1. Equations in which the accumulated plastic strain is calculated as a function of the first cycle of loading, the initial density, or the value of repeating stress level. 2. The relationship between the permanent strain in the referenced cycle, which is dependable on the state of stress, void ratio, the stress amplitude, which are constant. 3. The equations which are able to predict the permanent strain development as a function of the number of cycles in the stated of stress, and other factors. The most widely used equations among engineers are the equations which relate the permanent strain \(\varepsilon_{a}^{p}\) to the number of cycles. Such a relationship was proposed by [PERSON] [15], and it connects the accumulated plastic strain to the logarithm of a number of cycles \(N\) as in Equation (1):  
\[\varepsilon_{a}^{p}=a+b\cdot\log(N), \tag{1}\]  
where \(a\) and \(b\) are constants, the value of which value depends on the stress level and physical properties of the soil, respectively. This type of empirical model has undergone numerous extensions and modifications. One of the earliest was the modification, which was proposed by [PERSON] [16; 17] based on the observation of triaxial tests. The best fit with test results gave the empirical model where the log-log relationship was established in Equation (2):  
\[\log\bigl{(}\varepsilon_{a}^{p}\bigr{)}=a+b\cdot\log(N)=a\cdot N^{b}, \tag{2}\]  
The \(a\) and \(b\) parameters are regression parameters. Further repeating loading tests have shown that the stress level has a significant impact on the soil response. Therefore, the empirical models have been extended with the stress part. The tests performed by [PERSON] et al. [18; 19] have led to establishing the relationship between the permanent soil strain during cyclic loading and stress state. The assumption was that the permanent strain would increase asymptotically in the direction of a particular point defined as a function of stresses related to the static failure condition of the material as in Equation (3)  
\[\varepsilon_{a}^{p,\ 100}=a\cdot\biggl{(}1-\biggl{(}\frac{N}{100}\biggr{)}^{-b} \biggr{)}, \tag{3}\]  
where \(\varepsilon_{a}^{p,100}\) is the permanent strain value at cycle \(N=100\). One of the widely known equations is the [PERSON] and [PERSON] [20] model by Equation (4):  
\[\frac{\varepsilon_{a}^{p}\bigl{(}N_{ref}\bigr{)}}{\frac{L}{P_{0}}}=a\cdot \biggl{(}\frac{q}{p}\biggr{)}_{max}^{b}, \tag{4}\]  
This model relates the permanent strain to the \(\Bigl{(}\frac{q}{p}\Bigr{)}_{max}\) relationship. The idea of such a solution comes from the observation that the combined effect of a maximum stress potential is transferred to the effort to reach this maximum stress potential done by loaded soil [21]. In this equation, \(\varepsilon_{a}^{p}\bigl{(}N_{ref}\bigr{)}\) is accumulated permanent axial strain after \(N_{ref}\) number of cycles where the \(N_{ref}\) is the number of cycles higher than 100. The \(L\) parameter stands for the length of the stress path in Equation (5):  
\[L=\sqrt{\bigl{(}q_{max}-q_{min}\bigr{)}^{2}+\bigl{(}p_{max}-p_{min}\bigr{)}^{ 2}}=\sqrt{\bigl{(}2\cdot q_{a}\bigr{)}^{2}+\bigl{(}2\cdot p_{a}\bigr{)}^{2}}, \tag{5}\]  
where \(p_{0}\) is the reference stress (1 kPa) and \(\Bigl{(}\frac{q}{p}\Bigr{)}_{max}\) is the maximum shear to normal stress ratio. The permanent strain accumulation models were also developed for unbound granular materials. One of the most well-known models is [PERSON] et al. [22]'s proposition, which was derived from tests performed on two types of granular material. The tests were conducted for different stress paths, the ration of which was between 0 and 3. The [PERSON] et al.