# Fractal Pattern of Particle Crushing of Granular Geomaterials during One-Dimensional Compression

Authors: Jiru Zhang; Biwen Zhang
Journal: Advances in Civil Engineering
 (2018)

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## Abstract

This paper focuses on the effect of particle crushing on the behavior of granular geomaterials. Series of high-pressure one-dimensional compression tests were carried out on a quartz sand-gravel. A detail investigation was performed primarily on the compression behavior, the evolution of particle-size distribution (PSD), the fractal pattern of the grading curve, and the amount of particle crushing. It was found that both the yielding state and the state at the maximum compressibility are closely associated with the evolution of PSD and the fractal pattern of particle crushing. As the vertical stress increases, the fractal characteristic of the grading curve appears only within the finer part at first, evolves into bifractal within the overall measurable grading curve late, and translates into monofractal finally. Furthermore, a pair of particle crushing indexesBe1 and Be2 considering different particle size scales were proposed. The reasonability of using Be1 and Be2 to describe the amount of crushing corresponding to the scale of particles was discussed. Finally, it was found that the value of the ratio between the volumetric strain and the crushing index Be1 is constant and independent of the initial particle size and the initial PSD when the vertical stress is larger than the stress at the maximum compressibility or the coarser part of the grading curve is evolved into fractal.


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## Body

## 1. Introduction

Particle crushing (a confluent terminology describing all types of fragmentation and breakage), which is triggered when the applied stresses of particles exceed their strength, essentially generates a new complicated system with smaller sized and various shaped particles and consequently influences mechanical properties and behaviors of soils. This theme appears to play a vitally significant role associated with several natural disasters, such as landslides, earthquakes, debris flows, and surface collapses, and engineering applications; for instance, how to drive long offshore piles into bioclastic, calcareous soils or how to choose an applicable PSD of materials in constructing large earth and rockfill dams [1–9]. Hence, whatever the behaviors of single particle compression [10, 11], one-dimensional compression [12–14], ring shear tests [15–18], plane strain tests [19], or triaxial tests [6, 20–22], it attracts extensive and special attention in geotechnical engineering owing to the complex mechanism involved. Totally, it is important and necessary to improve our understanding of the internal mechanism between particle crushing and soil behavior from the perspective of scientific research and the practical view of engineering problems.By means of either laboratory tests or discrete element modeling [13, 15, 23–34], several studies focused on the significant role of PSD in the basic constitutive properties of granular materials, especially in the critical states. Particle crushing which produces a new PSD depends not only on the nature of the crushable particle, but also to a relatively great extent on the surroundings involving the sizes and shapes of adjacent particles and the force chains exerted to the focused objective. Although the particle-crushing-induced complicated system during compression or shearing depends on manifold factors, such as mineralogies, morphology, stress-strain state, void ratio, water content, and test methodologies, but it is consistent with fractal theory in many cases. Some previous studies confirmed in broad agreement that both the crushing of a single particle and the continuous crushing of granular aggregates result in an increasing tendency towards a monofractal PSD with a single fractal dimension D [10, 35–39]. Also, the bifractal behavior of geomaterials during crushing was observed [18]. Consequently, a question which arises is what role particle crushing plays in influencing the fractal behavior, especially in the development of fractal. This has not been detailed in studies yet. No researchers so far have examined the relationship between the crushing of particles and the varying fractal characteristics of grading curves, especially at different scales. A detailed investigation of particle-crushing-induced fractal development of grading curves will return us with a higher cognition of the sensibility of crushing of soils and of the nature of behavior of granular materials.According to the data of one-dimensional compression tests on quartz sand-gravel up to 204.8 MPa [39], the present paper examined the effects of particle size and initial PSD on the compression and crushing behaviors of granular geomaterials. In addition, the fractal dimensions of varying grading curves taking different scales into account were calculated to explore the fractal mechanism of particle crushing. Due to the fact that particle crushing generally exists in overall mechanical behaviors of crushable soils, even though the findings are obtained from one-dimensional compression tests, it still improves our understanding of the nature of particle crushing implicated in the fractal theory and may open the door to the intelligent interpretation of soil behaviors considering the effect of crushing.


## 2. Materials and Test Methods

Tests in this paper were performed on a type of crushable granular geomaterial which is a mixture of sand and gravel from the Yangtze River in China. The percent content of main component is 89.72% of SiO2 by means of X-ray diffraction experiments. Hence, it is named as quartz sand-gravel (QS). Before tests, the materials were dry-sieved into fractions to make various specimens involving uniformly graded specimens (marked as QS1 and QS2) and natural graded specimens (marked as QS3). The initial grading curves of all types of specimens are shown in Figure 1. As shown in Figure 2, a self-made confined compression device [37] with a 79.8 mm diameter and 20 mm height cylindrical specimen was used to reach very high vertical stress up to the maximum 204.8 MPa. All types of specimens were prepared by tamping using various energy levels to achieve a similar value of the initial void ratio e0 = 0.78 ± 0.03. The use of organo-silicic oil coating applied to the bottom of the piston minimized the effect of boundary friction. The dial indicators fixed to the piston were applied to measure compressive deformation of the specimens, where each value was the average of two measurements. In order to insight the evolution of PSD during crushing, these one-dimensional compression tests were terminated at nine vertical stress levels from 0.8 MPa to 204.8 MPa with one as the constant loading increment ratio. The details of the initial PSDs of all types of specimens and parameters of tests are summarized in Table 1.Figure 1
Initial grading curves.Figure 2
Test device.Table 1
Details of initial PSDs of the specimens and parameters of the tests.
Geomaterials
Specimen number
Initial PSD (%)
Specimen dimension
Initial void ratio
Specific weight
Vertical stress level
5000∼2000 (μm)
2000∼1000 (μm)
1000∼500 (μm)
500∼250 (μm)
250∼100 (μm)
100∼74 (μm)
D × H (mm × mm)
e
0
G
s
σ
V (MPa)
Quartz sand and gravel (QS)
QS1
100
0
0
0
0
0
79.8 × 20
0.78 ± 0.03
2.67
0, 0.8, 1.6, 3.2, 6.4, 12.8, 25.6, 51.2, 102.4, 204.8
QS2
0
60
40
0
0
0
QS3
34.2
33.3
22.3
9.4
0.3
0.5After one-dimensional compression tests, the grading curves of all precrushed specimens were determined by means of the sieve shaker and the laser particle analyzer. On one hand, the sieve shaker was used to obtain the first part of PSD, in which particles are ranged from 74μm to the largest particle size, from the part of specimens retaining on the 74 μm sieve pore. On the other hand, the laser particle analyzer was used to obtain the second part of PSD, in which particles are ranged from the minimum measurable size 1 μm to 74 μm, from the balance passing through the 74 μm sieve pore. The data obtained by these two methods were summed up, and the whole measurable grading curves of the specimens after tests were calculated. Because the method of the laser particle analyzer to obtain the corresponding PSD is based on the equivalent diameter analysis, the accuracy will not be enough if the shapes of the considered particles are flake-like or needle-like. Fortunately, according to the shape factor data of precrushed geomaterials, such as quartz sand and carbonate sand [40] and the scanning electron microscope (SEM) images of the generated particles after crushing [41], the ratios between the 3D of the created particles by crushing are not much different. Hence, it is reasonable to guarantee the validity of using the laser particle analyzer to describe that part of PSD. It is also shown that the ratios between the 3D of most particles generated from crushing, whose sizes are smaller than 74 μm, are similar.


