Modeling the fragmentation of rock grains using ...

Powder Technology 308 (2017) 388–397

Contents lists available at ScienceDirect

Powder Technology journal homepage: www.elsevier.com/locate/powtec

Modeling the fragmentation of rock grains using computed tomography and combined FDEM Gang Ma a, Wei Zhou a,⁎, Richard A. Regueiro b, Qiao Wang a, Xiaolin Chang a a b

State Key Laboratory of Water Resources and Hydropower Engineering Science, Wuhan University, Wuhan 430072, China Department of Civil, Environmental, and Architectural Engineering, University of Colorado Boulder, Boulder, CO 80309-0428, USA

a r t i c l e

i n f o

Article history: Received 20 August 2016 Received in revised form 6 November 2016 Accepted 26 November 2016 Available online 08 December 2016 Keywords: Rock grain Computed tomography Combined FDEM Fragmentation pattern Weibull statistics Fractal

a b s t r a c t The fracture and fragmentation of rock grains is attracting increasingly more research attention because it is relevant to today's challenges with particle raw materials. To contribute to understanding the fracture mechanisms of rock grains, the combined finite and discrete element method (FDEM) is adopted to simulate realistically shaped rock grains under single particle axial compression. Two groups of rock grains with different shape characteristics are considered to study their fracture processes under axial compression. Detained information of grain morphology was obtained using computed tomography (CT). The initial microstructure or heterogeneity within each rock grain is represented by a random distribution of the strength threshold. The influences of grain morphology on fragmentation patterns and fragmentation size distributions are systematically investigated. The results reveal that the dominant fracture mechanisms are related to the grain morphology. The more complex the grain shape, the more variable are the fracture patterns deviating from simple vertical splitting. Weibull statistics of grain crushing strength demonstrate that angular grains are characterized by both smaller characteristic strength and smaller Weibull modulus compared to those of rounded grains. This finding suggests that the crushing strength of angular grains may exhibit a more significant size hardening effect. The two-parameter Weibull equation can be used to fit the fragment size distribution, and the knowledge of the two fitting parameters is helpful for assessing the extent of fragmentation. The fragments of both angular and rounded grains satisfy the fractal distribution, and the fractal dimension narrowly distributes around 1.86 ± 0.16 irrespective of the initial morphology, which is close to the exponent τ of the analytical predication of the fragment size distribution results from merged crack branches in three dimensions, where τ≈1.67. © 2016 Elsevier B.V. All rights reserved.

1. Introduction Granular soils yield and exhibit irrecoverable deformation when compressed to high stresses. This macroscopic “yielding” has been shown to originate from grain breakage [1–3]. This grain scale behavior causes significant changes to the macroscopic response. Extensive laboratory investigations have been conducted to correlate the breakage behavior with the initial density, stress level, gradation characteristics, loading conditions, and grain characteristics [4–8]. Recently, some advanced in-situ measurement techniques, such as X-ray computed tomography (CT) scanning and synchrotron micro-computed tomography (SMT), have been adopted in experimental studies to visualize the development of grain breakage under axial compression [9–11]. As an effective complement to experimental studies, the discrete element method (DEM) has been widely used to study the mechanical behavior of crushable granular materials. Two groups of methods have been developed to simulate grain breakage in discrete element modeling, namely, the agglomerate method [12–16] and replacement method ⁎ Corresponding author. E-mail address: [email protected] (W. Zhou).

http://dx.doi.org/10.1016/j.powtec.2016.11.046 0032-5910/© 2016 Elsevier B.V. All rights reserved.

[17–19]. Both methods have their advantages and disadvantages [20]. One of the disadvantages of the agglomerate approach is the loss of individual grain surface continuity and angularity, which are necessary to properly simulate the contact behaviors. The DEM has proven to be efficient in investigating the micromechanics of crushable granular materials. However, for simplicity, many researchers have chosen to use spheres or disks in the replacement method. These models do not capture the true shapes and geometric orientations of granular soil particles [21]. The significant impact that real grain shape has on the stiffness and strength of sand was reported by Cho et al. [22]. Therefore, including the actual grain morphology in numerical modeling will enable us to better capture the behavior of a granular assembly. Various imaging techniques, including scanning electron microscopy, X-ray CT imaging, optical imaging, and neutron imaging, were employed to characterize the microstructure of granular materials [23,24]. Imaging techniques were also used for the particle shape acquisition and granular reconstruction [25–32], which can be further used in numerical simulations. For example, Matsushima et al. [29] used an agglomerate composed of many bonded spheres to approximate the irregular shape of a grain based on X-ray CT data; Cil and Alshibli [33] employed SMT and DEM

G. Ma et al. / Powder Technology 308 (2017) 388–397

to investigate the influence of particle morphology on the fracture behavior of silica sand. It is also now possible to obtain a finite element mesh of tomography image data for FEM simulation [21]; Latham et al. [28] employed 3D laser ranging (LADAR) to capture realistic rock grain geometries and established a shape library, which can be used in the non-spherical DEM and combined finite discrete element method (FDEM). Given the significant influence of varying grain morphologies and grain fracture and fragmentation on the “plastic” behavior of granular soils, it is therefore necessary that the detailed modeling of these processes should be involved in numerical simulations at the grain scale [21]. The influences of particle shape and breakage on the mechanical behavior of granular assemblies have been investigated by the same authors [34] [35]. In previous studies, the granular materials were represented by an assembly of polyhedral particles, which are randomly generated within ellipsoids by a specially designed and efficient algorithm [34]. This study continues the previous particle breakage research, and provides a more refined rock grain fracture modeling. In this paper, the combined FDEM modeling of uniaxial compression of rock grains is conducted, taking into account the actual grain morphology. Attempts are made to advance our understanding of the fracture behaviors of rock grains. Following a brief description of the combined FDEM and especially the particle breakage model, the reconstruction of rock grains using detailed CT information is illustrated. Then, single particle crushing tests on two groups of rock grains with different shape characteristics are simulated. The fracture processes of an angular grain and a rounded grain are graphically demonstrated and qualitatively compared. The governing fracture mechanisms leading to grain fragmentation are discussed. The influences of grain morphology on the fragmentation pattern, Weibull statistics of grain strength, and fragment size distribution are systematically analyzed and discussed. 2. Fracture modeling in the context of the combined FDEM The combined FDEM, a numerical method initially proposed by Munjiza et al. [36], permits the modeling of a large number of interacting bodies. The combined FDEM employs a combination of FEM techniques to assess the deformation and evaluate the failure criteria for fracturing and DEM concepts for detecting new contacts and addressing the translation, rotation, and interaction of discrete bodies. Combined FDEM solves the contact mechanics using a distributed contact force approach and a penalty function method, which are important for handling complex deformable shapes [37]. Some of the computational implementations of the combined FDEM are the Y-code developed by Munjiza [37] and its variants, e.g., VGW [38] and Y-Geo [39]. Other combinations of FEM and DEM have been commercialized to produce the codes UDEC [40] and ELFEN [41]. The explicit integration scheme and general contact capability of ABAQUS/Explicit also make it appropriate for capturing a large number of actual and potential contacts undergoing large deformation [42]. The fundamental processes of the combined FDEM, including the contact detection, contact interaction, and governing equations, are not repeated in this paper. Some relevant works are selected for readers' reference [43] [44]. In the combined FDEM, the initiation and propagation of fractures are explicitly simulated under the framework of nonlinear elastic fracture mechanics (NLEFM). The cohesive interface elements (CIEs) with zero thickness, which behave like the cohesive bonds in DEM, are inserted into the finite element discretization associated with each particle to consider the particle breakage. Specifically, non-thickness cohesive interface elements (CIEs) are inserted between the edges of all adjacent bulk element pairs from the very beginning of the simulation. Thus, arbitrary fracture trajectories can develop within the constraints imposed by the mesh topology. It should be noted that a remeshing technique can technically be implemented into the combined FDEM, but the computational cost will be extremely high when modeling crushable granular materials. As remeshing is not performed

