Size effect on the compression breakage strengths of glass particles

Powder Technology 268 (2014) 86–94

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Size effect on the compression breakage strengths of glass particles J. Huang, S. Xu ⁎, H. Yi, S. Hu University of Science and Technology of China, CAS Key Laboratory for Mechanical Behavior and Design of Materials, Hefei, Anhui, 230027, China

a r t i c l e

i n f o

Article history: Received 9 May 2014 Received in revised form 13 August 2014 Accepted 15 August 2014 Available online 25 August 2014 Keywords: Particle breakage Glass spheres Breakage strength Size effect

a b s t r a c t The quasi-static compression-breakage responses of the glass spheres with five different sizes (4–25 mm) are investigated. The breakage strength data are found to be highly scattered. Based on the Weibull model, a statistical approach is proposed to interpret the characteristics in the breakage strength of the glass particles. The Weibull stress concept is introduced to accurately define the breakage stress of the particles considering the effects of finite contact area and breakage modes. It is observed that the relationship between the cumulative survival probability of the particles and the breakage stress follows the Weibull distribution reasonably well. Moreover, the scaling law between the breakage strength and the particle size is consistent with the theoretical predication by the Weibull model. Finally, the energy consumption during the particle breakage process is discussed with a three-parameter Weibull distribution. The relationship of the characteristic energy and the particle size also follows a power law. © 2014 Elsevier B.V. All rights reserved.

1. Introduction Fracture and fragmentation of brittle materials are commonly encountered in engineering applications and of considerable scientific and industrial interest [1,2]. They are related to a wide range of phenomena, ranging from collisional evolution of large asteroids to disintegration of small agglomerates in the processing industries, e.g., pharmaceutical, food, and mining [3,4]. The breakage strength of single particles is a key parameter involved in several powder processing problems, including accidental breakage of the particles during transportation or storage [5], pulverization of catalytic particles [6], comminution efficiency of crushers and mills [7], and yielding transition of the particle beds under bulk compaction [8,9]. Extensive experimental studies have been dedicated to breakage properties of various brittle particles such as glass, ceramic, lime stone, and soil [10–12]. Two loading geometries are frequently used, i.e., the particle impact test and the symmetric compression test [11]. The particle impact tests have been widely employed to study the dynamic breakage process of brittle particles [10,13,14]. Effects of the impact velocity and the particle properties on the breakage modes and the fragmentation size distributions are discussed thoroughly. However, it is difficult to define and determine the breakage strength (the stress corresponding to failure) of the particles from the impact test. Only the

⁎ Corresponding author at: School of Engineering and Science, University of Science and Technology of China, Hefei, Anhui 230027, China. Tel.: +86 551 63601249; fax: +86 551 63606459. E-mail address: [email protected] (S. Xu).

http://dx.doi.org/10.1016/j.powtec.2014.08.037 0032-5910/© 2014 Elsevier B.V. All rights reserved.

threshold impact velocity which induces fragmentation is discussed concerning particle size or mineralogy [10,15]. Moreover, the inertial effects cannot be neglected under high-velocity impact. Compression test on spherical or disk-shaped particles is a common approach to measuring the tensile strength of brittle materials indirectly [16]. One can also use it to measure the breakage strength of brittle particles [5]. However, the definition of the stress for the spherical particles is still controversial [12]. The stress definition is critical for accurate measurements on the breakage strength and needs to be clarified from a physical basis. In addition, since particles of various sizes are handled in processing industries, the size effect on the breakage strength of the brittle particles should be investigated quantitatively. Analytical solutions to the stress distributions inside an isotropic sphere subject to symmetric compression have been derived based on the Hertzian contact mechanics [2,17]. The theoretical stress distribution calculations suggest that along the compression axis, tensile stress develops in the circumferential direction, which may induce meridian cracks and consequently splitting of the sphere. This was observed for concrete [18] and plaster [2] spheres. Thus, the bulk crushing strengths of these spheres can represent the tensile strength of the constituent materials. However, the breakage modes of glass particles are much more complex [10,14]. Hertzian ring cracks are observed around the contact area and the failure planes are not necessarily meridian [14]. Moreover, the stress distributions and the breakage modes of the sphere are sensitive to the contact area between the sphere and the wall. When the contact area becomes comparable to the cross-sectional area of the sphere, the surface tensile stress exceeds the bulk tensile stress [17] and the surface cracks may dominate the fragmentation of the sphere [19]. However, the effects of the finite contact area on the breakage strength of the brittle particles have not been fully investigated.

