# Modeling the Simultaneous Effects of Particle Size and Porosity in Simulating Geo-Materials

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

_{0}is the original porosity, E is the elastic modulus, υ is the Poisson’s ratio, e

_{0}is the original void ratio, Δ is the pore change/reduction, e

_{0}is the original void ratio, Θ = σ

_{x}+ σ

_{y}+ σ

_{z}, σ

_{x}, σ

_{y}, and σ

_{z}are the stress in the x, y, and z direction, θ

_{t}is the volumetric strain, and n

_{new}is the new porosity. The void is pressed and reduced, but the volume of the soil/rock particles does not diminish. So, the new porosity is calculated by the stress/strain. The method is not direct.

## 2. Joint Particle Model and Joint Particle Size

#### 2.1. Joint Model for Soils

^{n}= K

^{n}U

^{n}, F

^{s}= K

^{s}U

^{s}, where F

^{n}and F

^{s}are the normal force and tangential force between two particles or between a particle and a wall plane, respectively. U

^{n}is the compression displacement between particles or between a particle and the wall plane. K

^{n}is the elastic stiffness. U

^{s}is the sliding shear displacement between particles or a single particle and a wall plane. K

^{s}is the shear stiffness of this shear.

_{j}

_{1}, O

_{j}

_{2}, O

_{j}

_{3,}and O

_{j}

_{4}are the 1st, 2nd, 3rd, and 4th circular particles in the jth joint particle (Figure 2d). O

_{i}

_{1}is the first circular particle in the ith joint particle. The relationship between the distance of two circles’ centers and the sum of the two radii is needed to judge whether two joint particles are in contact; to calculate the coincidence degree of mutual extrusion, R

_{im}is the mth circular particle radius composed of the ith joint particle (as shown in Figure 2d). R

_{jn}is the radius of the nth circular particle composed of the jth joint particle. D is U in Figure 2a. When D

_{ijmn}> R

_{i}

_{m}+ R

_{jn}, two joint particles do not contact each other. When D

_{ijmn}= R

_{i}

_{m}+ R

_{jn}, two joint particles just contact. When D

_{ijmn}< R

_{i}

_{m}+ R

_{jn}, two joint particles squeeze against each other. m = 1, 2, …, n = 1, 2, …, the maximum value of m and n is the number of circular particles in the joint particle of the ith and jth joint particle.

#### 2.2. Joint Particle Size: Rotation Calculation Model

_{u}= D

_{60}/D

_{10}; the gradation coefficient is C

_{c}= (D

_{30})

^{2}/(D

_{60}D

_{10}), where D

_{60}, D

_{30,}and D

_{10}are the diameters through which 60%, 30%, and 10% of the total soil mass pass. One must obtain the particle-size distribution of the simulated soil particles constituting the soil to determine the parameters of D

_{60}, D

_{30}, D

_{10}, C

_{u}and C

_{c}.

_{β}, y

_{β}) after the coordinate system rotated an angle β (Figure 4c,d). The relationship between the coordinates of the two coordinate systems is as follows: x

_{β}= x × cosβ − y × sinβ, y

_{β}= x × sinβ + y × cosβ, where β∈ [0, π].

_{β}), min(x

_{β}), max(y

_{β}) and min(y

_{β}) of the coordinates of all points on the boundaries of all circles, where the two side lengths of rectangles were w

_{1β}= max(x

_{β}) − min(x

_{β}) and w

_{2β}= max(y

_{β}) − min(y

_{β}). When β∈(0, π), min(w

_{1β}) and min(w

_{2β}) can be obtained. These two values should be equal, i.e., min(w

_{1β}) = min(w

_{2β}) = w. Here, w is the minimum width of the joint soil particles and reflects the joint soil particles’ particle size. P(x, y) includes many points. P(x, y) is any point on the joint particles’ boundary as shown in Figure 4c.

_{1β}and w

_{2β}corresponding to the minimum rectangle covering a soil particle. When β∈(0, π), min(w

_{1β}) and min(w

_{2β}) can be obtained, the values of which should be equal, i.e., min(w

_{1β}) = min(w

_{2β}) = w, which is the particle size of the pixel soil particle. According to the image scale, obtain the joint soil particles’ actual particle sizes.

_{1}/w

_{2}and β. Here, w

_{1}and w

_{2}are the minimum width of the joint soil particles and are less than or equal to the sieve diameter in Figure 3d. β = β

_{i}is the rotation angle from the initial position when w = w

_{1(βi)}.

