects could be nonnegligible and should be taken into account. Then, extrapolated results from predictive methods (validated with experimental data) are suitable in the design procedures.

Validation of a theoretical predictive method

The predictive method proposed by Frossard [START_REF] Frossard | Scale effects in granular fill shear strength and in stability of large rockfill structures[END_REF] considers two homothetic granular materials (e.g., G1 the finer and G2 the coarser) under the same loading condition and with the same mineralogy and grain shape. Thus, if an identical survival probability is assumed for two grains under compression of characteristic sizes d 1 and d 2 in G1 and G2, respectively, the following relation for the particle crushing strength (σ Gi ) can be obtained from Eq. ( 3):

σ G2 = σ G1 d 2 d 1 -n d /m . ( 11
)
Then, according to Eq. ( 1), crushing forces of each particle ( f Gi ) are related by

f G2 = f G1 d 2 d 1 2-n d /m . ( 12
)
Under the same stress condition, it has been shown in this paper that the amount of particle breakage will be higher in G2, compared to G1, since coarse grains have lower strength. Frossard's method is based on the assumption that the breakage ratio is the same in both materials, which is possible only if the stress magnitude on G2 is lower. Then, focusing on contact micromechanics and based on Eq. ( 12), a condition for the intensity of the contact forces between grains n and p ( f (n/ p) ) is obtained:

f G2(n/ p) = f G1(n/ p) d 2 d 1 2-n d /m . ( 13
)
In parallel, the geometric scaling gives a relationship between branch vectors (l (n/ p) ) and grain volumes:

l G2(n/ p) = l G1(n/ p) d 2 d 1 and V G2 = V G1 d 2 d 1 3 . ( 14
)
The stress tensors σ i j for G2 and G1 are given by the following expressions [START_REF] Christofferson | A micromechanical description on granular material behavior[END_REF][START_REF] Rothenburg | Micromechanical definitions of the Cauchy stress tensor for particular media[END_REF]:

σ i j-G1 = 1 V G1 n< p≤N f G1(n/ p) ⊗ l G1(n/ p) and σ i j-G2 = 1 V G2 n< p≤N f G2(n/ p) ⊗ l G2(n/ p) . ( 15
)
Hence, combining Eqs. ( 13), ( 14) and ( 15), the stress tensor of G2 as a function of G1 is obtained:

σ i j-G2 = d 2 d 1 -n d /m 1 V G1 n< p≤N f G1(n/ p) ⊗ l G1(n/ p) . (16)
Therefore, in order to obtain the same breakage ratio in each granular material under shearing, the reduction factor (d 2 /d 1 ) -nd/m should be applied to the shear envelope (τ -σ n ) for the granular assembly:

τ G2 = τ G1 d 2 d 1 -n d /m and σ n-G2 = σ n-G1 d 2 d 1 -n d /m . ( 17
)
Finally, considering the curved failure envelope in Eq. ( 8) for G1, with fitted parameters A G1 and b G1 , a shear strength envelope can be predicted for G2:

τ G2 = A G1 • d 2 d 1 -n d (1-b G1 ) m • (σ n-G2 ) b G1 . ( 18
)
In this paper, the statistical data on rock aggregate crushing strength of CP and STV have been used in order to validate the prediction of Eq. [START_REF] Jaeger | Failure of rocks under tensile conditions[END_REF]. Table 4 presents the set of parameters used, and Fig. 18a, b shows the predicted envelopes for CP2 and STV2, respectively. It can be seen that Frossard's method reasonably predicts the size effects for both rockfills, even if it appears to fall on the safe side for STV. This could be explained because the method does take into account neither the magnitude of particle strength nor the effect of the initial GSD. It is well known that these material properties strongly affect the amount of grain breakage, which is amplified with relatively weak grains [START_REF] Billam | Some aspects of the behaviour of granular materials at high pressures[END_REF] and rather uniform GSD [START_REF] Hardin | Crushing of soil particles[END_REF]. In the case of STV, it has been shown that it is composed of relatively strong grains and a well-graded GSD, if compared to CP. Then, for an equivalent shear state, STV presents a lower breakage ratio (see Fig. 15a) and therefore a less marked size effect on the shear envelope.

Conclusions

As with any brittle material, the experimental evidence shows that there is a size effect on the crushing strength of rock aggregates, expressed as an inverse relation between size and strength. These results on rock aggregates can explain the size effects in crushable granular packings, such as rockfills. However, due to the necessity of large and costly experimental equipments, the data available on size scaling of the mechanical behavior of coarse granular materials are still quite scarce.

In this paper, new data on rock aggregates tensile strength and on the mechanical behavior of size scaled rockfills are presented. First, the size effect on particle strength has been tested for rock aggregates of a porous limestone and an anisotropic shale rock. The suitability of Weibull's theory in order to fit the experimental results on the crushing strength has been evaluated. As reported in the literature, the theory assumes a bulk tensile failure within homogeneous samples. Nevertheless, rockfills are composed by grinded aggregates which are randomly shaped (normally angular or subangular), and the rock could have quite heterogeneous structure. Even if under these conditions the hypothesis of Weibull's theory may not be satisfied, the statistical distribution can be fitted to experimental results and can be used as a purely phenomenological statistical tool for size effects.

Second, the experimental results on large triaxial tests showed that when grain shape and mineralogy are preserved between two granular materials with size scaled GSD (maximum particle sizes of 40 and 160 mm), size effects are observed as an increase in the amount of particle breakage and a slight decrease in the shear strength envelope for the coarser material. Quantitatively, for a particle size reduction factor of 4, a decrease in the maximum friction angle of about 2 • to 3 • was obtained. On the other hand, no size effect in the material compressibility has been observed in large triaxial tests, presumably due to a limited straining, which does not allow for full packing rearrangement after crushing.

