# Deformation Behavior of Saturated Soft Clay under Cyclic Loading with Principal Stress Rotation
## 4 Test Results and Discussion
### Backbone Curve  
The backbone curve is the line at the top of the stress-strain curve for each loop of the soil subjected to different dynamic stress amplitudes. It represents the relationship between the maximum dynamic stress and strain, as shown in Figure 12. Under cyclic loading, the  
Figure 11: Comparison of typical stress–strain hysteresis curves at different cycles. Figure 10: Axial dynamic strain versus number of cycles under different cyclic loading conditions: (a) \(\eta\) = 0.00; (b) \(\eta\) = 0.25; (c) \(\eta\) = 0.50.  
hysteresis loop is elongated, and the soil exhibits stiffness degradation. Therefore, there is no unique method for determining the backbone curve. The backbone curve is usually determined by taking the hysteresis loop data under the same number of loadings at each dynamic stress level. [PERSON] et al. [25] analyzed the hysteresis loops for different numbers of loops under the same cycle test. They proposed a dynamic backbone model that includes the dynamic stress amplitude, consolidation confining pressure, cycle number, and other influencing factors. [PERSON] et al. [24] shifted the center of the hysteresis loops at a certain stress amplitude to the origin to eliminate the accumulated residual strain and then used the average value of all hysteresis loops (10 loops in total) to fit the backbone curve model. The backbone curve model used by the above authors is the modified Hardin-Drnevich model, which is often used to construct backbone curves for soft clay, i.e., the modification of the parameters of the H-D model. The H-D model is usually expressed as a hyperbolic type, that is,  
\[\sigma_{\mathrm{z}}^{\mathrm{cyc}}=\frac{\varepsilon_{x,\mathrm{r}}}{a+b \varepsilon_{x,\mathrm{r}}} \tag{10}\]  
where \(\sigma_{\mathrm{z}}^{\mathrm{cyc}}\) and \(\varepsilon_{x,\mathrm{r}}\) denote the dynamic stress amplitude and resilient strain (\(\varepsilon_{x,\mathrm{r}}\) is the recoverable portion of axial strain, i.e., the difference between the maximum strain and the strain at the end of the specific loading cycle), respectively, and \(a\) and \(b\) are test-related parameters, where \(a\) is the reciprocal of the maximum resilient modulus \(\frac{1}{E_{\mathrm{max}}}\) and \(b\) is the reciprocal of the ultimate dynamic stress of the soil \(\frac{1}{\sigma_{x,\mathrm{max}}}\)[22], as shown in Figure 12.  
Figure 13 shows the backbone curve model for different cycles with different degrees of principal stress rotation. The backbone curve has a typical hyperbolic type. As the number of cycles increases, the backbone curve shifts to the left, the initial slope decreases, and the backbone curve pattern stabilizes. This indicates that the stiffness of the soft clay decreases with the number of cycles. After the backbone curve shape has gradually stabilized, it is found that with the increase in the principal stress rotation, the backbone curve lowers. This indicates that the effect of the principal stress rotation significantly weakens the dynamic properties of the soft clay. Equation (10) is used to fit the curve. The results are in good agreement with the original test data. The values of the backbone curve equations for the different number of cycles with different torsional cyclic stress ratios \(\eta\) are shown in Table 4.  
Figure 12: Schematic diagram of stress–strain backbone curve. Figure 13: Backbone curves with different \(\eta\) under different cycles: (**a**) \(\eta\) = 0.00; (**b**) \(\eta\) = 0.25; (**c**) \(\eta\) = 0.50.