## 2 Quantitative Index for Measuring Weathering Process and Particle Breakage
### Quantitative Index for Measuring Particle Breakage  
The relative breakage proposed by [PERSON] [14] was obtained from the variation in particle size distribution curves before and after loading. Relative breakage can reflect the overall change in particle size distribution before and after loading and overcomes the shortcoming of using the increase in percentage of particles passing a given sieve size [7; 8; 9] or the decrease in sieve size corresponding to a given percentage passing [11] as the evaluation index for particle breakage. As mentioned above, for soil elements with the same particle shape and initial conditions, but different particle size distributions, after loading along the same path, the relative breakage rate is basically the same. Relative breakage can be regarded as a basic attribute of the soil material, so it is more scientific to take relative breakage as an index to evaluate soil particle breakage characteristics.  
The breakage potential (\(B_{p}\)) defined by [PERSON] [14] can be seen as the possibility of particle breakage (shown in Figure 2). It is considered that the possibility of soil particle breakage increases with particle size. That is because the normal contact forces in a soil element increase with particle size and the probability that there will be a defect in a given particle increases with its size [14].  
Figure 2: Definition of [PERSON]’s breakage potential [14].  
Under high stress, large particles will be broken into silt particles 0.074 mm in size, while silt particles cannot be broken further. Therefore, the ultimate particle size is 0.074 mm in the process of breakage [14]. All particles larger than this size have the possibility of breakage. The potential for breakage of a particle of a given size, \(D\), can be represented by  
\[b_{p}=\log_{10}\left(\frac{D}{0.074}\right), \tag{9}\]  
where \(D\) is particle size (mm) and \(D\geq 0.074\) mm; when \(D<0.074\) mm, \(b_{\text{p}}=0\).  
According to the definition of [PERSON] [14], the desired integration of the breakage potential (\(B_{p}\)), shown as the hatched areas in Figure 2, can be given by  
\[B_{p}=\int_{0}^{1}b_{p}df, \tag{10}\]  
where \(df\) is a differential of percentage of particles passing divided by 100.  
[PERSON] [14] defined the difference between the breakage potential of soil samples before and after loading as the total breakage (\(B_{t}\)), shown by the hatched areas in Figure 3 and given by  
\[B_{t}=\int_{0}^{1}(b_{po}-b_{pl})df, \tag{11}\]  
where \(b_{po}\) is the original value of \(b_{p}\) and \(b_{pl}\) is the value of \(b_{p}\) after loading.  
The data measured by [PERSON] and [PERSON] [11] show that values of \(B_{r}\), for soil elements with the same particle material and shape, at a given void ratio and loaded along a specified effective stress path to the same state of effective stress, are approximately proportional to \(B_{p}\). In view of the above, [PERSON] [14] concluded that when particle size distribution is the only variable, \(B_{t}\) is proportional to \(B_{p}\) and defined the relative breakage index, \(B_{r}\), as a quantitative indicator of particle breakage, as follows:  
\[B_{r}=\frac{B_{t}}{B_{p}}\times 100\%, \tag{12}\]  
In other words, the relative breakage, \(B_{r}\), is approximately independent of particle size distribution when the only variable is particle size distribution.  
The lower limit of relative breakage is zero, which means no soil particles are crushed. The theoretical upper limit of relative breakage is 1, which means all soil particles are crushed to the extent that no particles larger than 0.074 mm remain.  
Figure 3: Definition of [PERSON]’s total breakage [14].