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Many experiments have shown that the breakage strengths of brittle particles, though of the same size and morphology, vary in a considerable range due to the random distribution and evolvement of the micro-defects [13,19]. Therefore, sufficient number of samples should be tested to obtain an effective breakage strength from a statistical point of view. The Weibull statistics [20] is usually utilized to interpret the scattering data points. A two- or three-parameter Weibull distribution is used to fit the relationship between the breakage strengths and the survival probability [19]. The breakage strength distributions of several brittle particles (e.g., ceramic particles [21], soil particles [12], and rock aggregates [19]) are found to follow the Weibull distribution. The Weibull modulus, which characterizes the scatter of the breakage strengths of ceramic particles, is much bigger than those of the rock and soil particles. Moreover, some researchers found that the particle size dependence of the breakage strengths is consistent with the theoretical prediction (power law) by the Weibull model [12], but some did not [19]. The uncritical use of the Weibull distribution to fit the strength data of the brittle particles has been questioned [21]. McDowell and Amon [12] proposed that three assumptions should be cautiously considered when applying the Weibull distribution: (a) the loading geometries of all particles are similar; (b) particles are subject to bulk fracture; and (c) the contact radius is negligible compared to the particle size. The effectiveness of these assumptions should be examined and some modifications may need to be included in the commonly used Weibull model. In the present work, the quasi-static compression-breakage properties of the glass spheres with five different sizes are investigated with the material test system 809 (MTS809). The compression force and displacement curves can be obtained from MTS809. The fragmentation process of the spheres is captured with the high-speed photography system. The size effect on the breakage strengths of the glass particles is discussed based on the Weibull model.

0.22 mm, respectively. The standard deviations are much smaller than the means, indicating that the spheres are of good uniformity in size. The experimental setups are presented in Fig. 1. Fig. 1a shows the compression setups, i.e., the material test system (MTS809) and the high-speed photography system. All the compression tests are performed with the MTS809 at a constant loading velocity of 0.002 mm/s, which is sufficiently low to make the inertial effects negligible. The high-speed camera is used to capture the catastrophic breakage process of the glass spheres. A steel box with three transparent PMMA windows is used to gather the fragmentation debris of the glass spheres and protect the operator and the camera in case of debris splash. The enlarged graph of the loading cell is shown in Fig. 1b. Two steel blocks (Φ14.5 × 10 mm) are stuck to the compression platens to avoid any damage to the platens. The glass sphere is sandwiched between the blocks with a thin layer of petrolatum in between to reduce friction. Upon loading, the lower platen compresses the sphere from the bottom up at a loading velocity of 0.002 mm/s while the upper platen is fixed. The loading stops when the sphere undergoes a catastrophic failure and the axial stress exhibits a sharp decrease from the maximum load. The axial force is measured by the force sensor embedded in the upper platen. The axial displacement, i.e., the displacement of the lower platen, is measured by the displacement sensor embedded in the lower platen. However, the axial displacement cannot be taken directly as the deformation of the sphere since it contains the deformation of the loading frame. Therefore, a compression test without a sphere between the blocks is performed to a maximum load of 60 kN. In this case, the axial displacement actually represents the deformation of the loading frame.

2. Experimental materials and setups

For each kind of glass spheres about 30 samples are tested to provide sufficient data for the subsequent statistical analyses of the breakage strengths of glass materials. The choice of this number of tests was based on the work of McDowell [23]. He proposed a statistical way to estimate the accuracy on the sample mean and standard deviation within specified confidence limits. He examined the specific case of a Weibull distribution of particle strengths (which is exactly the topic of this work) and concluded that for a Weibull modulus of 3 or greater (valid for glass [24]), 30 tests are adequate to accurately measure the mean particle strength. The typical compression curves of the spheres with three different sizes, i.e., Φ4.18 mm, Φ15.71 mm and Φ25.03 mm, are presented in Fig. 2a, b and c, respectively. In the figure, “Corrected” means that the axial displacement is corrected to give the real deformation of the

The K9 glass spheres are used as the experimental material. The spherical morphology eliminates the complications caused by the irregular shapes of real particles. K9 glass is widely used as an optical material and its fracture behavior is important for the manufacture of K9 optics. The bulk density of the K9 glass under normal conditions is 2.52 × 10 3 g/m 3 . The chemical composition is SiO 2 (69.13%), B2 O 3 (10.75%), K2 O (6.29%), Na 2 O (10.40%), As2 O 3 (0.36%), and BaO (3.07%) [22]. Spheres of five different sizes are used to investigate the size effect on the breakage strengths of the glass particles. The means and standard deviations of the sphere diameters calculated from 30 samples are 4.36 ± 0.08 mm, 8.02 ± 0.19 mm, 15.77 ± 0.19 mm, 19.78 ± 0.15 mm, and 24.70 ±

(a)

3. Experimental results and discussions 3.1. Compression responses

(b)

Flash High-speed camera Steel box

Steel blocks

Steel platens PMMA window

Glass sphere

To computer

Fig. 1. (a) The MTS809 compression setup and the high-speed photography system. (b) The enlarged graph of the loading cell.

