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1 Powder Technology 25 (211) Contents lists available at ScienceDirect Powder Technology journal homepage: Numerical simulation of particle breakage of angular particles using combined DEM and FEM A. Bagherzadeh Kh., A.A. Mirghasemi, S. Mohammadi School of Civil Engineering, College of Engineering, University of Tehran, Tehran, Iran article info abstract Article history: Received 21 January 28 Received in revised form 14 June 21 Accepted 29 July 21 Available online 16 August 21 Keywords: DEM FEM Particle breakage Rockfill Marsal's breakage factor q v p behavior One of the effective parameters of the behavior of rockfill materials is particle breakage. As a result of particle breakage, both the stress strain and deformability of materials change significantly. In this article, a novel approach for the two-dimensional numerical simulation of the phenomenon in rockfill (sharp-edge particles) has been developed using combined DEM and FEM. All particles are simulated by the discrete element method (DEM) as an assembly and after each step of DEM analysis, each particle is separately modeled by FEM to determine its possible breakage. If the particle fulfilled the proposed breakage criteria, the breakage path is assumed to be a straight line and is determined by a full finite element stress strain analysis within that particle and two new particles are generated, replacing the original particle. These procedures are carried out on all particles in each time step of the DEM analysis. Novel approach for the numeric of breakage appears to produce reassuring physically consistent results that improve earlier made unnecessary simplistic assumptions about breakage. To evaluate the effect of particle breakage on rockfill's behavior, two test series with and without breakable particles have been simulated under a biaxial test with different confining pressures. Results indicate that particle breakage reduces the internal friction but increases the deformability of rockfill. Review of the v p variation of the simulated samples shows that the specific volume has initially been reduced with the increase of mean pressures and then followed by an increase. Also, the increase of stress level reduces the growing length of the v p path and it means that the dilation is reduced. Generally, any increase of confining stress decreases the internal friction angle of the assembly and the sample fail at higher values of axial stresses and promotes an increase in the deformability. The comparison between the simulations and the reported experimental data shows that the numerical simulation and experimental results are qualitatively in agreement. Overall the presented results show that the proposed model is capable with more accuracy to simulate the particle breakage in rockfill. 21 Elsevier B.V. All rights reserved. 1. Introduction Particle breakage, designated to describe the fracture of the constituent components (grains) of a soil structure, has been frequently observed in various soil-rockfill masses such as rockfill dams. Several laboratory oriented research tests [1 4], have shown that many engineering characteristics of granular materials such as strength (stress strain), deformability, pore pressure distribution and permeability are greatly influenced by the level of breakage of materials [1,2]. Marsal [3,4], who was perhaps the first to deal with the concept of crushing of particles through large-scale triaxial tests, summarized the phenomenon of breakage in rockfills as, It seems that phenomenon of fragmentation is an important factor that impacts shear resistance and potentiality of compaction of grain Corresponding author. No.1, Khoddami Ave., Vanak Sq., Tehran, Iran , P.O. Box Tel.: , fax: address: A_bagher_kh@Yahoo.com (A. Bagherzadeh Kh.). materials and this phenomenon is effective on aforesaid parameters in different conditions of implementing stresses such as confining pressure stage or stage of divertive loading in triaxial test. 2. Breakage of particle In a granular medium, the interaction forces are transferred through the contact between particles. This phenomenon becomes more complicated because of the different geometrical shapes and various mineralogy of these particles. In 1921, Griffith [quoted from 4] suggested a theory for considering the breakage path within a brittle particle based on the main assumption that fracture occurs due to gradual expansion of pre-existing cracks. Studies of Joisel (1962) [quoted from 4] on crushing within a particle resulted in presenting a simple model for breakage based on the elastic modulus of different minerals of that particle. This model could only describe the breakage path under uniaxial pressure. In 1973, Marsal [4] presented an equation by comparing the results of the studies of Joisel and Griffith for calculation of a load required for crushing a particle /$ see front matter 21 Elsevier B.V. All rights reserved. doi:116/j.powtec

2 16 A. Bagherzadeh Kh. et al. / Powder Technology 25 (211) B C F x B C F n A A F s Fig. 1. Boundary and loading conditions on a particle with 3 contacts. Several researchers have studied the ratio of principal stresses (σ 1 / σ 3 ) imposed on different rock materials at failure by triaxial tests [5]. In an overall view, the ratio of principal stresses and the quantity of sin (ϕ), which is an indicator of shear strength, reduce by an increase in the amount of particle breakage. The mobilized internal friction angle of granular soils can then be calculated as follows [3 5]: σ 1 SinðφÞ = σ 1 σ 3 σ 1 = 3 σ 1 + σ σ : ð1þ 3 1 σ +1 3 Other studies have concluded that any increase of particle breakage leads to a reduction in void ratio and therefore the material becomes more deformable. Marsal believed that changes in void ratio were due to new arrangement of grains after breakage and filling up of void spaces with smaller broken pieces. Lade and Yamamuro came to the conclusion from tests on sand with different confining pressures (from to 7 MPa) that the breakage of particles played the major role in changing the volume of materials under high pressures [1,6]. 