Scale effects on strength of geomaterials, case study: coal

1 ELSEVIER Scale effects on strength of geomaterials, case study: coal Luc Scholtès, Frédéric-Victor Donzé *, Manoj Khanal Earth Science and Resource Engineering, QCAT, CSIRO, Pullenvale, Australia Elsevier use only: Received date here; revised date here; accepted date here Abstract Scale effects on the strength of coal are studied using a Discrete Element Model. The key point of the model is its capability to discriminate between the strictly sample size effect and the Discrete Fracture Network (DFN) density effect on the mechanical response. True triaxial compressive test simulations are carried out to identify their respective roles. The possible bias due to the discretization size distribution of the discrete element model has been investigated in detail by considering low-resolution models. The model is shown to be capable of quantitatively reproducing the dependency of the maximum strength on the size of the sample. This evolution mainly relies on the DFN density. For all given sizes, as long as the DFN density remains constant or if discontinuities are absent, the maximum strength of the sample does not change. 2001 Elsevier Science. All rights reserved Keywords: Scale effect; Strength properties; Discrete Fracture Network; Fracture density; Discrete Element Method. 1. Introduction Discontinuities are observed at all scale in coal and they greatly affect its properties. For instance, when the density of discontinuities increases, a decrease of the strength is observed. The unconfined compressive strength (UCS) of coal is also known to be dependent on the scale at which it is tested [1, 2, 3, 17]. A non-uniform distribution of discontinuities can also induce en important strength anisotropy within the rock mass [4, 5, 6]. Coal, like rock, is a cohesive frictional geomaterial structure of which usually contains complex networks of discontinuities (cleats). Cleat networks can present very different spatial distributions which depend on the coalification process, ranging from highly organized to disordered configurations. Generally, an organized cleat set can be decomposed into face and orthogonal butt cleats * Corresponding author. Tel.: +0-64-733274403; e-mail: frederic.donze@csiro.au. (see Figure 1). Face cleats are considered as extensional fractures oriented perpendicular to the least compressive principal stress applied on the coal during coalification, whereas butt cleats occur as non persistent fractures, usually perpendicular to face cleats, resulting from the relaxation of the original stress field. Face cleats Butt cleats Figure 1. An illustration of face and butt cleats in coal (after [7]).

2 In common bituminous coal, bright coal (vitrain rich) and dull coal (durain rich) are generally the main constituents. In bright coal, cleat spacing occurs at the millimeter scale, whereas it is generally much greater in dull coal (decimeter scale). Based upon the observation that cleating is often confined to the brighter bands in coal, the use of the terms bright and dull infers a measure of volumetric cleat density. The brightness profiles, therefore, enable an assessment to be made of coal seams structure [3]. This variation in density could also explain why dull coal is observed to have a greater strength than bright coal [3]. Master cleats, which are defined as vertically cutting both vitrain and durain layers, have been observed even more widely spaced than the other classes of cleat [4, 7]. In the present study, a numerical model based on the Discrete Element Method (DEM) [8] is used to investigate the contribution of discontinuities on strength properties of a generic bituminous coal. Cleats are represented as discontinuity planes (also referred to here as fracture or joint planes) for which both spatial distribution and mechanical properties are controlled. Simulations of true triaxial compression tests are performed on numerical samples. In order to verify the validity of the method, the predicted results are systematically compared to the experimental ones obtained by Medhurst and Brown on coal [3]. First, the study focuses on intact media, in order to assess the possible effect of discretization, that is, the resolution of the model, on the results. Then, the effect of an increasing density of discontinuities is characterized. Previous relationships between the strength and the sizescale have been proposed. For example, Evans and Pomeroy [21] found that the average compressive strength σ c of unconfined cubes of coal depend on the cube side β length a according to an exponential law σ c = kα, where β is a constant depending on the type of coal. Hoeck and Brown [15] suggested that in the case of an intact rock specimen, its strength can be related to a reference sample strength of 50 mm diameter by the formula σ ( ) 0. 