# Investigation of the Mechanical Behavior of Polypropylene Fiber-Reinforced Red Clay
## 3 Results and Discussion
### Constitutive Model for Fiber-Reinforced Red Clay
#### 3.2.2 Fitting of Model Parameters  
Figure 5 shows the test results for the specimens with different \(FL\) and \(FC\) with a cell pressure of 200 kPa. The curves basically exhibit a linear relationship. At a certain \(FL\), the curve slope decrease with the increase in \(FC\). At a certain \(FC\), the curve slope decreases with the increase in \(FL\). The values of parameters \(a\), \(b\), \(a_{1}\), and \(a_{2}\) were obtained through regression of the test data, as shown in Table 5. The regression results at different cell pressures, \(FL\), and \(FC\) showed strong correlations, with the coefficients of determination (\(R^{2}\)) being larger than 0.99, confirming that the hyperbolic model is suitable for describing the stress-strain relationship of fiber-reinforced red clay.  
A statistical analysis of the model parameters showed the following patterns: (1) For a given cell pressure, parameter \(a_{1}\) varies in a small range as \(FL\) and content vary. Thus, the mean value of parameter \(a_{1}\) is used as the approximate value of \(a_{1}\)*. (2) For a given cell pressure and \(FL\), the value of parameter \(a_{2}\) increases with \(FC\). Thus, the value of \(a_{2}\) at \(FC=0.15\%\) is used as the basic value of \(a_{2}\). The values of \(a_{2}\)* at the other five content levels are calculated as follows: \(a_{2}\)* (\(FC=0.25\%\)) = \(1.5a_{2}\)* (\(FC=0.15\%\)); \(a_{2}\)* (\(FC=0.35\%\)) = \(2.0a_{2}\)* (\(FC=0.15\%\)); \(a_{2}\)* (\(FC=0.50\%\)) = \(2.5a_{2}\)* (\(FC=0.15\%\)); \(a_{2}\)* (\(FC=0.80\%\)) = \(3.0a_{2}\)* (\(FC=0.15\%\)).  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \(\sigma_{3}\) & \(FL\) & \multirow{2}{*}{**\(\mathbf{(}\mathbf{(}\mathbf{(}\mathbf{(}\mathbf{k)}\mathbf{)}\mathbf{)}\)**} & \multirow{2}{*}{**\(\mathbf{(}\mathbf{(}\mathbf{(}\mathbf{(}\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{ \mathbf{\mathbf{0}}}}}}}}}}}}}\mathbf{)\mathbf{)}}\)\)**} & \multirow{2}{*}{**\(\mathbf{(}\mathbf{(}\mathbf{(}\mathbf{(}\mathbf{(}\mathbf{(}\mathbf{(}\mathbf{(}\mathbf{(}\mathbf{}\mathbf{(}\mathbf{}\mathbf{(}\mathbf{}\mathbf{(\mathbf{}}\mathbf{ (\mathbf{}}\mathbf{\mathbf{}}\mathbf{(\mathbf{}}\mathbf{\mathbf{}}\mathbf{(\mathbf{}}\mathbf{(\mathbf{}}\mathbf{(\mathbf{}}\mathbf{ }\mathbf{(\mathbf{}}\mathbf{(\mathbf{}}\mathbf{(\mathbf{}}\mathbf{(\mathbf{}}\mathbf{(\mathbf{}}\mathbf{(\mathbf{ }}\(\(\(\(\(\(\(\(\(\(((((((((3) For a given cell pressure and \(FC\), the value of parameter \(a_{2}\) increases with \(FL\). Thus, the value of \(a_{2}\) at \(FL=7\) mm is used as the basic value. The values of \(a_{2}\)* at the other \(FL\) are calculated as follows: \(a_{2}\)* (\(FL=13\) mm) = 1.5\(a_{2}\)* (\(FL=7\) mm); \(a_{2}\)* (\(FL=19\) mm) = 1.7\(a_{2}\)* (\(FL=7\) mm).  
Parameters \(a_{1}\)* and \(a_{2}\)* are estimated using the above-mentioned method. Table 5 shows the results. Thus, for fiber-reinforced red clay subjected to a given cell pressure, the parametric values of the constitutive model at the basic conditions (\(FL=7\) mm, \(FC=0.15\%\)) can be obtained by conducting a test under these basic conditions and performing a regression analysis on the test data; the parametric values for other conditions can be derived using the above-discussed relationships, thereby significantly reducing the number of tests that would otherwise be necessary to characterize the stress-strain relationship under different conditions.