## 3. Results and Discussion

### 3.1. One-Dimensional Compression Behavior

The data points in Figure3 show the decreasing tendency of the void ratio e with increasing logarithm of vertical stress σV for all types of specimens. The continuous reduction of the value of the void ratio e is mainly due to successive particle crushing and rearrangement. With almost the identical initial void ratio e0, the reduction of the value of the void ratio e is largest for QS1, intermediate for QS2, and lowest for QS3 at the same vertical stress levels. The reasonable interpretation of the above observation is as following [14, 36]. The crushing strength of the larger particle is lower because there are more interior flaws in it. Hence, QS1 with larger particle sizes produces more crushing and rearrangement for uniformly graded specimens. Meanwhile, the natural graded specimen QS3 is well graded comparing to the uniformly graded specimens. The particles in QS3 are protected by more surrounding contact points. Hence, the probability of the particle crushing of QS3 is lower and less void ratio produced by crushing is used for particle rearrangement.Figure 3
Relationship betweene and σV.On the other hand, as shown in Figure3, we are obliged to select the smooth B-spline to describe e versus log σV relationship with unimportant error within the maximum testing load because the decreasing tendency above is actually a continuous process. According to [14], two states on the one-dimensional compression are defined. One is the yielding state which corresponds to the state when the curvature of the relation between the void ratio e and the logarithmic of the vertical stress σV is maximum. Another is the state at the maximum compressibility, which corresponds to the state when the compression index Cc is maximum. Hence, in Figure 3, the yielding states are marked as the hollow legends and the states at the maximum compressibility are marked as the semihollow legends for all types of the specimens. The stress σVy corresponds to the stress at the yielding state, and the stress σVc corresponds to the stress at the state when the compression index Cc is maximum. With respect to these two indexes σVy and σVc, three situations can be observed from Figure 3: (i) for uniformly graded specimens, the specimen QS1 with larger particle size has higher σVy and σVc; (ii) both σVy and σVc of uniformly graded specimens QS1 and QS2 are lower than those of the natural graded specimen QS3; and (iii) for a given specimen, σVc is higher than σVy. The first situation manifests that specimens with smaller particle size are more difficult to yield and are more difficult to show the maximum compressibility for uniformly graded specimens. Analogously, the second situation indicates that uniformly graded specimens are easier to yield and are easier to show the maximum compressibility comparing with well-graded specimens. Hence, the yield state and the state at the maximum compressibility depend not only on the nature of particles but also on the packing of multitudinous particles. The last situation expresses that the state at the maximum compressibility is always after the yielding state. It also implied that the gradient in the e-log σV relationship increases after yielding and then gradually decreases to a relative steady value after showing the maximum compressibility, combining with the shapes of the B-splines in Figure 3.Figure4 shows the relationship between the vertical strain or the volumetric strain εV (in the one-dimensional compression tests of this paper, the vertical strain is identical with the volumetric strain because of the constant cross section of the specimen) and the vertical stress σV. It can be observed that the value of εV increases rapidly at relative small stress levels, then increases less rapidly after the vertical stress σV passes beyond σVc, and gradually approaches stabilization at relative high stress levels for all types of specimens. The implications of the yielding state and the state at the maximum compressibility are further investigated in the following part.Figure 4
Relationship betweenεV and σV.


### 3.2. Evolution of PSD

The grading curves of all the specimens after one-dimensional compression tests terminated at each preset vertical stress level are plotted in Figure5, with semilogarithmic coordinates on the left pictures and double-logarithmic coordinates on the right pictures. Meanwhile, it is assumed that the grading curve at the preset stress level which is closest to σVy or σVc can be approximately regarded as the grading curve at the stress equal to σVy or σVc, respectively, with unimportant error. The dashed dotted line is the referenced ultimate fractal grading curve with a fractal dimension D = 2.6 [23]. Examination of Figure 5 shows a similarity for all types of specimens that the proportion of particles finer than 74 μm increases markedly as the vertical stress σV increases; however, part of the largest particles of initial specimens still exist even at the maximum vertical stress level. In the semilogarithmic coordinates of Figure 5, the curve gradually rises up and the degree of concave upward of the curve gradually decreases, in particular, after the vertical stress σV exceeds σVy. It indicates that the degree of crushing increases markedly after yielding. Meanwhile, in the double-logarithmic coordinates of Figure 5, the curve is gradually transformed into approximately a straight line, especially after the vertical stress σV exceeds a definite extent to σVc. It indicates that the grading curve is gradually equipped with fractal characteristic, especially after the specimen shows the maximum compressibility. Therefore, it may be reasonable to regard σVc as a transition point of the grading curve to be equipped with the fractal characteristic. In a similar fashion, it should be noted that when the vertical stress σV passes σVy, the shape of the grading curve within the particle size interval finer than 74 μm becomes a straight line, which indicates that the finer part of the grading curve may be previously equipped with fractal characteristic. Hence, we can deduct that σVy can at least be regarded as another transition point to be equipped with fractal characteristic within the finer part of the grading curve. Also, the reasonability of the above deductions is verified in the next part.Figure 5
Grading curves after tests terminated at each preset stress level for (a) QS1, (b) QS2, and (c) QS3.
(a)
(b)
(c)