18 c = σ 50 D 50, with D being the section diameter of the sample. For jointed rocks, an empirical formula, using a geological strength index (GSI), has been set up. However, these formulae do not reflect in a quantitative way the overall relationships between the size-scale, the fracture density and the material strength. In this paper, a quantitative investigation on the scale effects on strength is proposed. The size of the sample is increased and the evolution of its strength is studied. The scale dependency of the strength is then discussed. 2. Formulation of the model The algorithm used in the present Discrete (or Distinct) Element Method (DEM) [8], involves two steps. First, based on constitutive laws, interaction forces between Discrete Elements (DE) are computed. Second, Newton second law is applied to determine, for each DE, the resulting acceleration, which is then time integrated to find its new position. This process is repeated until the simulation is finished. All developments and simulations described in this paper have been implemented into the YADE Open DEM platform [9]. 2.1. Constitutive law For small deformations, coal exhibits a linear elastic response characteristic of geomaterials. To reproduce this behavior, linear elastic interaction forces between the discrete elements are sufficient. In the present model, the interaction force F which represents the action of DE a on DE b may be decomposed into a normal force F n and a shear force F s which are related to relative normal and incremental shear displacements through the stiffnesses K n and K s, in the normal and the tangential directions respectively. Once the DE s packing has been generated to form the numerical sample, pairs of initially interacting Discrete Elements are identified within an interaction range γ int, such that, D = γ ( R + R ) (1) eq int a b where D eq is the initial equilibrium distance, R a and R b the radius of elements a and b respectively, and γ int 1 [19]. Contrary to classical DEM which only considers strictly contacting DE (i.e. γ int = 1 ), this feature provides the possibility of controlling the initial number of interacting links (also referred here to as the coordination number), whatever the granular packing, by simply varying γ int before the first timestep of the computation cycle [10, 18]. The normal interaction force is calculated through the local constitutive law (see Figure 2) and can be split into two parts, the compressive and the tensile components. In compression, F n is linear and it is given by, Fn = K n ( D Deq ) (2) where D is the distance between the centroids and K n the normal stiffness.

3 F s F n A int C φ b D rupture 1 K n D φ c F n K n ζ 1 TA int F n,max Figure 3. Rupture criterion used in the model. Figure 2. Normal interaction force between DE. In tension, the normal interaction force is also linear using the same stiffness for small deformations. The maximum acceptable tensile force F n,max is defined as a function of the tensile strength T such that, F = TA (3) n,max int where A int = π(min(r a,r b )) 2 is the interacting surface between a and b. After the tensile force reaches its maximum value, the stiffness may be modified by a softening factor ζ to control the energy released by the link breakage: Kn F n = ( D Drupture) (4) ζ When D > D rupture, rupture occurs and interaction forces are set to zero. The shear force F s is calculated by updating its orientation which depends on the orientation of the direction passing through the two centroids of the interacting DE, and by adding the increment of shear force F s as defined by Hart et al. [11]: F = K u (5) s s s with K s being the shear stiffness and u s the relative incremental tangential displacement. The maximum shear force F s,max is characterized by the normal force F n, the cohesion C, the local frictional angle φ b and the local residual frictional angle φ c. The maximum shear force is calculated for a bonded interaction according to, Fs, max = Fn tanϕ b + CAint (6) Only purely frictional new contacts can appear during the simulation, with a maximum shear force defined as, F tanϕ s, max = F n (7) c Since spherical discrete elements are used in this model, the interaction law must include a moment in order to control the rolling occurring between DE during shear displacements. The moment transferred between two interacting DE can be written as, M elast = Σθ K (8) r r where θ r is the relative rotation angle and K r is the rotational stiffness defined by β.k s.r avg [12], where β is a dimensionless coefficient and R avg the average radius of the two interacting DE. An elastic limit is introduced (Figure 4) and when it is reached, the plastic moment M plast defined by M ηf R plast = (9) n avg takes place, η being a dimensionless factor. To model the nonlinear behavior of geomaterials, a modified Mohr-Coulomb model is used (Figure 3).

