## 3 Elasto-Plastic Constitutive Model
### Key Points for the Revised SMP Model  
The implementation of the elasto-plastic constitutive model of UGM is introduced from the following four aspects:  
1. To determine the elasto-plastic yield surface of UGM under repeated loads, the steps are as follows:  
_Step 1_: Carry out repeated loads test of UGM with more than three kinds of loading strength. Before the construction of the revised SMP model, a series of laboratory tests were conducted to obtain the experimental data. The gradation the tested UGM was designed by the interlocking skeleton method [5; 11; 29; 30]. The optimum moisture content was 5.0%, and the maximum dry density was 2.263 g/cm\({}^{3}\). The coefficient of curvature (\(Cc\)) and coefficient of uniformity (\(Cu\)) was 26.95 and 2.72, respectively. The loading strength was determined depending on the axial stress range of UGM layer, the confining pressure was 10-50 kPa. For the long-term repeated load tests, four axial stresses (80, 160, 240 and 320 kPa) and four loading frequencies (1, 5, 8 and 10 Hz) were used and the loading cycles were up to 20,000 repetitions [11; 29; 30]. In addition, three compaction degrees (96%, 98% and 100%) and five moisture contents (0, 3%, 4%, 5% and 6%) were taken into account. Then the plastic strain rate and resilient strain was obtained, and which were calculated as follows:  
\[\dot{\varepsilon}_{p(N)}=\varepsilon_{p(N)}-\varepsilon_{p(N-1)} \tag{1}\]  
\[\varepsilon_{r(N)}=\varepsilon_{PK-PK(N)}-\dot{\varepsilon}_{p(N)} \tag{2}\]  
where, \(\dot{\varepsilon}_{p(N)}\) is the plastic strain rate which means the plastic strain generated in the loading cycle of number of \(N\); \(\varepsilon_{p(N)}\) is the cumulative plastic strain during the \(N\) times of loading cycle; \(\varepsilon_{p(N-1)}\) is the cumulative plastic strain during the \(N-1\) times of loading cycle; \(\varepsilon_{r(N)}\) is the resilient strain in the loading cycle of number \(N\); \(\varepsilon_{PK-PK(N)}\) is the amplitude of the maximum strain and the minimum strain in the loading cycle of number \(N\); \(N\) is the loading repetitions; \(p\) represents 'plastic' and \(r\) represents'resilient'. _Step 2_: Evaluate the shakedown state of UGM using shakedown theory. The requirement of plastic shakedown of UGM is that the cumulative plastic strain needs to meet the following condition:  
\[\left(\varepsilon_{p}^{5000}-\varepsilon_{p}^{3000}\right)<0.045\times 10^{-3} \tag{3}\]where, \(\varepsilon_{p}^{3000}\) is the cumulative plastic strain during the first 3000 times of loading; \(\varepsilon_{p}^{5000}\) is the cumulative plastic strain during the first 5000 times of loading. If this condition is satisfied, UGM is regarded as being in a stable state under the current loading strength. If not, the material category or loading strength of UGM should be selected again, and the loading strength should preferentially be changed first. _Step 3_: Establish the prediction models of plastic strain rate and resilient strain under plastic shakedown state, which consist of the loading strength, loading cycle, loading frequency, degree of compaction and moisture content. The models are as follows:  
\[\dot{\varepsilon}_{p}=K_{f}^{p}\cdot K_{d}^{p}\cdot K_{m}^{p}\cdot aN^{b} \tag{4}\]  
\[\varepsilon_{r}=K_{f}^{r}\cdot K_{d}^{r}\cdot K_{m}^{r}\cdot(A+B*C^{N}) \tag{5}\]  
where, \(a\), \(b\), \(A\), \(B\), \(C\) are the parameters of the models, which can be expressed as the function of axial deviator stress \(\sigma_{d}\), \(a=f(\sigma_{d})\), etc. ; \(K_{f}\), \(K_{d}\) and \(K_{m}\) are the correction coefficients of loading frequency, degree of compaction, and moisture content, respectively. _Step 4_: Analyze the relationship between yield stress and plastic strain during the typical loading-unloading cycle in stepwise and repeated loads tests under the shakedown load, obtaining Equation (6).