## 3 Test Results and Analysis
### Effect of CSR on the Development of Cumulative Generalized Shear Strain  
To predict the rate of cumulative strain, Ref. [30] developed an empirical model that is given in Equation (2), as follows:  
\[\gamma_{\rm g}=aN^{b} \tag{2}\]  
where \(a\) and \(b\) are the stress-specific fitting parameters. Figure 7 presents the test data points and the best-fit line of the Monismith model; in view of the \(\gamma_{\rm g}\) variations within a loading cycle, showing an absolute sine form wave-like growth, the value of \(\gamma_{\rm g}\) within the loading cycle shown in Figure 6 is the maximum value within a loading cycle. As illustrated in Figure 7, the Monismith model can appropriately describe the development law of \(\gamma_{\rm g}\). Under various stress paths, \(\gamma_{\rm g}\) increases with increasing \(N\), and \(CSR\) and \(\delta\) have significant effects on the rate of increase. For the circular stress path, when \(CSR\) = 0.03, \(\gamma_{\rm g}\) increases almost linearly with increasing \(N\). When \(CSR\) = 0.04 and 0.05, the growth rate of \(\gamma_{\rm g}\) decreases with the increase in \(N\). For elliptical stress paths, when \(CSR\) = 0.03, 0.05, and 0.065, \(\gamma_{\rm g}\) increases almost linearly with increasing \(N\). When \(CSR\) = 0.08 and 0.05, the growth rate of \(\gamma_{\rm g}\) gradually slows down with the increase in \(N\). It is observed that \(\gamma_{\rm g}\) increases with increasing \(CSR\), when \(\delta\) is the same. Of the many criteria used in the undrained dynamic tests, to determine when sand liquefaction occurs, the most commonly used is initial liquefaction, which is the state of the Ru at 100% [31], and the more widely used 2-3% SA or 5% DA axial strain [32]. However, in the axial-torsion coupling cyclic loading test, the sample will not only produce \(\gamma\), but also \(\varepsilon_{\rm z}\) and \(\varepsilon_{\rm r}\). It is somewhat one-sided to use the development of a single strain parameter as the criterion for liquefaction. Therefore, this study uses the generalized shear strain \(\gamma_{\rm g}\) to describe the deformation of the soil, and \"\(\gamma_{\rm g}\) = 5%\" is used as the criterion for the liquefaction failure of the specimen in this article. Figure 8 presents the relationship between \(\gamma_{\rm g}\) and \(N\) under various stress paths.  
Figure 6: Typical test results of saturated silt sample under cyclic axial–torsional combined loading. (**a**) The relationship between excess pore water pressure and number of cycles; (**b**) The relationship between axial strain and number of cycles; (**c**) The relationship between shear strain and number of cycles; (**d**) The relationship between radial strain and number of cycles.  
Note that the development of \(\gamma_{g}\) is different under different combinations of \(CSR\) and \(\delta\). According to Figure 8, when \(CSR\leq 0.05\), regardless of whether \(\delta=1\) or \(2\), \(\gamma_{g}\) is less than \(0.2\%\), the plastic deformation of the sample is very small, and \(\gamma_{g}\) does not meet the cyclic failure. When \(CSR=0.065\), \(\delta=1\) and \(N>460\), \(\gamma_{g}\) develops rapidly, and \(\gamma_{g}\) reaches \(5\%\) for \(N=478\). The critical cycle stress ratio \(CSR_{\text{th}}\) is defined as the minimum \(CSR\) required for the specimen to reach liquefaction failure. When \(\delta=1\), \(0.05<CSR_{\text{th}}\leq 0.065\); when \(\delta=2\), \(0.065<CSR_{\text{th}}\leq 0.08\); and when \(\delta=4\), no liquefaction occurs in the silt samples.