### 3.3. Fractal Pattern

A simplex fractal dimension is usually used to describe the PSD of soil, and this fractal parameter can be used as a constant in models of soil-water characteristic curves (SWCCs) or particle crushing [33, 42–47]. However, according to the data in this paper and other relevant literatures [18, 48], a simplex fractal dimension sometimes does not adequately describe the fractal characteristics of the whole PSD. The evolution pattern of fractals requires further study.By assuming the referenced particle size asd, the largest particle size as dM, the mass of particles finer than d as M(δ < d), and the total mass of a specimen as MT, if there is a linear relationship between log(d/dM) and log[M(δ < d)/MT] with a slope k, we can deem that the grading curve is fractal with the fractal dimension D = 3 – k. The results of linear fitting between lg[M(δ < d)/MT] and lg(d/dM) are shown in Figure 6. The left pictures of Figure 6 regard particle size intervals which are finer than 74 μm and coarser than 74 μm as fitting intervals, respectively. And the middle pictures of Figure 6 regard the whole measurable particle size interval as the only fitting interval. Actually, not all the grading curves after tests are equipped with fractal characteristics. Hence, we are obliged to assume a value of determination coefficients R2 of linear fitting as a transformation point to make a decision that whether or not the fractal of the grading curve is relative strict. By examination of the right pictures of Figure 6, which show the histograms of determination coefficients R2 of these three aforementioned linear fittings at each stress level, the value of R2 equal to 0.98 may be a plausible and relative high value to represent this transformation point. Therefore, in the left and middle pictures in Figure 6, fitting lines having relative high significance level with R2 > 0.98 are presented by the solid lines, and those having relative low significance level with R2 < 0.98 are presented by the dashed lines. In addition, the values of the fractal dimensions of the fitting lines are labeled beside with D1 for those when d is coarser than 74 μm, D2 for those when d is finer than 74 μm, and D for those of the whole measurable size interval.Figure 6
Linear fitting between lg[M(δ < d)/MT] and lg(d/dM) with the left picture regarding particle size finer than 74 μm and coarser than 74 μm as a fitting range, respectively, with the middle picture regarding the whole measurable particle size interval as the only fitting range, and with the right picture expressing the histograms of determination coefficients R2 of these three linear fittings at each preset vertical stress level for (a) QS1, (b) QS2, and (c) QS3.
(a)
(b)
(c)The detailed examination of the left and right pictures of Figure6 shows that the fractal characteristic of the finer particle size interval is not relatively strict when the vertical stress σV is lower than σVy for all types of specimens. However, the fractal characteristic of the above part is relatively strict when the vertical stress σV is greater than σVy. Nevertheless, the similar result of the coarser particle size interval appears when the vertical stress σV is greater than σVc. The narration above at least implies that the fractal characteristic previously appears on the finer particle size interval after yielding and subsequently appears on the coarser interval after the specimen shows the maximum compressibility. By means of the middle and right pictures of Figure 6, the whole measurable grading curves of all types of specimens are equipped with monofractal characteristic only when the vertical stress σV exceeds a certain degree to σVc. Hence, three particle size regions should be differentiated. In the first region ranged between 1 μm and an uncertain size which may be finer than 74 μm, the grading curve is fractal when the vertical stress σV passes through σVy. In the second region ranged between the largest particle size and another uncertain size which may be coarser than 74 μm, the grading curve is fractal when the vertical stress σV passes through σVc. And the third region ranged between these two uncertain sizes represents a transition region from the fractal characteristic in the coarser region to that in the finer region. The existence of this transition region makes the problem extremely complicated. In order to simplify the analysis, the existence of this transition region is disregarded, and the transition particle size between the two fractal regions is assumed to be 74 μm. In addition, as shown in Figures 7(a) and 7(b), despite the different initial PSDs and particle sizes between all types of specimens, the PSD data of the finer region in which the values of fractal dimensions exceed 2.29 are shown to be strictly self-similar, and those of the coarser region in which the values of fractal dimensions exceed 1.85 are shown to be strictly self-similar. Therefore, the values of 2.29 and 1.85 can be considered as the lower limits of the fractal dimensions corresponding to the finer and coarser regions, respectively. In a similar fashion, the value of 2.16 can be considered as the lower limit of the fractal dimension for the whole measurable PSD, as shown in Figure 7(c).Figure 7
Relationship between vertical stressσV and fractal dimensions (a) D1, (b) D2, and (c) D.
(a)
(b)
(c)In general, during one-dimensional compression, when the vertical stressσV is between σVy and σVc, the fractal characteristic exists only in the finer interval of the grading curve, in which particle sizes are finer than 74 μm. Subsequently, the bifractal characteristic exists in the whole measurable grading curve when the vertical stress σV is immediately larger than σVc. And 74 μm should be the transition particle size point between the different fractal characteristics of the finer and coarser intervals. Finally, the whole measurable grading curve is monofractal when the vertical stress exceeds a certain degree to σVc. The evolution of the fractal characteristic above consists of the fractal pattern during one-dimensional compression.


