On the breakage function for constructing the fragment replacement modes

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On the breakage function for constructing the fragment replacement modes Wei Zhou a,c , Kun Xu a,b,c,∗ , Gang Ma a,c , Xiaolin Chang a,c a b c

State Key Laboratory of Water Resources and Hydropower Engineering Science, Wuhan University, Wuhan 430072, China Changjiang Institute of Survey, Planning, Design and Research, Wuhan 430010, China Key Laboratory of Rock Mechanics in Hydraulic Structural Engineering, Ministry of Education, Wuhan University, Wuhan 430072, China

a r t i c l e

i n f o

Article history: Received 6 July 2017 Received in revised form 17 July 2018 Accepted 15 August 2018 Available online xxx Keywords: Particle breakage Single particle compression test DEM Fragment replacement mode Breakage function

a b s t r a c t The fragment replacement method (FRM), a particle breakage simulation method, is often used in discrete element simulations to investigate the particle breakage effect on the mechanical behavior of granular materials. The fragment size distribution of the fragment replacement mode of FRM, which is generally generated based on the fragmentation characteristics of single particles after uniaxial compression, affects the breakage process and the mechanical behavior of the particle assembly. However, existing fragment replacement modes are seldom generated based on experimental data analysis. To capture the fragmentation process and investigate the breakage function for the construction of the fragment replacement mode, 60 numerical single particle compression tests were implemented by DEM. The bonded-particle model was applied to generate the crushable rock particles. The numerical simulations were qualitatively validated by experimental results, and the fragment size of broken single particles was analyzed. The fractal dimension was used to describe the fragmentation degree of single particles after compression. The fragmentation degree was random, and the fractal dimensions of the 60 tests at the same loading displacement fit the Weibull distribution well. The characteristic fractal dimension increased with increasing loading displacement, indicating that the fragmentation of single particles is a gradual process. According to the overall breakage function of the 60 tests at the first bulk breakage, a two-stage distribution model with 4 parameters was proposed and validated by the numerical and experimental results. The various fracture patterns of a single particle at the first bulk breakage under compression tests were well captured by the two-stage distribution model. Finally, an initial application strategy using the two-stage distribution model to construct fragment replacement modes was discussed and presented. © 2019 Chinese Society of Particuology and Institute of Process Engineering, Chinese Academy of Sciences. Published by Elsevier B.V. All rights reserved.

Introduction Granular materials such as sand, gravel, ballast, and rockfill are ubiquitous in nature. They are widely used in civil engineering for the construction of earth dams and embankments. The brittleness and natural defects of grains often lead to particle breakage when the granular assembly is subjected to loading, especially under high stresses. Particle breakage dissipates energy and changes the particle size distribution and fabric structure of the granular assemblies, which affects the mechanical behavior of granular materials (Hardin, 1985; Indraratna, Lackenby, & Christie, 2005). Particle

∗ Corresponding author at: State Key Laboratory of Water Resources and Hydropower Engineering Science, Wuhan University, Wuhan 430072, China. E-mail addresses: zw [email protected] (W. Zhou), xukun [email protected] (K. Xu), [email protected] (G. Ma), [email protected] (X. Chang).

breakage has been intensively studied using laboratory tests and some numerical methods such as the discrete element method (DEM) (Khanal, Schubert, & Tomas, 2004, 2007; McDowell & Bolton, 1998; Tsoungui, Vallet, & Charmet, 1999), the combined finite discrete element method (Bagherzadeh, Mirghasemi, & Mohammadi, 2011; Ma, Zhou, Chang, & Chen, 2016), the extended finite element method (Druckrey & Alshibli, 2016), and the contact dynamic method (Cantor, Estrada, & Azéma, 2015; Nguyen, Azéma, Sornay, & Radjai, 2015). The advantages of DEM are its theoretical simplicity, clear physical meaning of its input parameters, and general level of acceptance. DEM provides a new way to bridge the macro-scale responses and the micro-scale characteristics of granular materials. Particle breakage modeling in DEM has been widely studied and many effective methods were developed. Generally, these methods can be classified into two categories. The first category uses an agglomerate of bonded spherical sub-particles to represent the crushable particle (Alonso, Tapias, & Gili, 2012; Bolton, Nakata, &

https://doi.org/10.1016/j.partic.2018.08.006 1674-2001/© 2019 Chinese Society of Particuology and Institute of Process Engineering, Chinese Academy of Sciences. Published by Elsevier B.V. All rights reserved.

