## 3 Results and Discussion
### Stiffness Characteristics  
#### 3.4.1 Structural Characterization Based on Tangential Stiffness  
Observing that ESR yields under shear, [PERSON] and [PERSON] [37] and [PERSON] et al. [38] indicated that the yield point can be determined from the curve of the tangential stiffness \(E_{\text{tan}}\) and axial strain in double logarithmic coordinates. The formula for calculating \(E_{\text{tan}}\) refers to the generalized [PERSON]'s law, as shown in Equation (2):  
\[E_{\text{tan}}=\left(\text{d}\sigma_{1}{}^{\prime}-2\
u\text{d}\sigma_{3}{}^{ \prime}\right)/\text{d}\epsilon_{1} \tag{2}\]  
where \(\text{d}\sigma_{1}{}^{\prime}\) is the first effective principal stress increment, \(\text{d}\sigma_{3}{}^{\prime}\) is the third effective principal stress increment, \(\text{d}\epsilon_{1}\) is the axial strain increment, and \(\
u\) is [PERSON]'s ratio.  
From the consolidated drained condition, \(\sigma_{3}{}^{\prime}\) is a constant, and \(\text{d}\sigma_{3}{}^{\prime}=0\). The shear tangential stiffness \(E_{\text{tan}}\), can be expressed as follows:  
\[E_{\text{tan}}=\frac{\text{d}\sigma_{1}{}^{\prime}}{\text{d}\epsilon_{1}} \tag{3}\]  
To eliminate the influence of the average effective stress on the tangential stiffness during the shear process, \(E_{\text{tan}}\) was normalized. Figure 9 compares the \(E_{\text{tan}}/p{}^{\prime}\)-\(\epsilon_{1}\) relationship curves for different rubber fiber contents at a confining pressure of 100 kPa with that of rubber-reinforced soil determined by other scholars [36; 39; 40]. From the figure, when  
Figure 8: The relationship between the deviator stress and effective stress for the unreinforced expansive soil and 10% rubber fiber-reinforced expansive soil.  
\(\varepsilon_{1}<1.2\%\), the ESR of \(E_{\text{tan}}/p^{\prime}\) remained at approximately 20, although it reduced marginally. However, the magnitude of the reduction was not apparent. When \(1.2\%<\varepsilon_{1}<15\%\), as the shear progressed, the samples began to yield structurally, and \(E_{\text{tan}}/p^{\prime}\) decreased gradually with the increase in \(\varepsilon_{1}\). When \(15\%<\varepsilon_{1}\), the rate of variation in the normalized stiffness with the axial strain further increased, the curve was essentially close to being vertical, and the samples underwent overall failure. The relationship between \(E_{\text{tan}}/p^{\prime}\) and \(\varepsilon_{1}\) reveals an apparent yield point in the unreinforced soil. It corresponds to the destruction of the original arrangement and cementation of soil particles. After the addition of rubber, there was no apparent yield point in the reinforced soil. [PERSON] and [PERSON] [38] indicated that the stretching of rubber would prevent the formation of face-to-face arrangement of soil particles.  
Although the addition of rubber increased the shear strength of the mixture, \(E_{\text{tan}}/p^{\prime}\) decreased gradually as the rubber content increased. This is attributed to the temporary compression of the rubber fibers when subjected to confining pressure or kinking. This is in agreement with the observations of [PERSON] et al. [39] and [PERSON] and [PERSON] [25]. Similarly, [PERSON] and [PERSON] [41] determined that fibers with a higher modulus tended to increase the shear strength and failed to increase the stiffness of the mixture. Meanwhile, [PERSON] and [PERSON] [42] demonstrated that fibers reduce the stiffness or modulus.