### 3.4. Particle Crushing

Based on the different PSDs before and after crushing, some scholars put forward series of quantitative indexes to represent the degree of particle crushing. As a whole, those indexes may be divided into two principal groups: one is based on the change of one or several characteristic particle sizes before and after crushing, such asB10, B15, and BM as depicted in Figure 8 [1, 3, 49]. The other one is based on the relative change of the whole PSD before and after crushing, which are primarily represented by Br and Be proposed by Hardin [50] and Einav [51], respectively, as depicted in Figure 9. The difference between Br and Be lies in the assumption of the ultimate state of particle crushing. Hardin [50] assumed that particle crushing eventually formed a uniform distribution system with particle sizes less than 74 μm. However, Einav [51] assumed that the ultimate distribution was a monofractal distribution with scale invariance or self-similarity. It should be noted that most properties of soil may depend on the overall PSDs, such as void ratio, volumetric strain, constitutive relation, and critical state. Actually, particle crushing is not an active process like gravity, which is always fully active and forever produces its total effect, but can be denoted by a passive quantity which can be measured by the relative distance to its ultimate state indicating the elimination of possibility for any further crushing. From this point of view, particle crushing, which resembles internal friction and cohesion of soils, has to be mobilized by external conditions until the potential of crushing approaches zero. Therefore, comparing with the indexes of the first group which depend on just one or several characteristic particle sizes, Br and Be chosen in this study may be more appropriate for representing the amount of crushing not only owing to their clear concepts and clearly defined assumptions but also due to considering the changes of distribution in almost overall particle size range.Figure 8
Definitions of the particle crushing indexesBM [1], B10 [3], and B15 [49].Figure 9
Definitions of the particle crushing indexesBr [50] and Be [51] and the division of Be.On the other hand, with the above fractal pattern of particle crushing in mind, it is advisable to separate the relative breakageBe proposed by Einav [51] into two parts considering different scales. These different scales are the two ranges of the overall measurable PSD corresponding to the aforementioned bifractal characteristic. To achieve this objective, as defined in Figure 9, the total potential breakage Bp′ is divided into two parts Bp1′ and Bp2′ to represent the corresponding potential breakage whose particle sizes are ranged coarser and finer than 74 μm, respectively. And the total breakage Bt′ is also divided into two parts Bt1′ and Bt2′ to represent the corresponding total breakage whose particle sizes are ranged coarser and finer than 74 μm, respectively. Then, defining Be1 = Bt1′/Bp1′ to represent the relative breakage of particles coarser than 74 μm and defining Be2 = Bt2′/Bp2′ to represent the relative breakage of particles finer than 74 μm. The aim of this division is to apply the divided indexes to investigate the amounts of crushing in different particle scales.Figure10 shows the relationship between vertical stress σV and Be1 or Be2. For all types of specimens, as the vertical stress σV increases, both of the values of Be1 and Be2 increase rapidly, when the vertical stress σV is smaller than σVc. Then, when the vertical stress σV is larger than σVc, both of the values of Be1 and Be2 gradually approach a relatively steady state. However, the increasing rate of Be2 is obviously faster than that of Be1. This is because the value of Be1 is closer to unity than that of Be2, such as the values of Be1 at the maximum vertical stress 204.8 MPa for QS1 and QS2 are even 0.84 and 0.78, respectively, but the values of Be2 at that state for QS1 and QS2 are merely 0.46 and 0.35, respectively. It implies that the breakage potential in the scale of particles larger than 74 μm is not much left; however, the breakage potential in the scale of particles finer than 74 μm is still a lot at that state. Hence, the probability of crushing in the finer scale is higher. This difference also implies the reasonability of the division of Be. Comparing to the natural graded specimen QS3, this increasing tendency of the value of Be1 is more significant for the uniformly graded specimens QS1 and QS2. This is because that the cushioning effect of surrounding contacts of coarser particles in the natural graded specimen QS3 is more significant; therefore, the higher degree of particle crushing of the coarser particles is induced in the uniformly graded specimens QS1 and QS2. As shown in Figure 9, with respect to Br and Be1, the value of Bt is equal to that of Bt1′, and the value of Bp is equal to the sum of Bp1′ and Area1. Hence, the difference between the values of Br and Be1 is only due to the difference between their denominators. With respect to Be and Be1, although the value of Bp′ is larger than that of Bp1′ by Bp2′, the value of Bt2′ is mostly only a very small fraction of the value of Bt′. The difference between the values of Be and Be1 is also due to the difference between their denominators with not much important error. Hence, the variation tendencies of Be and Br in Figure 11, which show the relationship between the vertical stress σV and Be or Br, are similar to those of Be1 in Figure 10.Figure 10
Relationship betweenσV and Be1 or Be2.Figure 11
Relationship betweenσV and Be or Br.Hence, we can deduce that, at an extremely large vertical stress, the coarser part of the uniformly graded specimens is faster than the finer part to reach the ultimate state of particle crushing, although this deduction can only be executed by mental operations due to the limitation of laboratory tests. In addition, the effect of the increasing vertical stressσV on the evolution of Be and Br should be identical with that on the evolution of Be1; hence, these three crushing indexes Be, Br, and Be1 are reasonable for describing the amount of crushing in which particles are coarser than 74 μm with unimportant error, but the most accurate index should be Be1.Nevertheless, as shown in Figure10, the increasing tendency of the value of Be2 in the uniformly graded specimens QS1 and QS2 is less rapid than that in the natural graded specimen QS3. This may be resulted from that the average number of contact points of finer particles in QS3 is less than those in QS1 and QS2, and the contact cushioning effect exceeds the crushing strength effect on particle crushing. Hence, the probabilities of crushing of the finer “parent” particles (i.e., the finer particles which have not been crushed) and the finer and finer particles produced by previous rounds of particle crushing in QS3 are higher than the probabilities of crushing of those in QS1 and QS2. In addition, the pronounced different increasing tendencies between Be2 and Be1 represent that the reasonable index of particle crushing for describing the corresponding relative breakage, in which particles are finer than 74 μm, should be Be2 which is essentially different from Be, Be1, and Br.Figure12 shows the relationship between the vertical stress σV and the ratio εV/Be1 or εV/Be2. Meanwhile, Figure 13 shows the relationship between the vertical stress σV and the ratio εV/Be or εV/Br. From ring shear tests [15] and triaxial tests [16], the phenomenon that the value of εV/Br has been constant is observed almost at the critical state of soils. However, from Figures 12 and 13, it can be observed that both the values of εV/Be1 and εV/Be are already constant, respectively, after the vertical stress σV exceeds σVc. The values of the constants of εV/Be1 and εV/Be are 0.391 and 0.461, respectively. From the previous part of this paper, when the vertical stress σV is larger than σVc, it marks that the coarser part of PSD, in which particles are larger than 74 μm, is fractal. Hence, it indicates that the coarser part of PSD is equipped with the fractal characteristic can be regarded as a token of that the values of εV/Be1 and εV/Be are constant. On the other hand, the values of εV/Be2 and εV/Br also have a tendency to be constant as the vertical stress σV increases; however, the prominence of this tendency is far inferior to that for the values of εV/Be1 and εV/Be. With respect to εV/Be1 and εV/Be, the constants for both are determined by the average of the ordinates of data points, which are all at a state after the maximum compressibility, as shown in Figures 12 and 13. The relevant statistical parameters are summarized in Table 2. It can be found that the square deviation of the values of εV/Be1 is very small and equal to 0.00287 which is even slightly lower than that of εV/Be. Hence, comparing to εV/Be, the more appropriate physical quantity to represent the physical phenomenon that the ratio of the volumetric strain to the relative breakage is a constant should be εV/Be1. Hence, during one-dimensional compression, once the coarser part of PSD has been reached to fractal or the vertical stress σV has been larger than σVc, the prediction of the relative breakage can be accomplished by means of the measurable volumetric strain εV and the constant of the value of εV/Be1 obtained from tests.Figure 12
Relationship betweenσV and εV/Be1 or εV/Be2.Figure 13
Relationship betweenσV and εV/Be or εV/Br.Table 2
Statistical parameters of the values ofεV/Be1 and εV/Be for all types of specimens when σV ≥ σVc.
Vertical stress,σV (MPa)
ε
V/Be1
ε
V/Be
QS1 (σVc ≈ 9.9 MPa)
QS2 (σVc ≈ 20.3 MPa)
QS3 (σVc ≈ 54.9 MPa)
QS1 (σVc ≈ 9.9 MPa)
QS2 (σVc ≈ 20.3 MPa)
QS3 (σVc ≈ 54.9 MPa)
12.8
0.50349
—
—
0.6019
—
—
25.6
0.41932
0.30881
—
0.49532
0.40419
—
51.2
0.42075
0.33716
0.38471
0.4859
0.42325
0.46193
102.4
0.42174
0.35402
0.3502
0.47581
0.42814
0.41262
204.8
0.4594
0.37241
0.35382
0.5031
0.43703
0.39979
Mean value
—
0.39049
—
—
0.46075
—
Square deviation
—
0.00287
—
—
0.00300
—


