## 3 Results and Discussion  
### Modulus and Phase Angle Are Affected by Frequency  
The dynamic shear modulus \(\mid G^{*}\mid\) of the fiber-modified asphalt mastic was found to exhibit a positive correlation with the loading frequency, as shown in Figure 3. The logarithmic value of \(\mid G^{*}\mid\) and the loading frequency were discovered to be in a logarithmic relationship. This is explained by the reduced contact time between the applied stress and the asphalt material at higher frequencies. Additionally, asphalt materials are viscoelastic by nature and can deform in three ways: elastic, viscoelastic, and vicious. As a consequence, the modulus of asphalt material rises as load action frequency rises. This is because at higher frequencies, the load action time is shorter, resulting in smaller deformation of the asphalt material and thus an increase in modulus. Conversely, at lower frequencies, the longer load action time leads to increased deformation and a decrease in modulus. Figure 6 shows that the dynamic modulus of the modified asphalt mixture steadily increased with the addition of basalt and bamboo fibers, a finding that is consistent with the reinforcing effect of fibers mentioned by [PERSON] et al. in their paper, a phenomenon that can be explained by the fact that natural fibers can increase the modulus of asphalt mastic by acting as reinforcement and adhesion [14]. The chemical adsorption and attachment of the fibers to the asphalt simultaneously reduces the amount of free asphalt and increases the amount of structural asphalt, increasing the stability of the asphalt mastic.  
Figure 5: Time–strain results model based on ER test.  
The phase angle is an important parameter for characterizing the viscoelastic behavior of a material. It represents the time difference between the stress and strain waveforms when a material is subjected to a load. The tangent of the phase angle reflects the relative contributions of the elastic and viscous components in the material during stress application. A phase angle of \(0^{\circ}\) indicates purely elastic behavior, while an angle of \(90^{\circ}\) indicates purely viscous behavior. If the phase angle falls between these values, the material is considered viscoelastic. Increasing the load frequency leads to a smaller phase angle in the asphalt mastic, as shown in Figure 7. The fiber-modified asphalt mastics exhibit lower phase angles compared to the matrix asphalt. Figure 8 illustrates the state of the fibers in the asphalt mastic. Increasing the content of basalt fibers and bamboo fibers effectively enhances the strength of asphalt mastic. The mechanisms and effects of basalt fibers and bamboo fibers in enhancing asphalt mastic differ slightly. Basalt fibers are inorganic fibers with high strength and rigidity. They possess excellent mechanical properties and can provide high tensile and flexural strength. The interface interactions between basalt fibers and asphalt are mainly physical adsorption and surface adhesion, and due to their inorganic nature, they exhibit strong binding ability with asphalt. Bamboo fibers, on the other hand, are natural organic fibers with a certain degree of flexibility and plasticity. The interface interactions of bamboo fibers in asphalt mastic are primarily attributed to the natural hydrophilicity of fiber surfaces and the compatibility with the asphalt's colloidal substances, resulting in good adhesion.  
Figure 6: Master curves displaying the dynamic modulus for various types of fiber-modified asphalt mastic.  
Figure 7: Master curves of phase angle for various types of fiber-modified asphalt mastic.