3. A brief review on simulation of particle breakage Cundall, a pioneer of using DEM (discrete element method) in studying the behavior of granular media and stability of rock slopes, developed the RBMC code in which the breakage mechanism of rock blocks was simulated similar to that of a Brazilian test [7,8]. In this code, in each cycle of simulation from the set of all point loads applied to each block, the application point and magnitude of the two maximum loads, which are applied in opposite directions, are determined. σ 1f σ c σ 1 σ 1 SF= σ 1f σ 1 Potapov and Campbell [9,1] have studied the breakage induced in a single circular particle that impacts on a solid plate and the brittle particle attrition in a shear cell. In both simulations, a breakable solid material is created by attaching unbreakable and non-deformable solid triangular elements. It is assumed that a cohesive joint can only withstand normal tensile stress up to some limit. If the tensile stress on any portion of the joint exceeds the limit, the cohesion along that portion is removed and can no longer bear any tensile stress; creating a crack along that portion. In an alternative approach and in order to study the process of fragmentation in two-dimensional brittle blocks, Kun and Hermann [11] considered each block as a mesh of inter-connected tiny cells located within that block. Such a cellular mesh is generated by the use of a random process (Voronoi Construction). Each cell is a rigid convex polygon that as the smallest component of the block neither breaks nor deforms and acts as a distinct element of other cells. Cells have one rotational and two linear degrees of freedom in the block plane and their behavior in contact is simulated by DEM. In order to study the influence of particle breakage on macro- and micro-mechanical parameters in two-dimensional polygon-shaped particles, Seyedi Hosseininia and Mirghasemi [12] have presented a simple DEM model, where each uniform (uncracked) particle (arbitrary convex polygon-shaped) is replaced with smaller interconnected bonded rigid sub-particles. If the bond between subparticles breaks, breakage will occur. Robertson and Bolton [13] and McDowell and Harireche [14] simulated three-dimensional crushable soils by using the DEM technique, as implemented in PFC 3D. In this method, agglomerates are made by bonding elementary spheres in crystallographic arrays. Stiffness bonding and slip models are included in the constitutive representation of contact points between the elementary spheres. It limits the total normal and shear contact forces by enforcing bondstrength limits. The bond breaks if either of these limits is violated. A slip model acts between un-bonded objects in contact, or between bonded objects when their contact breaks. It limits the shear force between objects in contact and allows for slip to occur at a limiting shear force, governed by the Coulomb's equation. In this approach, the shear and tensile bond strengths are set equal; much higher tensile strengths than the observed ones are assumed. Nevertheless, it has been accomplished for the simulation of silica sand grains and the results have been compared with the available test data [15]. The method can efficiently model the behavior of sands, whereas, it cannot be used for particles with sharp angles such as rockfills, since the proposed procedure for sand agglomerate consists of only smaller rounded spheres. σ t σ 3 Fig. 2. Definition of the safety factor. 4. Present methodology of particle breakage In this research, the phenomenon of particle breakage in a rockfill (sharp-edge) material is simulated under a biaxial test (pure shear)

3 A. Bagherzadeh Kh. et al. / Powder Technology 25 (211) (a) (b) θ' Directions of max. tensile stress θ ' = π/2 φ θ' Directions of shear failure surface Fig. 3. Failure surfaces of tensile and shear modes. (a) Tensile mode. (b) Shear mode. * Arrows show the direction of stress on the element. condition by a new methodology based on a model of combined DEM and finite element method (FEM). Successful applications of the combined discrete and finite element methods have already been reported by a number of researchers in other applications [16 19]. This novel approach has been proposed for improving the existing simulations of the phenomenon of the breakage of polygon-shaped particles, by removing the need for any definition or preliminary assumptions of the breakage path in particles. In the proposed method, all particles are simulated by DEM and after each step of DEM analysis, each particle is separately modeled by FEM to determine its possible breakage. The breakage analysis will be performed based on the loading conditions. If the particle is to break, the breakage path is assumed to be a straight line, determined by a full finite element stress strain analysis within that particle and two new particles are generated, replacing the original particle. These procedures are carried out on all particles in each time step of the DEM analysis. In this research, the POLY software [2,21] is used for DEM modeling of irregularly sharp-edge shaped particle assemblies under biaxial tests. Also, a new developed code (FEA) is used to analyse the breakage within a particle using FEM. The following two main criteria are then required to determine: - Onset of fracture in each particle. - Breakage line within each broken particle DEM simulation of particle assembly Due to the discontinuous nature of granular materials, the discrete element method has been widely adopted as an effective method for simulation of polygon-shaped particles (rockfill materials). A series of successive calculations in certain time intervals are carried out to obtain the stress/force equilibrium within the assembly. Time intervals of Δt have to be small enough to ensure numerical stability, (a) X=2. Cm (b) X= Y Cm Example A: beam under gravity-vertical load at the centre. Point A Point A X=1.25 Cm Example B: sample under unconfined compressive test. X= Y Cm Point A Point A Example C: sample under simple shear test. Y= X Cm Y= X Cm Point A Point A Fig. 4. Failure line for examples simulated by the FEA software to evaluate the effect of weak zones on the breakage line. (a) Breakage lines without modeling the weak zones. (b) Breakage lines with modeling the weak zones. * Point A is assumed as the coordinate center.

