# Shear Strength of Sand: Integrated Analysis of Initial Porosity and Stress Effects
## 3 Results
### Critical and Peak Friction Angles in Relation to Stress Range and Initial Porosity Index  
Considering Formula (2) within Formula (3), the critical friction angle depended only on the stress. This phenomenon was also observed by the authors of [24; 26]. This relationship is presented in Figure 9 by the sky-blue dashed line. The peak friction angle depended on both the critical friction angle and initial porosity index. Therefore, following Formulas (1), (3), and (5), the peak friction angle depended on normal stress in the soil sample and initial porosity index, as is presented in Figure 9 through dashed lines, respectively. As Figure 9 shows, both the critical and peak angles of internal friction decreased as the normal stress in the sand sample tested increased. The change was most pronounced at low stresses up to 50 kPa, where the critical soil friction angle decreased from a value close to 50 degrees to 36 degrees. A further increase in normal stress caused a less significant decrease in the critical angle of friction to 34 degrees at a stress of 300 kPa. The lower the initial porosity index, the higher the peak angle of internal friction. Figure 9 is a synthetic summary of the tests carried out, illustrating the effect of the initial soil porosity index and the normal stress acting on the shear plane on the value of the peak friction angle. An analysis of the graph indicates that both the critical and peak friction angles are not constant quantities but show a dependence on the level of normal stress. One of the key conclusions from the interpretation of the data presented is that  
Figure 8: The difference between the peak and critical friction angles versus dilation angle on the investigated sand. assuming a constant value for the internal friction angle in advanced numerical analyses can lead to significant computational errors, especially in areas with low stress levels. Figure 10 presents the peak shear strength and peak friction angle versus normal stress of sand with an initial porosity index of \(\epsilon_{0}=0.54\). The difference in the peak angle of internal friction of a medium-dense sand sample, measured under stress levels commonly used in engineering research, i.e., from 50 to 300 kPa, was 40 and 36 degrees, respectively. In standard direct shear tests, a friction angle was determined for multiple points (e.g., 4). This gave an average value of 35 degrees over the stress range studied and an additional cohesion of 8 kPa, as shown in Figure 9(b). In the case of low stress levels, as presented in Figure 9(a), the peak internal friction angle was higher and the cohesion was lower, which confirms the nonlinear relationship between soil shear strength and normal stress. This observation is further supported by the red curve, which illustrates the variation in the peak internal friction angle. Figure 11 presents the approach related to the critical internal friction angle, revealing a similar trend, albeit with lower values. The cohesion obtained with this approach is due to the use of the linear Coulomb-Mohr function for the nonlinear analysis of the peak shear strength envelope with respect to normal stress. Consequently, the cohesion obtained in  
Figure 10: Shear strength of soil and friction angle in relation to normal stress in soil sample \(e_{0}=0.54\). Peak shear strength and friction angle determined (**a**) under low normal stress conditions and (**b**) under high normal stress conditions. Figure 9: Peak and critical friction angles versus normal stress in soil sample and different initial porosity index values. this way was the result of a mathematical approximation and not a physical soil parameter. In soil shear tests, it was observed that when approximating the results in the low stress range (\(5\div 20\) kPa), the friction angle was higher than when approximating the shear results of the same soil (Figures 9(a) and 9(a)) in the higher stress range (\(50\div 300\) kPa) (Figures 9(b) and 9(b)).