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J. Huang et al. / Powder Technology 268 (2014) 86–94

(a) 2

Axial force (kN)

Hertz, Eq. (1)

Failure point

1

Corrected Original 0 0.00

0.05

0.10

0.15

Displacement (mm)

(b) 10

Failure point

Axial force (kN)

Hertz, Eq. (1)

5

Corrected Original 0 0.0

0.1

0.2

0.3

Displacement (mm)

(c)

Axial force (kN)

20 15

Failure point

Hertz Eq. (1)

10 5 0 0.0

Corrected Original

0.1

0.2

0.3

0.4

Displacement (mm) Fig. 2. The typical axial force-displacement curves of the glass spheres with different sizes: (a) Φ4.39 mm, (b) Φ15.71 mm and (c) Φ25.03 mm. Here Φ denotes diameter. In the figure, “Corrected” means that the axial displacement has been corrected to give the real deformation of the sphere by taking out the elastic deformation of the loading frame and the steel blocks. The elastic Hertzian contact model (Eq. (1)) is plotted for comparison.

glass sphere by subtracting the elastic deformation of the loading frame and the steel blocks. The normal elastic deformation of a sphere can be described by the Hertzian contact model as shown in Eq. (1). 4  pffiffiffiffiffi 32 E R U 3 !−1   1−ν 21 1−ν 22 1 1 −1   E ¼ þ ; R ¼ þ R1 R2 E1 E2 F¼

ð1Þ

Here F and U are the contact force and displacement, respectively. E⁎ and R⁎ are the effective modulus and radius, respectively. Ei, νi, and Ri are the elastic modulus, Poisson's ratio and radius of the curvature of the contacting objects, respectively. Here subscript 1 refers to the glass sphere and subscript 2 refers to the steel block. The Hertz contact curve is plotted in the figure for comparison with E1 = 67.25 GPa, ν1 = 0.21, E2 = 210 GPa, ν2 = 0.31, and R2 = ∞. The elastic deformation of the steel blocks can be estimated as the difference between the

Hertzian contact displacement when E2 = 210 GPa and E2 = ∞, respectively. Calculations show that the elastic deformation of the steel blocks is much smaller than that of the loading frame. Fig. 2 shows that the compression curves of the glass spheres can be divided into three stages, i.e., elastic, elastic–plastic, and brittle failure. The elastic deformation stage is consistent with the theoretical Hertz curve. The yield point where the force–displacement curve deviates from the Hertz curve corresponds to a contact displacement of about 40 μm. At the failure point, the sphere subjects to a catastrophic failure and the axial force exhibits a sharp decrease to zero. Therefore, the deformation of the glass sphere is dominantly elastic-brittle and the slight plasticity mainly concentrates around the contact area [11]. The corresponding breakage process of the glass spheres around the failure point (Fig. 2) is presented in Fig. 3a, b and c, respectively. The frame rate is 60,000 frames/s. The speckles on the surface of the spheres are intended to be used to calculate the surface strain field of the glass sphere during loading. However, the calculation results are unsatisfactory and will not be discussed in the present work. This has little influence on the following discussions which mainly focus on the bulk fragmentation of the spheres. The stress distributions across the sphere can be obtained directly from the theoretical solutions. Fig. 3a shows that the Φ4.39 mm sphere fragments into a lot of small pieces and there exist few large pieces comparable to the original sphere. The slight shape destruction of the sphere is caused by the Hertzian ring cracks splitting off a shallow slice before the release of the disintegrating cracks. However, the breakage modes of the larger spheres are quite different from those of the Φ4.39 mm sphere. The shape destruction of the larger spheres is much severer which even results in some small stress adjustments in the compression curves as is shown in Fig. 2b and c. The time interval from the crack initiation to the fragmentation becomes much longer than that for the Φ4.39 mm sphere and the crack propagation process can be observed clearly. With increasing loading, the cracks initiate from the contact region and propagate mainly along the loading direction. That produces big pieces in the fragmentation debris. However, crack branching is also observed on the sphere surface and small debris particles are produced on the joint of the branching cracks [25]. The breakage modes of the glass spheres are too complex to be distinguished and categorized clearly. The results indicate that the catastrophic failure of the glass spheres is mainly caused by the final bulk fragmentation of the sphere though surface damage is also observed. The following theoretical discussions are also based on the bulkfragmentation assumption. The maximum axial force and deformation of the glass spheres corresponding to the catastrophic failure are summarized in Fig. 4. The figure shows that the magnitudes of the breakage forces and maximum deformation of the glass spheres are highly scattered, especially for the larger spheres. Therefore, statistics should be adopted to interpret the characteristic breakage strengths of the glass spheres. The mean and standard deviations of the maximum contact ratios (i.e., the ratio of the contact radius to the sphere radius corresponding to the failure point) of the glass spheres are summarized in Table 1. The way for estimating the maximum contact radius qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi is shown in Fig. A.1, i.e., ac ¼ R2 −R2b, where ac is the contact radius, R is the initial radius of the sphere, and Rb is the sphere radius at the moment of particle breakage. As shown in the table, the contact ratios of the glass spheres with the same size do not vary significantly since the standard deviation is one order of magnitude smaller than the mean. However, the mean contact ratio of the glass spheres with different sizes decreases with the increasing particle size. 3.2. Weibull statistical analyses 3.2.1. Breakage strength and Weibull stress It has been theoretically and experimentally verified that the Weibull distribution can be used to describe the tensile strength

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Fig. 3. The corresponding breakage process of the spheres around the failure point of the sphere (Fig. 2): (a) Φ4.39 mm, (b) Φ15.71 mm, and (c) Φ25.03 mm. Here Φ denotes diameter. In (a), the frame interval is 16.7 μs. In (b) and (c), the frame interval is 50.1 μs. The white and black dots on the surface of the sphere are intended for calculating the surface strain field during loading.