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Fig 1. Illustration of the two particle breakage simulation methods in DEM.

Cheng, 2008; Cheng, Nakata, & Bolton, 2003; Cil & Alshibli, 2012; Khanal et al., 2004, 2007; Lim & McDowell, 2005; McDowell & Harireche, 2002; Wang & Yan, 2013), and the particle is broken into several parts through the failure of bonds based on the bondedparticle model (BPM). In the second category, once a pre-defined failure criterion is satisfied, the crushed particle is replaced with a set of unbonded smaller particles (Astrom & Herrmann, 1998; BenNun & Einav, 2010; Brosh, Kalman, & Levy, 2011; Ciantia, Arroyo, Calvetti, & Gens, 2015; Cil & Buscarnera, 2016; de Bono & McDowell, 2014; Lobo-Guerrero & Vallejo, 2006; McDowell & de Bono, 2013; Tsoungui et al., 1999; Zhou et al., 2015); this is generally referred to as the fragment replacement method (FRM). The two methods are illustrated in Fig. 1. When conducting particle breakage simulation based on FRM, two issues should be considered (Astrom & Herrmann, 1998; BenNun & Einav, 2010; Lobo-Guerrero & Vallejo, 2006; Tsoungui et al., 1999), namely, the particle breakage criterion and the fragment replacement mode. The particle breakage criterion triggers the fragment replacement event. The fragment replacement mode refers to the fragment packing configuration used to replace the broken particle, including the fragment number and the fragment size distribution. Various fragment replacement modes have been proposed (BenNun & Einav, 2010; Ciantia et al., 2015; Cil & Buscarnera, 2016; Lobo-Guerrero & Vallejo, 2006; Marketos & Bolton, 2009; McDowell & de Bono, 2013; Tsoungui et al., 1999; Zhou et al., 2015, 2016), some of which are presented in Fig. 2. The most notable differences among those fragment replacement modes are the fragment number and the fragment size distribution. The fragment number and fragment size distribution of these modes were generally obtained from the spatial geometrical relations (Ciantia et al., 2015) or assigned by the researchers based on experimental observations (Lobo-Guerrero & Vallejo, 2005; Zhou et al., 2015), rather than derived from experimental data. This may cause a large discrepancy between the estimated and real values. Åström and Herrmann (1998) found that the fragment replacement mode has a significant effect on the breakage process of particles, leading to different final particle size distributions. Ciantia et al. (2015) and Li, McDowell, and Lowndes (2014) also observed this phenomenon. Ben-Nun and Einav (2010) noted that the fragment allocation affects the ultimate topology through a change in the fractal dimension. Because the particle size distribution has an apparent effect on the mechanical behavior of granular assemblies (Li, Liu, Dano, & Hicher, 2015; Muir Wood & Maeda, 2008; Yan & Dong, 2011) and different fragment replacement modes can result in different final particle size distributions, the breakage function of realistic particle breakage events

should be considered to some extent when constructing the fragment replacement mode. Brosh et al. (2011) was the first to consider this issue and proposed a fragment spawning strategy, in which the fragment number is randomly distributed and the fragment size is selected from a pre-defined breakage function. The breakage function adopted by Brosh et al. (2011) was obtained from impact tests. It is clear that the breakage function estimated from a single particle compression test differs from that derived from an impact test, and the breakage function is also sensitive to the impact velocity (Kalman, Rodnianski, & Haim, 2009; Khanal, Schubert, & Tomas, 2008). To simulate particle breakage by FRM in quasi-static conditions such as a confined compression test and a triaxial compression test, an appropriate breakage function should be adopted to model the fragment replacement mode. In experimental studies (Li et al., 2014; Lobo-Guerrero & Vallejo, 2005, 2006; Tsoungui et al., 1999) a crushed particle was split into a group of different-sized fragments. For example, Takei, Kusakabe, and Hayashi (2001) conducted a series of single particle compression tests using various materials and found that glass beads always broke into 30–50 measurable pieces and quartz particles generally broke into less than 10 measurable pieces (tiny fragments due to abrasion were not calculated). Due to the complexity of the loading condition of a single particle in a granular assembly, a simplified approach was adopted, where the fragment replacement mode is generated using the breakage function of a single particle after the compression test (Li et al., 2014; Lobo-Guerrero & Vallejo, 2006; Zhou et al., 2015). However, even the breakage function of a single particle after the compression test is quite complex (Li et al., 2014; Zhao, Wang, Coop, Viggiani, & Jiang, 2015). To the best of our knowledge, a breakage function that can cover the complexity of the fracture patterns of a single particle following a compression test has seldom been considered in fragment replacement mode models. Therefore, in this paper we investigate the breakage function of a single particle after a compression test and present a simple predictive model for the fragment replacement modes. A series of numerical single particle compression tests were implemented, and the reliability of the simulation tests was qualitatively validated by published experimental results. Then, the fragmentation degree of each particle following the compression test was described by the fractal theory. The fragmentation degree of the broken single particles was analyzed and the statistical distribution of the fragmentation degree for all the broken particles was investigated. Then, the overall breakage function of all the fragments of the 60 tests was investigated, and a two-stage distribution model for modeling the fragment replacement modes was proposed. The pro-