#### 2.3. Porosity Estimation of Overlapping Particles: Pixel Counting Method

_{min}, x

_{max},) y∈ (y

_{min}, y

_{max}), x

_{min}, y

_{min,}and x

_{max}, y

_{max}are the minimum and maximum values of the x, y position coordinates in the picture. The total number of points of the color pixels is S

_{B}= ΣPI∣

_{(R}

_{≠ 255 or G}

_{≠ 255 or B}

_{≠ 255)}, and the total number of points of the white pixel is S

_{W}= ΣPI∣

_{(R = 255, G = 255, B = 255)}. The total number of pixels in the entire image is ΣPI = (x

_{max}− x

_{min}) × (y

_{max}− y

_{min}), where ΣPI = S

_{B}+ S

_{W}; thus, the percentage of pores in the whole soil is n = S

_{W}/ΣPI, where n is the porosity of the soil after being compressed by an external force. The porosity in Figure 5 is 0.1623 after calculation.

#### 2.4. Elastic Modulus and Poisson’s Ratio

_{x}and σ

_{y}, respectively. Fx and Fy are the force applied to the sample in x and y directions. σ

_{x}and σ

_{y}are the stress on the sample in x- and y-direction. The strain formulas are ε

_{x}= (σ

_{x}− υσ

_{y})/E and ε

_{y}= (σ

_{y}− υσ

_{x})/E. Because the problem is a plane strain problem, ε

_{x}= 0 and σ

_{x}= υσ

_{y}. Then, σ

_{x}+ Δσ

_{x}= υ(σ

_{y}+ Δσ

_{y}), and, thus, Δσ

_{x}= υΔσ

_{y}. ε

_{x}and ε

_{y}are the strain on the sample in x- and y-direction. υ is Poisson’s ratio. E is the elastic modulus.

_{y}to the soil sample when soils reached stability. Δσ

_{x}can be obtained after the soils reach stability again under Δσ

_{y}. In this way, υ can be obtained from Δσ

_{x}= υΔσ

_{y}. Every pair of Δσ

_{y}and Δσ

_{y}values can correspond to a new value ε

_{y}+ Δε

_{y}. We obtained Δε

_{y}from the soil deformation. Thus, we obtained the elastic modulus E according to Δε

_{y}= (Δσ

_{y}− υΔσ

_{x})/E.

## 3. Example

#### 3.1. Joint Particle Size

#### 3.2. Particle Gradation and Porosity under Pressure

_{s}(1 − n), P is the upper pressure, ρ is the soil density, ρ

_{s}is the soil particle density, g is the gravitational acceleration, h is the depth, and n is the soil porosity. Furthermore, we obtained the relationship between the upper pressure and porosity.

^{3}. The normal contact stiffness K

^{n}is 4.4 × 10

^{7}N/m, and the shear contact stiffness K

^{s}is 2.2 × 10

^{7}N/m. The ratio of the two stiffness K

^{s}/K

^{n}is 0.5. The gravitational acceleration g is 9.81 m/s

^{2}, the ball radius r is 2–3 mm, and the friction coefficient Fr is 0.5.

_{u}and C

_{c}were calculated according to Section 2.2. The results are listed in Table 1 and shown in Figure 8.

_{u}and C

_{c}in Table 1 of soils were made up of two types of particles. D

_{60}, D

_{30,}and D

_{10}were the sieve diameter, with 60%, 30%, and 10% of the total soil mass passing through the sieve (unit: mm). C

_{u}= D

_{60}/D

_{10}was the uniformity coefficient. C

_{c}= (D

_{30})

^{2}/(D

_{60}D

_{10}) was the gradation coefficient. C

_{cc}and C

_{cs}were C

_{c}of the combined particles and the single ball, respectively. C

_{uc}and C

_{us}are the C

_{u}of the combined particles and the single ball, respectively. | | denotes the absolute value. C

_{u}= D

_{60}/D

_{10}is the uniformity coefficient. C

_{c}= (D

_{30})

^{2}/(D

_{60}D

_{10}) is the gradation coefficient. C

_{u}, C

_{c}are important parameters in particle, soil, and sand research, and engineering [35]. We used the two parameters to distinguish between well graded and poorly graded coarse-grained soil using laboratory tests of the grain size distribution [36]. The two parameters satisfy the requirements in engineering projects or science research. We can say that the materials were qualified [37].

_{u}and C

_{c}was not significant. Therefore, the gradation curve was not closely related to the number of selected soil particles. Figure 8b is a gradation curve for ball particles and not joint particles. Figure 8b and Table 1 show that the five gradation curves were almost coincident. Figure 8a,b shows that the particle size in Figure 8a was larger than the particle size in Figure 8b and that the corresponding gradation curves were significantly different. Therefore, we did not take the ball particle gradation curve as the joint soil particles’ gradation curve, which is different from previous research methods [12].