distributions of brittle materials [12,26]. Based on the Weibull model, the cumulative probability that a brittle material of volume V0 survives a tensile stress σ can be described as   m  σ P s ðσ Þ ¼ exp − σ0

ð2Þ

where σ0 is the characteristic tensile strength of the brittle material and refers to the stress level at which the material has a survival probability of 0.37. m is the Weibull modulus which describes the variability in the tensile strengths of different samples. Then, for the same material with an arbitrary volume V, the relationship of the cumulative survival

Axial force (kN)

30

20

dm (mm)

h  m i V V0 P s ðσ ; V Þ ¼ exp − σσ 0   m  V σ ¼ exp − V0 σ0 h  m i ¼ exp − σσ

where σ0,d is the characteristic tensile strength of the sample of size d. The volume ratio can be expressed as V/V0 = (d/d0)3 if the volume V and V0 are geometrically similar. This is true for sphere samples. Therefore, the Weibullian size effect on the tensile strengths of the brittle materials can be described as 

d d0

−3=m

:

ð4Þ

Table 1 Summary of the maximum contact ratios for the glass spheres with five different sizes. Mean sphere diameter (mm)

0.1

ð3Þ

0;d

σ 0;d ¼ σ0

Φ 4.36 Φ 8.02 Φ 15.66 Φ 19.78 Φ 24.66

10

0 0.0

probability Ps and the tensile stress σ can be obtained from Eq. (2) based on the weakest-chain principle.

0.2

0.3

0.4

0.5

Maximum deformation (mm) Fig. 4. Summary of the maximum axial forces and the maximum deformation of the glass spheres corresponding to the fracture failure of the glass spheres with different sizes.

4.36 8.02 15.77 19.78 24.70 a b

Maximum contact ratioa Mean

Std.b

0.227 0.186 0.170 0.159 0.142

0.018 0.013 0.016 0.016 0.015

Test number

34 34 32 33 35

The ratio of the contact radius to the sphere radius corresponding to the failure point. Standard deviation.

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J. Huang et al. / Powder Technology 268 (2014) 86–94

In the present study, we will not consider the effect of contact plasticity. The stress distributions in an elastic isotropic sphere under static compression have been proposed by Chau et al. [2]. Only three stresses exist in the sphere due to geometrical symmetry, i.e., the radial normal stress σrr, the tangential normal stress σθθ and the shear stress σrθ. r, θ, and ϕ are defined in the spherical coordinates as is shown in Fig. A.1. For simplicity, it is assumed that the failure of the glass spheres is mainly caused by the tensile stress. The expressions of the radial and tangential normal stresses are presented in the Appendix A (see Eqs. (A.1) and (A.2)). The tensile stress distributions in the sphere can be described in a dimensionless form as

a r  σ

¼ f c ; ;θ 2 d d F=d

ð5Þ P s ¼ 1−

where d is the current diameter of the sphere. ac is the contact radius, which is dependent on the load applied by the platens. McDowell and Amon [12] derived a similar expression (Eq. (7) in their paper) through the dimensional analysis. However, the contact ratio was neglected. Eq. (3) can be differentiated to give the probability density distribution over volume V as ð6Þ

The survival probability of a sample in which the tensile stress vary over its volume V can be obtained by integrating Eq. (6) over the volume under tension Vt. P s ðσ ; V Þ ¼ exp

#   1 σ m − dV V0 σ0 Vt

ð7Þ

Substitute Eq. (5) into Eq. (7) and then the survival probability of a sphere of size d under a compressive force F can be described as " P s ðσ ; V Þ ¼ exp −

2

V V0

F=d σ0

!m Z f Vt

m

# a r  dV c ; ;θ : V d d

ð8Þ

Since f(ac/d, r/d, θ) is a dimensionless function and all the volumes (samples) are geometrically similar, the volume average of f m in the sphere is only a function of the contact ratio, provided that the Weibull modulus m is a material constant independent of particle  1=m m size, i.e. g ðac =dÞ ¼ ∫V f ðac =d; r=d; θÞdV=V . The assumption that

i nþ1

ð11Þ

where n is the number of particles, and i is the rank of a particle according to the strength data sorted in an ascending order. For the convenience of calculating the characteristic breakage stress and the Weibull modulus, Eq. (9) can be transformed into another form as ln ln

  1 ∂P s ðσ ; V Þ 1 σ m : ¼− P s ðσ ; V Þ V0 σ0 ∂V

"Z

 −1=m m where σ 0f ¼ σ 0 ∫V f ðac =d; r=d; θÞdV=V . σf0 is the characteristic t stress corresponding to the stress definition of F/d2. This approach neglects the complexity in calculating the Weibull coefficient g(ac/d). However, the stress definition is not based on the physical principles. The coefficient before F/d2 is controversial for various reasons, e.g., 1.0 [28], 0.7 [29], and 0.4 [19]. The controversy can be avoided and clarified when the definition of the Weibull stress is introduced. When the Weibull model is applied to discussing the breakage strength distributions of the brittle particles, a large number of samples should be tested. For a finite number of tested samples, the survival probability Ps of the particles can be calculated with the probability estimators [26]. The commonly used estimator for Ps is

