# An Image-Based Gradation Calculation Method considering Crushed Stone Morphology

Authors: Chong Jiang; Xin Ding; Li Pang; Liang Deng; Zexiong Shi
Journal: Geofluids
 (2022)

---

## Abstract

An image-based gradation calculation method considering crushed stone morphology is proposed in this paper. Eight kinds of crushed stones are prepared to figure out the effect of the particle shape on the gradation. All particles are pictured, and the photos are processed to obtain particle shape indices by using the image-based method. A minimum Feret diameter analysis method is used, which can reflect the influence of the particle shape. Then, the paper proposes a developed overall regularity to describe the morphology of particles, which is calculated by the weight of three shape indices: aspect ratio, convexity, and sphericity. The developed overall regularity is applied on gradation curve equations to provide an image-based gradation calculation method. And the provided method is validated to well fit the actual case. Furthermore, the method is also applied on two examples of gradation designation and the result proves to be feasible and practicable.


---

## Body

## 1. Introduction

Crushed stone is small pieces of the broken rock due to natural reasons or man-made destruction, which has irregular size, shape, and texture. Materials with different particle size distributions are often used to form coarse aggregate [1, 2], backfill material [3–6], and rockfill [7–9] in high fill, subgrade [10–12], and mining engineering. There are various parameters describing soil materials, among which the particle size and gradation are commonly used. Gradation is known to be defined as the distribution of aggregate particles of different sizes, which can be determined by a sieve analysis test. However, the pore of the sieves is generally composed by a lattice square according to the method of the British Standards Institute [13], which sometimes makes particles of quite different characteristics have the same size [14]. This brings about difficulties to design the gradation for different kinds of materials in engineering.Particle geometry is a result of the genesis of the parent rock, stages of transportation and depositional history, and chemical and biological effects [15]. For particulate materials including crushed stones, the characteristics of the particle shape govern their complex mechanical behavior at multiscales, about which the research interest has been growing in recent years [16, 17]. Since Wadell [18] first proposed the shape parameter: sphericity, the parameter evaluating the particle shape has been developed and the number of the parameter has been added. Traditional methods are to make use of charts such as the charts of Krumbein and Sloss [19] and Cho et al. [20]. The roundness and sphericity of a given particle are assessed by visual comparison with the reference shapes in these charts. Lee et al. [17] not only redefined sphericity but also selected elongation, slenderness, and convexity to describe the irregularity of sand grains. Berrezueta et al. [21] evaluated the representativity of four independent 2D particle parameters which included fractal dimension, Wadell’s roundness, a perimeter-area normalized ratio, and sphericity. Liu et al. [22] proposed an empirical small-strain shear modulus model including a Fourier-based particle shape factor modified by gradation through a series of shear wave velocity tests.Recently, with the application of the computer image processing technology, the accuracy of the particle shape measurements has been improved. An image-based method is known to be a relatively convenient and accurate method, which can obtain particle shape data by using image processing software [22–34]. Basically, particles are usually pictured finely or roughly and the photos are processed in specific software such as ImageJ [22, 31, 33] and OnDemand3D [34] to obtain the particle shape indices. Some researchers used professional instruments to measure the particle shape of sand and review the particle shape effect on mechanical behavior such as QICPIC [23, 24, 29], Sedimaging [26], and CX2 [32] systems. Moreover, some computational algorithm was proposed to be integrated into existing optical soil characterization systems [27, 28]. The software or apparatus used by these researchers is mainly focused on soil or sand materials. Different from these small particles, the shape measurement of crushed stone is easier and the requirement of apparatus is lower.The gradation of materials is usually described by nonuniformity coefficient (Cu) and curvature coefficient (Cc), which is determined by effective particle size (D10), mean particle size (D50), and controlling particle size (D60). However, the method cannot fully express the characteristics of the gradation. A gradation curve equation method is proposed by researchers to overcome the shortcomings. Since Fuller and Thompson [35] and Talbot and Richart [36] proposed the equations of maximum density gradation curves and fractal gradation curves, respectively, the studies on gradation curve equations have been developed. Zhu et al. [37] and Guo et al. [38] proposed a gradation equation mainly focused on soil and coarse-grained soil. Then, Wu et al. [39] improved the method to reduce the required parameters. Zhu et al. [40] provided a gradation design method for rockfill materials based on fractal theory. Sun et al. [41] studied the particle generation procedure in DEM and proposed a gradation algorithm based on the particle number. The gradation curve equations are used to design the required gradation and have been gradually applied on the engineering.To figure out the effect of particle shape when designing the gradation of aggregate, in this paper, eight kinds of crushed stones are prepared. The stones are divided into two groups which are the small particle group and the large particle group. The gradations for each group are processed to be the same by sieving. Meanwhile, the image processing software is used to obtain shape indices, including the minimum Feret diameter, aspect ratio, convexity, and sphericity. The minimum Feret diameter of particles is measured to form gradations which can reflect the particle shape effect. To quantitatively analyze this particle shape effect, a developed overall regularity is proposed, which is calculated by the weight of the aspect ratio, convexity, and sphericity. The developed overall regularity is applied on the gradation curve equations to provide an image-based gradation calculation method. The newly proposed gradation calculation method is validated to be well fitted, and the method is applied to two examples of gradation designation.