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Fig. 2. Commonly used fragment replacement modes. (a) and (b) are 2D cases; (c)–(f) are 3D cases.

posed model was validated by numerical and experimental results. Finally, an initial application strategy using the two-stage distribution model to generate fragment replacement modes was discussed and presented.

Table 1 Input parameters used in the numerical single particle crushing tests. Parameter Rock particle

Value

Radius, R (mm)

50 3

Subparticles

Density,  (kg/m ) Friction coefficient,  Minimum radius, Rmin (mm) Rmax /Rmin Young’s modulus, Ec (GPa) Stiffness ratio, kn /ks

2650 0.5 2 2.0 70 2.5

Parallel bond

Young’s modulus, E¯ c (GPa) Stiffness ratio, k¯ n /k¯ s Friction angle (◦ ) Tensile strength and cohesion

70 2.5 30 Gaussian distribution with a mean value of 60 MPa and a variation coefficient of 0.3

Numerical single-particle compression test The single-particle compression test on spherical or disk-shaped particles is a common approach to measure the tensile strength of brittle materials indirectly (Darvell, 1990). As the particle breakage process in a granular assembly is complex and hard to capture in a laboratory test, the single-particle compression test was deemed an effective way to investigate the breakage behavior of particles, mainly due to its simplicity and intuitiveness (Cheshomi & Sheshde, 2013; Zhao et al., 2015). Discrete element simulations of single-particle compression tests were conducted to capture the fragmentation evolutional process during the compression test. The particle shape effect was not considered in this numerical study, and all the numerical samples were spherical particles. A DEM code, PFC3D (Itasca, 2014), was employed in this work. The Mohr–Coulomb model with tension cut-off was used as the failure criterion in the bonded-particle model. Spherical agglomerates, composed of densely packed sub-particles, with a radius of 50 mm were generated to represent crushable rock particles. The radius of the sub-particles varied uniformly from 2.0 to 4.0 mm. The different packing configurations of the agglomerates were formed by using random seeds in the sample preparation. Many experiments have shown that the breakage strength of brittle particles of the same size and morphology varied considerably due to the random distribution and evolution of the micro-defects. To account for the randomness of natural defects and reproduce the Weibull distributed strength of brittle particles, the bond strength was randomly assigned based on a Gaussian distribution as suggested by Cil and Alshibli (2012). The first numerical single particle compression test is denoted S-1, with the rest deduced by analogy. The input parameters are listed in Table 1. A loading velocity sensitivity analysis was conducted. Considering the quasi-static condition and computational efficiency, the loading velocity applied at the top loading platen was set to 0.02 m/s. The validity of the numerical single particle crushing tests

Fig. 3. Force–displacement curves of test S-24.

can be demonstrated qualitatively by comparing the simulated behavior (Fig. 3) with the experimental results of platen compression tests on single quartz particles (Nakata, Hyde, Hyodo, &

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Fig. 4. The simulation data and the best-fit Weibull distribution curve.

Murata, 1999) and microcrystalline limestone particles (Cheshomi & Sheshde, 2013). The 60 crushing strength points are plotted in Fig. 4. The data  points are fitted using the Weibull distribution law ln ln(1/Ps ) = m ln  − m ln 0 , where Ps is the survival probability of a particle, m is the Weibull modulus,  is the crushing strength, and  0 is the characteristic stress. The crushing strength points satisfy the Weibull distribution well, which is in agreement with experimental results (Huang, Xu, Yi, & Hu, 2014; Lim, McDowell, & Collop, 2004; Weibull, 1951) and DEM simulation results (Ergenzinger, Seifried, & Eberhard, 2012; Ma et al., 2016; Wang & Yan, 2013). This indicates that the simulation strategy can reliably reflect the Weibullian behavior of natural particles, and is capable of mimicking the single particle compression test. Cheshomi and Sheshde (2013) observed that spherical microcrystalline limestone particles have two frequent fracture patterns, namely, the planar and triad fracture patterns. Zhao et al. (2015)