## 3.1. One-Dimensional Compression Behavior

The data points in Figure3 show the decreasing tendency of the void ratio e with increasing logarithm of vertical stress σV for all types of specimens. The continuous reduction of the value of the void ratio e is mainly due to successive particle crushing and rearrangement. With almost the identical initial void ratio e0, the reduction of the value of the void ratio e is largest for QS1, intermediate for QS2, and lowest for QS3 at the same vertical stress levels. The reasonable interpretation of the above observation is as following [14, 36]. The crushing strength of the larger particle is lower because there are more interior flaws in it. Hence, QS1 with larger particle sizes produces more crushing and rearrangement for uniformly graded specimens. Meanwhile, the natural graded specimen QS3 is well graded comparing to the uniformly graded specimens. The particles in QS3 are protected by more surrounding contact points. Hence, the probability of the particle crushing of QS3 is lower and less void ratio produced by crushing is used for particle rearrangement.Figure 3
Relationship betweene and σV.On the other hand, as shown in Figure3, we are obliged to select the smooth B-spline to describe e versus log σV relationship with unimportant error within the maximum testing load because the decreasing tendency above is actually a continuous process. According to [14], two states on the one-dimensional compression are defined. One is the yielding state which corresponds to the state when the curvature of the relation between the void ratio e and the logarithmic of the vertical stress σV is maximum. Another is the state at the maximum compressibility, which corresponds to the state when the compression index Cc is maximum. Hence, in Figure 3, the yielding states are marked as the hollow legends and the states at the maximum compressibility are marked as the semihollow legends for all types of the specimens. The stress σVy corresponds to the stress at the yielding state, and the stress σVc corresponds to the stress at the state when the compression index Cc is maximum. With respect to these two indexes σVy and σVc, three situations can be observed from Figure 3: (i) for uniformly graded specimens, the specimen QS1 with larger particle size has higher σVy and σVc; (ii) both σVy and σVc of uniformly graded specimens QS1 and QS2 are lower than those of the natural graded specimen QS3; and (iii) for a given specimen, σVc is higher than σVy. The first situation manifests that specimens with smaller particle size are more difficult to yield and are more difficult to show the maximum compressibility for uniformly graded specimens. Analogously, the second situation indicates that uniformly graded specimens are easier to yield and are easier to show the maximum compressibility comparing with well-graded specimens. Hence, the yield state and the state at the maximum compressibility depend not only on the nature of particles but also on the packing of multitudinous particles. The last situation expresses that the state at the maximum compressibility is always after the yielding state. It also implied that the gradient in the e-log σV relationship increases after yielding and then gradually decreases to a relative steady value after showing the maximum compressibility, combining with the shapes of the B-splines in Figure 3.Figure4 shows the relationship between the vertical strain or the volumetric strain εV (in the one-dimensional compression tests of this paper, the vertical strain is identical with the volumetric strain because of the constant cross section of the specimen) and the vertical stress σV. It can be observed that the value of εV increases rapidly at relative small stress levels, then increases less rapidly after the vertical stress σV passes beyond σVc, and gradually approaches stabilization at relative high stress levels for all types of specimens. The implications of the yielding state and the state at the maximum compressibility are further investigated in the following part.Figure 4
Relationship betweenεV and σV.