4 18 A. Bagherzadeh Kh. et al. / Powder Technology 25 (211) (a) 4. Cm (b) Cm Example A 7.5 Cm Example B 7.5 Cm Example C 2.5 Cm Fig. 5. Three examples simulated by the FEA software incorporating weak zones. (a) Location of weak zones. (b) Plastic zones after the analysis. while the velocity of particles in each interval can be assumed almost constant. Accordingly, if time steps are sufficiently short, a particle can only affect its immediate adjacent particles during each time interval. Therefore, in order to calculate the forces imposed on each particle at any time, only the particles that are in contact with that particle are taken into consideration. In this method, particle deformations remain too small compared with the deformation of assembly. As a result, particles are assumed to be rigid, and may only slightly overlap each other at contact points which generate corresponding contact forces of particles. In each cycle, DEM calculations include application of the second law of Newton for determination of particle displacement and the force displacement relation to calculate the contact force between two particles from their relative overlaps. The stress tensor of an assembly with area of A can be calculated based on the existing contact force f i C and the contact vector l j C as suggested by Rothenburg (198) [22]: σ ij = 1 A f C i 1 C j i; j =1; 2: ð2þ C A (a) With Breakage 7.51 Cm (b) Exerimental test Failure Surface (Line) RAT & IND (%) RAT & IND versus Poisson's Ratio y = -.811x RAT IND Linear (RAT) Linear (IND) y = x Cm Poisson's Ratio 5 Fig. 6. Comparison of the numerically predicted failure line and the observed failure line (in an experimental unconfined test). Fig. 7. Variations of plastic indicators versus Poisson's ratio in unconfined compression tests.

5 A. Bagherzadeh Kh. et al. / Powder Technology 25 (211) RAT & IND (%) RAT & IND versus UnConfined Strength y = -.6x y = -.8x RAT IND Linear (RAT) Linear (IND) 5 1 UCS (MPa) Fig. 8. Plastic indicators at failure versus unconfined compression strength in unconfined tests Analysis of particle breakage using FEM In the proposed model, in an assembly, each particle is considered intact without any voids and cracks. Each DEM time interval, each particle is analyzed by the developed FEA code (based on FEM) subjected to contact forces from neighboring particles. The resulting stresses from FEM analysis of a particle allow for determination of plastic elements of that particle using the popular Hoek Brown failure criterion [23,24]. Also, the probability of breakage in a particle is estimated based on the number of plastic elements. Details of the proposed breakage analysis procedure are now described in more detail Principles of finite element method Due to the geometrical shape of rockfill materials, the triangular linear element (3-node) was chosen for meshing each particle. In a finite element analysis, loadings and fixed points should be precisely defined which are determined on the basis of contact points of its adjacent particles. Fig. 1 shows the boundary condition and external loading for a sample typical particle with three contacts. For particles with more than three contacts, two contact points are assumed as fixed points and the remaining contacts points are considered as the points of external loadings. The external load is determined from the contact overlap area between the particles by the governing DEM contact law. In this research, a linear elastic model is used for stress strain finite element analysis. The conventional solution of linear elastic FEM model has been comprehensively presented during the past decades [25] and will not be reviewed here Determination of plastic elements within the particles The second key concept in the breakage analysis is the use of an appropriate rock failure model for determining the plastic elements RAT & IND (%) 1.8 RAT & IND versus Deviatoric Pressure TRIAXIAL TESTS SIMULATION y =.12x Deviatoric Pressure (MPa) y =.17x RAT IND Linear (RAT) Linear (IND) Fig. 9. Variations of plastic indicators at failure versus deviatoric stress in triaxial tests within the FEM mesh. Different constitutive models such as Mohr Coulomb, Hoek Brown, Griffith, Morel, Franklin, Hobs, etc. can be used as a rock failure criterion. The popular Hoek Brown failure criterion has been selected to determine the rock failure in this research. In 198, Hoek and Brown presented the following relation for the purpose of determination of failure in intact rocks [23]: m σ 1f = σ 3 + σ b σ :5 3 c +1 ; σ σ 3 N σ c c m b σ 1f = σ 3 ; σ 3 σ c m b : The coefficient m b is a characteristics constant value and σ c is the uniaxial compression strength of rock. The elements under tension