found that roughly spherical sand particles demonstrate a fracture pattern in which two fracture planes are nearly perpendicular to each other, separating the particle into four major fragments. Zhang, Buscarnera, and Einav (2016) noted three types of failure modes for single particles under uniaxial compression: the center crack, contact crack, and random crack failure modes. These fracture patterns are shown in Fig. 5. Some typical fracture patterns obtained at the first bulk breakage of the particles in this study are shown in Fig. 6. Almost all the fracture patterns observed in the laboratory tests can be reproduced by the numerical simulations. The fracture patterns of S-1, S-10, and S-32 are typical cases of the center crack failure mode; the failure patterns of S-44 and S-36 are similar to the contact crack failure mode and random crack failure mode, respectively. The comparison further indicates that the numerical test reproduces the experimental fracture patterns well and is therefore suitable for analyzing the fragment characteristics to some extent.

Statistical characteristics of single particle fragmentation Natural materials may be fragmented in a variety of ways such as weathering, explosions, impact, compaction, and geological loading. In many cases, fragmentation results in a fractal distribution (Altuhafi & Baudet, 2011; Turcotte, 1986; Zhao et al., 2015) which can reflect the characteristics of the grading curves. Moreover, the value of the fractal dimension can be used to describe the fragmentation degree of natural materials (Zhang, Li, Liu, & Zhou, 2017). Therefore, fractal theory was employed to estimate the fragmentation degree of broken single particles after compression tests. A fractal can be defined by the relationship between the particle number and particle size. Many experimental results indicate that the size-cumulative number distribution of fragments is given by N(d) ∼ d−D , where N(d) is the cumulative number of fragments having a diameter ≥ d, and D is the fractal dimension that can be estimated from the slope of a straight line in the double-log plot

Fig. 5. (a) Fracture patterns of single particles subjected to compression (Cheshomi & Sheshde, 2013). (b) Fracture patterns of LBS-3 and LBS-4 obtained from X-ray microtomography (Zhao et al., 2015). (c) Schematic diagrams of different failure modes, from left to right, the center crack, contact crack, and random crack failure modes (Zhang et al., 2016).

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Fig. 6. Typical fracture patterns of single-particle simulation compression tests: top view of S-1, S-10 and S-32, and lateral view of S-44 and S-36 (the colors represent different fragments).

Fig. 7. Fractal distribution of fragments of test S-24 at different compression displacements.

(Turcotte, 1986). The number of fragments N(d) cannot be accurately simulated by the numerical tests implemented in this paper, as the minimum size of the sub-particle is a fixed value that induces a limitation on the numerical breakage. Therefore, an equivalent approach for calculating the fractal dimension of particle crushing by the cumulative fragment mass is applied as follows (Einav, 2007; Turcotte, 1986; Xu, 2005): dM ∝ d3 dN(d) ∝ d2−D ,

(1)

where M is the mass of the fragments. Then, the total mass of the fragments with d < dm can be expressed as



M(dm ) =

3−D sd3 dN(d) ∝ dm ,

(2)

where s is the shape factor,  is the specific mass, and both s and  are assumed to be constant. Thus, the fragment size cumulative distribution by mass is M(dm ) ∝ M(d)

 d 3−D m

d

.

(3)

Taking the logarithm on both sides of Eq. (3),  the fractal dimen sion can be determined from the slope of log (M(dm )/M(d)) vs. log(dm /d). Fig. 7 shows the fitted fractal distributions of fragments at loading displacements of 1–5 mm of test S-24. It should be mentioned that in most of the compression tests bulk breakage was observed before and near the loading displacement of 1 mm. The volume equivalent fragment diameter (d) is defined as the diameter of a sphere that has the same volume as the fragment, and d0 is the original particle diameter. The volume equivalent diameter ratio is defined as  = d/d0 . In the numerical tests, the volume of each single fragment was obtained by adding the volume of the sub-particles of

Fig. 8. Fractal dimensions at different compression displacements.

