## 4 Discussion
### Developing Tendency Coefficient \(m\) Against Cyclic Amplitudes  
Developing tendency coefficient \(m\) is a more critical parameter than \(C\) to describe the evolving characteristics of plastic deformation. The value of \(m\) is plotted against cyclic amplitude level \(P_{d}/P_{cr}\) for the three testing models, as shown in Figure 17. Interestingly, the data points for the parameter \(m\) develop along three different sigmoidal curves for the three models. The sigmoidal curves originate from a mutual starting point at the initial stress level. As cyclic amplitude level increases, these three curves disperse from each other gradually. As a consequence, the curves form a unique sigmoidal belt, which can be adopted to demonstrate the evolving characteristics of plastic deformation rate. It is worth noting that two inflection points are existed on the sigmoidal belt. One locates approximately at \(P_{d}/P_{cr}\) = 0.2 (or \(m\) = 0.7), and the other corresponds to the condition of \(P_{d}/P_{cr}\) = 0.6 (or \(m\) = 0.4). When \(P_{d}/P_{cr}\) < 0.2, the tendency coefficient \(m\) decreases remarkably with the increased amplitude level. However, a diminution of decreasing rate for the parameter \(m\) occurred when \(P_{d}/P_{cr}\) ranges from 0.2 to 0.6. Finally, as \(P_{d}/P_{cr}\) increase beyond 0.6, \(m\) decreases rapidly again, and even reduces below zero. Therefore, the two plastic deformation evolving status corresponding to \(P_{d}/P_{cr}\) = 0.2 and \(P_{d}/P_{cr}\) = 0.6 respectively are so critical that the evolving characteristics of tendency coefficient \(m\) is changed when traverse these two critical values. On the other hand, the relational expression between plastic deformation \(s^{\prime}_{1}\) and loading times \(N\) can be obtained by carrying out integral operation on Equation (6). The relational expression can be shown as follows:  
\[s_{1}=\frac{C}{1-m}N^{1-m}+S \tag{9}\]  
where \(S\) is a constant produced in the integral process. It is obvious that plastic deformation tends to converge towards \(S\) with increased loading times when \(m\) < 1. With overall consideration, \(0.7<m<1\) can be regarded as the transitional zone of model fillings from stabilization to failure. With plastic deformation rate evaluated quantitatively, evolving states can be subdivided according to the developing rate of relevant status. In detail, both stabilization and failure status can be subdivided into the rapid and tardy one. It has been mentioned that the first inflection point when \(m\) = 0.7, together with the data point corresponding to \(m\) = 1, is the boundary from stabilization to failure. Then, the second inflection point when \(m\) = 0.4 can be regarded as the boundary of the tardy  
Figure 17: Classification of different evolving states according to the relationship between \(m\) and \(P_{d}/P_{cr}\). failure zone and the rapid one. When \(0.4<m<0.7\), although plastic deformation is inclined to develop continuously with increased loading times, the decreasing rate of tendency coefficient \(m\) is significantly slow. This phenomenon demonstrates that the evolving characteristics of plastic deformation rate just has very slight changes when \(P_{d}/P_{\sigma}\) ranges from 0.2 to 0.6. Thus, the relevant section can be defined as tardy failure zone. When \(m<0.4\), decreasing rate of the parameter \(m\) improves obviously, which indicates that plastic deformation increases at an accelerated speed when \(P_{d}/P_{\sigma}\) tends to unity. Therefore, this section can be termed as rapid failure zone. In addition to this, stabilization zone can also be subdivided when \(m>1.0\) in spite of the obvious decreasing trend of the parameter \(m\). A classification criterion can be arbitrarily set that fillings can be judged to fall into rapid stabilization zone on the condition of plastic deformation rate less than 0.001 mm under 30,000 loading times. This is a rigorous standard that only on the condition of \(P_{d}=50\) kPa can Model 3 with \(K=1.0\) fall into rapid stabilization zone.