larger than σc m b will fail in the tensile mode, while failure in elements with a high-compressive stress occurs when the major principal stress becomes equal or larger than σ 1f (shear mode). In order to define a failure (plastification) safety factor for an element, the following relations are used: For tensile mode: SF = σ t σ 3 For shearmode: Fig. 1. Initial generated assembly of particles. SF = σ 1f σ 1 : Parameters used in Eq. (4) are shown in Fig. 2. Safety factors equal to or greater than 1 introduce elastic elements, whereas safety factors smaller than 1 illustrate occurrence of failure in that element (plastic element). The plastic elements will then determine the breakage path within the particle. Since the linear elastic model is used as the constitutive model of rock, it is possible that generation of maximum and minimum principal stresses at failure may violate the Hoek Brown criterion Determination of breakage line There are two different methods for determination of a linear path based on a set of pre-determined points. They are either based on least squares of error for the horizontal distances (method X) or vertical distances (method Y) of points to the line. It is noted that, in these methods, all points have a similar level of effect on the fitted line because of the similar weighting coefficient. Also, it should be noted that stress strain analyses in this research have been carried out on the basis of a linear elastic model, and the assignment of plastic elements (points of breakage line) are basically different from a full ð3þ ð4þ

6 2 A. Bagherzadeh Kh. et al. / Powder Technology 25 (211) (a) (b) Fig. 11. Isotropically compacted assembly. (a) Assembly of particles during compaction. (b) Displacement trajectories of all particles during compaction. plastic analysis. However, it is numerically acceptable that elements with minimum safety factors are the elements with possible failure usually occuring around them (in plastic analyses). Therefore, definition of a weighting coefficient (W i ) effectively enhances the accuracy of determination of a breakage path. The weighting coefficient of point i with the safety factor of SF i can be defined as: W i = 1 SF i : With determination of the weighting coefficients of plastic points, elements that have been turned into plastic state faster shall have greater weighting coefficients. As a result, they are expected to have greater effect on the breakage line and the line will remain closer to these points. Weighting coefficients are implemented in both X and Y least squares methods. If the least square method in Y direction is considered and the equation of the best line is assumed to be from n ð5þ points with accurate coordinates (x i,y i ) (Y=mX+b), then the total sum of error squares is: Δ i =y iðaccurateþ y iðcalculatedþ y iðcalculatedþ =mx iðaccurateþ +b n S = W i Δ 2 i i =1 where, W i is the weighting coefficient related to point i, and Δ i is the associated distance error. Minimization of S with respect to m and S S b, = and =, respectively, allows for evaluation of the m b optimum values of m and b: m = w : i w i x i y i w i x i : w i y i w i : w i x 2 i w ð ix i Þ 2 b = w ix 2: i w i y i w i x i : w i x i y i : w i : w i x 2 i w ð ix i Þ 2 ð6þ ð7þ

7 A. Bagherzadeh Kh. et al. / Powder Technology 25 (211) (a) (b) No Breakage With Breakage 39 2 Approximate location of the shear path Fig. 12. Properties of particles assembly after shearing for a confining pressure of 14. MPa. (a) Assembly of particles at failure. (b) Displacement trajectories of all particles after shearing. A similar approach can be adopted for the X-weighted least squares method: X=m Y+b m = w : i w i x i y i w i y i : w i x i w i : w i y 2 i w ð i y iþ 2 b = w iy 2: i w i x i w i y i : w i x i y i : w i : w i y 2 i w ð iy i Þ 2 As mentioned above, both weighted least square methods can be used to determine the breakage line within a broken particle. Thus, from the two independent potential breakage lines for each particle one should be selected as the final breakage path of that particle. Therefore, a new criterion is required to make this selection. Numerical studies have shown that the slope of the failure surface in the first plastic element can be considered as a proper criterion for selection of the final breakage line. As the first crack is created in and propagated along the first plastic element, a breakage line with the Table 1 Parameters used in simulations. WB NB Normal and tangential stiffness (N/m) Unit weight of particles (kg/m 3 ) Friction coefficient Strain rate.5.5 Rock parameters Modulus of elasticity (E) (MN/m 2 ) Poisson's ratio (ν) 7 Compressive strength (MN/m 2 ) 3 m b 25. S b 1. a ð8þ closet slope to