## 3.2. Evolution of PSD

The grading curves of all the specimens after one-dimensional compression tests terminated at each preset vertical stress level are plotted in Figure5, with semilogarithmic coordinates on the left pictures and double-logarithmic coordinates on the right pictures. Meanwhile, it is assumed that the grading curve at the preset stress level which is closest to σVy or σVc can be approximately regarded as the grading curve at the stress equal to σVy or σVc, respectively, with unimportant error. The dashed dotted line is the referenced ultimate fractal grading curve with a fractal dimension D = 2.6 [23]. Examination of Figure 5 shows a similarity for all types of specimens that the proportion of particles finer than 74 μm increases markedly as the vertical stress σV increases; however, part of the largest particles of initial specimens still exist even at the maximum vertical stress level. In the semilogarithmic coordinates of Figure 5, the curve gradually rises up and the degree of concave upward of the curve gradually decreases, in particular, after the vertical stress σV exceeds σVy. It indicates that the degree of crushing increases markedly after yielding. Meanwhile, in the double-logarithmic coordinates of Figure 5, the curve is gradually transformed into approximately a straight line, especially after the vertical stress σV exceeds a definite extent to σVc. It indicates that the grading curve is gradually equipped with fractal characteristic, especially after the specimen shows the maximum compressibility. Therefore, it may be reasonable to regard σVc as a transition point of the grading curve to be equipped with the fractal characteristic. In a similar fashion, it should be noted that when the vertical stress σV passes σVy, the shape of the grading curve within the particle size interval finer than 74 μm becomes a straight line, which indicates that the finer part of the grading curve may be previously equipped with fractal characteristic. Hence, we can deduct that σVy can at least be regarded as another transition point to be equipped with fractal characteristic within the finer part of the grading curve. Also, the reasonability of the above deductions is verified in the next part.Figure 5
Grading curves after tests terminated at each preset stress level for (a) QS1, (b) QS2, and (c) QS3.
(a)
(b)
(c)


## 3.3. Fractal Pattern

A simplex fractal dimension is usually used to describe the PSD of soil, and this fractal parameter can be used as a constant in models of soil-water characteristic curves (SWCCs) or particle crushing [33, 42–47]. However, according to the data in this paper and other relevant literatures [18, 48], a simplex fractal dimension sometimes does not adequately describe the fractal characteristics of the whole PSD. The evolution pattern of fractals requires further study.By assuming the referenced particle size asd, the largest particle size as dM, the mass of particles finer than d as M(δ < d), and the total mass of a specimen as MT, if there is a linear relationship between log(d/dM) and log[M(δ < d)/MT] with a slope k, we can deem that the grading curve is fractal with the fractal dimension D = 3 – k. The results of linear fitting between lg[M(δ < d)/MT] and lg(d/dM) are shown in Figure 6. The left pictures of Figure 6 regard particle size intervals which are finer than 74 μm and coarser than 74 μm as fitting intervals, respectively. And the middle pictures of Figure 6 regard the whole measurable particle size interval as the only fitting interval. Actually, not all the grading curves after tests are equipped with fractal characteristics. Hence, we are obliged to assume a value of determination coefficients R2 of linear fitting as a transformation point to make a decision that whether or not the fractal of the grading curve is relative strict. By examination of the right pictures of Figure 6, which show the histograms of determination coefficients R2 of these three aforementioned linear fittings at each stress level, the value of R2 equal to 0.98 may be a plausible and relative high value to represent this transformation point. Therefore, in the left and middle pictures in Figure 6, fitting lines having relative high significance level with R2 > 0.98 are presented by the solid lines, and those having relative low significance level with R2 < 0.98 are presented by the dashed lines. In addition, the values of the fractal dimensions of the fitting lines are labeled beside with D1 for those when d is coarser than 74 μm, D2 for those when d is finer than 74 μm, and D for those of the whole measurable size interval.Figure 6
Linear fitting between lg[M(δ < d)/MT] and lg(d/dM) with the left picture regarding particle size finer than 74 μm and coarser than 74 μm as a fitting range, respectively, with the middle picture regarding the whole measurable particle size interval as the only fitting range, and with the right picture expressing the histograms of determination coefficients R2 of these three linear fittings at each preset vertical stress level for (a) QS1, (b) QS2, and (c) QS3.
(a)
(b)
(c)The detailed examination of the left and right pictures of Figure6 shows that the fractal characteristic of the finer particle size interval is not relatively strict when the vertical stress σV is lower than σVy for all types of specimens. However, the fractal characteristic of the above part is relatively strict when the vertical stress σV is greater than σVy. Nevertheless, the similar result of the coarser particle size interval appears when the vertical stress σV is greater than σVc. The narration above at least implies that the fractal characteristic previously appears on the finer particle size interval after yielding and subsequently appears on the coarser interval after the specimen shows the maximum compressibility. By means of the middle and right pictures of Figure 6, the whole measurable grading curves of all types of specimens are equipped with monofractal characteristic only when the vertical stress σV exceeds a certain degree to σVc. Hence, three particle size regions should be differentiated. In the first region ranged between 1 μm and an uncertain size which may be finer than 74 μm, the grading curve is fractal when the vertical stress σV passes through σVy. In the second region ranged between the largest particle size and another uncertain size which may be coarser than 74 μm, the grading curve is fractal when the vertical stress σV passes through σVc. And the third region ranged between these two uncertain sizes represents a transition region from the fractal characteristic in the coarser region to that in the finer region. The existence of this transition region makes the problem extremely complicated. In order to simplify the analysis, the existence of this transition region is disregarded, and the transition particle size between the two fractal regions is assumed to be 74 μm. In addition, as shown in Figures 7(a) and 7(b), despite the different initial PSDs and particle sizes between all types of specimens, the PSD data of the finer region in which the values of fractal dimensions exceed 2.29 are shown to be strictly self-similar, and those of the coarser region in which the values of fractal dimensions exceed 1.85 are shown to be strictly self-similar. Therefore, the values of 2.29 and 1.85 can be considered as the lower limits of the fractal dimensions corresponding to the finer and coarser regions, respectively. In a similar fashion, the value of 2.16 can be considered as the lower limit of the fractal dimension for the whole measurable PSD, as shown in Figure 7(c).Figure 7
Relationship between vertical stressσV and fractal dimensions (a) D1, (b) D2, and (c) D.
(a)
(b)
(c)In general, during one-dimensional compression, when the vertical stressσV is between σVy and σVc, the fractal characteristic exists only in the finer interval of the grading curve, in which particle sizes are finer than 74 μm. Subsequently, the bifractal characteristic exists in the whole measurable grading curve when the vertical stress σV is immediately larger than σVc. And 74 μm should be the transition particle size point between the different fractal characteristics of the finer and coarser intervals. Finally, the whole measurable grading curve is monofractal when the vertical stress exceeds a certain degree to σVc. The evolution of the fractal characteristic above consists of the fractal pattern during one-dimensional compression.


