## 4 Results
### Deformability Characteristics
Figure 11: Total degradation of Young modulus in cyclic loading \(E_{cyc}\) versus axial strain (**a**), and versus total strain normalization parameter \(T_{x}\) (**b**), the relation between the \(a_{TX}\) parameter and the normalized cyclic Young modulus \(\mathrm{E_{cyc}}\)/\(\mathrm{E_{max}}\). (**c**) the impact of the normalized Young modulus and cyclic stress ratio (CSR) on the \(a_{TX,1000}\) parameter value. The relationship between \(a_{TX}\) parameter has to depend on the stress parameter, which can be represented by CSR, and on initial stiffness of the sample during cyclic loading, which is represented by the normalized young modulus \(E_{cyc,1}/E_{max}\) in the first cycle presented in Equation (24). The relationship between \(a_{TX}\) and \(b_{TX}\) parameters is represented by Equation (25):  
\[\delta=\frac{G_{SN}}{G_{S1}}=\frac{\frac{\delta_{u}}{\epsilon_{a}^{pN}}}{ \frac{\delta_{u}}{\epsilon_{a}^{pN}}}=\frac{\epsilon_{a}^{p,1}}{\epsilon_{a} ^{p,1}}[-], \tag{20}\]0  
\[\delta=\frac{G_{SN}}{G_{S1}}=\frac{\frac{\delta_{u}}{\epsilon_{a}^{pN}}}{ \frac{\delta_{u}}{\epsilon_{a}^{pN}}}=\frac{\epsilon_{a}^{p,1}}{\epsilon_{a} ^{p,1}}[-], \tag{20}\]1  
where \(\alpha\), \(\beta\) and \(\gamma\) are constants equal to 2.079599896, 0.100418643, 0.003033555, respectively. For Equation (24), the coefficient of determination is equal to 0.996, and for Equation (25), R\({}^{2}\) is equal to 0.999. Based on these relationships, the value of \(T_{x,1000}\) can be calculated in the desired cycle. From the relationships presented in Figure 11b, we know that in Equation (26):  
\[\delta=\frac{G_{SN}}{G_{S1}}=\frac{\frac{\delta_{u}}{\epsilon_{a}^{pN}}}{ \frac{\delta_{u}}{\epsilon_{a}^{pN}}}=\frac{\epsilon_{a}^{p,1}}{\epsilon_{a} ^{p,1}}[-], \tag{20}\]2  
the \(T_{x,1}\) can be calculated based on the relationship between the \(b_{TX,1}\) and \(b_{TX,1000}\), which seems to be constant despite the cycle reference, so we made the assumption that \(b_{TX,1}=b_{TX,1000}\). Therefore, for T\({}_{\rm x}\) calculated from cycle 1, the \(a_{TX,1}\) is equal to 1.0 and finally, the relationship becomes as presented in Equation (27):  
\[\delta=\frac{G_{SN}}{G_{S1}}=\frac{\frac{\delta_{u}}{\epsilon_{a}^{pN}}}{ \frac{\delta_{u}}{\epsilon_{a}^{pN}}}=\frac{\epsilon_{a}^{p,1}}{\epsilon_{a} ^{p,1}}[-], \tag{20}\]3  
Since relationship presented in Equation (27) describes the degradation of Young modulus during cyclic loading, the total strain can be calculated based on the \(E_{cyc}\) parameter and \(q_{max}\) value as in Equation (28):  
\[\delta=\frac{G_{SN}}{G_{S1}}=\frac{\frac{\delta_{u}}{\epsilon_{a}^{pN}}}{ \frac{\delta_{u}}{\epsilon_{a}^{pN}}}=\frac{\epsilon_{a}^{p,1}}{\epsilon_{a} ^{p,1}}[-], \tag{20}\]4  
The plastic strain in the following cycles of loading can be calculated based on the abation parameter A\({}_{\rm x}\), of which Figure 12a presents the characteristics. The presented relationship shows the \(A_{x}\) parameter change versus the change in cyclic loading Young modulus which shows that the soil abation parameter will rise during the test with a decrease in the \(E_{cyc}\) value. This characteristic shows that the abation parameter achieves the value of 1.0 when the \(E_{cyc}\) will be equal to 0.0. This dependence can be modeled with the following Equation (29):  
\[\delta=\frac{G_{SN}}{G_{S1}}=\frac{\frac{\delta_{u}}{\epsilon_{a}^{pN}}}{ \frac{\delta_{u}}{\epsilon_{a}^{pN}}}=\frac{\epsilon_{a}^{p,1}}{\epsilon_{a} ^{p,1}}[-], \tag{20}\]5  
The \(\alpha_{AX}\) constant for each test is the modulus of the function \(A_{x}-E_{cyc}\). The value of each modulus was a plot against the initial total strain in the first cycle (Figure 12b). The \(\alpha_{AX}\) modulus value was set to \(-0.00001\) for three tests where no clear tendency of \(A_{x}\) and no significant value change was observed.