4 n j t j M r M plast K r ηf n R avg n c t c 1 θ r Figure 4. Rolling moment law considered between interacting DE. n j t j Doing so, the sliding process increases and the resulting macroscopic internal friction angle Φ can reach values corresponding to those measured for geomaterials [12]. 2.2. Modeling Discontinuities In order to model discontinuities in the DEM, a specific contact formulation, inspired by the work of Cundall and co-authors on the development of the Synthetic Rock Mass model [13], has been implemented into YADE Open DEM. This joint contact logic is based on the identification of each interaction crossing the plane which represents the discontinuity plane (see Figure 5). Normal and tangential directions ( n r j, t r j ) of these interactions are updated to new orientations ( n r c, t r c ) which depend on the orientation of the joint. With this formulation, it becomes possible to get rid of the interlocking process (due to the spherical geometry of the DE) and of the discretization of the joint surface occurring during shearing displacements [14]. The interactions forces concerned by this modified contact logic use the same formulation as defined in section 2.1, except that they do not transmit any moments. Their mechanical properties (K n, K s, C, T, φ b and φ c ) can be chosen equal to or different from the ones used to represent the intact material. In addition, a dilatation component has been added to the normal force, in order to simulate the aperture of the joint, usually observed with shearing [14]. When sliding occurs between two DE on both sides of a discontinuity, F n is updated such that, F n = K D + u tanψ (10) n s where ψ is the dilatation angle. All the simulations performed in this paper use this joint contact logic when discontinuities are taken into account. (c) Figure 5. Joint contact logic: identification of the interactions crossing the discontinuity plane, DE are colored depending on the side of the plane to which they belong. Contact orientation and displacements induced by shearing displacement: classic contact orientation and modified contact orientation (c). 3. Model setup Within the proposed DEM, intact coal material is represented by a set of spherical DE. In this study, numerical samples are generated using a polydisperse distribution size for the DE in order to avoid any artifacts related to regular packings. A wide range of coal properties can be found in the literature depending on site or testing specificities (see for example [2, 3]). Since the experimental results obtained by Medhurst and Brown [3] provide an often referenced data base, they are used here to setup the mechanical properties of the model (see Table 1). Due to the intrinsic presence of cleats at all scales in coal, it is difficult to assess directly its intact mechanical properties from triaxial tests, even if coal specimens are very small. Measurements made on small dull coal specimens by Medhurst and Brown [3] are thus used to provide reference properties for the intact coal s mechanical properties. Average values of 4 GPa for Young s modulus E, 0.25 for Poisson s ratio ν, 43 for the internal friction angle Φ and 7 MPa for the cohesion C have been estimated by [3] based on the generalized Hoek and Brown criterion parameters [15]. Corresponding local properties used in the DEM to reproduce this generic coal are summarized in Table 2.

5 Table 1. Mohr-Coulomb properties for generic coal. Mechanical properties Values Young s modulus E (GPa) 4 Poisson s ratio ν (-) 0.25 Internal friction angle Φ ( ) 48 Cohesion C (MPa) 7 moved vertically at a constant deformation rate while keeping the confining pressure P by adjusting lateral wall positions. Strain and stress variations are computed through wall displacements and resultant forces on walls respectively. ε 1 varies Table 2. Input parameters used to model generic coal. Input parameters Values Normal stiffness K n (MN.m -2 ) 48.30 Tangential stiffness K s (MN.m -2 ) 9.66 Local friction angle φ= φ b =φc ( ) 40 Tensile strength T (MPa) 11.5 Local cohesion C (MPa) 115 Coefficient of rolling stiffness β (-) 1.0 Coefficient of rolling elastic limit η (-) 0.15 Coefficient of damaging law in release mode (-) 0.0 Damping coefficient η 0.4 Mechanical characteristics of cleats are not straight forward to determine because, in vitrain layers for example, both mineralized and unmineralized cleats can be found in the same seam. A local strength value close to the average tensile strength for coal is thus chosen here (see Table 3). The friction angle is chosen to be equal to 30 whereas cohesion C and tensile strength T are taken to be a tenth of the intact ones. In addition, the dilatation angle ψ is set to 5. Table 3. Input parameters for discontinuities in the DEM. Input parameters for discontinuities Friction angle φ ( ) 30 Values Tensile strength T (MPa) 1.15 Local cohesion C (MPa) 11.5 Dilation angle ψ ( ) 5 4. Triaxial loading tests A series of numerical tests have been performed simulating true triaxial testing apparatus. The loading is applied on the numerical specimen through six rigid frictionless boundary walls with servo-controlled positions depending on the chosen loading path (see Figure 6). The numerical samples are subjected to an isotropic compression up to a predefined confining pressure P. Once this confining pressure is reached, top and bottom plates are σ 2 = σ 3 =P Figure 6. Triaxial testing configuration. σ 2 = σ 3 =P Like the number of elements, the density or the strain rate applied on the specimen need to be selected correctly [12]. The density of the model is chosen to be equal to an average value for coal material (1450 kg.m -3 ), while the strain rateε& 1 is set sufficiently low to limit the inertial contribution of the DE. Quasi-static conditions are systematically verified by checking that dividing ε& 1 by any factor greater than 1 does not produce significant changes in the mechanical response. Since the scope of this article is the scale effect on strength, it is crucial to first identify the possible contribution of the discretization of DE sets on the response of an intact numerical specimen. For this purpose, four different cubical samples with identical volumes V=L 3 are considered. They are made of different numbers of DE with different average diameters d but the distribution size ratio is kept the same, approximately 2 (see Figure 7). Each specimen is generated according to the same procedure, which consists in growing particle s radii iteratively starting from an initial set of non overlapping DE contained within a box of volume V, until the pressure applied on the walls has reached the predefined value P. The four samples are made of a number of DE varying from 400 to 50 000, resulting in different L/d ratio, ranging from 7 to 37. Figure 7. Four cubic samples of identical volume V=L 3, composed with discrete elements of different average diameters d, such that L/d = 7, 15, 22, 37.