the direction of the failure surface within the first plastic element is selected as the final breakage line. As mentioned before, the two main failure modes of rock are tensile and shear ruptures. It is clear that in a rock element under the tensile mode, the failure is mobilized along the direction of minor principal stress. Fig. 3-a shows a schematic view of this mode. In contrast, if the rock reaches to failure under the shear mode, two different directions can be anticipated for the failure surface. In these elements, the angle of one of the failure directions with respect to the direction of the major principal stress is: θ f = π 4 + ϕ 2 : Since the angle between the two failure directions is equal to π/2 ϕ(fig. 3-b); in contrary to the tensile mode, both failure surfaces in the shear mode become a function of the internal friction angle of rockfill materials. In general, while the direction of principal stress is obtained from the stress strain analysis based on the finite element method, the internal friction angle of rock is required for determination of the failure surface of sheared elements. In 22, Hoek defined the relation between parameters of the Hoek Brown criterion and ϕ, based on the comparison of a linear Hoek Brown assumption and Mohr Coulomb relation in principal stresses [26]: " # ϕ = Sin 1 6a:m b ðs b + m b σ 3n Þ a 1 21+a ð Þð2+aÞ +6a:m b ðs b + m b σ 3n Þ a 1 ð9þ ð1þ where parameters a, m b and S b are the Hoek Brown parameters for rock. a= and S b =1. are used for an intact rock, and m b can be obtained for various rocks from the experimental tests [24]. To show the efficiency of the proposed approach to select the final breakage path, three simple examples: A, B and C, illustrated in Fig. 4,

8 22 A. Bagherzadeh Kh. et al. / Powder Technology 25 (211) (a) 7. Confining Pressure 2. MPa (b).7 Confining Pressure 2. MPa σ1-σ3 (MPa) With Breakage No Breakage Sin(φ) - Mobilized Friction Angle No Breakage (c) Volumetric Strain(%) Confining Pressure 2. MPa No Breakage With Breakage (d) Percent of Particle Breakage (%) Confining Pressure 2. MPa (e) Volumetric Strain (%) Confining Pressure 2. MPa Percent of Particle Breakage (%) Fig. 13. Results of biaxial test simulations under a confining pressure of 2. MPa. are considered. In example A, an elastic beam under vertical loading at the beam center has been simulated, while examples B and C have studied uniaxial compression and direct shear tests on a rectangular specimen of intact rock, respectively. Properties of quartzite were assumed for the rock. Two separate breakage analyses have been carried out; with and without the presence of weak elements within the mesh. At first, all three examples were simulated by the FEA code to determine the final breakage line (Fig. 4(a)). Then, some of elements within the mesh were replaced by a weak type rock (2% of the rock strength) in order to simulate a weak zone (Fig. 5(a)). The patterns of plastic elements for these cases are given in Fig. 5(b). Fig. 4(b) shows the final breakage line for the examples with the weak zone. Comparison of the determined breakage line in two series of examples (Fig. 4(a) and (b)) presents the good efficiency of the proposed mechanisms to determine the final breakage path. As expected, the breakage line is located along the weak zone Criteria of particle breakage An important step in accomplishing the proposed modeling of particle breakage is the selection of a proper criterion for breakage and consequently, determination of geometrical specifications of newly generated particles, if a particle is broken. So far, a series of mechanisms have been discussed to determine the final breakage path for a particle. In this stage of modeling, however, the question of

9 A. Bagherzadeh Kh. et al. / Powder Technology 25 (211) Fig. 14. Variations of maximum principal stress ratio versus the Marsal breakage factor (%); obtained from simulated biaxial test and experimental tests reported by different researches [5]. whether the particle under study crushes or not, should be answered. If the answer is positive, breakage occurs in the direction of the final breakage line and new particles are produced, whereas if the answer is negative, no breakage will occur. The following three criteria are suggested to determine the particle breakage: Plastic indicator along the final breakage line, Plastic indicator within the particle, and Simultaneous evaluation of the two above criteria. These criteria basically compare plastic indicators along the breakage line or within the particle with the pre-defined values. The plastic indicator along a breakage line (Ind) is defined as the ratio of the number of plastic elements along the line to the total