389

and the mesh topology is not updated during the simulation, the particles therefore cannot continue to become smaller during the loading process. For brevity, only key details of the fracture model will be repeated; for full details of the model, as well as a critical discussion of the choice of model parameters, readers are referred to prior publications [35] [45]. Depending on the local tractions of a CIE, fracture can occur in mode I, mode II, and mixed-mode I/II conditions. A mode I fracture initiates when the normal traction, tn, reaches the tensile strength of the material, ft. As the damage progresses, tn is assumed to gradually decrease to zero, and a traction-free surface is formed. According to a slip-weakening model, a mode II fracture is initiated when the tangential traction, ts, reaches the shear strength of the material, fs. The tensile strength, ft, is assumed to be a constant, while fs is defined by the Mohr-Coulomb failure criterion with a tension cut-off, such that. f s ¼ c−t n tanφi t n ≤ f t

ð1Þ

where c is the internal cohesion, φi is the material internal friction angle, and tn is the normal stress perpendicular to the shear direction. Note that here, tensile stress is positive, and compressive stress is negative. The tension cut-off is automatically activated when tn N ft. Upon undergoing the shear strength, ts is assumed to gradually reduce to a purely frictional resistance, fr, which is given by fr = − tn tan φf, where φf is the fracture friction angle after the breakage of the embedded CIE. Besides pure mode I or mode II fracture, CIEs often undergo a combination of the two. Although the normal and tangential tractions of CIE may be less than the corresponding strength threshold, the resultant tractions could also lead to fracture. In such cases, fracture is assumed to initiate if the following coupled criterion involving tensile strength and shear strength is satisfied:

2 t ht n i 2 þ shear ≥1 fs ft

ð2Þ

where 〈〉 is the Macaulay bracket considering that compressive normal traction does not affect damage initiation. The post-peak gradual reduction of tractions acting on CIEs are governed by an energy-based linear softening law [35]. Following breakage, the broken CIE is removed from the model, and a physical discontinuity is formed. The contact detection and contact interaction algorithms in the DEM formation are then used to simulate the interaction between newly created fracture surfaces. 3. Grain shape library The utilization of actual grain morphology, obtained with either Xray CT or 3D laser ranging (LADAR), allows for the more realistic representation of granular materials. LADAR is well suited for the surface capture of coarse aggregate-sized particles (1– 10 cm) [28,46]. To achieve a balance between computational cost and accuracy of the results, the resolution selection is of paramount importance. In this study, it will be necessary to apply mesh coarsening to obtain more computationally manageable meshes while retaining an optimum of edge and triangle quality to preserve the topology of the grain surfaces [28]. The image data went through a series of image processing procedures and were finally converted to a surface mesh. The finite element meshes of triangulated surfaces and mesh connectivity can be imported into various preprocessors for meshing and visualization. Different sources of rock grains have different shape characteristics. Two sets of rock grains are selected to study their fracture patterns and fragmentation size distributions: the first group consists of primarily angular to subangular rock aggregates obtained by crushing granite from a quarry (Fig. 1); the second group includes quartzite beach pebbles, which are generally rounded to subrounded (Fig. 2). The two groups are denoted as “Q” (quarry) and “B” (beach). The grain shape

390

G. Ma et al. / Powder Technology 308 (2017) 388–397

Fig. 1. Angular to subangular rock grains in group Q.

library used in this study was first established by Latham et al. [28], who employed LADAR with a camera focal length set up for scanning at a 100–500 μm resolution. However, the number of grains is not sufficient for the subsequent statistical analysis. We expanded the grain shape library using a scanning technique similar to that of Latham et al. [28]. In total, 63 rock grains in group Q and 64 rock grains in group B are considered. The surface area can be obtained directly from the surface mesh of the scanned grains, which consists of a set of non-overlapping triangles. The volume is then the sum of the sub-volumes of all tetrahedra in the volumetric mesh. The differences between grains in terms of their shapes can be classified and quantified using different shape descriptors. Blott and Pye [47] divided the shape descriptors into three levels from the geometric point of view, i.e., form, roundness, and surface texture. Relatively simple shape descriptors used for the assessment of engineering materials are employed. By constructing the bounding box of each grain with principal axes S b I b L, the form is conventionally characterized by using the elongation L/I or flatness S/I ratio or the combination of the three axes (see reference [47] for details). At the intermediate scale between form and surface roughness, the grain morphology is described by sphericity or roundness [48]. Using the above results for the grain volume and surface area, the sphericity, ψ3D, can be calculated as: ψ3D ¼

As ¼ A

p ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 36πV 2 A

ð3Þ

where V and A are the actual volume and surface area of the scanned grain, and As is the surface area of a sphere having the same volume as the scanned grain. The sphericity for a perfectly spherical grain has a maximum value of 1.0. A few important shape descriptors relating to form and roundness are outlined in Table 1, where the numbers outside and inside the bracket represent the mean and standard deviation of each shape descriptor, respectively.

