## 4 Results
### Model Validation  
The presented results of the test show that the deformability of the soil can be described in terms of the equations derived from the previous considerations. We use undrained cyclic triaxial tests conducted on the samples presented in this study to derive model quality with _ex-post_ evaluation. For calculations, the required data include the information about CSR, maximum shear modulus \(E_{max}\) and total strain in the first cycle caused by cyclic loading \(\varepsilon_{a}^{T,1}\). The CSR is calculated here as the ratio of the maximal deviator stress \(q_{max}\) and the initial effective confining pressure \(\sigma^{{}^{\prime}}_{3}\). In total, four parameters are required to calculate the soil deformability in such conditions. Notably, the presented model uses the parameters which can be derived from the static triaxial tests. Figure 13 presents the results of the cyclic triaxial test and the results of the total and plastic strain value calculations versus the number of cycles. We evaluated the SHLM quality of strain forecasting with the use of _ex-post_ methods. In order to do so, we used cycle 10,000 as a reference. In Table 3, the data concerning the value of strain in the 10,000 th cycle are presented. The _ex-post_ analysis answers the question of how well the model would perform if it was used before the tests. This approach helps us to evaluate model reliability. Based on the calculations performed in Table 3 the SHLM percentage error was between 0.0% and 51.72%. These values concern the 10,000 cycle. For the error of the whole model, the mean error (ME), mean percentage error (MPE), mean absolute error (MAE), mean absolute percentage error (MAPE) and Theil index values are presented in Table 4.  
Figure 12: The ablation parameter characteristics: **(a)** the ablation parameter \(A_{x}\) relationship to cyclic Young modulus \(E_{cyc}\) (**b**) and the relation between the ablation constant and total strain in the first cycle of loading (**b**). Figure 13: _Cont._ \begin{table}
\begin{tabular}{c c c c c} \hline \hline \(\epsilon_{a}\)\({}^{T,10000}\) **Test (-)** & \(\epsilon_{a}\)\({}^{T,10000}\) **Prognosis (-)** & **Sample No.** & **Error (-)** & **Procentage Error (\%)** \\ \hline
0.00134 & 0.00139 & 1.1 & \(-\)0.00005 & \(-\)4.06\% \\
0.00687 & 0.00705 & 1.2 & \(-\)0.00018 & \(-\)2.62\% \\
0.00844 & 0.00407 & 1.3 & 0.00436 & 51.72\% \\
0.00857 & 0.00889 & 1.4 & \(-\)0.00032 & \(-\)3.71\% \\
0.00544 & 0.00532 & 1.5 & 0.00012 & 2.21\% \\
0.00368 & 0.00335 & 1.6 & 0.00033 & 9.05\% \\
0.00659 & 0.00364 & 1.9 & 0.00296 & 44.83\% \\
0.00316 & 0.00336 & 1.10 & \(-\)0.00020 & 0.00\% \\ \hline \hline \end{tabular} Error is the difference between the test result and prognosis, Error = (\(\epsilon_{a}\)\({}^{T,10000}\) test) - (\(\epsilon_{a}\)\({}^{T,10000}\) prognosis).  
\end{table} 
Table 3: _Ex-post_ analysis of the 10,000 th cycle: error and percentage error calculation results. The ME and MPE show that the SHLM will deliver the strain value with 12.18% error and this value should be accounted as the standard error of model prognosis. The MAE gives information about the absolute error after 10,000 cycles. The conclusion can be made that for every 10,000 cycles, the value prognoses might be different according to the MAE value. Finally, we calculated the Theil parameter \(\Gamma^{2}\) in order to estimate the relative error of the model results. The Theil parameter can be decomposed into three categories, \(\rm{I_{1}}^{2}\), \(\rm{I_{2}}^{2}\) and \(\rm{I_{3}}^{2}\). The first parameter shows how well the model delivers the value \begin{table}
\begin{tabular}{c c} \hline
**Index** & **Value** \\ \hline ME & 0.00087 \\ MPE & 12.18\% \\ MAE & 0.001066 \\ MAPE & 15.57\% \\ \(\rm{I^{2}}\) & 0.0976 \\ \(\rm{I_{1}}^{2}\) & 21.41\% \\ \(\rm{I_{2}}^{2}\) & 1.23\% \\ \(\rm{I_{3}}^{2}\) & 76.83\% \\ \hline \end{tabular}
\end{table} 
Table 4: Ex-post analysis results for the Simple Hysteresis Loop Model (SHLM) error calculations and Theil parameter values.