# Resilient Modulus--Physical Parameters Relationship of Improved Red Clay by Dynamic Tri-Axial Test
## 4 The Dependence of Resilient Modulus on Material Properties
### New Comprehensive Prediction Model  
From the analysis of the existing prediction models, it can be concluded that the Andrei model can better predict the resilient modulus of the red clay chosen for this test. However, the Andrei model and other existing prediction models reflect only the influence of the stress state on resilient modulus, but the influence of moisture content, compactness, and additive content were not considered. [PERSON] et al. proposed a new methodolgy to convert the band shaped drain into an annular instead of a circle to keep the sectional area or perimeter of the band shaped drain unchanged before and after the convertion [72]. From Figure 4 through 7, it can be seen that the resilient modulus is correlated negatively with moisture content, meaning that the higher the moisture content, the smaller the resilient modulus. Additionally, the resilient modulus is correlated positively with the compactness and the additive content, which means that the greater the compactness and the additive content, the larger the resilient modulus. It can be concluded that the influence of compactness, moisture content, and additive content on the resilient modulus is remarkable. In this paper, considering the above factors affecting the resilient modulus, the relationship between moisture content (\(W\)), compactness (\(P\)), additive content (\(C\)), octahedral shear stress (\(\tau_{oct}\)), bulk stress (\(\theta\)), atmospheric pressure (\(P_{a}\)) and resilient modulus (\(M_{R}\)) are established, which is expressed as a Equation (7). \[F(W,P,C,\tau_{oct},\theta,P_{a},M_{R})=0 \tag{7}\]  
Figure 6: Relationship between resilient modulus (\(M_{r}\)) and lime content for various stress states. (**a**) \(OMC\), \(P93\%\), \(\sigma_{d}=30\) kpa; (**b**) \(OMC\), \(P93\%\), \(\sigma_{d}=55\) kpa. Figure 7: Relationship between resilient modulus (\(M_{r}\)) and cement content for various stress states. (**a**) \(OMC\), \(P93\%\), \(\sigma_{d}=30\) kpa; (**b**) \(OMC\), \(P93\%\), \(\sigma_{d}=55\) kpa. As the resilient modulus is correlated positively with the compactness and the additive content, and negatively with the moisture content, based on dimensional analysis [73], Equation (7) can be transformed into Equation (8). \[F(P/W,C/W,\tau_{oct}/P_{a},\theta/P_{a},M_{R})=0 \tag{8}\]  
On this basis, combined with the above existing prediction models, to eliminate the influence of dimension, the compactness and the additive content were normalized by the moisture content, respectively. The following prediction model was given (Equation (9)). In Equation (9), to eliminate the problem that the predicted value of resilient modulus is 0, \(\frac{C}{W}\) and \(\frac{\tau_{oct}}{P_{a}}\) add 1, respectively. \[M_{R}=k_{1}P_{a}\bigg{(}\frac{P}{W}\bigg{)}^{k_{2}}\bigg{(}\frac{C}{W}+1\bigg{)} ^{k_{3}}\bigg{(}\frac{\theta}{P_{a}}\bigg{)}^{k_{4}}\bigg{(}\frac{\tau_{oct}}{ P_{a}}+1\bigg{)}^{k_{5}} \tag{9}\]  
where \(\tau_{oct}\) is the octahedral shear stress, \(P_{a}=\) atmospheric pressure, \(k_{1},k_{2},k_{3},k_{4}\) and \(k_{5}\) are regression coefficients, \(\theta\) is the bulk stress, \(P\) is compactness, \(C\) is additive content, \(W\) is moisture content, and \(P_{a}\) is atmospheric pressure. Regression analysis of the testing results on the red clay samples improved with cement and/or lime was carried out by using Equation (9). Presented in Table 8 are the parameters that the proposed prediction model requires, the determination coefficient of improved red clay samples prepared at different moisture content, and the compactness and additive content. For prediction models, the greater the determination coefficient, R\({}^{2}\), the better the regression effect, indicating a higher rationality and reliability of the prediction model. The determination coefficient of the proposed comprehensive prediction model for the additive of cement and lime is 0.84, and 0.86, respectively. The comparison between the predicted and the measured values are presented in Figures 8 and 9.