to investigate the grain crushing quantitatively. The objective of both the physical tests and numerical simulations has been to better understand the link between the fracture behavior of individual grains and the overall mechanical response of crushable granular materials. The most influential factors in determining the fragmentation patterns of rock or sand grains were found to be the grain morphology and initial microstructure [67]. The fracture process was also confirmed to be affected by the mineralogy. However, to reduce the complexity of this study and highlight the influence of morphology, all grains with different morphologies possess the same mineralogy and are represented by the same set of input parameters. The equivalent diameter d⁎, which is defined as the diameter of a sphere with a volume equal to that of the irregularly shaped grain, is 60 mm. As described in Section 3, the crushability of a grain is characterized by the embedded CIE's strength and is quantified by three parameters: the uniaxial tensile strength, ft, internal friction angle, φi, and cohesive strength, c. The tensile strengths of the CIEs are randomly assigned based on a lognormal distribution to consider the material heterogeneity. For simplicity, the internal friction angle, φi, and the ratio of the uniaxial compressive strength (UCS) to the tensile strength, fc/ft, are set to 40° and 15, respectively. The cohesive strength can be calculated from the Mohr-Coulomb strength criterion as c = 15ft(1 − sin φi)/(2 cos φi). The parameters summarized in Table 2 are assumed to represent medium crushable rock grains. The grain is discretized with tetrahedral elements, and zero-thickness CIEs are inserted between the edges of all adjacent tetrahedral element pairs from the very beginning of the simulation. To achieve accurate results in fracture simulations using the combined FDEM method, the size of the finite elements close to the fracture tip needs to be much smaller than the length of the fracture process zone (FPZ) [68]. Based on the previous analytical and experimental studies, the upper and lower values of the fracture zone length for a tensile fracture can be estimated as: upper

lfpz

4. Single grain crushing simulations Experimental studies on the crushing of individual grains, in which grains were placed between two rigid platens, are often used to investigate the fracture behavior of various types of grains, such as sand [50, 52], rock aggregate [53–55], and glass [56]. DEM simulations of single particle compression tests have been conducted by many researchers [12,57–64]. Numerical simulations by molecular dynamics (MD) [65] and the finite- and discrete element method [66] also provide a means

lower

lfpz

¼

¼

3EG f 2

4f t

3πEG f 2

32f t

ð4Þ

ð5Þ

and llower are the upper and lower bounds of the fracture where lupper fpz fpz zone length estimated from Muskhelishvili's solution and Westergaard's solution, respectively. E is the Young's modulus of the continuum, ft is the tensile strength, and Gf is the fracture energy.

Fig. 2. Rounded to subrounded rock grains in group B.

G. Ma et al. / Powder Technology 308 (2017) 388–397

391

Table 1 Shape descriptors of form and roundness. Parameter

Formula

Granite quarry (Q)

Beach pebbles (B)

Wentworth flatness index

LþI 2S

1.52(0.207)

1.40(0.180)

0.77(0.094)

0.82(0.096)

Krumbein intercept sphericity Corey shape factor Maximum projection sphericity

qffiffiffiffi

3 IS L2 pSffiffiffi LI

qffiffiffiffi S2 LI

3

Aschenbrenner working sphericity

12:8

p ffiffiffiffiffiffiffi 3 2 Q pPffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2

1þPð1þQ Þþ6

0.68(0.091)

0.73(0.083)

0.77(0.068)

0.81(0.062)

0.55(0.112)

0.61(0.105)

1þP ð1þQ Þ

where P ¼ S=I and Q ¼ I=L

Aschenbrenner shape factor

LS I2

1.03(0.296)

1.00(0.210)

Janke form factor

S pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2

1.26(0.151)

1.35(0.143)

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi ð1=S þ 1=I þ 1=LÞ S2 þ I 2 þ L2 = 3

3.26(0.175)

3.09(0.153)

See Eq. (3)

0.82(0.023)

0.92(0.040)

Domokos shape parameter (see reference [49]) Sphericity

ðL þI þS Þ=3

Using the material properties given in Table 2, the theoretical estimates of the upper and lower values of the fracture zone length can ≈ 24.8 mm and lupper ≈ 9.7 mm, respectively. be obtained as lupper fpz fpz Based on the theoretical estimation, for an average mesh size of 3 mm, the FPZ can be discretized by 3– 8 finite elements, which is sufficient to give a correct numerical representation of the FPZ ahead of the fracture tip [68]. Therefore, an average mesh size of 3 mm is used in the finite element discretization of all scanned grains. 5. Fracture and fragmentation patterns Fragmentation can be defined as the process of the reduction of a solid's size by breaking it into smaller parts [28]. The irregular morphology of rock grains, together with the loading conditions, define the boundary conditions, which influence the stress distribution inside the grain and lead to different fracture patterns. Two typical grains in group Q and group B are selected for the comparison of fragmentation patterns. The load on the top plate together with the broken CIEs for every 0.1 mm of loading displacement is recorded and illustrated in Fig. 3 for an angular particle in group Q, which shows a typical sawtooth load-displacement relationship. Several peaks are marked on the figure. Small peaks in force represent the fracture of small asperities and are not considered to correspond to failure, while the maximum peak corresponds to the major splitting of the particle having occurred. The loaddisplacement curve indicates the main stages in the compression of a typical angular particle: Chipping or yielding of the asperities, volumetric strain of the bulk, failure and loss of parts of the bulk, and catastrophic collapse. As noted by Cavarretta and O′Sullivan [69], these stages are not linearly sequential and may overlap partially or entirely. As shown in Fig. 3, the fragmentation is often followed by a new stage of elastic response, in which the mechanical behavior might depend on the new

post-fragmentation geometry and will be sensitive to new internal flaws that may have developed during crushing. Fig. 4 shows the fracture process of the grain from the top view. As a result of the fragmentation, a number of fragments are produced. The letters in the figure indicate the different loading stages, which are labeled in Fig. 3. Different fracture mechanisms, including chipping, tension, bending, and any combination of these mechanisms, may occur inside the grain due to the direct compression by the loading platen and the interaction between fragments. There are many intersecting and branching cracks that propagate through the material, forming crack planes that reach the grain surface, thus resulting in fragmentation. This pattern is similar to that summarized by Paluszny et al. [66], in which the mechanism of fragmentation involves the activation of preexisting flaws, the propagation of activated cracks, crack branching, and the separation of fragments. Fig. 5 shows the 10 largest fragments from the top at the end of compression. The fragments are marked with different colors to distinguish them from each other, which can also allow the fracture pattern to be viewed more intuitively. The majority of the grain is fragmented into several main wedge-shaped fragments, accompanied by a large number of small fragments in the crushed zones. The fragment size distribution will be analyzed in the next section. The typical load-displacement curve of a rounded grain is shown in Fig. 6. The letters in the figure indicate the different loading stages. The figure shows that the deformation of the rounded grain is dominantly elastic-brittle and the compression of a rounded grain can be divided into two stages, i.e., elastic and brittle failure. Before reaching the peak force, the grain deforms linearly, with slight fluctuation due to a series of minor perturbations caused by local micro-cracking. Near the peak point b, the grain is subjected to a catastrophic failure that is followed by a sharp drop in the load-bearing capability. Fig. 7 shows

Table 2 Input parameters used in combined FDEM modeling.