number of elements through which that breakage path passes. Also, the plastic indicator within a particle (Rat) is defined as the ratio of the total number of plastic elements to the total number of elements within that particle. A series of laboratory results for rock failure tests, used in the dam construction projects of Iran, were collected and re-analyzed by the σ 1 -σ 3 (MPa) MPa 8. MPa 4. MPa No Breakage With Breakage 2. MPa 5 1. MPa MPa Fig. 15. Effect of the confining stress level on deviatoric stress in a biaxial test. FEA code to calibrate the above-mentioned indicators. Mechanical parameters of rock were fully determined by appropriate laboratory tests. Then, the laboratory tests were simulated by the proposed 2-D FEM model at the failure and the plastic indicators were determined; ignoring the 3D condition of the results. For this purpose, 51 unconfined and 3 triaxial tests together with 6 cases of Brazilian tensile strength tests, all performed on different intact rock specimens, were collected. They included rocks with different specifications such as diorite, basalt, quartz and limestone. Fig. 6 shows a sample result of the comparison between the laboratory unconfined test and its numerical simulation. Good agreement was observed in determination of the breakage line. In general, the values of Ind and Rat at the failure (i.e. IndF and RatF) for simulated samples were determined. Figs. 7 and 8 illustrate the effect of Poisson's ratio and the unconfined strength (UCS) of rock on the values of plastic indicators at failure. The relation between the values of indicators at failure and the mechanical parameters of rock is apparently opposite. For example, as elasticity modulus of rock increases, simulated rock samples fail at lower plastic indicators, an indication of higher fragile behavior which causes crushing of such materials before developing major plastic zones. The same variation is also observed with the increase of UCS. The average value of RatF is 4 and the average value of IndF is computed 2 for unconfined compressive tests. Fig. 9 shows the plastic indicators which were determined by the simulations of triaxial tests. This figure shows a direct relation between plastic indicators at failure with the deviatoric stress or confining pressure in triaxial tests. If the confining stress is increased, the breakage path cannot be easily formed and therefore more plastic elements will be created before failure. For this reason, variations that resulted from numerical simulations of triaxial tests are logical. The values of plastic indicators at failure in triaxial tests are within the range of 6 and.96. The average determined values of RatF and IndF for triaxial tests are.8 and.75, respectively. The average values of RatF and IndF for Brazilian tests are 8 and 9, respectively. In addition, the critical values of the proposed indicators (i.e. RatF and IndF) remain close together. As a result, it is acceptable to use only one of them to confirm the occurrence of breakage within a particle; the plastic indicator within a particle has been adopted in this research. Also, if in each rockfill particle, unconfined conditions are established, then the critical plastic indicator is proposed to be about 2 to 9. If a full confined condition is applied on a particle such as a triaxial test, the critical plastic indicator will be selected within the range of 6 and.93, depending to the intensity of the confining pressure. No full confined condition presents in most particles of DEM assemblies of this study, therefore the critical value of the plastic indicator at failure is assumed to be. Although a parametric study is presented in this paper, generally, it can be strongly recommended to perform laboratory tests, such as the unconfined test, on rocks to determine the values of critical plastic indicators. The critical value can then be readily selected by comparing the results of FEM simulation and experimental test on rock samples at failure. With this model, it is possible to study the influence of particle breakage on macro- and micro-mechanical behavior of simulated angular materials. The developed mechanism has been recently presented from the microscopic view and the effects of particle breakage on the microstructure of sharp-edge materials are discussed [27]. 5. Simulations and results To investigate the effect of particle breakage on the behavior of sharp-edge (rockfill) assemblies, several biaxial tests under different confining pressures were simulated on an initial assembly of 5 particles. Under each confining stress, two tests were simulated by the developed software with no possibility of particle breakage (NB) and with breakable particles (WB).