_{1}and σ

_{2}, respectively. H

_{σ}

_{1}and H

_{σ}

_{2}are the pressure conversion depths of σ

_{1}and σ

_{2}, respectively, where H

_{σ}

_{1}= σ

_{1}/(ρg) and H

_{σ}

_{2}= σ

_{2}/(ρg); h is the depth; ρ is the soil density; g is the acceleration due to gravity.

_{σ}

_{1}was more significant than H

_{σ}

_{2}, and both forces were greater than the natural depth h. The slope of Line h is 1. The slopes of Line H

_{σ}

_{2}and Line H

_{σ}

_{1}gradually increase with the depth increasing. After 70 m, the slopes of Line H

_{σ}

_{2}and Line H

_{σ}

_{1}tend to decrease, at approximately 80 m, Line H

_{σ}

_{2}begins to fall below the h line.

## 4. Discussion

_{0}in an entirely pressure-free and loose state. At that time, a = n

_{0}− c is related to the loose state of the material. B represents the degree of porosity increase with the depth increasing. B has a direct relationship with the elastic modulus E of the material and the density ρ of the particles and is, thus, related to the material’s mechanical parameters.

^{n}= K

^{n}U

^{n}, U

^{n}must be less than R. As the pressure increased, K

^{n}remained stable during the calculation; U

^{n}gradually increased. When U

^{n}> R, the black shaded area in Figure 2b exceeded half the circle; the circle center passed through the wall. The force between the ball and the wall changed to F

^{n}= K

^{n}(2R − U

^{n}), and the direction changed by 180°. The ball immediately passed through and away from the wall.

_{x}= υΔσ

_{y}and Δε

_{y}= (Δσ

_{y}− υΔσ

_{x})/E [12], the elastic modulus and Poisson’s ratio change drastically. However, space opened up by particles’ ejection is immediately occupied by the elastic release and deformation recovery of other elastically deformed particles. As a result, the superimposed elastic area between the particles (as shown in Figure 2a,b) was diminished. The release of the particle superposed area makes up for the space left by escaping particles. The particle area in the soil does not change much; thus, the soil porosity does not change significantly.

_{y}in y direction. Similarly, the force in the x direction produced a strain ε

_{x}in the y direction. The y direction was the primary stress direction. Poisson’s ratio is the ratio of the lateral strain ε

_{x}to the active strain ε

_{y}, i.e., υ = ε

_{x}/ε

_{y}. When the pressure in the active direction was slight, the specimen was not prone to the lateral strain; Poisson’s ratio was relatively small. However, as the pressure increased and the particles were pressed and superimposed, the entire soil became more compact. After the strain occurred in the active direction, the lateral direction was more prone to strain. As a result, Poisson’s ratio increased. Therefore, Poisson’s ratio increased with the burial depth and the upper load applied to the soils (as shown in Figure 11).

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Conflicts of Interest

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**Figure 1.**Polarized microscopic marble slice and crystal particle simulation (

**a**) Single-particle/circular particle/ball particle; (

**b**) one crystal that is a joint particle/crystal particle; (

**c**) partial enlargement of a joint particle; (

**d**) crystal in marble slice. 1, 2, 3, 4, and 5 are five crystals in a marble slice, this scale is approximate scale.

**Figure 2.**Mechanical relationship between ball particles and walls (

**a**). mechanical model of interaction between single ball particles; (

**b**) mechanical model of interaction between particle and wall plane. (

**c**) the relationships. R

_{1}, R

_{2}are the radii of single particles; K

^{n}is the elastic stiffness, and K

^{s}is the shear stiffness. The black shaded area overlaps the ball particles between the ball particles and the wall plane. Three joint particles are red, green, and black, which contact each other. There is a wall plane at the bottom.

**1-a, 2-a**and

**3-a**are three contact points between two particles: the contact relationship in (

**a**).

**4-b**is the contact point between particle and wall, the contact relationship in (

**b**); (

**d**) determination of distance, contact, and extrusion of joint particles.

**Figure 3.**Joint particle size and screening process. (

**a**) Four ball particles composite a joint particle; (

**b**) the maximum width rectangle covering the joint particle; (

**c**) some joint particles with its own the maximum width rectangle; (

**d**) joint particles passing through the sieve by rotation at the maximum width of its rectangle; the solid ball is the elementary ball; the solid black block is the soil particle. Soils consist of many joint soil particles; joint soil particles consist of solid balls. Width and height denote the joint soil particle’s width and height, respectively, where Height ≥ Width.