Size effects on shear strength of rockfills seem to be limited in the laboratory. However, they could be more significant in full-scale materials, which can easily have particles sized around one order of magnitude coarser than the largest one in a large laboratory sample. Therefore, predictive methods are needed in order to take into account size effects in engineering designs. In this work, the experimental results were used for validation of the only existing theoretical method that links the rock aggregate with the strength of the granular assembly, obtaining a good agreement. However, the method makes this link based only on the size effect on particle strength, without considering other materials properties that affect grain breakage. Consequently, when the amount of breakage is relatively low, the method gives a conservative (safe) prediction.

Further work may be done on the influence of particle shape and the mineralogy of rock aggregates. These properties could change along size fractions due to the geological morphology of the rock, adding more uncertainties to size effect predictions. Thus, a representative scaled sample may not always be obtained by simply sieving and dividing a granular material by particle sizes, but also by considering the size-dependent geological composition and angularity of rock aggregates.

Fig. 1

 1 Fig.1Comparison of theoretical prediction of the size effect exponent with the empirical fitted value for different ballast materials (results from[START_REF] Lim | The application of Weibull statistics to the strength of railway ballast[END_REF])

Fig. 3

 3 Fig. 3 CP rock samples: a microscopic observations on thin sheets, each image represents an area of about 5 × 5 mm, b rock aggregates of sizes 5 to 40 mm

Fig. 4

 4 Fig. 4 STV rock samples: a angular rock aggregates of sizes 10 to 160 mm, b microscopic observations on fine crushed grains sieved between 0.08 and 0.1 mm

3. 1

 1 Large triaxial testsFor CP rockfills, Hu et al.[START_REF] Hu | Effect of sample size on the behavior of granular materials[END_REF] performed a series of drained triaxial tests on a dry material using two homothetic grain size distributions (GSD) with d max = 40 and 160 mm and uniformity coefficient C u = d 60 /d 10 = 2

Fig. 5 Fig. 6 Fig. 7 Fig. 8 Fig. 9 Fig. 10

 5678910 Fig. 5 Computation of Weibull parameters m and σ o for CP

Fig. 11

 11 Fig. 11 Initial and final GSD of homothetic materials

Fig. 14

 14 Fig. 14 Stress-strain curves for a CP and b STV rockfills

Fig. 15 Fig. 16

 1516 Fig. 15 Triaxial results: a breakage ratio and b maximum friction angles of CP and STV rockfills

Fig. 17 Fig. 18

 1718 Fig.[START_REF] Hu | Effect of sample size on the behavior of granular materials[END_REF] Shear strength envelopes of Marsal's[START_REF] Marsal | Mechanical properties of rockfill dams[END_REF] tests, as well as CP and STV rockfills

Fig. 19

 19 Fig.[START_REF] Lee | The angles of friction of granular fills[END_REF] Dimensionless stability numbers from circular slope failures[START_REF] Charles | Stability of compacted rockfill slopes[END_REF]

Table 1

 1 Weibull modulus of soils grains and rock aggregates reported in the literature

	Author(s)	Material

Table 2

 2 Weibull

		41	10.9	1.55	3.35
	15 < d < 25	35	19.7	2.14	3.33
	25 < d < 50	41	44.6	2.51	2.68
	50 < d < 80	27	62.0	2.52	1.95
	Quartzite shale rockfill from Trois Vallées (STV)
	15 < d < 30	37	25.4	1.53	16.99
	30 < d < 40	29	34.5	1.26	14.21
	40 < d < 70	34	48.4	1.82	12.16

statistical analysis of particle crushing tests by size fraction Size fraction (mm) No. of crushing tests d o (mm) m σ o (MPa) Calcareous rockfill from Prefontaines (CP) 7 < d < 15

Table 3

 3 Large triaxial tests

	Material	d max (mm)	σ 3 (kPa)	Initial void ratio
	Calcareous rockfill from Prefontaines (CP)
	CP1	40	100	0.84
			200	0.83
			400	0.85
	CP2	160	200	0.89
			400	0.86
	Quartzite shale rockfill from Trois Vallées (STV)
	STV1	40	100	0.49
			200	0.53
			300	0.54
			400	0.55
	STV2	160	200	0.51
			300	0.53
			400	0.53

Table 4

 4 Summary of parameters for the prediction of Frossard's method

	Parameter	Material
		CP1	CP2	STV1	STV2
	A	0.80	0.68	0.87	0.81
	b	0.82	0.72	0.77	0.78
	d max (mm)	40	160	40	160
	m avg	2.18		1.54
	n d	0.63		0.81

Acknowledgments

This work was part of the French research project ECHO (size effects in Civil Engineering Granular Fill Works), sponsored by the French National Agency for Research (ANR) and involving a wide partnership, including: EIFFAGE Travaux Publics; Electricité de France-CIH, Laboratoire de Mécanique des Solides from Ecole Polytechnique, Centre d'Etudes Techniques de l'Equipement-Laboratoire Régional des Ponts et Chaussées Lyon, Coyne et Bellier Ingénieurs-Conseils (now incorporated in Tractebel Engineering), Institut de Recherche en Génie Civil et Mécanique from Ecole Centrale Nantes. Carlos Ovalle gratefully acknowledges the financial support of CONICYT Chile, the Embassy of France in Chile and Pontificia Universidad Católica de Chile. The authors would like to thank the anonymous reviewers for their valuable comments and suggestions to improve the quality of the paper.