model considers the stress state, number of cycles and physical state of soil as an influencing factor and according to this model, the permanent deformation can be calculated as in Equation (6):  
\[\varepsilon_{a}^{p}=g(\Delta p_{max},\Delta q_{max})f(N), \tag{6}\]  
The idea of this model is to separate the effect from the number of cycles of applied stress from the stress intensity as the function of maximum stress. The function of the number of cycles is the [PERSON] et al. model (3) and the \(g(\Delta p_{max},\Delta q_{max})\) can be defined as follows (7):  
\[g(\Delta p_{max},\Delta q_{max})=\varepsilon_{a}^{p,0}\bigg{(}\frac{l_{max}}{ p_{a}}\bigg{)}^{n}\frac{1}{\Big{(}m+\frac{s}{\Delta p_{max}}-\frac{\Delta q_{max}}{ \Delta p_{max}}\Big{)}}, \tag{7}\]  
where the \(\varepsilon_{a}^{p,0}\), \(m\), \(n\), and \(s\) are the parameters, \(l_{max}=\sqrt{\Delta p_{max}^{2}+\Delta q_{max}^{2}}\), \(p_{a}=100\) kPa and \(p_{a}\) is the atmospheric pressure equal to 100 kPa. The [PERSON] et al. model was further developed and adapted to different kinds of geotechnical applications. For example, [PERSON] et al. [23; 24] applied this model to calculate the settlement of pile-supported ballastless track-bed. These calculations were conducted based on the full-scale model, where the settlement of the embankment was observed. The accumulated settlement curve of subgrade presents in this study had a parabolic-shaped characteristic. The modification of [PERSON] et al. model concerned the \(g(\Delta p_{max},\Delta q_{max})\) function where the modified one was as follows in Equation (8):  
\[g(\Delta p_{init},\Delta q_{init},\Delta p_{max},\Delta q_{max})=\frac{1}{\Big{(} m\cdot\Big{(}1+\frac{\Delta p_{init}}{\Delta p_{max}}\Big{)}+\frac{s}{ \Delta p_{max}}-\frac{\Delta q_{init}+\Delta q_{max}}{\Delta p_{max}}\Big{)}}, \tag{8}\]  
where the \(\Delta p_{init}=(\sigma_{z}+2\sigma_{x})/3\) and \(\Delta q_{init}=\sigma_{z}-\sigma_{x}\) are parameters induced by the soil unit weight. The second application of the [PERSON] et al. model was considering the mechanical characterization of the fouled ballast in the railway track substructure. The large-triaxial tests conducted by [PERSON] et al. [25] showed a significant effect of water content on the soil mechanical behavior where, in low water content, the shear strength was higher and a permanent axial strain was equal to 0.4% for 4% water content, while for water content equal to 6%, the permanent strain was equal to 1.4%. Based on the test results, the modification of the [PERSON] et al. model was proposed. The extension of the model is as follows in Equation (9):  
\[\varepsilon_{a}^{p}=t(w,\Delta q_{max})f(N), \tag{9}\]  
where the \(t(w,\Delta q_{max})\) relates to the permanent axial strain to the water content and applied deviator stress as in Equation (10):  
\[t(w,\Delta q_{max})=\varepsilon_{a}^{p,0}(w+a)\bigg{(}\frac{\Delta q_{max}}{p _{a}}\bigg{)}^{\alpha}, \tag{10}\]  
where \(\varepsilon_{a}^{p,0}\), \(a\), and \(\alpha\) are parameters. The estimation method of permanent deformation developed by [PERSON] et al. [26] is able to combine two successive cyclic levels, namely M and M+1, into one characteristic by application of the following Equation (11):  
\[\log\bigl{(}\varepsilon_{a}^{p}\bigr{)}=a+b\cdot\log(N)=a\cdot N^{b}, \tag{2}\]0  
where \(\varepsilon_{a}^{p,M}\) is the measured permanent deformation of loading level M, \(\delta\varepsilon_{a}^{p,M+1}\) is the translated permanent deformation from the measured curve of loading level M+1 (the initial loading \(\varepsilon_{a}^{p,N=1}\) is set to 0 in this case). In this method, the slope \(\theta\) is defined by the quotient of permanent strain accumulation to the number of cycles and it is kept constant in M+1 loading phase. When the elastic-plastic structure is under repeated loading, three phenomena can occur [27]. When the magnitude of the applied load is low enough (lower than the yield limit of the material),plastic deformation will not happen, and the behavior of this structure will be entirely elastic.