## 3 Results and Discussions
### Initial Stiffness and Ductility  
Stiffness and ductility are important mechanical properties in the constitutive behaviors of sand. Stiffness reflects the deformation resistance of soil under external loads. Since soil is a type of discontinuous granular material, its stiffness is related to the stress level. In this section, the tangent stiffness of sand specimens at the displacement of 0.1 mm was calculated, which can be regarded as the initial stiffness. The data points of initial stiffness of sand specimens under normal pressure of 200 kPa are plotted in Figure 7. It was found that a fiber content of 0.04% did not obviously influence the initial stiffness of sand specimens. For the sand specimens with a higher fiber content (0.2% and 0.4%), the initial stiffness decreased with the increase in initial orientation angle, which was consistent with the change rule of peak strength mentioned above. However, there was a negative correlation between the initial stiffness and fiber content, and the influences of fiber content on the initial stiffness were obviously larger than those of fiber orientation. Two possible reasons may have contributed to this. First, the density of the sand close to fibers may have been slightly lower than that of the sand far away from fibers due to the high difficulty of compaction in specimen preparation. The initial stiffness may have been sensitive to the density inhomogeneity of specimens. Ref. [7] proposed another possible reason for the stiffness loss. Upon application of the vertical normal pressure, the fiber strips may be pre-compressed because of their low stiffness modulus. In this situation, at the beginning of the shearing process, sufficient shear distortion has to occur to overcome the pre-compression of fibers. As a result, the initial stiffness of reinforced sand specimens may be lower than that of unreinforced sand specimens of the same density.  
Figure 6: Vertical displacement of sand specimens with fibers initially orientated at 60\({}^{\circ}\) under normal pressures of (**a**) 400 and (**b**) 800 kPa.  
Ductility reflects the deformation that can be sustained by soil from yield to a failure state. To intuitively illustrate the influences of fiber strips on sand ductility in this study, the failure displacements of the sand specimens, i.e., the displacement of the peak strength point, are shown in Figure 8 versus the fiber content. It can be observed that the failure displacement increased with fiber content, indicating that the introduction of fiber strips could improve the ductility of dense sand in this study. For the sand specimens with the same fiber content, the fibers initially orientated at 60\({}^{\circ}\) provided the largest ductility increments while the fibers initially orientated at 120\({}^{\circ}\) provided the smallest. Fiber strips may influence the ductility of dense sand via the same mechanism as residual strength. As discussed above, fibers can restrain the movement among sand particles and then reduce the volume dilation. Therefore, compared with unreinforced sand, the reinforced sand specimens may need larger macroscopic displacement in the direct shear tests for the degree of microscopic deformation of their internal structures to reach the failure degree.  
Figure 8: Failure displacement versus fiber content of sand specimens under normal pressures of (**a**) 100, (**b**) 200, (**c**) 400, and (**d**) 800 kPa.  
Figure 7: Initial stiffness of sand specimens under normal pressure of 200 kPa.