## 2 Introduced the Improved Particle Breakage Index \(B_{C}\)  
### Improved the Definition of the Breakage Index  
The particle breakage index is a quantitative measure used to characterize the degree and properties of particle breakage. With regard to the breakage index \(B_{W}\) suggested by [PERSON] et al. [5]:  
\[B_{W}=\frac{S_{1}-S_{0}}{S_{0}} \tag{1}\]  
The modified curve captures the characteristics of the original grading curve and minimizes the need for additional experimental calculations of the limit grading curve, making it a simple and practical solution. However, because the breakage index \(B_{W}\) ranges from 0 to infinity, it is not suitable for comparing or evaluating the degree of particle breakage. To address this limitation, the following improvements are made to the \(B_{W}\) suggested by [PERSON] et al. [5]. Based on this, the breakage index \(B_{C}\) is proposed:  
\[B_{C}=\frac{S_{1}-S_{0}}{S_{1}} \tag{2}\]  
where the starting grading curve is \(S_{0}\), the grading curve after the test is \(S_{1}\), and the area enclosed by the maximum particle size is \(d=d_{max}\). As shown in Figure 1.  
The theoretical value range of \(B_{C}\) is defined as between 0 and 1. A value of \(B_{C}=0\) indicates an unbroken state, while \(B_{C}=1\) signifies full breakage and reflects the complete shift in gradation. This revised index is straightforward and practical. [PERSON] et al. [19] introduced the breakage index \(I_{G}\), which is calculated as the ratio of the area between the shear particle-grading curve and the maximum particle size line to the area between the limit particle-grading curve and the maximum particle size line. Although the forms of \(B_{C}\) and \(I_{G}\) may differ, their underlying cores are identical. Figure 1: Schematic diagram of the definition of the current breakage, \(B_{W}\) and \(B_{C}\). Improve the Quantitative Correlation between the Breakage Index \(B_{c}\) and Fractal Dimension \(D\)  
The concept of fractals was introduced by [PERSON] (1963) to explain the complex geometric patterns observed in nature. In the context of a mixture of earth and rock, internal particles experience uneven breakage when subjected to external loads. As a result, fractal dimension, a measure of complexity within fractal geometry, is used to objectively characterize particle breakage and determine the relationship between breakage and strength [PERSON] (1963). [PERSON] et al. [PERSON] et al. (2018) assumed that particle groups of varying sizes shared the same density and developed a mathematical model for particle fractal dimension. \[lg\Big{[}\frac{M(\rho<d_{i})}{M_{T}}\Big{]}=(3-D)lg(\frac{d_{i}}{d_{max}}) \tag{3}\]  
where \(M(\rho<d_{i})\) represents the particle mass below a specific scale; \(M_{T}\) is the total mass of particles in the sample; and \(d_{max}\) is the largest particle size in the sample. \(lg(\frac{d_{i}}{d_{max}})\) is plotted on the x-axis, and \(lg\Big{[}\frac{M(\rho<d_{i})}{M_{T}}\Big{]}\) is plotted on the y-axis, based on linear regression fitting a straight line with slope \(k\) in a type \(3-D\) graph. The slope \(k\) is determined to calculate the score dimension \(D\) Mandelbrot (1963). The particle size \(d_{i}\) ranges from a minimum value of 0 mm to a maximum value that represents 100% soil content. When calculating the area using integration, the integral of \(P\) is taken over the interval [\(i\), 1]. Then, let \(k\) approach 0 and find the limit, as shown in Figure 2. The grading curve area is represented in \(P\)-lg\(d\) coordinates:  
\[S=\underset{i\to 0}{lim}S_{A}=\int_{i}^{1}(lgd_{max}-lgd_{i})dP \tag{4}\]  
where \(d_{i}\) is the particle size that corresponds to \(P=i\).