## 2. Test Materials and Particle Characteristics

Eight kinds of materials of different particle shapes are involved in the experiment as Figure1. Table 1 shows the summary of the geological classification and characteristics and particle shape classification of the materials.Figure 1
Eight kinds of materials were used in the experiment.Table 1
Summary of the geological classification and characteristics and particle shape classification.
StoneGeological classification and characteristicsParticle shapeWhite pebblesNearly pure silica stone with smoother surfaceSubroundedBlack stoneCarbonate stoneAngularBasaltPorous stone about 40% void ratio, 45% silica, and 30% Fe2O3, CaOAngularPerliteFragile vitreous stone composed of 70% silica and 12% aluminaSubroundedMedical stoneAluminosilicate stone with dense compositionSubangularGuangxi stoneCarbonate stone with quartz grainAngularSepenggianteHard marble stone with the grain of a treeSubroundedLimestoneCarbonate stone with roughness surface, CaCO3 over 90%Subangular


### 2.1. Particle Size and Gradation

Because the size of the crushed stone prepared in the experiment is around20mm, the circular sieve aperture diameter is 10mm to 30mm with 1mm interval. To obtain representative and repeatable data for each material, about 5kg for each kind of crushed stone with hundreds of particles is used to establish cumulative distribution curves. The particle size distribution curves of the eight crushed stones are controlled by sieving to be exactly the same, giving the mean particle diameter D50sieve=15.5mm and uniformity coefficient Cusieve=1.34 for smaller particles as the black line in Figure 2(a); D50sieve=20.5mm and uniformity coefficient Cusieve=1.15 for larger particles as the black line in Figure 2(b).Figure 2
Particle size cumulative distribution for sieve analysis andDFmin analysis.
(a)
Small particles(b)
Large particlesBesides the conventional sieving results, an imaging-based method is also used to obtain particle size and grading. Several indices measured by this method can describe the size of the particle, along which the minimum Feret diameter (DFmin) is used in this paper, and DFmin means the smallest distance between two tangents on opposite sides of the particle, as shown in Figure 3(a). The gradation curve provided by DFmin as the size index has been found to be close to but different from that determined by sieve analysis [23], and the influence of the particle shape can be reflected with the DFmin analysis [29]. Figure 2 shows the similar results. Additionally, Table 2 shows more details about the size distribution curves.Figure 3
Definition of shape and size indices [29].Table 2
The shape data of the eight materials for the minimum Feret (DFmin) analysis.
MaterialD10Fmin (mm)D50Fmin (mm)D60Fmin (mm)CuFminWhite pebbles11.514.915.61.35Black stone12.216.016.81.37Basalt12.916.817.71.37Perlite11.415.015.61.38Medical stone12.115.816.31.35Guangxi stone18.121.622.41.24Sepenggiante18.321.422.11.21Limestone18.722.122.81.22Note:D10 is the particle size of the 10% size distribution curve; D50 is the mean particle size; D60 is the particle size of the 60% size distribution curve; Cu is the uniformity coefficient defined by D60/D10.