  1 1 ¼ ln m þ m ln σ w Ps σ 0; d

ð12Þ

where σ0, d is defined in Eq. (4). Substituting the definition of σw into Eq. (12), we obtain ln ln

    h i 1 2 ¼ m ln F b =db þ m ln g ðac =dÞ− lnσ 0; d Ps

ð13Þ

where Fb and db are the force and sphere radius corresponding to the failure point (Fig. 2) of the sphere. They can be determined from the compression curves. As shown in Eq. (13), only the strength data Fb/d2b and the corresponding survival probability Ps of the glass spheres are needed to calculate the Weibull modulus. The survival probability and the strength data arrays are linearly fitted according to Eq. (13) and the slope of the line is the Weibull modulus. Then, the Weibull coefficient g(ac/d) can be calculated using the Weibull modulus and the stress distribution functions. Finally, the characteristic stress can be calculated from the intercept in the horizontal axis, i.e., ln ln(1/Ps) = 0. When calculating the Weibull coefficient, the average value of the Weibull moduli of five glass spheres is used. The experimental data points and the corresponding fitting curves according to Eq. (13) for the glass spheres with five different sizes are shown in Fig. 5. The data points satisfy the Weibull distribution well

t

m is independent of the particle size is generally acceptable [23] and will also be confirmed by our experimental results. Moreover, since the contact ratios of the samples with the same size do not vary significantly, g(ac/d) will be assumed to be a constant for the samples of the same size for simplicity, and defined as the Weibull coefficient. Therefore, Eq. (8) can be changed to a simpler form as ð9Þ

  1=m m is defined as the where σ w ¼ F=d2 ∫V f ðac =d; r=d; θÞdV=V

1

ln (ln (1/P s))

    V σw m P s ðσ ; V Þ ¼ exp − V0 σ0

2

0 -1

dm (mm) Φ 4.36 Φ 8.02 Φ 15.66 Φ 19.78 Φ 24.66

-2

t

Weibull breakage stress of the samples [27]. However, since McDowell and Amon [12] did not take into account the effect of the contact ratio, the integral remains a constant even for particles with different sizes. This constant was then grouped into the characteristic stress, and the stress corresponding to the failure of the particles was defined as F/d2. "

V P s ðσ ; V Þ ¼ exp − V0

F=d2 σ 0f

!m # ð10Þ

-3 -4 2.5

3.0

3.5

4.0

4.5

5.0

ln σ (MPa) Fig. 5. Calculation of the Weibull moduli and the characteristic stresses of the glass spheres with different sizes. In the figure, the scatter points represent the experimental data and the dashed lines are fitted according to Eq. (13). Ps refers to the cumulative survival probability of a sphere subject to a loading stress of σ.

J. Huang et al. / Powder Technology 268 (2014) 86–94 Table 2 Summary of the Weibull moduli and the characteristic stresses of the glass particles calculated from the two-parameter Weibull distribution. m

σf0,d(MPa)

σθ0,d(MPa)

σr0,d(MPa)

R2

4.36 8.02 15.77 19.78 24.70

5.62 4.24 5.66 5.04 4.5

109.9 60.0 37.5 32.2 27.3

187.4 107.9 69.1 60.3 52.4

64.3 35.8 23.2 18.3 7.6

0.99 0.94 0.93 0.90 0.95

except for small strength ranges. The calculated Weibull moduli of the glass spheres are presented in Table 2. The Weibull modulus does not show obvious dependence on the particle size, which seems to support the assumption that the Weibull modulus is a material constant. However, the Weibull modulus for different particle sizes shows great variability. The reason is that the standard deviation of the particle strength cannot be accurately measured with about 30 tests. Using the chi-square distribution estimation, the standard deviation can only be estimated to within approximately 25% of the true population value with 95% confidence for a sample size of 30 [23]. The Weibull modulus is related to the ratio of the standard deviation Σ to the mean μ of the Þ population as 1 þ Σμ 2 ¼ ΓΓ2ðð1þ2=m , where Г is the gamma function. Then, 1þ1=mÞ 2

for a Weibull modulus of 5, the variation in the measured values of the sample Weibull modulus is expected to be 4 to 7, which can explain the results presented in Table 2. Increasing the number of tests may reduce the variability of the Weibull modulus. The average Weibull modulus of the K9 glass material is calculated to be 5.01. Then, the evolution of the Weibull coefficient g(ac/d) with the contact ratio can be calculated with the stress distribution functions. Since the tangential and the radial normal stress can both result in a tensile failure of the glass spheres, two Weibull coefficients can be obtained assuming that the failure of the glass sphere is completely attributed to the tangential or the radial normal stress. The results are presented in Fig. 6. The figure shows that the tangential Weibull coefficient decreases approximately linearly with increasing contact ratio. However, the radial Weibull coefficient decreases sharply from 2.5 to 0.5 when the contact ratio increases from 0.1 to 0.2. After that, it also decreases linearly with increasing contact ratio. Then, three characteristic stresses can be obtained depending on the definition of the Weibull breakage stress and the results are summarized in Table 2. σf0,d refers to the normal characteristic stress corresponding to the stress definition of F/d2 (Eq. (10)). σθ0,d is the tangential characteristic stress calculated from the tangential Weibull