10 24 A. Bagherzadeh Kh. et al. / Powder Technology 25 (211) Sin (φ) - Mobilized Friction Angle.7 MPa 1. MPa 8. MPa 14. MPa 2. MPa 4. MPa Breakage Disabled Sin (φ) - Mobilized Friction Angle.7 MPa 1. MPa 8. MPa 14. MPa 2. MPa 4. MPa Breakage Enabled Fig. 16. Comparison between the mobilized friction angle for both groups of simulations (WB & NB). From the macroscopic point of view, the influence of particle breakage on strength, deformability and Marsal breakage factor are discussed. Also, the effects of stress level and rock strength on the particle breakage phenomenon are explained. The biaxial tests are carried out under drained condition and the tests are simulated in four continuing stages including compaction of initially generated assembly, relaxation of compacted assembly, application of hydrostatic pressure and finally shearing of the assembly. As Fig. 1 illustrates, the initial generated assembly of particles is loose due to existing large voids between the particles. To compact the assembly, under a strain control boundary, the boundary particles are moved towards the center of the assembly with a constant strain rate. This procedure has been shown in Fig. 11, illustrating the displacement trajectories of particles during this stage. As shown, movements of boundary particles move the internal particles towards the center of the assembly and the model is finally compacted. When the assembly is sufficiently compacted, a zero rate strain control loading is applied on the assembly's boundary particles. In other words, the boundary of the assembly is kept fixed in its place and the inside particles are allowed to slowly move and rotate in order to reach the state of minimum contact forces. Due to the displacement and rotation of particles within the assembly, particles are placed in new positions with minimum contact overlaps with their adjacent particles. Then, a stress control loading is used for applying the confining pressure, while the applied strain is controlled in such Volumetric Strain (%) MPa MPa 4. MPa MPa MPa 8. MPa 4. MPa MPa 14. MPa 14. MPa Breakage Disabled MPa 1. MPa Volumetric Strain (%) Breakage Enabled Fig. 17. Relationship between volumetric and axial strains at different confining pressures ( to 14. MPa).

11 A. Bagherzadeh Kh. et al. / Powder Technology 25 (211) Table 2 Maximum internal friction angle of samples. Confining pressure (MPa) manner that the average amount of internal stresses of particles reaches the pre-defined confining pressure. Accordingly, if the average amount of internal stresses is smaller than the applied hydrostatic pressure, the boundary particles approach to the center of the assembly, and get far from the center otherwise. This stage will continue until a balance is achieved between the pre-defined external hydrostatic pressure and internal stresses. For simulation of a 2-D model of a triaxial test, the deviatoric axial strain is applied in direction 2-2 (Fig. 12) under the constant confining pressure in the direction of 1-1. Simulations are continued until the axial strain of about 2% is reached. Fig. 12(a) shows the status of sheared sample in the final stage of simulated biaxial test (ε a =18%) under 14 MPa confining pressure. Fig. 12(a) illustrates the location of particles after the failure of a sample for the two simulated groups (NB and WB). The displacement trajectories of particles during the shear are demonstrated in Fig. 12(b) which shows that the shear paths within the assembly have been mobilized along four lines and more particle breakage has occurred along these. In both series of tests (WB and NB), the friction coefficient between particles is set to and particles are assumed cohesionless and weightless. In order to compare the results between the test groups, the parameters are kept the same for both test series. Simulations have been carried out under, 1, 2, 4, 8 and 14 MPa confining pressures. Table 1 shows the parameters used in numerical simulations Results and discussions WB group (degree) NB group (degree) Effect of particle breakage on the behavior of a rockfill material Fig. 13 shows variations of deviatoric stress(a), mobilized friction angle(b), volumetric strain(c) and percentage of particle breakage(d and e) for the test under a confining pressure of 2 MPa. These diagrams show a reduction in deviatoric stress for the test with breakable particles. This reduction causes the strength of assembly of breakable particles to be lower than the other one (Fig. 13(b)). Comparison of deformability shows that the particle breakage reduces Marsal's Particle Breakage Factor MPa 1. MPa 2. MPa 4. MPa 8. MPa 14. MPa Fig. 18. Variations of the Marsal's particle breakage factor during shear under different confining pressures. the sample's dilation (Fig. 13(c)); leading to more contraction. The breakage percentage of an assembly is simply defined as the ratio of the number of broken particles to the total number of initial particles. According to Fig. 13(d), variation of breakage percentage with respect to the axial strain in a biaxial test can be assumed as linear. The diagram of volumetric strain versus the breakage percentage of samples (Fig. 13(e)) indicates that the particle breakage during dilation stage is more than its value during the initial contraction of sample. This deference may be attributed to the mobilization of shear and tensile failures during the dilation of samples in comparison to the contraction phase. Marsal [3] presented a breakage factor, called B g, for the estimation of crushed particles. In this method, the value of breakage is calculated from the sieve analysis of rockfill samples as follows. Before testing, the sample is sieved using a set of standard sieves and the percentage of particles retained in each sieve is calculated. Due to the breakage of particles, the percentage of particles retained in large size sieves will decrease, whereas the percentage of particles retained in small size sieves will increase. The sum of decreases in percentage retained will be equal to the sum of increases in percentage