### 2.2. Particle Shape Measurements

All particles in the experiment are pictured by a camera. Because the particle sizes are around 20 mm, a camera with a normal configuration (80 megapixels) is adequate to capture the particle shape characteristics [21]. Then, an image processing software Image-Pro Plus 7.0 [42] is used to convert the real image to a binary image to obtain shape indices. The experiment procedure is shown as Figure 4.Figure 4
The experiment procedure to obtain the data of shape indices.Three shape indices are used in this paper and have been widely used in recent years [23, 24, 29, 30], namely, the aspect ratio (AR), the convexity (C), and the sphericity (S), as shown in Figure 3: AR is defined as the ratio between DFmin and DFmax; C is defined by the ratio between the area of the particle (A) and the area of the convex hull of the particle (A+B); S is defined as the ratio between the perimeter of a circle with the same area as the projected area of the particle and its real perimeter.Figure5 presents the cumulative distribution of shape measurements of eight crushed stones. Different from the results of the glass beads by Yang and Luo [29], the range of the convexity (C) is normally from 0.9 to 1 for crushed stone, but the range of the aspect ratio (AR) and the sphericity (S) are similar. For a given material, it is reasonable to choose the shape value corresponding to the 50% cumulative distribution as a representative value [29]. In this method, shape data of the eight crushed stones are provided in Table 3.Figure 5
Cumulative distribution for (a) aspect ratio, (b) convexity, and (c) sphericity.
(a)(b)(c)Table 3
The 50% cumulative distribution data of the eight materials.
MaterialAspect ratio (AR50)Convexity (C50)Sphericity (S50)OR of Yang and LuoOR of the present studyWhite pebbles0.7080.9770.8420.8420.648Black stone0.6830.9710.7920.8150.562Basalt0.6860.9490.6810.7720.367Perlite0.7010.9880.8690.8530.720Medical stone0.7000.9580.8130.8240.523Guangxi stone0.6590.9670.7400.7890.486Sepenggiante0.6810.9680.7880.8120.543Limestone0.6960.9760.7810.8180.588Yang and Luo [29] provided a calculation method to describe the morphology of particles, referred to as overall regularity (OR), and it can be calculated as the average of the value of three shape indices AR50+S50+C50/3. This means the weights of the three-particle shape indices are equal. However, as for crushed stone in this paper, the value of OR will reflect more characteristics of S and C in this method, because there are narrower range and larger value for S and C than AR. To better reflect the weight of each index, the calculation of the overall regularity (OR) is modified as the following equation:
(1)OR=AR50−ARmin/ARmax−ARmin+C50−Cmin/Cmax−Cmin+S50−Smin/Smax−Smin3,where the subscript “max” and “min” means the upper and lower bound of the 95% confidence interval, respectively, as shown in Figure 6. The figure shows the counting number distribution for the three shape indices. It is obvious that the contribution of each particle shape index is concentrated in a certain range. Equation (1) makes the weights of the three indices the same through the normalization method. Thus, OR with Equation (1) can describe the particle shape in a collective manner.Figure 6
Counting number distribution for (a) aspect ratio, (b) convexity, and (c) sphericity.
(a)(b)(c)As shown in Figure7, the difference is obvious for the rather different kinds of particles, comparing with the method of Yang and Luo [29]. The calculation results of Yang and Luo [29] for the used particles all concentrate at 0.8. By using the present method, the range of OR are from 0.3 to 0.8. For example, OR=0.367 for basalt which is irregular and porous, and OR=0.720 for perlite which is subrounded and smooth. This proves that for different kinds of particles, the proposed developed method can distinguish the morphology more obviously.Figure 7
The results of the method of Yang and Luo [29] and the present study.


