## 4 Discussion
### Modified Deformation Rate  
Modified deformation rate is quantitatively evaluated on a basis of modified deformation curve for the three models in this study, as shown in Figures 12-14, respectively. Scattered points in the co-ordinates represent modified deformation rates, which are obtained from testing data. It can be seen in Figures 12-14 that plastic deformation rate levels off to a small magnitude rapidly when amplitude of cyclic loading stays at a low level. As cyclic amplitude increases, deformation rate also increases. However, deformation rate also converges as long as the amplitudes remain less than a critical value. Nevertheless, once amplitude rises up beyond the critical value, plastic deformation rate is inclined to keep beyond some constant value in spite of its decreasing trend at initial stage. That is to say, plastic deformation will develop continuously at a constant rate till filling structure breaks down. If the amplitude of cyclic loading increases abidingly, plastic deformation rate starts to increase with loading times, which leads to the model collapse at an overwhelming speed.  
Figure 11: Modified plastic deformation against loading times for Model 1.  
Figure 14: The relationship between modified deformation rate and loading times for Model 3 (\(K=1.0\)) when cyclic amplitude ranges from (**a**) 50 kPa to 350 kPa; (**b**) 400 kPa to 700 kPa.  
Figure 12: The relationship between modified deformation rate and loading times for Model 1 (\(K=0.9\)) when cyclic amplitude ranges from (**a**) 50 kPa to 200 kPa; (**b**) 250 kPa to 300 kPa.  
Figure 13: The relationship between modified deformation rate and loading times for Model 2 (\(K=0.95\)) when cyclic amplitude ranges from (**a**) 50 kPa to 250 kPa; (**b**) 300 kPa to 500 kPa.  
Following an examination of a great deal of experimental data for both coarse-grained gravels and fine-grained soils, a basic principle is proposed as follows: Under the effect of different amplitudes of cyclic loading, the relationship between accumulative deformation rate and loading times (or elapsed time) can be expressed by a negative power function. This principle implies that plastic behavior of gravel fillings can be expressed by the following equation:  
\[{v^{\prime}}_{1}=CN^{-m} \tag{6}\]  
where \({v^{\prime}}_{1}\) is modified deformation rate, \(N\) is loading times, and \(C\) and \(m\) are constants under a certain amplitude for a given model. The parameter \(C\) reflects average deformation rate for a given amplitude of cyclic load. The larger average level of deformation rates at different moments, the higher value of \(C\). The parameter \(C\) can be termed as comprehensive evaluation coefficient for plastic deformation rate. Furthermore, the parameter \(m\) is an indicator to describe the successional trend for deformation rate curve. Thus, \(m\) can be defined as developing tendency coefficient.  
Equation (6) can be adopted to describe the modified deformation rate under different amplitudes of cyclic loading for these three models. In the process, the value of the parameters \(C\) and \(m\) can be determined by regression analysis method, as shown in Table 3. Then, fitting curves can be plotted with modified deformation rate against loading times, as shown in Figures 12-14. All of the fitting curves develop well along the testing data, which indicates that the parameters \(C\) and \(m\) are reasonably set in the fitting process. Moreover, almost every regression analysis coefficient is higher than 0.90, averaging at 0.919. Obviously, for a given model, there is a group of values for \(C\) and \(m\) corresponding to a certain amplitude of cyclic loading.