each single fragment directly. The crushed single particle comprises a few large fragments and a large amount of debris, which is shown with a gap-graded distribution in Fig. 7. This phenomenon is clearer at a small loading displacement. Therefore, the fractal dimension obtained from the fractal analysis in this section is used only to describe the fragmentation degree of the broken single particle to some extent. As shown in Fig. 7, the fractal dimension D increases with loading displacement and the fractal distribution no longer changes with further increase of the loading displacement. The fractal dimension stays nearly constant when the loading displacement increases from 3 to 5 mm. The fractal distribution performs better when the fragmentation is extensive, i.e., with a larger loading displacement. These results demonstrate that the fractal dimension can appropriately describe the evolution of the degree of particle fragmentation with increasing loading displacement to some extent. The fractal dimensions of 60 tests at different loading displacements were calculated and analyzed. As 60 compression tests were implemented in this study, 60 fractal dimensions were obtained at each compression displacement. The 60 fractal dimensions are plotted in Fig. 8 with their box plots. The distribution of the fractal dimension is non-uniform, which can be explained by the different micro structures and the distribution of the micro defects of each simulated rock particle, which result in different breakage functions. It also reflects the different fragmentation degrees as well as the different failure types of the broken particles. The phenomenon is consistent with the experimental tests implemented by Wang, Dan, Xu, and Xi (2015) and Nakata et al. (1999), in which marble pebbles and sand grains were broken with various failure types in single particle crushing tests. The range of the fractal dimension decreases with increasing loading displacement, and the mean value increases noticeably, while the upper boundary of the fractal dimension remains almost constant with increasing loading displacement. This indicates that the fractal dimension of the

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Fig. 9. Fractal dimensions obtained for different compression displacements and their best-fit Weibull distribution curves.

fragments of a single particle may have a normalized trend when undergoing complete fragmentation. The fractal dimensions of the 60 tests at each loading displacement can be fitted by the Weibull distribution as ln[ln(1/Ps )] = m ln D − m ln Dc ,

(4)

where Ps is the cumulative probability of the fractal dimension, and m is the Weibull modulus, which describes the variability of the fractal dimension. The larger the m value, the more clustered the distribution of the fractal dimension. The characteristic fractal dimension (Dc ) is the value of the fractal dimension when ln[ln(1/Ps )] = 0, namely Ps equals 37%. As shown in Fig. 9, at loading displacements of 1–3 mm, m = 6.596, 9.869, and 10.825, respectively. The corresponding Dc values, which increase with increasing loading displacement, are 2.04, 2.51, and 2.59. This result indicates that the fragmentation of a single particle is a gradual process and a particle will generally not be crushed completely at the first bulk breakage. This agrees with the single particle fragmentation process in particle assemblies observed in experimental tests (Takei et al., 2001) and resembles the evolution of the particle size distribution (or fractal dimension) of crushable particle assemblies (Coop, Sorensen, Bodas Freitas, & Georgoutsos, 2004; Takei et al., 2001; Wang & Yan, 2013; Zhang et al., 2017). According to the continuous spawning feature of FRM, we hold that the appropriate fragment replacement mode should be considered as the fragment distribution of a single particle at the first bulk breakage (e.g., 1 mm in this study). Therefore, the fragment data at compression displacement of 1 mm was further investigated in the next section. Breakage function for constructing the fragment replacement modes As mentioned in the previous section, the fractal dimension obtained from the fitted fractal distribution is only used to describe the fragmentation degree of a single particle to some extent. It is not possible and rational to use a fractal distribution with only one fractal dimension to predict the breakage function of a single particle at the first bulk breakage. The breakage of single particles presents an apparent randomness; therefore, it seems difficult to construct a predictive model for modeling the fragment replacement mode directly. To investigate the fracture patterns comprehensively, we collected all the fragments of the 60 tested particles to plot the overall breakage function and described the various fracture patterns of a single particle after compression. The overall breakage function