Bulk elements

Cohesive elements

Contact law

Parameter

Value

Mass density, ρ (kg/m3) Young's modulus, E (GPa) Poisson's ratio, v Normal penalty of CIE, kn (N/m3) Shear penalty of CIE, ks (N/m3) Tensile strength of CIE, ft (MPa)

2700 40 0.2 3.0 × 1013 1.25 × 1013 Lognormal distribution with a mean value of 11 MPa and a variation coefficient of 0.5 40° 30° c = 15ft(1 − sinφi)/(2 cos φi) 100 500 0.577

Friction angle of intact material, φi(°) Friction angle of fractures, φf(°) Cohesion of CIE, c (MPa) Mode-I fracture energy, GI (N/m) Mode-II fracture energy, GII (N/m) Inter-particle sliding friction, μ

Fig. 3. Typical load-displacement curve of an angular grain in group Q.

392

G. Ma et al. / Powder Technology 308 (2017) 388–397

Fig. 6. Typical load-displacement curve of a rounded grain in group B.

Fig. 4. Visualization of the fracture process of an angular grain in group Q from the top view.

the fracture process of the grain from the top view. Upon uniaxial compression, the zone beneath the contact area is fully crushed, pushing into the structure and resulting in a slight lateral displacement of the structure. Then the crack initiates from the contact zone between the grain and platen, where the tensile circumferential stress exceeds its threshold value, and then runs through the grain during the softening stage. More macro-cracks form due to the excessive radial tensile stress, and they gradually connect to each other and propagate through the weakest pathways, which is roughly parallel to the direction of loading, to form meridional cracks, leading to the final splitting of the grain [59, 71–73]. It can be seen from Fig. 8 that the orange-like slices or

meridional fracture planes are nearly orthogonal to the platens, and they separate the grain into several wedge-shaped fragments. This mechanism was reported experimentally in the fragmentation of brittle and spherical grains made of rock [55] and glass [56], and chemical powders [73]. Similar mechanisms of fracture have been observed in the impact loading of spheres made of glass, concrete, rock, aluminum oxide, and plaster [66]. This primary fracture mechanism is robust with respect to the internal structure of the grain and repeatable between different materials. The differences in the load-displacement relationships and fracture patterns are similar to those observed by Zhao et al. [67], in which the highly decomposed granite (HDG) and Leighton Buzzard sand (LBS) particles correspond to the Q and B groups, respectively, whereas the crack morphology and fracture pattern are more complicated due to the difference in the initial morphology, mineralogy, and microstructure of the natural sand particles. 6. Weibull statistics of crushing strength A series of simulations were conducted to investigate the statistical characteristics of the grain fracture strength. We also performed extensive Monte Carlo simulations on a perfectly spherical grain to eliminate

Fig. 5. Shape of the 10 largest fragments resulting from the axial compression of an angular grain in group Q.

Fig. 7. Visualization of the fracture process of a rounded grain in group B from the top view.

G. Ma et al. / Powder Technology 308 (2017) 388–397

393

for group Q, group B and a perfectly spherical grain are plotted in Fig. 9. The insets of each figure show the frequency distributions of the grain crushing strength. The data points satisfy the Weibull distribution well, except at a lower strength range, where the data deviate from the Weibull best fit line. The Kolmogorov Smirnov (K-S) goodness of fit test is performed, and the returned value of 0 indicates a failure to reject the Weibull hypothesis at the 0.05 significance level. This result indicates that the rock grain has a structure that is dominated by the presence of flaws or weak zones, which always fail at their weakest points [54]. The Weibull modulus and 37% characteristic strength are 5.593 and 2.028 MPa for group Q, 6.796 and 2.787 MPa for group B, and 12.156 and

Fig. 8. Shape of the 10 largest fragments resulting from the axial compression of a rounded grain in group B.

the influence of grain shape on crushing behavior. The strength thresholds of individual simulations are randomly sampled from a lognormal distribution, as depicted in Section 4. Due to small asperities that may fracture before the final failure of the grain, care is needed in adopting a suitable definition of failure. In this paper, the failure is taken to correspond to the largest force such that the current size of the particle is still greater than 50% of the initial size [50]. In all cases, this time was found to correspond to catastrophic splitting of the particle. For a given grain of average size d loaded diametrically, an induced characteristic strength σ is defined as the diametral force at failure F divided by the square of the grain diameter, which may be taken to be the distance between the platens at failure: σ¼

F

ð6Þ

2

d

It has been theoretically and experimentally verified that the grain crushing strength follows a Weibull distribution [50,53]. Based on the Weibull model, the cumulative probability Ps that a grain of size d survives a tensile stress σ can be described as:    m  d σ P s ¼ exp − d0 σ0

ð7Þ

where d0 is a reference size, σ0 is the characteristic stress for a grain of size d0 to exhibit a survival probability of 37%, and m is the Weibull modulus that describes the variability in the tensile strength of different grains. For a finite number of tested grains, the survival probability Ps is calculated using the mean rank position: P s ¼ 1−

i Nþ1

ð8Þ

where N is the total number of grains, and i is the rank of the grain sorted in an ascending order. By rewriting Eq. (7) with d =d0, a linear relationship can be obtained: ln ½ ln ð1=P s Þ ¼ m lnσ −m lnσ 0

ð9Þ

Following Eq. (9), by plotting ln[ln(1/Ps)] against lnσ, the Weibull modulus m can be determined from the slope of the best fit line, and the value of σ0 is the value of σ when ln[ln(1/Ps)] = 0. The simulated data points and the corresponding fitting curves according to Eq. (9)

Fig. 9. Weibull plots for each group of grains: (a) group Q; (b) group B; (c) 60 random simulations of a perfectly spherical particle.