6 The behaviors of the four samples subjected to triaxial compressions under a 200 kpa confining pressure are presented in Figure 8. It can be observed that the stressstrain responses as well as the volumetric deformations are different depending on the discretization size. In addition to the less pronounced variations in the sample response around the peak stress, the strength, as well as Young s modulus, tends to increase when the L/d ratio increases. discontinuities. In that case, the local properties of the fabric may be altered. To get rid of these resolution effects on Young s modulus and the strength when the discretization is low, the local values of the stiffness and tensile strength could be modified. Another alternative without modifying the micromechanical parameters is to control the initial interacting distance for which links between particles are created (see section 2.1). The interaction factor γ int has therefore been modified in order to keep the initial coordination number constant for all samples. The results are presented in Figure 9. (c) (d) Figure 8. The evolution of the differential stress q, volumetric deformation dv/v, porosity (c) and coordination number (d) during triaxial compressions performed for 200 kpa confinement on four cubic samples composed of 400, 3200, 10 800 and 50 000 DE respectively. (c) (d) Yet, local interaction laws have been carefully implemented to be size independent and the respective fabric of each sample should be equivalent since they have been generated using the same procedure. One could therefore expect a discretization independency. The porosities and, more particularly, the initial values of the coordination number (average number of interacting links per element) of each sample, are specific to each assembly and thus show an evolution of the internal fabric which contributes to the global response discrepancy. This evolution can be formulated using the ratio between the number of DE which composes the inner part of the sample N int and the number of DE in contact with boundary walls N bound. At the considered discretization levels, N bound /N int varies dramatically (from 0.7 to 0.16). Given an identical volume V, fewer DE induces, proportionally, more frictional contacts with boundaries than cohesive links within the whole specimen, resulting in a lower Young s modulus as well as a weaker strength at the scale of the assembly. Nevertheless, it is worth noting that the differences tend to decrease for higher values of L/d. However, it was of utmost importance in this paper to show the consequences on the mechanical response of using low resolution models. This needs to be taken into consideration when representing highly discontinuous media which may then have a low resolution in between Figure 9. Evolution of the differential stress q, volumetric deformation dv/v, porosity (c) and coordination number (d) during triaxial compressions performed under 200 kpa confinement on four cubic samples composed of 400, 3200, 10 800 and 50 000 DE respectively with an identical initial coordination number (d). Contrary to the case where initial coordination numbers were not kept constant (Figure 8), Young s modulus and the differential peak strength converge to the same value whatever the L/d ratio is (except for the low discretization case which is strongly dependent on the spatial distribution of the DE). Thus, the coordination number can be used as a key parameter in DEM simulations when low discretization levels are needed, for example, to keep a low calculation cost. This ratio can counter balance boundary effects when L/d <20. 5. Contribution of the density of pre-existing discontinuities As the purpose of the present study is to deal with the effect of discontinuities on the overall mechanical properties of cohesive frictional materials, several series of triaxial compression tests have been performed on numerical samples containing a pre-existing set of