presents a two-stage distribution for  > 0.1, in which the slope of the fitting line of the large fragments ( ≥ 0.54) is different from that of the small fragments (Fig. 10(a)). The slopes of the large and small fragments are 2.855 and 0.31, respectively. There is a relatively large gap between these two slopes. This can be explained by the experimental phenomenon observed in some published studies (Cheshomi & Sheshde, 2013; Ma, Zhou, Regueiro, Wang, & Chang, 2017; Zhao et al., 2015), in which extensive fragmentation appeared around the contact point while large fragments accounting for most of the mass did not experience significant fragmentation during single particle compression tests. To validate this result, experimental data (Li et al., 2014) were extracted and analyzed from the mass classification data of the failure particles subjected to the diametrical compression test. In the experimental tests, the mass of each fragment was weighed by a laboratory electronic scale. Then, the volume of each single fragment was obtained by dividing the mass by the density. The volume equivalent diameter and the volume equivalent diameter ratio could then be calculated. The overall breakage function shows the same pattern as the numerical one obtained in this paper (Fig. 10(b)). In addition, 30 natural marble pebble particles were used in single particle compression tests to validate the numerical result (Fig. 10(c)). These particles have an initial diameter, namely, the initial distance between the two loading plates, of 19.5–25 mm. The overall breakage function of the implemented experimental data has the same pattern as that presented by Li et al. (2014) as well as that of the numerical model. These indicate that the overall breakage function of single particles at the first bulk breakage can be described by a two-stage linear distribution in a double-log plot. It is interesting to note that the experimental tests implemented in this study and by Li et al. (2014) used rounded or sub-rounded particles with various shapes, which suggests that the pattern of the overall breakage function of particles at the first bulk breakage is not sensitive to shape when the particles are rounded or subrounded. One reason for this may be that the fracture mechanism of a single particle under a compression test causes this condition, namely, regardless of the particle shape, the fracture mechanism of a single particle causes the shatter around the contact point and the cleavage of the main body during the single particle compression test. Ahmed and Drzymala (2005) observed that a single fractal for a wide size range is frequently not sufficient, and usually there are two separate fractal dimensions for the coarse and fine fractions. Zhang and Baudet (2014) concluded that a mono-fractal dimension could not represent the grading in a wide size range and carried out multi-fractal analysis, especially for the incompletely crushed natural granular assemblies. According to their results and the numerical and experimental results obtained in our study, a fragment size predictive model based on multi-fractal analysis was proposed. We assume that the overall breakage function of single particles at the first bulk breakage under the compression test obeys a two-stage linear distribution in a double-log plot as shown in Fig. 11. To determine the turning point, we first obtain the best fitting line for the large-fragment data in the double-log plot. The fitting distribution width of the large fragments should be as large as possible. Then, the turning point (t ) is defined as the lower boundary of the fitting distribution width of the large fragments. The turning point separates the fragmentation into two regimes: the large fragments ( ≥ t ) and the small fragments ( < t ). The fitting line of the large fragments has a slope of ˛, and the fitting line of the small fragments has a slope of ˇ. Apparently, ˛ is larger than ˇ. The experimental phenomenon whereby the shatter appears around the contact point and cleavage of the main body during the single particle compression test can be clearly demonstrated by this model.

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Fig. 10. The overall breakage function. (a) Numerical results; (b) experimental results adopted from Li et al. (2014); (c) experimental results of marble pebbles (this study).

According to Eq. (3), the equations used to obtain the fragment size configuration can be written as follows: (1) Calculating the equivalent diameter ratio of the large fragments ( ≥ t ) Mi =

Fig. 11. The two-stage distribution for fragments obtained from single particle compression tests at the first bulk breakage.

max

−



i+1

max

˛

, i = 1,2,3..., Mr ≥ Mt ,

(5)

where max is the volume equivalent diameter ratio of the largest fragment, and i and i+1 are the volume equivalent diameter ratios of the ith and (i + 1)th ranked fragment, respectively. We defined i ≥ i+1 ; therefore, max = 1 . ˛ is the slope of the fitting line of the large fragments. Mi , Mr , and Mt are the cumulative fragment mass percent between i and i+1 , the residual total fragment mass percent and the total fragment mass percent of the fragments with  < t , respectively. Therefore, Mi = di3 ,Mr = 1−

Åström and Herrmann (1998) pointed out that the number of fragments should be kept low so that many breakings can be made in each simulation when modeling the fragment replacement mode in DEM. This suggests that the fragment size distribution of a fragment replacement mode is mainly based on the size distribution of the large fragments. A fragment with  = 0.1 has a mass ratio of 0.001, which is too small to consider as a sub-particle in the fragment replacement mode. A cut-off ratio of 0.1 can sufficiently satisfy the purpose of our study. Therefore, fragments with  < 0.1 are not considered in this study.

  ˛ i

i

Mi = 1 −

i

1

di3 . In addition, Mt = (t /max )˛ .