## 2.1. Particle Size and Gradation

Because the size of the crushed stone prepared in the experiment is around20mm, the circular sieve aperture diameter is 10mm to 30mm with 1mm interval. To obtain representative and repeatable data for each material, about 5kg for each kind of crushed stone with hundreds of particles is used to establish cumulative distribution curves. The particle size distribution curves of the eight crushed stones are controlled by sieving to be exactly the same, giving the mean particle diameter D50sieve=15.5mm and uniformity coefficient Cusieve=1.34 for smaller particles as the black line in Figure 2(a); D50sieve=20.5mm and uniformity coefficient Cusieve=1.15 for larger particles as the black line in Figure 2(b).Figure 2
Particle size cumulative distribution for sieve analysis andDFmin analysis.
(a)
Small particles(b)
Large particlesBesides the conventional sieving results, an imaging-based method is also used to obtain particle size and grading. Several indices measured by this method can describe the size of the particle, along which the minimum Feret diameter (DFmin) is used in this paper, and DFmin means the smallest distance between two tangents on opposite sides of the particle, as shown in Figure 3(a). The gradation curve provided by DFmin as the size index has been found to be close to but different from that determined by sieve analysis [23], and the influence of the particle shape can be reflected with the DFmin analysis [29]. Figure 2 shows the similar results. Additionally, Table 2 shows more details about the size distribution curves.Figure 3
Definition of shape and size indices [29].Table 2
The shape data of the eight materials for the minimum Feret (DFmin) analysis.
MaterialD10Fmin (mm)D50Fmin (mm)D60Fmin (mm)CuFminWhite pebbles11.514.915.61.35Black stone12.216.016.81.37Basalt12.916.817.71.37Perlite11.415.015.61.38Medical stone12.115.816.31.35Guangxi stone18.121.622.41.24Sepenggiante18.321.422.11.21Limestone18.722.122.81.22Note:D10 is the particle size of the 10% size distribution curve; D50 is the mean particle size; D60 is the particle size of the 60% size distribution curve; Cu is the uniformity coefficient defined by D60/D10.


## 2.2. Particle Shape Measurements

All particles in the experiment are pictured by a camera. Because the particle sizes are around 20 mm, a camera with a normal configuration (80 megapixels) is adequate to capture the particle shape characteristics [21]. Then, an image processing software Image-Pro Plus 7.0 [42] is used to convert the real image to a binary image to obtain shape indices. The experiment procedure is shown as Figure 4.Figure 4
The experiment procedure to obtain the data of shape indices.Three shape indices are used in this paper and have been widely used in recent years [23, 24, 29, 30], namely, the aspect ratio (AR), the convexity (C), and the sphericity (S), as shown in Figure 3: AR is defined as the ratio between DFmin and DFmax; C is defined by the ratio between the area of the particle (A) and the area of the convex hull of the particle (A+B); S is defined as the ratio between the perimeter of a circle with the same area as the projected area of the particle and its real perimeter.Figure5 presents the cumulative distribution of shape measurements of eight crushed stones. Different from the results of the glass beads by Yang and Luo [29], the range of the convexity (C) is normally from 0.9 to 1 for crushed stone, but the range of the aspect ratio (AR) and the sphericity (S) are similar. For a given material, it is reasonable to choose the shape value corresponding to the 50% cumulative distribution as a representative value [29]. In this method, shape data of the eight crushed stones are provided in Table 3.Figure 5
Cumulative distribution for (a) aspect ratio, (b) convexity, and (c) sphericity.
(a)(b)(c)Table 3
The 50% cumulative distribution data of the eight materials.
MaterialAspect ratio (AR50)Convexity (C50)Sphericity (S50)OR of Yang and LuoOR of the present studyWhite pebbles0.7080.9770.8420.8420.648Black stone0.6830.9710.7920.8150.562Basalt0.6860.9490.6810.7720.367Perlite0.7010.9880.8690.8530.720Medical stone0.7000.9580.8130.8240.523Guangxi stone0.6590.9670.7400.7890.486Sepenggiante0.6810.9680.7880.8120.543Limestone0.6960.9760.7810.8180.588Yang and Luo [29] provided a calculation method to describe the morphology of particles, referred to as overall regularity (OR), and it can be calculated as the average of the value of three shape indices AR50+S50+C50/3. This means the weights of the three-particle shape indices are equal. However, as for crushed stone in this paper, the value of OR will reflect more characteristics of S and C in this method, because there are narrower range and larger value for S and C than AR. To better reflect the weight of each index, the calculation of the overall regularity (OR) is modified as the following equation:
(1)OR=AR50−ARmin/ARmax−ARmin+C50−Cmin/Cmax−Cmin+S50−Smin/Smax−Smin3,where the subscript “max” and “min” means the upper and lower bound of the 95% confidence interval, respectively, as shown in Figure 6. The figure shows the counting number distribution for the three shape indices. It is obvious that the contribution of each particle shape index is concentrated in a certain range. Equation (1) makes the weights of the three indices the same through the normalization method. Thus, OR with Equation (1) can describe the particle shape in a collective manner.Figure 6
Counting number distribution for (a) aspect ratio, (b) convexity, and (c) sphericity.
(a)(b)(c)As shown in Figure7, the difference is obvious for the rather different kinds of particles, comparing with the method of Yang and Luo [29]. The calculation results of Yang and Luo [29] for the used particles all concentrate at 0.8. By using the present method, the range of OR are from 0.3 to 0.8. For example, OR=0.367 for basalt which is irregular and porous, and OR=0.720 for perlite which is subrounded and smooth. This proves that for different kinds of particles, the proposed developed method can distinguish the morphology more obviously.Figure 7
The results of the method of Yang and Luo [29] and the present study.


