# Study on Stress-Strain Relationship of Coir Fiber-Reinforced Red Clay Based on Duncan-Chang Model
## 4 Duncan-Chang Model and Parameter Analysis of Coir Fiber-Reinforced Red Clay
### Establishment of the Model  
In 1963, [PERSON] [44] proposed that the hyperbolic curve can be used to fit the curve according to the stress-strain relationship curve of a large number of soil triaxial tests. The hyperbolic equation is shown in Equation (1). [PERSON] et al. proposed a widely used incremental elastic model based on this hyperbolic stress-strain relationship, which is called Duncan-Chang model.  
\[\sigma_{1}-\sigma_{3}=\frac{\varepsilon_{\text{a}}}{a+b\varepsilon_{\text{a}}} \tag{1}\]  
In the formula: \(\sigma_{1}\) and \(\sigma_{3}\) are large and small principal stresses, respectively; \(a\) and \(b\) are parameters related to soil properties; \(\sigma_{\text{a}}\) is axial strain. For the conventional triaxial test, \(\sigma_{\text{a}}=\sigma_{1}\).  
By transforming Equation (1), we can obtain:  
\[\frac{\varepsilon_{1}}{\sigma_{1}-\sigma_{3}}=a+b\varepsilon_{1} \tag{2}\]  
In conventional triaxial tests, since d\(\sigma_{1}\) = d\(\sigma_{2}\) = 0, the tangent modulus is  
\[E_{\text{t}}=\frac{\text{d}(\sigma_{1}\text{-}\sigma_{3})}{\text{d}\varepsilon _{1}}=\frac{a}{\left(a+b\varepsilon_{1}\right)^{2}} \tag{3}\]  
According to Equation (3), when \(\varepsilon_{1}\)=0 \(\varepsilon_{1}\), the initial tangent modulus \(\text{E}_{\text{i}}\)\(E_{\text{t}}\) = \(E_{\text{i}}\) can be obtained. Then, the initial deformation modulus \(E_{\text{i}}\) is  
\[E_{\text{i}}=E_{\text{t}}\bigg{|}_{\varepsilon_{1}=0}=\frac{a}{\left(a+b \varepsilon_{1}\right)^{2}}\bigg{|}_{\varepsilon_{1}=0}=\frac{1}{a} \tag{4}\]  
The formula shows that the parameter \(a\) represents the reciprocal of the initial deformation modulus \(E_{\text{i}}\).  
According to Equation (2), when \(\varepsilon_{1}\rightarrow\infty\), the limit deviatoric stress difference \(\left(\sigma_{1}\text{-}\sigma_{3}\right)_{\text{ult}}\) is  
\[\left(\sigma_{1}\text{-}\sigma_{3}\right)_{\text{ult}}=\lim_{\varepsilon_{1} \rightarrow\infty}(\sigma_{1}\text{-}\sigma_{3})=\lim_{\varepsilon_{1} \rightarrow\infty}\frac{\varepsilon_{1}}{a+b\varepsilon_{1}}=\frac{1}{b} \tag{5}\]  
This formula shows that the parameter \(b\) represents the reciprocal of the limit deviatoric stress \(\left(\sigma_{1}\text{-}\sigma_{3}\right)_{\text{ult}}\) corresponding to the asymptote of the \(\left(\sigma_{1}\text{-}\sigma_{3}\right)\sim\varepsilon_{1}\) hyperbola.  
Figure 9: The failure mode of specimen under triaxial test.  
In the hyperbolic stress-strain relationship, the strength \(\left(\sigma_{1}\text{-}\sigma_{3}\right)_{\text{f}}\) of the specimen at failure usually does not reach the ultimate deviatoric stress \(\left(\sigma_{1}\text{-}\sigma_{3}\right)_{\text{ult}}\), so the failure ratio \(R_{\text{f}}\) can be defined as  
\[R_{\text{f}}=\frac{\left(\sigma_{1}\text{-}\sigma_{3}\right)_{\text{f}}}{\left( \sigma_{1}\text{-}\sigma_{3}\right)_{\text{ult}}} \tag{6}\]  
Substituting Equations (4)-(6) into Equation (3), the expression of tangent deformation modulus \(E_{\text{t}}\) is  
\[E_{\text{t}}=E_{\text{i}}\left[1-R_{\text{f}}\frac{\sigma_{1}-\sigma_{3}}{ \left(\sigma_{1}-\sigma_{3}\right)_{\text{f}}}\right]^{2} \tag{7}\]