**Figure 4.**Rotation calculation model of the coordinate system to determine the soil particle size. (

**a**) The calculation model for digital joint soil particles composed of balls; (

**b**) the method for calculating the particle size of pixel joint particles of soil; (

**c**) the points on the boundary of one joint particle; (

**d**) programming flowchart.

**Figure 5.**The joint soil particles form the porosity of the soil. (

**a**) The overlap and relationship between balls compose one example joint particle; A, B and C are the areas of balls. ball

_{i}is i st ball area. W and L are the width and length of the example. ψ and Φ are the rate of the projected and the total area to the example area. n is the porosity. (

**b**) compression diagram;

**c**partial enlargement, with n = 0.1623.

**Figure 6.**Different numbers of circular particles compose a joint particle and joint particle sizes (

**a**–

**d**) are the crystals 1 in Figure 1d generated by 400, 600, 800, and 1000 circular particles, respectively. (

**e**–

**i**) are the crystals 2, 1, 5, 3, and 4 in Figure 1 and are composed of 1000 particles. The width, height and counterclockwise rotation angle of the minimum width rectangle are, respectively (5.216, 8.346, 124.905), (3.707, 5.441, 107.143), (1.712, 4.737, 89.381), (2.499, 4.429, 26.929) and (2.337, 4.535, 11.4590).

**Figure 7.**Particle size calculation of joint particles under different rotation angles. The graph of the rotation angle and the height/width of the corresponding rectangle with minimum width. The average length is (Height + Width)/2.

**Figure 8.**The uniformity coefficient, gradation coefficient, and gradation curve of soils. The particle min-width unit is mm. (

**a**) Combined particle soil gradation curves calculated by the rotation calculation model; (

**b**) single-ball soil gradation curves, where 400, 800, 1200, 1600, and 2000 denote the number of combined particles and single balls. For example, 400~1594 indicates that 1594 single balls constitute the 400 combined particles. The min-width unit is mm. The X coordinate is ln(min-width).

**Figure 10.**The soil porosity curve for different depths P1, P2, P3, P4, and P5 are soil particles’ morphologies under pressure due to the upper soil weight. The blue areas represent the soil particles, while the white areas depict the pores between the particles. P1: h = 10 m, n = 0.2694; P2: h = 30 m, n = 0.1772; P3: h = 50 m, n = 0.1165; P4: h = 70 m, n = 0.0769; and P5: h = 90 m, n = 0.0763.

Number of Particles | Combined Particles | Single Ball | Error | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

D_{10} | D_{30} | D_{60} | C_{c} | C_{u} | D_{10} | D_{30} | D_{60} | C_{c} | C_{u} | |C_{cc}-C_{cs}|/C_{cc}% | |C_{uc}-C_{us}|/C_{uc}% | |

400 | 5.87 | 6.692 | 7.744 | 0.985 | 1.319 | 4.342 | 4.842 | 5.427 | 0.995 | 1.250 | 1.016 | 5.245 |

800 | 5.881 | 6.605 | 7.605 | 0.975 | 1.293 | 4.340 | 4.840 | 5.430 | 0.994 | 1.251 | 1.951 | 3.243 |

1200 | 5.934 | 6.731 | 7.758 | 0.984 | 1.307 | 4.349 | 4.854 | 5.435 | 0.997 | 1.250 | 1.322 | 4.375 |

1600 | 6.009 | 6.885 | 7.904 | 0.998 | 1.315 | 4.344 | 4.864 | 5.441 | 1.001 | 1.252 | 0.301 | 4.766 |

2000 | 6.082 | 6.961 | 7.943 | 1.003 | 1.306 | 4.344 | 4.864 | 5.435 | 1.002 | 1.251 | 0.100 | 4.206 |

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**MDPI and ACS Style**

Sun, J.; Huang, Y.
Modeling the Simultaneous Effects of Particle Size and Porosity in Simulating Geo-Materials. *Materials* **2022**, *15*, 1576.
https://doi.org/10.3390/ma15041576

**AMA Style**

Sun J, Huang Y.
Modeling the Simultaneous Effects of Particle Size and Porosity in Simulating Geo-Materials. *Materials*. 2022; 15(4):1576.
https://doi.org/10.3390/ma15041576

**Chicago/Turabian Style**

Sun, Jichao, and Yuefei Huang.
2022. "Modeling the Simultaneous Effects of Particle Size and Porosity in Simulating Geo-Materials" *Materials* 15, no. 4: 1576.
https://doi.org/10.3390/ma15041576