394

G. Ma et al. / Powder Technology 308 (2017) 388–397

2.80 MPa for a perfectly spherical grain. The Weibull modulus defines the variability of the grain crushing strength, which increases with decreasing Weibull modulus. There appears to be a relationship wherein the particle sphericity decreases such that the variability of the crushing strength increases. In other words, as the angularity of the grains increases, so does the variability of crushing strength. The variation of the 37% characteristic strength with particle sphericity has a reverse trend. The variations in the Weibull modulus and 37% characteristic strength with particle sphericity are consistent with experimental observations [74]. This result serves as the basis for further exploration of the quantitative relationship between the crushing strength of grains and their initial and evolving morphologies during the crushing process. It was found that the particle yield stress increased with decreasing particle size [50,51,53]. This relationship occurs because the larger grains have more and larger flaws than do the smaller ones [54]. A similar result was produced by Marsal [75] and Hu et al. [76], who found that the relationship between the average failure force F and particle diameter d can be fitted by a power function F = ηdλ, where η and λ are fitting parameters. The parameter λ describes the size hardening effect. Through the derivation, the relationship between the Weibull modulus m and parameter λ is expressed as λ =2 − 3/m, where a larger m yields a larger λ and indicates a smaller size hardening effect. Combined with the above analysis, the size hardening effect of the grain crushing strength increases with increasing grain angularity. In the range of rockfill materials investigated by Marsal [75], the parameter λ was between 1.2 and 1.8, corresponding to a Weibull modulus m between 4 and 15. Such a range is typical of many rockfill materials encountered in earthen dam engineering. 7. Fragment size distribution after crushing Because of the intense local crushing together with oblique and radial cracking, each grain is fragmented into several larger fragments and a number of smaller fragments. The sizes of the final fragments depend on the morphology and internal structure of the grain. The volume-based cumulative fragment size distribution can be fitted using a two parameter Weibull equation. This distribution, also named the Rosin– Rammler distribution, has been successfully used in characterizing the fragment size of spherical glass grains [77] and rock grains [66]. The two-parameter Weibull distribution can be expressed as:

y ¼ 1− exp −ðx=xc Þn

ð10Þ

where y is the total volume of the fraction corresponding to fragment sizes less than x, and n and xc are the fitting parameters. n indicates the width of the distribution, and larger values of n correspond to narrower distributions. xc is the fragment size that corresponds to a volume fraction of 1 − 1/e (0.368). Hereafter, the grain or fragment size is referred to as the equivalent diameter, which is defined as the diameter

of a sphere that has the same volume as the fragment. For all grains, the coefficient of determination R2 of the least squared fit are larger than 0.95. Fig. 10 shows the volume-based cumulative fragment size distributions and the corresponding fitting curves of an angular grain and a rounded grain. It can be seen that the two-parameter Weibull equation provides a good description of the simulated fragment size distribution. The average values of xc and n are 37.49 mm and 3.55 for group Q, and 36.45 mm and 4.81 for group B. As indicated before, xc can be used as an index to quantify the level of fines in the fragmenting system. Thus, a low xc value is expected if the fragmenting system consists of a high level of fines. The parameter n is a measure of the spread of the fragment size distribution, where the distribution is narrower for a larger n. Compared with angular grains, the fragment size distributions of rounded grains are characterized by a larger n. The difference is mainly due to the governing fracture mechanisms that occurred in the two sets of grains. Close examination of Fig. 5 and Fig. 8 reveals that the rounded grains tend to produce similarly sized major fragments [73]. The similarly sized major fragments resulted from the meridian cracks, which cause the size distribution to span a narrower size range compared with that of the angular grain. This explains the steeper middle section in the volume fraction of the rounded grain, as highlighted in Fig. 10. As already mentioned, the fracture mechanisms of angular grains consist of chipping, bending, and tension. Therefore, fragments obtained from irregular-shaped grains display sufficient fracture, with a small number of large fragments and a larger amount of chipping. This result is in agreement with the observation that severe local damage yields a crushed cone in addition to debris around the contact area [67]. The volume of the largest fragment, V1st, normalized by the initial volume of the grain, V0, is statistically analyzed. Fig. 11 shows the histograms of the frequency count of V1st/V0 for the two groups. The K-S test indicates that the V1st/V0 values for both groups follow a Gaussian distribution. It is evident that the largest fragment becomes smaller and narrower for the rounded grains. The fragments produced by weathering, abrasion, impact and geological loading often satisfy a fractal condition over a wide range of scales [78]. Using the combined FDEM modeling of real shaped grain crushing tests, we want to examine the fractal characteristics of differently shaped grains. A fractal can be defined by the power-law relationship between the number and size N ~ d−D, where N is the number of fragments with a characteristic linear dimension larger than d, and D is the fractal dimension [67] [78] [79]. Fig. 12 shows the accumulative number of fragments N(Nd) plotted against d in log-log scale for eight angular and eight rounded grains. It appears that, in all cases, the middle part of the fragment size distribution is almost linear within the range of 2 mm to 18 mm. This linearity indicates that the distribution of the fragment sizes has a fractal structure. Some larger fragments that have not experienced significant fragmentation define the upper end of the fractal region. The lower end of the fractal range may be associated with the characteristic mesh size.

Fig. 10. Volume-based size distributions of fragments for two grains in (a) group Q and (b) group B.

G. Ma et al. / Powder Technology 308 (2017) 388–397

395

Fig. 11. Frequency distributions of the volume of the largest fragments normalized by the initial volume for (a) group Q and (b) group B.

Previous research on the statistics of fragment size distributions have revealed a power law relationship N ∝ d−τ, which is characterized with a high degree of robustness of the exponent τ, irrespective of the materials properties and loading conditions [49]. Under rather general assumptions for the fragment size distribution, the power law distributed fragment sizes are obtained with the exponent τ = (2D − 1)/D, where D is the dimensionality of the system [81]. Fig. 12 demonstrates that the power law form prevails in the intermediate size regime of all the distributions. As demonstrated in Fig. 13, the fractal dimension follows a Gaussian distribution, and most of the fragment sizes distribute with a fractal dimension of approximately 1.86 ± 0.16 in spite of the strongly different governing fracture mechanisms and grain morphologies. The fractal dimension value is close to the exponent τ of the analytical predication of the fragment size distribution results from merged crack branches in three dimensions, where τ≈ 1.67. Zhao et al. [67] also found that the fragments of both LBS and HDG grains have a fractal distribution with a fractal dimension of approximately 2.0. Thus, it can be assumed that grain morphology has a slight influence on the fractal characteristics.