retained. The sum of decreases (or increases) is the value of the breakage factor (B g ). The values of the maximum principal stress ratio (σ 1 /σ 3 ) max in biaxial simulation and experimental tests (collected from Varadarajan et al. [5]) are compared in Fig. 14 for various degrees of breakage (B g ). It is observed that the simulation results fall inside the lower bound of experimental data. Also, the degree of breakage increases with the decrease of ratio (σ 1 / σ 3 ) max in both numerical and experimental tests Effect of stress level on particle breakage Figs. 15 and 17 show the effect of stress level on deviatoric stress and mobilized friction angle in all simulated tests. As expected, with increasing confining pressure, deviatoric stresses are increased for both groups of simulations, but any increase of confining pressure increases the effect of particle breakage on reduction of deviatoric stresses. As shown in Fig. 16, any increase of confining stress decreases the internal friction angle (sin ϕ) of assemblies and the assemblies fail at higher values of axial stresses. These effects are more intensive in larger stress levels. These results have already been reported in laboratory tests and numerical simulations as well [5,12]. Table 2 shows the maximum angle of mobilized internal friction of the simulated assemblies. Table 2 shows that an increase in confining stress causes the reduction of ϕ max for both groups of simulations. As a result, particle breakage reduces the internal friction of rockfill materials at all stress levels. Three samples with breakable particles showed almost similar maximum friction angle under low-stress levels (, 1 and 2 MPa). It means that the increase of confining pressure at low-stress levels has no substantial effect on the maximum mobilized internal friction angle in WB group tests. At these stress levels, due to particle breakage, small broken particles fill the voids between the larger particles of the samples and so the maximum internal friction angle should not be reduced. In addition to filling the voids, these small broken particles are also placed between the larger particles. Hence, the internal texture of the sample is influenced by smaller particles and the reduction of ϕ is clearly noticed due to their activity in transfer of load within the larger particles. The same trend was reported by Marsal [3] in the laboratory large-scale triaxial tests on rockfills. It can be concluded that such a phenomenon is a result of a simultaneous effect of particle breakage and confining pressure on the shear strength of rockfill materials [12]. Fig. 17 illustrates that dilations of samples reduce and initial contractions increase with the increase of confining pressure for both series of tests. It generally seems that particle breakage phenomenon limits the dilation of samples at failure while preventing high probable contraction of samples due to creation of smaller particles at low-level strains.

12 26 A. Bagherzadeh Kh. et al. / Powder Technology 25 (211) Breakage Disabled Breakage Enabled 14. MPa q (MPa) MPa MPa 1 2. MPa MPa MPa P (MPa) Fig. 19. Effect of the confining pressure level on the stress path of rockfill materials in numerical simulations Particle breakage factor Fig. 18 shows variations of Marsal's particle breakage factor versus axial strain. The growing rate of this factor is higher at the beginning of biaxial tests. By increasing the axial strain, which is associated with the increase of imposed stresses on the assembly, more particles are crushed and voids become smaller. As a result, fine particles play a more important role in the transfer of stress to their adjacent particles. Since more particles participate in the transfer of force around a particle, lower contact forces are generated and the rate of breakage is continually reduced at higher strains. These results are in good agreement with the results reported by Marsal [3] on rockfill materials Effect of particle breakage on the q p v behavior of rockfill In this section, the behavior of rockfill materials as a function of specific volume (v), mean stress (p) and shear stress (q) is investigated. For this purpose, the following two variations are discussed. At first the q p diagram is presented to illustrate variations of the shear stress (q) versus the mean stress (p), in which p and q are defined in term of the functions of σ 1 and σ 2 principal stresses: q = σ 1 σ 2 p = 1 ð 3 σ 1 +2σ 2 Þ: ð11þ Also, the v p diagram is illustrated which shows variations of the specific volume of the sample (v) versus the mean stress of the sample. The specific volume is defined based on the void ratio of the sample (e) as: v =1+e: ð12þ Figs. 19 and 2 clearly show the effect of stress level on q p and v p variations. Fig. 19 shows that the shear strength of an assembly is increased by the increase of confining pressure. Also, particle breakage causes more reduction in the strength of assemblies at higher stress levels. This is in agreement with the findings of a number of researchers based on experimental tests [1,6]. The v q diagrams obtained from simulations (Fig. 2) indicate that simulated assemblies have behaved similar to pre-consolidated soils in all tests because of the high density of the samples. The specific volume has initially been reduced with the increase of mean pressures and then followed by an increase. This is in close conformity with the contractive dilative behavior of samples in popular stress strain views. Fig. 2 demonstrates the effect of confining pressure on variations of the specific volume. It is observed that the increase of stress level reduces the growing length of the v p path, which means that the dilation is reduced Effect of rock strength on the behavior of the assembly To study the effect of rock type in the phenomenon of particle breakage, simulations of three biaxial compression tests on samples 1.5 MPa 1.45 v MPa 2. MPa 4. MPa 8. MPa Breakage Disabled Breakage Enabled MPa P (MPa) Fig. 2. Influence of the confining stress on the v p graphs of simulated materials.