1

(2) Calculating the equivalent diameter ratio of the small fragments (0.1 ≤  < t ) When Mr < Mt , we obtain  < dcut . From Fig. 11, the equation to obtain i < t is deduced as follows: lg (Mr ) = lg



i+1

t

ˇ

+ lg(Mt ), i = 1,2,3..., 0 < Mr < Mt ,

(6)

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Table 2 Calculated fragment size construction with different d1 (dcut = 0.54). Maximum equivalent diameter ratio, d1

Mass percent of the maximum equivalent diameter ratio, M1

Prediction of the fragment size construction

Number of the fragments have equivalent diameter ratio larger than 0.1

0.95

0.857

11

0.90

0.729

0.85

0.614

0.80

0.512

0.75

0.421

0.70

0.343

0.65

0.275

0.95 0.183, 0.159, 0.144. . . 0.90, 0.570 0.022 × 9. . . 0.85, 0.609 0.096, 0.094, 0.092. . . 0.8, 0.622 0.222, 0.192, 0.174. . . 0.75, 0.619 0.346, 0.2278, 0.20. . . 0.7, 0.604 0.406, 0.237, 0.211. . . 0.65, 0.581 0.383, 0.267, 0.234. . .

2 2 21 26 32 39

Fig. 12. First six equivalent fragment size ratios from 60 samples sorted by the largest fragment size. (a) Numerical results; (b) results calculated using Eq. (7).

where ˇ is the slope of the fitting line of the small fragments. Then, Eqs. (5) and (6) can be rewritten as

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨

 1 3 ˛ ˛ , M ≥ M ; i+1 = ˛ −  r t i i 1 ⎡



i

˛ 3i ⎪ ⎢ 1 1 − ⎪ ⎪ ⎢ ⎪ 1 ⎪ ⎪ ⎪ i+1 = t ⎢ ⎢ ⎪ ˛ t ⎪ ⎣ ⎪ ⎪ ⎩

⎤1

ˇ

⎥ ⎥ ⎥ , 0 < Mr < Mt ; ⎥ ⎦

i = 1, 2, 3, ... (7)

According to Eq. (7), given a set of 1 , ˛, ˇ, and t , a group of volume equivalent diameter ratios can be obtained. Table 2 lists several sets of fragment sizes deduced from Eq. (7) with different 1 values and ˛, ˇ, and t values from Fig. 10(a). It should be mentioned that hundreds of tiny fragments will be obtained from Eq. (7) to conserve the mass; this agrees to some extent with the fact that the experimental single particle breakage phenomenon can generate a large number of tiny fragments. Validation of the two-stage distribution model To validate the reliability of the prediction model, the  values of the fragments of the 60 samples were presented and sorted by the diameter of the largest fragment from each test (Fig. 12(a)). The ratios derived by Eq. (7) are shown in Fig. 12(b).

A similar trend of fragment size distribution can be observed for Fig. 12(a) and (b). Three fragment categories can be identified by the number of fragments with  > 0.3. For samples no. 1–20, no. 21–33 and no. 33–60, there are approximately 3 fragments, 2 fragments, and only one fragment with  > 0.3, respectively (Fig. 12(a)). Although the predicted fragment size distribution of the 60 samples has some error, the same three categories can also be found in Fig. 12(b). This validates the proposed fragment size construction method to some extent. The fragment data of the 30 marble pebbles after the compression test and the predicted results are shown in Fig. 13; only fragments with  > 0.1 were plotted. The predicted result describes the average trend of the experimental result. Four fragment categories were found: pebbles with two pieces, three pieces, four pieces, and more than four pieces of main fragments. This accurately captures the fragmentation phenomena observed in the experimental tests (Fig. 14). In addition, as Li et al. (2014) considered in their study, we set d1 = 0.843, i.e., the largest fragment has 60% of the original particle mass. The fitted parameters ˛, ˇ, and t of the experimental result shown in Fig. 10(b) were used in Eq. (7). The subsequent three  values calculated by Eq. (7) are 0.669, 0.323, and 0.175. The second largest fragment ( = 0.669) has 29.94% of the original particle mass, which is consistent with the mean value of 30% observed by Li et al. (2014). This also indicates that the proposed fragment size construction method is appropriate to some extent, at least for the large fragments ( ≥ t ). These comparisons demonstrate the reliability and capability of the proposed fragment size prediction model to model the fragment replacement mode.

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Fig. 13. First six equivalent fragment size ratios of 30 marble pebbles sorted by the largest fragment size. (a) Experimental results; (b) results calculated using Eq. (7).

Fig. 14. Fragment types observed in the experimental tests, categorized according to number of main fragments; from left to right: two main fragments, three main fragments, four main fragments, and more than four main fragments (very fine particles are not shown in the figures).

