# Effects of Relative Density and Grading on the Particle Breakage and Fractal Dimension of Granular Materials

^{*}

## Abstract

**:**

## 1. Introduction

_{cs}, e

_{Γ}, and λ

_{c}) decrease with increasing aspect ratio, sphericity and convexity [8].

_{c}) are less affected by relative density and particle grading [7,13]. However, the critical state parameters (e

_{Γ}) decrease with increasing relative density. The relative breakage index decreases with increasing relative density [14]. Extensive research has also been carried out on the impact of granular materials particle gradation [23,24,25]. For example, model parameters varied linearly with the coefficient of uniformity. The larger the coefficient of uniformity, the smaller the peak intensity and the larger volumetric strain.

## 2. Laboratory Tests

_{u}) and curvature coefficient (C

_{c}) are listed in Table 1.

## 3. Analysis of Test Results

#### 3.1. Particle Breakage under Different Relative Density

_{0}is the total mass of granular materials, d

_{M}is the particle maximum diameter, F(d) is the mass ratio of granular materials with diameter less than d and D is the fractal dimension.

_{3}= 0.6 MPa). However, the residual shear stresses remain approximately the same for the different Rd for a given confining pressure. The stress–strain behaviour of granular materials transforms from a strain hardening type to a strain softening type for all the tested specimens. As the shear strain further increases, the residual shear stresses corresponding to the rockfill with different Rd become stable. All the granular materials tested in this study exhibit characteristics similar to those observed by Lade [28] for sands. This can be attributed to the observation that the greater the relative density, the greater the interlocking between the particles; and the greater the loading, the greater was the peak strength of the specimen under the same confining pressure. As the load increased, particle breakage occurred, the interlocking force between particles would decrease and the shear strength would decrease.

_{M}) after loading under different confining pressures. The values of R

^{2}are larger than 0.97, and the Root mean squared Error (Re) is smaller than 0.23. The fractal dimension increases with confining pressure during of particle breakage. For example, the fractal dimension increases from 2.33 (σ

_{3}= 0.2 MPa) to 2.37 (σ

_{3}= 0.8 MPa) when relative density Rd = 0.6. However, relative density has little influences on the fractal dimension, especially under higher relative density. For example, the fractal dimensions are almost the same for Rd = 0.8 and 0.9.

_{test}is percentage by mass of particles after the test, and P

_{ini}is percentage by mass of particles before the test.

_{1}and n are model parameters, determined to be 6.36 and 0.46, respectively; p

_{a}(= 101 kPa) is the atmospheric pressure.

#### 3.2. Particle Breakage under Different Grading Curve

_{M}) after loading at different confining pressures. Compared with those under different relative densities, the initial fractal dimension has great influences on the final fractal dimensions obtained under different confining pressures, as shown in Figure 7. It can also be observed that under low confining pressures, granular materials can undergo significant particle breakage. The smaller the fractal dimension is, the more significant the particle breakage extent will be. The larger the fractal dimension is, the greater the content of fine particles, the smaller the particle crushing rate during shearing and the smaller the effect of confining pressure on particle crushing. As the confining pressure increases, it tends to the final limit fractal dimension 2.6 [30].

_{2}and n are model parameters, determined to be 19.1 and 0.46, respectively.

_{50}is an important soil grading curve index. The larger the median diameter, the higher the shear strength and the more obvious is the shear expansion effect [31]. Figure 9 shows the relationship between the particle breakage ratio with d

_{50}under different confining pressures after monotonic loading. The greater d

_{50}, the greater is the particle breakage ratio. According to previous research, the confining pressure effect can be expressed by a power function. A normalized power function is also found to fit well the relationship between breakage ratio, d

_{5}and confining pressure. The expression is shown as follows:

_{2}and n

_{3}are model parameters, determined to be 16.5 and 0.46 (which is the same as the previous value of the coefficient of uniformity).

## 4. Conclusions

- (1)
- The particle size distribution exhibited good fractal characteristics after monotonic loading of rockfill at different confining pressures. The fractal dimension increased with the increase in confining pressure. The coefficient of uniformity exhibited a greater effect on the fractal dimension than relative density.
- (2)
- During the shearing process, the main occurrence of breakage was found to be in large particles. The extent of particle breakage increased with the increase of confining pressure and relative density, whereas it decreased with an increase in the coefficient of uniformity, which can be well described by a normalized power function. The relationship between the breakage ratio and the median diameter can be described by a linear function.
- (3)
- The conclusions are mainly based on the results after the test. In fact, fractal dimension and particle breakage change with axial loading. Future research should focus on the results during the shear test. The relationship between fractal dimension and particle breakage with shear modulus and volume strain should be investigated.

## Author Contributions

## Funding

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 2.**(

**a**) Grading curve; (

**b**) Triaxial apparatus. Grain size distribution and triaxial apparatus for tested granular materials.

**Figure 3.**(

**a**) confining pressure of 0.2 MPa; (

**b**) confining pressure of 0.4 MPa; (

**c**) confining pressure of 0.6 MPa; (

**d**) confining pressure of 0.8 MPa. Stress–strain behaviour of granular materials under different confining pressures.

**Figure 4.**(

**a**) Rd = 0.6; (

**b**) Rd = 0.7; (

**c**) Rd = 0.8; (

**d**) Rd = 0.9. Grading curves of different relative densities.

**Figure 5.**(

**a**) Breakage ratio vs. confining pressure; (

**b**) Breakage ratio vs. normalized confining pressure. Particle breakage of different relative densities.

**Figure 6.**(

**a**) Grading curves of Gc1; (

**b**) Grading curves of Gc2; (

**c**) Grading curves of Gc3; (

**d**) Grading curves of Gc4; (

**e**) Grading curves of Gc5. Grading curves of different grading curves.

**Figure 8.**(

**a**) Breakage ratio vs. confining pressure; (

**b**) Breakage ratio vs. normalized confining pressure. Particle breakage of different grading curves.

**Figure 10.**(

**a**) Different relative density; (

**b**) different grading curves. Particle percentage increment (ΔF > 0, particle percentage increasing).

Gc1 | Gc2 | Gc3 | Gc4 | Gc5 | |
---|---|---|---|---|---|

Coefficient of uniformity | 2 | 5 | 10 | 20 | 40 |

Curvature coefficient | 1.17 | 1.44 | 1.68 | 1.97 | 2.30 |

Maximum dry density (g/cm^{3}) | 1.71 | 1.78 | 1.94 | 2.12 | 2.23 |

Minimum dry density (g/cm^{3}) | 1.47 | 1.57 | 1.68 | 1.78 | 1.79 |

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**MDPI and ACS Style**

Yang, G.; Chen, Z.; Sun, Y.; Jiang, Y.
Effects of Relative Density and Grading on the Particle Breakage and Fractal Dimension of Granular Materials. *Fractal Fract.* **2022**, *6*, 347.
https://doi.org/10.3390/fractalfract6070347

**AMA Style**

Yang G, Chen Z, Sun Y, Jiang Y.
Effects of Relative Density and Grading on the Particle Breakage and Fractal Dimension of Granular Materials. *Fractal and Fractional*. 2022; 6(7):347.
https://doi.org/10.3390/fractalfract6070347

**Chicago/Turabian Style**

Yang, Gui, Zhuanzhuan Chen, Yifei Sun, and Yang Jiang.
2022. "Effects of Relative Density and Grading on the Particle Breakage and Fractal Dimension of Granular Materials" *Fractal and Fractional* 6, no. 7: 347.
https://doi.org/10.3390/fractalfract6070347