8. Conclusions Computed tomography (CT) and the combined finite discrete element method (FDEM) are used to investigate the fragmentation of rock grains. Fracture-driven fragmentation is simulated by the failure of inserted cohesive interface elements (CIEs). To reveal the role of grain morphology in the fragmentation process, axial compression of two sets of rock grains with different shape characteristics was performed. The first group consists of angular to subangular grains, and the second group is composed of rounded grains. The initial heterogeneity of each grain is considered using a random distribution of the

strength threshold (tension and shear). The effects of grain morphology on the fragmentation patterns, Weibull statistics of grain crushing strength, and fragment size distribution are systematically analyzed. The novel insights acquired from this study about the grain fracture and fragmentation processes lie in the statistical information of the fragmentation process of rock grains with different morphologies. The wealth of information will deepen our understanding of grain-scale fracture mechanisms of natural rock grains. Although the results presented are obtained from numerical simulations, many observations and insights have a solid physical background and are in qualitative agreement with experimental results. These results can also be extended to the fracture behavior of quasi-brittle geomaterials over a range of initial morphologies, intrinsic microstructures and length scales, e.g., sand, gravel, and rockfill. The main conclusions of the study are summarized as follows: (1) The dominant fracture mechanisms are related to grain morphology. The typical fragmentation pattern for rounded grains is caused by the circumferential tensile stress in the ring-shaped region around the contact area and the tensile radial stress on the meridian plane. These cracks connect to each other and propagate through the weakest pathways to form meridional fracture planes that result in similar-sized large fragments. The more complex morphologies in angular grains lead to a much richer array of fragmentation patterns. Chipping, tension, and bending may occur inside the angular grains due to the axial compression by the loading platen and the interaction between fragments. Many intersecting and branching cracks propagate through the material, forming crack planes that reach the grain surface, thus resulting in fragmentation. (2) The grain crushing strength satisfies the Weibull distribution, except that at a lower strength range, the data deviate from the

Fig. 12. Fractal distributions of fragment sizes for (a) group Q and (b) group B.

396

G. Ma et al. / Powder Technology 308 (2017) 388–397

References

Fig. 13. Distribution of the fractal dimension of 127 grains.

Weibull best fit line. The Weibull modulus and 37% characteristic strength are 5.593 and 2.028 MPa for group Q, 6.796 and 2.787 MPa for group B, and 12.156 and 2.80 MPa for a perfectly spherical grain, respectively. The variations of the Weibull modulus and 37% characteristic strength with particle shape characteristic are consistent with experimental observations. The size hardening effect of grain crushing strength increases with increasing grain angularity. (3) The two-parameter Weibull equation provides an adequate description of the simulated fragment size distributions. The Weibull distributions of the grain crushing strengths and fragment sizes indicate that the behavior of the rock grains is approximately Weibullian. Compared with an angular grain, the fragment size distribution of a rounded grain is characterized by a larger n. The difference is mainly due to the governing fracture mechanisms that occur in the two sets of grains. The fracture mechanisms of rounded grains produce some similarly sized major fragments, which cause the size distribution to span a narrower size range compared with that of the angular grain. On the other hand, the fragmentation of irregular-shaped grains involves much more complex fracture mechanisms, such as chipping, tension, bending, and any combination of these mechanisms, which generally produce a smaller number of large fragments and a larger amount of debris. (4) The fragments of both angular and rounded grains are found to have a fractal distribution with a fractal dimension of approximately 1.86 ± 0.16 in spite of the strongly different governing fracture mechanisms and grain morphologies. The fractal dimension value is close to the exponent τ of the analytical predication of the fragment size distribution in three dimensions. The fractal size distribution indicates that the fragmentation is a scaleinvariant process. The simulated results and the existing experimental findings indicate that the grain morphology has a slight influence on the fractal structure. However, whether the fractal dimension is an intrinsic material property of rock grains requires through experimental validation.

Acknowledgments The authors gratefully acknowledge financial support by the National Natural Science Foundation of China (Grant No. 51379161, No. 51579193 and No. 51509190), China Postdoctoral Science Foundation (Grant No. 2016T907272), and the Fundamental Research Funds for the Central Universities (Grant No. 2042016kf1077). Thanks are due to Dr. John-Paul Latham of Imperial College London for the scanning data. The useful suggestions given by Prof. Tang-tat Ng are also acknowledged.

