## 3 Hysteresis Curve Characteristics of GRS
### Hysteresis Curve Parameters  
To a certain extent, the hysteresis curve's evolution law reveals the soil's internal structure change rule. By extracting the parameters of the shape change from the hysteresis curve, the influence of external conditions on soil damage can be obtained. This study uses the hysteresis curve area \(S\), the slope of the long axis \(k\), the center offset of hysteresis curve \(d\) and residual strain \(\varepsilon_{sp}\) to analyze the hysteresis curve characteristics of GRS. The calculation methods of \(S\), \(k\), \(d\) and \(\varepsilon_{sp}\) and their variations will be introduced separately below.  
(1) Hysteresis curve area \(S\)  
The variation of \(S\) normally indicates the degree of soil compaction. A large \(S\) shows that the soil structure is loose and easily compacted and that a large amount of energy is lost during compaction. On the contrary, a small \(S\) indicates that the soil structure is dense and difficult to compact, and less energy is lost during compaction.  
The area of the arc from the initial point of loading, to the final point of unloading is the area of the hysteresis curve. The hysteresis curve area \(S\) is expressed as follows:  
\[S=\int\sigma_{di}d\varepsilon_{i} \tag{1}\]  
where \(\sigma_{di}\) and \(\varepsilon_{i}\) denote the deviator stress and the axial strain under the ith loading cycle.  
(2) Major axis gradient \(k\)  
The stiffness and elasticity modulus of the soil can be reflected by the major axis gradient \(k\) of the hysteresis curve. When \(k\) grows, the soil stiffness and elasticity modulus increase and vice versa. The major axis gradient \(k\) is calculated by:  
\[k=\frac{\sigma_{N,\max}-\sigma_{N,\min}}{\varepsilon_{N,\max}-\varepsilon_{N, \min}} \tag{2}\]  
where \(\sigma_{N,\max}\), \(\varepsilon_{N,\max}\), \(\sigma_{N,\min}\) and \(\varepsilon_{N,\min}\) are the maximum and the minimum stress and strain of the \(N\)th hysteresis curve, respectively.  
(3) Center offset \(d\)  
The center offset \(d\) is the distance between the centers of the previous and subsequent hysteresis curves, which reflects the soil deformation and internal structural damage. The larger the \(d\), the greater the irrecoverable deformation of the soil sample and the greater the damage to the internal structure of the soil, and vice versa. Since the graph enclosed by the hysteresis curve is not a regular shape, it is not trivial to pinpoint the exact location of the center point. This work primarily studies the trend of its center distance with loading and D-W cycles. The difference between the strain maximum of the previous hysteresis curve and the latter hysteresis curve is proportional to the center distance. Therefore, the center offset \(d\) is computed as follows:  
\[d=\varepsilon_{N,\max}-\varepsilon_{N-1,\max} \tag{3}\]  
where \(\varepsilon_{N,\max}\) and \(\varepsilon_{N-1,\max}\) represent the maximum strain of the \(N\)th and the \((N-1)\)th hysteresis curve, respectively.  
(4) Residual strain \(\varepsilon_{sp}\)  
The residual strain is the strain difference between the end point of loading and the starting point in one cycle, namely, the degree of non-closure or the size of the opening of the hysteresis curve. Thus, the residual strain \(\varepsilon_{sp}\) is given as follows:  
\[\varepsilon_{sp}=\varepsilon_{e}-\varepsilon_{s} \tag{4}\]  
where \(\varepsilon_{s}\) and \(\varepsilon_{e}\) are the strain values corresponding to the starting and ending points of the \(N\)th loading, respectively.