7 discontinuities (i.e. a Discrete Fracture Network, DFN). The first of these tests concerns the effect of the DFN density on the mechanical properties of a sample with a given size and discretization. In a numerical sample of L = 60 mm with a discretization ratio of L/d=22, different sets of discontinuities have been initially introduced (see section 2.2 for implementation details). In order to represent coal in a realistic but simple manner, persistent discontinuities are are positioned in two directions (bedding planes and face cleats respectively perpendicular to the Y and Z axis), whereas non persistent discontinuities (butt cleats) are placed in the third direction (perpendicular to the X axis) (see Figure 10). A constant ratio between cleat spacing and cleat height is chosen here based upon observations done by Dawson and Esterle [7] on Australian Permian coal. Y X Z Figure 10. Four cubic samples of identical size (L=60 mm) and discretization (L/d=22) with an increase of the density of discontinuities. Four different samples have been built up with an increasing fracture density which can be characterized by the number of cleats per sample length L (from 0 to 4 cleats per length, see Figure 10). Triaxial compressions as described previously have then been performed at several confining pressures (along the vertical Y axis) and model predictions are compared to the Medhurst and Brown results obtained on same size coal samples with different brightness levels [3]. For clarity reasons and since the experimental results have also shown a good agreement with the generalized Hoek and Brown Criterion (HBC) [15], the present numerical results are also compared in terms of this criterion which is defined as: a complement to the HB Criterion. According to the GSI, the strength of rock type materials decreases with the increase of the fracture density or blockiness of the considered rock mass, which is the case here. In addition, it is remarkable that, despite its relative simplicity, the idealized cleat network defined here seems to be relevant to model the change of brightness in coal, at least for the considered case. Table 4. Hoek and Brown Criterion parameters for both experimental [3] and DEM results which show the effects of cleat density. Sample Hoek and Brown parameters σ c (MPa) m s Medhurst and Brown C5 32.7 19.4 1 DEM Intact (no cleat) 31.5 20 1 Medhurst and Brown - C4/C3 27.5 15.6 0.708 DEM - 1 cleat/l 26.4 15 0.7 Medhurst and Brown - C3/C2/C4 16.7 11.7 0.261 DEM - 2 cleats/l 22.3 12 0.5 Medhurst and Brown - C2 9.9 8.8 0.0902 DEM - 4 cleats/l 14.1 5 0.2 Figure 12 shows the stress-strain responses as well as the volumetric deformations predicted by the DEM for triaxial compressions performed on all four specimens under a 200 kpa confining pressure. 0.5 mσ 3 1 = σ3 + σc + s σ c σ (11) where σ c is the uniaxial compressive strength of the intact material, and m and s are chosen to fit Medhurst results. The HBC parameter values for both experimental and DEM results are presented in Table 4, whereas the comparison between the experimental data and the numerical results are presented in Figure 11. Numerical results show a decrease of the strength with an increase in the DFN density which is in good agreement with experimental data. This result also agrees with the Geological Strength Index (GSI) approach which is commonly used in rock and mining engineering [15, 16] as Figure 11. Medhurst and Brown [3] experimental results obtained on coal samples with different brightness levels compared to DEM predictions for an increasing fracture density in terms of the generalized Hoek and Brown criterion [15].