## 3. Gradation Equations

Traditional calculation of particle gradation needs to draw an initial gradation curve first and interpolate to get the value of effective particle size (D10) and controlling particle size (D60). Then, calculate the uniformity coefficient (Cu) and curvature coefficient (Cc) to check whether the result meets the engineering requirements and adjust the initial curve continuously until it finally meets. Comparing the complicated process shown above, the calculation process can be simplified by using the gradation equation method.The continuous gradation equation proposed by Zhu et al. [37] and Guo et al. [43] is used in this paper:
(2)P=11−bDmax/Da+b×100%,where D is any particle size within the gradation range; P is the percentage of particles smaller than D; Dmax is the maximum particle size; a and b are dimensionless parameters that need to be determined. For minimum Feret diameter (DFmin) analysis, Equation (2) can be expressed as follows:
(3)P=11−bFminDmaxFmin/DFminaFmin+bFmin×100%,where the superscript “Fmin” means the parameter is obtained by minimum Feret diameter analysis, and the superscript “sieve” used in the following part means the parameter is obtained by sieve analysis.Zhu et al. [37] indicated that the gradation curve shape was like “S” for a higher value of b. Because the shapes of the gradation curves in Figure 2 are all like “S,” b=bFmin=0.95 is considered to be reasonable. Fitting DFmin curves with Equation (3), the values of aFmin for each kind of particles are as shown in Table 4. And fitting sieve curves with Equation (2), the values of asieve for sieve analysis of small and large particles are calculated to be 10.37 and 15.28, respectively. The ratios between aFmin and asieve are also shown in Table 4. Figure 8 shows the relationship between overall regularity (OR) and parameter aFmin/asieve. A linear equation is considered to be reasonable and is provided as follows:
(4)aFminasieve=0.4OR+0.5.Table 4
The values ofaFmin and DFmin for each kind of gradation curves.
Gradation curvesaFminaFmin/asieveDmaxFmin (mm)DmaxFmin/DmaxsieveWhite pebbles7.7350.74622.2821.114Black stone7.5890.73224.0151.201Basalt6.9120.66726.1691.308Perlite8.4380.81421.3301.066Medical stone7.2680.70124.0711.203Guangxi stone10.520.68829.9701.199Sepenggiante10.940.71629.5411.182Limestone11.240.73629.0711.163Figure 8
Curve fitting for overall regularity (OR) versusaFmin/asieve.It is obvious that a certain range of OR can be found for particles with different particle shapes in Figure8. In engineering, particle shape indices are needed to calculate OR, sometimes making the work relatively complex and inconvenient. Thus, the paper provides an approach to obtain the value of OR as shown in Figure 8: nearly spherical particles are classified as “rounded” and OR=0.8~0.9; elliptical or well-shaped particles are classified as “subrounded” and OR=0.6~0.8; particles with some edges and corners are classified as “subangular” and OR=0.4~0.6; crushed and irregular particles are classified as “angular” and OR=0.3~0.4.The value ofDmaxFmin is also related to OR, as shown in Figure 9. The figure shows that more angular particles have larger DmaxFmin comparing to rounded particles. It is noted that the DFmin size of most rounded particles is exactly the same as the sieve pore size, referred to DmaxFmin=Dmaxsieve. Then, an asymptotic relationship is more suitable between DmaxFmin/Dmaxsieve and OR. The equation can be expressed as follows:
(5)DmaxFminDmaxsieve=e−3OR+1.Figure 9
Curve fitting for overall regularity (OR) versusDmaxFmin/Dmaxsieve.Then combining Equation (3) with Equations (4) and (5), the following equation can be provided to calculate the gradation considering the effect of particle morphology:
