# Strength and Compressibility of Ammonia-Soda Residue from the Solvay Sodium Plant
## 3 Results and Discussion
### Drained Triaxial Tests
#### 3.3.3 Stress Paths  
The stress paths for two ASR are shown in Figure 6. The established failure and critical state lines are presented in the plots, defined by a line in the q-p\({}^{\prime}\) plane, with a slope of M. The indices p and cs denote failure (peak strength) and the critical state, respectively. In particular, the critical state concept is the idea that soil and other granular materials, if continuously distorted until they flow as a frictional fluid, will come into a well-defined critical state [51].  
Figure 5: Typical failure modes of samples observed from the present study.  
Figure 6: Effective stress paths in the drained triaxial compression tests with respect to the peak deviator stress and the critical state: (**a**) ASR I, (**b**) ASR II.  
The slope of the critical state line (CSL) is commonly called the critical state frictional constant M or critical state ratio and should be treated as a function of the intermediate principal stress, expressed by the lode angle (\(\theta\)) [52]. Parameter M defines the critical state stress ratio at failure conditions and is comparable to the critical friction angle (\(\varphi^{\prime}{}_{\text{cs}}\)) for the Mohr-Coulomb failure line using the following expression:  
\[M=\frac{q}{p^{\prime}}=\frac{6 sin\varphi^{\prime}_{\text{cs}}}{3-sin\varphi^{ \prime}_{\text{cs}}} \tag{3}\]  
Note that the correlation between M and \(\varphi^{\prime}{}_{\text{cs}}\) is not unique but depends on the stress conditions [53], and Equation (3) is valid only for the compression case (i.e., when \(\sigma^{\prime}{}_{\text{a}}>\)\(\sigma^{\prime}{}_{\text{r}}\)). Figure 6 shows the stress paths for two ASRs. For the ASR I, the peak points of the stress path in the drained condition were lower than for ASR II over the whole range of confining pressures. The slope of critical state lines in the q-p\({}^{\prime}\) plane is M\({}_{\text{cs}}\) = 1.56 and M\({}_{\text{f}}\) = 1.36 for ASR I. In turn, the values of M\({}_{\text{cs}}\) and M\({}_{\text{f}}\) for ASR II are 1.40 and 1.46, respectively. Overall, it can be seen that the data points drawn in Figure 6 can accurately be fitted by a linear function (one for failure and one for the critical state). In this regard, the failure locus and the critical state locus are independent of the drainage conditions.  
Figure 7 shows Mohr stress circles drawn for the peak deviator stress or for the axial strain of 25% as obtained from the results of CD tests on ASR I and II. Note that the solid lines indicate the critical state, and the dashed lines the failure condition corresponding to the peak shear strength. These lines are tangent to the Mohr stress circles over the entire range of applied confining pressures (Figure 7). The values of effective cohesion, effective friction angles at peak \(\varphi^{\prime}{}_{\text{p}}\), and critical friction angles \(\varphi^{\prime}{}_{\text{cs}}\) for ASR I and ASR II (Figure 7) were comparable with the values for alluvial clayy silts [54] and many soft alluvial deltaic clays [55].  
Figure 7: Mohr-Coulomb failure envelope based on the maximum deviator stress and the critical state from the drained triaxial tests: (**a**) ASR I, (**b**) ASR II.  
Moreover, the values of the effective friction angles from the present study were consistent with the reported numbers by [PERSON] [56] for biosolids, sewage sludge, and water treatment residue materials. Considering the stability calculations for the intermediate and long-term conditions, the derived values are far too high for the design purposes.