4 θ σ 0,d r σ 0,d f σ 0,d

3

2 1.0

1.5

2.0

2.5

3.5

Fig. 7. Three characteristic stresses obtained depending on the definition of the breakage stress. σf0,d refers to the normal characteristic stress of the particles of size d corresponding to the stress definition of F/d2 (see Eq. (10)). σθ0,d and σr0,d refer to the tangential and radial characteristic stresses which are calculated with the tangential and radial Weibull coefficients, respectively.

coefficient. σr0,d denotes the radial characteristic stress calculated from the radial Weibull coefficient. As shown in Table 2, σθ0,d is much bigger than σf0,d and σr0,d corresponding to the same particle size. Thus, the stress definition of F/d2 underestimates the breakage strength of brittle particles. The results indicate that the breakage strength of the brittle particles is actually dependent on their breakage modes. According to the definition of the corresponding Weibull coefficient, σθ0,d and σr0,d can be viewed as the upper and the lower limit of the characteristic breakage strength of the glass spheres under quasi-static compression. However, since there are many hemispherical (or approximately hemispherical) chunks in the fragmentation debris and σ θθ is generally higher than σrr in the tensile region of the sphere, it can be concluded that σθθ plays the dominant role in the fragmentation of the glass spheres. Therefore, the real characteristic breakage strength of the glass spheres is closer to the tangential characteristic stress σθ0,d and can be approximated by it. The relationship between the characteristic stress and the particle size is presented in Fig. 7. The figure shows that in the double logarithmic coordinate, the three characteristic stresses σf0,d, σθ0,d, and σr0,d all decrease approximately linearly with the increasing particle size. However, the correlation coefficients for σf0,d and σθ0,d are much higher than that for σr0,d. The slopes of the lines are − 0.79, − 0.72 and − 1.04, respectively. The slope for σθ0,d is slightly bigger than that for

1.0

dm (mm)

Tangential Radial

2.0

Φ 4.36 Φ 8.02 Φ 15.66 Φ 19.78 Φ 24.66

0.8

0.6

Ps

1.5

1.0

0.4

0.5

0.2

0.0 0.0

3.0

ln d

2.5

g(ac/d)

Eq. (4)

5

ln σ0,d

Particle size Φ (mm)

91

0.2

0.4

0.6

0.8

1.0

0.0

0

50

100

150

200

250

ac/d Fig. 6. Evolution of the Weibull coefficient g(ac/d) with the contact ratio ac/d. In the figure, “Tangential” or “Radial” means that the curve is calculated using the tangential or the radial normal stress distribution function.

Fig. 8. The fitting results of the experimental data with the three-parameter Weibull distribution. In the figure, the scatter points represent the experimental data and the dashed lines represent fitting with Eq. (14).

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J. Huang et al. / Powder Technology 268 (2014) 86–94

Table 3 Summary of the Weibull moduli, the threshold stress and the net characteristic stress of the glass particles calculated from the three-parameter Weibull distribution. Particle size Φ (mm)

m⁎

σth(MPa)

σ0,d⁎(MPa)

R2

4.36 8.02 15.77 19.78 24.70

3.37 1.91 1.95 1.71 2.52

60.2 48.3 39.3 34.1 18.9

126.8 56.8 28.1 24.2 32.7

0.985 0.987 0.993 0.986 0.974

σf0,d since the contact ratio decreases with the particle size and the Weibull coefficient also results in a size effect on the characteristic stress. The theoretical value of the slope predicted by Eq. (4) is −3/m, i.e., −0.6, slightly larger than the experimental results. A probable reason is that the pre-failure caused by the Hertzian ring cracks increases with the particle size and thus the breakage strength of the glass particles decreases faster than the theoretical prediction. The Weibull model can indeed be applied to characterizing the size effect on the breakage strengths of the glass particles. The experimental scaling law can be implemented in the numerical modeling concerning the fracture and fragmentation of brittle materials. As shown in Fig. 5, the data points in the small stress ranges deviate obviously from the two-parameter Weibull distribution. Most plots show a downward curvature, suggesting that there may exist a critical strength below which no failure occurs (i.e., Ps = 0) [19,30,31]. This threshold stress actually implies an upper limit to the flaw size in the brittle material [31]. Lim et al. [19] also observed the similar phenomenon, and a three-parameter Weibull distribution (Eq. (14)) was adopted to fit the experimental results. In the following, only the σw = σθw case is discussed. 2 σ w −σ th P s ðσ ; V Þ ¼ exp4− σ 0;d