8 One can note that Young s modulus and the dilatation angle are equally affected by the presence of discontinuities. Both of them tend to decrease with an increase in material cleating, which also confirms experimental observations [3]. The presence of pre-existing discontinuities induces modifications into the fabric of the material which produce dramatic changes at the sample scale. To get more insights on the microscopic mechanisms (i.e. local mechanisms here) induced by cleating, it is possible to compare the deformation and fracturing processes which occur during loading for both, an intact and a discontinuous samples (see Figures 13 and 14). Figure 14. Displacement fields for both intact sample (left) and sample with pre-existing discontinuities (right): global view (top) and Y view (bottom). Figure 12. Effect of cleat density. Differential stress q, volumetric deformation dv/v, obtained during triaxial compressions performed under 200 kpa confinement on four specimens of identical size and discretization but different cleat density. Figure 13. Crack distribution for both intact sample (left) and sample with pre-existing discontinuities (right): global view (top) and Y view (bottom). Obviously, deformations as well as propagation of cracks (i.e. when a local interaction link breaks) are strongly driven by the presence of pre-existing discontinuities which act as planes of weakness inside the material. Due to their lower cohesive properties, the interacting links crossing the discontinuities tend to fail before the ones located in the intact zones of the material. It should also be noted that, as for the intact part of the material, micro-cracking mainly occurs in the direction parallel to the main applied stress (the Y axis here), which is also in agreement with experimental observations made on rocks as long as the confining pressure remains sufficiently low. In order to assess once again, the influence of discretization on DEM predictions, the same series of triaxial compressions on samples have been performed on samples with different L/d ratios while keeping the same networks of discontinuities (Figure 15). As expected, a discretization effect can be observed. The amplitude of the strength decrease is not constant, with a higher strength decrease for the lower L/d values. As noted in section 5, in the case of poor discretization levels, the properties of the intact part of the material can vary dramatically. Here, with the presence of discontinuities, this effect is reported at the scale of each intact cell separated by the discontinuities. Thus, for different L/d, values, each cell has a different fabric, inducing a variation of the global strength of the sample. Thus, attention must be paid when a network of discontinuities is introduced into a DEM and comparisons need to be made between results using different discretization sizes.

9 Maximum strength (MPa) Figure 15. Maximum strength values obtained from triaxial compressions test performed on samples with different discretization size, 3200, 10800, 5000 ED (i.e. L/d = 15, 22 and 37 respectively) at 200 kpa confinement. 6. Scale effect on strength Fracture density [m -1 ] A test configuration has been set up to assess whether or not the DEM model can reproduce the scale effect on strength, by comparing its predictions to the Medhurst and Brown results [3] obtained on coal samples of diameters ranging from 60 to 300 mm. Using a given fixed distribution size for the DE (d=8.2 mm), four cubic samples of different edge lengths L i =60, 120, 180 and 300 mm, have been generated (Figure 16), and subjected to triaxial loadings according to the previous procedures. A slight decrease of the strength, which converges to a limit value, can be observed when the sample size increases (see also Figure 20). This slight decrease of the strength is of the same nature as the one observed in section 5. It is due to the low resolution configuration of the smallest sample. Despite this artifact, it can be concluded that the intact material does not show any scale effect regarding its strength, which is confirmed by previous observations [19, 20]. A discrete fracture network (DFN) is now defined and introduced into the numerical samples (see Figure 18). This DFN has the same characteristics as the ones used in section 6 in terms of its spatial distribution (bedding planes, face and butt cleats) and mechanical properties. The difference is that the DFN is kept constant for all the samples, i.e. the space length between the discontinuities is kept constant and equal to the edge length of the smallest sample (60 mm), which is considered to be the reference intact length. Doing so, the numerical simulation procedure is consistent with the Medhurst and Brown study, which considered the smallest sample of 60 mm length to be the intact one (see values in Table 5) when using the HB criterion to quantify scale effects for the considered case. The stress-strain and volumetric curves obtained for the tests performed under a 200 kpa confinement are presented in Figure 19. The corresponding peak strength values are plotted as a function of the sample size in Figure 20. Z Y X Z Y X Figure 16. Four intact cubic samples of different sizes L i=60, 120, 180 and 