## 3.4. Particle Crushing

Based on the different PSDs before and after crushing, some scholars put forward series of quantitative indexes to represent the degree of particle crushing. As a whole, those indexes may be divided into two principal groups: one is based on the change of one or several characteristic particle sizes before and after crushing, such asB10, B15, and BM as depicted in Figure 8 [1, 3, 49]. The other one is based on the relative change of the whole PSD before and after crushing, which are primarily represented by Br and Be proposed by Hardin [50] and Einav [51], respectively, as depicted in Figure 9. The difference between Br and Be lies in the assumption of the ultimate state of particle crushing. Hardin [50] assumed that particle crushing eventually formed a uniform distribution system with particle sizes less than 74 μm. However, Einav [51] assumed that the ultimate distribution was a monofractal distribution with scale invariance or self-similarity. It should be noted that most properties of soil may depend on the overall PSDs, such as void ratio, volumetric strain, constitutive relation, and critical state. Actually, particle crushing is not an active process like gravity, which is always fully active and forever produces its total effect, but can be denoted by a passive quantity which can be measured by the relative distance to its ultimate state indicating the elimination of possibility for any further crushing. From this point of view, particle crushing, which resembles internal friction and cohesion of soils, has to be mobilized by external conditions until the potential of crushing approaches zero. Therefore, comparing with the indexes of the first group which depend on just one or several characteristic particle sizes, Br and Be chosen in this study may be more appropriate for representing the amount of crushing not only owing to their clear concepts and clearly defined assumptions but also due to considering the changes of distribution in almost overall particle size range.Figure 8
Definitions of the particle crushing indexesBM [1], B10 [3], and B15 [49].Figure 9
Definitions of the particle crushing indexesBr [50] and Be [51] and the division of Be.On the other hand, with the above fractal pattern of particle crushing in mind, it is advisable to separate the relative breakageBe proposed by Einav [51] into two parts considering different scales. These different scales are the two ranges of the overall measurable PSD corresponding to the aforementioned bifractal characteristic. To achieve this objective, as defined in Figure 9, the total potential breakage Bp′ is divided into two parts Bp1′ and Bp2′ to represent the corresponding potential breakage whose particle sizes are ranged coarser and finer than 74 μm, respectively. And the total breakage Bt′ is also divided into two parts Bt1′ and Bt2′ to represent the corresponding total breakage whose particle sizes are ranged coarser and finer than 74 μm, respectively. Then, defining Be1 = Bt1′/Bp1′ to represent the relative breakage of particles coarser than 74 μm and defining Be2 = Bt2′/Bp2′ to represent the relative breakage of particles finer than 74 μm. The aim of this division is to apply the divided indexes to investigate the amounts of crushing in different particle scales.Figure10 shows the relationship between vertical stress σV and Be1 or Be2. For all types of specimens, as the vertical stress σV increases, both of the values of Be1 and Be2 increase rapidly, when the vertical stress σV is smaller than σVc. Then, when the vertical stress σV is larger than σVc, both of the values of Be1 and Be2 gradually approach a relatively steady state. However, the increasing rate of Be2 is obviously faster than that of Be1. This is because the value of Be1 is closer to unity than that of Be2, such as the values of Be1 at the maximum vertical stress 204.8 MPa for QS1 and QS2 are even 0.84 and 0.78, respectively, but the values of Be2 at that state for QS1 and QS2 are merely 0.46 and 0.35, respectively. It implies that the breakage potential in the scale of particles larger than 74 μm is not much left; however, the breakage potential in the scale of particles finer than 74 μm is still a lot at that state. Hence, the probability of crushing in the finer scale is higher. This difference also implies the reasonability of the division of Be. Comparing to the natural graded specimen QS3, this increasing tendency of the value of Be1 is more significant for the uniformly graded specimens QS1 and QS2. This is because that the cushioning effect of surrounding contacts of coarser particles in the natural graded specimen QS3 is more significant; therefore, the higher degree of particle crushing of the coarser particles is induced in the uniformly graded specimens QS1 and QS2. As shown in Figure 9, with respect to Br and Be1, the value of Bt is equal to that of Bt1′, and the value of Bp is equal to the sum of Bp1′ and Area1. Hence, the difference between the values of Br and Be1 is only due to the difference between their denominators. With respect to Be and Be1, although the value of Bp′ is larger than that of Bp1′ by Bp2′, the value of Bt2′ is mostly only a very small fraction of the value of Bt′. The difference between the values of Be and Be1 is also due to the difference between their denominators with not much important error. Hence, the variation tendencies of Be and Br in Figure 11, which show the relationship between the vertical stress σV and Be or Br, are similar to those of Be1 in Figure 10.Figure 10
Relationship betweenσV and Be1 or Be2.Figure 11
Relationship betweenσV and Be or Br.Hence, we can deduce that, at an extremely large vertical stress, the coarser part of the uniformly graded specimens is faster than the finer part to reach the ultimate state of particle crushing, although this deduction can only be executed by mental operations due to the limitation of laboratory tests. In addition, the effect of the increasing vertical stressσV on the evolution of Be and Br should be identical with that on the evolution of Be1; hence, these three crushing indexes Be, Br, and Be1 are reasonable for describing the amount of crushing in which particles are coarser than 74 μm with unimportant error, but the most accurate index should be Be1.Nevertheless, as shown in Figure10, the increasing tendency of the value of Be2 in the uniformly graded specimens QS1 and QS2 is less rapid than that in the natural graded specimen QS3. This may be resulted from that the average number of contact points of finer particles in QS3 is less than those in QS1 and QS2, and the contact cushioning effect exceeds the crushing strength effect on particle crushing. Hence, the probabilities of crushing of the finer “parent” particles (i.e., the finer particles which have not been crushed) and the finer and finer particles produced by previous rounds of particle crushing in QS3 are higher than the probabilities of crushing of those in QS1 and QS2. In addition, the pronounced different increasing tendencies between Be2 and Be1 represent that the reasonable index of particle crushing for describing the corresponding relative breakage, in which particles are finer than 74 μm, should be Be2 which is essentially different from Be, Be1, and Br.Figure12 shows the relationship between the vertical stress σV and the ratio εV/Be1 or εV/Be2. Meanwhile, Figure 13 shows the relationship between the vertical stress σV and the ratio εV/Be or εV/Br. From ring shear tests [15] and triaxial tests [16], the phenomenon that the value of εV/Br has been constant is observed almost at the critical state of soils. However, from Figures 12 and 13, it can be observed that both the values of εV/Be1 and εV/Be are already constant, respectively, after the vertical stress σV exceeds σVc. The values of the constants of εV/Be1 and εV/Be are 0.391 and 0.461, respectively. From the previous part of this paper, when the vertical stress σV is larger than σVc, it marks that the coarser part of PSD, in which particles are larger than 74 μm, is fractal. Hence, it indicates that the coarser part of PSD is equipped with the fractal characteristic can be regarded as a token of that the values of εV/Be1 and εV/Be are constant. On the other hand, the values of εV/Be2 and εV/Br also have a tendency to be constant as the vertical stress σV increases; however, the prominence of this tendency is far inferior to that for the values of εV/Be1 and εV/Be. With respect to εV/Be1 and εV/Be, the constants for both are determined by the average of the ordinates of data points, which are all at a state after the maximum compressibility, as shown in Figures 12 and 13. The relevant statistical parameters are summarized in Table 2. It can be found that the square deviation of the values of εV/Be1 is very small and equal to 0.00287 which is even slightly lower than that of εV/Be. Hence, comparing to εV/Be, the more appropriate physical quantity to represent the physical phenomenon that the ratio of the volumetric strain to the relative breakage is a constant should be εV/Be1. Hence, during one-dimensional compression, once the coarser part of PSD has been reached to fractal or the vertical stress σV has been larger than σVc, the prediction of the relative breakage can be accomplished by means of the measurable volumetric strain εV and the constant of the value of εV/Be1 obtained from tests.Figure 12
Relationship betweenσV and εV/Be1 or εV/Be2.Figure 13
Relationship betweenσV and εV/Be or εV/Br.Table 2
Statistical parameters of the values ofεV/Be1 and εV/Be for all types of specimens when σV ≥ σVc.
Vertical stress,σV (MPa)
ε
V/Be1
ε
V/Be
QS1 (σVc ≈ 9.9 MPa)
QS2 (σVc ≈ 20.3 MPa)
QS3 (σVc ≈ 54.9 MPa)
QS1 (σVc ≈ 9.9 MPa)
QS2 (σVc ≈ 20.3 MPa)
QS3 (σVc ≈ 54.9 MPa)
12.8
0.50349
—
—
0.6019
—
—
25.6
0.41932
0.30881
—
0.49532
0.40419
—
51.2
0.42075
0.33716
0.38471
0.4859
0.42325
0.46193
102.4
0.42174
0.35402
0.3502
0.47581
0.42814
0.41262
204.8
0.4594
0.37241
0.35382
0.5031
0.43703
0.39979
Mean value
—
0.39049
—
—
0.46075
—
Square deviation
—
0.00287
—
—
0.00300
—