(6)P=11−bFmine−3OR+1Dmaxsieve/DFmin0.4OR+0.5asieve+bFmin×100%.In the engineering, some parameters are required to design or calculate the gradation: uniformity coefficientCu, curvature coefficient Cc, effective particle size D10, mean particle size D50, controlling particle size D60, or the maximum particle size Dmax. These parameters are enough to determine a certain gradation curve or a series of gradation curves. Then, with Equation (2), asieve and bsieve can be obtained (the process can be referred to in Section 5). The overall regularity (OR) can be assumed or be calculated more precisely with Equation (1). Because Dmaxsieve is usually provided first, the gradation curve of DFmin analysis can be drawn in DFmin−P plane with Equation (6). The above steps can be accomplished in paper, then using methods to measure DFmin for these particles to form the drawn DFmin gradation curve. At that time, these particles are composed of the required gradation considering the effect of particle shape. This process is provided as in Figure 10.Figure 10
Process of obtaining gradations considering the effect of the particle shape.The key part of the proposed method is to obtainDFmin of the particles. However, different from the sieve analysis, measuring the minimum Feret diameter DFmin is relatively more difficult. If there are few particles, calipers can be used to simply measure DFmin. At present, there is no convenient way to obtain DFmin for a lot of particles. It is time-consuming to measure DFmin one by one by using calipers. The image-based method is a relatively practicable approach. Some apparatus are used to measure particle shape parameters in laboratory tests [16, 23, 44, 45]. Moreover, after the particles are pictured by a clear enough camera, the image processing software and numerical calculation software can also be used to obtain the shape parameters. Further studies and more apparatus need to be discussed and developed in the future.


## 4. The Validation of the Method

The image-based apparatus, known as QICPIC, have been widely used in recent years to characterize soil particle shape [16, 23, 44, 45]. The detailed description of how the apparatus works can be found in Sympatec [45]. Three kinds of particles with different shapes are used to figure out the influence of the particle shape on gradation by Yang and Luo [29]: glass beads (well-rounded), Fujian sand (subangular), and crushed glass beads (angular). As shown in Figure 11, the particles are sieved as the same gradation with Cu=1.20 and D50=512.5μm, together with gradations by DFmin analysis with apparatus QICPIC. Using the method of Equation (2), for b=0.95, the gradation curve of sieve analysis is fitted and the result proves to be asieve=9.450. The ranges of the aspect ratio (AR), the convexity (C), and the sphericity (S) are 0.6~0.1, 0.9~0.1, and 0.6~0.1, respectively. The 50% cumulative distribution of the shape indices provided by Yang and Luo [29] and the values of the calculated OR with Equation (1) are shown in Table 5. The maximum sieve diameter (Dmaxsieve) is 700 μm; then, the DFmin gradation curves can be drawn with Equation (6) in Figure 11, which are well fitted with the results provided by the test of Yang and Luo [29].Figure 11
Validation for three different particles.Table 5
Shape indices and calculated OR.
Material kindsAspect ratio (AR50)Convexity (C50)Sphericity (S50)Overall regularity (OR)Glass beads0.9740.9740.9440.845Fujian sand0.7450.9560.8910.550Crushed glass beads0.6530.9290.8220.324It is noted that Equations (2) and (3) focus on the continuous gradation with “S” shape when b approaches 1. For a smaller value of b, the gradation curve shape proves to be hyperbolic, and Equations (2) and (3) are still applicable. However, further research is needed when the curve shape is not continuous, and some researchers have studied the related contents [46, 47].