300 mm, with constant particle size distribution (d=0.82 mm). Results in terms of stress-strain responses and volumetric deformations of the intact material (i.e. no DFN inside) are presented in Figure 17. Figure 17. Differential stress q and volumetric deformation dv/v obtained during triaxial compressions performed for a confinement of 200 kpa, on four intact samples with length L i ranging from 60 to 300 mm. Figure 18. Four cubic samples of different sizes with constant Discrete Element size distribution and constant DFN (i.e. same spacing between discontinuities). Table 5. Hoek and Brown criterion parameters for Moura coal depending on sample size, from [3]. Sample size Hoek and Brown parameters σ c (MPa) m s 60 mm 32.7 19.4 1 101 mm - 13.3 0.555 146 mm - 10.0 0.236 300 mm - 5.7 0.184

10 Figure 19. Differential stress q and volumetric deformation dv/v evolutions obtained during triaxial compressions performed under 200 kpa confinement on four pre-fractured numerical samples with length L i ranging from 60 to 300 mm. Maximum strength (MPa) Figure 20. Scale effect on the maximum strength of the numerical model. Comparison between intact and pre-fractured samples. In the presence of discontinuities, the decrease in the maximum strength is now well pronounced and it is consistent with the experimental observations provided by Medhurst and Brown and expressed in terms of the generalized Hoek and Brown criterion (see Figure 21). A remarkable aspect concerns the contribution of the DFN density with the sample size. When representing the evolution of this density (computed here as the ratio between the total surface S of the discontinuities and the total volume V of the sample, which is also known to be the P 32 value [19]) versus the length L of the sample, it can be observed that the DFN density converges towards a limit value (see Figure 22). This convergence is associated to the convergence observed for the maximum strength value of the cleated sample (Figure 20). Figure 21. Medhurst and Brown [3] experimental results on coal samples of different sizes compared to DEM predictions for an increasing sample size in terms of the generalized Hoek and Brown criterion [15]. Thus, when the size of the sample increases, as long as the density of the DFN increases, a decrease of the maximum strength is observed. One could extrapolate that given any sample size, as soon as the density of the DFN remains constant, the maximum strength of this sample will no longer change. These results suggest that the scale effect on strength mainly relies on the evolution of the DFN density rather than directly on the size of the sample. Fracture density [m -1 ] Figure 22. Evolution of the density of fractures with the sample length. Comparable numerical results have been recently obtained by Esmaieli et al. [19]. When using disordered DFNs based on observations from a volcanic rock, their model exhibits a decrease of the sample s strength when the size of the sample increases. Depending on the DFN, for the same number of fractures, different densities of fractures (P 30 ) could be also generated. As long as the density of fractures increases, the strength of the sample decreases. An interesting point is that for the same number

11 of fractures, different values of P 30 could be obtained. Thus, even if the quantity of fractures increases when the size of the sample increases, a decrease of the strength is obtained only when an increase of the density P 32 occurs. These results fully corroborate our results in terms of scale effect, i.e. the preeminent role of the density of fractures. Based on the 2D method of fractal geometry provided by Zengchao et al. [20] it is suggested that the scale effect on the rock mass strength can be formulated using the fracture quantity, the fractal dimension of fracture quantity distribution and rock mass size. For 3D fracture networks, the 2D information such as the distribution of the fracture length cannot give a unique solution for the total fracture surface. Hence, eventhough the present model provides results comparable to Zengchao et al. [20], one can wonder whether or not their model suffers from the lack of a more complete 3D formulation. 7. Conclusion The effect of discontinuities on strength was numerically investigated using a Discrete Element Method. The model integrates structural effects induced by discontinuities which are known to produce strong behavior changes in rock-like materials, scale dependency above all. Triaxial test simulations were performed on intact samples and samples with pre-existing discontinuities. The results were systematically compared to experimental ones in order to estimate the reliability of the method. Emphasis was put on possible numerical bias. Resolution effects (i.e. Discretization size effects) have been shown for low resolutions which impose special care when discrimination studies and quantitative results are expected. Controlling the coordination number is one of the possible solutions to get rid of this artifact. The study focused on the relationship between the strength and the set of pre-existing discontinuities (DFN) in a numerical sample. Comparisons with experimental data have shown that the model can reproduce a strength