## 4. Conclusion

To clarify the influence of particle crushing on the behavior of granular geomaterials, a series of one-dimensional compression tests were conducted on quartz sand-gravel up to 204.8 MPa. The detailed investigations were mainly on the compression behavior, the evolution of PSD, the fractal pattern of grading curve, and the degree of particle crushing. The major findings can be drawn as follows.As the vertical stress increases, both the yielding state and the state at the maximum compressibility of uniformly graded specimens are easier to be reached than those of the well-graded specimens. For uniformly graded specimens, the smaller the particle sizes are, the more difficult these two states are to be reached. For a given specimen, the yielding state is always previous to the state at the maximum compressibility. In addition, these two states are closely associated with the evolution of PSD and the fractal pattern of particle crushing.During one-dimensional compression, the evolution of PSD induced by particle crushing is described as follows: for each type of specimens, the finer part, in which particles are finer than 74μm, of the grading curve tends to be approximately a straight line after the yielding state in the double-logarithmic coordinates. However, the coarser part, in which particles are coarser than 74 μm, of the grading curve tends to be approximately another straight line after the state at the maximum compressibility. The whole grading curve tends to be a straight line when the vertical stress exceeds a definite extent to the stress at the maximum compressibility state.On one-dimensional compression, the fractal pattern of particle crushing is described as follows: the fractal first appears on the finer part of a grading curve after the yielding state. Then the bifractal appears on the whole grading curve with 74μm as the transition particle size, which is utilized to distinguish the different fractal characteristics between the finer and coarser parts, after the state at the maximum compressibility. Finally, the entire grading curve evolves into monofractal when the vertical stress exceeds a certain degree to the stress at the maximum compressibility. In addition, the values of 2.29, 1.85, and 2.16 are considered as the lower limits of the fractal dimensions for the finer, the coarser, and the whole parts of a grading curve, respectively.With the fractal pattern of particle crushing in mind, a pair of crushing indexesBe1 and Be2 by means of the division of the relative breakage Be are proposed for representing the relative breakages of particles coarser than 74 μm and finer than 74 μm, respectively. The division of the relative breakage Be is reasonable due to the relatively large difference between the values of Be1 and Be2 during crushing.Either that the vertical stress is larger than the stress at the maximum compressibility or that the coarser part of PSD is evolved into fractal can be regarded as a token of the physical phenomenon that the ratio between the volumetric strain and the relative breakage is a constant. Hence, the prediction of the relative breakage can be accomplished by means of the volumetric strain and the constant which are available from one-dimensional compression tests.



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*Source: 2153971-2018-09-25.xml*