particles. However, the Weibullian scaling law cannot be reconstructed. The calculated Weibull moduli m⁎ and characteristic stress σ0,d⁎ are both much lower than those (m and σ0,d) calculated from the two-parameter Weibull distribution. However, it can be observed that the real characteristic stress σ0,d = σ0,d⁎ + σth. It is interesting to note that the threshold stress also decreases with the increasing particle size. This is consistent with the experimental results of Keshavan et al. [32]. Since the threshold stress is the breakage strength corresponding to a zero survival probability, it must also be size dependent as the characteristic stress is. Therefore, the assumption that the threshold stress is independent of particle size is questionable [19]. 3.2.2. Energy consumption In comminution industry, the energy consumption is also important [5,11]. Under quasi-static loading, the work input from the platens is firstly transformed into the elastic energy of the glass sphere and the loading system (including the loading frame and the steel blocks). When the glass sphere breaks, the stored elastic energy is transformed into the kinetic energy of fragmentation debris, frictional dissipation (by contact sliding among the debris), surface energy and acoustic emissions. Therefore, the input work is completely dissipated during the particle breakage process except for the elastic energy stored in the loading system. As discussed in Section 3.1, the force–displacement curve of the loading frame can be obtained from the compression test without samples, and that of the steel blocks can be estimated from the Hertzian contact theory. Then, the energy consumed in the elastic deformation of the loading system can be estimated and deducted from the input work. In fact, the energy consumption in breaking a glass sphere Eb can also be calculated by integrating the corrected force–displacement curves directly (Fig. 2). This is physically self-evident and a brief mathematical demonstration is presented in Eq. (15). Z

!m 3 5

ð14Þ

where σth is the threshold stress below which no failure occurs. m* is the Weibull modulus, and σ0,d⁎ can be called the net characteristic stress of the particles of size d. They are different from those defined in the two-parameter Weibull distribution. σθw = gθ(ac/d)F/d2, where gθ(ac/d) is the tangential Weibull coefficient as presented in Fig. 6. Then, a nonlinear regression procedure is used to fit the experimental data points to Eq. (14). The fitting results are presented in Fig. 8 and the calculated parameters are summarized in Table 3. As shown in the table, the correlation coefficients become much higher than those presented in Table 2. Therefore, the three-parameter Weibull distribution seems to be more suitable for describing the compression-breakage strengths of the glass

(a)

Z Z f b p Fdu − Fdu − Fdu Z Z   Z p f b ¼ F up −u f −ub −ð u dF− u dF− u d FÞ Z δ b s Fdu ¼

Eb ¼

0

where F is the axial force. up is the platen displacement. uf, ub and us are the deformation of the loading frame, the blocks and the sphere, respectively. δb is the pure deformation of the sphere at the moment of breakage. As shown in Table 3, a threshold stress exists for material failure, and it is natural to assume that there is an energy barrier for breaking the glass sphere. This is also verified by the particle impact test where the particles rebound intact when the impact energy is low [10]. Therefore, a three-parameter Weibull distribution similar to Eq. (14) is used to

(b) 1

1.0 0.8

ln E0,d (J)

Ps

dm (mm)

0.2 0.0 0.01

R2=0.979

0

0.6 0.4

ð15Þ

Φ 4.36 Φ 8.02 Φ 15.66 Φ 19.78 Φ 24.66

1.73 -1

1

-2

0.1

1

Eb (J)

10

-3 1.0

1.5

2.0

2.5

3.0

3.5

ln d (mm)

Fig. 9. (a) The relationship of the input energy Eb and the cumulative survival probability Ps of the glass spheres with different sizes. Eb is plotted in a logarithmic scale. (b) Evolution of the characteristic breakage energy E0,d with the particle size.

J. Huang et al. / Powder Technology 268 (2014) 86–94 Table 4 Summary of the Weibull modulus, the threshold energy and the net characteristic energy of the glass particles. Particle size Φ (mm)

m⁎E

Eth (J)

E0,d⁎(J)

R2

4.36 8.02 15.77 19.78 24.70

1.94 0.98 1.83 1.26 1.83

27.1 87.6 350.6 682.5 623.2

100.3 154.7 678.1 783.1 1741.4

0.982 0.962 0.981 0.981 0.958

describe the relationship of the breakage energy and the cumulative survival probability. 2

!m 3 E E −E 5 P s ðE; V Þ ¼ exp4− b  th E0;d

93

Appendix A The schematic diagram of the loading geometry is shown in Fig. A.1. In the spherical coordinates, the tangential normal stress in the sphere can be expressed as   r 2n  r 2n‐2  Bn A2n þ 4nμC 2n R R   r 2n  r 2n‐2  ∂2 P 2n ð cosθÞ Dn A2n þ μC 2n þ2 R R ∂θ2

∞ X σ θθ ¼ P 2n ð cosθÞ  2 F=d n¼0

 ∞  r 2n  r 2n‐2  X σ rr ¼ P 2n ð cosθÞ  F n A2n þ4nð2n−1ÞμC 2n 2 R R F=d n¼0

ðA:1Þ

ðA:2Þ

ð16Þ

where Eth is the threshold energy below which no failure occurs. E0,d⁎ is the net characteristic breakage energy of the particles of size d. mE⁎ is the Weibull modulus. The fitting results are presented in Fig. 9a and the calculated parameters are presented in Table 4. Fig. 9a shows that the three-parameter Weibull distribution can fit the experimental data well though the physical significance is not clear. Table 4 shows that the threshold energy and the net characteristic energy both increase with the increasing particle size. The real characteristic breakage energy E0,d is calculated as the sum of the threshold energy and the net characteristic energy. Evolution of the real characteristic energy E0,d with the particle size d is presented in Fig. 9b. The figure shows that the relationship of the real characteristic energy and the particle size also follows the power law with a positive power of 1.73. The characteristic breakage energy of glass spheres is much higher than that of the irregular particles, e.g., NaCl and sugar [5].

