## Interpretation of Results
### Cyclic and Permanent Strain
#### 5.4.2 Qualitative Evaluation
* For a constant value of \(\tau_{\text{c}}\)/\(\sigma^{\prime}_{\text{ov}}\) and a different \(\tau_{\text{o}}\)/\(\sigma^{\prime}_{\text{ov}}\), the symmetry of the loops with respect to the origin of the vertical axis is broken; this occurs to the extent that \(\tau_{\text{o}}\)/\(\sigma^{\prime}_{\text{ov}}\) increases. * For a constant value of \(\tau_{\text{c}}\)/\(\sigma^{\prime}_{\text{ov}}\) and \(\tau_{\text{o}}\)/\(\sigma^{\prime}_{\text{ov}}+\tau_{\text{c}}\)/\(\sigma^{\prime}_{\text{ov}}\), and a different \(\tau_{\text{c}}\)/\(\sigma^{\prime}_{\text{ov}}\), the relationship between permanent and cyclic strains increased significantly as \(\tau_{\text{o}}\)/\(\sigma^{\prime}_{\text{ov}}\) increased. For example, for \(\tau_{\text{o}}\)/\(\sigma^{\prime}_{\text{ov}}\) = 5% and \(\tau_{\text{c}}\)/\(\sigma^{\prime}_{\text{ov}}\) = 25%, \(\gamma_{\text{P}}\)/\(\gamma_{\text{c}}\) = 0.57; while, for \(\tau_{\text{o}}\)/\(\sigma^{\prime}_{\text{ov}}\) = 25%  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Sample** & \(\sigma^{\prime}{}_{\rm ov}\) & \(\tau_{\rm c}/\sigma^{\prime}{}_{\rm ov}\) & \(\gamma_{\rm o}\) & \(\gamma_{\rm p}\) & \(\gamma_{\rm c}\) & **N** \\ \hline
**Ident. ** & **kPa** & **\%** & **\%** & **\%** & **\%** & **Cycles** \\ \hline S1A-1 & 277 & 10 & 0.0 & 0.4 & 0.7 & 1310 \\ S1A-1 & 277 & 15 & 0.0 & \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Sample** & \(\sigma^{\prime}_{\text{ov}}\) & \(\tau_{\text{c}}/\sigma^{\prime}_{\text{ov}}\) & \(\gamma_{\text{o}}\) & \(\gamma_{\text{P}}\) & \(\gamma_{\text{c}}\) & **N** \\ \hline
**Ident.** & **kPa** & \% & \% & \% & \% & **Cycles** \\ \hline S1A-1 & 277 & 10 & 1.1 & 2.6 & 2.4 & 1300 \\ S1A-1 & 277 & 15 & 1.3 & 18.0 & 3.5 & 18 \\ S1A-6 & 349 & & 0.4 & 6.3 & 1.8 & 1300 \\ S1A-10 & 413 & & 1.1 & 17.8 & 6.5 & 87 \\ S2A-1 & 294 & & 1.2 & 15.3 & 5.0 & 96 \\ S2A-4 & 343 & & 1.7 & 17.2 & 7.0 & 139 \\ S2A-5 & 347 & & 0.9 & 14.9 & 5.5 & 686 \\ S2A-6 & 364 & & 0.9 & 11.2 & 2.4 & 120 \\ S2A-7 & 373 & & 0.9 & 16.6 & 5.7 & 251 \\ S2A-8 & 384 & & 1.2 & 16.3 & 4.8 & 367 \\ S1A-4 & 311 & 20 & 1.0 & 15.5 & 8.3 & 11 \\ S1A-6 & 349 & & 0.8 & 16.8 & 8.9 & 31 \\ S1A-7 & 366 & & 1.2 & 20.4 & 10.9 & 11 \\ S1A-10 & 413 & & 1.4 & 15.5 & 7.7 & 28 \\ S2A-1 & 294 & & 1.0 & 14.6 & 6.1 & 39 \\ S2A-2 & 315 & & 1.3 & 11.9 & 6.4 & 6 \\ S2A-3 & 328 & & 1.1 & 11.5 & 6.6 & 20 \\ S2A-4 & 343 & & 1.5 & 15.8 & 7.9 & 5 \\ S2A-6 & 364 & & 0.9 & 8.9 & 4.6 & 12 \\ S2A-7 & 373 & & 1.2 & 12.7 & 10.8 & 31 \\ S2A-8 & 384 & & 0.7 & 15.6 & 7.6 & 42 \\ S2A-9 & 401 & & 1.4 & 18.2 & 7.9 & 10 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Results of cyclic simple shear test with \(\tau_{\text{o}}/\sigma^{\prime}_{\text{ov}}=10\%\). \begin{table}
\begin{tabular}{l c c c c c c} \hline \hline
**Sample** & \(\sigma^{\prime}_{\text{ov}}\) & \(\tau_{\text{c}}\)/\(\sigma^{\prime}_{\text{ov}}\) & \(\gamma_{\text{o}}\) & \(\gamma_{\text{P}}\) & \(\gamma_{\text{c}}\) & **N** \\ \hline
**Ident. ** & **kPa** & \% & \% & \% & \% & **Cycles** \\ \hline S1A-1 & 277 & 10 & 1.5 & 17.4 & 1.1 & 1000 \\ S1A-6 & 349 & & 1.3 & 8.0 & 0.5 & 1300 \\ S1A-10 & 413 & & 1.7 & 10.5 & 0.6 &  
Figure 10: Variation of the cyclic shear strain as a function of the number of cycles and of the permanent shear strain corresponding to \(\tau_{0}/\sigma^{\prime}{}_{\rm ov}=0\). Figure 11 was obtained by plotting vertical sections in Figure 10 through abscissa N = 1, 10, 100, and 1000. For modeling purposes, Figure 11 shows the cyclic and permanent strains for each number of cycles (N) and can be approximately correlated by means of a power function type (4). \[\gamma_{\rm c}=\rm A\left(\gamma_{\rm p}\right)^{\rm B} \tag{4}\]  
where: \(\gamma_{\rm c}\), cyclic strain (%), \(\gamma_{\rm p}\), permanent strain (%). The constants A and B can be expressed in terms of N by a power function (5) and a logarithmic function (6), as shown in Figure 12.  
\[\rm A=D\cdot N^{\rm E} \tag{5}\]  
\[\rm B=F\cdot ln(N)+H \tag{6}\]  
where D, E, F, and H are empirical constants.