[1] M.R. Coop, I.K. Lee, The behaviour of granular soils at elevated stresses, in: GT Houlsby, A.N. Schofield (Eds.), Predictive soil mechanics, Predictive Soil Mech., Thomas Telford, London, UK 1993, pp. 186–199. [2] J.A. Yamamuro, P.A. Bopp, P.V. Lade, One-dimensional compression of sands at high pressures[J], J. Geotech. Eng. 122 (2) (1996) 147–155. [3] Y. Nakata, M. Hyodo, A.F.L. Hyde, et al., Microscopic particle crushing of sand subjected to high pressure one-dimensional compression [J], Soils Found. 41 (1) (2001) 69–82. [4] B.O. Hardin, Crushing of soil particles[J], J. Geotech. Eng. 111 (10) (1985) 1177–1192. [5] M.M. Hagerty, D.R. Hite, C.R. Ullrich, et al., One-dimensional high-pressure compression of granular media[J], J. Geotech. Eng. 119 (1) (1993) 1–18. [6] P.V. Lade, J.A. Yamamuro, P.A. Bopp, Significance of particle crushing in granular materials[J], Am. Soc. Civil Eng. 122 (4) (1996) 309–316. [7] M.R. Coop, Particle breakage during shearing of a carbonate sand[J], Geotechnique 54 (3) (2004) 157–164. [8] M.R. Coop, F.N. Altuhafi, Changes to particle characteristics associated with the compression of sands[J], Géotechnique 61 (6) (2011) 459–471. [9] J. Otani, T. Mukunoki, K. Sugawara, Evaluation of particle crushing in soils using Xray CT data[J], Soils Found. 45 (1) (2005) 99–108. [10] K.A. Alshibli, M.B. Cil, 3D evolution of sand fracture under 1D compression[J], Géotechnique 64 (5) (2014) 351–364. [11] Y. Shi, W.M. Yan, T. Mukunoki, et al., A microscopic investigation into the breakage behavior of calcareous origin grains in 1D compression[J], Jpn. Geotech. Soc. Spec. Publ. 2 (16) (2016) 630–634. [12] Y.P. Cheng, Y. Nakata, M.D. Bolton, Discrete element simulation of crushable soil[J], Geotechnique 53 (7) (2003) 633–642. [13] M.D. Bolton, Y.P. Cheng, Crushing and plastic deformation of soils simulated using DEM[J], Géotechnique 54 (2) (2004) 131–142. [14] E.S. Hosseininia, A.A. Mirghasemi, Numerical simulation of breakage of two-dimensional polygon-shaped particles using discrete element method[J], Powder Technol. 166 (2) (2006) 100–112. [15] J. Wang, H. Yan, On the role of particle breakage in the shear failure behavior of granular soils by DEM[J], Int. J. Numer. Anal. Methods Geomech. 37 (8) (2013) 832–854. [16] D.H. Nguyen, E. Azéma, P. Sornay, et al., Effects of shape and size polydispersity on strength properties of granular materials[J], Phys. Rev. E 91 (3) (2015) 032203. [17] S. Lobo-Guerrero, L.E. Vallejo, L.F. Vesga, Visualization of crushing evolution in granular materials under compression using DEM[J], Int. J. Geosci. 6 (3) (2006) 195–200. [18] G.R. McDowell, J.P. de Bono, On the micro mechanics of one-dimensional normal compression[J], Géotechnique 63 (11) (2013) 895. [19] W. Zhou, L. Yang, G. Ma, et al., Macro–micro responses of crushable granular materials in simulated true triaxial tests[J], Granul. Matter 17 (4) (2015) 497–509. [20] D. Shi, L. Zheng, J. Xue, et al., DEM modeling of particle breakage in Silica Sands under one-dimensional compression[J], Acta Mech. Solida Sin. 29 (1) (2016) 78–94. [21] A.K. Turner, F.H. Kim, D. Penumadu, et al., Meso-scale framework for modeling granular material using computed tomography[J], Comput. Geotech. 76 (2016) 140–146. [22] G.C. Cho, J. Dodds, J.C. Santamarina, Particle shape effects on packing density, stiffness, and strength: natural and crushed sands[J], J. Geotech. Geoenviron. 132 (5) (2006) 591–602. [23] S.A. Hall, M. Bornert, J. Desrues, et al., Discrete and continuum analysis of localised deformation in sand using X-ray μCT and volumetric digital image correlation[J], Géotechnique 60 (5) (2010) 315–322. [24] F.H. Kim, D. Penumadu, J. Gregor, et al., High-resolution neutron and X-ray imaging of granular materials[J], J. Geotech. Geoenviron. 139 (5) (2012) 715–723. [25] X. Fu, M. Dutt, A.C. Bentham, et al., Investigation of particle packing in model pharmaceutical powders using X-ray microtomography and discrete element method[J], Powder Technol. 167 (3) (2006) 134–140. [26] K.E. Thompson, C.S. Willson, W. Zhang, Quantitative computer reconstruction of particulate materials from microtomography images[J], Powder Technol. 163 (3) (2006) 169–182. [27] L. Wang, J.Y. Park, Y. Fu, Representation of real particles for DEM simulation using Xray tomography[J], Constr. Build. Mater. 21 (2) (2007) 338–346. [28] J.P. Latham, A. Munjiza, X. Garcia, et al., Three-dimensional particle shape acquisition and use of shape library for DEM and FEM/DEM simulation[J], Miner. Eng. 21 (11) (2008) 797–805. [29] T. Matsushima, J. Katagiri, K. Uesugi, et al., 3D shape characterization and imagebased DEM simulation of the lunar soil simulant FJS-1[J], J. Aerosp. Eng. 22 (1) (2009) 15–23. [30] R. Moreno-Atanasio, R.A. Williams, X. Jia, Combining X-ray microtomography with computer simulation for analysis of granular and porous materials[J], Paraplegia 8 (2) (2010) 81–99. [31] B. Zhou, J. Wang, B. Zhao, Micromorphology characterization and reconstruction of sand particles using micro X-ray tomography and spherical harmonics[J], Eng. Geol. 184 (2015) 126–137. [32] W.J. Xu, L.M. Hu, W. Gao, Random generation of the meso-structure of a soil-rock mixture and its application in the study of the mechanical behavior in a landslide dam[J], Int. J. Rock Mech. Min. Sci. 86 (2016) 166–178. [33] M.B. Cil, K.A. Alshibli, Modeling the influence of particle morphology on the fracture behavior of silica sand using a 3D discrete element method[J], C. R. Méc. 343 (2) (2015) 133–142. [34] G. Ma, W. Zhou, X.L. Chang, et al., Combined FEM/DEM modeling of triaxial compression tests for rockfills with polyhedral particles[J], Int. J. Geosci. 14 (4) (2013) 04014014.