dependency on the density of the DFN. The effect of structural features on the fabric of the material can be simulated. It has been shown that, starting from an intact material, increasing the DFN density tends to diminish the strength properties in a way quantitatively comparable to the experimental observations. When increasing the size of the numerical sample, it has been observed that the strength is kept constant when discontinuities are absent or when the DFN density (P 32 ) is not modified. As soon as the DFN density varies, the strength is modified as well. This conclusion is in good agreement with the results recently obtained by Esmaieli et al. [19], where the role of the density of fractures has been explicitly considered in the scale effects on the strength of disordered fractured rock mass. Thus, the formulation of a relationship between the distribution of fractures and the size of the rock mass would need to integrate an accurate value for P 32. Moreover, the contributions of the persistency of the fractures and their interconnectivity would also have to be quantified in this formulation. Acknowledgments Authors would like to acknowledge the Large Open Pit Project for its participation in this work. References [1] Z.T. Beniawski, The effect of specimen size on compressive strength of coal, International Journal of Rock Mechanics and Mining Sciences, 5, pp. 325-335, 1968. [2] N. Deisman, D. Mas Ivars, C. Darcel, R.J. Chalaturnyk, Empirical and numerical approaches for geomechanical characterization of coal seam reservoirs, International Journal of Coal Geology, doi:10.1016/j.coal.2009.11.003, 2009. [3] T.P. Medhurst and E.T. Brown, A study of the mechanical behaviour of coal for pillar design, International Journal of Rock Mechanics and Mining Sciences and Geomechanics Abstracts, 35(8), pp. 1087 1105, 1998. [4] G.S. Esterhuizen, The effect of structural discontinuities on coal pillar strength as a basis for improving safety in the design of coal pillar systems, Project report, University of Pretoria, 1998. [5] A.B. Swilski, Determination of the anisotropic elastic moduli of coal, International Journal of Rock Mechanics and Mining Sciences and Geomechanics Abstracts, 21(1), pp. 3-12, 1984. [6] S. Okubo, K. Fukui and Qi Qingxin, Uniaxial compression and tension tests of anthracite and loading rate dependence of peak stregth, International Journal of Coal Geology, 68, pp. 196-204, 2006. [7] G.K.W. Dawson and J.S. Esterle, Controls on coal cleat spacing, International Journal of Coal Geology, DOI: 10.1016/j.coal. 2009.10.004, 2009. [8] F.V., Donzé, V. Richefeu and S.-A. Magnier, Advances in Discrete Element Method applied to Soil, Rock and Concrete Mechanics, in: State of the art of geotechnical engineering, Electronic Journal of Geotechnical Engineering, p. 44, 2009 [9] J. Kozicki and F.V. Donzé, A new open-source software developed for numerical simulations using discrete modeling methods, Computer Methods in Applied Mechanics and Engineering, 197, pp. 4429-4443, 2008. [10] F. V. Donzé, J. Bouchez, S. A. Magnier, Modeling fractures in rock blasting, International Journal of Rock Mechanics and Mining Sciences, 34, pp. 1153-1163. 1997. [11] R. Hart, P.A. Cundall and J. Lemos, Formulation of threedimensional distinct element model Part II - mechanical calculations of a system composed of many polyhedral blocks. International Journal of Rock Mechanics and Mining Sciences and Geomechanics Abstracts, 25, pp. 3117-125.1988. [12] J.P. Plassiard, N. Belheine and F.V. Donzé, A spherical discrete element model: calibration procedure and incremental response, Granular Matter, 11, pp. 293-306, 2009. [13] M. Pierce, D. Mas Ivars, P.A. Cundall and D. Potyondy, A synthetic rock mass model for jointed rock. Proceedings of the first CA-US rock mechanics symposium, Vancouver, BC, pp. 341-349, 2007. [14] D. Mas Ivars, D. Potyondy, M. Pierce and P.A. Cundall, The smoothjoint contact model. Proceedings of the eighth world congress on computational mechanics, fifth European congress on computational

12 methods in applied sciences and engineering, paper no. a2735, Venice, Italy, 2008. [15] E. Hoek and E.T. Brown, Practical estimates of rock mass strength. Int J Rock Mech Min Sci, 34(8), pp. 1165-1186, 1997. [16] J.L. Justo, E. Justo and J.M. Azanon, The use of rock mass classification systems to estimate the modulus and strength of jointed rock, Rock Mech Rock Eng, 43(3), 287-304, 2009. [17] F. Heuze, Scale effects in the determination of rock mass strength and deformability, Rock Mech, 12, 167-192, 1980. [18] S. Hentz, L. Daudeville and F.V. Donzé, Identification and validation of a discrete element model for concrete, J Eng Mech, 130(6), pp. 709-719, 2004. [19] K. Esmaieli, J. Hadjigeorgiou and M. Grenon, Estimating geometrical and mechanical REV based on synthetic rock mass models at Brunswick Mine, International Journal of Rock Mechanics and Mining Sciences, DOI: 10.1016/j.ijrmms.2010.05.010, online 2010. [20] F. Zengchao, Z. Yangsheng and Z. Dong, Investigating the scale effects in strength of fractured rock mass, Chaos, Solitons & Fractals, Volume 41, Issue 5, 15 September 2009, pp. 2377-2386, [21] I. Evans and C.D.Pomeroy Strength fracture and workability of coal. London: National Coal Board; 1966.