Discussion As the fragments with  < 0.1 were not considered in this study, we cannot determine their size distribution; thus, the reliability of the obtained fragments with  < 0.1 from Eq. (7) cannot be guaranteed. Moreover, the small fragment sizes deduced from Eq. (7) have some discrepancies with the real data (Fig. 12) since a fragment will not be split infinitely as defined by fractal theory. Therefore, the minimum fragment diameter obtained from Eq. (7) should have a limit. For example, the minimal  could be set as 0.1 or 0.2 to limit the total number of fragments. Then, the remaining mass can be conserved by generating a finite number of fragments with  = 0.1 or 0.2. To generate an efficient fragment replacement mode and guarantee sufficient breaking in FRM-based simulations, some simplifications should be implemented to reduce the total fragment number of the established fragment replacement mode (Åström & Herrmann, 1998). Li et al. (2014) only considered the large fragments, ignoring the small fragments after the second mass percent ranking and assigning the remaining mass to an assumed fragment, namely, the third-ranking fragment, based on the fragmentation characteristics of experimental tests. According to this compromise, Eq. (7) can be revised as a two-stage distribution model with

Fig. 15. Modified two-stage distribution for fragments at the first bulk breakage of a single particle compression test.

a size cut-off for generating the fragment packing configuration of the fragment replacement mode. The illustration of the revised distribution is shown in Fig. 15, where Lim (≤t ) represents the lim-

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itation of the minimal volume equivalent diameter ratio. One can set Lim according to the main fragment distribution characteristics obtained from experimental tests when establishing the fragment replacement mode based on the proposed model. One simple way of modifying the governing equation sets is shown in Eq. (8).

⎧  1 ⎪ ⎪ ⎪ i+1 = ˛i − 3i ˛1 ˛ , Mr ≥ Mt ; ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎤1 ⎪ ⎡  ⎪ i ˇ ⎨ ˛ 3i ⎢ 1 1 − ⎥ ⎪ ⎢ ⎥ ⎪ 1 ⎪ ⎢ ⎥ , 0 < Mr < Mt , i+1 > Lim ; ⎪ ⎪ i+1 = t ⎢ ⎥ ˛t ⎪ ⎪ ⎣ ⎦ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

(3) A modified version of this model, namely, the two-stage distribution model with size cut-off, was proposed to construct the fragment replacement modes. The fragment replacement modes can be constructed from this model based on a series of experimental single particle compression tests. However, further investigation and simplifications should be made when constructing an efficient and appropriate fragment replacement mode for DEM simulation. Acknowledgements

i = 1, 2, 3....

i+1 = Lim , 0 < Mr < Mt , i+1 ≤ Lim ;

The authors gratefully acknowledge the financial support of the National Key R&D Program of China (grant No. 2017YFC0404801) and the National Natural Science Foundation of China (grant No. 51579193). The authors also acknowledge the anonymous reviewers for their constructive comments.

(8)

References However, as the fragment number has a significant influence on the efficiency of the DEM study of crushable particles by FRM, further investigation and simplifications should be made according to a large number of experimental results when applying this prediction model to construct an efficient and appropriate fragment replacement mode. Conclusions The fragment size construction of the fragment replacement mode has an obvious influence on the breakage process of granular material in the FRM-based DEM simulation. A simple way to establish the fragment replacement mode is based on the fragmentation characteristics of single particles after compression. To investigate the breakage function of a single particle after a compression test and present a simple predictive model for estimating the fragment replacement mode, 60 numerical single particle compression tests were conducted by DEM. The numerical single particle compression tests were simulated using the DEM. The force–displacement curve, statistical distributions of the peak breakage stress and failure patterns of the numerical tests were all validated by experimental results. The main conclusions of this study are as follows. (1) The fragmentation degree of single particles subjected to compression was described by the fractal dimension of the fitted fractal distribution according to the breakage function of a single particle after the compression tests. The fractal dimensions of the 60 tests had a distinct randomness, indicating the various breakage functions of single particles. In addition, the fractal dimensions of the 60 tests at the same loading displacement showed a good fit to the Weibull distribution, and the characteristic fractal dimension, Dc , increased with increasing compression displacement, demonstrating that the fragmentation of single particles is generally gradual. (2) A two-stage distribution model to describe the breakage function of single particles at the first bulk breakage after a quasi-static compression test was proposed. The model has four parameters: the maximal volume equivalent diameter ratio of the fragment (1 ), the slope of the fitting line of the large fragments (˛), the slope of the fitting line of the small fragments (ˇ), and the volume equivalent diameter ratio of the turning point (t ). The performance of this model was validated by numerical and experimental results, confirming that the different fragment types of a single particle at the first bulk breakage under compression test can be comprehensively captured by the two-stage distribution model.

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