# Research on the Accumulated Plastic Strain of Expansive Soil under Subway Loading
## 4 Classical Accumulated Plastic Strain Modeling Analysis  
Currently, researchers from both domestic and international backgrounds have extensively investigated the accumulation of strain in soils under cyclic loading, and have proposed different strain models based on their research results, which can be summarized as explicit or implicit models related to accumulated plastic strain of soil samples and the number of cycles and experimental parameters. Classical accumulated plastic strain calculation models can roughly be divided into the following two types.  
1. Exponential Model represented by the Monismith Model  
The exponential model proposed by [PERSON] et al. [23] is one of the widely used models, and subsequent scholars mostly used this model as a foundation and made refinements to develop improved models. The formula is expressed as follows:  
\[\varepsilon_{p}=aN^{b} \tag{1}\]  
In the equation: \(\varepsilon_{p}\)--accumulated plastic strain; \(a\), \(b\)--related test parameters; \(N\)--number of cyclic vibrations.  
[PERSON] et al. [24] based their research on the [PERSON] exponential model, and considering factors such as confinement pressure and deviatoric stress, they proposed a modified model.  
\[\varepsilon_{p}=a\bigg{(}\frac{\sigma_{oct}}{p_{a}}\bigg{)}^{m}N^{b} \tag{2}\]  
In the equation: \(\sigma_{oct}\)--shear stress; \(p_{a}\)--atmospheric pressure, with a value of 101 kPa; \(a\), \(b\), \(m\)--test parameters; \(N\)--number of cyclic vibrations.  
[PERSON] et al. [25] developed a model that comprehensively considers the effects of initial static deviatoric stress, dynamic deviatoric stress, and static failure deviatoric stress. The formula is expressed as follows:  
\[\varepsilon_{p}=a\bigg{(}\frac{\sigma_{d}}{\sigma_{f}}\bigg{)}^{m}\bigg{(}1+ \frac{\sigma_{is}}{\sigma_{f}}\bigg{)}^{n}N^{b} \tag{3}\]  
In the equation: \(\varepsilon_{p}\)--accumulated plastic strain; \(a\), \(b\), \(m\), \(n\)--test parameters; \(\sigma_{d}\),\(\sigma_{f}\), \(\sigma_{is}\)--dynamic stress amplitude, static failure deviatoric stress, and initial static deviatoric stress, respectively; \(N\)--number of cyclic vibrations.  
2. Tang Yiqun Logit Model  
[PERSON] [26] proposed using logarithmic relationship curves to reflect the strain accumulation characteristics of reinforced soils around tunnels under Shanghai subway loads. The model comprehensively considers the influences of dynamic stress amplitude, consolidation ratio, confining pressure, and soil properties. The specific calculation model is as follows:  
\[\varepsilon_{p}=AlnN+B \tag{4}\]  
In the equation: \(\varepsilon_{p}\)--accumulated plastic strain; \(A\), \(B\)--parameters related to influencing factors; \(N\)--number of cyclic vibrations.  
Fitting of the experimental results was carried out using both the Monismith index model and the Tang Yiqun model. However, due to the congruence of the conclusions drawn from both models, only specimens S8 and S29 were chosen for detailed illustration, with the omission of further elaboration on the remaining samples. This is demonstrated in Figures 9 and 10.  
In comparison with the experimental data, prior to the inflection point of the fitting curves, the logarithmic model proposed by [PERSON] exhibits closer alignment with the experimental values. However, beyond this inflection point, both fitted curves deviate significantly from the experimental data. Consequently, for lower cycle numbers, [PERSON]'s logarithmic model provides a better fit to the experimental data than the Monismith index model. Nevertheless, as the cycle numbers increase, both models continue to exhibit a rising trend, whereas the experimental values in this study tend to stabilize and show incremental growth. Ultimately, it can be determined that neither the Monismith index model nor the [PERSON] logarithmic model adequately captures the behavior observed in this study's experimental results.