## 5. Application of the Method

Usually, there are two cases to calculate the particle gradations according to the particle parameters: ① several particle parameters (e.g.,D50, Cu, and Cc) have been determined and a specific gradation needs to be designed; ② a particle parameter is fixed and other parameters are changed in a certain interval, and a series of gradations need to be designed. The nonuniformity coefficient (Cu) and the curvature coefficient (Cc) are defined by the following equation:
(7)Cu=D60D10,(8)Cu=D302D10D60,where D10, D30, and D60 are the particle sizes of 10%, 30%, and 60% by weight, respectively. The calculation method by using the new image-based gradation calculation method is shown in detail in the following subsections.


### 5.1. Calculation for a Specific Gradation Curve

An example of three different kinds of particle shapes of sands is used to design gradation curves by Ren et al. [48]. Calcareous sand, standard sand, and glass beads are used and can be classified as “angular,” “subangular,” and “rounded.” The parameters Cu=8 and 10 and Cc=1.5 are needed and Dmaxsieve=8mm. Based on Equation (2), for Cu=8 and Cc=1.5, the following equations can be provided as
(9)Cu=D60D10=61−0.1b1−0.6b1/asieve=8,Cc=D302D10D60=31−0.1b1−0.6b21−0.3b21/asieve=1.5.The result of Equation (9) is asieve=0.241 and b=0.925. Then, using the same method, asieve=0.237 and b=0.870 for the case of Cu=10 and Cc=1.5. Taking the median value of the gradation range of “angular,” “subangular,” and “rounded,” OR is assumed to be 0.35, 0.5, and 0.85, respectively. By the method of Equation (6), the calculation results are shown in Figure 12, then using methods to measure DFmin for these particles to form the drawn DFmin gradation curve. At that time, these particles are composed of the required gradation considering the effect of particle shape. This process is the same as Figure 10.Figure 12
Calculation of a case of the specific gradation.
(a)(b)


### 5.2. Calculation for a Series of Gradation Curves

An example of coarse-grained soil to study the permeability by Peng [49]. The Dmaxsieve=20mm for the aggregate of the coarse-grained. The percentage of particles smaller than 10mm needs to be 55%~75%. Then, the nonuniformity coefficient Cu=2.2 and a series of changing Cc is required. Based on Equation (2), the following equations can be provided as
(10)55%<11−bDmaxsieve/10asieve+b<75%,Cu=D60D10=61−0.1b1−0.6b1/asieve=2.2.The sieve analysis result of Equation (10) is plotted on the plane of b−asieve, as shown in Figure 13(a). Because the used particles can be classified as “subregular,” OR=0.5 is assumed in this paper. Then calculating DmaxFmin to be 24.46mm with Equation (5) and replacing asieve with aFmin/0.4OR+0.5, the DFmin analysis result of Equation (10) is plotted on the plane of b−aFmin, as shown in Figure 13(b). Four points are chosen for each analysis of b=0.9,0.92,0.94,and0.96 as the red circle in the figure; then, the required Cc is calculated in Equation (6) to be 1.054, 1.049, 1.044, and 1.040 and 0.738, 0.734, 0.731, and 0.728 for sieve analysis and DFmin analysis, respectively.Figure 13
Calculation of a case of a series of gradations.
(a)
Sieve analysis(b)
Present study