G. Ma et al. / Powder Technology 308 (2017) 388–397 [35] G. Ma, W. Zhou, X.L. Chang, et al., A hybrid approach for modeling of breakable granular materials using combined finite-discrete element method[J], Granul. Matter 18 (1) (2016) 1–17. [36] A. Munjiza, D.R.J. Owen, N. Bicanic, A combined finite-discrete element method in transient dynamics of fracturing solids[J], Eng. Comput. 12 (2) (1995) 145–174. [37] A. Munjiza, The Combined Finite–Discrete Element Method, Wiley, New York, 2004. [38] A. Munjiza, J. Xiang, X. Garcia, et al., The virtual geoscience workbench, VGW: open source tools for discontinuous systems[J], Paraplegia 8 (2) (2010) 100–105. [39] O.K. Mahabadi, A. Lisjak, A. Munjiza, et al., Y-geo: new combined finite-discrete element numerical code for geomechanical applications[J], Int. J. Geosci. 12 (6) (2012) 676–688. [40] Itasca Consulting Group Inc, UDEC (universal distinct element code). Minneapolis, USA, 2013. [41] Rockfield Software Ltd, ELFEN 2D/3D Numerical Modelling Package. Swansea, UK, 2004. [42] Dassault Systèmes Simulia Corp, Abaqus 6.14–1: Analysis user's Manual. Providence, USA, 2014. [43] B.S.A. Tatone, G. Grasselli, A calibration procedure for two-dimensional laboratoryscale hybrid finite–discrete element simulations[J], Int. J. Rock Mech. Min. Sci. 75 (2015) 56–72. [44] L. Guo, J.P. Latham, J. Xiang, Numerical simulation of breakages of concrete armour units using a three-dimensional fracture model in the context of the combined finite-discrete element method[J], Comput. Struct. 146 (2015) 117–142. [45] G. Ma, W. Zhou, X.L. Chang, Modeling the particle breakage of rockfill materials with the cohesive crack model[J], Comput. Geotech. 61 (9) (2014) 132–143. [46] F. Lanaro, P. Tolppanen, 3D characterization of coarse aggregates[J], Eng. Geol. 65 (1) (2002) 17–30. [47] S.J. Blott, K. Pye, Particle shape: a review and new methods of characterization and classification[J], Sedimentology 55 (1) (2008) 31–63. [48] H. Wadell, Volume, shape, and roundness of rock particles[J], J. Geol. (1932) 443–451. [49] G. Domokos, F. Kun, A.Á. Sipos, et al., Universality of fragment shapes[J], Sci. Report. 5 (2015). [50] G.R. McDowell, A. Amon, The application of Weibull statistics to the fracture of soil particles[J], Soils Found. 40 (5) (2000) 133–141. [51] G.R. McDowell, On the yielding and plastic compression of sand[J], Soils Found. 42 (1) (2002) 139–145. [52] I. Cavarretta, M. Coop, C. O'sullivan, The influence of particle characteristics on the behaviour of coarse grained soils[J], Géotechnique 60 (6) (2010) 413–423. [53] W.L. Lim, G.R. McDowell, A.C. Collop, The application of Weibull statistics to the strength of railway ballast[J], Granul. Matter 6 (4) (2004) 229–237. [54] S. Lobo-Guerrero, L.E. Vallejo, Application of Weibull statistics to the tensile strength of rock aggregates[J], J. Geotech. Geoenviron. 132 (6) (2006) 786–790. [55] A. Cheshomi, E.A. Sheshde, Determination of uniaxial compressive strength of microcrystalline limestone using single particles load test[J], J. Pet. Sci. Eng. 111 (2013) 121–126. [56] J. Huang, S. Xu, H. Yi, et al., Size effect on the compression breakage strengths of glass particles[J], Powder Technol. 268 (2014) 86–94. [57] W.L. Lim, G.R. McDowell, Discrete element modelling of railway ballast[J], Granul. Matter 7 (1) (2005) 19–29.

397

[58] M.D. Bolton, Y. Nakata, Y.P. Cheng, Micro-and macro-mechanical behaviour of DEM crushable materials[J], Géotechnique 58 (6) (2008) 471–480. [59] H.A. Carmona, F.K. Wittel, F. Kun, et al., Fragmentation processes in impact of spheres[J], Phys. Rev. E 77 (5) (2008) 051302. [60] K.J. Hanley, C. O'Sullivan, J.C. Oliveira, et al., Application of Taguchi methods to DEM calibration of bonded agglomerates[J], Powder Technol. 210 (3) (2011) 230–240. [61] X. Wang, W. Xu, B. Zhang, Breakage characteristics and crushing mechanism of cement materials using point-loading tests and the discrete-element method[J], J. Mater. Civ. Eng. 26 (5) (2013) 992–1002. [62] Y. Yan, J. Zhao, S. Ji, Discrete element analysis of breakage of irregularly shaped railway ballast[J], Geol. Geofiz. 10 (1) (2015) 1–9. [63] H.A. Carmona, A.V. Guimarães, J.S. Andrade Jr., et al., Fragmentation processes in two-phase materials[J], Phys. Rev. E 91 (1) (2015) 012402. [64] K. Zheng, C. Du, J. Li, et al., Numerical simulation of the impact-breakage behavior of non-spherical agglomerates[J], Powder Technol. 286 (2015) 582–591. [65] D.H. Nguyen, E. Azéma, P. Sornay, et al., Bonded-cell model for particle fracture[J], Phys. Rev. E 91 (2) (2015) 022203. [66] A. Paluszny, X.H. Tang, M. Nejati, et al., A direct fragmentation method with Weibull function distribution of sizes based on finite-and discrete element simulations[J], Int. J. Solids Struct. 80 (2016) 38–51. [67] B. Zhao, J. Wang, M.R. Coop, et al., An investigation of single sand particle fracture using X-ray micro-tomography[J], Géotechnique 65 (8) (2015) 625–641. [68] L. Guo, J. Xiang, J.P. Latham, et al., A numerical investigation of mesh sensitivity for a new three-dimensional fracture model within the combined finite-discrete element method[J], Eng. Fract. Mech. 151 (2016) 70–91. [69] I. Cavarretta, C. O'sullivan, The mechanics of rigid irregular particles subject to uniaxial compression[J], Geotechnique 62 (8) (2012) 681–692. [71] A. Russell, P. Müller, H. Shi, et al., Influences of loading rate and preloading on the mechanical properties of dry elasto-plastic granules under compression[J], AICHE J. 60 (12) (2014) 4037–4050. [72] A. Russell, S. Aman, J. Tomas, Breakage probability of granules during repeated loading[J], Powder Technol. 269 (2015) 541–547. [73] A. Russell, J. Schmelzer, P. Müller, et al., Mechanical properties and failure probability of compact agglomerates[J], Powder Technol. 286 (2015) 546–556. [74] Y. Nakata, Y. Kato, M. Hyodo, A.F.L. Hyde, et al., One-dimensional compression behaviour of uniformly graded sand related to single particle crushing strength [J], Soils Found. 41 (2) (2001) 39–51. [75] R.J. Marsal, Large-scale testing of rockfill materials[J], J. Soil Mech. Found. Div. 93 (2) (1967) 27–43. [76] W. Hu, C. Dano, E. Merliot, F. Derkx, et al., Effect of sample size on the behavior of granular materials[J], ASTM. Geotech. Test. J. 34 (3) (2011) 186–197. [77] Y.S. Cheong, G.K. Reynolds, A.D. Salman, et al., Modelling fragment size distribution using two-parameter Weibull equation[J], Int. J. Miner. Process. 74 (2004) S227–S237. [78] D.L. Turcotte, Fractals and fragmentation[J], J. Geophys. Res. Solid Earth 91 (B2) (1986) 1921–1926. [79] H. Nagahama, Fractal fragment size distribution for brittle rocks[J], Int. J. Rock Mech. Min. Sci. Geomech. Abstr. 30 (4) (1993) 469–471. [81] P. Kekäläinen, J.A. Åström, J. Timonen, Solution for the fragment-size distribution in a crack-branching model of fragmentation[J], Phys. Rev. E 76 (2) (2007) 026112.