# Shear Strength of Sand: Integrated Analysis of Initial Porosity and Stress Effects
## 2 Materials and Methods
### Test Procedure  
The tests were carried out using a direct shear apparatus (Figure 3) on 21 samples with different densities, from loose to compacted. The vertical load applied was \(\sigma\in\{4.87,9.81,14.75,50.00,100.00,200.00\}\) kPa. A compaction index of \(e_{\max}=0.753\) was  
Figure 1: Grain size distribution of soil sample. Figure 2: Microscopic images of fractionated sand sample. Zoom 43\(\times\) for \(d\geq 0.2\) mm and 162\(\times\) for \(d<0.2\) mm. determined for the loose sample, and \(e_{\text{min}}=0.432\) for the most compacted sample. Table 1 shows the results of the physical tests on the sand. Figure 3 presents a schematic and photographs of the apparatus used for the experiments. The testing procedure is described below, followed by a discussion of the results summarised in Table 1.  
Each soil specimen was prepared by placing the material into a shear box with internal dimensions of 8 cm \(\times\) 8 cm and a height of 4.55 cm, followed by layer-by-layer compaction. The prepared sample was then mounted in a direct shear apparatus. The vertical load was applied using a gravitational loading system consisting of calibrated weights. After configuring the measurement system, including strain gauge and optoelectronic displacement sensors, the shearing process was initiated at a constant displacement rate of \(v=1\) (mm/min) applied to the upper part of the shear box. The verified consolidation conditions enabled the selection of a sliding speed that prevented the generation of excess pore water pressure during shearing, thereby maintaining drained conditions  
\begin{table}
\begin{tabular}{c c c c c c c} \hline
**No. ** & **nr** & \(\sigma\) **[kPa]** & \(e_{0}\)**[- ]** & \(\tau_{p}\) **[kPa]** & \(\tau_{cv}\) **[kPa]** & \(e_{cv}\) **[- ]** \\ \hline
1 & ln-1-A & 4.87 & 0.708 & 5.06 & 5.06 & 0.629 \\
2 & ln-1-B & 9.81 & 0.657 & 9.18 & 9.18 & 0.639 \\
3 & ln-1-C & 14.75 & 0.675 & 12.05 & 12.05 & 0.639 \\
4 & zg-1-A & 4.87 & 0.472 & 10.44 & 6.2 & 0.602 \\
5 & zg-1-B & 9.81 & 0.494 & 16.21 & 9.23 & 0.54 \\
6 & zg-1-C & 14.75 & 0.484 & 21.12 & 12.99 & 0.553 \\
7 & szg-1-A & 4.87 & 0.486 & 8.22 & 5.62 & 0.636 \\
8 & szg-1-B & 9.81 & 0.522 & 12.69 & 9.18 & 0.623 \\
9 & szg-  
throughout the test. The test involved the continuous monitoring of shear force, horizontal displacement, and vertical deformation. Upon completion of the test, the mass of the soil sample was measured to determine the initial porosity index. The critical porosity index was subsequently calculated based on the initial porosity and the volumetric change inferred from the final specimen height. The change in volume was determined from the change in height of the specimen represented by the stabilised value of the specimen height after reaching the critical shear strength from the graph \(u_{y}=f(u_{x})\) shown in Figure 4. The remaining results are presented in Appendix A in the form of plots. Based on the data obtained from these plots, Table 1 was prepared. Figure 4 presents the results for the loose and compacted sample. The graph shows how the shear stress \(\tau\) (kPa) increased during the test with the displacement of the upper part of the box \(u_{x}\) (mm). At the same time, the vertical displacement of the piston \(u_{y}\) (mm) was plotted against the horizontal displacement of the box \(u_{x}\) (mm). The test was carried out at a sliding speed of the upper part of the box \(v=1\) (mm/min). The influence of the initial compaction of the sample was very important for the generation of shear stresses. Two completely different patterns of shear stress mobilisation were shown as an example. In the loose sample (Figure 4a), the shear stress increased slowly until it reached its maximum value. At the same time, a decrease in the height of the sample was observed. This behaviour was typical for a soil sample in a contractive state. In the compacted sample (Figure 4b), the shear stresses increased rapidly until the peak value was reached, after which they decreased until they stabilised at the critical value.