## 5.1. Calculation for a Specific Gradation Curve

An example of three different kinds of particle shapes of sands is used to design gradation curves by Ren et al. [48]. Calcareous sand, standard sand, and glass beads are used and can be classified as “angular,” “subangular,” and “rounded.” The parameters Cu=8 and 10 and Cc=1.5 are needed and Dmaxsieve=8mm. Based on Equation (2), for Cu=8 and Cc=1.5, the following equations can be provided as
(9)Cu=D60D10=61−0.1b1−0.6b1/asieve=8,Cc=D302D10D60=31−0.1b1−0.6b21−0.3b21/asieve=1.5.The result of Equation (9) is asieve=0.241 and b=0.925. Then, using the same method, asieve=0.237 and b=0.870 for the case of Cu=10 and Cc=1.5. Taking the median value of the gradation range of “angular,” “subangular,” and “rounded,” OR is assumed to be 0.35, 0.5, and 0.85, respectively. By the method of Equation (6), the calculation results are shown in Figure 12, then using methods to measure DFmin for these particles to form the drawn DFmin gradation curve. At that time, these particles are composed of the required gradation considering the effect of particle shape. This process is the same as Figure 10.Figure 12
Calculation of a case of the specific gradation.
(a)(b)


## 5.2. Calculation for a Series of Gradation Curves

An example of coarse-grained soil to study the permeability by Peng [49]. The Dmaxsieve=20mm for the aggregate of the coarse-grained. The percentage of particles smaller than 10mm needs to be 55%~75%. Then, the nonuniformity coefficient Cu=2.2 and a series of changing Cc is required. Based on Equation (2), the following equations can be provided as
(10)55%<11−bDmaxsieve/10asieve+b<75%,Cu=D60D10=61−0.1b1−0.6b1/asieve=2.2.The sieve analysis result of Equation (10) is plotted on the plane of b−asieve, as shown in Figure 13(a). Because the used particles can be classified as “subregular,” OR=0.5 is assumed in this paper. Then calculating DmaxFmin to be 24.46mm with Equation (5) and replacing asieve with aFmin/0.4OR+0.5, the DFmin analysis result of Equation (10) is plotted on the plane of b−aFmin, as shown in Figure 13(b). Four points are chosen for each analysis of b=0.9,0.92,0.94,and0.96 as the red circle in the figure; then, the required Cc is calculated in Equation (6) to be 1.054, 1.049, 1.044, and 1.040 and 0.738, 0.734, 0.731, and 0.728 for sieve analysis and DFmin analysis, respectively.Figure 13
Calculation of a case of a series of gradations.
(a)
Sieve analysis(b)
Present study


## 6. Conclusion

This paper proposed an image-based gradation calculation method considering crushed stone morphology. The pictures of the eight kinds of crushed stone particles are processed to obtain the particle shape indices. The gradation determined by the minimum Feret diameter analysis proves to be different from what is determined by sieve analysis. The modified overall regularity is used to describe the particle shape, which is calculated by three shape indices: aspect ratio, convexity, and sphericity. The developed overall regularity is applied on the gradation curve equations to provide an image-based gradation calculation method. The proposed gradation calculation method can fit well with real cases. Additionally, some other conclusions are also obtained as follows:(1)
Comparing with the sieve gradation, the minimum Feret diameter gradation can reflect the effect of the particle morphology. The difference between the two gradation curves is relatively small for rounded particles. Conversely, it is large for angular particles(2)
There is a relatively fixed range for each shape indices of aspect ratio, convexity, and sphericity, but not distributed in the range of 0~1. This means to reflect the real weight of these shape indices, the range needs to be processedFor continuous gradation equation, the relationship between OR and parametera is linear, and a increases with increasing OR; the relationship between OR and parameter DmaxFmin is progressive, and DmaxFmin approaches 1 with increasing OR.Some problems should also be further studied. The image-based method in this paper is focused on 2D, which is a relatively effective method. The 3D methods have been developed in these years [50, 51], which is a potential method in the future. Moreover, to apply the shape-effect gradation calculation method to the engineering, some related apparatus and equipment need to be manufactured.



---

*Source: 7291223-2022-07-20.xml*