4. Conclusions

2ð2n þ 3Þλ−2ð2n−2Þμ ð2n þ 3Þλ þ ð2n þ 5Þμ ; Dn ¼ − 8n þ 6 ð2n þ 1Þð8n þ 6Þ   8n2 −4n−6 λ þ 2ð2n þ 1Þð2n−2Þμ Fn ¼ − 8n þ 6

Bn ¼

" A2n ¼ E2n

ð2n þ 1Þð4n þ 3Þ



8n2 þ 8n þ 3 λ þ 8n2 þ 4n þ 2 μ

#

  3 2 4nðn þ 1Þλ þ 4n þ 4n−1 μ 4





5 ¼ E2n 2μ ð2n−1Þ 8n2 þ 8n þ 3 λ þ 8n2 þ 4n þ 3 μ 2

C 2n

pffiffiffiffiffiffiffiffiffiffiffiffiffi Z 6ð4n þ 1Þ 1−α 2 πα 3 pffiffiffiffiffiffiffiffiffiffiffiffiffi cosθ0 ¼ 1−α 2 ; α ¼ ac =R

E2n ¼ −

1 cosθ0

!1=2 x2 −1 P 2n ðxÞdx cos2 θ0

where F is the compression force. d is the current diameter of the sphere. Pn(x) is the Legendre polynomial of order n. Bn, Dn, Fn, A2n, C2n, and E2n

Experiments show that the breakage strength data of the glass spheres are highly scattered. Therefore, the general Weibull statistical approach is modified to interpret the characteristic breakage strength of the glass particles on a physical basis. The Weibull stress concept is introduced to accurately define the breakage stress of the particles considering the effects of the finite contact area and the breakage modes (or tensile stress distributions). It is observed that the breakage strength distributions of the particles of the same size follow the two- and three-parameter Weibull distribution reasonably well. The scaling law between the characteristic breakage strengths and the particle size is approximately consistent with the theoretical predication by the Weibull model. Moreover, the characteristic breakage strength of the glass particles is dependent on their breakage modes, which can be well characterized by the Weibull coefficient. The energy consumption during the particle breakage process is calculated from the compression curves of the glass spheres. The breakage energy also satisfies the threeparameter Weibull distribution, and the relationship between the characteristic breakage energy and the particle size follows a positive power law.

Acknowledgments This work is supported by the National Natural Science Foundation of China (Grant No. 11472264, 11272304), and the China Scholarship Council (No. 201306345008). We thank Professor S. N. Luo for the helpful discussions.

Fig. A.1. The schematic diagram of the loading geometry. The spherical coordinates (r, θ, ϕ) are presented in the figure. F is the load. ac is the contact radius. r is the initial radius of the sphere. rb is the sphere radius at the moment of particle breakage.

94

J. Huang et al. / Powder Technology 268 (2014) 86–94

σθθ / (F/d 2), σrr / (F/d 2)

3

σθθ 0

-3

-6 0.0

n 8 15 20 23

σrr

0.2

0.4

0.6

0.8

1.0

r/R Fig. A.2. The convergence analysis of the theoretical solutions (with E = 82.31 GPa, ν = 0.21, ac/R = 0.15, θ = 0). F is the load and d is the current diameter of the sphere. n refers to the number of terms used to calculate the stress σθθ and σrr.

are the constants determined from the boundary conditions. R is the initial radius of the sphere and ac is the contact radius. λ and μ are the Lame's constants. As shown in Eqs. (A.1) and (A.2), σθθ and σrr are the sum of infinite terms which should be truncated in calculation. Thus, a convergence analysis of the theoretical solutions should be performed to select an appropriate number of terms that participates in calculation. The results are presented in Fig. A.2. It can be seen that σθθ converges when the number of calculation terms exceeds 15, and σrr converges even faster than σθθ. Therefore, 20 terms are used to calculate the σθθ, σrr stress distributions in the present study. References [1] H.A. Carmona, F.K. Wittel, F. Kun, H.J. Herrmann, Fragmentation processes in impact of spheres, Phys. Rev. E. 77 (2008) 051302. [2] K.T. Chau, X.X. Wei, R.H.C. Wong, T.X. Xu, Fragmentation of brittle spheres under static and dynamic compressions: experiments and analyses, Mech. Mater. 32 (2000) 543–554. [3] C. Shang, I.C. Sinka, B. Jayaraman, J. Pan, Break force and tensile strength relationships for curved faced tablets subject to diametrical compression, Int. J. Pharm. 442 (2013) 57–64. [4] C. Thornton, K.K. Yin, M.J. Adams, Numerical simulation of the impact fracture and fragmentation of agglomerates, J. Phys. D. Appl. Phys. 29 (1996) 424–435. [5] Y. Rozenblat, D. Portnikov, A. Levy, H. Kalman, S. Aman, J. Tomas, Strength distribution of particles under compression, Powder Technol. 208 (2011) 215–224. [6] Z. Chen, C.J. Lim, J.R. Grace, Study of limestone particle impact attrition, Chem. Eng. Sci. 62 (2007) 867–877.

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